FAQ-Learning in Matrix Games: Demonstrating Convergence Near Nash Equilibria, and Bifurcation of Attractors in the Battle of Sexes

This article studies Frequency Adjusted Q-learning (FAQ-learning), a variation of Q-learning that simulates simultaneous value function updates. The main contributions are empirical and theoretical support for the convergence of FAQ-learning to attractors near Nash equilibria in two-agent two-action matrix games.The games can be divided into three types: Matching pennies, Prisoners' Dilemma and Battle of Sexes. This article shows that the Matching pennies and Prisoners' Dilemma yield one attractor of the learning dynamics, while the Battle of Sexes exhibits a supercritical pitchfork bifurcation at a critical temperature, where one attractor splits into two attractors and one repellent fixed point. Experiments illustrate that the distance between fixed points of the FAQ-learning dynamics and Nash equilibria tends to zero as the exploration parameter of FAQ-learning approaches zero.