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• The Calculus of Consent: Logical Foundations of Constitutional Democracy
• Part I.: The Conceptual Framework
• 1.: Introduction
• Foreword
• Preface
• 2.: The Individualistic Postulate
• 3.: Politics and the Economic Nexus
• 4.: Individual Rationality In Social Choice
• Part II.: The Realm of Social Choice
• 5.: The Organization of Human Activity
• 6.: A Generalized Economic Theory of Constitutions
• 7.: The Rule of Unanimity
• 8.: The Costs of Decision-making
• Part III.: Analyses of Decision-making Rules
• 9.: The Structure of the Models
• 10.: Simple Majority Voting 7
• 11.: Simple Majority Voting and the Theory of Games
• 12.: Majority Rule, Game Theory, and Pareto Optimality
• 13.: Pareto Optimality, External Costs, and Income Redistribution
• 14.: The Range and Extent of Collective Action
• 15.: Qualified Majority Voting Rules, Representation, and the Interdependence of Constitutional Variables
• 16.: The Bicameral Legislature
• 17.: The Orthodox Model of Majority Rule
• Part IV.: The Economics and the Ethics of Democracy
• 18.: Democratic Ethics and Economic Efficiency
• 19.: Pressure Groups, Special Interests, and the Constitution
• 20.: The Politics of the Good Society
• Appendix
• 1.: James M. Buchanan, Marginal Notes On Reading Political Philosophy
• 2.: Gordon Tullock, Theoretical Forerunners

Part III.

Analyses of Decision-Making Rules

9.

The Structure of the Models

The theory of individual constitutional choice developed in Part II is very general. Problems that arise in the individual’s estimates of expected costs must be introduced before more useful applications of the theory can be made. Before the individual can estimate accurately the external costs that a given collective-choice rule will impose on him, he must have some idea as to how the rule itself will work. Our next step, therefore, is to analyze some of the more important decision-making rules. Most of the discussion will be concerned with a single rule—that of simple majority. However, the analysis of this rule, once completed, may be modified slightly and extended without difficulty to other more or less inclusive rules for social choice.

Before commencing the analysis proper, the underlying assumptions of our models must be stated. The restricted nature of these assumptions, their “unrealism,” must appear to limit sharply the relevance of our conclusions to real-world political institutions. We shall argue, however, that such limitation is largely apparent and that, fundamentally, the conclusions are generally applicable to a wide variety of collective institutions and that they help us to understand and to explain many real-world phenomena.

We shall continue to focus our attention on the calculus of the single individual, but here we are no longer placing him at the stage of constitutional choice. We assume the existence of a constitution that lays down the rules for amalgamating individual choices into social decisions. The individual participates in taking direct collective action with a knowledge of the fixed decision-making rules. As before, he is assumed to be motivated by a desire to further his own interest, to maximize his expected utility, narrowly or broadly defined. In this stage, which we have called and shall continue to call the operational as opposed to the constitutional, the individual’s interest will be more readily identifiable and more sharply distinguishable from those of his fellows than was the case at the constitutional level of decision.

Direct Democracy and Representative Government

The approach proceeds from the calculus of the individual, and it is, therefore, more concise and understandable if the individual is presumed to choose directly among the alternatives of collective action. That is to say, the analysis is sharper if we assume that collective decisions are made by rules of direct democracy. Quite clearly, this model has direct applicability only to an extremely limited set of real-world institutions. The New England town is the exceptional rather than the normal form of democratic organization. It is necessary to explain the operation of various rules at this most simple organizational level before proceeding to the more complex organizational forms contained in larger political units.

Our analysis of direct democracy can, we think, be extended to almost any set of political institutions while still retaining much of its explanatory and predictive value. We shall discuss this extension in Chapter 15, but now we shall proceed to analyze the operation of decision-making rules in terms of simple models involving individual participation in collective choices at the operational level. We shall occasionally refer to the action of legislative assemblies which seems to conform to the implications of our analysis. In one sense, these phenomena confirm the hypothesis that our model is of general relevance.1

The Time Sequence of Collective Decisions

Perhaps the most distinctive feature of our models, in comparison with other analyses of collective choice-making, is the central place assigned to the plurality of collective decisions over time. The analysis is not designed to explain the operation of decision-making rules on single, isolated issues. The analytical problem posed is that of examining comparative rules for choice as these apply to many decisions spread over “time.” Any rule must be analyzed in terms of the results it will produce, not on a single issue, but on the whole set of issues extending over a period of conceptually finite length.

The individual participant’s recognition that issues for collective choice are not unique and isolated events imposes severe limitations on any analysis of single decisions. Issues may be wholly unrelated in their descriptive characteristics, but the rational participant will recognize the time sequence of political choice. Moreover, this will cause him to seek “gains from trade,” when possible, by exchanging his vote on one issue for reciprocal support of his own interest by other participants on other issues. Thus, the time sequence of collective choice is very important in that it allows us to introduce an economic dimension to individual votes somewhat more handily than would otherwise be the case.

The difficulty of attributing such an economic dimension to votes in the political process has long been one of the stumbling blocks in the extension of economic reasoning to political models. The economic value of votes is confirmed by the selling and buying activities of individuals in “corrupt” circumstances, but models based on this “immoral” behavior pattern have not been considered to be useful in analyzing accepted political behavior. In the latter the essential requirement of scarcity has not been incorporated in the models, with the result that the applicability of an economic approach has been sharply limited. The individual participant normally has a single vote on each separate issue; votes do not “run out” or get “used up” as do the allegedly analogous “dollar votes” of individual participants in market choices. There seems to have been present a rather common failure to recognize the simple fact that if political votes did not have economic value, “corruption” would be impossible.

Individual votes result in collective decisions that exert economic effects. Each decision can be described in terms of its effects on individual incomes and wealth. So defined or described, the collective decision assumes a time dimension; it can be located in time and its impact can be measured over time. The political vote that assumes economic value can only refer to the vote exercised when decisive action is taken. The opportunity for the decision-making group to modify and change a provisionally approved decision through various forms of repeat voting represents yet another factor that has caused the application of an economic dimension to the political vote to be neglected.

Individuals’ votes have economic value. Moreover, for any commodity or service having economic value, a market will tend to emerge from the ordinary self-seeking behavior of men unless there are strong legal or moral prohibitions against trade. Such prohibitions are, of course, present to prevent the development of open markets in individual votes,2 but this does nothing toward removing the economic content. The absence of open markets serves only to prevent the full utilization of the pricing mechanism in allocating the scarce elements among competing alternative uses. Moreover, if pricing cannot be employed, some substitute means of rationing must be introduced. There are an almost infinite number of schemes that could be devised, and each scheme can be described by a set of voting rules. In each case valuable individual votes will be distributed on some basis, and this basis may be wholly unrelated to individual evaluations.

Let us look briefly at an example. Suppose that the group is required to make only one collective decision. It must decide how to divide up the one and only lot of manna that has fallen from heaven. There are five members of the group, and the constitution dictates that all collective decisions are to be made by simple majority rule. This means that three, any three, of the five members must agree. Since buying and selling votes is ruled out, and since there is only one decision to be made, the first three individuals who form a voting coalition will secure the manna. The two in the minority may place a much higher value on the manna than any one of the three winners, but this is irrelevant to the decision. We shall discuss models similar to this one in much greater detail later. Our purpose here is to indicate not only that any voting rule acts as a means of rationing, but that this rationing may cause a distribution of collective “goods” that is wholly unrelated to individual evaluations.

We note, however, that the introduction of a time sequence of political choices allows a market of sorts to be developed without the necessity of changing the rules for decision on single issues. If the individual participant recognizes the economic value of his own vote to others on certain issues and, in turn, recognizes the economic value of others’ votes to him on separate issues, he will be motivated to engage in “trade.” Moreover, if ways of “trading” can be found that do not clearly conflict with accepted standards of behavior, individuals will seek mutual advantages in this way. The possibility of exchanging votes on separate issues opens up such trading prospects. The individual may effectively, but imperfectly, “sell” his vote on a particular issue, securing in return the votes of other individuals on issues of more direct interest. This process of “logrolling” will be carefully analyzed in the following chapter, but some preliminary points should be made here.

With relatively few exceptions logrolling phenomena have been viewed as deviations from the orderly working of the democratic process. This view seems to have been adopted for two separate reasons. First, and more important, the economic motivation for political behavior reveals itself most clearly in the occasional examples of Congressional logrolling legislation. Students of the political process, who adopt the view that, at base, political behavior is not motivated by economic interest, must explain such action in terms of aberrations from more orthodox behavior. Secondly, and related to the first, there has been a failure to recognize that logrolling phenomena are much more pervasive than the more obvious examples would indicate. The phenomena surely occur at several levels of political sophistication, and the fact that the cruder instances occur at all should give the student of political process cause for looking somewhat carefully for more “acceptable” means of accomplishing similar purposes.3

It seems clear that, insofar as divergent interests affect the political choices of individuals and groups, the logrolling process provides the general model for analyzing the various choice-making rules.4 Surely the individual participant in collective choice recognizes the time sequence of events requiring collective action, and, just as surely, he will be motivated to engage in mutually advantageous “trades” or “compromises” with his fellows. The cruder models, in which the trade is made explicit, are useful in that they are more readily subject to analysis, but the more important cases probably occur beneath the outwardly visible surface of “politics.” The assumption that these crude models provide a general approach to the operation of political rules seems considerably more acceptable than the contrary one which assumes that the analysis of rules on the basis of single issues is a more satisfactory approach to a general theory of collective choice.

Perfect and Imperfect Markets

When a time sequence of issues is allowed for, some trading of votes takes place. No longer does the decision-making rule alone serve as the rationing device. An illustrative analogy may be helpful. Suppose that all rents on dwelling accommodations are strictly controlled, and at levels much below hypothetical “market” values. Individual landlords are subject to prosecution if they accept direct money payments (“bribes”) above the controlled rents from prospective tenants. On the other hand, they are not prevented from entering into other “exchanges” with tenants at freely determined and mutually advantageous terms of trade. Landlords may “sell” furniture to tenants, or they may “purchase” other commodities. Under circumstances such as these, the expected results would be less arbitrary than under the alternative system in which no free “exchanges” between landlord and tenant are allowed, that is, in which housing is rationed solely on a nonprice basis. On the other hand, the nonprice aspects of the “market” system would make the expected results diverge significantly from that which could be predicted to emerge from a completely free market in rental units.

In our rent-control analogy, to which we shall return in a later chapter, the combination of price and nonprice rationing appears as a special institutional pattern. In the political-vote case, however, this in-between or “imperfect” model represents, perhaps, the most general model of democratic process. This “imperfection,” however, makes the analysis especially difficult.5 For this reason we shall find it necessary, in the chapters that follow, to employ extremely simplified models.

Some predictions concerning the results to be expected from the operations of the in-between model may also be derived by considering the alternative models that bracket the logrolling or imperfect-vote marketing model. As we have suggested, other scholars have analyzed the nonprice model, being forced to do so by their concentration on single issues. To our knowledge, however, the full price-rationing model has not been fully developed: that is, the model in which political votes are freely marketed for money has not been subjected to rigorous analysis, even for simple voting rules. The tools supplied by modern game theory are helpful in this respect, and in Chapters 11 and 12 we analyze the operation of simple majority-rule games under the assumption of full side payments. By relaxing the full side-payments assumption, we may also compare this model with one more closely approximating the logrolling model.

The Intensity of Individual Preference

Much of the traditional discussion about the operation of voting rules seems to have been based on the implicit assumption that the positive and negative preferences of voters for and against alternatives of collective choice are of approximately equal intensities. Only on an assumption such as this can the failure to introduce a more careful analysis of vote-trading through logrolling be explained. If all intensities of preference are identical over all individuals and over all issues, no trading of votes is possible. In this case the individual feels as strongly on one issue as on any other, and he will never rationally agree to exchange his vote for reciprocal favors.

An example may be helpful. Consider a society confronted with three issues in sequence. The group must choose between A and Ā, between B and , and between C and . Let us assume that the constitution dictates that each of these issues shall be decided by simple majority voting rules. Assume that, in each case, 51 per cent of the voters favor the first alternative and 49 per cent favor the second alternative, but assume also that the majorities and the minorities are not uniformly composed over the three issues. If all preferences are equal in intensity, no bargains can be struck, and A, B, and C will be chosen. Consider Voter I who favors A, B, and , and Voter II who favors Ā, , and C. Neither would be willing to trade his vote on two issues for the other’s vote on one issue, and a one-for-one trade would not be mutually advantageous.

Intuitively the assumption of equal intensity of preference seems unacceptable. Clearly the more general assumption is that individual “tastes” for collectively obtained “goods” vary in both object and intensity. In the extremes there would seem to be no question of such variance. If the issue to be decided is whether or not Voter I will or will not be executed, the intensity of preference of Voter I against this action will clearly, in some circumstances, be greater than the desires of other voters in favor of the action. As with certain other aspects of political theory, there seems to have been a failure here to distinguish between positive analysis and normative theory. Implicit in much of the discussion of majority rule has been the idea that individual votes should be treated as reflecting equal intensities of preference, quite independently of whether or not the norms agree with the facts in the case. This idea, in turn, probably stems from the more fundamental norm of democratic organization—that of political equality. Political equality may be fully accepted as essential to any form of democratic process, but this does not imply that individual votes on particular issues should be considered as if they reflect equal intensities of preferences over all participants.

The assumption of equal intensity of preference for all voters over all issues really amounts to imputing to each individual a most restricted utility function, and one that is wholly different from that which is employed in economic analysis. Not only is utility measurable; it is directly comparable among separate individuals. To the modern economist this approach to individual calculus seems anachronistic and sterile.

Equal Intensities and Majority Rule

Although we do not propose to discuss the equal-intensity assumption in detail here, a brief digression on the relationship between it and simple majority rule may be worthwhile. When all individual preferences are of assumed equal intensity, simple majority rule will insure that the summed “benefits” from action will exceed the summed “losses.” In this way simple majority rule appears to assume a unique position in terms of a very restricted “welfare” criterion.

Consider our earlier example. Recall that 51 per cent of the voters favor A and that 49 per cent favor Ā, and that positive and negative intensities are equal. Let us interpret this equal intensity specifically as indicating that any voter would be willing to give up his preference (to accept the reverse) for $100.00. Thus, passage of the legislation in question will benefit 51 per cent of the voters by $100 each, and it will harm 49 per cent of the voters by $100 each. In the hundred-man model, A would be selected by simple majority voting, and total benefits of $5100 exceed total losses of $4900.

Note that other voting rules need not produce this result, unless compensation of some sort is allowed. For example, under a 53 per cent voting rule the project could not be approved, and, in the additive sense employed above, the community would “lose” the potential $200 in benefits. However, if individual intensities of preference are not equal over all voters, this unique feature of simple majority rule disappears. If minorities feel more strongly on particular issues than majorities, then any rule short of unanimity may lead to policies that will produce net “harm,” even if the comparability of utilities among separate persons is still accepted as legitimate.

If vote-trading or compensation in any other form is allowed to be introduced, however, even this extremely restricted uniqueness of simple majority rule disappears. Let us continue to accept the equal intensity assumption. If compensation is introduced, any rule will cause A to be selected over Ā in the foregoing example. If the unanimity rule were in force, for example, the 51 citizens who would be the potential gainers would have to compensate the 49 potential losers by at least $4900 in order to insure the passage of the legislation.6 The demonstration that the same results would be produced under simple majority rule and the unanimity rule can be extended to apply also to less-than-majority rules. Suppose, for example, that we reverse the arithmetical model and consider the case in which 51 voters oppose the measure while 49 voters approve, and that each voter is willing to give up his preference for $100. If, in this situation, the community operates under a rule in which any person, individually, can order collective action, the potentially damaged majority will be able, out of the opportunity “benefit” they receive from not having the action taken, to fully compensate the members of the minority who might otherwise impose the change. Thus, even with equal intensities assumed from the outset, any voting rule will produce “desirable” results as measured by the comparative utility scales that are implicit in the assumption, provided only that compensation is allowed. However, if no compensation is allowed, either directly or through vote-trading, this restricted “welfare” conclusion no longer holds, and each rule must be analyzed anew for its welfare-producing properties.

As we have suggested, moral restraints may prohibit open buying and selling of votes. However, compensations may be arranged through vote-trading over a sequence of issues. If this is allowed to take place, the uniqueness of simple majority rule disappears, even on the equal-intensity assumption. The unique features reappear only when the equal-intensity assumption is extended to apply over all issues as well as over all voters. If all individual preferences are equally intense over a single issue, and if the preferences of each single individual are equally intense over all the separate issues in which he might participate as a voter, no vote-trading will take place (as we have shown above). Under these circumstances, and under these only, can simple majority rule be said to take on particular characteristic features that distinguish it from other decision-making rules.

Some of these points will be clarified in later chapters. The main purpose here is to emphasize the overly restrictive nature of the equal-intensity assumption. In our models we propose to place no such restrictions on individual preferences for the alternatives of political choice.

Equal Intensity and Random Variation of Preferences

The equal-intensity assumption may be employed, without great distortion, in the analysis of the situations in which the intensities of individual preference vary symmetrically among the separate and identifiable subgroups in the population and over all issues. In effect, this situation simply translates the equal-intensity case from the individual to the group level. This situation seems rather special. Normally, an act of government will either markedly harm or markedly benefit at least one specific and identifiable group which will, accordingly, feel more strongly about the issue than will the masses of voters. There are some measures undertaken by governments, however, which are relatively general in nature, that is, which apply in a relatively nondiscriminatory fashion to all individuals and groups. For such measures, individual preferences for and against may vary, but there seems to be no particular reason to expect that such variation would systematically reflect differential intensity. If this variation is distributed in some random fashion among all groups, the employment of the equal-intensity assumption may be reasonably appropriate.

Specific minorities on issues of this sort cannot readily arrange trades to secure favorable action. Majorities will tend always to be able to secure desired action under simple majority rule, and even under other rules if compensations are allowed. The constitutional calculus discussed in previous chapters is not changed significantly in application to this case. The decision-costs function might be changed somewhat, but the appropriate method of choosing decision rules is not modified. Insofar as the equal-intensity assumption is accepted as appropriate, the low-cost point on the aggregate “cost curve” would tend to be that represented by simple majority voting. If intensities of preference are assumed equal, anything desired by a majority, by sheer arithmetic, represents, when approved, a shift to the Pareto-optimality surface. The prevention of the implementation of the will of the majority, in this special case, is never to the “interest” of “society as a whole.” If simple majority rule is allowed to prevail, then “optimal” policy will always be selected.

This does not, of course, mean that majority rule will produce results that will be “optimal” for each individual in each particular case. In the case of equal intensity of preferences, the incremental payments that might be needed to obtain any qualified majority are simply transfer payments. The money would go from one man’s pocket into the next man’s, but there is no mutual gain from trade. In fact, there would be mutual loss when the costs of negotiating agreements are taken into account. Thus, at the time of constitutional choice, if an individual could feel confident that there would be a large number of such “equal intensity” issues to be put up for decision in the future, and if he felt that these issues would be such that his own position would fluctuate randomly between majority and minority without predictable differential intensity in the two cases, then he would expect any rule requiring compensation from the simple majority to a part of the whole of the minority to involve payments by him in some cases and payments to him by others in other cases. Over time, these could be expected to balance out. He might, therefore, wish to save himself the negotiating costs by accepting simple majority rule.

In order for this constitutional decision to be made, however, several conditions would be necessary. In the first place, there must be enough general (“equal intensity”) issues expected to arise to insure that they will, with respect to the individual, be mutually canceling. Secondly, the individual must feel fairly confident that he will not tend to be in the minority more than the average number of times. Thirdly, and most restrictive, there must be some method of distinguishing these “general” cases from those clearly characterized by differential intensities of individual preference. Little comment need be added on the first two conditions, but the third may be subjected to analysis. We might try two approaches: first, we might attempt to classify legislative activities that do not seem likely to generate differential intensities of preference among separate groups, and allow decisions on these activities to be made by simple majority rule; secondly, the constitution itself might be so designed that it automatically distinguishes among issues on this basis. The first approach is clearly feasible, and to some extent it is reflected in the constitutions of Western democracies.

Designing a constitution so that it will discriminate automatically between legislation potentially affecting intense minorities and legislation on which the intensity of desires is more or less equal, or can appropriately be assumed so, may not initially seem feasible, but this is, in fact, practicable. As discussed in Chapter 16, a properly designed bicameral legislature does make this distinction automatically.

10.

Simple Majority Voting7

In this chapter we propose to examine the operation of a single collective decision-making rule, that of simple majority, under certain highly restricted assumptions. Theorists of the democratic process have, traditionally, paid little attention to the actual operation of voting rules, and they seem, by and large, to have been uninterested in making generalized predictions regarding the results of actual political decision-making. This relative neglect is explained, at least in part, by the implicit assumption that participants in collective choice seek to further the “public interest,” although, as we have suggested, this concept is never defined.

Quite recently a few pioneers have tried to introduce a more positive approach in political theory. Two of these, Anthony Downs and Duncan Black, have tried to develop theories of the political voting process that are based on behavioral assumptions similar to ours.8 These contributions have been important ones, but the political process has been drastically simplified by concentration on single issues, taken one at a time and separately. Such an approach appears to have only a limited value for our purpose, which is that of analyzing the operation of voting rules as one stage in the individual’s constitutional-choice problem, that of choosing the voting rules themselves. The working of a voting rule can be analyzed only as it produces results over a series of issues.

Majority Voting without Logrolling

Once it is recognized that the political process embodies a continuing stream of separate decisions, the most general model must include the possibility of vote-trading, or, to use the commonly employed American term, “logrolling.” The existence of a logrolling process is central to our general analysis of simple majority voting, but it will be helpful, by way of comparison, to consider briefly a model in which logrolling is not permitted to take place, either by legal institutions or by certain widely acknowledged moral precepts. There are certain relatively rare institutional situations in which logrolling will not be likely to occur, and in such situations the contrasting analytical model may be explanatory. The best example is the standard referendum on a simple issue. Here the individual voter cannot easily trade his own vote on the one issue for reciprocal favors on other issues because, first, he is uncertain as to when other issues will be voted on in this way, and, second, he and his immediate acquaintances represent such a small part of the total electorate that such trading effort may not be worthwhile. Furthermore, the secret ballot, normally employed in such cases, makes it impossible for any external observer to tell whether voting commitments are honored or not. Under circumstances such as these, the individual voter will make his voting decision in accordance with his own preferences on the single question posed.

In this model each voter indicates his preference, and the preference of the majority of the whole group is decisive. The defect in this procedure, a serious one that has already been mentioned in Chapter 9, is that it ignores the varying intensities of preference among the separate voters. A man who is passionately opposed to a given measure and a man who is slightly favorable but does not care greatly about it are given equal weight in the process of making final decisions. It seems obvious that both of these individuals could be made better off, in terms of their own expressed preferences, if the man strongly opposed should be permitted in some way to “trade” or exchange something with the relatively indifferent supporter of the proposed measure. Applying the strict Pareto rules for determining whether one social situation represents an improvement over another, almost any system of voting that allows some such exchange to take place would be superior to that system which weights all preferences equally on each issue. By way of illustration, it is conceivable that a proposal to prohibit Southern Democrats from having access to free radio time might be passed by simple majority vote in a national referendum should the issue be raised in this way. Such a measure, by contrast, would not have the slightest chance of being adopted by the decision-making process actually prevailing in the United States. The measure would never pass the Congress because the supporters of the minority threatened with damage would, if the issue arose, be willing to promise support on other measures in return for votes against such discriminatory legislation. In the complete absence of vote-trading, support for specific legislation may reach 51 per cent without much of this support being intense. In such cases a minimal introduction of vote-trading will insure defeat.

Without some form of vote-trading, even those voters who are completely indifferent on a given issue will find their preferences given as much weight as those of the most concerned individuals. The fact of voting demonstrates that an individual is not wholly indifferent, but many voters may, on referendum issues, be led to the polls more by a sense of duty or obligation than by any real interest in the issue to be determined. Interestingly enough, this “duty of a citizen to vote” is much emphasized as an essential feature of effective democratic process.9 Even the smallest preference for one side or the other may actually determine the final choice. Permitting those citizens who feel strongly about an issue to compensate in some way those whose opinion is only feebly held can result in a great increase in the well-being of both groups, and the prohibition of such transactions will serve to prevent movement toward the conceptual “social optimality” surface, under almost any definition of this term.

Note that the results under logrolling and under nonlogrolling differ only if the minority feels more intensely about an issue than the majority. If the majority is equal or more intense in its preferences, its will must prevail in either model. It is only when the intensity of preferences of the minority is sufficiently greater than that of the majority to make the minority willing to sacrifice enough votes on other issues to detach marginal voters from the majority (intense members of the majority group may, of course, make counteroffers) that the logrolling process will change the outcome. As we have suggested, the assumption of possible differences in intensity of preferences seems more acceptable than any assumption of equal intensities, and it seems clear that on many issues specific minorities may be much more interested in the outcome of political decisions.

The above discussion suggests that a reasonably strong ethical case can be made for a certain amount of vote-trading under majority-rule institutions. We emphasize, however, that our model, which incorporates the logrolling model as the general case, is not chosen because of the ethical desirability of the institutions analyzed. Positive theory must always analyze those institutions that are, in fact, general (the test of generality being the validity of the predictions made), quite independently of ethical or moral considerations. Therefore, even if vote-trading should be viewed as morally reprehensible behavior, it might still be necessary to analyze the phenomenon carefully if it were observed in the operation of real-world political processes.

Two Types of Logrolling

Logrolling seems to occur in many of the institutions of political choice-making in Western democracies. It may occur in two separate and distinct ways. In all of those cases where a reasonably small number of individuals vote openly on each measure in a continuing sequence of measures, the phenomenon seems pervasive. This is normally characteristic of representative assemblies, and it may also be present in very small governmental units employing “direct democracy.” The applicability of our models to representative assemblies has already been mentioned. Under the rules within which such assemblies operate, exchanges of votes are easy to arrange and to observe. Such exchanges significantly affect the results of the political process. It seems probable that this fact provides one of the major reasons for the widespread use of representative democracy.

Logrolling may occur in a second way, which we shall call implicit logrolling. Large bodies of voters may be called on to decide on complex issues, such as which party will rule or which set of issues will be approved in a referendum vote. Here there is no formal trading of votes, but an analogous process takes place. The political “entrepreneurs” who offer candidates or programs to the voters make up a complex mixture of policies designed to attract support. In so doing, they keep firmly in mind the fact that the single voter may be so interested in the outcome of a particular issue that he will vote for the one party that supports this issue, although he may be opposed to the party stand on all other issues.10 Institutions described by this implicit logrolling are characteristic of much of the modern democratic procedure. Since the analysis is somewhat more incisive in the first type of logrolling, we shall not discuss the second type at this point.

A Simple Logrolling Model

Let us consider a simple model. A township inhabited by one hundred farmers who own similar farms is cut by a number of main highways maintained by the state. However, these are limited-access highways, and the farmers are permitted to enter this primary network only at the appropriate intersections with local roads. All local roads are built and maintained by the township. Maintenance is simple. Any farmer who desires to have a specific road repaired is allowed to present the issue to the whole group for a vote. If the repairing proposal is approved by a simple majority, the cost is assessed against all of the farmers as a part of the real property tax, the rate of which is automatically adjusted upward or downward so as to make revenues always equal to expenditures. The principal use of the local roads by the farmers is getting to and from the major state highways. Since these major highways cut through the whole district, there are four or five farmers dependent on each particular piece of local road, and each farmer requires at least one local road to provide him with access to the main network.

In this model the simple referendum system would result in no local road being repaired because an overwhelming majority of the farmers would vote against the repairing of any given road, considered separately. A logrolling system, however, permits the local roads to be kept in repair through the emergence of bargains among voters. The actual bargaining may take a number of forms, but most of the “solutions” will tend to be unstable. In any case, “equilibrium” involves some overinvestment of resources.

One form that an implicit bargain might take is the following: each individual might determine, in his own mind, the general standard of maintenance that should be set for all local roads. That is to say, he would balance, according to his own scale of preferences, the costs of maintaining his own road at various levels of repair with the benefits expected, and try to reach a decision at the point where expected marginal costs equal marginal benefits. Generalizing this, he could then vote on each separate project to repair a given road in the same way that he would vote for repairs on his own road. If all voters would follow this rule of reaching decisions, we would find a schedule of voting behavior such as that shown below in Figure 12. Each mark or dot on the horizontal line represents the “idealized” standard of maintenance on all roads for a single voter. If a proposal for repairing a given road falls to the left of his own position on this scale, the individual will support it; if a proposal falls to the right of his own position, he will vote against it. If each road has at least one farmer living along it whose preference for general road repairs falls to the right of the median (A in Figure 12), then a proposal for road repair will be advanced as soon as any given road falls below this farmer’s standard of maintenance. Successive further proposals would be made as the road deteriorated further. When the deterioration of any road reached the median level, a repair project would secure approval by simple majority vote. Hence, all local roads would, in this model, tend to be maintained up to the standard indicated by the median preference.

>Figure 12

This result will not represent a fully “efficient” solution in any Pareto sense,11 but it is possible to support this procedure on ethical grounds. In fact, this solution seems to be the one that most of the proponents of majoritarian democracy have in mind when they discuss democratic process. In any event, we propose to use this solution, which we shall call the “Kantian,”12 as a more or less “correct” solution against which we shall contrast our more realistic result.13

If the farmers of the township generally follow such a policy in voting, then any single farmer could benefit himself simply by voting against all proposals to repair roads other than his own and by voting to repair his own road at each opportunity. This single departure from the general pattern of behavior would shift the median of the schedules slightly so that the taxes on the farmer concerned would be reduced or his road kept in better-than-average repair. If the other farmers living along this road should follow the first farmer’s example (we shall call such farmers “maximizers”), they would be able to shift the standards of repair so that the road on which they live would be repaired at level B’ while reducing the standard on all other roads to B in Figure 12. Since the largest share of the costs of keeping their own road in repair would fall on other taxpayers, while the largest share of their own taxes would go to the repair of other roads, this change in behavior would be greatly to the advantage of the maximizers and greatly to the disadvantage of the “Kantians,” although in the initial stages the disadvantages would not be concentrated to the same degree as the advantages.

If the farmers located on a second local road should also switch to a maximizing pattern of behavior, this action would have the effect of bringing the level of road-repairing on the two roads particularly affected down toward that which would prevail under the generalized Kantian system, while still further lowering the standards on the remaining “Kantian” roads. However, it seems probable that, finding themselves in this situation, the two groups of maximizers could benefit by forming a coalition designed to raise the standards of maintenance on the two roads. Let us consider the situation that would be confronted by an individual maximizer when he tries to decide whether or not to enter into such a coalition with other maximizers. Since he will pay only about 1/100 of the cost, almost any proposal to repair his own road will be supported by him. If, however, in order to obtain support for some repair project for his own road, he must also vote for the repair of another road, the individual must also count the cost to him of other repair projects. In weighing costs and benefits, he must consider not only the tax cost to himself from a proposal to repair his own road but also the tax cost to him of the other repair jobs which he must support in order to get his own proposal adopted. In the particular situation under discussion, when the farmers on all of the local roads except two are still Kantians, this added cost consideration would put few restraints on feasible projects, but some recognition of the incremental costs of securing agreement would have to be taken into account. Furthermore, as more and more farmers became tired of being exploited by the maximizers and shifted to the maximizing pattern of behavior, this cost consideration would become more and more important.

Let us now examine a rather unlikely, but theoretically important, special case. Suppose that exactly 51 of the 100 farmers follow a maximizing policy, while 49 are pure “Kantians.” Let us further suppose that all of the maximizers live on some local roads, while all of the Kantians live on other roads. Under these circumstances, the Kantians clearly would never be able to get their roads repaired at all, but the level of repairs on the maximizers’ roads is more difficult to determine. In order to simplify the issue somewhat, let us assume (plausibly) that these roads are maintained on such a high level that all of the Kantian farmers would vote against all further repair proposals. In this case, it would be necessary to attain the approval of all of the maximizers to carry any single repair project. A maximizing farmer, considering the repair of his own road, would necessarily be forced to take into account his share in the costs of repairing the roads of all maximizers. He would have to consider the incremental taxes that he must pay in order to repair the roads of all other parties to the bargain. His calculus requires, however, only that he compare his own marginal benefits against his own marginal costs. No knowledge of anyone else’s utility function is required. The individual need only decide whether the total bargain is or is not to his advantage.14

For the Kantians, note that, while no roads leading to their own farms will be repaired, they will be required to contribute toward the repair of the roads leading to the farms of the maximizers. Thus, a part of the total repair costs in the township will be paid by persons who are not parties to the decisive bargain, and, since the maximizers count only the costs to themselves when they make voting decisions, the general standard of road maintenance on the roads of the maximizers will tend to be higher than it would be if the Kantians were also included in the calculus. Under such conditions as these, where “virtue” so conspicuously would not pay, it seems likely that at least some of the Kantians would decide to switch to a maximizing policy. For simplicity, let us assume that they all do so at the same time. Since these reluctant maximizers would still be in a minority, their changes of heart would not immediately redound to their private benefit. However, it might be relatively easy for this minority, acting as a coalition, to find two of the original maximizers who would, in return for a promise of very good maintenance on their own roads, desert their former colleagues. It is again obvious, however, that the new majority would now be equally susceptible to similar desertions. A permanent coalition of 51 farmers formed for the purpose of exploiting the remaining 49 could not be considered to be stable in the usual sense of this term. In the terminology of game theory, which we shall use in the following chapter, any combination of 51 voters dominates any combination of less than this number, but no combination of 51 dominates all other combinations of 51.15

The outcome is clearly indicated. Each farmer would enter into bilateral agreements with enough other farmers on other roads to insure that his own road is repaired. The individual farmer would then be forced to include as a part of the cost of getting his own road repaired the cost (to him) of repairing the roads of 50 other farmers. These bilateral agreements would overlap, however. Farmer A (more precisely, the group of farmers living on Road A) would bargain with Farmers B, C, ..., M. Farmer M, on the other hand, might make up a majority bargain from an agreement with Farmer A and Farmers N, O, ..., Z.

In counting the costs to himself involved in the repair of other roads necessary to secure the repair of his own road, each farmer would consider only the repair of those roads which he agrees to support. In this way his expenditure pattern would include as a free gift the tax payments of 49 voters. The fiscal institutions postulated insure that all 100 voters share in the costs of each repair project approved, but a minimum participation of only 51 voters in the net benefits is required by simple majority voting. The natural result would be that each road in the township would be maintained at a level considerably higher and at a greater expense than is rational from the individual standpoint of the farmers living along it. Each individual in the group would be behaving quite rationally, but the outcome would be irrational. This apparent paradox may be explained as follows: each voter pays enough in support for the repair of other roads to attain a position of equivalence between estimated individual marginal costs and individual marginal benefits, but the payments included in his private calculus make up only a part of the costs of total road repair that he must, as a taxpayer in the community, support.16 There are other roads which will be repaired because of successful bargains to which he is not a party. Taken as a group, the road-repair projects for which he votes represent a good bargain for the individual; but other ad hoc bargains will also take place. The individual will, of course, vote against all projects included in these outside bargains, but he will be in the minority. Therefore, he will have to bear a part of the costs.

Any individual farmer who followed another course of action would be worse off, however, than the individual whose behavior is considered here. For example, a Kantian farmer would never have his own road repaired, but he would have to pay taxes for the support of other local roads. In any practical situation the whole decision-making process would tend to become one of elaborate negotiations, open and concealed, taking place at several levels of discourse. The man who is the most effective bargainer would have a considerable advantage. However, the general pattern of results may be less than optimal for all parties (optimal being defined here in terms of either the Kantian or the Paretian solution).

Possible Objections

We may now consider certain possible objections that may be raised against the reasoning implicit in our simple logrolling model. It may be argued that those individuals whom we have called maximizers would be behaving wickedly and that ethical considerations will prevent a majority of the population in the real world from following such a course of action. Ethical and moral systems vary greatly from culture to culture, and the strength of moral restraints on private action is not readily predictable. We do not want to preclude the possible existence somewhere of a system of human behavior which could effectively restrain logrolling, but surely the American behavior pattern contains no such restraints. Under our system open logrolling is normally publicly characterized as “bad,” but no real stigma attaches to those who participate in it. The press describes open logrolling arrangements without apparent disapproval, and, in fact, all of our political organizations operate on a logrolling basis.17 Moreover, no stigma at all attaches to implicit as opposed to open logrolling.

A second argument asserts that each farmer in our model community would soon realize that if he adopted a maximizing pattern of behavior, this would lead all other farmers to do the same thing. Since the “maximizing equilibrium” is worse for almost all farmers18 than the “Kantian median,” each farmer would, on the basis of his own cold and selfish calculation, follow the Kantian system. This argument is familiar, and it is precisely analogous to the one which holds that no single labor union will force wage rates up for its own members because it will realize that such action will lead other unions to do the same and that the eventual outcome will simply be higher prices and wages without any increase in real incomes. There seems to be overwhelming empirical evidence that men do not act in this way.19 The argument overlooks the fact that there will, of course, be short-run gains to the individuals or groups who initiate action first. In addition, the argument seems to contain a logical flaw. It is based on the observation that, in any series of actions by a number of men, there must be a first step. If this can be prevented, then the whole series can be prevented. This observation is, in itself, correct; but there must also be a second, a third, and a fourth step, etc., in each series. If any one action in the series is prevented, then the whole series cannot be completed. If all of our maximizing farmers should refrain from following a maximizing course of action because each one felt that his own personal adoption of such behavior would lead to a switch to a position of “maximizing equilibrium,” then, if only one of them had done so, we could construct an exactly similar argument “proving” that none of the remaining 99 would follow his example. However, if the second argument is true, the first is false; hence, the chain of reasoning contains an inconsistency.

Note that our refutation of this argument does not preclude an individual’s taking the attitude: “If no one else acts, I shall not act.” However, not only must all members of the group assume this attitude if the argument is to be valid, but each member of the group must also believe that all other members will take this attitude. This combination of attitudes, which would amount to complete mutual trust, seems highly improbable in any real-world situation. The argument that all individuals in the group will be worse off than if they all adopted Kantian norms of behavior does have some relevance for the support of constitutional changes in the decision-making rules or institutions for choice. While it may never be to the interest of the individual to refrain from adopting a maximizing attitude, given the rules as laid down, it may well be to his long-range interest to support a change in these rules themselves, which, by definition, will be generally applicable.

Alternatives

One means through which the separate farmers in our model might enter into a bargain so as to insure results somewhat closer to the Kantian median would be the development of a specific formula that would determine when a road should be repaired. Yet another means would be the delegation of decision-making authority to a single individual or small group. These become practicable institutions, however, only within the confines of a set of closely related issues that may be expected to arise: in our model, separate proposals for road repair. In the more general and realistic case where governmental units must consider a continuing stream of radically different projects, neither an agreed-on formula nor a single expert or group of experts would seem feasible. A formula that would permit the weighing of the costs and the benefits of such diverse programs as building irrigation projects in the West to increase agricultural production, paying farmers in the Midwest to decrease agricultural production, giving increased aid to Israel, and dredging Baltimore’s harbor, is inconceivable. There could not, therefore, be any real agreement on any automatic or quasi-automatic system of allocating collective resources, and the delegation of authority to make such decisions would mean the abandonment of the legislative process as such. We are reduced to the reaching of separate decisions by logrolling processes, given the constitutional rules as laid down in advance.

Majority Rule and External Costs

This is by no means so much a tragedy as our simple model may have appeared to suggest. Implicit in the comparison of the logrolling solution with the Kantian solution has been the idea that the external costs imposed on the individual by the “maximizing equilibrium” exceed those resulting from the Kantian “equilibrium.” This will be true if individual farmers are primarily interested in the repair of their own roads, as our model postulates. If, by contrast, some or all of the farmers should be genuinely and intensely interested in the standards of general road repair over the whole township, the Kantian solution might be worse than the maximizing one. This is because the Kantian solution under simple majority rule can take no account of varying intensities in individual standards. For example, if there should exist a minority of farmers who feel very intensely that much more should be spent on road repairs than the majority of other voters, whose standards are somewhat indifferently held, the maximizing solution, which does result in a standard of general repair above the Kantian median, may be more “desirable” on certain commonly acknowledged welfare grounds than the Kantian solution. In this case the introduction of logrolling into the Kantian model could be beneficial to all parties.20

A central feature of our analysis is the demonstration that the operation of simple majority rule, quite independently of any assumption about individual motivation, will almost always impose external costs on the individual. If more than a simple majority is required for decision, fewer resources will be devoted to road-building in our model, and the individual comparison of marginal benefits and marginal costs would tend to approach more closely the calculus required by the economists’ standard criteria for attaining a Pareto-optimality surface. As the analysis of Part II has shown, however, when any consideration of more inclusive voting rules is made, the incremental costs of negotiating bargains must also be taken into account.

Generalizations

Some of these points will be discussed later. We shall now inquire as to what extent our simple logrolling model can be generalized. It would appear that any governmental activity which benefits specific individuals or groups in a discriminatory fashion and which is financed from general taxation would fit our model well. It is not, of course, necessary that the revenues employed in paying for the projects be collected equally from all voters, either in terms of tax rates or tax collections. The minimum necessary condition is that the benefits from public activity be significantly more concentrated or localized than the costs. This is a very weak condition, and many budgetary patterns seem to meet it. If the taxes are collected by indirect methods so that individuals cannot really tell how much they individually pay for each specific public-service project, this accentuates the distortions described by our analytical model. In the marginal case the individual may be indifferent about projects benefiting others, the costs of which seem slight to him and also difficult to measure. Under these circumstances he would be particularly likely to trade his support for such projects, which may appear costless or nearly so, for reciprocal support for his own pet proposals.

Additional types of governmental activity may also be fitted into the analysis. Other forms of taxation-expenditure problems are most easily incorporated. First, we may suppose that there is some governmental activity that provides general benefit to all voters, e.g., police protection, which is financed out of general taxation. In this case the maximizing solution and the Kantian solution will tend to be identical to the extent that the benefits and the taxes are truly general. However, as soon as general taxation is departed from, parallel reasoning to that above demonstrates that special tax exemptions and favors to individuals and groups will be introduced.

On the tax side of the fiscal account, if a given sum of money is to be raised, we should expect the revenue-raising pattern to include general taxes that are, relatively, “too heavy,” but which are riddled with special exemptions for all sorts of groups. The result is that of greatly reducing the efficacy of any generally accepted norms for fiscal organization (such as progression in taxes) that are supposedly adopted. The pattern that we are able to predict as a result of our analysis thus seems to be descriptive of existing fiscal institutions, quite independently of the moral justification of the behavior that our model incorporates. General and diffuse taxes, characterized by many special exemptions, finance budgets in which public services are designed, at least to a large degree, to benefit particular groups in the society. There is clearly no apparent conflict between the predictions that emerge initially from our model and fiscal reality as it is commonly interpreted.

If our analysis is to be applied even more generally to all public activity, it must be radically generalized. For any individual voter all possible measures can be arrayed according to his intensity of interest. His welfare can be improved if he accepts a decision contrary to his desire in an area where his preferences are weak in exchange for a decision in his favor in an area where his feelings are stronger. Bargains among voters can, therefore, be mutually beneficial. Potentially, the voter should enter into such bargains until the marginal “cost” of voting for something of which he disapproves but about which his feelings are weak exactly matches the expected marginal benefits of the vote or votes secured in return for support for issues in which he is more interested. Thus, he will expect to benefit from the total complex of issues which enter into his set of bargains with his fellows. In making such bargains, however, the individual must try to gain the assent of only a bare majority of other voters, not of all of them. On any given issue he can simply ignore 49 per cent of the individual decision-makers. This means that he can afford to “pay” more for other support because a part of the inconvenience caused by the measure will fall on parties who are not members of the decisive bargaining coalition.

Unfortunately, from the point of view of the individual voter, the converse also holds true. Bargains will certainly be concluded in which the single voter does not participate. Yet he will have to bear a part of the costs of action taken. As a result, the whole effect of the measures which result from his bargains and on which he votes on the winning side will be beneficial to him; but this will tend, normally, to be only slightly more than one-half of all “bargained” measures passed, and the remainder will be carried out adverse to his interest. The same result would hold true for the average voter under a pure referendum system. The whole problem analyzed here can be eliminated by changing the rule which compels the minority to accept the decisions of the majority without compensation. So long as this rule is employed to make collective decisions, the individual voter must expect to incur external costs as a result of public or collective action.

11.

Simple Majority Voting and the Theory of Games

We shall now examine the contributions that modern game theory can make toward an analysis of simple majority voting. In one sense we shall be discussing the same problems considered in Chapter 10, but we shall use here a slightly different set of analytical tools. As will become evident to those who are even moderately sophisticated in the field, our constructions will be reasonably elementary. Our purpose is, however, not that of making any contribution to game theory itself, but rather that of applying the relevant theory to our particular problems.21

The application of game theory to majority voting is relatively straightforward and simple, but the limited extent to which game theory can be helpful for our purposes should be acknowledged at the outset. Most of the refinements in this theory have been developed in the analysis of two-person, zero-sum games. Quite clearly, the analysis of such games will not take us very far in predicting the outcomes of simple majority voting rules in the political process. For assistance here, we must look to the developments in the theory of n-person games, a theory that is considerably less sophisticated and more speculative than is that for two-person games. The zero- or constant-sum restriction is also bothersome, but, to some extent, this hurdle can be surmounted.22

A Three-Person, Constant-Sum Game

As was the case with our model in the preceding chapter, it will be useful to “idealize” the institution under consideration, that is, to construct a model which will embody the essential characteristics of the institution without the complicating features. The model to be employed here must be even more restricted than the one used earlier. We shall initially assume that the total group is composed of three persons, equally situated. In order to relate the analysis to that of the preceding chapter, we may also assume that the individuals are farmers in a township interested in road repair. We shall assume further that the repair of one man’s road produces no external or spillover effects on other members of the group.

We assume that a decision has already been made to spend a total of $1 (additional zeros will not modify our analysis) on road repair in the whole township. For simplification, let us suppose also that this sum is not raised from general taxes but is instead received in the form of an earmarked grant from some higher-level governmental unit. This assumption assures us that the game we shall consider will be one of constant-sum at $1. We continue to assume that all decisions concerning the allocation of road-repair funds are to be made by simple majority vote, and that this is the only accepted way of making collective decisions. In our first model, we analyze the operation of this rule in an isolated, single action: that is to say, the $1 grant is received only once and it must be allocated once and for all and in complete abstraction from other collective issues that may arise.

This “game” may now be normalized and put in characteristic-function form as follows:

This characteristic function states the values of the various possible coalitions that may be formed. The function clearly shows that no “coalition” composed of less than two members of the group will have value, while all coalitions of two or more members will have a value of one. If the members of a winning two-person coalition choose to share their gains symmetrically, the following three imputations become possible “solutions”:

(½, ½, 0)    (½, 0, ½)    (0, ½, ½).

This set of imputations will be called F, or the F set. This set, and this set only, satisfies the Von Neumann-Morgenstern requirements for “solution” to n-person games, and may, in a restricted sense, be called the solution. The first of these requirements is that no single imputation in F either dominates or is dominated by any other imputation in the same set. (Domination is defined in terms of the effective decision-making subgroup or coalition: two in the model under analysis.) The second requirement is that any imputation not in F is dominated by at least one imputation in F.23

The dominance aspects of the imputations in F may be illustrated with reference to proposed shifts to imputations not in F. Suppose that the imputation (0, ½, ½) is proposed by a majority coalition (2, 3). Individual 1 can propose an alternative imputation (¼, ¾, 0), which the coalition (1, 2) can carry (which dominates the first imputation). Individual 2 might be led to abandon the first coalition with 3 and support the modified proposal since his position will be improved (¾ > ½). However, this second imputation, which is not in F, will, in turn, be dominated by the imputation (½, 0, ½), which is in F for the majority (1, 3). Individual 2 may be wary about any initial departure from the coalition with 3 if he foresees the prospect of more than one move before action is finally taken.24 Because of this fact, the imputations in F are presumed to be more stable than those not in F, although game theorists recognize and acknowledge the limitations on the ideas of “solution” and “stability” in the n-person game.

The set of imputations, F, contains the imputations that we could predict from the operation of majority voting in isolated actions. Two persons would tend to secure all of the benefits while the third person would secure nothing, assuming that each individual approaches the collective decision with a view toward maximizing his own expected utility, and assuming that individual utility functions are independent. Note that the set F includes imputations that dominate the “equitable” imputation (1/3, 1/3, 1/3).25Any one of the three imputations in F dominates the equitable imputation with respect to a required number of individual voters. The equitable imputation would seem, therefore, to be the most “unstable” of all imputations since any majority can upset it. Compare this with another “weak” imputation not in F, say, (¼, ¾, 0). This imputation is dominated only by the imputation (½, 0, ½) in F, and by a limited subset of other nonstable imputations. Hence, to change from (¼, ¾, 0) to a solution in F, a particular majority (1, 3) is needed, whereas to shift from (1/3, 1/3, 1/3) to a solution in F, any majority will be sufficient. Thus, the “equitable” imputation may be stabilized only by significant departures by many individuals from utility maximization.

A Five-Person, Constant-Sum Game

Let us now extend this analysis to a five-person group, with the same initial conditions assumed. We continue to assume simple majority rule so that three persons are now sufficient for decision. The characteristic function is now as follows:

For the solution, set F, developed as before, we get:

Note that any one of these imputations in F dominates what we have called the equitable imputation (1/5, 1/5, 1/5, 1/5, 1/5) for the required decisive coalition of three persons. On the assumption of individual utility maximization, therefore, the equitable imputation would never be chosen.

It is clear that the analysis can be extended to a group of any size. The F-set, or “solution,” imputations will always contain only those involving the symmetric sharing of all gains among the members of the smallest effective coalition. In the game of simple majority rule the smallest effective set will approach 50 per cent of the total number of voters as the group is increased in size. Imputations within the solution set can always be found which will dominate, for an effective coalition, any imputation outside the set. As the size of the group is increased, however, the stability properties of the imputations in the set F seem to become less strong. In our earlier example of the three-person game, we found that the solution within the F set tends to be more stable than any similar set of imputations outside F because successful individuals might be able to foresee the consequences of departing initially from a coalition formed within F, which dictated that the gains be shared symmetrically among the members of the coalition. These consequences are, of course, that members of an apparently effective coalition might, before action is finally taken, be replaced by outsiders in a newly formed coalition.

It is perhaps useful to note that the argument for symmetry in the sharing of the gains among members of the dominant coalition rests on slightly different grounds than it does in the case with two-person co-operative games or in n-person games requiring that all participants must agree on a sharing arrangement. Schelling, in his recent argument for abandonment of symmetry, confined his discussion largely to these latter games.26 If, as in the “majority-rule game” that we are considering here, the rules dictate that only a certain share of the total group need agree, the case for effective-coalition symmetry is stronger. The individual in the winning coalition will tend to be satisfied with a symmetrical share in total gains, not because he expects no member to concede him a larger share due to a general attitude of “fairness,” but because he knows that, if he does demand more, alternative individuals stand ready and willing to join new coalitions which could effectively remove his gains entirely.

As the total group grows in size, these effective restraints on individual action are weakened. The individual will reckon his own contribution to an effective coalition at a lower value, and he will be more tempted to depart from imputations within the “solution.” The outcome of the majority-rule game in large groups seems likely to be that predicted by our model of Chapter 10. Coalitions will be formed, but any single winning coalition will be relatively unstable and impermanent. On the other hand, it should also be emphasized that as the size of the group becomes larger, any tacit adherence to moral or ethical restraints against individual utility-maximizing behavior also becomes much more difficult to secure. The deliberate exploitation of the third member by any two members of a three-man social group may be difficult to conceive, but the individual’s interest in his fellow man falls off quite sharply as the group is enlarged. In this sense, therefore, the basic assumptions of the game-theory model become more relevant for large groups than for small ones. The concept of “solution” may be considerably more fuzzy in large-group situations, but the direction of effect that may be predicted to emerge seems to be of significant relevance for any study of real-world political decision-making.

The Limitation of Side Payments

We have analyzed the operation of majority voting in the simplest of models. We have assumed the group to be confronted with a single issue that was to be decided once and for all. As applied to real-world institutions, the limitations of this model must be carefully kept in mind. Many of these have been obscured in the analysis above, and some of them must now be mentioned. In the first place, as we have suggested in Chapter 10, logrolling or vote-trading processes would tend to arise when more than a single issue is presented to voters. We propose, however, to leave this complication aside for the time, and to assume that all forms of vote-trading are prohibited in some way. If we want to employ the terminology of game theory here, we may say that all side payments are prohibited. This prohibition effectively prevents the individual voter from being able to express his intensity of preference for or against the specific measure proposed. All that he may register is the direction of this preference, not the intensity. Implicit in the support of decision-making institutions and rules which do serve, wholly or in part, to limit side payments seems to be the psychological assumption that individual preferences are essentially symmetrical.27

Let us see precisely what this complete prohibition of all side payments implies for our “solution” imputations. Consider the same three-person game discussed above, in which the $1 grant is to be divided among the three roads, with each repair project benefiting only one individual. Let us assume that, in actuality, road repair is highly productive on only one of the three roads, moderately productive on the second, and not worth the cost on the third. The values resulting from one-half (50¢) of the total expenditures on each road, respectively, are as follows: $1, 50¢, 25¢, or to use fractions: 1, ½, ¼ (note that these are not imputations). Simple majority voting, with all side payments (open and concealed) being prohibited, will convert all such “political games” into a fully normalized form. The solution set of imputations will be the same as before. Quantified or measured in terms of input or cost values, this set is:

(½, ½, 0)    (½, 0, ½)    (0, ½, ½).

It is now necessary, however, to distinguish between input or cost values and output or benefit (utility) values. The latter become, in the same set of imputations:

(1, ½, 0)    (1, 0, ¼)    (0, ½, ¼).

The important conclusion here is obvious. In benefit or productivity terms, the “game” is not constant-sum, and, with all side payments prohibited, there is no assurance that collective action will be taken in the most productive way. There is no more likelihood that the first imputation will be chosen than the second or third. The rule is as likely to select the least “productive” imputation as it is the most “productive.”28

The prohibition of all side payments also prevents any imputation being selected which directly benefits less than a simple majority of the voting population, regardless of the relative productivities of public investment. For example, let us now suppose that the $1 grant, if expended exclusively on the first road, would yield a benefit value of $10, on the second road $5, and on the third road only $1. If, in fact, all funds were expended on the first road, the imputation would be (10, 0, 0). However, note that any imputation such as (0, 2½, ½) would dominate the more concentrated, but more productive, investment. The set of imputations having the solution properties under the conditions outlined would be:

(5, 2½, 0)    (5, 0, ½)    (0, 2½, ½).

These rudimentary elements of game theory have helped us to demonstrate in a somewhat different, and perhaps more decisive, manner the effects of simple majority rule that were already discussed in Chapter 10. If some vote-trading is not introduced, no allowance can be made for possible variations in individual intensities of preference, a point that is rather dramatically shown in a quantitative way in the last simple model.

Allowance of Side Payments

The apparent distortions that may be produced by the operation of simple majority rule without side payments suggest that the model with side payments be examined. Side payments may “improve” the results. We propose, therefore, to examine this prospect more carefully. Let us now suppose that there exists complete freedom for individuals to make all of the side payments or compensations that they choose to make. No restrictions are placed on the methods of payments, but we may think of them as being made in generalized purchasing power, or money. Such behavior of individuals is assumed not to be prohibited by either legal or moral restraints. This model allows us to introduce something akin to vote-trading in the model without departing from the confines of a single, simple issue.

Let us assume the existence of the last benefit schedule mentioned above: that is, if the whole grant were to be expended on each road, the “productivities” would be, respectively, $10, $5, and $1. Simple majority voting, with full side payments, will now produce a “solution” set of imputations as follows:

(5, 5, 0)    (5, 0, 5)    (0, 5, 5).

In the first imputation, Individual 1 gets all of the grant expended on the repair of his own road, but he must pay Individual 2 one-half of the monetary value of the net gains for his political support. In the second imputation, Individuals 2 and 3 simply trade places. The third imputation in the solution set is most interesting. Here all road repairs are still carried out on the first road, where investment is far more productive than on the other roads, but Individuals 2 and 3 form the political majority which forces Individual 1 to pay full compensation for the road repair that he secures. Despite the fact that only his road is repaired, Individual 1 is no better off after collective action is taken than he is before.29

We see that the results of simple majority voting in the model where full side payments are allowed differ in several essential respects from the results of this rule when such payments are not allowed. First of all, side payments insure that the funds will be invested in the most productive manner. Secondly, there is no requirement that the projects undertaken provide physical services to more than a majority of the voters. As in all of the earlier models, the solution will embody a symmetrical sharing of total gains among the members of the smallest effective coalition, but note that the introduction of side payments tends to insure a symmetric sharing of gains measured in benefit or productivity terms.

In contrast to a logrolling model, the model which does include open buying and selling of votes (that is, full side payments in money) does not seem characteristic of modern democratic governments. We do not want to prejudge the ethical issues introduced by this model at this time, but commonly accepted attitudes and standards of behavior, as well as established legal institutions, prevent any approach to full side payments being carried out in actuality. In spite of this, the model is a highly useful one in that it does point to the type of solution attained under the more complex models which allow indirect side payments to be made.

Simple Logrolling and Game Theory

We refer, of course, to those vote-trading or logrolling models that have been discussed in Chapter 10. The simple logrolling model falls halfway between that containing no side payments and that which allows full side payments. In order to introduce logrolling, we must depart from single issues and assume that the group confronts a continuing series of separate measures. In game-theory terms, logrolling is simply an indirect means of making side payments. Individuals are unable to “purchase” voter support directly with money, but they are able to exchange votes on separate issues.

Let us continue to employ the road-repair example, with the prospect of a $1 grant from external sources being made available to the community for disposition in each of a successive number of time periods. Let us also assume the same payoffs as before: namely, that the productivity of a $1 investment on Road 1 is $10, and on Road 2, $5, and on Road 3, $1. We must also now make some assumption about the marginal productivity functions in this model. We shall assume that, over the range of decisions considered in any bargain, the marginal productivity of investment on each road is constant: that is to say, the productivity of any $1 investment on Road 1 is $10, regardless of the amount of incremental investment undertaken on that road in previous periods.

Recall that under simple majority voting without side payments the solution set of imputations, measured in benefit terms, was:

(5, 2½, 0)    (5, 0, ½)    (0, 2½, ½),

while in the model with full payments, this set was:

(5, 5, 0)    (5, 0, 5)    (0, 5, 5).

In the first case, the repairs would be carried out on any two of the roads represented in an effective coalition, not necessarily those roads most in need of repair. In the second case, the repairs would tend to be made where the investment is most productive, with a side payment or payments being made to insure sufficient support in the voting process.

In our simple logrolling model, the only way in which the first individual can “purchase” support for repairs on his road is by agreeing to vote for the repair of some road other than his own. He cannot substitute for this the more “efficient” transfer of money. It is difficult to present the results here in terms of a single set of benefit imputations because we must include a whole series of issues, but clearly these results must approach more closely those of the first rather than those of the second alternative model. Since some funds must be devoted to relatively unproductive investment, in some periods, the greater “efficiency” of the second model cannot be secured. We may convert simple logrolling into a political game by considering a single road-repair project in which the individual beneficiary secures majority support by giving promises of reciprocal support on future proposals, with these “promises” commanding some current economic value. The general logrolling model can then be thought of as consisting of a sequence of such games. There are, however, some differences between the simple logrolling model or its game analogue and the basic games discussed earlier. Simple logrolling, even if the issues are closely related to each other, can introduce minimal improvements in “efficiency.” The process removes the necessity of insuring some physical benefits to an absolute majority for each single piece of legislation. Road repairs could, in any one period, be devoted exclusively to one road. Moreover, if there should exist important returns to scale of single-period investment, this could produce significant efficiencies.

Our general logrolling model can best be interpreted on the assumption that the political process embodies a continuing series of issues: in specific reference to the illustration, separate road-repair proposals. If, however, all road-repair projects must be voted on a single omnibus proposal, the results become equivalent to those demonstrated in the elementary games previously discussed. In this case, a minority of farmers will secure no road repairs, whereas in the general logrolling model, even under majority rule, each road would tend to be repaired because of the multiplicity of issues allowing for many separate coalitions. This difference between these two majority-rule models, however, will not affect the individual constitutional evaluation of majority voting as a means of making political decisions. In the one case, external costs will be expected because of the excessive road repairs generally carried out; in the other, external costs will be expected because of the fact that the individual might occasionally find himself in the losing coalition on a single, large, omnibus issue.

Complex Logrolling

In our example, we have discussed the game theory aspects of logrolling phenomena that are confined to closely related issues. Instead of this, logrolling may actually take place by the trading of votes over a wide range of collective decisions, which may or may not bear physical resemblance to each other. As the “bargains” expand to include more heterogeneous issues, it seems clear that the results will begin to approach those emerging from the model which allows unrestricted side payments. If there is a sufficient number of issues confronted by voters at all times, and if the range and distribution of the individual intensities of preference over these issues are sufficiently broad, the complex logrolling process may approximate unrestricted side payments in results. Insofar as this is true, the full extent of the differential benefits from public outlay, or the differential costs of general-benefit legislation (that is, the differential intensities of individual preferences), can be exploited. The individual voter who is either strongly opposed to or strongly in favor of certain measures may, if necessary, “sell” his vote on a sufficient number of other issues to insure victory for his side in the strongly preferred outcome. His “purchasing power” is determined by the value of his support on all issues considered by other voters. Of course, the individual voter will rarely want to use up all of his purchasing power on any single measure, just as the individual consumer in the marketplace rarely uses up all his purchasing power on a single commodity or service. Complex logrolling of this type remains a “barter” system, but it merges into a pure “monetary” system (that is, one with full side payments) as the range of issues undertaken collectively is broadened. Implicit logrolling (discussed in Chapter 10), in which the voter is presented with a complex set of issues at the same time, is one form of the complex logrolling discussed here. If the voter is enabled to choose from among a sufficiently large number of alternative sets, his effective “purchasing power” approaches the limit that would be available to him under a “monetary” system.

The “Individual Rationality” Condition

To this point our models have been simplified by the assumption that the choice or choices facing the group involve only the final sharing of an earmarked grant or grants received from external sources. We now propose to make the models somewhat more realistic by dropping the external-grant features. Let us now suppose (just as we did in Chapter 10) that all funds for road repair are to be raised from general taxes levied uniformly on all citizens. We return to the simplest three-person game initially analyzed. This “new” game can also be discussed in the normalized form. To do so requires only that we attribute a fixed monetary sum to the various individuals at the outset. In the three-person game let us suppose that each person retains, at the beginning of “play,” $1/3; the beginning imputation is (1/3, 1/3, 1/3). Now assume further that “play” is to involve, in every case, the disposition of $1. The form of the characteristic function is not changed:

As in the earlier game, the individuals acting jointly as a group, [v(1, 2, 3) = 1], for example, under a rule of unanimity, cannot receive more than the gainers receive from the formation of coalitions under simple majority rule. There is, however, one major difference between the game now under consideration and the simpler one discussed earlier. In the previous game there could exist complete individual freedom to withdraw from the group. Since the funds to be expended there were assumed to come from outside the group itself, the withdrawal of a member would not serve to reduce the total gains to be secured. In other words, the earlier game satisfied a condition which may be represented as an adaptation of what Luce and Raiffa call the condition of individual rationality.30 They define this condition as follows:

v({i}) ≤ x_i for every I in I_n.

This condition states that no individual in the whole group, I_n, will ever receive less by being in the “game,” regardless of whether or not he is in the winning or losing coalition, than he would if he “played alone” against all other members of the group. Applied to our particular problem, “playing alone” ({i}) may be interpreted as withdrawal from the game altogether.

The relevance of this condition is obvious when the purpose is that of analyzing “voluntary” games, and when it is further recognized that most of the game situations in which the individual finds himself do, in fact, represent such voluntary games. The extension of game-theory models to any analysis of political decision-making requires some consideration of “coercive” games. The condition of individual rationality, as we have stated it above, need not be satisfied at all. The individual participant in collective decision-making may, in many of the actual choices made through the political process, prefer to withdraw from “play.” This does not suggest that the individual necessarily would want to withdraw from participation in the whole set of games represented by state action (although, conceptually, he could also want to do this). In any case, the individual can normally neither choose the political “games” in which he desires to participate nor can he withdraw from the ultimate social contract readily. He must remain as a participant on each issue that the group confronts.

Returning to the simple game before us, the individual, if he should be allowed to withdraw, could always retain his original value of $1/3. It follows that he would not voluntarily accept an expected value of less than 1/3 in any game if he were offered the alternative of not playing. However, in political groups, such action is not normally possible. Individuals cannot refuse to pay taxes even though they find themselves in a minority.

The solution set of imputations, in cost values, will be equivalent to that in the initial three-person game:

(½, ½, 0)    (½, 0, ½)    (0, ½, ½).

In each of these imputations, one of the three persons will be made worse off than when play begins. However, as a member of the political unit for whom decisions are being made, he is forced to submit to the results indicated by the operation of the rules.

The Limits to “Social” Waste

The majority-rule game considered here results in a net transfer of real income from one member of the three-person group to the other two members. Such transfers could, of course, take place directly without any necessity that tax revenues be expended in the provision of public services. In constitutional democracies, however, some limitations on majority action are almost always to be found. Moreover, since the individuals in our model are assumed equal in fiscal capacity at the outset, directly redistributive transfers would probably be prevented by constitutional provisions and traditions. If such transfers are prohibited, the majority coalition may effectively exploit the minority only through levying general taxes to provide special benefits, or through financing general benefits by special taxes. With this in mind, we shall now consider the extent to which the operation of simple majority voting rules can produce “social” wastage of resources.

If the solution set of imputations shown above is assumed to represent the imputed sets of individual evaluations of the public services (road repairs), note that there is no over-all wastage of resources. No “inefficiency” is introduced by the combined taxing-spending operation. The imputation (½, ½, 0), for example, means, in this sense, that an expenditure of $½ on the first person’s road yields to him an estimated value of $½; similarly, for the second man. The total additions to utility created by the expenditure of the $1 are valued at the same total as are the total subtractions from utility caused by the necessary taxes (½ + ½ = 1/3 + 1/3 + 1/3). The “productivity” of the public expenditure is exactly equal to the alternative “productivity” of the resources should they have been left available for private disposition. This means that no introduction of side payments could modify the results, which are identical to those of purely redistributive transfers. Such transfers, by definition, involve no “social wastage” in the sense considered here, assuming, of course, that the supplies of the productive factors are not affected.

Let us now suppose, however, that the expenditure of $½ on the first person’s road yields to him an incremental utility that he values at $5/12, and similarly for the second and third man. Under this modified assumption about the productivity of road repairs, we get a set of possible solution imputations as follows:

(5/12, 5/12, 0)    (5/12, 0, 5/12)    (0, 5/12, 5/12).

Note that it will still be profitable for the members of the winning coalition to play the game (5/12 > 1/3), but the total estimated value of the “gains” is less than the “losses” (10/12 < 1), or, in net terms, (1/3 > 1/6). If these individual evaluations can be compared in some way, then clearly “social wastage” of resources must be involved in the carrying out of the majority decision. One means of allowing some comparison of individual utilities is, of course, that of allowing side payments. If these are introduced, the set of imputations above cannot be said to represent any solution. Instead, in each imputation the person in the minority could always offer to compensate at least one of the others in order to get him to refrain from playing. For example, the imputation (11/24, 11/24, 2/24) outside the set above is dominated by no imputation in the set. Hence, the set of possible solution imputations,

(5/12, 5/12, 0)    (5/12, 0, 5/12)    (0, 5/12, 5/12),

does not satisfy the Von Neumann-Morgenstern requirements. In this situation it does not seem likely that the “game,” which must be negative-sum, will be played at all. No road repairs will be undertaken.

It should be remarked, however, that this result follows only if side payments are allowed. If neither purely redistributive income transfers nor side payments are possible, there is nothing that can arise to prevent the social process from proceeding, even if, translated into game-theory concepts, the game is one of negative-sum. Under the same productivity assumptions as before, the set of imputations,

(5/12, 5/12, 0)    (5/12, 0, 5/12)    (0, 5/12, 5/12),

now takes on all of the characteristics of the Von Neumann-Morgenstern “solution.” The person in a minority position can offer a maximum of 1/3 to another to refrain from playing.

It is reasonably clear from this analysis that the limits to resource wastage that could possibly result from the operation of simple majority rule will be determined by the size of the group. In our model three-person group, a “total productivity” of public investment must be at least two-thirds as great as that sacrificed in the private sector. In a five-person group this fraction becomes three-fifths. The maximum limits to resource wastage are defined by the fraction M/N, where M is the minimum number of voters required to carry a decision, and N is the number of voters in the whole group for which choices are to be made. Thus, at the limit, a public-investment project need only be slightly more than one-half as productive as the private-investment projects that are sacrificed, productivity in each case being measured in terms of the individual evaluation of benefits.31

This analysis is not intended to suggest that majority-rule “games” will tend to be constant- or negative-sum. In many cases, the game will, of course, be positive-sum. By altering the productivity assumptions of our simple models here, the results of positive-sum games are readily attainable. Let us suppose that the investment of $½ on each road yields $1 in benefits, as estimated by the individuals themselves. The “solution” set of imputations becomes:

(1, 1, 0)    (1, 0, 1)    (0, 1, 1).

Note that here, as in the constant-sum case, the introduction of side payments will not change this solution. Under the conditions outlined, the introduction of side payments will change the solution only if the game is negative-sum.

This limitation is no longer present, however, if we introduce some asymmetry in the benefit schedules, that is, if we assume that the productivity of public investment may vary from road to road in our model. We can, of course, conceive of games with asymmetrical benefit schedules which are positive-, constant-, or negative-sum. Moreover, a game may be switched from positive- to negative-sum within a single “solution” imputation. Consider the following set:

(11/12, ½, 0)    (11/12, 0, 1/12)    (0, ½, 1/12).

Let the imputed values represent the estimated individual evaluations of the public investment of $½ on each road. Thus, the set takes on the properties of a solution unless side payments are allowed to take place. No imputation in the set is more likely to be chosen than another. If the first imputation is chosen, the game, for the whole group considered as a unit, is positive-sum (17/12 > 1); if the second imputation is chosen, constant-sum (1 = 1); if the third imputation, negative-sum (7/12 < 1).

The introduction of side payments will insure that the second and the third imputations would never be produced, and even the first imputation would not exhibit the required stability properties required for solution. The F set would in this case become

(11/12, 11/12, 0)    (11/12, 0, 11/12)    (0, 11/12, 11/12),

assuming constant returns to investment on the first road.

The General Benefit-Special Taxation Model

The previous models have incorporated the assumption that public projects providing differential benefits to individual citizens are financed by general taxes imposed equally on all citizens. The elementary propositions of n-person game theory applied to these models enable us to predict that serious resource wastage can result from the operation of simple majority rule. The reasons are the same as those discussed in Chapter 10. Majority rule allows members of the decisive coalition to impose external costs on other individuals in the group, costs that are not adequately taken into account in the effective decisions. Aggregate marginal costs exceed the aggregate marginal benefits from public investment. Relatively too many resources are invested in the type of public projects analyzed in the model—relatively too many as compared with both alternative private employments of resources and with alternative public employments.

The assumption that general taxation is levied to finance special benefits is clearly more descriptive of real-world fiscal institutions than the converse case. Ethically accepted principles which have long been espoused and which have found expression in modern tax institutions stress the importance of generality in the distribution of the tax burden among members of the social group. No such principles have guided the distribution of public expenditure among the several possible uses. However, in order to make our analytical models complete, it will be useful to modify our assumptions and to consider the reverse situation. Let us try to apply the elementary game-theory constructions used above to the model in which collective goods, providing general (equal) benefits to all citizens, are financed by discriminatory taxation. The analysis is relatively straightforward, but, interestingly enough, this model is not symmetrical in all respects with the one previously considered, as we shall demonstrate.

We begin, as before, with an initial imputation (1/3, 1/3, 1/3), which represents asset values held by the individuals. We now introduce a general-benefit situation. Suppose that the group is confronted with the opportunity to purchase a genuinely collective good, the benefits from which are not divisible; if one individual secures these benefits, each individual in the group must secure them in like amounts. As a first example, let us assume that each individual estimates his own benefits from the good to be 1/12. Assume further that the total costs of the collective good are 4/12 or 1/3. If the good is purchased, the final imputation of benefits, from the collective good alone, must be (1/12, 1/12, 1/12). However, what is relevant in this case is the “net” imputation that will result from the purchase of the collective good and the retention of shares of the initial assets.32 The effective coalition will tend to impose special taxes on the minority, producing a “solution set” as follows,

(5/12, 5/12, 1/12)    (5/12, 1/12, 5/12)    (1/12, 5/12, 5/12)

assuming that side payments are not allowed. The over-all investment is not worth the cost (3/12 < 4/12); but, if taxes can be imposed in a discriminatory manner, it will still be an advantageous project from the point of view of the members of the effective coalition (5/12 > 4/12). The game in our illustrative example is negative-sum. Positive- or constant-sum games can also be constructed in this framework. Our purpose in this illustration is to demonstrate the possibility of negative-sum games being played and, thus, the possible wastage of resources. In the illustrative example here, the public investment should not have been undertaken since the resources employed are more productive if left in the private sector of the economy.

It can readily be seen that there are no effective limits to the possible extent of resource wastage under the assumptions of this model. Any project yielding general benefits, quite independently of cost considerations, will be supported by the dominating majority if they are successful in imposing the full tax financing of the project onto the shoulders of the minority. This feature differs substantially from the general-taxation model, where some quantitative limits could be estimated for the degree of resource wastage made possible under majority rule. Note that this feature also differs from the general implications of the logrolling analysis of Chapter 10. The analysis there implies that general-benefit projects would tend to be slighted in favor of special-benefit projects. This implication must be carefully constrained; it remains clearly true only if the assumption of general taxation is retained. If discriminatory taxation is allowed, there seems to be no a priori reason to expect special-benefit projects to take a dominating role in the operation of majority rule, except for the general presupposition that individuals may be more interested in special-benefit projects.

There is another important respect in which the general-benefit model is asymmetrical with the general-taxation model. In the latter, we have been able to demonstrate that, under the operation of simple majority rule, relatively too many resources are likely to be devoted to special-purpose public-investment projects. To be fully symmetrical with this, the general-benefit model might appear to require the conclusion that relatively too few resources be devoted to general-purpose public projects. This conclusion, however, cannot be supported. It can be demonstrated that relatively too many resources will be devoted to both special-benefit and general-benefit public projects under the operation of simple majority rule. This is an especially significant implication that emerges from our application of game theory to this voting rule, and the demonstration deserves to be carefully presented.

We shall show that every general-benefit project that is worth its cost will tend to be adopted by simple majority voting: that is to say, we shall try to prove that all possible projects involving resource investment more “productive” than the alternative investment in the private sector of the economy will tend to be adopted by majority rule. If this can be demonstrated to be true, our main point will have been established because, in the illustrative model first employed in this section, we have shown that some unproductive projects (negative-sum games) will be selected.

The proof is almost intuitive. If the dominant majority is able to impose the full costs of general-benefit projects on the minority, it follows that all projects yielding any benefits at all to the majority coalition members, and costing no more than the maximum taxable capacity of the minority, will be adopted without question. In our current example, any general-benefit project (any pure collective good) that costs up to 1/3 will surely be adopted. This is because, if discriminatory taxation is allowed, a sum up to this amount may be collected from the single minority member of the group. Hence, for all such projects a member of the majority coalition may secure some net benefit without cost to himself.

As the costs of collective goods move beyond the maximum taxable capacity of the minority member of the group, beyond 1/3 in our example, the individual members of the majority will be able to balance off gains against costs. Since they are the residual taxpayers, their own calculus will insure that a more than satisfactory balancing off will be achieved. Any project will be adopted that provides the group with general benefits valued more highly than the alternative private investments. While it is true that in making their final decisions they do not include in their calculus the full marginal benefits of the collective goods, because, by definition, these goods provide benefits to all members of the group equally, neither do the members of the majority include the full marginal costs. Moreover, the calculus will always reflect a more accurate estimate of marginal benefits (since the minority members will receive only an equal share) than of marginal costs (of which the minority members will bear more than an equal share).

In our analysis of the general-benefit model, we have not introduced side payments. If these are introduced, the effects are similar to those traced in the general-taxation models. Side payments will insure that no negative-sum games will ever be played: that is to say, “unproductive” public investments could never be undertaken if full side payments were to be permitted. If indirect side payments in the form of logrolling are allowed, some mitigation of the resource wastage involved in the operation of majority rule decision-making is to be expected. The extent of this mitigation will be dependent on the extent and range of the logrolling that takes place.

The General Taxation-General Benefit Model

Many of the modern activities of governments can be classified as falling within one of the two models previously discussed or in some combination of the two. For completeness, however, there remains the examination of those activities undertaken by governments that provide general benefits and are financed from general taxation. Let us assume that a community of identical individuals is faced with the task of providing a genuinely collective good. Benefits from this good are to be distributed equally among all citizens. This good is to be financed by taxation that is also equally distributed among all citizens.

It is immediately clear in this model that the collective-choice process does not take on the attributes of a game, regardless of the rules that may be adopted for decision-making. In this model the political process offers to the individual participant no opportunity to gain differential advantage at the expense of fellow participants. When the individual makes a decision, under any rule, he must try to compare the advantages that he will secure from the availability of the collective good and the costs that he will undergo from the increase in the general tax. His behavior can exert no external effect, either in costs or benefits, on third parties.

Communities are not, of course, made up of identical individuals. Moreover, once differences among individuals in tastes, capacities, endowments, etc., are admitted, the model for general taxation and for general benefits becomes much more difficult to discuss. It remains possible to imagine a collective decision in which the benefits from the public services provided are distributed among the membership of the group in such a manner as to precisely offset the distribution of the tax burden for this particular extension of service. In this case, where public expenditure is financed solely on some principle of marginal-benefit taxation, the conclusions reached above will hold. The individual cannot benefit at the expense of his fellows through the political process, and the game analogy breaks down. It is clear, however, that this model cannot be observed in the real world. We know that public services provided by governmental units do exert differential benefits and that these services are financed by taxation that is not general in the sense required for this extreme conceptual model.

The usefulness of this model lies in its implication that, insofar as collective action takes on such characteristics of generality (that is, nondiscrimination), the applicability of the game-theory conclusions is reduced. As we have emphasized elsewhere, the trend away from general legislation toward special legislation in modern democracies makes the conclusions drawn from the game-theory analogues more applicable than they might have been a century past.

Conclusions

The generalized conclusion that may be reached as a result of the application of elementary game theory to the institution of simple majority voting is evident. There is nothing inherent in the operation of this voting rule that will produce “desirable” collective decisions, considered in terms of individuals’ own evaluations of possible social alternatives. Instead, majority voting will, under the assumptions about individual behavior postulated, tend to result in an overinvestment in the public sector when the investment projects provide differential benefits or are financed from differential taxation. There is nothing in the operation of majority rule to insure that public investment is more “productive” than alternative employments of resources, that is, nothing that insures that the games be positive-sum. Insofar as the vote-trading processes which emerge out of the sequence of separate issues confronted produce something akin to side payments, this resource-wasteful aspect of majority voting will tend to be reduced in significance.

The whole question of the relationship between the operation of simple majority voting rules and the “efficiency” in resource usage, within the context of the game-theory models, can best be discussed in terms of the constructions of modern welfare economics. In the following chapter we shall introduce these constructions in specific reference to the analysis of this chapter.

12.

Majority Rule, Game Theory, and Pareto Optimality

At several points in this book we have found, and shall find again, occasion to relate our analysis to that of modern welfare economics. This seems to be particularly useful following our application of elementary game theory to the operation of majority voting rules. By examining our results in comparison with the criteria of efficiency or optimality employed by the welfare economist, a somewhat better appreciation of the constitutional-choice problem may be achieved. To this point we have, in several instances, made reference to the Paretian criteria for efficiency. In Chapter 7 we discussed these criteria briefly. Additional discussion is wholly unnecessary for some readers, but even at the risk of introducing some redundancy, we shall first try to clarify the meaning of the fundamental Paretian construction.

Pareto Optimality

The criterion that the modern welfare economist employs in determining whether or not a given situation is “efficient” or “optimal” and whether or not a given move or change is “efficient” or “optimal” was developed by Vilfredo Pareto. We shall first define this criterion carefully, and we shall then distinguish two separate applications of the criterion.

The underlying premise of the modern Paretian construction is the purely individualistic one. The individual himself is assumed to be the only one who is able to measure or to quantify his own utility or satisfaction. No external observer is presumed able to make comparisons of utility among separate individuals. It is possible, however, even within these limits, to develop a means of evaluating either “situations” or “changes in situations” in terms of their “efficiency.” To do this, a very weak ethical postulate is advanced. The “welfare” of the whole group of individuals is said to be increased if (1) every individual in the group is made better off, or (2) if at least one member in the group is made better off without anyone being made worse off. Clearly this postulate must be accepted by those who accept any form of individualistic values, that is, those who consider the individual rather than the group to be the essential philosophical entity. The ambiguities in the terms “better off” and “worse off” are removed by equating these to the individual’s own preferences. If an individual shifts to position A from position B when he could have freely remained in B, he is presumed to be “better off” at B than at A.

On the basis of this construction, it becomes possible to define the property of a “social state” or “situation” that is necessary to insure its qualification as a Paretian P-point, that is, a point on a conceptual “optimality surface,” a surface that will contain an infinity of such points. If, in any given situation, it is found to be impossible to make any change without making some individual in the group worse off, the situation is defined as Pareto-optimal or Pareto-efficient. On the other hand, if, in a given situation, it is found possible to make at least one individual better off by a change while making no individual in the group worse off, this situation is defined as nonoptimal. The first use of the Pareto norm is, therefore, to provide a means of classifying all possible social states or situations into the Pareto-optimal set and the nonoptimal set. Central to this approach is the idea that no single “most efficient” situation can be located or defined.

The second application of the Paretian construction lies in the development of a rule for classifying changes in social situations. A change is defined to be Pareto-optimal if, in the transition from one situation to another, either (1) every individual in the group is made better off, or (2) at least one individual in the group is made better off and no one is made worse off. It is important to note carefully just what this rule states, since much confusion has arisen in its application. It does not state that any shift from a nonoptimal to a Pareto-optimal situation is itself Pareto-optimal. The rule describes the characteristics of a change and does not relate directly to the characteristic of a situation or state either before or after change. A change away from an established Pareto-optimal situation cannot be itself Pareto-optimal, by definition. However, any other change may or may not be Pareto-optimal in itself. A change from one nonoptimal position to another may be Pareto-optimal, and a change from a nonoptimal position to a Pareto-optimal position may not be itself Pareto-optimal. These points can be easily illustrated in a simple diagram (see Figure 13). On the ordinate and the abscissa is measured the “welfare” or “utility” of individuals Y and X, measured in terms of their own expressions of preference. Any point along the frontier curve Y_mX_m represents a Pareto-optimal situation or state. Any movement from such a point to another point on or inside the frontier must reduce the expressed utility of one of the two individuals. Assume an initial position at A. A change from A to any point on the frontier between B and C is clearly Pareto-efficient since both parties are made better off. However, a change from A to D is not itself Pareto-efficient since Y is made worse off in the process, even though the change represents a shift from a nonoptimal position A to a Pareto-optimal position D. On the other hand, a change from A to G is Pareto-optimal in itself, although it represents a shift from one nonoptimal position to another.33

>Figure 13

This very elementary review of the Pareto criterion has been developed here because it will prove helpful to us in subsequent stages of our analysis. In the remaining parts of this chapter we shall use the Paretian conceptual apparatus in examining the results of the application of game theory to majority voting rules.

Imputations and Pareto Optimality

Let us recall the initial three-person game of Chapter 11, which involved the sharing of a fixed-sum external grant among three separate road-repair projects. The solution set of imputations was:

(½, ½, 0)    (½, 0, ½)    (0, ½, ½).

Note that all of the imputations in F are Pareto-optimal: this is to say, there is no imputation outside the set which dominates any imputation in the set for all three individuals; there is no change from one of the imputations in F which could be made on the approval of all members of the group. This Pareto-optimality condition is imposed through the definition of the characteristic function which makes the return to the whole group,

v(1,2,3) = 1,

along with that to any two-person majority coalition, such as

v(1,3) = 1.

In more general terms, the condition required for an imputation to exhibit Pareto optimality is that the sum of the gains to all individuals be at least as much as the whole group could gain if the members chose to act as a grand all-inclusive coalition. In more formal terms, Pareto optimality is insured by

where x_i is the return to an individual member of the group in a “solution” imputation, and v(I_n) is the expression for the return to all individuals acting jointly as an all-inclusive coalition.34 In our particular example, Pareto optimality is guaranteed by the assumption that a positive-valued grant is received from some outside agency. The game here consists wholly of dividing this fixed-sized gain, and, unless wastage is involved in the process, the whole amount must be disposed. Therefore, any imputation, whether in the F set or not, is Pareto-optimal. Once divided, there is no way that side payments or compensations could possibly be arranged so as to move all members of the group to preferred or indifferently valued positions. This reflects the familiar point that the Pareto-optimality surface contains an infinity of points, each reflecting a separate distribution of “welfare” among members of the group.

In this initial example, the playing of the game is also Pareto-optimal, as distinct from the characteristic of the final solution: that is to say, the change in situation represented by the shift from the position prior to “play” to that after “play” is also Pareto-optimal. The preplay imputation is (0, 0, 0); thus, any final imputation represents individual positions which are either improvements or no worse than initial positions. The assumption that the grant is received from external sources also insures that the game itself will be Pareto-optimal. The individual-rationality condition,

v({i}) ≤ x_i for every i in I_n,

as we have interpreted it in Chapter 11, is satisfied.

If we now modify the game and consider that one introduced in the last part of Chapter 11 in which road-repair funds are to be raised from general taxes, the individual-rationality condition no longer holds. The majority-rule game under these circumstances is no longer Pareto-optimal. The initial imputation in this case is (1/3, 1/3, 1/3), and, in any final imputation after “play,” one member of the group is moved to a less preferred position. Hence, the change represented by the game itself is nonoptimal in Pareto terms.

The solution imputation will continue to be Pareto-optimal, however, so long as condition (4) holds: that is, so long as the sum of the individual gains in any solution imputation is as much as the whole group could gain by acting through an all-inclusive coalition. However, so long as the solution imputation qualifies as a Pareto-optimal point, the playing of the game itself, in an expectational sense, may be considered “optimal.” That is to say, this restriction on the solution insures that the payoffs to the winners of the majority-rule game are at least equal to the losses incurred by the losers. Therefore, the expected payoffs to each individual, at the start of play, must be at least equal to the value of the initially held assets. Although the game itself, as finally played, must reduce the utility of some of the players and hence be nonoptimal, the game does not involve the reduction in the expected utility of any player at the time of the participation decision, provided only that the solution imputation qualify as Pareto-optimal. We are neglecting here the possible utility or disutility from play itself, as well as the possibility of diminishing marginal utility of income.

Need Solution Imputations Be Pareto-Optimal?

The results to this point are perhaps obvious, especially after the analysis of Chapter 11. The more interesting question is the following: Does a “solution” to the majority-rule game embody only imputations that are Pareto-optimal?

The game theorists seem to be rather unhappy about imposing this restrictive requirement on any solution to n-person games.35 We may be able to shed some light on this question by a re-examination of our simple models. Suppose that the initial endowment is, as before assumed, (1/3, 1/3, 1/3). Further, let us assume that there exists no spending opportunity through which the group can increase its net real income. There are no “productive” public investments, and, in the private sector, opportunities are equalized at the appropriate margins of expenditure. In other words, the local roads simply do not need further repair, and, considered in additive cost-benefit terms, any repair project will yield less in benefits than it costs. To be specific, let us assume that the benefits yielded by repairing a road amount to only 5/6 of the costs. We shall assume full symmetry in benefit schedules: that is, public investment is equally productive on every road.

As we have put the problem, the initial imputation is Pareto-optimal. Will the group remain at this imputation? Or will majority voting move the group from an optimal to a nonoptimal position? Or from one optimal position to another?

Consider now the imputation (5/12, 5/12, 0) used before. Clearly, a shift to this imputation brings the group below the Pareto-optimality surface, but the imputation also dominates the initial one for the effective majority coalition, (1, 2) in this case. For the time being, let us label as D the set of imputations:

(5/12, 5/12, 0)    (5/12, 0, 5/12)    (0, 5/12, 5/12).

This set seems to yield “solution” imputations although no single imputation in the set is Pareto-optimal. By proposing the imputation (0, 7/12, 3/12), the third man can form a new coalition with the second, and they could carry decision. However, as in our earlier discussion, one and three may then combine and shift to (5/12, 0, 5/12) which is in D. The stability properties of imputations in D seem to be identical to those in F.

Luce and Raiffa state that the D set, which does not contain Pareto-optimal imputations, does not represent a set of stable imputations. They argue that only that set containing Pareto-optimal imputations will exhibit the required stability of solutions. Their argument seems worth examining in some detail.

They suggest that group rationality (Pareto optimality), expressed in condition (4), is immaterial since all solutions that are stable must lie within the set of Pareto-optimal imputations. Basing their discussion on the work of Shapley and Gillies, they isolate four classes of n-tuples of payments:

This states that Ē is the set of imputations for which the aggregate gains resulting from an all-inclusive coalition are greater than or equal to the summation of the gains received by the separate individuals through participation in the game, that is, in the imputations in X. In our numerical example here, the imputations listed fall within Ē since, by hypothesis, the aggregate real income of the group is lowered by the action taken. In numerical units, the value of the left-hand side of condition (5) would be 5/6 and the value of the right-hand side would be 1.

which is the same as condition (4) above. This is the set of Pareto-optimal imputations. The first three-person game yielded imputations that necessarily fell within E, regardless of their location within or without F. Games that are purely redistributional must yield imputations in E.

This is the subset of Ē which represents final imputations in which all individuals have either improved their position by participating in the game or have not been made worse off. This is the condition of individual rationality, as interpreted, which we have discussed earlier. In slightly different terminology, this condition, if satisfied, insures that the game itself is Pareto-optimal, even though a position on the optimality surface may not be achieved.

This is a subset of the Pareto-optimal set of imputations. In particular, it is the subset of the Pareto-optimal set that may be attained in a Pareto-optimal manner from the initial no-play position. In other words, this set of imputations, on the Pareto frontier, can be reached by playing only “optimal” games.

In a two-person model, which can be represented on a two-dimensional surface, each set of these n-tuples can be shown readily. Refer to Figure 14, which is similar to Figure 13. Ē is represented by the whole area enclosed by the two axes and the frontier Y_mX_m. Any point along the frontier or inside the frontier satisfies the weak requirements of condition (5). E, the Pareto-optimal set, is represented by the set of points along Y_mX_m, that is, on the frontier. Note that E is a subset of Ē. If A is defined to be the initial position, then Ī includes the set of points enclosed by the area ABC. I is that set of points falling along the frontier between B and C, being a subset of Ī.

>Figure 14

Luce and Raiffa (pp. 216-18) accept a proof by Shapley to the effect that a stable solution in E must lie within E. Hence, they conclude that no real restriction is placed on the results by assuming group rationality (Pareto optimality) in the first place. A commonsense approach may reveal the reasoning here. Why are the imputations

(5/12, 5/12, 0)    (5/12, 0, 5/12)    (0, 5/12, 5/12),

suggested as “solutions” to the particular problem considered, unstable? No element in this set, which we have called D, dominates any other element; but is every imputation not in D dominated by one in D? This second requirement is the crucial one, and D clearly does not satisfy it. Consider the imputations

(½, ½, 0)    (½, 0, ½)    (0, ½, ½),

which we recognize as the F set. One of the elements or imputations in this set dominates each imputation in D, yet no element in D dominates all of the imputations in F. This suggests that D could not represent a set of stable imputations.

Let us consider the real-world implications of this proof. Note that the imputations in F are Pareto-optimal. However, in order to attain an imputation in this set, the playing of the game must result in a shift that is equivalent to a purely redistributive transfer of real income among individual members of the group. That is to say, the game must be constant-sum, as defined by condition (6). However, given the requirement that collective decisions must involve the employment of general tax revenues to finance public services, this constant-sum restriction disappears. Moreover, when this happens, the F set of imputations remains as the solution only if full side payments are allowed. If both purely redistributive transfers and side payments are excluded, the game is severely constrained. There is no need whatever for the solution to exhibit the Pareto-optimality property. Condition (6) need not be met. The conclusion here is clearly that, if a majority is to exploit a minority, the most “efficient” means of so doing is the imposition of simple redistributive transfers (lump-sum taxes) instead of the indirect means of general-tax financing of special public-service benefits (or, conversely, special-tax financing of general-benefit public services), which may, as in our example here, involve a net cost for the group considered as a unit.

In the more constrained game without side payments, the imputations in F cannot be said to dominate those in D. Dominance has meaning only if the coalition is effective in shifting from one imputation to another. The set of imputations, F, simply does not exist in the constrained model. The D set embodies the solution with the same stability properties as the F set in the more general model, unless the human proclivities to make side payments are considered to be so strong as to rule out meaningful discussion of such constrained games.

Geometrical Illustration

The essential points may be clarified by geometrical illustration. In Figure 15 below we measure on the ordinate the position of the dominant or the effective majority. The gains are added over the two members since we must use two-dimensional surfaces. On the abscissa we measure the position of the minority member of the three-man group. In the restricted model that we have been discussing, we assume that no investment in the public sector is productive. This makes the initial imputation Pareto-optimal; this imputation is (1/3, 1/3, 1/3), which becomes the point (2/3, 1/3) when plotted on the two-dimensional diagram as point I, which is, by definition, on the Pareto frontier. Since any division of one unit is also, by definition, on the frontier, the line AB in Figure 15 represents the whole set of Pareto-optimal points. Since we do not identify the members of the majority in the diagram and since the benefit schedules are symmetrical, if we allow individuals to shift from membership in the majority to membership in the minority, all points that are Pareto-optimal in the three-person model can be represented on AB. The set of imputations, F, the solution to the generalized game, is shown at A. At this point the member of the minority is deprived of all assets, and the two members of the majority coalition symmetrically share the gains, which are equivalent to the whole product. Again, by shifting separate individuals, A can be taken to represent all three of the imputations in F. As we have noted, if purely redistributive transfers should be allowed, a majority would immediately shift the group from I to A. Nothing would be modified except the distribution of the fixed-sum among the members of the group.

>Figure 15

If redistributive action is excluded, the majority might still find it advantageous to take action, even though, by hypothesis, such action will be unproductive for the whole group. The point C represents the set of imputations D, defined as the solution to the more constrained model. Here the combined “gains” of the majority are 5/6, while the assets of the minority are confiscated. C is clearly beneath the optimality frontier. This suggests that, conceptually, all of the members of the group could be made better off by some change. The range of such changes is shown by the heavily shaded area in Figure 15. A shift or change from C to any point in this area would itself be Pareto-optimal. If side payments are allowed, the minority member of the group could, for example, “afford” to offer the majority IK, valued at KH by the majority, in order to allow all the group to shift to H instead of undertaking the action shown at C. The majority would, if allowed, accept this offer, but they need not stop there. They could, instead, try to outbargain the minority member and to force him to concede sufficient side payments to allow the group to move to A. The precise outcome of the actual bargaining process is unimportant; the relevant point is that such payments will insure that a final solution somewhere along the frontier will be reached. Under the specific conditions of this example, where the public project yields a total benefit value of 5/6, the relevant range on the frontier is AG. Side payments will be paid to the majority to prevent the investment from being undertaken.

The limits to resource wastage discussed in the last chapter can also be shown readily in this diagram. If all redistributive transfers and side payments are ruled out, any collective project that yields more than 2/3 to the effective majority will be selected. Any position on the vertical axis above L becomes a possible solution to the constrained game of majority rule.

Symmetry in Benefit Schedules

We have demonstrated clearly that majority voting rules may result in a shift of the group from a Pareto-optimal to a nonoptimal position in the constrained form of the game. It will now be useful to demonstrate geometrically that, if the initial position is nonoptimal and if an optimal position can be attained by collective action, majority voting will move the group to a Pareto-optimal position only if the benefit schedules are symmetrical over the whole group. Benefit schedules were assumed to be symmetrical in the previous example, where it was demonstrated that majority voting may shift the group off an initial position on the Pareto frontier. Symmetry in benefit schedules may be at most, therefore, a necessary condition for attaining a Pareto-optimal position. It can never be sufficient to insure the attainment of such a position. Refer to Figure 16. As before, we assume an initial (before play) position at I. However, let us assume that public investment in all three roads is equally productive, and highly productive. An investment of $½ on each road is assumed to yield a benefit value of $1. In this case the F set becomes

>Figure 16

(1, 1, 0)    (1, 0, 1)    (0, 1, 1),

represented in Figure 16 as the single point A. Majority decision will tend to shift the group to A, which is on the Pareto frontier. The majority-rule game, as actually played, is not, of course, Pareto-optimal, since the minority member of the group is shifted to a lower utility level in the process of paying taxes to support the public projects beneficial to other members of the social group. In an expectational sense, however, the game is “optimal,” provided, of course, that the rules are “fair.” Note that, in this case, no introduction of purely redistributive transfers or side payments will change the result. The majority can reach the position shown at A only by undertaking the projects, and there is no way that the minority can make an effective counteroffer.

Note carefully, however, that this conclusion follows only when we assume symmetry in benefit schedules over all individuals.36 If this assumption is dropped, the operation of majority-rule decision-making will not necessarily shift the group from an initial nonoptimal to a Pareto-optimal position without the introduction of side payments. For purposes of illustrating this point, we now assume that the investment of $½ on each road project will yield, respectively, $1, $½, $¼. The solution set becomes

(1, ½, 0)    (1, 0, ¼)    (0, ½, ¼),

assuming no side payments. Let us assume that there exist no investments in the private economy that are more productive than investment on the first road. If individuals 1 and 2 form a dominant majority, the group will shift to point a₁ in Figure 16; if 1 and 3 form the majority, to point a₂; if 2 and 3, to point a₃. In none of these cases does majority voting shift the group to the optimality frontier, which could only be reached if all investment should be made on the first road.

When the benefits are asymmetrical, the frontier will be attained only if full side payments are allowed to take place. In the example here, Pareto optimality will be attained, after side payments, in the solution set:

(1, 1, 0)    (1, 0, 1)    (0, 1, 1).

If side payments are allowed, the first man can afford to pay the second man more than ½, the value of repairs to the second road, for his support of a policy of exclusive investment on the first road; and the first man could clearly pay the third man more than ¼, the value of his own estimated benefits from local road repair.

This introduces an extremely interesting point that we have deliberately neglected in the discussion of Chapter 11. In a purely formal sense, the imputations in F, written above, satisfy the Von Neumann-Morgenstern requirements for solution when full side payments are allowed. Moreover, since all three of the imputations satisfy the requirements jointly (that is, as a set of imputations), nothing further can be stated in terms of the formal construction. However, we have noted previously that the notions of stability and solution in n-person games generally are not fully satisfactory. Many games contain numerous solutions in the simple mathematical sense. Intuitively, we may see that these ideas of solution and stability are considerably less applicable to those games where benefit schedules are not symmetrical than to those in which such schedules are symmetrical. Let us consider the set F, above, more carefully. It seems clear that, of the three imputations in F, the second is more likely to emerge, or, to state this somewhat more correctly, the coalition represented by the second imputation seems more likely to emerge. Nor do some of the imputations in F seem more stable, under the restrictions of this model, than others outside the set. The second-imputation coalition between the first and the third person in the group seems more probable because the support of the third man for repair on the first road can be secured more “cheaply,” even with full money payments, than the support of the second man. This is because the relevant alternative, as considered by the third man, may be, not his combination with the second to exploit the first to the maximum, but his combination with the second to finance repairs to their own roads. If the third imputation is not considered by 2 and 3 to be a genuine alternative, then any imputation (c₁, 0, c₃), where 7/4 ≥ c₁ ≥ 1, and 1 ≥ c₃ ≥ ¼, would be equally stable with (1, 0, 1). This point, which amounts to the denial that full side payments would be carried out in situations like the ones postulated, suggests the probable emergence of coalitions between those individuals and groups who are the direct beneficiaries of the most productive public projects and those individuals and groups for whom public investment is the least productive. This result will emerge, of course, only if some side payments are allowed. However, even if only limited forms of vote-trading are permitted, this general conclusion does not seem at all implausible and appears to be in accord with those reached in Chapter 10.

Side Payments and Pareto Optimality

In a very real sense, the introduction of full side payments serves to create a marketable property right in the individual’s political vote, his power of collective decision. If this power is marketable (that is, if it is to command a price or a market value), some element of scarcity must be present. On single issues such as those discussed in our models, the scarcity of decision-making power is evident. Only one decision can ultimately be made; only one majority can be effective. The aggregate payoff function is reduced to the (1, 0) form. If collective decisions affect the disposition of economic resources, and if resources are used up over finite time, the decision-making power over any disposition of resources is scarce indeed. Decisions become irrevocable once made.

We have shown that only if side payments are introduced is there any assurance that majority-rule decision-making will lead to positions on the Pareto-optimality frontier. It will now be shown that this property depends solely on the introduction of side payments and that it has no specific connection with majority rule. In order to demonstrate this, we must prove that any decision-making rule, with full side payments, will produce only Pareto-optimal situations.

We may take two extreme decision-making rules, those of individual dictatorship and unanimity. First, we assume that all decisions for the group are to be made by a single individual, the dictator, who is interested only in maximizing his own utility. Let us keep within the limits of our simple three-person model, and again let us assume that the group receives a grant from external sources. The benefit schedules are as follows: if all funds are spent on the first road, $10; if all are spent on the second road, $5; if all are spent on the third road, $1. If Individual 1 is dictator, no question arises. However, if Individual 2 is dictator, he will find that his own utility can be maximized by “selling” his power of disposition over the external grant to Individual 1 for something in the bargaining range of $5 to $10. He will sell to the highest bidder, and it is evident in this model that Individual 1, for whom road repairs are the most productive, can bid highest. Similar conclusions follow if Individual 3 is dictator. A Pareto-optimal position is always attained. If the assumption of an external grant is dropped and general-tax financing assumed, this conclusion is not modified. The only difference here is that, with general-tax financing, the game itself is not Pareto-optimal. Under dictatorship, two of the individuals will tend to be made worse off as a result of any political action, always under the behavioral assumptions implicit in all of our models.

Let us now go to the opposite extreme and show that, even if a unanimity rule is adopted for collective decisions, all solution imputations will be Pareto-optimal when side payments are allowed. In the external grant case, any possible n-tuple or imputation dominating (0, 0, 0) for all three individuals can be attained through unanimous approval; or, if we are assuming tax financing from an initial position (1/3, 1/3, 1/3), any imputation dominating this may be a “solution.” Again, however, note that all repairs will be made on the first road, if side payments can take place. The set of possible solution imputations is extremely large here. The following three imputations represent the limits in the “negotiation set”:

(9 1/3, 1/3, 1/3)    (1/3, 9 1/3, 1/3)    (1/3, 1/3, 9 1/3).

If symmetry in gain is held to be characteristic of solution, a single imputation (3 1/3, 3 1/3, 3 1/3) emerges, but, as we have noted, the argument for symmetry seems much less convincing in games of this sort where all participants must agree on the sharing than it does in games such as that of majority rule. The final outcome will depend on the relative bargaining strengths of the parties in negotiation, but the bargaining will take place only to determine in what proportions the gains are to be shared. The Pareto frontier will tend to be reached, and it will be reached in a Pareto-optimal manner. The latter is the unique feature of the unanimity rule. The “game” itself is Pareto-optimal. Only with the unanimity rule will collective decision-making produce changes that are necessarily Pareto-optimal.

If side payments are not allowed, neither dictatorship nor the unanimity rule will produce imputations on the Pareto frontier in all cases. The unanimity rule will always result in movement toward the frontier, but there is no assurance that the frontier or surface will be reached. Thus, we find that the Pareto criterion suggests the paradoxical conclusion that open buying and selling of political votes may actually lead to an “improvement” for the group, measured in the extremely weak ethical sense of making everyone in the group better off as a result. This conclusion deserves more careful attention, but we propose to delay this to a later point. What has been demonstrated is that, without side payments, there is nothing in any particular voting rule to insure that collective decisions will move the group to the Pareto-optimality surface or that such decisions will keep the group on this surface if it is once attained.

13.

Pareto Optimality, External Costs, and Income Redistribution

We have shown that, if full side payments are allowed to take place, any decision-making rule for collective action will lead to positions that may properly be classified as Pareto-optimal, although Pareto optimality may not characterize the process or processes through which the positions are attained. Because of the latter, nothing can be said concerning the “desirability” or the “undesirability” of the changes embodied in the operation of any given decision-making rule short of unanimity. Recall that the definition of a Paretian P-point is as follows: a position from which no change can be made without harming at least one individual in the group. This suggests that, when such a position is attained, no external costs are being imposed on the individual by other individuals. Economists are familiar with the fact that one of the necessary conditions for Pareto optimality is the absence of such externalities. Moreover, as we have previously shown, the presence of external costs is equivalent to the existence of “mutual gains from trade,” which can, by definition, be secured to the advantage of all parties.

The introduction of full side payments into the model of collective choice seems to imply, therefore, some restrictions on the applicability of the external-costs function developed in Chapter 6. This function, you will recall, relates the expected external costs on the individual to the decision-making rules. The value of the function decreases as the rule becomes more inclusive, but this value remains positive throughout the range. The relevance of this construction has been demonstrated for the individual constitutional calculus when full side payments are not present. Any rule for collective choice embodying less than full consensus must impose some external costs on the individual since resources will tend to be allocated “inefficiently” because of the choice mechanism. If, however, the introduction of full side payments should negate the relevance of this external-costs function, our analysis of constitutional choice would be rather severely limited.

In this chapter we shall try to show that the individual, at the stage of constitutional choice, will expect collective activity to impose some external costs on him, even if full side payments are allowed to take place in the process of reaching decisions, given any decision-making rule other than unanimity. The apparent contradiction between the existence of external costs and the satisfaction of the orthodox conditions for Pareto optimality, which side payments will tend to produce, must be resolved. In so doing, we shall also be able to relate the introduction of side payments generally to the constitutional-choice models of Chapter 6. A by-product of our discussion will be the integration of income redistribution into our model of collective activity. In one sense, this chapter represents a digression from the main stream of our analysis. It seems necessary, however, to avoid certain logical pitfalls, and the material which follows will provide some foundation for the analysis of later chapters.

Redistributive Elements in Majority Decisions

Under the behavioral assumptions of our models, majority decision-making (or any decision-making with less-than-unanimity rules for choice) will tend to produce some asymmetry in gain-sharing among the individual members of the group for which the choices are made. The members of the effective coalition will receive differentially larger shares of the benefits expected to result from collective action and/or they will bear differentially smaller shares of the costs of collective action providing general benefits for the whole group. This amounts to saying that redistributive elements must be a part of any collective decision reached by a less-than-unanimity rule.

What the introduction of side payments accomplishes is the conversion of all collective decisions to these purely redistributive elements. Unless a public investment project is “worthwhile” in a market-value sense, side payments (“bribes”) will arise to prevent action from being taken, regardless of the rule for choice. What side payments cannot prevent are the net transfers of real income among the separate individuals and groups. With full side payments, the decision-making rules determine the structure of the net income transfers only; they do not influence the extent of “productive” collective activity. The latter will always be extended to the limits defined by the satisfaction of the Paretian conditions.37 It is his inability to say anything about the distributive problem that has inhibited the modern welfare economist. Since he cannot presume to make interpersonal comparisons of utility, he cannot adjudge one Pareto-optimal position to be better than any other or even adjudge one optimal position to be superior to all nonoptimal positions. A move from one point to another on the conceptual optimality surface must remain outside the analytical framework of the welfare economist. Since all decisions, public and private, leading to a point on the optimality surface must be made by a proper comparison of marginal costs with marginal benefits, no external effects of the ordinary sort can be present in the final Pareto “equilibrium.” From this the inference seems clear that, under a regime with full side payments, since different decision-making rules act only to effect the location of the position on the optimality surface, the external-costs function of Chapter 6 is not applicable. This function appears from this approach to be meaningless for the analysis of purely redistributive transfers. The geometrical inference is that, for such transfers, the external-costs function would lie along the abscissa. External costs would appear to be zero under any rule.

Let us see precisely what the acceptance of such an inference would imply for the constitutional calculus of the individual. Recall that, under our assumptions, the individual, at the time of constitutional choice, is uncertain as to his own role on particular issues in the future. If the inference suggested here is correct, the individual, because of this uncertainty, will not expect positive external costs to be imposed on him by purely redistributive transfers of real income. The reason is evident: he will see that the external benefits which he may secure through imposing external costs on others on certain occasions will tend to equal the external costs which others will impose on him on different occasions. In any single action, the external costs imposed on those from whom income is taken are equal to the external benefits received by those to whom income is transferred. Since, at the constitutional stage, the individual will identify himself with neither of these groups, he will see that the effects tend to balance out as he considers the whole sequence of possible redistributive transfers.

Note carefully, however, just where this line of argument is leading us. If correct, the argument suggests that the individual, at the constitutional level, would never choose to collectivize the redistribution of real income among members of the group. If the external-costs function does not exist for such transfers, then clearly cost minimization of this activity is achieved only by allowing purely private activity. Only in this way will the decision-making costs (the costs of reaching agreement between two or more persons required to form an effective coalition for decisive collective action) be eliminated. If the distribution of real income among members of the society really does not matter, as would be implied by the argument, the most efficient way of organizing “redistribution” is to do nothing about it.

An Alternative Explanation

There seems to be decisive empirical evidence that individuals do not behave as the above argument would indicate. In almost every society some collectivization of income redistribution is to be found; some efforts are made to accomplish real-income transfers among members of the group by collective intervention. How is this observed phenomenon to be explained in terms of our analytical approach? We shall propose an explanation which will incorporate the existence of external costs into a model restricted to purely redistributive transfers. In this explanation the extension of our analysis beyond the limits of orthodox welfare economics can be most easily made apparent.

We may assume that the marginal utility of income declines as the individual receives more income in any particular time period and that the individual recognizes this. We do not require further restrictions on the shape of the individual’s utility function. If the individual recognizes that, in any given period, the marginal utility of income will decline as more is received, he will see that, over a succession of periods, his total utility would be increased if some means of “exchange through time” could be arranged. If some institution could be established which would add to his income during periods of bad fortune and subtract from his income during periods of good fortune, the individual’s total utility over time could clearly be increased. If, in fact, he could assume that the years of good fortune would be matched by years of bad fortune within his life span, the individual could, conceptually, purchase such “income insurance” from privately organized sellers. However, at the stage of constitutional choice, the single individual cannot make this required assumption. He will recognize that, individually, he may suffer a succession of low-income periods or, alternatively, he may enjoy a succession of high-income periods. Moreover, since income is the primary economic magnitude to be considered in his over-all life planning, the individual will rarely have sufficient wealth at the outset of his life to purchase the “income insurance” that utility-maximizing considerations would dictate to be rational. Nor will potential private sellers of such insurance be in a position to enforce the sort of contracts that might be required to implement such a program in the real world. All of these obstacles to a private “income insurance” would be present even if the most fundamental obstacle were overlooked. This is the fact that the risk in question would be essentially uninsurable by ordinary standards. Since the private individual, by modifying his current behavior, is able to affect his claims for compensations, a privately organized insurance plan might be impossible.

By such considerations as these, the individual may be led to examine the prospects of collectivizing the redistribution of real income to the extent that is indicated to be rational by his utility function. In order to prevent the possibility of his falling into dire poverty in some unpredictable periods in the future, the individual may consider collective organization which will, effectively, force him to contribute real income during periods of relative affluence. Such collective redistribution of real income among individuals, viewed as the working out of this sort of “income insurance” plan, may appear rational to the utility-maximizing individual at the stage of constitutional decision. The essential “uninsurability” of the risk will not, of course, be eliminated by collectivization, but the individual may be more willing to accept the costs of such uninsurability if he knows that all members of the group are to be included in the plan.

Before committing himself, however, the individual must try, as best he can, to analyze the operation of the decision-making rules that may be adopted in carrying out the collective activity of redistribution. Once the constitution is established, the individual actor operates within the predefined rules; no longer must he try to reach full agreement with his fellows. Moreover, in the implementation of income redistribution through collective action, external effects become the essence of private behavior.

Let us suppose that a constitution is adopted which openly and explicitly states that net-income transfers among individuals and groups will be carried out by simple majority voting. In this situation it seems clear that the maximum possible departure from rational behavior in choosing the amount of redistribution could be present. The individuals in a successful majority coalition could impose net taxes on the minority and receive net subsidies for themselves. In the calculus of the individual participant in a majority coalition, a symmetrical share of the coalition gains will be treated as the marginal benefits of action and balanced off against zero marginal costs. It seems certain that “redistribution,” considered as an activity, will be carried relatively “too far” under these conditions.

But “too far” relative to what? This is the difficult step in the analysis. Pareto criteria can be drawn in for ordinary collective action, but they are useless here. Nevertheless, the constitutional-choice model is helpful, and it allows us to answer this question, at least conceptually. Redistribution, under the circumstances postulated, will proceed “too far” relative to the amount that the individual, in the role of constitution-maker, could choose to be rational on the basis of long-run utility-maximizing considerations. In one sense, we may translate this into Pareto-optimality terms at a different level of decision-making. The amount of redistribution that unrestrained majority voting will generate will tend to be greater than that which the whole group of individuals could conceptually agree on as “desirable” at the time of constitutional choice. Since conceptual unanimity is possible on this degree of income redistribution, we may, in a certain sense, call this a Pareto-optimal amount of redistribution. The more orthodox Paretian construction applies only to the operational level of decision, that is, within the confines of established constitutional rules. If we are to discuss the formation of the rules themselves, something quite similar to the Pareto criterion emerges when we consider the “optimal” rules. However, it seems best to avoid using the same terms in both cases.

If, in fact, voting rules are expected to result in real income redistribution being extended “too far” relative to that which the individual would rationally choose, we may clearly say that the organization of this activity will be expected to impose some external costs on the individual. The external-costs function of Chapter 6 is equally as relevant in analyzing this activity as all other collective activities. In our model of collective action which allows full side payments to take place, the external costs that are expected from the operation of any decision-making rule are solely those resulting from the overextension of redistribution. Side payments will insure that the orthodox Pareto-optimality surface will be reached, but the redistribution that will take place through the collective-choice process will not represent the “optimal” shifting among positions on this orthodox optimality surface. Note that we do not require an interpersonal comparison of utility in the usual sense to be able to reach this conclusion. We require only that the individual be able to make decisions based on some presumption about his utility function in different periods of time. In a sense, of course, this does represent an interpersonal comparison of utility, but it is of a sort that individuals must, in fact, make in many everyday decisions.

We reach the conclusion that the attainment of an orthodox Pareto-optimal position is not sufficient to insure that there exist no external effects from an activity. The external costs of redistribution will remain, even if perfectly operating side payments arise to insure that the more familiar externalities are eliminated.

“Income Insurance” and Individual Behavior

The expected external costs from redistributive collective action become more pronounced when it is recognized that the form of the transfers may not be at all similar to that which the rational individual, in the role of constitutional chooser, would select as the “optimal” plan of income insurance. Under the assumptions of our model, there is no reason to expect that simple majority voting, for example, would result in a net transfer of real income from the rich to the poor. There is no assurance that the dominant coalition will, in fact, be such that the transfers will provide the “insurance” considered in the constitutional calculus.

This suggests that the expected external costs of purely redistributive action may, in fact, be so high that the individual, at the constitutional level of choice, may decide that any collectivization of direct redistribution is undesirable. Because of this, he may seek to “institutionalize” the “income insurance” plan via constitutional processes.

An analogy that frequently appears in bargaining theory may prove helpful. At the outset of a hunt each of two hunters may consider that his expected utility will be maximized by agreeing on a predefined rule for sharing the day’s catch. Each might realize that, only by agreeing to such a rule, could a “fair” sharing be assured. Otherwise, without rules, the hunter securing the major share of the game would probably think that his good fortune was due to his exceptional skill, and he would be extremely reluctant to part voluntarily with a share of the size that he might otherwise have agreed to under a predefined sharing rule.

Empirical evidence points strongly toward some such explanation as that developed here. Not only do most societies with democratically organized governments undertake some collective action with a view toward redistribution of real income, but the manner in which this action is taken suggests clearly that the external effects are sensed acutely by the framers of political constitutions. In the first place, arbitrary and discriminatory redistributive transfers of income and wealth among individuals and groups are normally prohibited. For direct transfers to be effected, some general bases for classifying individuals are usually required. Secondly, the whole constitutional emphasis on securing and guaranteeing the basic human rights and civil liberties can be broadly interpreted as aiming toward an equalization of opportunities rather than an equalization of rewards. If the legal and institutional framework is such that the distribution of emerging rewards is tolerably acceptable, the direct collective intervention to effect the redistribution that may be dictated is reduced. Insofar as the “income insurance” can be provided by improving the rules within which the “economic game” is played, the individual, at the stage of constitutional choice, may be spared the expected external costs of too much and possibly wrongly directed redistribution through collective action. This point was recognized by Knut Wicksell, himself a genuine humanitarian, when he suggested that efforts toward improving distributive results should be centered on reforms of the institutions of property instead of on the redistributive potential of the fiscal system.

Finally, and most importantly, redistribution of real income, per se, is rarely collectivized, in spite of the almost universal acceptance of some collective effort to intervene in the distribution process. Surely there must exist some explanation for the continuing reluctance of societies in the Western world to throw open the redistributive potential of the fiscal system to the ordinary mechanism of collective choice-making. The most plausible explanation seems to be found in the very real fear of the external effects that such an unrestricted collectivization of redistribution might generate. Instead of following this path, Western governments have opened the way for more and more effective redistribution which is accomplished indirectly through the tax financing of public goods and services. By incorporating highly progressive, but nominally general, taxes with special-benefit public services in the fiscal process, the redistribution that is carried out far exceeds that which could be accomplished directly.

This points up the difficulty of putting to practical use the conceptual separation of the allocational and the distributional aspects of the budget, a separation urged recently by R. A. Musgrave.38 If such a separation were, in fact, required, much less effective redistribution would be carried out since the individual, fearful of the external costs of unrestricted redistribution, would not allow governments as much power as they now possess indirectly.

Allocational and Redistributional Externalities

From the operation of any collective decision-making rule short of unanimity, therefore, the individual normally expects two distinct sorts of external costs to be imposed on him as he considers his possible role over an extended series of issues in a sequence of time periods. If side payments (“bribes”) are not allowed, or if only partially effective substitutes are sanctioned, there can be expected to arise some allocational externalities. That is to say, the collective-choice process will cause resources to be employed “inefficiently.”

The effects of introducing logrolling or side payments into the collective-choice mechanism are those of “squeezing” out these allocational inefficiencies. If side payments are conceived to be perfectly organized, all such allocational inefficiencies will tend to be eliminated. There will remain only the redistributional “inefficiencies,” which can also be called “externalities,” with which we have been primarily concerned in this chapter.

The impact of these expected redistributional externalities (these redistributional external costs) on the individual constitutional calculus could scarcely be overemphasized, for it seems to be this expectation which causes the individual to refrain from assigning to the collective sector many activities which he would tend to collectivize if such externalities were absent. Examples are easy to come by. Full efficiency in resource usage in the United States might require the co-ordinated development of the water resources of each regional watershed. The full range of externalities in the allocational sense cannot be exploited except through the co-ordination of development extending over a geographic area encompassing several states. If we accept these presumptions as being true, does it follow that “nationalization” of this function should be supported by the rational, utility-maximizing “average” citizen of the United States, as he might be assumed to adopt a rule of making such a choice? The answer is not nearly as clear as some modern welfare economists, and applied cost-benefit analysts, would like to make it. If such projects are to be financed, or if the individual expects them to be financed, out of general tax revenues collected from the whole population of the country, the redistributional externalities expected may well be sufficiently large to offset the allocational externalities that may be continued by failure to undertake co-ordinated development.

Conclusions

As suggested at the beginning, this chapter has represented somewhat of a digression from our main line of argument. It has been designed to show that our analysis of the constitutional-choice problem (contained centrally in Chapter 6) is applicable to the collective redistribution of real income among persons, despite the apparent contradiction between the attainment of the orthodox Pareto-optimality surface and the continuing existence of net external costs. The contradiction was resolved by showing that our analytical model, extending as it does to the choice of rules for choice, is more extensive than the standard Paretian construction. External costs, in our model of constitutional choice, are made up of two elements: those resulting from what we have called allocational externalities, and those resulting from what we have called redistributional externalities.

14.

The Range and Extent of Collective Action

Implications concerning the relative size of the public and the private sectors of the economy have been suggested at several points in our analysis. These implications have not been fully explored, nor have they been related to each other. In this chapter we shall try to answer the questions: What can be said about the relative size of the public sector as a result of our analysis? Does the analysis suggest that the public sector will be “too large” with respect to the private sector, given certain decision-making rules for collective choice? Or “too small”? What criteria are to be employed in judging whether or not the sphere of collective activity is “too large” or “too small”? How do these criteria and these results compare with those that have been utilized in more orthodox or standard analyses?

Majority Voting and External Costs

The analysis of Chapters 10, 11, and 12 demonstrated that the organization of collective action through simple majority voting tends to cause a relative overinvestment in the public sector if the standard Paretian criteria are accepted. Note that the effects are always in this direction under the behavioral assumptions employed in our models. This is because the majority-voting rule allows the individual in the decisive coalition to secure benefits from collective action without bearing the full marginal costs properly attributable to him. In other words, the divergence between private marginal cost and social marginal cost (the familiar Pigovian variables) is always in the same direction.39

Recognition that the rule will result in such relative overinvestment will make the individual, at the time of constitutional choice, anticipate some net external costs as a result of the operation of majority voting. A simple extension of the majority-voting model to apply to qualified majority voting yields similar results, the only difference being that expected external costs are reduced as the voting rule becomes more inclusive.

We have shown that majority voting will tend to cause overinvestment in the public sector relative to the private sector on the basis of the orthodox or standard criterion of Pareto optimality. This is a meaningful criterion for static analysis, but it is severely limited in certain important respects. In the first place, Pareto optimality, taken alone, cannot be used to assess the effects of purely redistributive transfers of real income among persons. Moreover, as we have demonstrated in Chapter 13, almost all collective decisions embody certain redistributive elements as well as allocational elements. Redistributive action can also impose external costs, costs which the orthodox Paretian criterion cannot take into account.

The Bench-Mark Criterion

A more comprehensive criterion is provided by the bench mark or zero point used in the construction of the models of Chapter 6. With respect to any given activity, the bench mark is defined as that situation or position which would be achieved when all external costs are absent. In a sense, this represents an “ideally efficient” solution to the problem of organization. In those cases where decision-making costs can be neglected and where no restrictions are placed on the form that collective action is to take, this ideally efficient solution can be attained under the rule of unanimity and the characteristics of the solution are identical with Pareto optimality. Even this limited unanimity test fails, however, when we consider purely redistributive transfers of real income. This is because all members of the group could hardly be expected to agree on an amount of net redistribution considered “optimal” by the individual at the time of constitutional choice. Whereas majority decision-making would tend to involve redistributive “externalities” because redistribution would be extended relatively too far, the requirement of unanimity would tend to involve redistributive “externalities” because redistribution would not be extended far enough.40 The conceptual unanimity test is helpful, therefore, only in analyzing the allocational aspects of collective action; it is not helpful in analyzing the redistributive aspects. In any case, the test is directly useful only if decision-making costs are neglected.

These costs cannot, however, be neglected. Hence our bench-mark criterion becomes a purely hypothetical standard of achievement. For all purely allocational decisions, the bench mark becomes that position which could be attained by the operation of the rule of unanimity, with compensations as appropriate, if individuals did not invest resources in strategic bargaining. The position is identical to that defined more rigorously by Paul Samuelson and R. A. Musgrave in their development of the pure theory of public expenditure.41 For such allocational decisions, the bench-mark position may be conceptualized on the assumption that individual-preference fields are fully known at a single point in time. However, for redistributive decisions, this sort of conceptualization is not possible. A hypothetical position characterized by the absence of all external effects may be imagined, but its more precise conceptualization requires the knowledge of individual utility functions at the stage of constitutional choice as well as at the stage of operational collective decision-making.

This difficulty in conceiving the existence of a bench-mark situation is actually helpful to us instead of providing a barrier to our understanding. This is because one of the main points to be emphasized is the fact that an independent criterion for determining the appropriate allocation of resources between the public sector and the private sector does not exist. Even if all external effects could be eliminated, the costs of agreement required might be so large that the costs-minimizing organization of the activity in question would require the presence of some positive external costs. If this is the case, there must be an overextension of the activity, that is, too many resources utilized relative to that organization presented by the hypothetical ideal. However, these external costs, which measure the distortions caused by the relative overextension, may be more than offset by the reduction in decision-making costs below the level that full unanimity might entail. All of these points were made in Chapter 6; they are repeated here in order to show their relevance in answering the questions posed at the beginning of this chapter.

In one sense, therefore, we can quite properly say that all decision-making rules embodying less than full consensus will tend to cause relatively too many resources to be devoted to the public sector—too many relative to that idealized allocation of resources that the omniscient observer, knowing all utility functions over time, might be able to describe. In another sense, however, if we leave such omniscience out of account, no such conclusion can be reached. The alternative organization of activity—either a removal from the public sector or a change in the collective decision-making rules—might increase rather than decrease the necessary interdependence costs of the activity in question. At this more meaningful level of discussion, when we consider realizable organizational alternatives, no normative judgment can be formed concerning the extent of the public sector from a simple comparison of an existing organization with the bench-mark or ideal solution. Such meaningful judgments can be made only on the basis of a comparison with realizable and relevant alternatives. To say, for example, that majority rule tends to overextend the public sector relative to some idealized and unattainable bench-mark allocation of resources is descriptively meaningful, but the statement is useless in answering the only important question that must confront the individual in framing constitutional decisions. The only meaningful overextension of the public sector must refer to realizable alternatives, and unless interdependence costs can be shown to be reduced under these alternatives, normative statements cannot be made. As Frank Knight has often remarked, “To call a situation hopeless is equivalent to calling it ideal.”

The organization of an activity can be classified as “ideal,” even though it will be overextended relative to some hypothetical ideal, only if the appropriate constitutional decisions have been made. If the organization is not that which effectively minimizes the interdependence costs, realizable alternatives are possible and normative judgments can be made. If, for those activities that have been shifted to the public sector, the costs-minimization decision-making rule has not been chosen, normative statements can be made about certain changes in organization. External costs imposed on individuals through the operation of the activity may be higher than they need be, and these costs can be reduced only by a change in the decision-making rules.

The “overextension” of collective activities relative to the hypothetical ideal is precisely equivalent in normative content to the existence of externalities resulting from individual behavior in activities appropriately organized in the private sector. As we pointed out in Chapter 5, the existence of such external effects provides neither a necessary nor a sufficient condition for a change in institutional organization.

The Range and Extent of Collective Action

When we discuss the allocation of economic resources between the public or collective sector and the private sector of the economy, it is essential to distinguish between the range of activities that may be collectivized and the extent to which collectivized activities may be pushed. This important distinction is often overlooked. We may clarify the distinction by a single example. Water-resource development and the provision of telephone services are two separate “activities,” either of which may be organized privately or collectively. Let us assume that, as in the United States, the first is largely collectivized while the second is primarily organized in the private sector. In the terminology above, the range of collective action will include the activity of water-resource development but not that of telephone service.

What our analysis of the decision-making rules has shown is that, with less-than-unanimity rules, water-resource development, as a single activity, will tend to be “overextended” relative to the hypothetical bench mark. Relatively “too many” resources may be devoted to the development of water-resource projects, even though it may be “ideally” organized in a more meaningful sense. The main point is that our analysis of the operation of decision-making rules says nothing about the range of collectivization. This may or may not be “overextended” relative to the bench-mark criterion. This range of activities will depend on the constitutional decision that has been made concerning the organization of the activities in question.

Such constitutional decisions may or may not be appropriately made. If these decisions are made correctly, the range of collective action will be the “ideal” one, and within this range the separate activities will be organized by the costs-minimizing decision-making rules. External effects will normally be present, which is the same as saying that these activities will be “overextended” relative to some hypothetical ideal, but this sort of inefficiency will be necessary to achieve an organization which will minimize over-all interdependence costs. However, if constitutional decisions are not appropriately made, either the range or the extent of collective action, or both, may be modified in the direction of improved social organization. The set of activities organized through the public sector may be either unduly restricted or unduly expanded, while the extension of the separate activities collectivized may fall short of or exceed that which would be present under more efficient costs-minimizing decision-making rules.

Several of these points may be illustrated clearly with reference to Figure 17, which is similar to Figure 6 employed earlier. The figure depicts expected costs for a single activity—external costs plus decision-making costs. For this activity collective organization is indicated. If 0A represents the expected external costs from private organization, then any collective decision-making rule between P/N and Q/N will allow collective organization to reduce interdependence costs. The appropriate constitutional decision would be to collectivize the activity and to specify that all decisions relating to it shall be taken under the rule R/N. This “ideal” organization will still involve interdependence costs of RR’, a portion of which must consist of expected external costs resulting from an overextension of activity relative to the bench-mark position.

>Figure 17

Assume now that the constitutional decision dictates collective organization of the activity under a decision rule Q/N. External costs are clearly expected to be lower (since the external-costs function slopes downward throughout its range), but decision-making costs are expected to be much higher than under the rule R/N. Under the Q/N rule, relatively fewer resources will be devoted directly to employment in the activity, say road repairs, and, measured in this dimension only, the allocation of resources would more closely approximate some “optimal” allocation. However, under Q/N, far “too many” resources will be devoted to investment in strategic bargaining. A shift from the rule Q/N to the rule R/N will cause relatively more resources to be employed directly in the carrying out of the function involved (more roads repaired to excess) and, if decision-making costs are neglected, this will represent a shift away from the “optimality” surface. However, the incremental external costs involved in this shift will be more than offset by the reduction in decision-making costs that is expected to take place.

Collective Action and Rules for Decision

One of the most important conclusions stemming from our whole analysis is that the decision as to whether or not any specific activity should or should not be organized in the public sector will depend on the decision-making rules that are to be chosen. It is almost completely meaningless to discuss seriously the appropriateness or the inappropriateness of shifting any particular activity from private to public organization without specifying carefully the rules for decision that are to be adopted if the shift is made. If the rules for decision in the collective sector are assumed to be exogenously determined by constitutional provisions and by convention, the choices concerning the organization of activities will be directly dependent on these independent variables, and the whole constitutional-choice process will be severely constrained. As suggested in Chapter 6, it may be quite sensible to shift certain activities to the public sector provided certain rules for decision are adopted, and quite irrational to shift the same activities to the public sector under the expectation of still other rules. Figure 17 is again illustrative. If any decision-making rule less inclusive than P/N should be assumed to be fixed independently of the organizational decision, the individual should rationally reject all attempts to place the activity depicted in Figure 17 in the public sector. Only if the rules for decision fall within the range P/N to Q/N will collectivization of the activity be desirable.

Institutional Variables as Analogues for Decision Rules

As we have previously suggested, it will be possible in many cases to organize the operation of an activity in such a manner that analogues to decision rules may be built into the activity itself. For example, if the activity depicted in Figure 17 is expected to impose some external costs on the individual because of the differential or discriminatory nature of the benefits provided, a differential pricing or taxing scheme may be constitutionally adopted. This institutional change would, of course, modify completely the nature of the activity as conceived by the individual at the stage of constitutional choice, and, other things being equal, this would make the individual much more willing to accept both the collectivization of the activity and the operation of the activity under less-than-unanimity rules for decision-making. For our purposes, it seems best to treat activities organized through different institutional arrangements as different activities. For example, a postal system organized wholly on the basis of user pricing becomes a different activity from a postal system designed to be financed from general taxation. To the individual considering these at the stage of constitutional choice, the shape of the expected-costs functions would be so different in the two cases that it seems best to consider them as wholly distinct activities.42

Side Payments and the Size of the Public Sector

We have previously said that any form of vote-trading, extending from simple logrolling to full monetary side payments (open buying and selling of votes), tends to allow individual intensities of preference on political issues to be more fully expressed. Any of these institutional modifications in the operation of voting rules will tend, therefore, to lower somewhat the external costs that the activity is expected to impose on the individual. If the individual knows in advance that he can, on an issue about which he feels very strongly, take some action to secure the support of less interested voters, he will expect the external costs of the activity to be less severe. In terms of our diagrammatic construction, the introduction of vote-trading in any form serves to shift downward the combined costs function shown in Figure 17.

Since the introduction of vote-trading under consideration applies only to political votes or political support, the expected external costs from private organization should not be modified by such an institutional difference. From this it follows that the constitutional decision as to the organization of the activity will depend also on the extent to which vote-trading is permitted and the extent to which such trading is expected to approximate perfect side payments in final results. The direction of this effect is clear. The more perfect the vote-trading “market,” the wider the range of collective activities that will tend to be selected at the stage of constitutional choice. The less perfect the “market,” the more restrictive must be the range and scope of collective action. The society that is characterized by strong and effective ethical and moral restraints, which prevent vote-trading, will find it more essential to place constitutional curbs on the political decisions of the majority than will the society in which these restraints are less effective.43

The Choice of Rules

The discussion continued in this and the preceding chapters emphasizes clearly the ambiguity that is necessarily introduced when reference is made to the “ideal” allocation of resources or, in our particular case, to the size of the public sector as being “too large” or “too small.” We have demonstrated that the criteria against which the size of the public sector is usually measured are not fully appropriate. In many instances “optimal” positions represent hypothetical ideals impossible of attainment. Normative judgments can be made only after a comparison of realizable alternatives.

An important, and closely related, point is also illustrated here. The individual, in his role as constitution-maker, does not choose directly the size and the scope of the public sector, “the allocation of resources.” Individuals choose, first of all, the fundamental organization of activity. Secondly, they choose the decision-making rules. In a somewhat broader context both of these choices can be conceived in terms of rules, and rational decisions must always be based on some comparison of the working out of alternative rules of organization over a sequence of issues. This emphasis on the fact that policy-makers always choose among organizational rules and not among “allocations” is often forcefully made by Professor Rutledge Vining. Our discussion of the constitutional calculus makes Vining’s criticism of the orthodox or standard discussion of policy norms quite meaningful. To make normative statements concerning whether or not governments undertake “too much” or “too little” activity seems to be rather wasted effort unless one is prepared to suggest some possible modifications in the organizational rules through which decisions are made, aside, of course, from the purely propagandist and nonscientific effects of such pronouncements.

15.

Qualified Majority Voting Rules, Representation, and the Interdependence of Constitutional Variables

The analysis of the simple majority voting rule can be extended without difficulty to cover more or less inclusive rules for reaching collective decisions. The results from this sort of extension will be apparent to those who have understood and accepted the analytical models of the preceding chapters. If less than a simple majority should be required for carrying a decision, the expected external costs would be greater, but the costs of reaching the necessary agreement among members of the effective coalition would be lower than under the operation of simple majority rules. If more than a simple majority should be required for decision, the expected external costs would be reduced, but the decision-making costs would be increased.

Given the behavioral assumptions of our models, individuals will tend to make collective decisions by organizing themselves in the smallest coalitions defined as effective by the decision-making rules, and, for members of dominant coalitions, the gains will tend to be shared symmetrically. Larger coalitions than those necessary for decision will not tend to emerge for two reasons. First, a larger-than-necessary individual investment in strategic bargaining will be required. Secondly, a smaller individual share of the gains from collective action will result in the larger-than-necessary coalition. If we relax our behavioral assumptions or if we introduce specific uncertainties about individual bargains into the analysis, these results will be modified. However, it seems useful to remain for the time being within the strictest limits of the original analysis.

As we prohibit full side payments on single issues and introduce logrolling as an imperfect system of vote-trading, the analysis of simple majority voting can also be applied to other voting rules. Coalitions will be formed embodying reciprocal support over a sequence of issues, and these coalitions will also tend to be of the minimum effective size.

Only one interesting analytical point seems worth raising. Intuitively, it seems plausible to expect that the more inclusive voting rules will tend to produce “solutions” that are somewhat more stable than less inclusive rules. For example, a rule which requires a three-fourths majority may appear to produce more stable solutions than one which requires one-fourth. Such an inference may not, however, be correct. While larger investment in bargaining will be required the larger the coalition that is needed for decision, the reward to the individual member will also be less the larger the coalition. The “price” at which individuals can be induced to abandon the coalition will tend to be lower in the larger coalition than in smaller ones. There are thus two opposing effects on the stability of the solutions produced by the operation of voting rules, and any general conclusions relating the stability properties to the rules themselves would probably be premature.44

We do not propose to discuss further the extension of our analysis to simple voting rules, that is, to rules representing merely changes in the fraction of the total population required to reach collective decisions. The remainder of this chapter and the following chapters will be devoted to a discussion of two somewhat more complex modifications of our models. In this chapter we shall discuss the applicability of our analysis in moving beyond direct democracy to representative government. As we introduce representation, we shall find it necessary to consider four basic constitutional variables and their interrelationships. In Chapter 16 we shall consider the effects of introducing dual representation in two-house legislatures while retaining simple majority voting rules in each house. From these two still elementary models it should be clear that the basic analysis can be extended to a rather bewildering and complex set of possible institutional structures, many of which are to be found in real-world political systems. We do not, however, propose to make such extensions in this book.

Representative Government

Direct democracy, under almost any decision-making rule, becomes too costly in other than very small political units when more than a few isolated issues must be considered. The costs of decision-making become too large relative to the possible reductions in expected external costs that collective action might produce. If direct democracy were required, the individual, in his presumed role as constitutional choice-maker, would leave many traditional activities of the State to be organized in the private sector, and, for those few activities that he chose to collectivize, he would tend to adopt the less inclusive decision-making rules. In terms of our models, one means of reducing the interdependence costs generally is through the introduction of representative government. This step serves to shift downward the decision-costs function that we have previously employed several times in analyzing constitutional decisions.

If we utilize the models developed in Part II, it becomes relatively easy to construct a conceptual normative theory for the “optimal” degree of representation. At the one extreme, we have direct democracy in which the number of individuals directly participating in collective choice (the number of “representatives”) and the number of individuals in the total voting population stand in a one-to-one correspondence. At the other extreme, we have a single individual who “represents” or chooses for the whole group. In either of these two extreme cases, the constitutional-choice problem is greatly simplified. In the case of direct democracy, the single choice to be made, once a basic organizational decision is assumed, concerns the rules under which collective action shall be taken. Under the other extreme dictatorship model, the rules for collective action are set; the only choice facing the conceptual constitution-maker concerns the rules for choosing the dictator. In any of the models falling between these two extremes, both of these choices must be faced. Rules for choosing representatives must be determined, and rules for deciding issues in legislative assemblies must also be laid down. In addition, there is a third choice that must be faced, a choice that is assumed to be resolved in the two extreme models. The degree of representation must be chosen: that is to say, the proportion of the total population to be elected to the representative assembly must be selected. Finally, to all of these choices a fourth must be added: namely, the selection of the basis for representation. We shall refer to these as the four essential constitutional variables.

Consideration of the complexities introduced by these several constitutional-choice problems reveals the abstract and highly simplified nature of our direct democracy models, in which we were able to eliminate all of the choices except the one relating directly to decision-making rules. In a more general context it is evident that the four constitutional problems are interrelated, and, ideally, the individual should reach a decision on all four variables simultaneously. The basis of representation and the degree of representation indicated to be most “efficient” will depend surely on the rules through which representatives are to be selected and the rules which are to be required to carry decision in the legislative assembly. The separate variables can only be discussed individually in partial terms: that is, we may assume three of the variables to be fixed while discussing the fourth. Essentially this is what we have done in our earlier chapters. If we assume that the rules for selecting representatives are given, and that the degree of representation and the basis of representation are predetermined, our models may be applied directly to the setting of the rules for decision in legislative assemblies. On the other hand, if we assume these latter rules to be given, along with the degree and the basis of representation, we may apply our analysis to the selection of rules for selecting representatives without major analytical changes being required. The problems of determining the degree and the basis of representation are similar, but they seem sufficiently different to warrant some detailed consideration.

The Degree of Representation

We now want to consider only the choice concerning the degree of representation. Let us assume that representatives are to be chosen by simple majority voting rules, that the basis of representation is geographical, and that the unicameral legislature is to reach all decisions by majority voting. All of the constitutional variables are thus fixed except that which defines the proportion of the population that will sit as “representative” for the whole population in the assembly.45

Within the restrictions of this model, we can derive costs functions that are quite similar, but not identical, to those which we have previously employed. Figure 18 illustrates. On the ordinate we measure expected costs, as before, but on the abscissa the quantity measured is different from that of earlier models. Here we measure the proportion of the group to be selected as members of the representative assembly. As before, we may now derive an external-costs function and a decision-making-costs function. They will have the same general shape as before. Let I represent the expected external-costs function. This will tend to slope downward because surely the individual will recognize that his own interests will be represented more adequately and more faithfully the more closely the representation approaches the full membership of the group. Note that, even at N/N, external costs are expected to be positive. This is because we have assumed a single rule, majority voting, in the legislative assembly. The positive value of the function at N/N, therefore, suggests that even with direct democracy the individual will expect to be in the losing coalition on some occasions.

>Figure 18

Let J represent the decision-making-costs function. This will rise as the legislative assembly becomes larger because, given any rule, the costs of securing agreement increase. For example, let us suppose that the total group is made up of 100 persons. If one representative in 20 is selected, we should have a legislature composed of 5 persons, and, under simple majority rule, the agreement of 3 persons would be required for decision. On the other hand, if one representative in 10 is selected, we should have a legislature of 10 members, and a majority of 6 persons would be needed for decision. Clearly, the costs of securing agreement among 6 persons are greater than those of securing similar agreement among 3. As before, we may now add these two costs functions vertically, securing the curve I + J in Figure 18. The “optimal” degree of representation is shown where K/N of the total group are chosen to sit in the legislative assembly.

This analysis is simple and straightforward, but unfortunately it is also rather useless as it stands. Nevertheless, some interesting implications do emerge. First of all, the functional relationships described above are clearly affected by the size of the total group. As N becomes larger, the decision-costs function in Figure 18 will tend to shift upward. By comparison, the external-costs function, I, seems likely to be more directly influenced by the proportion of the population sitting in the assembly than by the size of the total population. If this is true, this function will be less affected by shifts in the over-all size of the group than the decision-costs function. The implication seems to be that the costs-minimizing solution is reached at a lower fraction of the total group in larger groups than in smaller groups. This implication seems intuitively obvious, but it does provide us with a quasi-empirical check on the conceptual validity of our general analytical models. It also helps to rationalize the common practice of democratic governments to lower the fraction of the population in the representative assembly as the population grows. They tend to do this by maintaining approximately fixed-sized representative assemblies.

A second, and less obvious, implication follows directly from the first. Since decision-making costs increase as the group grows larger, and since there seems to be no reason to expect that external costs will decrease, the total costs expected to arise from collective organization of activity, under any given rules for legislative decision-making, will tend to be higher in large groups than in small groups. This suggests that the basic organizational decisions will be affected by the size of the group; ceteris paribus, the larger the size of the group, the smaller should be the set of activities undertaken collectively.

The Basis of Representation

The constitutional variable that we have called “the basis of representation” is difficult to analyze in precise quantitative terms. Meaningful analysis does seem possible, however. First of all, let us “freeze” the other three constitutional variables. We shall assume that a simple majority of constituents is required to elect a representative who can normally be expected to act in a manner that will please a majority of his constituents. We shall also assume that the number of representatives in the legislature is fixed, and that a simple majority rule is to be adopted for decision-making in the legislature. The only variable left free for determination is the one that defines the basis upon which the representatives are to be selected from among the whole population.

We may proceed by examining the extreme cases. Conceptually we can think of a basis for representation that embodies a deliberate attempt at randomizing individual variations of political interest. For example, suppose that individuals should be classified into constituent groups solely on the basis of beginning letters of their surnames. Each group, appropriately adjusted in size with other groups, would be authorized to elect a single representative to the legislative assembly. Under this or any other roughly similar basis for representation, we should expect little or no convergence of special-interest groups behind particular representatives on any continuing or permanent pattern. Relatively, the most important stage for coalition formation in these circumstances would be at the level of electing the representative. The individual would anticipate significant external costs at this level of the political process; his own “representative” would effectively support his interest (would “represent” him) only if the individual voter should belong to the winning or majority coalition. Different coalitions would, of course, emerge in different constituencies, and some external costs would be expected to be produced by the actions of the legislative assembly. However, under the circumstances postulated, the individual citizen should be, relatively, more interested in the rules under which representatives are to be selected and in the degree of representation than in the rules for final legislative decision.

In this model (which we will call the “randomized-basis” model) vote-trading would take place at all levels, but it would be most pronounced at the level of electing representatives and would take the form of implicit logrolling. The individual who sought to be elected to the representative assembly would find it necessary to offer a “package” program sufficiently attractive to encourage the support of a majority of his constituents. Since, by hypothesis, the separate interests of his constituents correspond in range to those of the whole social group, he will include in the “package” many special programs designed to appeal to the strongly expressed interests of minority groups.

In the simplest “randomized-basis” model, there would be no assurance that similar “packages” would even be presented to each group of constituents, and very slight probability that the elected representatives to the assembly could be grouped readily into identifiable positions. Each representative might reflect a wholly different configuration of interests.

Certain statements can be made concerning the over-all characteristics of such a system of representation. By and large, it would seem that the expected external costs of collective action should be lower than under alternative bases of representation. The randomized basis would probably offer somewhat greater protection against the deliberate exploitation of specific minority interests, assuming fixed values for the other three constitutional variables. On the other hand, the costs of reaching collective decisions would probably be quite high in this model. Bargains of complex nature would have to be arranged at the level of selection of representatives, and exceedingly complex bargains might be required for the functioning of the legislative process.

Let us now consider a model at the opposite extreme. Assume that a purely functional basis for representation is selected. That is, assume that each definable interest group in the population is allowed to select a representative or representatives as members of the legislative assembly. The contrast with the first model is sharp and clear. If individual interests are homogeneous over reasonably large groups of individuals by identifiable functional characteristics, there will be relatively little difference in the various rules for electing representatives. The individual, in making constitutional choices, will only be interested in seeing that a member of his group (union, trade association, or professional society) sits in the assembly and that the membership of the latter is distributed over the different groups so that “adequate” representation is provided his own group. The expected external costs in this model will be concentrated on the prospects of adverse legislative decisions, not on the prospects of electing representatives who will not effectively act on behalf of individual voters. From this it follows that the rules for legislative decision will be the important constitutional variable under this basis for representation.

It seems obvious that decision-making costs will be considerably lower in this than in the randomized-basis model. On the other hand, expected external costs will surely be higher, assuming, of course, that the rules for selection and for decision are fixed. If we should want to diagram the selection of a basis for representation in terms of two costs functions similar to those employed several times before, we could, conceptually, think of starting at the left with the functional representation basis and proceeding to the right as we approach the purely randomized basis. If this were done, the curves so drawn would slope in the same directions as in the earlier problems, and, conceptually, an “optimal” basis of representation could be chosen—“optimal” being defined here in terms of the “ideal” mix of random and functional elements in the basis.

Geographic representation, the standard basis for at least one house in the legislatures of most Western democratic countries, falls somewhere between the two extreme models discussed above—between purely randomized representation and purely functional representation. If, in fact, individuals and groups were distributed randomly over space with reference to their political interests, geographical representation would approximate the first model. On the other hand, if separate political interests should prove to be primarily geographical, the second model would be more closely approached. We know, of course, that elements of both random and functional representation are present in the geographical basis. Within single constituencies there is normally to be found a reasonably wide range of voter interests, but there also remain many political issues which involve differential geographical impact. On such issues the geographical basis becomes similar to the purely functional in effect. Geographical representation is similar to majority voting in that, a priori, there is nothing that can be said for it as regards superiority over other possible bases.

The Structure of Control in Representative Democracy

The costs implicit in the substitution of representative democracy for direct democracy are of the category that we have denominated “external costs.” Bargaining costs are reduced by the use of the representatives. The costs which would arise from attempting to govern the whole United States through direct majority voting are so extreme that the representative system is acceptable even though it does markedly increase the external costs. In order to examine the external costs created by the representation device, let us construct a simple model. Consider a society composed of 25 voters who organize themselves into 5 constituencies of 5 each for the purpose of appointing representatives to conduct their mutual affairs (Figure 19).46

As a first approximation, let us suppose that the representatives, r₁ ... r₅, simply vote as the majority of their constituents want them to. Under these circumstances a measure favored by nine voters, arranged like those marked X in the diagram, will be adopted. In the real world, as the number of voters and constituencies increases, the minimum-sized coalition required for dominance under simple majority voting approaches ¼ of all voters as a limit. For example, if there should be 39,601 voters arranged in 199 constituencies of 199 voters each, only 10,000 voters would have to favor an issue to secure passage (only 100 more than ¼ of all the voters). Thus, a logrolling bargain to obtain benefits from the political process need only involve about ¼ of the voters under a representative system. Therefore, representative institutions of this type are almost equivalent to permitting any group of ¼ of the voters in direct democracy to form a logrolling coalition empowered to determine what roads will be repaired, which harbors dredged, and which special interest groups will receive government aid. At this stage in the book it should not be necessary to point out how great the external cost imposed by such a procedure would be.

>Figure 1947

These external costs imposed by representative voting would be moderated by two factors. In the first place, not just any group of ¼ of the voters could win. It would be necessary for the group to be approximately equally distributed among a bare majority of the constituencies and absent in the remainder of the constituencies. This fact (which has already been discussed) would presumably put some, although not very onerous, restrictions on the bargains which could be struck. The type of project which is traditionally associated with the pork barrel—a small item benefiting a small group of voters, most of whom are within one constituency—would be little handicapped by this factor. Bargains intended to benefit groups spread through several constituencies, however, will be harder to negotiate, and groups spread through more than a majority of the constituencies will find profitable bargains extremely hard to arrange.

The second limiting factor lies in the organization of the bargaining process. Instead of each voter entering into bargains with other voters, the bargains are negotiated entirely by the representatives. This undoubtedly reduces the total-bargaining cost as compared with attempting to make bargains directly among millions of voters, but it also introduces sizable imperfections in the “market,” and these may affect (either positively or negatively) the external costs. In offering themselves for election, representatives offer to the voters in their constituencies a “platform” embodying that which they propose to accomplish. The individual voter then judges which of the competing candidates’ platforms is most to his liking, discounting this judgment by his estimate of the likelihood of the various candidates’ succeeding in making their promised bargains in the representative body, and casts his vote accordingly. The result is not precisely equivalent to that which would be expected under direct bargaining, but we do not propose to consider the differences in this work.

In general, legislative bodies are designed with two chambers (a subject discussed in the following chapter), but there are some countries which have either a one-chamber legislature or a two-chamber legislature with one chamber having greatly restricted powers. We might expect governments depending on this device to be highly inefficient, but an examination indicates that they frequently have mechanisms which, in essence, change the nature of the system enough to avoid the consequences that we have been discussing. Most of the small North European democracies, for example, follow a voting system under which the voter opts for a party and then the parties are given votes in the legislature in proportion to their respective totals. Although this system has its disadvantages,48 it does have the advantage of providing what amounts to a unanimity system in selecting members of the representative body. All voters, not just the majority of each constituency, are represented in the legislature. Consequently, a majority of the legislature represents a majority of the voters, not just ¼+ as may be the case in a logrolling or party coalition when the members are elected from single-member constituencies.

Interdependence among Constitutional Variables

We have emphasized that the four basic constitutional variables introduced by representative government are interdependent. The “optimal” or “equilibrium” value for any one variable will depend on the values for the remaining variables, and, conceptually, the fully rational constitutional choice will embody the results of a simultaneous determination of all four variables, along with the more fundamental organizational decision concerning whether or not an activity or a set of activities is to be collectivized at all. We know, of course, that the variables may not be set simultaneously at their “optimal” values. Even at the highly abstract level of analysis characterizing our discussion, it will be useful to examine more carefully the interdependence among these variables. This examination will be helpful in demonstrating that our basic model may be applied to a wide range of constitutional-choice problems. We should be able to indicate some of the directions of change in the “equilibrium” values for remaining variables that would result from exogenous or externally imposed changes in single variables. In terms of a specific illustration, we should try to predict the direction of change in, say, the legislative-assembly rules for decision that would be suggested as a result of an externally imposed shift from a randomized basis to a functional basis of representation in the assembly. Or, to introduce a second illustration, we may be able to suggest the “desirable” change in the degree of representation indicated as a result of changing the rules for electing representatives.

In order to discuss these interrelationships carefully, we shall find it useful to define the separate constitutional variables:

Our whole analysis here is normative in the sense that we are considering the calculus of the individual as he faces constitutional choices. The four variables are interdependent in this rational calculus. There is no necessary interdependence in any other institutional sense. This individual, as he considers these variables, will be able to construct four independent relationships which will, in turn, enable him to solve the system for four unknowns. We may summarize this set of relationships by (9) given below.

F (X₁, X₂, X₃, X₄)     (9)

We may assume that the individual whose calculus we consider is initially in full “constitutional equilibrium.” This means simply that we assume that he has selected values for the four variables that seem most suitable from his own point of view. In mathematical terms, he has minimized total interdependence costs as a function of the four variables.

min y = F (X₁, X₂, X₃, X₄)     (10)

This function is, of course, minimized when the set of simultaneous equations represented by (11) is solved.

We want now to examine the effects on these “equilibrium” values that will be exerted by imposing exogenous changes on the variables, one at a time. That is to say, let us suppose that an exogenous change forces X₁ to take on some value other than its “equilibrium” value. Let us label this exogenously determined, nonequilibrium value for X₁ as ₁. We ask the question: Granted this change in the value for X₁, what values should the other variables, X₂, X₃, X₄, take in order to minimize total interdependence costs in the new situation, that is, in that situation where ₁ cannot be modified? The problem is the same as before. We seek to minimize total interdependence costs; but, since one of our four constitutional variables is fixed exogenously, we must solve a system of simultaneous equations in only three variables.

min Z = F (₁, X₂, X₃, X₄)

This is accomplished when the following set of equations is solved.

What we want to determine now is the difference in the solution values for X₂, X₃, and X₄ in equations (11) and in equations (13). Since these differences are generated by the initial exogenously imposed change on X₁, we may represent them in the following form.

These symbols represent the changes in the “equilibrium” values for X₂, X₃, and X₄ that are generated when X₁ is exogenously changed from its initial “equilibrium” value, X₁, to its new value, ₁.

To bring this discussion back to our basic constitutional problem, suppose that a satisfactory constitution exists but that the migration of persons over space shifts the established geographical basis of representation from one that was close to the randomized-basis model to one that is significantly more functional in nature. What should the rational individual, if he were confronted with the opportunity to choose, do as regards the possible changes in the rules for selecting representatives, the possible changes in the size of the representative assembly, and the possible changes in the rules for decision in the assembly?

The whole set of effects that we want to examine may be summarized in the form of the following matrix, (15) below, using the symbols as developed in (14).

Each element in this matrix represents the effects on one variable that will result from changing the value of one other variable, assuming that the individual whose calculus we are considering reacts to the exogenous change by seeking to minimize total interdependence costs. For example, let us look at the last entry in the first row, dX₄/d₁. This represents the change in the equilibrium or optimal value for X₄ that would result from the exogenous change represented by shifting X₁ to some arbitrarily determined value, ₁. In terms of the specific meaning attached to these symbols, dX₄/d₁ indicates the change in the rules for decision in the legislative assembly that the individual might consider desirable as a result of an exogenously imposed change in the rules for electing representatives.

It is clear that we cannot expect to do more with this analysis than to indicate the directions of change: that is, we cannot do more than to insert the signs for the symbols in matrix (15). However, this in itself can possibly provide us with a significant amount of information.

Let us now concentrate on the first row. The elements, dX₂/d₁, dX₃/d₁, dX₄/d₁, represent the changes that would be generated in X₂, X₃, and X₄, respectively, by externally imposed changes in X₁, defined as the rule for electing representatives to the legislative assembly. As this rule is made more inclusive (for example, as X₁ increases in value from (N/2 + 1)/2 to 2N/3), the decision-making costs at this level of collective action will increase.

We may note first of all that any exogenously imposed change from the initially assumed “equilibrium” set of values for the constitutional variables must result in an increase in over-all interdependence costs. This follows from the fact that the initial situation is, by definition, “optimal” for the individual in question. In responding to the exogenously imposed change in the single variable under consideration, the individual will, however, attempt again to minimize interdependence costs, within the limits of the new set of constraints. As we have suggested above, the increase in X₁, defined as the inclusiveness of the rule for electing representatives to the legislative assembly, will increase decision-making costs. The change will also reduce external costs,49 but not to the extent that decision-making costs are increased. If no change in the other constitutional variables is allowed to occur, the individual will find himself devoting more resources to the making of collective decisions than he would choose if given the opportunity. While he will be somewhat more protected than before the change from the dangers of adverse collective action, he will want to consider how he might modify those constitutional variables remaining within his control. Specifically, what changes will the individual desire to make in X₂, X₃, and X₄ in response to the change imposed on X₁?

Note that we have specifically defined each of the constitutional variables in such a manner that an increase involves an addition to decision-making costs and a reduction in external costs. We are now inquiring about the changes in X₂, X₃, and X₄ that will result from an increase in X₁. The direction of change in the three variables will depend on the type of relationship that exists among the separate variables. It seems reasonable to suppose that these variables are mutually compensating in the individual’s calculus: that is to say, he will try to shift to a new position of equilibrium by changing those variables remaining within his power of choice in such a manner as to compensate or to offset the initial change imposed on X₁. More specifically, he will try to shift the values for the variables X₂, X₃, and X₄ in the directions that will represent decreases in decision-making costs and increases in external costs. For a decrease in X₁, changes in the other directions would be suggested. As we have defined the four variables, the direction of change in X₂, X₃, and X₄ would, in each case, be opposite to the change imposed on X₁. Thus, we fill in the first row of matrix (15) with minus signs.

(-)    (-)    (-)

These signs indicate that, if the rule for the election of representatives to the assembly becomes more inclusive (if X₁ increases), the basis of representation will tend to become somewhat more functional (X₂ will be decreased), the degree of representation will tend to be decreased, that is, the assembly can be made smaller (X₃ will be decreased), and the rule for decision-making in the assembly itself will tend to be made less inclusive (X₄ will be decreased).

In a similar fashion we may examine the remaining rows in matrix (15). Look at the second row. Here we examine the effects on X₁, X₃, and X₄ that might be predicted to result from a change imposed on X₂, which measures the basis of representation. As the earlier discussion has suggested, a shift from a functional basis for representation to one that contains more randomized elements (an increase in X₂) probably increases decision-making costs but decreases expected external costs. If this is correct, and if the variables are related in a compensating rather than a complementing way, the appropriate changes in the other variables will involve decreases in decision-making costs and increases in external costs. The signs in the second row of the matrix will also be negative. As the basis for representation in the assembly is increasingly randomized (as X₂ is increased), the rational constitutional choice will tend to embody less inclusive rules for selecting representatives (lower values for X₁), smaller representative assemblies (lower values for X₃), and less inclusive rules for decision-making within the assembly itself (lower values for X₄). Accordingly, two rows in the matrix can now be filled in, at least as to sign.

(-)    (-)    (-)

(-)    (-)    (-)

We now move to the third row, which relates to the effects on the “equilibrium” values for X₁, X₂, and X₄ that are produced by independent changes imposed on X₃, defined as the degree of representation. As X₃ increases, that is, as direct democracy is approached, decision-making costs increase sharply, but, of course, expected external costs decrease. The rational individual, assumed to have some opportunity to choose values for the remaining variables, will tend to bear additional external costs (expected) at the other stages of the collective-decision process in order to “save” some decision-making costs (expected). He will tend to select some less inclusive rule for electing representatives (lower values for X₁), a more functional basis for representation (lower values for X₂), and some less inclusive rule for decision-making in the assembly (lower values for X₄). The signs in the third row of the matrix are also negative.

(-)    (-)    (-)

(-)    (-)    (-)

(-)    (-)    (-)

The last row involves changes exogenously imposed on X₄, the variable that describes the rules for making choices in the legislative assembly itself. For the same reasons as before, the signs of the symbols in the row will be negative. As the decision-making rule is made more inclusive (as X₄ increases), rational constitutional choice should dictate a somewhat smaller assembly (lower values for X₃), a somewhat more functional basis for representation (lower values for X₂), and somewhat less inclusive rules for selecting representatives to the assembly (lower values for X₁). The whole sign matrix may now be filled in.

(-)    (-)    (-)

(-)    (-)    (-)

(-)    (-)    (-)

(-)    (-)    (-)

If the relationships among the constitutional variables are those that we have assumed in constructing this matrix, the information contained in the matrix is of considerable importance.50 The fact that all of the elements in the matrix should prove, on the basis of reasonable assumptions about the relationships among the variables, to have negative signs is relevant, methodologically, for our whole analysis of the constitutional-choice process.

The negative signs arise because we have been able to define each of the four constitutional variables in such a manner that an increase in each variable must involve higher decision-making costs and lower external costs—both of these cost elements being considered in an expected sense. This, in turn, depends on our ability to describe each variable (and others that might be potentially considered) in terms of these two basic cost functions. We conclude, therefore, that the highly abstract and simplified analytical model of Chapter 6 is far more powerful than might have been anticipated at first. At the outset the model may have appeared to be applicable only to direct democracy; but, because the other constitutional variables can be readily translated into the same functional variables, the basic analytical model can be employed as the general model for constitutional choice.51 We have shown that the four constitutional variables introduced by representative government can be reduced in form to a single model that embodies the two essential cost functions.

This point may be clarified if we introduce an analogy with economic theory. Economists recognize that, in the real world, most business firms produce and market several products simultaneously. A full and complete analysis of the firm’s calculus would require an examination of many variables, and, conceptually, the fully rational firm must arrive at a determination of all of the variables under its control simultaneously. In spite of this recognition, economists can explain a great deal about the decision-making process of business firms by simplifying this process. By assuming that the firm produces and markets a single product, all of the analysis needed for a broad general understanding of the operation of business firms can be presented. Our model of the constitutional-choice process seems quite similar in this respect. In the real world there are many constitutional-institutional variables which the individual must rationally consider when he is given the opportunity of reflecting on the prospects of alternative political organizations. However, if our purpose is the relatively limited one of analyzing the essential decision-making processes through which all constitutional choices must be made, the simplified construction that we have emphasized seems quite helpful. Perhaps the absence of such models in the literature of political science is to be explained, in part at least, by an overconcentration on the apparent complexities of real-world political processes.

16.

The Bicameral Legislature

The two-house or bicameral legislative assembly is a common institution in Western democracies. This institution represents a particular configuration of the constitutional variables discussed in Chapter 15, and it may be analyzed, up to a point, in terms of our models. We shall proceed first to postulate an extreme case. Let us assume that a social group is composed of 9 persons, whom we shall designate by numbers 1 to 9. Further, we assume that these persons may be easily classified into three distinct interest or pressure groups, which, for convenience, we shall call: Labor, Property, and Trade. We shall use the subscripts L, P, and T to classify the numbered individuals.

Let us assume that the group has adopted a political constitution. All constitutional decisions have been made. (After analyzing the operation of the two-house system, we shall return to discuss the constitutional issue concerning the “efficiency” of this system.) The constitution calls for a bicameral legislature. There are to be three representatives in each house, and simple majority decision is required for action in each house. Final collective decision requires the approval of both houses.

Representatives to the first house, which we shall call the “House,” are to be elected on a functional basis. The three interests are each allowed to elect a single representative by simple majority vote. We may diagram the constituents of each representative to the House in the following way:

In the second house of the legislature, which we shall call the “Senate,” the basis of representation is fully randomized, that is, each constituency includes within it each of the defined interest groups. We may diagram the constituents of each representative to this house as follows:

The question is that of determining how this two-house legislature will work in producing collective decisions. To carry decision, a majority of each house is required. The minimum effective coalition would be composed of four members, two from each house. Let us initially confine our attention to a single, isolated issue. Suppose that R_L and R_P form a majority in the House, and S₁ and S₂ form a majority in the Senate. Let us look carefully at the combined coalition: R_L, R_P, S₁, S₂. No difficulty arises when we consider the first two members. These representatives will try to further the interests of Labor and Property, which, for current purposes, we assume to be well-defined and homogeneous over individuals in the groups represented. The interests represented by S₁ and S₂, however, will depend on the effective voting coalitions that have been successful in local elections. In order for the two-house legislature to yield results similar in nature to the single-house legislature, both S₁ and S₂ must represent coalitions of Labor and Property interests. In specific terms, S₁ must be elected by the coalition of 1_L and 2_P, and S₂ must be elected by the coalition of 4_L and 5_P. Under these highly restricted conditions, collective action would tend to promote the interests of Labor and Property at the expense of Trade. This result is identical to that which would arise from the operation of a single legislative body operating under the same decision-making rules. To be generally true, however, this requires that a majority of the representatives in the randomized-basis house, the Senate, be elected by the same coalition of interests that forms the majority in the functional-basis House. This requirement would appear to be rarely met, especially as we move beyond the abstract models and consider a world in which interests are many, changing, and ill-defined.

Returning to the coalition R_L, R_P, S₁, S₂, now assume that either S₁ or S₂ should be elected by a majority that includes a voter from the Trade group. In this case no legislation could find majority support in both houses unless it was genuinely to the “general” interest of the whole social group. “Class” or “discriminatory” legislation, such as that which could be predicted to arise under the previously discussed configuration, is no longer possible. If, in order to pass both houses, the “representative” of each interest group must participate in an effective coalition, the two-house system introduces a qualified rule of unanimity into the collective-choice process.

It seems clear that the two-house system of representation introduces an element of uncertainty that was not present in our other models. Whereas we could not, in the analysis of a single group, predict the identity of the members of the winning and the losing coalitions in single issues, we were able to indicate the size of the minimum effective coalition that would be required to carry legislation. Moreover, from this limited amount of information some predictions could be made about the degree of minority exploitation and the degree of possible social waste. This is no longer possible under the two-house system, even when we continue to employ the same basic behavioral assumptions. As our examples have shown, the two-house legislature may produce results ranging from those equivalent to simple majority voting in a single house to those equivalent to the operation of the unanimity rule in a single house. The precise results will depend in each case on the overlapping of the interest-group coalitions in each house.

A few points seem worth noting. It is evident that the two-house system will involve considerably higher decision-making costs than the single-house system, given the same rules for choice under each alternative. From this it follows that, unless the two-house system is expected to produce some offsetting reduction in external costs, there is little reason for its rational support. Translated into more practical terms, this means that unless the bases for representation are significantly different in the two houses, there would seem to be little excuse for the two-house system. On the other hand, if the basis of representation can be made significantly different in the two houses, the institution of the bicameral legislature may prove to be an effective means of securing a substantial reduction in the expected external costs of collective action without incurring as much added decision-making costs as a more inclusive rule would involve in a single house. For example, to produce the same results in a single-house legislature, a rule of three-fourths majority might be required under certain circumstances. However, the decision-making costs involved in the operation of this majority might be significantly greater than those involved in the two-house legislature with each house acting on simple majority-voting principles. A priori, it does not seem possible to make such comparisons readily.

Vote-trading will, of course, take place in the two-house legislature, as we all must recognize. The process of vote-trading through logrolling becomes somewhat more complex and its analysis considerably more difficult. In order to undertake this analysis, let us consider briefly a group of 49 voters who have organized themselves in 7 constituencies of 7 voters each for the purpose of electing one house of a legislature, and in another set of 7 constituencies of 7 each for the purpose of electing the other. Let us suppose the constituencies consist, respectively, of the columns and rows of the following square (Figure 20).

This is a system which follows the organizational principle which we may call “complete diversity.” Although complete diversity is unknown in political practice, it provides an excellent starting point for further analysis. The system, of course, is not limited to a group of 49 members. The 9-man electorate discussed above was also organized according to this rule, and a group that may be shown by a square of 199 by 199 will be used later in the chapter. Nor is it necessary that the illustrative diagram be a square; an oblong rectangle, with more representatives in one house than in the other, would be perfectly acceptable. Finally, our reasoning would not be changed if there were more than one voter reflected in each square of the diagram. Thus, we can consider a situation in which each square contains, say, 10,000 voters as one of complete diversity. The only requirement for complete diversity is that the members of the constituency of a representative in one house be distributed evenly among all of the constituencies for the other house.

>Figure 20

The smallest bargain which could enact a group of measures in this type of legislature would involve a coalition of 16 voters, arranged generally like the X’s in Figure 21. The coalition must include 4 voters in each of 4 constituencies of each legislative chamber. At first glance, it might appear that voting under a two-house legislative system leads to the same results as a one-house legislature, since this coalition is also that necessary to get a measure through a one-house legislature.52 In fact, this coalition would get a measure through either of the two houses which are elected by the completely diverse electorates shown on our diagram. A little further consideration, however, indicates that this form of bargaining would not be feasible.

>Figure 21

Suppose, for example, that voter X’ on the diagram decided that he was not being fairly treated and asked for a change which would lead to higher compensation for himself. The remaining members of the coalition would either have to give in or else construct a radically different bargain. If X’ were left out of the bargain, it would be necessary to drop either the row s₆ or the column r₃ and substitute another row or column for it. In other words, any member of such a coalition can be replaced only by radically changing the form of the coalition. In the mathematically convenient 199 by 199 square, a coalition of 10,000 voters organized like the X’s in Figure 21 could control the votes of both houses. However, if one member of the coalition demanded more compensation, then his coalition partners would have the choice of either giving in to his demands or of dropping him and 99 other members of the coalition. This situation is one in which substantial unanimity among a specified group is required to form the coalition, and the difficulties of getting unanimity in practice have been previously discussed. For each individual member of the coalition, investment of resources in strategic bargaining with the objective of getting much more than an equal share of the total returns from the coalition would be rational. In situations where large investments in strategic bargaining are rational, the cost of bargaining becomes prohibitively high. Thus we have an interesting situation in which, in essence, there are two costs-of-higgling functions. In addition to the decision-costs curve associated with changing voting rules, there is also a cost-of-higgling curve associated with the type of bargain to be struck. Although a minimum-membership bargain of the sort shown in Figure 21 would be the most economical from the standpoint of its members, the bargaining costs involved in making it up are prohibitive and this type of coalition can, therefore, be ruled out.

If X’ decides that he is not receiving favorable enough consideration from his coalition partners, they have yet another alternative to paying him what he asks or radically reconstructing the organization of the bargain. They could replace X’ by two other voters, who are located like the two O’s in Figure 21. A coalition constructed by this method, however, will be larger than one composed of people in the arrangement of the X’s and will also be composed of two classes of voters: those whose favorable consideration of the bargain is necessary to obtain approval in each one of both chambers, and those like the O’s whose vote is necessary only to obtain a majority in one or the other of the two chambers.

Leaving aside, for the time being, the question of the size of the new coalition, let us consider the bargaining problems raised by the existence of two classes of members of the coalition. There are two possible ways of dealing with the matter. Leaders may try to treat all members of the coalition equally, or they may choose to “compensate” the members of the two classes differently. The first leads to impossible difficulties. For example, if a policy were adopted of compensating the O’s equally with the X’s, then any X would know that the cost of replacing him would be two times the current “payment” received by the members of the coalition. It would only be rational for him to insist on receiving, say, 1.9 times the amount that others were receiving. If this offer were refused by the other members of the coalition, then they would have to obtain two replacements, and this is even more expensive than meeting his offer. Thus, each voter whose vote is required for approval of the measure in two houses would, if he were rational, hold out for about twice the standard “rate” of compensation. However, it is obviously impossible for a coalition to pay all of its members equally and at the same time pay some of them twice as much as others. The result would be that coalitions which attempted to stick to the system of making equal payments would find themselves, once again, confronted with members who invested sizable amounts in strategic bargaining, and the costs of bargaining would be too high for such a system to be feasible.

The contrary system of “paying” the members of the two classes differently does not raise this kind of problem. If each member of the coalition whose vote is necessary in both houses gets twice what a member whose vote is necessary in only one house does, then members of the coalition should get merely the marginal value of their votes. Any member withdrawing from the coalition can be replaced readily by one or two other voters, and there is, therefore, no incentive to invest excessive resources in strategic bargaining. However, if this two-category system is adopted, then there is no particular reason why coalition managers should favor voters whose votes are necessary in two houses, and who cost twice as much, over voters whose votes are necessary in only one house. The coalition can be made up just as “cheaply” from one type or from the other. This being so, there is no particular reason to expect that people trying to make up such a coalition will concentrate on voters who are necessary in both houses. Moreover, if they do not follow a conscious policy of trying to get such voters into the coalitions, then there would be only a random overlap between the voters in the coalitions which control the majority in each house.

This may be illustrated in Figure 22. The crosshatched squares represent the minimum-sized coalition (5 by 5 in the 81-voter group with two houses of 9 constituencies each) that would be necessary to secure a majority in both houses. This coalition, however, would be no more likely to arise than that shown by the squares marked “O” if the support of the “less powerful” voters (those marked “O” which fall outside the 5 by 5 crosshatched matrix) can be secured at a lower bargaining “price” than the “more powerful” voters. This suggests that in the two-house system the minimum-sized coalition (in terms of numbers) need not arise, even on the assumption of fully rational behavior on the part of all members. Instead, the agreement finally reached will represent the minimum number of voters required to form that effective coalition which involves a minimization of bargaining costs.

>Figure 22

We have no historical experience with systems which involve representation through two houses that are completely diverse in their constituencies, and therefore we cannot check our conclusions by examining data from the real world. However, it is possible to get the same general result by another line of reasoning, which may serve as a partial check. In representative government the negotiating is done by the representatives. Each representative should vote for any measure or combination of measures which will be approved by a majority of his constituents and should attempt to arrange bargains satisfactory to such a majority. Given the arrangements of the constituencies with complete diversity, this simple policy on the part of each representative would lead to the same result that we obtained by analyzing the coalition formation in the two-house legislature. This is because the constituents for a single representative in each house include members of all constituencies in the second house, randomly distributed. The end result, in a system in which the representation is like that shown in Figure 21 but in which the square is 199 by 199, would be that in the mean case approximately 17,500 out of the 39,601 voters would have to approve a measure before it was passed. Of these about 2500 would be situated so that their votes would be necessary in both houses, and these voters would tend to be suitably rewarded for their luck.

Compared with the 10,000 voters necessary to control a one-house representative assembly, 17,500 is a distinct “improvement”—although it is still less than a majority of the voters; 17,500, in fact, is the number of voters that would be needed to pass a measure through a one-house legislature if a 7/8 legislative majority were required. Requiring a ¾ legislative majority in both houses would mean that a little over 24,000 voters would be necessary to pass a measure, of whom almost 6000 would be required in both houses. This is more than a majority and better than could be obtained by requiring unanimity in one house. That is to say, the over-all result would reflect a more inclusive “rule” than would the requirement of legislative unanimity in a one-house legislature, where each representative is elected by a simple majority of constituents.

In Chapter 9 it was stated that the bicameral legislative system automatically discriminates between measures in which the intensities of the desires or antipathies of the voters are equal and measures in which the minority has stronger feelings than the majority. We have thus far been discussing the latter case; let us now turn to the equal-intensity situation. The reader will remember from the discussion in Chapter 9 that, although equal intensities of feelings are most unlikely, the situation could arise if the differences in intensities among the voters were to be symmetrically distributed among subgroups of voters. Studies of the equal-intensity situation, therefore, are useful for such issues as were involved in the traditional idea of general legislation. In matters concerned with foreign policy, the criminal code, and promotion of scientific discovery, etc., it is possible that differences of opinion may well exist, and there is no reason to believe that all opinions will be held with equal intensity, but there is also no particular reason to expect the differences in intensity to be systematically distributed among particular groupings. Although such matters are a relatively minor part of the activities of most modern governments, they are of considerable importance and may well deserve special handling.

In this chapter we have thus far been discussing the intense-minority case; let us now turn to an equal-intensity case. Suppose that in a representative government which uses a single-house legislature, the members of which are elected by simple majority vote from separate constituencies, some issue comes up in which the intensities of the feelings of the voters are equal. Given that the electorate in each constituency is large and that there are quite a number of constituencies (which is the situation in real life), it is highly likely that a majority of the constituencies will have a majority reflecting a majority of the whole electorate. If this is so, then the representative assembly should vote in accordance with the wishes of the majority of the people, which is the “correct” decision in this case. In those cases (and they would be much less common) where the majority was concentrated in a minority of the constituencies, the representatives of those constituencies would be motivated to enter into bargains with the representatives of other districts with the result that the measure would still be disposed of as the majority wished. All of this follows from the fact that, in the equal-intensity case, minorities are unable to compensate members of the majority for changing their votes, while the members of the majority can readily compensate the minority for such changes.

If we consider the changes in this picture which would result from a bicameral legislature with complete diversity of representation, they turn out to be small. Again, if the number of voters is very large and the number of constituencies quite large, the laws of combinations and permutations would result in a majority of constituencies in both houses being in agreement with the majority of the whole population, so in most cases the two houses would simply enact the will of the majority. Cases in which the voters were distributed in such a way that they failed of a majority in one house or the other would be commoner than with a one-house legislature, but still relatively uncommon. As in the one-house system, bargains would not be particularly hard to arrange in such a case. Thus the switch from single-house to two-house representative government makes only a very slight difference in the way that equal-intensity issues are treated. There is a small increase in the cost of higgling, but that is all.

This contrasts sharply with the results for cases where the minority is more intense in its desires than the majority. As we have seen, in such cases logrolling leads to only a little more than ¼ of the voters being able to control a one-house legislature, while over 7/16 are necessary to control a two-house legislature. A rule which required the organizers of a logrolling coalition to obtain the approval of 7/16 of the voters in a one-house legislative system of representative government would require that the legislature, if it were elected by simple majority in individual constituencies, operate on a 7/8 majority rule, i.e., pass only bills which are approved of by 7/8 of its members. The 7/8 rule, however, would impose quite a heavy bargaining cost on equal-intensity measures. The two-chamber legislature, by automatically distinguishing between the two cases and imposing much greater restraints on the erection of coalitions by members of intense minorities than on majorities in equal-intensity cases, can perform a very valuable function.

The advantage gained by the use of the two-house legislature, however, is rather dissipated by the simple majority method of voting. Even in the two-house legislature the intense minority can pass its measures with less popular support than can an equal-intensity majority. This appears the opposite of what should be the case, but given the simple-majority voting rule nothing can be done about it. Departures from the simple majority rule, however, can improve the situation. For example, if methods of election of the representatives should insure that each house represents the whole people, not just a majority in each constituency, then a two-house legislature with simple majority voting in each house would require ¾ of the people to approve bargains of intense minorities, while still permitting passage of equal-intensity measures which were approved of by only simple majorities. This sounds utopian, but conceivable practical arrangements to obtain comparable results would be possible.

So far we have been discussing a two-house legislature in which there is “complete diversity” in the constituencies of the representatives in the two houses. In practice this situation is never found; however, partial diversity is almost universal in governments which use the two-house system. Partial diversity takes many forms, and for purposes of analysis we shall divide it into two subtypes: “arrangement” and “number” diversity. Number diversity is fairly common in its pure form in the real world (the United States legislative branch is an example), while arrangement diversity is almost never found in its pure form. In most cases the two are intermixed in two-house legislatures. We shall examine them in their pure form largely for simplicity, and we shall start with “arrangement” diversity for the same reason.

We have covered a completely diverse system of constituencies for a two-house legislature. At the other extreme we can easily imagine a completely nondiverse system. If the members of each house were elected from the same constituencies (that is, if each constituency sent a representative to each house), then the two houses would be identically constituted, and the situation, from our present standpoint, would not differ from a one-house legislature.53 Using our diagram, it is possible to construct systems of representation which form a continuum from complete diversity to complete nondiversity. To illustrate “arrangement” diversity, see Figure 23. In this square matrix representing 49 voters, the columns denote constituencies in one house. A particular configuration can then be chosen to represent each constituency in the second house. Each senator shares 2, 3, 4, ... 7 voters with some given representative. In Figure 23 we have chosen to give each senator 4 constituents in common with some single representative. For example, the blank squares represent the first senate constituency. Here the senator shares 4 voters with r₁ and 1 voter each with r₂, r₃, and r₄.

>Figure 23

Obviously, as we proceed by small steps from complete diversity to complete nondiversity, the features of completely diverse systems which we have discussed gradually fade away. Semidiverse systems, however, have a special feature which neither completely diverse nor completely nondiverse systems share. Such systems, in effect, classify the voters into categories. For example, in Figure 23 the voters in the bottom four rows are represented in the Senate and in the House by representatives elected from the same constituencies, while the voters in the upper three rows are represented by diversely based legislators. The result is that it is much easier to work out coalitions which will benefit the people in the lower four rows than in the upper three. The costs of bargaining are lower because part of the bargain is already implicitly made by the arrangement of constituencies. Further, bargains involving only voters in the lower four rows will operate on a basis similar to that of the single-house legislature, while those involving voters in the top three rows will have to operate on the same basis as in a completely diverse system. Clearly, this system greatly favors the voters who are so arranged as to have the advantage of a sort of prefabricated bargain.

Although this situation is never exactly duplicated in real-life political organizations, something very like it is quite common. The American farmers, for example, possess what amounts to a built-in coalition in the two houses of our legislature. This gives them a great advantage over less fortunately situated groups.

Our second type of partial diversity is “number” diversity. Under the American constitution many Western voters are much more heavily represented in the Senate than the inhabitants of the more populous states. In the House, on the other hand, people from different parts of the country are more or less equally represented. This situation arises from the fact that each state has two senators, regardless of how sparse its population, while the representatives are distributed among the states according to population. The system has been criticized for giving the voters in the thinly populated states an unfair advantage. This “unfairness,” however, is not intrinsic in number diversity as a concept. It is easy to conceive of a system under which area A elects 5 representatives to the “House” and 1 to the “Senate,” while area B elects 1 to the “House” and 5 to the “Senate,” thus obtaining the advantages of number diversity without giving any voter more power than any other.

The system to which we are accustomed, however, does give the voters in some states an advantage over those in others. In the American system the constituency of most senators “includes” the constituencies of a number of representatives. As illustrated in Figure 24, the constituency for senator s₁ includes the constituencies of representatives r₁, r₂, and r₃. This type of diversity also leads to some improvements over the single-chamber legislature. Many coalitions which would pass in the House will fail in the Senate. For example, the voters marked X in Figure 24 could maneuver their bill through the House, but it would fail in the Senate. On the other hand, there would still be some bills that would be passed by this type of two-chamber legislature which would require only the very minimum of voter support in a one-chamber legislature, e.g., the one shown by the O’s on the diagram.

>Figure 24

Two chambers differing from each other only in this way offer much less of a safeguard against the imposition of excessive “external cost” on the citizen than organization in accordance with what we have called “arrangement” diversity. Further, if the number of constituents varies from “senator” to “senator,” it may introduce an element of discrimination among the voters. Those who are in small constituencies have an advantage over those who are in large ones. Nevertheless, the device does, to some extent, improve the operating characteristics of a system of representative government.

There is also another phenomenon in the real world which can be regarded as an extreme version of number diversity. The President of the United States and many other “executives” are equipped with the veto power. This, in effect, constitutes them as a third house of the legislature. In this case, however, the “third house” represents the entire body of voters in one grand constituency. The President should, insofar as he uses his veto power as a simple legislative tool, follow the preferences of the majority of the voters. Therefore, he would accept only bargains which meet the approval of the majority of the populace, and hence could considerably raise the minimum size of the logrolling coalitions. Normally, of course, the President tends simply to sign most bills, and vetoes only a minority. Nevertheless, he has the power to constitute himself as a third legislative house, and the exercise of this power, whether explicit or implicit, materially improves the functioning of the American Constitution.

17.

The Orthodox Model of Majority Rule

The crux of the question is not whether the majority should rule but what kind of majority should rule.

—Walter Lippmann, The Washington Post, 5 January 1961

We have made no attempt to relate our analysis directly to the history and development of political theory. We propose to leave this for somewhat extended development in an appendix. The “economic” approach to both the problem of constitutional choice and the analysis of decision-making rules is perhaps sufficiently differentiated from what has been the mainstream of political scholarship to warrant independent treatment before the doctrinal setting has been completed. Moreover, in this respect the preliminary and exploratory nature of our whole analytical inquiry must be doubly emphasized.

Nevertheless, it will be useful at this stage to try to compare and contrast our models with the orthodox models of modern political theory, as we conceive the latter. We take this step, not for the purpose of comparison per se, but because in this way the content of our own analysis may be more clearly demonstrated, especially to noneconomist readers. We stand, of course, in danger of having our descriptions of the orthodox models labeled as straw men. Whether the constructions are straw or stone (and we are willing to leave this decision to others), we observe merely that, methodologically, straw men may also be useful.

As implied in Chapters 1 and 2, our approach to collective action is avowedly individualist, rationalist, and secular. At the ultimate stage of constitutional choice, when decisions must be made among alternative means of organizing human activity and among rules for collective decision-making, full consensus of unanimity among all members of the social group seems to us to be the only conceivable test of the “rightness” of the choices made. This postulated unanimity rule for ultimate constitutional decisions allows us to divorce much of our analysis from the long and continuing debate concerning the validity of majority rule as an absolute doctrine of popular sovereignty.

Unanimity and “Political Exchange”

In our view, both at the level of ultimate constitutional choice and at the level of analyzing the operation of particular rules, the issues have often been posed in terms of false alternatives. The alternatives are not those of majority rule or minority rule. One of the great advantages of an essentially economic approach to collective action lies in the implicit recognition that “political exchange,” at all levels, is basically equivalent to economic exchange. By this we mean simply that mutually advantageous results can be expected from individual participation in community effort. Much of the debate surrounding the majority-rule doctrine seems to deny this possibility implicitly, even if such a denial is not explicitly stated. In this sense the discussion seems closely akin to the medieval arguments about the “just price.” If, in market or exchange transactions, the loss to one trader must be offset by gains to the other, some rational basis for philosophical argumentation about the “justice” of prices would be present. However, the simple fact is, of course, that in normal trade all parties gain; there exist mutual gains from trade. The great contribution of Adam Smith lay in his popularization of this simple point, but the full import of this conception for democratic political theory does not seem to have yet been appreciated.

Insofar as participation in the organization of a community, a State, is mutually advantageous to all parties, the formation of a “social contract” on the basis of unanimous agreement becomes possible. Moreover, the only test of the mutuality of advantage is the measure of agreement reached. Modern political theorists have perhaps shrugged off the unanimity requirement too early in their thinking. By noting that the attainment of unanimity is infeasible or impossible, they have tended to pose the false dilemma mentioned above. The early theorists (Hobbes, Althusius, Locke, and Rousseau) did assume consensus in the formation of the original contract. They did so because the essence of any contractual arrangement is voluntary participation, and no rational being will voluntarily agree to something which yields him, in net terms, expected damage or harm. The categorical opposition of interests that many theorists assume to arise to prevent unanimity is much more likely to characterize the operational as opposed to the constitutional level of decision, and it is essential that these two levels of decision be sharply distinguished. It is at the operational level, where solidified economic interests of individuals and groups are directly subjected to modification and change by State action, that violent conflicts of interest can, and do, arise. At the “higher” constitutional level the problem confronted by the individuals of the group is that of choosing among alternative rules for organizing operational choices, and the discussion at this level will be concerned with the predicted operation of these rules. By a careful separation of these two levels of decision, much of the confusion inherent in modern interpretations of the contract theory of the State can be removed. Conceptually, men can reach agreement on rules, even when each party recognizes in advance that he will be “coerced” by the operation of agreed-on rules in certain circumstances. A potential thief, recognizing the need for protecting his own person and property, will support laws against theft, even though he will anticipate the probability that he will himself be subjected to punishment under these laws. Individuals at the level of operational decisions may accept results that run contrary to their own interest, not because they accept the will of the decision-making group as their own in some undefined, metaphysical manner, but simply because they know that the acceptance of adverse decisions (in our terminology, the bearing of external costs) is inherent occasionally in the “bargain” or “exchange” which is, in the long run, beneficial to them. The expected external costs caused by adverse decisions may fall short of the added costs that would be involved in the participation in the more complex political bargaining process that might be required to protect individual interests more fully. In our construction, therefore, there is no necessary inconsistency implied in the adoption of, say, simple majority rule for the making of certain everyday decisions for the group with respect to those activities that have been explicitly collectivized, and the insistence on unanimity of consensus on changes in the fundamental organizational rules. The organizing principle or theme of our whole construction is the concentration on the individual calculus, and it is easy to see that both the unanimity rule at the constitutional level and other less-than-unanimity rules at the operational level of decisions may be based directly on this calculus.54

While it is clear that something akin to the doctrine of inalienable rights—institutionally embodied in constitutional provisions limiting the authority of legislative majorities—can easily be reconciled with our construction, we should emphasize that this doctrine is not central to our construction. The fact that much of our construction can be reconciled with a strain of orthodox democratic theory, and vice versa, should not obscure the profound differences between our approach and the one which has been implicit in much political theory and philosophy, both ancient and modern. The most basic difference lies in the incorporation into our models of the economic meaning of the unanimity rule, a part of our construction previously discussed in Chapter 7.

Much political discussion seems to have proceeded as follows: “If the interests of two or more individuals conflict, unanimity is impossible. Some interests must prevail over others if action is not to be wholly stifled.” This line of reasoning seems quite plausible until one confronts ordinary economic exchange. Note that in such an exchange the interests of the two contracting parties clearly conflict. Yet unanimity is reached. Contracts are made; bargains are struck without the introduction of explicit or implicit coercion. In this case, no interest prevails over the other; both interests are furthered. Our continued repetition of this simple analogy stems from our conviction that, at base, it is the failure to grasp fully the significance of this point that has retarded progress in political theory.

The “social contract” is, of course, vastly more complex than market exchange, involving as it does many individuals simultaneously. Nevertheless, the central notion of mutuality of gain may be carried over to the political relationship. When it is translated into individual behavior, mutuality of gain becomes equivalent to unanimous agreement among the contracting parties. The only test for the presence of mutual gain is agreement. If agreement cannot be reached, given adequate time for discussion and compromise, this fact in itself suggests the absence of any mutuality of gain. Moreover, where mutuality of gain is not possible, no criteria consistent with the individualistic philosophical conception of society can be introduced which will appropriately weight gains and losses among the separate parties to the institution taking the place of a contract (clearly, a relationship that does not embody unanimous consent is not a contract).

There may, of course, exist situations in which the formation of a “social contract” is not possible. When the negotiating parties are divided into groups that are classified on bases which seem reasonably certain to remain as permanent, independently of the decision-making rules that might be adopted, a “constitution” (in the sense that we have used this term throughout) may not be possible. The individual may never get the opportunity to participate (at the level of the Nation-State) in the choice process that we have been discussing. Under such conditions societies will tend to be controlled by some groups which will tyrannize over other groups. Such a situation must continue to exist, so long as genuinely mutual arrangements cannot be made.

Mutually Exclusive Alternatives

Situations such as these are not, however, what the orthodox theorist seems to have in mind when he makes statements like the one which we have attributed to him above. Implicitly, the orthodox theorist conceives all relevant political choice to take the form of selection between two mutually exclusive alternatives. An appropriate analogue is the choice confronted by the traveler at a fork in the road. He must either take one road or the other; the only other alternative is to stop. If, indeed, political decisions should assume this form, the statement imputed to the orthodox theorist above would be somewhat meaningful; but are the decisions that are confronted in the political process properly conceived as choices among mutually exclusive alternatives? Once more, let us turn to the analogy with market exchange. Such exchange could be converted into choices among mutually exclusive alternatives only if one partner to the bargain or contract should be required to secure gains at the direct expense of the other party. If such a rule were laid down in advance, any “solution” would require that the interests of one or the other of the parties prevail and the interests of the “loser” be subjected to “defeat.” In game-theoretic terms, the assumption of mutually exclusive alternatives is equivalent to assuming that the game is zero-sum. The winnings must match the losings. If, in fact, this is the appropriate conception of the political “game,” it is relatively easy to see that, once several persons (several players) are introduced, and if symmetry in preferences among individuals is postulated, the interests of the larger number (the majority) “should” or “ought” to prevail over the interests of the lesser number.

Clearly this would represent a wholly incorrect and misleading way of analyzing economic or market transactions. The implication of the approach would be that no exchange should take place at all because gainers balance losers in two-person trades and symmetry in preference is to be assumed present. Is this approach, by contrast, the appropriate means of analyzing political transactions? By now it is perhaps obvious that we do not think that political choices should, at base, be conceived in terms of selection among mutually exclusive alternatives. The essence of the contractual conception of the collectivity, quite independently of the empirical validity of this construction, involves the mutuality of gain among all members of the group. However, all participants in a zero-sum game cannot win simultaneously. Games of zero-sum are played, and political choices on many occasions do reduce to mutually exclusive alternatives; but why do we observe zero-sum games being played in the real world? The answer is that such games are played because each and every participant has, implicitly, accepted the “contract” embodied in the rules of the game when he chooses to play. The zero-sum characteristic applies to the “solution” of the game; it does not apply to the “contract” through which all participants agree on the rules. At this second level there must be mutual gains, and the rule of unanimity must apply. At this level there is no way in which a zero-sum solution could apply; the game simply would not be played unless all participants expected some individual benefit at the time of entry.

This reference to game theory may be helpful, but we have not yet clearly shown the statement of the hypothetical orthodox theorist to be demonstrably false. Let us turn to a simple model of a three-man society, engaged in the formation of a “social contract.” Let us call the three men A, B, and C. Suppose that the discussion is proceeding on the fundamental organizational rules that entering into a community or group life might involve. Let us assume that A is very interested in insuring that fishing is collectively organized, because he likes fish and because he also realizes that joint effort is much more productive than individual effort. If we limit our attention to this decision, we may reduce it to a yes-or-no question. Either the catching of fish will be collectively organized or it will not. These appear to be mutually exclusive alternatives, and it seems impossible that agreement could be reached unanimously if, say, C, who does not like fish anyway, does not agree to collective organization of this activity. This is the point at which our hypothetical orthodox theorist of the constitutional process seems to have stopped, but this represents the central error of his interpretation. Let us say now that C, in turn, is interested in insuring that the group allow the gathering of coconuts to be privately organized because he thinks he is a much better climber of trees than A or B. On the other hand, A and B both want to collectivize this activity as well as fishing. Suppose that B, in contrast to A and C, is really more interested in securing some defense against external attack than he is in either fish or coconuts. He wants, first of all, to organize some standing patrol or watch. Under these circumstances it becomes conceivable that the group can reach unanimous agreement of a “constitution” or contract. They can do so by making the appropriate compromises or “trades” among themselves. The process would be equivalent to the logrolling process discussed in Chapter 10, and the only test for determining whether or not the organization of the community is or is not mutually desirable to all parties lies in the possibility that such an agreement can be reached. Our hypothetical orthodox theorist, therefore, errs in not following through beyond the confines of a single issue. Once several issues are introduced, and the variance of interests among individuals and over separate issues is allowed for, trades become possible. Moreover, when trade can take place, the analogy with economic or market exchange is appropriate. No longer must the group reach yes-or-no decisions at the constitutional level; no longer must alternatives be mutually exclusive. The existence of conflicts of interest does not preclude the attainment of unanimity; this merely makes it necessary that discussion proceed until the appropriate compromises are found.

If direct side payments among individuals are allowed for, even this modification is not needed. Return to our illustrative model. Suppose that the only decision confronted is that concerning the organization of fishing. A and B desire collectivization because of the greater efficiency, but C, not liking fish, is opposed. If side payments are allowed, the support of C for the collectivization of fishing for the group may be secured by the transfer of some item possessing real value to C by A and B (e.g., a few cigarettes); and only if C can be so convinced to support the collectivization of fishing will the entering into the agreement with A and B be worthwhile to him.

The Meaning of “Majority Rule”

We have shown that the attainment of unanimity is always possible if there exist mutual gains from entering into the “social contract,” and that the orthodox theorist has tended to dismiss unanimity as a possible alternative to majority or minority rule too hurriedly because of a concentration on mutually exclusive alternatives. Our earlier models have shown, however, that the group may rationally choose less-than-unanimous decision-making rules for the carrying out of operational decisions for the collectivity. We now want to isolate a second major fallacy in the orthodox position. Even in these cases when unanimity is either not possible or not chosen as the rule by the group, we shall try to demonstrate that the dilemma posed by a a majority rule-minority rule dichotomy remains a false one.

Recall that the unique feature of our models for constitutional choice was the demonstration that, unless equal intensity of preferences is postulated, there are no particular characteristics attributable to the 51 per cent rule for choice. This is only one out of many possible decision-making rules. The peculiar position that this proportion has assumed in orthodox thinking seems to be due to the idea that if less than 50 per cent are allowed to make a decision, the more than 50 per cent will be “concluded” or “coerced” into acceptance. Thus, the requirement of a qualified majority really amounts to allowing a minority to rule. If we may again put the words into the mouth of our hypothetical orthodox theorist, he might say: “If more than 51 per cent are required for political decision, this will really allow the minority to rule since the wishes of the 51 per cent, a majority, can be thwarted.” In this construction there is no difference between the qualified majority of, say, 75 per cent and the simple minority rule of 26 per cent. Whereas in our constitutional models there may be a great difference in the external costs expected to be incurred under the 26 per cent minority rule and the 75 per cent majority rule, the orthodox theorist would deny this difference. Moreover, he would claim that the existing provisions for amending the United States Constitution embody the rule of the minority.

Does the requirement of a qualified majority amount to the rule of the minority? Here, as before, the error of the orthodox theorist seems to reflect his emphasis on reducing all decisions to the yes-no, mutually exclusive type, and the implied failure to put quantitative significance on alternatives confronted. If we come to an issue analogous to the fork in the road mentioned above, and if this is the only issue, and if no side payments are allowed, the orthodox theorist would seem to be on reasonably safe ground in saying that the requirement of 75 per cent agreement would allow the 26 per cent to be really controlling for decisions.

However, if the requirement of a qualified majority of, say, 75 per cent is really equivalent to the minority “rule” of 26 per cent, what sort of decisions must be involved? Not only must the alternatives of choice be conceived as being mutually exclusive, but the alternative of inaction must be counted as equivalent to action. The fork-in-the-road analogy mentioned above becomes too general because the alternative represented by stopping the journey is precluded. One way or the other must presumably be chosen. Suppose that there are 100 persons on a hayride and such a fork in the road looms ahead. Suppose that 74 of these persons choose to take the right-hand fork; 26 of them want to go to the left. With the 75 per cent rule in effect, neither road could be taken until and unless some compromises were made. With a 27 per cent rule in effect, the right-hand road would be taken without question in these circumstances. Surely these two rules produce different results. Failure to secure the required 75 per cent is not equivalent to granting “rule” to 27 per cent. If the third alternative of stopping the journey is allowed for, the 75 per cent rule will not allow action to be taken. The orthodox theorist would argue that such inaction, in this case, amounts to “victory” for the “recalcitrant” 26 persons making up the minority. Taken individually, however, these persons are thwarted in their desires in precisely the same way that the individual members of the larger group are thwarted. These individuals must also bear the costs of inaction. The argument may be advanced that, in such hypothetical situations as this, the interests of the greater number should be counted more heavily; but this, presumably, is a question that is appropriately answered only at the time when the decision-making rules are chosen. In our construction it seems wholly inappropriate to introduce this essentially irrelevant ethical issue at the stage of operational decision-making. When it is recognized at the ultimate constitutional stage that the larger the majority required for decision, the lower are the expected external costs that the individual expects to incur as a result of collective decisions being made adversely to his own interest, we may discuss the operation of the various rules quite independently of all attempts to measure utilities and to compare these interpersonally.

The orthodox theorist will not, however, accept this line of reasoning. He will say that the question to be decided in our illustration should be put as follows: Shall the group take the right-hand road or not?—Vote yes or vote no. In this way the qualified majority rule is made to appear equivalent to “minority rule.” A minority of 26 per cent is empowered to block action desired by 74 per cent.

This argument is more sophisticated than the one considered previously, and it is more difficult to refute convincingly. To do so, we must, first of all, clarify the meaning of the terms “majority rule” and “minority rule.” We have used these terms throughout our analysis to describe decision-making processes. Such general usage is no longer sufficient. We must sharply differentiate between two kinds of decisions: (1) the positive decision that authorizes action for the social group, and (2) the negative decision that effectively blocks action proposed by another group. If a group is empowered to make decisions resulting in positive action by/for the whole group, we shall say that this group effectively “rules” for the decisions in question. It does not seem meaningful to say that the power to block action constitutes effective “rule.”

This relevant distinction between the power of determining action and the power of blocking action has not been sufficiently emphasized in the literature of political science.55 The reason for this neglect seems to be an overconcentration on the operation of simple majority rule. If a simple majority is empowered to determine positive action, there can be no other simple majority empowered to block the action proposed. Two simple majorities cannot simultaneously exist. The distinction becomes clear only when we consider “minority rule.” If we adopt the meaning of this term suggested above, a group smaller than one of simple majority must be empowered to make positive decisions for the collectivity. For example, suppose we choose to consider a 40 per cent decision-making rule. This rule, under our definition, would be operative when 40 per cent of the voters, any 40 per cent, are empowered to take action positively for the whole group. It is clear that the same rule could not also be applied to blocking action. If 40 per cent were also required to block action, then 40 per cent could not be defined as the “rule of the minority” at all. The rule for blocking action must always be [(N + 1) - X] per cent, X being the percentage of the total group empowered to institute or to conclude positive action. Effective minority rule, therefore, must require a majority to block legislation proposed by the minority. Effective “rule” by the 40 per cent minority must involve the requirement that 61 per cent of the voters are required to veto action proposed by a minority.

When the orthodox theorist suggests that qualified majority voting amounts to “rule” by the minority, he is referring to the rule for blocking action. If this line of reasoning is carried to its logical conclusion, we get the paradoxical result that the rule of unanimity is the same as the minority rule of one. Thus the rule of requiring unanimity among members of a jury to acquit or to convict becomes equivalent to the rule that would permit any individual juror to convict or to acquit. Instead of being at the opposing ends of the decision-making spectrum, as our whole construction suggests, the unanimity rule and the rule of one become identical. This paradoxical result suggests clearly that the power of blocking action is not what we normally mean, or should mean, when we speak of “majority rule” or “minority rule.”56

The distinction between the power of taking action and of blocking action proposed by others is an essential one; it represents the difference between the power to impose external costs on others and the power to prevent external costs from being imposed.

We may illustrate with reference to our familiar road-repair example. Let us assume that the constitution of our model township dictates that all road-repair decisions are to be made by a two-thirds majority. Under these conditions the power to institute action, lodged in any effective coalition of two-thirds of the voters, involves the power to impose external costs on the other one-third, either through the levy of taxes or the failure to repair certain roads to standard. One-third of all voters plus one have the power to veto or block any proposed repair project, but this power is effective only in the sense that a group of this size can prevent the additional taxes being levied. In no way can this minority group impose additional external costs on the other members of the group.57

The Problem of Biased Rules

We have not yet satisfied the hypothetical orthodox theorist.58 He may conceivably accept all of our previous arguments but still try to stop us short by saying: What about the situation in which the issue confronted is whether or not a change in the rules should be made? Here the alternatives are mutually exclusive: change or no change. Moreover, should the established order (the status quo) operate in such a way as to benefit special minority interests, then surely the qualified majority rule for changing the “constitution” will allow the blocking power of the minority to be controlling. In effect, the maintenance of things as they are amounts to genuine “minority rule.”

This argument gets us to the heart of the whole discussion of majority rule as a doctrine of popular sovereignty, to which we referred earlier in this chapter to some extent. We have discussed the applicability of the unanimity rule at the stage of making original constitutional decisions. At any point in time subsequent to the formation of the original “contract,” the social organization must be presumed to be operating within the framework of certain established rules. These organizational rules define the way in which certain collective decisions will be made, including decisions to change the “contract.” If these basic rules suggest that, for some decisions, more than a 51 per cent majority is required for positive action, it is surely the established order of affairs that may be said to be “ruling,” and not the particular minority that may or may not be securing “benefits” through the continuance of this established order.

This is not to suggest that the established order must prevail for all time, once it is accepted, nor that, either at its beginning or at any particular moment in time, this order is necessarily “optimal.” The “social contract” is best conceived as subject to continual revision and change, and the consent that is given must be thought of as being continuous. However, the relevant point is that change in this “contract,” if it is desirable at all, can always find unanimous support, given the appropriate time for compromise.

Again we revert to the game analogy. We may, if we like, think of players as being continually engaged in two kinds of mental activity. First, they are trying to figure out moves or strategies with which their own interests can be advanced within the context of a well-defined game. Secondly, and simultaneously with this activity, they can be conceived as trying to figure out a possible change in the rules that would make for a better game. In this second activity they will realize that they must choose rule changes on which all players can agree if the game is to continue. A proposed change in the rules (or in the definition of the game) designed especially to further individual or group interests, majority or minority, would be recognized to be impossible. The other players could simply withdraw from the game.

Our conception of the constitutional-choice process is a dynamic one quite analogous to the game mentioned above. We do not conceive the “constitution” as having been established once and for all. We conceive the contractual aspects to be continuous, and the existence of a set of organizational rules is assumed to embody consensus. We think of the individual as engaging continuously both in everyday operational decisions within the confines of established organizational rules and in choices concerned with changes in the rules themselves, that is, constitutional choices. The implicit rule for securing the adoption of changes in these organizational rules (changes in the structure of the social contract) must be that of unanimity. This is because only through the securing of unanimity can any change be judged desirable on the acceptance of the individualistic ethic.

This does not imply, as is so often suggested, that the requirement of unanimity on changes in the rules (on the constitution) embodies an undue or unwarranted elevation of the status quo to a sacrosanct position. In the first place, the idea of status quo in terms of established organizational rules is hazy at best. The stability of the established rules for organizing public and private decisions does not, even remotely, tend to stabilize the results of these decisions measured in terms of the more standard variables such as income, wealth, employment, etc. The municipal-zoning ordinance may be accepted by all parties until someone has the opportunity to sell his own property to a developer at a huge capital gain. At this point in the sequence, the individual standing to gain would certainly desire a change in the rules to allow him to exploit this unforeseen opportunity, but it is precisely because this sort of thing is unforeseen that the zoning ordinance can be adopted in the first place. Ex post, the individual faced with the opportunity to gain is likely to object strenuously to the status quo (that is, to the zoning ordinance), but securing a variance for one individual alone is equivalent to changing the rules of the ordinary game to the strategic advantage of one player. In the continued playing of the “social game,” individuals will each confront situations in which they desire, strategically, to change the rules; but it is because these situations are distributed stochastically that agreement becomes possible. If a change in the rules (a change in the status quo) is mutually beneficial, it will, of course, be adopted. Empirical evidence from the operation of voluntary organizations suggests that rules are often changed.

An individual need not, of course, accept the “contract” that exists. He may rationally consider the rules to be undesirable. Faced with this conclusion, two choices remain open to him. He may seek to convert others to his point of view, and, if arrangements can be worked out through which all others come to agree, the “constitution” can be changed. Secondly, the individual may choose to reject the “contract” entirely; he may revert to a “state of nature”—in this case a revolt against established social order. On ethical grounds the individual must always be granted the “right” to make such a choice, but, once he has done so, the remaining members of the group have no contractual obligation to consider the revolutionary to be subject to the protections of the “contract.” This “right of revolution” is not modified as it extends beyond the single individual to a minority or even to a majority of the population. In this, as in other aspects of our construction of the constitutional implications of a consistent individualistic philosophy, the shifts in the fraction of the population approving or disapproving certain changes are not of central importance.

[1. ]Our approach is fundamentally different in this respect from that employed by Downs. He also adopts an “economic” approach to democratic process, but, instead of starting at the individual level, he starts with two-party representative democracy and analyzes the political process in terms of the attempts of governments to maximize voter support. See Anthony Downs, An Economic Theory of Democracy (New York: Harper and Bros., 1957).

[2. ]At this point in our analysis we do not imply either praise or condemnation of any behavior of the individual on moral or ethical grounds. Language conventions force us to use the words “moral” and “ethical,” and moral principles must be discussed later in the book, but we do not want to prejudice the analysis by moralizing at this early stage.

[3. ]An interesting recent novel about Washington politics, written by an observing journalist, includes logrolling at several levels as an important part of the political picture. See Allen Drury, Advise and Consent (New York: Doubleday, 1959). For the reactions of an “orthodox” liberal student of politics to this approach, see the review of the book that appeared in The Reporter for 11 November 1959.

[4. ]On this point, we agree with the view of Arthur Bentley. His statement on the issue is worth noting: “Log-rolling is a term of opprobrium. This is because it is used mainly with reference to its grosser forms. But grossness as it is used in this connection merely means that certain factors which we regard as of great importance are treated by the legislator as of small importance and traded off by him for things which we regard as a mess of pottage, but which he regards as the main business of his activity. Log-rolling is, however, in fact, the most characteristic legislative process. When we condemn it ’in principle,’ it is only by contrasting it with some assumed pure public spirit which is supposed to guide legislators, or which ought to guide them, and which enables them to pass judgment in Jovian calm on that which is best ’for the whole people.’ Since there is nothing which is best literally for the whole people, group arrays being what they are, the test is useless, even if one could actually find legislative judgments which are not reducible to interest-group activities. And when we have reduced the legislative process to the play of group interests, then log-rolling, or give and take, appears as the very nature of the process. It is compromise, not in the abstract moral form, which philosophers can sagely discuss, but in the practical form with which every legislator who gets results through government is acquainted. It is trading. It is the adjustment of interests.” (Arthur Bentley, The Process of Government [Bloomington: The Principia Press, 1935 (first published 1908)], pp. 370-71.)

[5. ]“Imperfection” is used here only in its purely economic sense. Nothing in the discussion should be taken to suggest that a “perfect” market in political votes would be, in any sense, “perfect” in respect to some set of ideals for the organization of a political system.

[6. ]We are ignoring the costs of decision-making in this example.

[7. ]A preliminary version of this chapter has been published. See Gordon Tullock, “Some Problems of Majority Voting,” Journal of Political Economy, LXVII (December 1959), 571-79. We are grateful to the editors of this journal for allowing us to reprint those parts of the earlier version that are relevant here.

[8. ]See Anthony Downs, An Economic Theory of Democracy (New York: Harper and Bros., 1957) and Duncan Black, The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1958).

[9. ]Owl: “We didn’t hear all yo’ speech—just heard yo’ say git on out and vote.”

Pogo: “That’s enough—as long as you do that, you cannot go wrong.”

(Walt Kelley, The Pogo Papers [New York: Simon and Schuster, 1952], p. 58.) For an excellent general comment, see Christopher Martin, “In Praise of Political Apathy,” The Listener (23 June 1960).

[10. ]An interesting example of this is presented in the comparison of voter support for education in local communities where educational expenditures are presented along with other issues for voter approval with those communities where the educational function is organized and financed through separate decision-making units. This comparison was discussed by Julius Margolis in a paper presented before the Conference on Public Finances, Universities—National Bureau of Economic Research Committee, held at Charlottesville, Virginia, on 10 and 11 April 1959. See National Bureau of Economic Research, Public Finances: Needs, Sources, and Utilization (Princeton: Princeton University Press, 1961).

[11. ]No solution which embodies general tax financing of public services valued differently by different individuals can be Pareto-optimal, unless, of course, fully offsetting compensations are allowed.

[12. ]Critics have objected to our usage of the word “Kantian” in this sense. We have no desire to raise complex philosophical issues here, and we point out only that the word is used solely for want of a more suitable single word describing the behavior that is adequately defined in the text.

[13. ]As suggested in footnote 5, the postulated institutions of the model will prevent the emergence of the fully “efficient” solution in any economic sense. The Kantian solution seems, therefore, to be the most nearly “correct” one that can be attained in the model as postulated.

Note that this Kantian solution is not equivalent to our “bench mark” employed in analyzing the individual constitutional calculus in Part II, which does represent a Pareto-efficient point. The Kantian solution of this model becomes equivalent to the bench-mark solution (that is to say, it eliminates all external costs) only if one of the two following conditions is satisfied.

(1) All voters have the same conception of the idealized standard of road repair: that is, all of the dots along the horizontal line in Figure 12 fall at the median point. In this case, no one is ever disappointed by a decision. “Consensus” is automatically achieved, and, given Kantian behavior for all individuals, the actual voting rule is unimportant.

(2) The distribution of the total costs of road repair among individuals is allowed to vary to correspond with differences in “tastes” concerning the idealized standard of repair. This second condition is, of course, prevented by the assumption that general taxation is employed as the revenue-producing device.

[14. ]In practice, the problem of securing the unanimous consent of the required 51 persons might be insoluble. However, since we are discussing a rather unique special model, we may ignore this possibility.

[15. ]In his paper, “The Theory of the Reluctant Duelist” (American Economic Review, XLVI [December 1956], 909-23), Daniel Ellsberg contends that accepted game-theory notions really apply only to “reluctant” players. Our case of voters is a particularly pure example. The voter must “play the game” by entering into bargains with 50 of his fellows, even though this leads to rather unsatisfactory results, simply because, given the rules, any other course of action would be worse.

This is not to suggest, however, that, given the fiscal institutions postulated in our model, simple majority rule is necessarily less desirable than some other decision-making rule. As the analysis of Part II demonstrates, this may or may not be the most “efficient” rule. What is clear from the analysis of our model is that the fiscal institutions postulated cannot produce “efficient” results under any collective decision-making rule short of unanimity.

[16. ]The fact that he is taxed for other roads not counted in his bargain reduces his real income and, hence, to some extent, reduces his desire for the consumption of road-repair services.

[17. ]See P. W. Bridgeman, The Way Things Are (Cambridge: Harvard University Press, 1959), pp. 268-69.

[18. ]Not necessarily for all. There might be one or more farmers whose personal preferences for road-repairing called for such a large investment as to make the “maximizing equilibrium” preferable to the “Kantian median.”

[19. ]The late C. O. Hardy once referred to this argument as the one which assumes the operation of “Dr. Nourse’s invisible left hand”: that is, men will further their own interest by acting in the public interest.

[20. ]As a practical example, assume that all Easterners should be intensely interested in general programs of water-resource development. Southerners are assumed to be wholly indifferent, and Westerners, by contrast, are, we assume, interested only in their own particular area projects. In this case Easterners should welcome the introduction of logrolling among the Western maximizers, since only in this way can over-all programs of water-resource development be approved.

[21. ]The treatment will be based directly on the constructions contained in the helpful survey of Luce and Raiffa. See R. Duncan Luce and Howard Raiffa, Games and Decisions (New York: John Wiley and Sons, 1957).For our particular purpose, we have not found the specific attempts to relate game theory and political theory to be useful, although these contributions may be helpful in a somewhat more general sense. See Karl Deutsch, “Game Theory and Politics: Some Problems of Application,” Canadian Journal of Economics and Political Science, XX (1954), 76-83; Martin Shubik, ed., Readings in Game Theory and Political Behavior (New York: Doubleday, 1954); and Richard C. Snyder, “Game Theory and the Analysis of Political Behavior,” contained in Research Frontiers in Politics and Government (Brookings Institution, 1955).

[22. ]As William Riker has pointed out in his comment on an earlier version of this book, all political situations that take on genuine “game” characteristics can, for some purposes, be analyzed under the zero-sum restriction. Through the interpretation of individual payoffs in a relative rather than an absolute sense, any positive-sum game can be converted into a zero-sum game. Since our purpose, however, is that of examining the economic meaning of the solutions to the various games analyzed, this conversion to a zero-sum model is not suitable.

[23. ]See J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behavior (3d ed.; Princeton: Princeton University Press, 1953), p. 264.

[24. ]Note that this does not contradict our argument of the last chapter in which it was suggested that individual farmers would not remain Kantians. The difference between the two cases is that there we were considering a whole series of separate but related actions, while here we are considering the possible shifting of coalitions prior to the taking of a single action.

[25. ]In this particular model, the “equitable” solution is equivalent to the “Kantian” solution discussed in the preceding chapter. We shall employ the different term here, however, because these two imputations will not be the same under different circumstances.

[26. ]T. C. Schelling, “For the Abandonment of Symmetry in Game Theory,” Review of Economics and Statistics, XLI (August 1959), 213-24. Reprinted as Appendix B in The Strategy of Conflict (Cambridge: Harvard University Press, 1960), pp. 267-91.

[27. ]This property attributed to simple majority rule has been called that of anonymity. May also calls it the equality condition. This terminology seems to be especially misleading since the psychological equality assumed is something quite different from the political equality insured by the fact that each individual has one vote. Cf. K. O. May, “A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decisions,” Econometrica, XX (October 1952), 680-84.

Note also that Dahl’s conception of political equality requires that each individual’s preference be given equal weight. See Robert A. Dahl, A Preface to Democratic Theory (Chicago: University of Chicago Press, 1956), p. 37.

[28. ]Assuming, of course, that the objective values imputed reflect accurate subjective estimates of the relative values of road repair.

[29. ]See the discussion in Chapter 12 for some questions about this particular “solution.”

[30. ]Luce and Raiffa, Games and Decisions, p. 193. Note that this is a much more limited usage of the term “individual rationality” than that which we have employed in Part I.

[31. ]In the terminology of some of the commonly used criteria for determining the allocation of public funds among separate projects, a minimum benefit-cost ratio of ½ would be required for a project to secure approval in a collective-decision process embodying simple majority rule.

[32. ]This adjustment was not necessary in the earlier models because we assumed, in each case, that the total initial assets were collected in general taxation: that is, we assumed that $1 was disposed of in each case.

[33. ]We should emphasize that the graphical construction of Figure 13 is wholly conceptual. A point inside, on, or outside the frontier has no descriptively physical meaning. The graphical apparatus is employed solely to assist the reader in making a conceptual separation among three sets of situations or points: (1) those that are nonoptimal by the Pareto criterion, (2) those that are optimal by the same criterion, and (3) those that are unattainable. The situations or points are classified only on an observed agreement or a failure to agree among the individuals in the group. It is essential that these qualifications be kept clearly in mind, especially by those economist readers who may have been accustomed to discussions of the Pareto criterion in units of measure embodying specifically observable physical dimensions (income, goods, etc.) independent of observed agreement.

[34. ]R. Duncan Luce and Howard Raiffa, Games and Decisions (New York: John Wiley and Sons, 1957), p. 193.

[35. ]Ibid., p. 195.

[36. ]The symmetry in benefit schedules referred to here is not equivalent to the equal or symmetrical intensities of preference referred to in Chapter 9. The discussion at that point was similar to this, but note that there we postulated that the preference, negative or positive, of each individual was valued equally.

[37. ]This does not imply that the same amount of productive collective activity will be undertaken under all rules if side payments are fully effective. The distribution of real income itself influences the final allocation between public and private goods that will satisfy the full Paretian conditions. On this point, see Paul A. Samuelson, “The Pure Theory of Public Expenditure,” Review of Economics and Statistics, XXXVI (1954), 387-89, and “Diagrammatic Exposition of a Theory of Public Expenditure,” Review of Economics and Statistics, XXXVII (1955), 350-56.

[38. ]R. A. Musgrave, The Theory of Public Finance (New York: McGraw-Hill, 1959).

[39. ]The direction of effect is, of course, just the opposite of that which results from the private organization of genuinely collective activities. It has been commonly recognized that, in such cases, the individual decision-maker will not be able to take into account the full benefits to the whole group when he makes his own private decisions. Therefore, the standard Pigovian analysis proceeds: there will tend to be relatively too little investment in such activities. The fact that collective decision-making, as it is organized by less-than-unanimity voting rules, has not been recognized to produce precisely the opposite results seems to be due to the implicit assumption that collective decisions are made, if not explicitly by some voting rule of unanimity, as if unanimity prevails.

[40. ]The difficulty in treating redistributive transfers (even at the purely conceptual level) within the framework of our model lies in the fact that the “income insurance” calculus, outlined in Chapter 13, involves essentially an “exchange through time” among individuals rather than any “exchange among individuals at a point in time,” which is central to the orthodox Paretian construction. The elimination of all external costs under the requirement of unanimity discussed in Chapter 7 follows directly from the consideration of collective allocational decisions only.

[41. ]Paul A. Samuelson, “The Pure Theory of Public Expenditure,” Review of Economics and Statistics, XXXVI (1954), 387-89, and “Diagrammatic Exposition of a Theory of Public Expenditure,” Review of Economics and Statistics, XXXVII (1955), 350-56; R. A. Musgrave, The Theory of Public Finance (New York: McGraw-Hill, 1959), chap. 4.

[42. ]For an example of a brief but interesting discussion of some of the activities (currently undertaken by the federal government) that might be reorganized through the introduction of user pricing, see “Picking Up the Check,” The New Republic (9 January 1961).

[43. ]This analytical conclusion is supported by historical experience. The comparison between the government of Robespierre and the one that followed in 9 Thermidor is instructive. Of the latter it has been noted: “These men had howled with the wolves while the Reign of Terror lasted, but since their sole aim was to acquire money and to keep their skins, they were the kind of men with whom respectable people can do business.” (J. Christopher Herold, Mistress to an Age: The Life of Madame de Staël [London: Hamish Hamilton, 1959], p. 151.)

[44. ]Note that this relationship between voting rules and the stability of solutions is not identical to the relationship between the size of the total group and the stability of solutions. As suggested earlier, game theorists argue that the stability properties of solutions to n-person games become less pronounced as the total group is enlarged in size. Within any group of given size, however, the change from less inclusive to more inclusive voting rules would not seem, a priori, to exert any clearly predictable effect on the stability of solutions obtained.

[45. ]The following reference to Washington’s position on this issue is revealing: “On the final day, after the constitution had been engrossed, and the printers had begun printing 500 copies, a motion was made to reduce the congressional constituencies from 40,000 to 30,000. ’When the President rose,’ as Madison’s notes record, ’for the purpose of putting the question,’ he said that ’although his situation’—as president—’had hitherto restrained him from offering his sentiments on questions pending in the House, and it might be thought, ought now to impose silence on him, yet he could not forbear expressing his wish that the alteration proposed might take place. It was much to be desired that the objections to the plan recommended might be as few as possible.—The smallness of the proportion of representatives had been considered by many members of the Convention, an insufficient security for the rights and interests of the people. He acknowledged that it had always appeared to himself among the exceptionable parts of the plan; and late as the present moment was for admitting amendments, he thought this of so much consequence that it would give him much satisfaction to see it adopted.’ “ (Carl Van Doren, The Great Rehearsal [New York: Viking Press, 1948], p. 170.)

[46. ]Actually, with such a small group, the costs of bargaining would be quite modest and direct democracy would be more efficient. Larger and more realistic models, however, are harder both on the draftsman and the reader.

[47. ]The voters do not, of course, necessarily form the square submatrix shown. Any combination of nine voters distributed three each in three constituencies is sufficient to constitute a dominant coalition.

[48. ]See Anthony Downs, An Economic Theory of Democracy (New York: Harper and Bros., 1957), pp. 142-64.

[49. ]The more inclusive rule for selecting representatives will guarantee to the individual that his own interest will be more likely to be “represented” in the assembly. This being true, his own interest stands a greater chance of being represented in any decisive coalition in the assembly. The danger of adverse collective actions is clearly reduced.

[50. ]It should be emphasized that the derivation of the sign matrix depends strictly upon the relationships among the variables that we have assumed to be present. These relationships are based on what seem to be reasonable assumptions about individual constitutional calculus. Essentially, we have assumed that the constitutional variables, as defined, are compensating rather than complementing. It seems rather difficult to imagine the complementary relationship as applying generally, although it would not, of course, be difficult to imagine a complementary relationship between two narrowly defined constitutional variables. However, it should be noted that there is no mathematical reason why the general relationship among the variables considered need be compensatory.

[51. ]Research now in progress suggests that the general analytical model is also useful in application to other problems of social organization. For example, the problems of the usage of common-property resources and of the degree of decentralization within a unified organizational structure can both be analyzed with essentially the same model as that introduced in this book.

[52. ]See Figure 19 and the discussion relevant to it.

[53. ]This is not to deny that such a system might have some advantages over a single house. In particular, it might provide for more careful consideration of issues.

[54. ]Note that essentially the same position has recently been taken by Morton A. Kaplan: “Thus social rules may often be considered as game payoffs, and we are willing to take lower contract payoffs at any particular time or to take risks of lower payoffs that we would be unwilling to accept if we had not internalized these social rules. And most men also have an interest in supporting the socialization process, for, although it constrains them, they are better off than if none was constrained.” (Morton A. Kaplan, Some Problems in the Strategic Analysis of International Politics, Research Monograph No. 2, Center of International Studies, Princeton [12 January 1959], p. 9.)

[55. ]This is not to suggest that the distinction has not been clearly understood by political theorists and that its recognition has not affected political institutions. The executive power to veto legislation adopted by a representative assembly finds its basic rationale in the recognition of this distinction. Cf. Benjamin Constant, Réflexions sur les Constitutions (Paris: Nicolle, 1814), pp. 50f.

[56. ]The tendency to slip inadvertently from one meaning into the other is well illustrated by a recent statement made by Anthony Downs: “... it is better for more voters to tell fewer what to do than vice versa. The only practical arrangement to accomplish this is simple majority rule. Any rule requiring more than a simple majority for passage of an act allows a minority to prevent action by the majority. ...” (Italics supplied.—Anthony Downs, “In Defense of Majority Voting,” unpublished mimeographed paper written as a general critique of Gordon Tullock’s paper, “Some Problems of Majority Voting,” which was an early version of Chapter 10. In Downs’ favor, however, it might be noted that he supports present procedures for the amending of the Constitution.)

On this general issue, see also Willmoore Kendall, John Locke and the Doctrine of Majority Rule (Urbana: University of Illinois Press, 1941), p. 116.

[57. ]This conclusion assumes that individuals of the blocking coalition are rationally motivated. If, instead, these individuals should all be irrational, they might refuse to enter into “bargains” advantageous to them, and, by this refusal, they might be said to impose “external costs” on others. For example, suppose that a two-thirds majority rule is in operation. Suppose now that 66 out of 100 voters propose a project that will be genuinely beneficial to all 100 voters. To prevent this project from being adopted all 34 other voters must be irrational. If only a few are irrational, the project will be carried. This example suggests that the rationality assumption is not important for the conclusions reached. Individuals will, by and large, tend to approve all proposals that provide them with expected net benefits. This relatively weak version of the rationality assumption seems to be all that is required.

[58. ]Note that we refer to the orthodox theorist, not orthodox institutions. In the real world the overwhelming majority of democratic constitutions cannot, in fact, be amended by simple majorities. Many theorists simply refuse to apply their theoretical structure to constitutions.