Generalized Logit Dynamics Based on Rational Logit Functions

In generalized logit dynamics, optimal probabilistic actions of agents in iterative games with perturbed utilities are described using partial integro-differential equations. Any equilibrium of a (generalized) logit dynamic is usually expected to converge to a Nash equilibrium of the original iterative game as the perturbation approaches zero. It has also been pointed out that using different perturbations, and in particular different entropies, approximates different equilibria. We explain this phenomenon by focusing on the tail behavior of the logit functions and demonstrate mathematically and numerically that different equilibria are obtained for different q-exponential type logit functions for different values of q. We also demonstrate that the generalized logit dynamic admits a unique measure-valued solution with theoretical estimates of the difference between the solutions corresponding to different values of q. Finally, we apply generalized logit dynamics to a problem inspired by hydropower generation and fishery resource management, both of which are key to sustainable and comprehensive development, to analyze the differences between different dynamics.