Convergence Analysis of Log-Linear Learning of Games With Irrationality

This article investigates the effect of irrational behaviors on the expected convergence speed to the Nash equilibrium (NE) set within the log-linear learning (LLL) model. First, a significant augmentation for the classical LLL model is proposed by reconstructing the strategy update probability, merged with the effect of irrational behaviors. By incorporating such irrationality, it proves that under certain conditions, underweighting probabilities improves the expected convergence speed while overweighting probabilities reduces it. Second, these new results are applied to discuss the expected convergence speed to the stochastically stable Nash equilibrium (SSNE) set in potential games with a typical irrationality model, namely, probability sensitivity. Finally, the effectiveness of the proposed methods is illustrated by a sensor deployment example.