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

TL;DR

This paper introduces an $oldsymbol{ extalpha}$-potential game framework to analyze distributed games with jumps, revealing how equilibria depend on network structure and interaction rules, with applications to crowd motion and portfolio games.

Contribution

It extends the $ extalpha$-potential game approach to distributed jump diffusion games, providing explicit equilibrium characterizations and computational methods for complex network interactions.

Findings

$ extalpha=0$ for all symmetric and asymmetric crowd motion networks.

Quantified decay rates of $ extalpha$ in relation to network parameters.

Constructed explicit Nash equilibria for a portfolio selection game.

Abstract

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.