We study stochastic voting models where the candidates are allowed to have any smooth, strictly increasing utility functions that translate vote shares into payoffs. We find that if a strict Nash equilibrium exists in a model with an infinite number of voters, then nearby equilibria should exist for similar large, but finite, electorates. If the votes are independent random events, then equilibria will not depend on the utility functions of the candidates. Our results have implications for existing models of redistributive politics and spatial competition, as the properties of pure-strategy equilibria in such games carry over to equilibria in games with arbitrary candidate preferences. On the other hand, candidate utility functions will matter if the individual voting decisions are correlated. In the presence of aggregate uncertainty, such as changing economic conditions or political scandals, the preferences of parties and candidates with respect to shares of votes will have an effect on political competition.