Equilibrium in the Spatial ‘Valence’ Model of Politics

It has been a standard result of the stochastic, or probabilistic, spatial model of voting that vote maximizing candidates, or parties, will converge to the electoral mean (the origin). This conclusion has appeared to be contradicted by empirical studies. Here, a more general stochastic model, incorporating ‘exogeneous’ valence, is constructed. Contrary to the standard result, it is shown in Theorem 1 of this paper that a potentially severe domain constraint (determined by the electoral and stochastic variance, valence as well as the dimension of the space) is necessary for the existence of equilibrium at the electoral mean. A more stringent condition, independent of the dimension of the space, is shown to be sufficient. An empirical study of Israel for 1992 shows that the necessary condition failed. This suggests that, in proportional electoral systems, a pure strategy equilibrium will almost always fail to exist at the electoral mean. Instead, in both the formal and empirical models, each party positions itself along a major electoral axis in a way which is determined by the valence terms. A second empirical analysis for Britain for the elections of 1992 and 1997 shows that, in fact, the necessary and sufficient condition for the validity of the ‘mean voter theorem’ was satisfied, under the assumption of unidimensionality of the policy space. Indeed the low valence party, the Liberal Democrat Party, did appear to locate at the electoral center. However, the high valence parties, Labour and the Conservatives, did not. This suggests that, in polities based on plurality rule, valence is a function of activist support rather than a purely exogenous factor. Theorem 2 shows, as in Britain, that exogeneous and activist valence produce opposite effects.