Abstract
The Dyson game is an explicitly solvable N player dynamic game that admits Dyson Brownian motion as a Nash equilibrium. The game is motivated by the real world phenomenon found in the spacing of buses, parked cars and perched birds, which exhibit random matrix statistics (i.e. Dyson Brownian motion). We find the optimal repulsion parameter (universality class) of the equilibrium depends on the information available to the players, furthering the understanding of an open problem in random matrix theory proposed by Deift. The limiting mean field game has a local cost term, which depends on the optimal universality class due to the nontrivial asymptotic behavior of the players. We solve the mean field game master equation and the associated Hamilton-Jacobi equation on Wasserstein space exactly, and we discuss how generalizing our results will require answering novel questions on the analysis of these equations on infinite dimensional spaces.