{"ID":2887438,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.01929","arxiv_id":"2508.01929","title":"Distributed games with jumps: An $α$-potential game approach","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 $α$-potential game framework. We demonstrate that analyzing the $α$-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 $α$-Nash equilibria in these distributed settings. For crowd motion network games, we show that $α= 0$ for all symmetric interaction networks, and or asymmetric networks. We quantify the precise polynomial and logarithmic decays of $α$ in terms of the number of players, the degree of the network, and the decay rate of interaction asymmetry. We also exploit the $α$-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 $α$-potential game framework in computing Nash equilibria for general dynamic games.","short_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 $α$-potential game framework. We demonstrate that analyzing the $α$-Nash equilibria reduces to solving a finite-dimensional control problem. Beyond the v...","url_abs":"https://arxiv.org/abs/2508.01929","url_pdf":"https://arxiv.org/pdf/2508.01929v2","authors":"[\"Xin Guo\",\"Xinyu Li\",\"Yufei Zhang\"]","published":"2025-08-03T21:45:10Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"cs.MA\",\"math.PR\"]","methods":"[\"Diffusion Model\"]","has_code":false}