Potential Games on Unimodular Random Graphs

TL;DR

This paper investigates potential games on unimodular random graphs, establishing a connection between the potential's minimizers and Nash equilibria, and providing explicit equilibrium descriptions in specific cases.

Contribution

It introduces a well-defined global potential for games on unimodular graphs and links its minimizers to Nash equilibria, including explicit solutions in linear-quadratic and convex cases.

Findings

Potential minimizers coincide with Nash equilibria under convexity.

Equilibria in linear-quadratic games are expressed via the Green kernel.

Solutions to nonlinear Poisson equations characterize equilibria in convex settings.

Abstract

We study potential games on unimodular random graphs of bounded degree, where players interact through the underlying network. Using the unimodular measure, we define a well-posed global potential that captures both finite- and infinite-player games. A key observation is that the mass-transport principle identifies the first variation of this potential with the first-order condition of a representative (root) player. Under suitable convexity assumptions, we prove that minimizers of the potential coincide with quenched Nash equilibria, and conversely. We also establish the thermodynamic limit of the potential along weakly convergent sequences of unimodular measures. Finally, we present examples with semi-explicit equilibrium descriptions. In linear-quadratic games on unimodular graphs, equilibria are expressed in terms of the Green kernel of the simple random walk operator, while in convex settings, equilibria are characterized by solutions to nonlinear Poisson equations.