Generalized Quantal Response Equilibrium: Existence and Efficient Learning

Abstract: We introduce a new solution concept for bounded rational agents in finite normal-form general-sum games called Generalized Quantal Response Equilibrium (GQRE) which generalizes Quantal Response Equilibrium~\citep{mckelvey1995quantal}. In our setup, each player maximizes a smooth, regularized expected utility of the mixed profiles used, reflecting bounded rationality that subsumes stochastic choice. After establishing existence under mild conditions, we present computationally efficient no-regret independent learning via smoothened versions of the Frank-Wolfe algorithm. Our algorithm uses noisy but correlated gradient estimates generated via a simulation oracle that reports on repeated plays of the game. We analyze convergence properties of our algorithm under assumptions that ensure uniqueness of equilibrium, using a class of gap functions that generalize the Nash gap. We end by demonstrating the effectiveness of our method on a set of complex general-sum games such as high-rank two-player games, large action two-player games, and known examples of difficult multi-player games.