Recent works on political competition incorporate a valence dimension into the standard spatial model. The analysis of the game between candidates in these models is typically based on two assumptions about voters' preferences. One is that valence scores enter the utility function of a voter in an `additively separable' way, so that the total utility can be decomposed into the `ideological utility' from the implemented policy (based on the Euclidean distance) plus the valence of the winner. The second is that all the voters identically perceive the platforms of the candidates and agree about their valence score.

The goal of this paper is to axiomatize collections of preferences that satisfy these assumptions. Specifically, we consider the case where only the ideal point in the policy space and the ranking over candidates are known for each voter. We characterize the case where there are policies

x1,...,xm for the m candidates and numbers v1,...,vm representing valence scores, such that a voter with an ideal policy y ranks the candidates according to vi - ||xi - y||^2.