2020/05/28 by Daniel Lacker, Lacker, Daniel, Agathe Soret +1 · 2 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Economic theories and models #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2005.14102
openalex publication_date 2020/05/28 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We study a class of linear-quadratic stochastic differential games in which\neach player interacts directly only with its nearest neighbors in a given\ngraph. We find a semi-explicit Markovian equilibrium for any transitive graph,\nin terms of the empirical eigenvalue distribution of the graph's normalized\nLaplacian matrix. This facilitates large-population asymptotics for various\ngraph sequences, with several sparse and dense examples discussed in detail. In\nparticular, the mean field game is the correct limit only in the dense graph\ncase, i.e., when the degrees diverge in a suitable sense. Even though\nequilibrium strategies are nonlocal, depending on the behavior of all players,\nwe use a correlation decay estimate to prove a propagation of chaos result in\nboth the dense and sparse regimes, with the sparse case owing to the large\ndistances between typical vertices. Without assuming the graphs are transitive,\nwe show also that the mean field game solution can be used to construct\ndecentralized approximate equilibria on any sufficiently dense graph sequence.\n