Quantal Correlated Equilibrium in Normal Form Games

Abstract

Correlated equilibrium is an established solution concept in game theory describing a situation when players condition their strategies on external signals produced by a correlation device. In recent years, the concept has begun gaining traction also in general artificial intelligence because of its suitability for studying coordinated multi-agent systems. Yet the original formulation of correlated equilibrium assumes entirely rational players and hence fails to capture the subrational behavior of human decision-makers. We investigate the analogue of quantal response for correlated equilibrium, which is among the most commonly used models of bounded rationality. We coin the solution concept the quantal correlated equilibrium and study its relation to quantal response and correlated equilibria. The definition corroborates with prior conception as every quantal response equilibrium is a quantal correlated equilibrium, and correlated equilibrium is its limit as quantal responses approach the best response. We prove the concept remains PPAD-hard but searching for an optimal correlation device is beneficial for the signaler. To this end, we introduce a homotopic algorithm that simultaneously traces the equilibrium and optimizes the signaling distribution. Empirical results on one structured and one random domain show that our approach is sufficiently precise and several orders of magnitude faster than a state-of-the-art non-convex optimization solver.