Unanimity and majority rule: the calculus of consent reconsidered*

Abstract

Economists and political scientists have long debated the socially optimal proportion of a collective decision-making body whose consent should be required in order to reach a decision. In their classic contribution to this debate, Buchanan and Tullock [Buchanan, J.M., Tullock, G., 1962. The Calculus of Consent. Univ. of Michigan Press, Ann Arbor] argued that, in the absence of what they called `decision-making costs', the unanimity rule is socially optimal. The present paper shows that their approach actually leads to the conclusion that the unanimity rule is (almost always) suboptimal. In contrast, simple majority rule—the rule most commonly observed in collective decisions—is found to be socially optimal under plausible conditions.

Introduction

A long-debated issue in the literature of economics and political science is the socially optimal proportion of a collective decision-making body whose consent should be required in order to reach a decision. Buchanan and Tullock (1962)argued in a classic contribution that, in the absence of what they called `decision-making costs', the unanimity rule is socially optimal. Yet casual observation suggests that the simple majority rule is the rule adopted in most committees, legislatures, and popular elections.¹ The present paper provides a formal analysis of the optimal collective decision rule, based on the approach of Buchanan and Tullock. It is shown that their approach actually leads to the conclusion that the unanimity rule is (almost always) suboptimal. In contrast, simple majority rule is found to be socially optimal under plausible conditions.

It should be noted at the outset that two different criteria have been used (by two distinct literatures) in the study of collective decision rules. The first criterion is efficiency—this is the criterion used by Buchanan and Tullock (1962). An efficient decision rule maximizes the likelihood that all Kaldor–Hicks efficient proposals are accepted, while all inefficient proposals are rejected. The second criterion may be called stability or `decisiveness': the decision rule should not permit Arrow-type cycles. In binary choices (in which Arrow-type cycles are impossible), the rule should produce a well-defined preference ordering over the two alternatives open to the decision-making body. The criterion of stability is the criterion studied by the voluminous social choice literature.

The present paper combines these two criteria. It is postulated that the socially optimal rule must be efficient, subject to the constraint that it be stable. As Arrow (1951)proved, it is impossible, in general, for a democratic decision rule to satisfy the stability criterion. However, if the number of alternatives is restricted to two, then the stability problem reduces to the problem of pairwise decisiveness (May, 1952). Requiring at least a simple majority suffices to guarantee that a decision rule is pairwise decisive. In order to simplify the analysis and still insure that the stability criterion can be met, we will restrict attention to binary (two-alternative) choices.

The analysis can be extended to decision problems having more than two alternatives, by increasing the lower bound on the required majority rule—provided that voters vote for the proposal closest to their ideal point in Euclidean issue space, and that the density of voters' ideal points is concave. Caplin and Nalebuff (1988)proved that, under these two conditions, there is a lower bound on the required majority rule which guarantees stability; this lower bound rises to a limit of just under 64% as the number of dimensions in the issue space rises to infinity. Note, however, that the Caplin–Nalebuff result is concerned only with the stability criterion.² In contrast, the present analysis focuses [as Buchanan and Tullock (1962)did] on the relative efficiency of alternative decision rules; my departure from the approach of Buchanan and Tullock is only in requiring that the recommended rule be stable as well as efficient.

Section 2lays the groundwork of the analysis, and presents an intuitive, diagrammatic argument for the inefficiency of the unanimity rule. Section 3formally develops a sufficient condition for the optimality of simple majority rule in binary collective choices. Section 4shows that this sufficient condition is plausible in a simplified model of legislative allocation of public funds for the provision of collective goods demanded by interest groups. Section 5shows that the fact that Buchanan and Tullock (1962)did not arrive at the same result is not due to any difference in assumptions, but rather stems from a subtle inconsistency in their analysis. Once this inconsistency is removed, their model would produce the same result. Section 6concludes the paper.