Generalized Pair-Wise Logit Dynamic and Its Connection to a Mean Field Game: Theoretical and Computational Investigations Focusing on Resource Management

Logit dynamics are evolution equations that describe transitions to equilibria of actions among many players. We formulate a pair-wise logit dynamic in a continuous action space with a generalized exponential function, which we call a generalized pair-wise logit dynamic, depicted by a new evolution equation nonlocal in space. We prove the well-posedness and approximability of the generalized pair-wise logit dynamic to show that it is computationally implementable. We also show that this dynamic has an explicit connection to a mean field game of a controlled pure-jump process, with which the two different mathematical models can be understood in a unified way. Particularly, we heuristically discuss that the generalized pair-wise logit dynamic is derived as a myopic version of the corresponding mean field game, and that the conditions to guarantee the existence of unique solutions are different from each other. The key in this procedure is to find the objective function to be optimized in the mean field game based on the logit function. The monotonicity of the utility is unnecessary for the generalized pair-wise logit dynamic but crucial for the mean field game. Finally, we present applications of the two approaches to fisheries management problems with collected data.