Symmetric Spatial Games Without Majority Rule Equilibria

The assumptions imposed in spatial models of election competition generally are restrictive in that they require either unidimensional issue spaces or symmetrically distributed electorate preferences. We attribute such assumptions to the reliance of these models on a single concept of a solution to the election game—pure strategy equilibria—and to the fact that such equilibria do not exist in general under less severe restrictions. This essay considers, then, the possibility that candidates adopt mixed minimax strategies. We show, for a general class of symmetric zero-sum two-person games, that the domain of these minimax strategies is restricted to a subset of the strategy space and that for spatial games this set not only exists, but if preferences are characterized by continuous densities, it is typically small. Thus, the hypothesis that candidates abide by mixed minimax strategies can limit considerably our expectation as to the policies candidates eventually advocate. Additionally, we examine the frequently blurred distinction between spatial conceptualizations of two-candidate elections and of committees, and we conclude that if pure strategy equilibria do not exist, this distinction is especially important since committees and elections can produce entirely different outcomes.