The Calculus of Consent: Logical Foundations of Constitutional Democracy

Simple Majority Voting*7

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.

Figure 12

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.

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 farmers*18 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.