A case study on stochastic games on large graphs in mean field and sparse regimes

TL;DR

This paper analyzes stochastic differential games on large graphs, deriving equilibrium strategies and asymptotic behaviors in both dense and sparse regimes, highlighting the limitations of mean field approximations.

Contribution

It provides a semi-explicit equilibrium characterization for transitive graphs and extends the analysis to non-transitive graphs using mean field approximations.

Findings

Mean field game is valid only in dense graph regimes.

Equilibrium strategies depend on the graph's eigenvalue distribution.

Propagation of chaos holds in both dense and sparse regimes.

Abstract

We study a class of linear-quadratic stochastic differential games in which each player interacts directly only with its nearest neighbors in a given graph. We find a semi-explicit Markovian equilibrium for any transitive graph, in terms of the empirical eigenvalue distribution of the graph's normalized Laplacian matrix. This facilitates large-population asymptotics for various graph sequences, with several sparse and dense examples discussed in detail. In particular, the mean field game is the correct limit only in the dense graph case, i.e., when the degrees diverge in a suitable sense. Even though equilibrium strategies are nonlocal, depending on the behavior of all players, we use a correlation decay estimate to prove a propagation of chaos result in both the dense and sparse regimes, with the sparse case owing to the large distances between typical vertices. Without assuming the graphs are transitive, we show also that the mean field game solution can be used to construct decentralized approximate equilibria on any sufficiently dense graph sequence.