Bounds for Mixed Strategy Equilibria and the Spatial Model of Elections

We prove that the support of mixed strategy equilibria of two-player, symmetric, zero-sum games lies in the uncovered set, a concept originating in the theory of tournaments, and the spatial theory of politics. We allow for uncountably infinite strategy spaces, and as a special case, we obtain a long-standing claim to the same effect, due to R. McKelvey (Amer. J. Polit. Sci.30 (1986), 283–314), in the political science literature. Further, we prove the nonemptiness of the uncovered set under quite general assumptions, and we establish, under various assumptions, the coanalyticity and measurability of this set. In the concluding section, we indicate how the inclusion result may be extended to multiplayer, non-zero-sum games.