Mean field games: A toy model on an Erdös-Renyi graph.

Delarue, François

doi:10.1051/proc/201760001

Cited by 50 publications

(25 citation statements)

References 15 publications

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“…A mathematically rigorous study of MFG systems with state values in finite graphs is provided in [21], and MFG systems where the agent subsystems are defined at the nodes (vertices) of finite random Erdös-Rényi graphs are treated in [11]. The system behaviour in [21] is subject to a fixed underlying network.…”

Section: Introductionmentioning

confidence: 99%

“…The system behaviour in [21] is subject to a fixed underlying network. The random graphs in [11] have unbounded growth but do not create spatial distinction of the agents due to symmetry properties of the interactions. However, graphon theory gives a rigorous formulation of the notion of limits for infinite sequences of networks of increasing size, and the first application of graphon theory in dynamics appears to be in the work of Medvedev [34,35], and Kaliuzhnyi-Verbovetskyi and Medvedev [26].…”

Section: Introductionmentioning

confidence: 99%

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Graphon Mean Field Games and the GMFG Equations

Caines

Huang

2018

2018 IEEE Conference on Decision and Control (CDC)

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The emergence of the graphon theory of large networks and their infinite limits has enabled the formulation of a theory of the centralized control of dynamical systems distributed on asymptotically infinite networks [16, 19]. Furthermore, the study of the decentralized control of such systems was initiated in [6, 7], where Graphon Mean Field Games (GMFG) and the GMFG equations were formulated for the analysis of noncooperative dynamic games on unbounded networks. In that work, existence and uniqueness results were introduced for the GMFG equations, together with an ǫ-Nash theory for GMFG systems which relates infinite population equilibria on infinite networks to finite population equilibria on finite networks. Those results are rigorously established in this paper.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Graphon Mean Field Games and the GMFG Equations

Caines

Huang

2018

2018 IEEE Conference on Decision and Control (CDC)

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“…Further, finite state mean field control, risk-sensitive control, and zero-sum games are treated in [20,18,19] which cover cases of unbounded jump intensities. Graphon games [24,9,8,50,10,31,4] have recently been receiving an increasing research interest. The motivation is the study of strategic decision making in the face of a large non-complete but dense networks of distinguishable agents.…”

Section: Graphon Games and Finite State Space Mean Field Gamesmentioning

confidence: 99%

Finite State Graphon Games with Applications to Epidemics

Aurell¹,

Carmona²,

Dayanıklı³

et al. 2021

Preprint

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We consider a game for a continuum of non-identical players evolving on a finite state space. Their heterogeneous interactions are represented by a graphon, which can be viewed as the limit of a dense random graph. The player's transition rates between the states depend on their own control and the interaction strengths with the other players. We develop a rigorous mathematical framework for this game and analyze Nash equilibria. We provide a sufficient condition for a Nash equilibrium and prove existence of solutions to a continuum of fully coupled forward-backward ordinary differential equations characterizing equilibria. Moreover, we propose a numerical approach based on machine learning tools and show experimental results on different applications to compartmental models in epidemiology.

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“…This powerful idea was extended in [27,28] to derive a central limit theorem, large deviations, and non-asymptotic concentration bounds for µ n . This versatile approach has been adapted to models with finite state space [24,9], a major player [61], local couplings [15], and graph-based interactions [26]. Proving well-posedness of the master equation, however, is a notoriously difficult task, especially when common noise is involved.…”

Section: Introductionmentioning

confidence: 99%

Closed-loop convergence for mean field games with common noise

Lacker¹,

Flem²

2021

Preprint

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This paper studies the convergence problem for mean field games with common noise. We define a suitable notion of weak mean field equilibria, which we prove captures all subsequential limit points, as n → ∞, of closed-loop approximate equilibria from the corresponding n-player games. This extends to the common noise setting a recent result of the first author, while also simplifying a key step in the proof and allowing unbounded coefficients and non-i.i.d. initial conditions. Conversely, we show that every weak mean field equilibrium arises as the limit of some sequence of approximate equilibria for the n-player games, as long as the latter are formulated over a broader class of closed-loop strategies which may depend on an additional common signal.

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Mean field games: A toy model on an Erdös-Renyi graph.

Cited by 50 publications

References 15 publications

Graphon Mean Field Games and the GMFG Equations

Graphon Mean Field Games and the GMFG Equations

Finite State Graphon Games with Applications to Epidemics

Closed-loop convergence for mean field games with common noise

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