Computational analysis on a linkage between generalized logit dynamic and discounted mean field game

TL;DR

This paper explores the connection between generalized logit dynamics and discounted mean field games, providing theoretical explanations, numerical methods, and applications to resource and environmental management problems.

Contribution

It offers a novel interpretation of generalized logit dynamics through discounted mean field games and develops numerical methods for their computation.

Findings

Large discount limit yields logit dynamics.

Mean field games can produce classical and generalized logit dynamics.

Numerical methods successfully applied to resource management problems.

Abstract

Logit dynamics are dynamical systems describing transitions and equilibria of actions of interacting players under uncertainty. An uncertainty is embodied in logit dynamic as a softmax type function often called a logit function originating from a maximization problem subjected to an entropic penalization. This study provides another explanation for the generalized logit dynamic, particularly its logit function and player's heterogeneity, based on a discounted mean field game subjected to the costly decision making of a representative player. A large discount limit of the mean field game is argued to yield a logit dynamic. Further, mean field games that lead to classical and generalized logit dynamics are clarified and their well posedness is discussed. Additionally, numerical methods based on a finite difference discretization for computing generalized logit dynamics and corresponding mean field games are presented. Numerical methods are applied to two problems arising in the management of resources and environment; one involves an inland fisheries management problem with legal and illegal anglers, while the other is a sustainable tourism problem. Particularly, cases that possibly lack the regularity condition to be satisfied for the unique existence of stationary solutions are computationally discussed.