Nash equilibrium in a spatial model of coalition bargaining

In the model presented here, it parties choose policy positions in a space Z of dimension at least two. Each party is represented by a "principal" whose true policy preferences on Z are unknown to other principals. In the first version of the model the party declarations determine the lottery outcome of coalition negotiation, The coalition risk functions are common knowledge to the parties. We assume these coalition probabilities are inversely proportional to the variance of the declarations of the parties in each coalition. It is shown that with this outcome function and with three parties there exists a stable, pure strategy Nash equilibrium, z* for certain classes of policy preferences. This Nash equilibrium represents the choice by each party principal of the position of the party leader and thus the policy platform to declare to the electorate. The 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 principals' preferred policy points are close to colinear. Conversely, divergence in equilibrium party positions occurs if the bliss points are close to symmetric. If private benefits (the non-policy perquisites from coalition membership) are sufficiently large (that is, of the order of policy benefits), then the variance in equilibrium party positions is less than the variance in ideal points. The general model attempts to incorporate party beliefs concerning electoral responses to party declarations. Because of the continuity properties imposed on both the coalition and electoral risk functions, there will exist mixed strategy Nash equilibria. We suggest that the existence of stable. pure strategy Nash equilibria in general political games of this type accounts for the non-convergence of party platforms in multiparty electoral systems based on proportional representation. (C) 2000 Elsevier Science B.V. All rights reserved.