Symmetric Spatial Games Without Majority Rule Equilibria

Spatial models of election competition now constitute a major part of the application to political science of the rational choice paradigm. Critics, of course, may disagree over the empirical adequacy of a particular model's assumptions. Nevertheless, the path of spatial theory's development-commencing with Downs's seminal analysisl-exhibits a concerted effort to accommodate numerous theoretical and empirical considerations (with varying degrees of success), including multidimensional issue spaces, variations in the functional form of citizens' utility functions, alternative hypotheses about abstention and candidate objectives, risky strategies, sequential elections, the electoral college, sophisticated voting, and the proper conceptualization of spatial analysis as a substantively interpreted application of game theory and social choice theory. Undoubtedly, we should anticipate further developments, including models that address elections other than two party majority rule contests. Despite the achievements of spatial theory, however, all models that seek an explicit prediction about candidate strategies focus on a single solution concept-pure strategy equilibria-and thereby suffer from a common and seemingly devastating shortcoming: they assume either that the election concerns a single issue or that citizens' preferences are distributed with perfect mathematical symmetry.2 Even if every other restrictive assumption heretofore considered is satisfied, abandoning these two restrictions on preference renders spatial theory incapable of revealing propositions about the strategies candidates should adopt. We attribute this state of affairs to the fact that unless the assumption of unidimensional or symmetrically distributed preference is satisfied, the solution to the election game posited by spatial theory does not generally exist. In this essay we consider the game-theoretic extension of pure strategy equilibria-mixed minimax solu-