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Title: 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 in a given graph. We find a semiexplicit 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, that is, when the degrees diverge in a suitable sense. Although 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.  more » « less
Award ID(s):
2045328
NSF-PAR ID:
10333518
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Mathematics of Operations Research
Volume:
47
Issue:
2
ISSN:
0364-765X
Page Range / eLocation ID:
1530 to 1565
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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