EXISTENCE OF NASH EQUILIBRIUM IN A SPATIAL MODEL OF n-PARTY COMPETITION

In the model presented here, $n$ parties choose policy positions in a space $Z$ of dimension at least two. Each party has true preferences on $Z$ that are unknown to other agents. In the first version of the model considered the party declarations determine the lottery outcome of coalition negotiation. The lottery outcome function is common knowledge to the parties and is determined by probabilities of coalition formation inversely proportional to the variance of the declarations of coalition members. It is shown that with this outcome function and with three parties there exists a stable, pure strategy Nash equilibrium in the game of party choices of declarations. The Nash equilibrium can be explicitly calculated in terms of the preferences of the parties and the scheme of private benefits from coalition membership. In particular, convergence in equilibrium party positions is shown to occur if the party bliss points are close to colinear. Conversely, divergence in equilibrium party positions occurs if the bliss points are close to symmetric. If private benefits are sufficiently large (that is, of the order of policy benefits), then the variance in equilibrium party positions is less than the variance in bliss points. The general model attempts to incorporate party beliefs concerning electoral responses to party declarations. A mixed strategy Nash equilibrium is shown to exist. It is conjectured that generically there exists a unique stable, pure strategy Nash equilibrium.