Downsian competition with four parties

Our purpose in this article is to study a unidimensional model of spatial electoral competition with four political parties. We assume that the voters are distributed along [0,1] in such a way that the density δ of this distribution is continuous on [0,1] and strictly positive on (0,1). The parties engage in a Downsian competition which is modeled as a non-cooperative four-person game G(δ) with [0,1] as the common strategy set. If ξi stands for the ith quartile of the above-mentioned distribution, then we prove that G(δ) has a pure Nash equilibrium, if and only if ∫ξ1+t2t+ξ32δ(x)dx≤14 for every t ∈ (ξ1,ξ3). Moreover, if this condition is satisfied, then G(δ) has exactly six pure Nash equilibria, which are characterized by the fact that two of the parties put forward the policy that corresponds to ξ1 and the other two of them put forward the policy that corresponds to ξ3.