Discontinuity and non-existence of equilibrium in the probabilistic spatial voting model

This paper shows that in the simplest one-dimensional, two-candidate probabilistic spatial voting model (PSVM), a pure strategy Nash equilibrium may fail to exist. The existence problem studied here is the result of a discontinuity in the function mapping the candidates' platforms into their probabilities of winning. Proposition 1 of the paper shows that, whenever this probability of winning function satisfies a certain monotonicity property, it must be discontinuous on the diagonal. As an immediate consequence of the discontinuity in the probability of winning function, the candidates' objective functions are discontinuous as well. It is therefore impossible to invoke standard theorems guaranteeing the existence of a pure strategy equilibrium, and an example is developed in which in fact there is no pure strategy equilibrium. Finally, however, it is demonstrated that, for a large class of probability of winning functions, the PSVM satisfies all the conditions of a theorem of Dasgupta and Maskin (1986a) which guarantees that it will always have an equilibrium in mixed strategies.