Convergence to the mean field game limit: A case study

Nutz, Marcel; Martı́n, Jaime San; Tan, Xiaowei

doi:10.1214/19-aap1501

Cited by 38 publications

(19 citation statements)

References 36 publications

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“…In recent years, the study of MFG of optimal stopping or other "singular" controls have been the subject of a growing number of researches, namely because such games have natural applications in Economy. Concerning the case of optimal stopping, let us mention [7,19,35,36], for impulse control we refer to [8] and to [27] for optimal switching. Let us also mention that the approach of [7,8] has been used by P.-L. Lions in [33] to study a case of MFG of singular controls.…”

Section: Introductionmentioning

confidence: 99%

Monotone solutions for mean field games master equations: finite state space and optimal stopping

Bertucci¹

2021

Journal De L’École Polytechnique — Mathématiques

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Section: Introductionmentioning

confidence: 99%

Monotone solutions for mean field games master equations: finite state space and optimal stopping

Bertucci¹

2021

Journal De L’École Polytechnique — Mathématiques

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“…, Y N (t)) the optimal trajectories of the N + 1-player game, i.e. when agents play the Nash equilibrium given by (26). Also, denote by X t the i.i.d process in which players choose the local control α(t, ±1) := [Z(t, m * (t))] ∓ , where Z is the entropy solution to (16) and m * is the unique mean field game solution induced by Z, if m 0 = 0 (µ 0 = 1 2 ), that is the one which does not change sign (see Proposition 6).…”

Section: Propositionmentioning

confidence: 99%

“…Let us mention three recent preprints that are related to our paper. In [26], Nutz, San Martin, and Tan address the convergence problem for a class of mean field games of optimal stopping. The limit model there possesses multiple solutions, which are grouped into three classes according to a qualitative criterion characterizing the proportion of players that have stopped at any given time.…”

Section: Introductionmentioning

confidence: 99%

On the Convergence Problem in Mean Field Games: A Two State Model without Uniqueness

Cecchin¹,

Pra²,

Fischer³

et al. 2019

SIAM J. Control Optim.

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We consider N -player and mean field games in continuous time over a finite horizon, where the position of each agent belongs to {−1, 1}. If there is uniqueness of mean field game solutions, e.g. under monotonicity assumptions, then the master equation possesses a smooth solution which can be used to prove convergence of the value functions and of the feedback Nash equilibria of the N -player game, as well as a propagation of chaos property for the associated optimal trajectories. We study here an example with anti-monotonous costs, and show that the mean field game has exactly three solutions. We prove that the value functions converge to the entropy solution of the master equation, which in this case can be written as a scalar conservation law in one space dimension, and that the optimal trajectories admit a limit: they select one mean field game soution, so there is propagation of chaos. Moreover, viewing the mean field game system as the necessary conditions for optimality of a deterministic control problem, we show that the N -player game selects the optimizer of this problem. 1 2 ALEKOS CECCHIN, PAOLO DAI PRA, MARKUS FISCHER, AND GUGLIELMO PELINO are shown to be concentrated on weak solutions of the corresponding mean field game. This concept of solution is also used in another, more recent work by Lacker; see below.Here, we are interested in the convergence problem for Nash equilibria in Markov feedback strategies with full state information. A first result in this direction was given by Gomes, Mohr, and Souza [19] in the case of finite state dynamics. There, convergence of Markovian Nash equilibria to the mean field game limit is proved, but only if the time horizon is small enough. A breakthrough was achieved by Cardaliaguet, Delarue, Lasry, and Lions in [7]. In the setting of games with non-degenerate Brownian dynamics, possibly including common noise, those authors establish convergence to the mean field game limit, in the sense of convergence of value functions as well as propagation of chaos for the optimal state trajectories, for arbitrary time horizon provided the so-called master equation associated with the mean field game possesses a unique sufficiently regular solution. The master equation arises as the formal limit of the Hamilton-Jacobi-Bellman systems determining the Markov feedback Nash equilibria. It yields, if well-posed, the optimal value in the mean field game as a function of initial time, state and distribution. It thus also provides the optimal control action, again as a function of time, state, and measure variable. This allows, in particular, to compare the prelimit Nash equilibria to the solution of the limit model through coupling arguments.If the master equation possesses a unique regular solution, which is guaranteed under the Lasry-Lions monotonicity conditions, then the convergence analysis can be considerably refined. In this case, for games with finite state dynamics, Cecchin and Pelino [11] and, independently, Bayraktar and Cohen [3] obtain a central limit theorem and lar...

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“…For general mean field games without uniqueness, the majority of the literature on the convergence analysis consider open loop controls. We refer to Camona-Delarue [10], Feleqi [22], Fischer [23], Lacker [26], Lasry-Lions [28], Nutz-San Martin-Tan [30]. In particular, [26] provides the full characterization for the convergence: any limit of approximate Nash equilibriums of N -player games is a weak mean field equilibrium, and conversely any weak mean field equilibrium can be obtained as such a limit.…”

Section: Introductionmentioning

confidence: 99%

Set Values for Mean Field Games

Iseri

Zhang

2021

Preprint

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When a mean field game satisfies certain monotonicity conditions, the mean field equilibrium is unique and the corresponding value function satisfies the so called master equation. In general, however, there can be multiple equilibriums which typically lead to different values. In this paper we study the set of values over all mean field equilibriums, which we call the set value of the game. We shall establish two main properties of the set value: (i) the dynamic programming principle; (ii) the convergence of the set values of the corresponding N -player games. We emphasize that the set value is very sensitive to the choice of the admissible controls. For the dynamic programming principle, one needs to use closed loop controls (instead of open loop controls). For the convergence, one has to restrict to the same type of equilibriums for the N-player game and for the mean field game. We shall investigate three cases, two in finite state space models and the other in a diffusion model.

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Convergence to the mean field game limit: A case study

Cited by 38 publications

References 36 publications

Monotone solutions for mean field games master equations: finite state space and optimal stopping

Monotone solutions for mean field games master equations: finite state space and optimal stopping

On the Convergence Problem in Mean Field Games: A Two State Model without Uniqueness

Set Values for Mean Field Games

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