Electoral competition in 2-dimensional ideology space with unidimensional commitment

We study a model of electoral 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: one where both candidates are committed to the same dimension, and the other where each candidate is committed to a different dimension. We focus on characterisation and existence of pure strategy Nash equilibria when the core is empty. We show that if the distribution of voters' ideal policies is continuously differentiable and has a bounded support, then an equilibrium exists if the candidates are differentiated enough. Our results for the case where the candidates have a common committed issue have implications for the literature on valence.