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

Understand

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.