Plurality distortion and majority rule

Abstract

Discussions of the implications of Arrow's theorem for majority rule have typically focused on calculating the probability of the paradox of voting. In doing so, these analyses have ignored another significant problem of preference aggregation in n-way (n > 2) elections, the existence of non-Condorcet winners, when a Condorcet alternative is available, under a simple plurality criterion. At the level of the democratic political system, e.g., a proposed national presidential primary or any electoral subsystem, e.g., state presidential primaries, gubernatorial, senatorial elections, the simple plurality criterion may distort electoral preferences by failing to select an available Condorcet candidate. An algorithm for calculating the probability of such plurality distortion is offered, and probability calculations for three-way elections for small electorates are presented. Most notable is the discovery that single-peaked preferences, while reducing or eliminating the probability of the paradox, lead to a concomitant increase in the probability of plurality distortion. In instances in which no candidate receives half the vote and preferences are single-peaked, the probability of plurality distortion is actually greater than the probability of plurality-Condorcet coincidence. These results cast some doubt on the validity of the plurality criterion as a voting procedure as well as indicating some of the difficulties inherent in the Condorcet criterion. Alternatives to the plurality criterion are discussed, and some preliminary tests of these alternatives are made.