Two Candidate Competition on Differentiated Policy Sets

Abstract

In the classic spatial model of office-motivated candidate competition, equilibrium exists only if the distribution of voter ideal points has a total median, a condition which is virtually never met. However, in the classic model, both candidates can propose policies anywhere in policy space, and this is a crucial element in proving the necessity of that condition. If each candidate may only propose policies from a subset of policy space, does an equilibrium exist more generally? I consider a model in which each of two office-motivated candidates proposes a policy from a distinct policy set. When the policy sets are convex and one does not contain the proposed equilibrium policy, it need not be the case that the proposed policy is a total median, because any competing proposal must be at least some positive distance away, and may only lie in certain directions; hence, the conditions of Plott do not apply. Instead, it is only necessary for each median hyperplane to lie in a given halfspace formed by the hyperplane separating the proposed equilibrium from the closest policy the opponent can propose. The intersection of all of the halfspaces is the set of guaranteed supporters of the equilibrium policy, that is, the set of voters for whom the equilibrium policy is closer than any alternative the opposing candidate can propose. Hence, the requirement on the median hyperplanes for the proposed policy to be an equilibrium is that one can choose a median voter on each median hyperplane to be a guaranteed supporter of the proposed policy.