A general characterization of the mean field limit for stochastic differential games

By Daniel Lacker. 2014

206 citations recorded by the source corpus.

The mean field limit of large-population symmetric stochastic differential games is derived in a general setting, with and without common noise, on a finite time horizon. Minimal assumptions are imposed on equilibrium strategies, which may be asymmetric and based on full information. It is shown that approximate Nash equilibria in the n-player games admit certain weak limits as n tends to infinity, and every limit is a weak solution of the mean field game (MFG). Conversely, every weak MFG solution can be obtained as the limit of a sequence of approximate Nash equilibria in the n-player games. Thus, the MFG precisely characterizes the possible limiting equilibrium behavior of the n-player games. Even in the setting without common noise, the empirical state distributions may admit stochastic limits which cannot be described by the usual notion of MFG solution.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/b621e94b7c0d946b1a02181d60134b4c2b7c2a83

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Citations in the corpus

On the connection between symmetric $N$-player games and mean field games (2014, background): “In Lacker (2016), which appeared as preprint three months after submission of the present paper, a general characterization of the limit points of N -player Nash equilibria is obtained through probabilistic methods.”
The Master Equation and the Convergence Problem in Mean Field Games (2015, background): “Let us finally quote the nice results by Fischer [25] and Lacker [45] on the convergence of open loop Nash equilibria for the N´player game and the characterization of the possible limits.”
Mean field and n‐agent games for optimal investment under relative performance criteria (2017, background): “We refer the reader to [11] for a detailed discussion of mean field games with common noise, and to [32, 8] for general results on limits of n-player games.”
Selection of equilibria in a linear quadratic mean-field game (2018, methodology): “[11] R. Carmona, F. Delarue, and D. Lacker.”
On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games (2017, unknown): “The “illuminating example” in Lacker [38], Section 3.3, also provides an MFG with explicitly known multiple solutions; cf.”
Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, background): “General results on existence, uniqueness, and limit theory require working with a notion of weak equilibrium which is well-suited for weak convergence arguments [18, 53, 17].”
A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria (2014, unknown): “2See however the recent advances in [13, 22].”
Mean Field Games of Timing and Models for Bank Runs (2016, unknown): “Many different names and characterizations are associated to this property of a filtration enlargement, such as the H-hypothesis [8], immersion [24], very good extensions [23], and natural extensions [15], while we borrow the term compatible from Kurtz [25], to be consistent with other works on mean”
From the master equation to mean field game limit theory: a central limit theorem (2018, background): “A law of large numbers has only recently come into focus [6, 31, 30, 19], clarifying how a MFG arises as the limit of a suitable sequence of n-player games as n→∞, in the presence of both idiosyncratic and common sources of noise.”
$N$-player games and mean-field games with absorption (2016, background): “The proof of Theorem 6.2 on existence of a feedback solution for the limit MFG is based on results on BSDEs with random terminal horizon as in Darling and Pardoux [12], Briand and Hu [4] and the use of the BrouwerSchauder-Tychonov fixed point theorem applied to a suitable map in the same spirit as C”
From the master equation to mean field game limit theory: Large deviations and concentration of measure (2018, background): “The papers [28, 23] provide laws of large numbers for open-loop equilibria, with the key advantage of addressing the non-unique regime, that is, when there are multiple mean field equilibria.”
Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource (2016, background): “The first largely comprehensive account of this problem for mean field games with common noise was given by Lacker in [44].”
Mean field games via controlled martingale problems: Existence of Markovian equilibria (2014, background): “For example, concurrently with the finalization of this manuscript for publication, recent papers exploited relaxed control theory to study mean field games with common noise [14] as well as the convergence of equilbria of finite-player games [20, 34].”