Graphon Mean-Field Logit Dynamic: Derivation, Computation, and Applications

We present a graphon mean-field logit dynamic, a stationary mean-field game based on logit interactions. This dynamic emerges from a stochastic control problem involving a continuum of nonexchangeable and interacting agents and reduces to solving a continuum of Hamilton-Jacobi-Bellman (HJB) equations connected through a graphon that models the connections among agents. Using a fixed-point argument, we prove that this HJB system admits a unique solution in the space of bounded functions when the discount rate is high (i.e., agents are myopic). Under certain assumptions, we also establish regularity properties of the system, such as equi-continuity. We propose a finite difference scheme for computing the HJB system and prove the uniqueness and existence of its numerical solutions. The mean-field logit dynamic is applied to a case study on inland fisheries resource management in the upper Tedori River of Japan. A series of computational cases are then conducted to investigate the dependence of the dynamic on both the discount rate and graphon.