A generalized Hotelling–Downs model with asymmetric candidates

We investigate an extension of the Hotelling–Downs model to the case where the preferences of the voters do not have to be single peaked. In the case where candidates only care about winning or losing, assuming that a voter elects each candidate symmetrically with equal probability when indifferent, previous works by Fisher and Ryan and by Laffond, Laslier, and Le Breton have shown that the equilibrium outcome of the model is unique. Uniqueness also holds in the case where candidates care about the strength of their majority. Our paper shows that furthermore the equilibrium is unique for small asymmetries in voter's behavior but not necessarily for large ones. In particular, if voters always vote for the incumbent rather than a newcomer when indifferent, the equilibrium policy outcome may fail to be unique. Moreover, we provide a further sufficient condition for uniqueness. Namely, suppose all voters vote for a given candidate with the same probability when indifferent. Then, if there is a Nash equilibrium in completely mixed strategies between the candidates, it is the only Nash equilibrium.