Electoral Competition in 2-Dimensional Ideology Space with Unidimensional Commitment

Abstract

We study a model of political competition between two candidates with two orthogonal issues, where candidates are office motivated and committed to a particular position in one of the dimensions, while having the freedom to select (credibly) any position on the other dimension. We analyse two settings: a homogeneous one, where both candidates are committed to the same dimension and a heterogeneous one, where each candidate is committed to a different dimension. We characterise and give necessary and sufficient conditions for existence of convergent and divergent Nash equilibria for distributions with a non-empty and an empty core. We identify a special point in the ideology space which we call a strict median, existence of which is strictly related to existence of divergent Nash equilibria. We show that if the distribution of voters' ideal points is smooth enough, then this point always exists and is the centre of the mass of the median line bisecting the policy spaces of the candidates. A central conclusion of our analysis is that for divergent equilibria, strong extremism (or differentiation) seems to be an important equilibrium feature.