Equilibrium Existence in Discontinuous Zero-Sum Games with Applications to Spatial Models of Elections

A theorem on existence of mixed strategy equilibria in discontinuous zero-sum games is proved and is 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 (1978) existence result for the multidimensional model with a continuum of voters. Third, existence follows for a class of multi-dimensional probabilistic voting models with discontinuous probability-of-winning functions.