Electoral competition with policy-motivated candidates

In the multi-dimensional spatial model of elections with two policy-motivated candidates, we prove that the candidates must adopt the same policy platform in equilibrium. Moreover, when the number of voters is odd, if the gradients of the candidates' utility functions point in different directions, then they must locate at some voter's ideal point and a strong symmetry condition must be satisfied: in particular, it must be possible to pair some voters so that their gradients point in exactly opposite directions. If the number of dimensions is more than two, then our condition is knife-edge. When the number of voters is even, the situation is worse: such equilibria never exist, regardless of the dimensionality of the policy space.