Downsian competition with four parties

• Published in: Mathematical social sciences 2005-11, Vol.50 (3), p.331-335
• Main Author: Sofronidis, Nikolaos Efstathiou
• Format: Article
• Language: English
• Subjects: Economic models of elections; Economic research; Elections; Four political parties; Mathematical methods; Modelling; Political economy; Political parties
• ISSN: 0165-4896; 1879-3118
• DOI: 10.1016/j.mathsocsci.2005.05.003

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 ξ i 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 ∫ ξ 1 + t 2 t + ξ 3 2 δ ( x ) d x ≤ 1 4 for every t ∈ ( ξ 1, ξ 3). 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 ξ 1 and the other two of them put forward the policy that corresponds to ξ 3.