The Dyson and Coulomb games

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.

arXiv comment40 pages, 2 figures. v3 adds results on the two dimensional extension for comparison, adds an early section reviewing results and stating main theorems in detail, adds a figure, and includes many other minor improvements

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Cited by in corpus (1)

• Duality for optimal couplings in free probability