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" who se true policy preferences on Z are unknown to other principals. In the fir st version of the model the party declarations determine the lottery outcom e of coalition negotiation, The coalition risk functions are common knowled ge to the parties. We assume these coalition probabilities are inversely pr oportional to the variance of the declarations of the parties in each coali tion. It is shown that with this outcome function and with three parties th ere 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 pl atform to declare to the electorate. The equilibrium can be explicitly calc ulated in terms of the preferences of the parties and the scheme of private benefits from coalition membership. In particular, convergence in equilibr ium party positions is shown to occur if the principals' preferred policy p oints are close to colinear. Conversely, divergence in equilibrium party po sitions occurs if the bliss points are close to symmetric. If private benef its (the non-policy perquisites from coalition membership) are sufficiently large (that is, of the order of policy benefits), then the variance in equ ilibrium party positions is less than the variance in ideal points. The gen eral model attempts to incorporate party beliefs concerning electoral respo nses to party declarations. Because of the continuity properties imposed on both the coalition and electoral risk functions, there will exist mixed st rategy Nash equilibria. We suggest that the existence of stable. pure strat egy Nash equilibria in general political games of this type accounts for th e non-convergence of party platforms in multiparty electoral systems based on proportional representation. (C) 2000 Elsevier Science B.V. All rights r eserved.