On the smooth unfolding of bifurcations in quantal-response equilibria

Abstract

We report on the topological structure of logit quantal response equilibria for asymmetric, two-player, two-choice games in normal form, via catastrophe theory. We present three main outcomes: first, a novel, smooth potential function for the underlying dynamics, relative to which logit equilibria arise as stationary points comprising a parameterised hyper-surface; secondly, a proof that all catastrophes within this manifold are of order at most three, i.e. “folds and cusps”; thirdly, discovery of a new topological phenomenon, the “pleated loop”, corresponding to a pair of opposed pitchfork bifurcations. This we exhibit for games, such as Prisoner's Dilemma, that have a unique Nash equilibrium, with multiple logit equilibria for finite precision parameters. These results extend work at the intersection of stochastic decision theory and catastrophe theory with economic games. Their application to the problem of bifurcations in the tracing procedure for logit solutions, proposed by McKelvey and Palfrey, is discussed throughout.