Graphon Mean Field Games and the GMFG Equations

Caines, Peter E.; Huang, Minyi

doi:10.1109/cdc.2018.8619367

Cited by 52 publications

(40 citation statements)

References 27 publications

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“…Here the second inequality follows from (13) and (14) and the final equality is by definition in (59).…”

Section: Appendix Dmentioning

confidence: 99%

“…Under assumptions (A1)-(A5), there exists a solution of the fixed-point equations (13) and (17) for every t.Proof. Under the assumption (A1)-(A5), it has been shown in[14] that there exists a solution to the GMFG equations. Concurrently, Theorem 2 and Theorem 4 show that all GMFE can be found using backward recursion for the finite and infinite horizon problems.…”

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confidence: 99%

“…t , µ G t ; g α ) + δV t+1 (µ G t+1 , X α t+1 ) µ G 1:t , x α 1:t (41d) = V t (µ G t , x α t ),(41e)(41c) follows from the induction hypothesis in (40), (41d) follows because the random variables involved inexpectation, X α t , A α t , µ G t , µ G t+1 , X α t+1 do not depend onσ α t+1:T σ −α t+1:T and (41e) follows from the definition of V t in(14).APPENDIX CProof. We prove this by contradiction.…”

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confidence: 99%

“…where (37a) follows from definition of V t in (14), (37c) follows from definition of γ α t and (37d) follows from (35). However this leads to a contradiction.…”

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confidence: 99%

“…where (39b) follows from the definition of V t in (14). Suppose the claim is true for t + 1, i.e., ∀i ∈ [N ], t ∈ 15 Then…”

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confidence: 99%

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Sequential Decomposition of Graphon Mean Field Games

2020

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In this paper, we present a sequential decomposition algorithm to compute graphon mean-field equillibrium (GMFE) of dynamic graphon mean-field games (GMFGs). We consider a large population of players sequentially making strategic decisions where the actions of each player affect their neighbors which is captured in a graph, generated by a known graphon. Each player observes a private state and also a common information as a graphon mean-field population state which represents the empirical networked distribution of other players' types. We consider non-stationary population state dynamics and present a novel backward recursive algorithm to compute GMFE that depend on both, a player's private type, and the current (dynamic) population state determined through the graphon.Each step in this algorithm consists of solving a fixed-point equation. We provide conditions on model parameters for which there exists such a GMFE. Using this algorithm, we obtain the GMFE for a specific security setup in cyber physical systems for different graphons that capture the interactions between the nodes in the system.

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“…Here the second inequality follows from (13) and (14) and the final equality is by definition in (59).…”

Section: Appendix Dmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…where (37a) follows from definition of V t in (14), (37c) follows from definition of γ α t and (37d) follows from (35). However this leads to a contradiction.…”

mentioning

confidence: 99%

“…where (39b) follows from the definition of V t in (14). Suppose the claim is true for t + 1, i.e., ∀i ∈ [N ], t ∈ 15 Then…”

mentioning

confidence: 99%

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