A Case Study on Stochastic Games on Large Graphs in Mean Field and Sparse Regimes

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