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

## Daniel Lacker – Columbia University, United States

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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. Joint work with Agathe Soret.

**Bio**: Daniel Lacker is an assistant professor in Industrial Engineering & Operations Research at Columbia University. He obtained his PhD from the Operations Research & Financial Engineering department of Princeton University in 2015, and he was an NSF postdoctoral fellow in the Division of Applied Mathematics at Brown University from 2015-2017. Daniel's research is in applied probability and mathematical finance, with a primary focus on mean field game theory and interacting particle systems.