Stochastic Games on Large Sparse Graphs

TL;DR

This paper develops a comprehensive framework for analyzing stochastic games on large, sparse graphs, establishing existence, uniqueness, and approximation of Nash equilibria, and demonstrating convergence properties on graph sequences.

Contribution

It introduces a unified framework for stochastic games on large sparse graphs, proving key properties and convergence results, including local approximation and reconstruction of equilibria.

Findings

Existence and uniqueness of Nash equilibria under contraction conditions

Exponential decay of correlations with graph distance

Convergence of equilibria on graph sequences, including hyperfinite unimodular random graphs

Abstract

We introduce a framework for stochastic games on large sparse graphs, covering continuous-time and discrete-time dynamic games as well as static games. Players are indexed by the vertices of simple, locally finite graphs, allowing both finite and countably infinite populations, with asymptotics described through local weak convergence of marked graphs. The framework allows path-dependent utility functionals that may be heterogeneous across players. Under a contraction condition, we prove existence and uniqueness of Nash equilibria and establish exponential decay of correlations with graph distance. We further show that global equilibria can be approximated by truncated local games, and can even be reconstructed exactly on subgraphs given information on their boundary. Finally, we prove convergence of Nash equilibria along locally weakly convergent graph sequences, including sequences sampled from hyperfinite unimodular random graphs.