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

Although there exist extensive results concerning equilibriain spatial models of two-party elections with probabilisticvoting, we know far less about equilibria in multiparty elections—i.e.,under what conditions will equilibria exist, and what are thecharacteristics of equilibrium configurations? We derive conditionsthat guarantee the existence of a unique Nash equilibrium anddevelop an algorithm to compute that equilibrium inmultipartyelections with probabilistic voting, in which voters chooseaccording to the behaviorists' fully specified multivariatevote model. Previously, such computations could only be approximatedby laborious search methods. The algorithm, which assumes aconditional logit choice function, can be applied to spatialcompetition for a variety of party objectives including vote-maximizationand margin-maximization, and can also encompass alternativevoter policy metrics such as quadratic and linear loss functions.We show that our conditions for an equilibrium are plausiblegiven the empirically-estimated parameters that behavioristsreport for voting behavior in historical elections. We alsoshow that parties' equilibrium positions depend not only onthe distribution of voters' policy preferences but also on theirnonpolicy-related attributes such as partisanship and sociodemographicvariables. Empirical applications to data from a recent Frenchelection illustrate the use of the algorithm and suggest thata unique Nash equilibrium existed in that election.