Mean Field Games and Systemic Risk

Carmona, René; Fouque, Jean‐Pierre; Sun, Li-Hsien

doi:10.48550/arxiv.1308.2172

Cited by 13 publications

(39 citation statements)

References 2 publications

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“…and equation (16) in equation ( 9), thus recovering the equation above. Similarly, it can be shown that if p(t, x) is a solution of equation ( 9) then g(t, x) = p f , with f (t, x) given by (15), is a solution to equation (18). We summarize this fact in the following theorem.…”

Section: A Cole-hopf Type Transformmentioning

confidence: 83%

“…From the preceeding discussion, it can be easily verified that using the transforms (15,17), the corresponding MFG model constituted by the time-varying optimality system (8, 9) and q := q has the following NLS representation:…”

Section: B Nonlinear Schrödinger Equation and Mean Field Gamesmentioning

confidence: 99%

“…Mean field games (MFGs) ( [12], [13]) utilize PDE representation to model such continuum systems and provide a game-theoretic optimal control interpretation of emergent behaviour in self-organized systems. Most works on MFGs consider explicit interactions between agents through the dependence of their dynamics or cost function on the population density ( [14], [15], [16], [17], [18]). The corresponding optimality system consists of a backward-in-time semilinear Hamilton-Jacobi-Bellman (HJB) equation governing the value function and a forward-in-time Fokker-Planck (FP) equation governing the density, wherein the HJB equation depends on the density and the FP equation depends on the value function.…”

Section: Introductionmentioning

confidence: 99%

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Schrödinger Approach to Optimal Control of Large-Size Populations

Bakshi,

Fan,

Theodorou

2018

Preprint

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Large-size populations consisting of a continuum of identical and non-cooperative agents with stochastic dynamics are useful in modeling various biological and engineered systems. This paper addresses the problem of designing mean-field type optimal state-feedback controllers which guarantee closed-loop stability of the stationary density of such agents with nonlinear Langevin dynamics, under the action of their individual steady state controls.We represent the corresponding optimality system which consists of coupled forward-backward PDEs as decoupled Schrödinger equations, by introducing a novel variable transform. Spectral properties of the Schrödinger operator which underlie the stability analysis are used to obtain explicit control design constraints, in the case that the agents do not interact explicitly via density dependent dynamics or cost function. Our interpretation of the Schrödinger potential as the cost function of a closely related optimal control problem motivates a quadrature based algorithm to compute the finite time optimal control 1 . We show the deep connection between the nonlinear Schrödinger equation and Mean Field Games for agents with nonlinear Langevin dynamics and explicit interactions.

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Section: A Cole-hopf Type Transformmentioning

confidence: 83%

Section: B Nonlinear Schrödinger Equation and Mean Field Gamesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Schrödinger Approach to Optimal Control of Large-Size Populations

Bakshi,

Fan,

Theodorou

2018

Preprint

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“…The MFG approach offers a promising route to investigate systemic risk. For instance, in the recent work [13], the MFG framework has been proposed to model systemic risk associated with inter-bank borrowing and lending strategies.…”

Section: Introductionmentioning

confidence: 99%

Mean-field games and model predictive control

Degond

Herty

Liu

2017

Communications in Mathematical Sciences

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Abstract. Mean-Field Games are games with a continuum of players that incorporate the time-dimension through a control-theoretic approach. Recently, simpler approaches relying on the Best Reply Strategy have been proposed. They assume that the agents navigate their strategies towards their goal by taking the direction of steepest descent of their cost function (i.e. the opposite of the utility function). In this paper, we explore the link between Mean-Field Games and the Best Reply Strategy approach. This is done by introducing a Model Predictive Control framework, which consists of setting the Mean-Field Game over a short time interval which recedes as time moves on. We show that the Model Predictive Control offers a compromise between a possibly unrealistic Mean-Field Game approach and the sub-optimal Best Reply Strategy.

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“…Mean field game theory has been intensively studied in the past few years, and in spite of its relative youth, a very large number of results have been obtained in the mathematical [4][5][6][7][8] and socio-economic community [9][10][11][12][13]. A recent overview is given by Gomes and Saúde in [14].…”

Section: Introductionmentioning

confidence: 99%

“Phase diagram” of a mean field game

Swiecicki

Gobron

Ullmo

2016

Physica A: Statistical Mechanics and its Applications

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Mean field games were introduced by J-M.Lasry and P-L. Lions in the mathematical community, and independently by M. Huang and co-workers in the engineering community, to deal with optimization problems when the number of agents becomes very large. In this article we study in detail a particular example called the "seminar problem" introduced by O.Guéant, J-M Lasry, and P-L. Lions in 2010. This model contains the main ingredients of any mean field game but has the particular feature that all agent are coupled only through a simple random event (the seminar starting time) that they all contribute to form. In the mean field limit, this event becomes deterministic and its value can be fixed through a self consistent procedure. This allows for a rather thorough understanding of the solutions of the problem, through both exact results and a detailed analysis of various limiting regimes.For a sensible class of initial configurations, distinct behaviors can be associated to different domains in the parameter space . For this reason, the "seminar problem" appears to be an interesting toy model on which both intuition and technical approaches can be tested as a preliminary study toward more complex mean field game models.

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Mean Field Games and Systemic Risk

Cited by 13 publications

References 2 publications

Schrödinger Approach to Optimal Control of Large-Size Populations

Schrödinger Approach to Optimal Control of Large-Size Populations

Mean-field games and model predictive control

“Phase diagram” of a mean field game

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