Distributed games with jumps: An $\alpha$-potential game approach

Motivated by game-theoretic models of crowd motion dynamics, this paper analyzes a broad class of distributed games with jump diffusions within the recently developed $\alpha$-potential game framework. We demonstrate that analyzing the $\alpha$-Nash equilibria reduces to solving a finite-dimensional control problem. Beyond the viscosity and verification characterizations for the general games, we examine explicitly and in detail how spatial population distributions and interaction rules influence the structure of $\alpha$-Nash equilibria in these distributed settings. For crowd motion network games, we show that $\alpha = 0$ for all symmetric interaction networks, and or asymmetric networks. We quantify the precise polynomial and logarithmic decays of $\alpha$ in terms of the number of players, the degree of the network, and the decay rate of interaction asymmetry. We also exploit the $\alpha$-potential game framework to analyze an $N$-player portfolio selection game under a mean-variance criterion. We show that this portfolio game constitutes a potential game and explicitly construct its Nash equilibrium. Our analysis allows for heterogeneous preference parameters, going beyond the mean-field interactions considered in the existing game literature. Our theoretical results are supported by numerical implementations using policy gradient-based algorithms, demonstrating the computational advantages of the $\alpha$-potential game framework in computing Nash equilibria for general dynamic games.