Equilibrium existence for zero-sum games and spatial models of elections

Abstract

A theorem on existence of mixed strategy equilibria in discontinuous zero-sum games is proved and applied to three models of elections. First, the existence theorem yields a mixed strategy equilibrium in the multidimensional spatial model of elections with three voters. A nine-voter example shows that a key condition of the existence theorem is violated for general finite numbers of voters and illustrates an obstacle to a general result. Second, the theorem provides a simple and self-contained proof of Kramer's existence result for the multidimensional model with a continuum of voters [Kramer, G., 1978. Existence of electoral equilibrium. In: Ordeshook, P. (Ed.), Game Theory and Political Science. NYU Press, New York]. Third, existence follows for a class of multidimensional probabilistic voting models with discontinuous probability-of-winning functions.