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Simple Majority Voting, Game Theory, and Resource Use

Published online by Cambridge University Press:  07 November 2014

James M. Buchanan
James M. Buchanan*
Affiliation:
University of Virginia
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I propose in this paper to examine majority voting with the tools of elementary game theory. This, in itself, represents nothing new. The “majority game” has been analysed by numerous game theorists. I propose, however, to derive certain implications from the analysis for the utilization of resources in the public or collective sector of the economy.

The models of collective decision-making that must be employed are necessarily abstract; so much so that the analyses may seem to appear to some observers as caricatures of actual political processes. Analysis must start somewhere, however, and even the most abstract of models, by isolating specific features of real-world institutions, may prove helpful to our over-all understanding. I shall introduce a model of pure democracy. That is to say, I shall assume that all collective decisions are to be made by a simple majority voting process in which all citizens participate. All problems of representation, leadership, parties, and coalitions shall be left out of account. Furthermore, I shall assume that the individual citizens in the model are motivated by utility-maximizing considerations; that is, each individual is assumed to vote in such a manner as to maximize his own utility. This assumption allows numerical values to be placed in the expected pay-offs that individuals receive through political processes. If the expected pay-offs can be numerically correlated with observable economic quantities or magnitudes, some provisional and tentative implications may be drawn concerning the tendency of majority voting rules to “over-extend” or to “starve” the public sector of the economy relative to the private sector.

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Articles
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Canadian Journal of Economics and Political Science/Revue canadienne de economiques et science politique , Volume 27 , Issue 3 , August 1961 , pp. 337 - 348
Copyright
Copyright © Canadian Political Science Association 1961

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References

1 As will become evident to readers who are even moderately sophisticated in the field, my constructions will be quite elementary. My purpose is, however, not that of making any contribution to game theory itself, but rather that of applying the relevant portions of developed theory to a particular set of problems. My treatment will be based directly on the constructions contained in Luce, R. Duncan and Raiffa, Howard, Games and Decisions (New York; 1957).Google Scholar

2 Tullock, Gordon, “Problems of Majority Voting,” Journal of Political Economy, LXVII, no. 6, 12, 1959, 571–9.CrossRefGoogle Scholar

3 See Neumann, J. Von and Morgenstern, O., Theory of Games and Economic Behavior (3rd. ed., Princeton, 1953), 264.Google Scholar

4 It is perhaps useful to note that the argument for symmetry in the sharing of gains among members of the dominant coalition rests on slightly different grounds in the n-person, majority-rule game than it does in the two-person, co-operative game or in n-person games which require that all participants must agree on the sharing arrangement. Schelling, in his recent argument for the abandonment of symmetry, confined his discussion largely to these latter games. If, as in the simple majority-rule game considered here, the rules dictate that only a certain portion of the group need agree, the case for effective-coalition symmetry is stronger than in the fully co-operative game. The individual in the winning coalition will tend to be satisfied with a symmetrical share of the total gain, not because he expects no member of the coalition to concede more to him out of a general attitude of “fairness,” but rather because he knows that, if he does demand more, alternative persons in the dominated minority stand ready and willing to join new coalitions which can remove the grains entirely. (Cf. Schelling, T. C., “For the Abandonment of Symmetry in Game Theory,” Review of Economics and Statistics, XLI, no. 3, 08, 1959, 213–24.CrossRefGoogle Scholar)

5 This property attributed to simple majority voting has been called that of “anonymity.” May also calls it the “equality condition.” This latter terminology seems to be especially misleading since the psychological equality assumed is something that is quite different from the “political equality” that is ensured so long as each individual has one vote. See May, K. O., “A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decisions,” Econometrica, XX, no. 4, 10, 1952, 680–4.CrossRefGoogle Scholar

Note also that Dahl's conception of political equality requires that each individual's preferences be given equal weight. See Dahl, Robert A., A Preface to Democratic Theory (Chicago, 1956), 37.Google Scholar

6 Luce, and Raiffa, , Games and Decisions, 193.Google Scholar

7 In the discussion here, I shall assume that majority-rule games are “fair” in the sense that each member of the group has, at the start of play, an equal expectation of being in the dominant coalition. This assumption is necessary, since it is the expected value of the pay-off that is relevant to the satisfaction of the individual rationality condition. I shall also assume that the utility value from participation in the game itself is not significant.

8 In the terminology of the commonly used criterion for determining the allocation of public funds among separate projects, a minimum benefit: cost ratio of one-half would be required for a project to secure approval in a collective decision process embodying majority rule.

9 This qualification is not important because separate projects can always be grouped together into a sufficiently large “package” to ensure that aggregate net benefits extend at least over a majority of the members of the group.

10 Musgrave, R. A., The Theory of Public Finance (New York, 1959).Google Scholar

11 The individual calculus that is involved for this sort of comparison is the central problem that is discussed in the larger work by Gordon Tullock and this writer now in press. See The Calculus of Consent: Logical Foundations of Constitutional Democracy (Ann Arbor, forthcoming).Google Scholar