The Dyson and Coulomb games

TL;DR

This paper introduces N-player dynamic games with Coulomb gas equilibria, revealing how local limits depend on information models in 1D but not in 2D, and demonstrating convergence to mean field equations.

Contribution

It presents new Coulomb gas-based N-player game models and analyzes their local limits and convergence properties, highlighting the role of information structure.

Findings

Local limits are sensitive to information models in 1D.

Players can achieve symmetry through selfish behavior.

Convergence to mean field equations is established for certain ensembles.

Abstract

We introduce and investigate certain $N$ player dynamic games on the line and in the plane that admit Coulomb gas dynamics as a Nash equilibrium. Most significantly, we find that the universal local limit of the equilibrium is sensitive to the chosen model of player information in one dimension but not in two dimensions. We also find that players can achieve game theoretic symmetry through selfish behavior despite non-exchangeability of states, which allows us to establish strong localized convergence of the N-Nash systems to the expected mean field equations against locally optimal player ensembles, i.e., those exhibiting the same local limit as the Nash-optimal ensemble. In one dimension, this convergence notably features a nonlocal-to-local transition in the population dependence of the $N$-Nash system.