Convergence to the Mean Field Game Limit: A Case Study

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

doi:10.48550/arxiv.1806.00817

Cited by 5 publications

(6 citation statements)

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“…The question of if or when this is true for weak MFE remains open. 1 We give some examples of weak MFE which are not strong but which do arise as limits of approximate n-player equilibria by exploiting an interesting connection with the 1 The recent work [50] provides a remarkably detailed analysis of nearly the same questions in the context of a specific MFG of optimal stopping, showing that certain MFE arise as the limits of n-player equilibria while others do not; while [50] notably does not consider approximate equilibria for the n-player games, it is still a good source of intuition for what can go wrong. regularization-by-noise or Peano phenomenon, which can be described as follows: Adding ǫdW t can turn a non-unique ODE into a well-posed SDE, and in certain cases the limit in distribution of the SDE solution as ǫ ↓ 0 is a particular mixture of the ODE solutions [1].…”

Section: Introductionmentioning

confidence: 99%

On the convergence of closed-loop Nash equilibria to the mean field game limit

Lacker¹

2018

Preprint

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This paper continues the study of the mean field game (MFG) convergence problem: In what sense do the Nash equilibria of n-player stochastic differential games converge to the mean field game as n → ∞? Previous work on this problem took two forms. First, when the n-player equilibria are open-loop, compactness arguments permit a characterization of all limit points of n-player equilibria as weak MFG equilibria, which contain additional randomness compared to the standard (strong) equilibrium concept. On the other hand, when the n-player equilibria are closed-loop, the convergence to the MFG equilibrium is known only when the MFG equilibrium is unique and the associated "master equation" is solvable and sufficiently smooth. This paper adapts the compactness arguments to the closed-loop case, proving a convergence theorem that holds even when the MFG equilibrium is non-unique. Every limit point of nplayer equilibria is shown to be the same kind of weak MFG equilibrium as in the open-loop case. Some partial results and examples are discussed for the converse question, regarding which of the weak MFG equilibria can arise as the limit of n-player (approximate) equilibria. DANIEL LACKER 6.1. Proof of Theorem 2.8 43 6.2. Proof of Theorem 2.12 43 7. Constructing n-player equilibria from mean field equilibria 43 7.1. Proof of Theorem 3.10 43 7.2. Uncontrolled models and ill-posed ODEs 46 7.3. A game-theoretic example 47 7.4. Proof of Proposition 7.2 49 Appendix A. SDEs with random coefficients 51 A.1. Deterministic well-posedness 52 A.2. Stochastic well-posedness 53 A.3. Stability 56 A.4. Forward equations 57 Appendix B. Joint measurability of regular conditional laws 58 Appendix C. Conditional means of random measures 60 References 61

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Section: Introductionmentioning

confidence: 99%

On the convergence of closed-loop Nash equilibria to the mean field game limit

Lacker¹

2018

Preprint

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“…See also [51, Sections 2.4 and 7] for additional discussion of the difficulties. If one restricts attention further and only considers the true equilibria (rather than approximate equilibria) for the n-player games, then one cannot even expect to obtain all of the strong MFE as limit points; see the recent case studies [23,29,60].…”

Section: 5mentioning

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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“…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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Convergence to the Mean Field Game Limit: A Case Study

Cited by 5 publications

References 0 publications

On the convergence of closed-loop Nash equilibria to the mean field game limit

On the convergence of closed-loop Nash equilibria to the mean field game limit

Closed-loop convergence for mean field games with common noise

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

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