Comparison of Scoring Rules in Poisson Voting Games

Scoring rules are compared by the equilibria that they generate for simple elections with three candidates and voters drawn from large Poisson distributions. A calculus for comparing pivot probabilities in Poisson voting games is applied. For a symmetric Condorcet cycle, nonsymmetric discriminatory equilibria exist under best-rewarding scoring rules like plurality voting. A candidate who is universally disliked may still not be out of contention under worst-punishing scoring rules like negative-plurality voting. In elections where two of three candidates have the same position, symmetric equilibria coincide with majority rule only for scoring rules that are balanced between best-rewarding and worst-punishing. When voters also care about continuous functions of vote shares, equilibria may still depend on pivot probabilities. Department of Managerial Economics and Decision Sciences, J. L. Kellogg Graduate School of * Management, Northwestern University, Evanston, IL 60208-2009. E-mail: [email protected] 1 COMPARISON OF SCORING RULES IN POISSON VOTING GAMES by Roger B. Myerson 1. From social choice theory to analysis of voting games The great impossibility theorems of social choice theory teach us that no democratic voting rule can guarantee the existence of a unique pure-strategy equilibrium in all social choice situations (Muller and Satterthwaite, 1977). Thus, any democratic voting rule must sometimes generate randomized equilibria or multiple equilibria in which the chosen alternative will not depend uniquely on the voters' preferences. But such impossibility results do not imply that all voting rules generate the same sets of equilibria for a given social choice situation. In fact, the set of equilibrium outcomes in a social choice situation (where voters' preferences over the alternatives are held fixed) may depend substantially on the voting rule. To develop a practical theory of social choice and constitution design, we need to understand this dependence. That is, we need to move from the negative impossibility theorems to a positive research agenda of characterizing and comparing the equilibria that are generated by different voting rules. In this paper, we consider some simple social choice situations involving just three alternatives, and with these examples we probe the ways that sets of equilibria may change when the voting rule is changed. The example that is considered first and longest in this paper is a version of the Condorcet cycle. Beginning with this example can help us to see more clearly the relationship between the impossibility theorems of social choice theory and the analysis of voting games, because the simple Condorcet cycle offers the easiest way to prove a social-choice impossibility theorem for