Computing Nash Equilibria in Probabilistic, Multiparty Spatial Models with Nonpolicy Components

Although there exist extensive results concerning equilibria in spatial models of two-party elections with probabilistic voting, we know far less about equilibria in multiparty elections—i.e., under what conditions will equilibria exist, and what are the characteristics of equilibrium configurations? We derive conditions that guarantee the existence of a unique Nash equilibrium and develop an algorithm to compute that equilibrium inmultiparty elections with probabilistic voting, in which voters choose according to the behaviorists' fully specified multivariate vote model. Previously, such computations could only be approximated by laborious search methods. The algorithm, which assumes a conditional logit choice function, can be applied to spatial competition for a variety of party objectives including vote-maximization and margin-maximization, and can also encompass alternative voter policy metrics such as quadratic and linear loss functions. We show that our conditions for an equilibrium are plausible given the empirically-estimated parameters that behaviorists report for voting behavior in historical elections. We also show that parties' equilibrium positions depend not only on the distribution of voters' policy preferences but also on their nonpolicy-related attributes such as partisanship and sociodemographic variables. Empirical applications to data from a recent French election illustrate the use of the algorithm and suggest that a unique Nash equilibrium existed in that election.