Pure-strategy Nash equilibrium in the spatial model with valence: existence and characterization

Pure-strategy Nash equilibria almost never exist in spatial majority voting games when the number of positional dimensions is at least two, as the majority core is typically empty when more than one positional dimension is modeled. In the general setting of proper spatial voting games, we study the existence of equilibrium when one candidate has a valence advantage over the other. When we consider such games, a valence equilibrium can exist that is not in the core, and a point in the core need not be a valence equilibrium. In this paper, we characterize the entire set of valence equilibria for any proper spatial voting game. We complement the analysis by deriving a simple inequality that is both necessary and sufficient for the existence of a valence equilibrium. The inequality refers to the radius of a new concept, the b-yolk, and we prove that a b-yolk always exists in a spatial voting game.