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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 s