Collective choice : a probabilistic analysis

Abstract

A group of N members wishing to select one of m alternative courses of action, A1, A2, . . . . , Am, may reach a decision either with the aid of a collective choice rule or through the informal operation of an implicit social decision scheme. Under investigation are three collective choice rules, viz. the plurality, Condorcet, and Borda procedures, and seven social decision scheme models, viz. majority, proportionality, equiprobability, highest expected value, majority if Ag but proportionality otherwise, majority with proportionality, and majority with equiprobability. A probabilistic choice model is adopted which provides the likelihood of a given preference orderings of the alternatives. By means of the model, expressions are derived, in each of the ten collective choice procedures, for the probability that Ai, I = 1, 2, . . . . , m, is selected by the group. Additionally, in the case of the Condorcet procedure, a recursion relation is developed which expressed the probability the Ai is the winning alternative when the group consists of N members in terms of the probability of the same event when the group contains N – 1 members. These results form the basis of a study encompassing both normative and descriptive aspects of social choice. The examination of collective choice rules, which is primarily normative in character, concentrates on two central issues. Firstly, it is proposed that decisiveness, i.e. the tendency to yield unambiguous, clear-cut outcomes, is a desirable property of a collective choice rule, and hence may be adopted as one of the criteria in terms of which rival social choice functions may be evaluated. To this end, expressions are developed for the likelihood of plurality, Condorcet, and Borda indecision, and a comparative analysis of these likelihoods is undertaken. Secondly, collective choice rules may produce inconsistent or anomalous outcomes. The plurality and Borda procedures may select an alternative other than the one preferred by a majority of the group members; the Condorcet procedure may give rise to the paradox of voting; and the Borda procedure may generate the reversed-order paradox. In order to gauge the seriousness of the problem facing a given collective choice rule, solutions are obtained for the likelihood of each type of anomalous result. In the analysis of social decision scheme models descriptive xondiderations predominate. The implications of each model are studied, and an experimental approach is suggested which provides effective discrimination between competing models.