Learning strict Nash equilibria through reinforcement

This paper studies the analytical properties of the reinforcement learning model proposed in Erev and Roth (1998), also termed cumulative reinforcement learning in Laslier et al (2001). This stochastic model of learning in games accounts for two main elements: the law of effect (positive reinforcement of actions that perform well) and the law of practice (the magnitude of the reinforcement effect decreases with players' experience). The main results of the paper show that, if the solution trajectories of the underlying replicator equation converge exponentially fast, then, with probability arbitrarily close to one, all the realizations of the reinforcement learning process lie within an e band of that solution. As the property of exponential convergence is shown to hold in proximity of any strict Nash equilibrium, the paper improves upon results currently available in the literature by showing that, whenever a strict Nash equilibrium exists, a reinforcement learning process started sufficiently close to it, will reach it with probability one.