Sufficient conditions for weak reciprocal upper semi-continuity in mixed extensions of games

We provide sufficient conditions for a game with discontinuous payoffs to be weakly reciprocally upper semi-continuous in mixed strategies. These conditions are imposed on the individual payoffs and not on their sum, and they can be readily verified in a large class of games even when the sum of payoffs in such games is not upper semi-continuous. We apply our result to establish the existence of mixed strategy equilibria in probabilistic voting competitions where candidates have very general utility functions as well as heterogeneous beliefs about the distribution of the voters.