PIMS - SFU Applied & Computational Math Seminar: Aaron Palmer

Multiplayer games model the behavior of competing agents, and we are interested in the structure that emerges when the number of agents is large. The Dyson and Coulomb games are N-agent dynamic games that admit well studied models of statistical physics (Dyson Brownian motion and 2D Coulomb gas) as Nash equilibria.

The Dyson game is motivated by the real world phenomena of the inter-arrival times of the buses of Cuernavaca, México, as well as the spacing of parked cars and perched birds, all of which consist of ordered interacting agents and have been observed to exhibit the universality of random matrix statistics (i.e. Dyson Brownian motion). We find the optimal repulsion parameter (corresponding to the universality class) of the Nash equilibrium for the Dyson game depends on the information available to the agents. We analyze the limit as N goes to infinity, where the Dyson game is approximated by a mean field game, which depends on the universality class through the nontrivial asymptotic behavior of the agents.

The Coulomb game is a natural extension to two dimensions. However, the leading term of the cost is an interaction of three agents that vanishes in one dimension and has an astonishing geometric interpretation: the reciprocal diameter squared of the circumcircle that passes through the three points. In contrast to the Dyson game, the mean field limit of the Coulomb game does not depend on the player information, and only features the three agent interaction.

This is joint work with René Carmona (Princeton) and Mark Cerenzia (U of Chicago).