Hidekazu Yoshioka

Hidekazu Yoshioka

We extend classical evolutionary game dynamics based on the momentary action choices of agents by accounting for two elements: forward-looking behavior and exploration cost. We focus on pairwise comparison protocols that cover major evolutionary game dynamics, such as replicator and logit models. In the proposed mathematical framework, agents update their actions by paying a cost so that a utility or its relative difference is maximized. We show that forward-looking behavior can be modeled as a coupling between the evolutionary game dynamic and static Hamilton-Jacobi-Bellman equation: a mean field game. The exploration cost and its constraint are naturally related to these equations as a function of the optimal Lagrangian multiplier serving as a relaxation parameter, and it is incorporated into the game as a constraint. We show that under certain conditions, our evolutionary game dynamic admits a unique solution. Finally, we computationally investigate one- and two-dimensional problems.

Hidekazu Yoshioka

Jacobi diffusion is a representative diffusion process whose solution is bounded in a domain under certain drift and diffusion coefficient conditions. However, the process without such conditions has not been thoroughly investigated. We explore a Jacobi diffusion whose drift coefficient is affected by another deterministic process, causing the process to hit the boundary of a domain in finite time. The Kolmogorov equation (a degenerate elliptic partial differential equation) for evaluating the boundary hitting of the proposed Jacobi diffusion is then presented and analyzed, with several conditional arguments, some of which are addressed computationally. We also investigate a related mean-field-type (McKean-Vlasov) self-consistent model arising in tourism management, where the drift depends on the index for sensor boundary hitting, thereby confining the process to a domain with higher probability. We propose a finite difference method for the linear and nonlinear Kolmogorov equations, which yields a unique numerical solution because of discrete ellipticity if the discount is positive. The accuracy of the finite difference method critically depends on the regularity of the boundary condition, and the use of high-order discretization is not always effective. Finally, we computationally investigate the mean field effect.

Hidekazu Yoshioka

This article shows that the unconditional stability of the Dual-Finite Volume Method, which is at least valid for linear problems, is not true for generic nonlinear differential equations including the PMEs unless the coefficient appearing in the numerical fluxes are appropriately evaluated. This article provides a theoretically truly isotone numerical fluxes specialized for solving the PMEs presented, which is still as simple as the conventional fully-upwind counterpart.