## Annals of Probability

### From the master equation to mean field game limit theory: Large deviations and concentration of measure

#### Abstract

We study a sequence of symmetric $n$-player stochastic differential games driven by both idiosyncratic and common sources of noise, in which players interact with each other through their empirical distribution. The unique Nash equilibrium empirical measure of the $n$-player game is known to converge, as $n$ goes to infinity, to the unique equilibrium of an associated mean field game. Under suitable regularity conditions, in the absence of common noise, we complement this law of large numbers result with nonasymptotic concentration bounds for the Wasserstein distance between the $n$-player Nash equilibrium empirical measure and the mean field equilibrium. We also show that the sequence of Nash equilibrium empirical measures satisfies a weak large deviation principle, which can be strengthened to a full large deviation principle only in the absence of common noise. For both sets of results, we first use the master equation, an infinite-dimensional partial differential equation that characterizes the value function of the mean field game, to construct an associated McKean–Vlasov interacting $n$-particle system that is exponentially close to the Nash equilibrium dynamics of the $n$-player game for large $n$, by refining estimates obtained in our companion paper. Then we establish a weak large deviation principle for McKean–Vlasov systems in the presence of common noise. In the absence of common noise, we upgrade this to a full large deviation principle and obtain new concentration estimates for McKean–Vlasov systems. Finally, in two specific examples that do not satisfy the assumptions of our main theorems, we show how to adapt our methodology to establish large deviations and concentration results.

#### Article information

Source
Ann. Probab., Volume 48, Number 1 (2020), 211-263.

Dates
Revised: February 2019
First available in Project Euclid: 25 March 2020

https://projecteuclid.org/euclid.aop/1585123327

Digital Object Identifier
doi:10.1214/19-AOP1359

Mathematical Reviews number (MathSciNet)
MR4079435

Zentralblatt MATH identifier
07206757

#### Citation

Delarue, François; Lacker, Daniel; Ramanan, Kavita. From the master equation to mean field game limit theory: Large deviations and concentration of measure. Ann. Probab. 48 (2020), no. 1, 211--263. doi:10.1214/19-AOP1359. https://projecteuclid.org/euclid.aop/1585123327

#### References

• [1] Bayraktar, E. and Cohen, A. (2018). Analysis of a finite state many player game using its master equation. SIAM J. Control Optim. 56 3538–3568.
• [2] Ben Arous, G. and Brunaud, M. (1990). Méthode de Laplace: étude variationnelle des fluctuations de diffusions de type “champ moyen”. Stoch. Stoch. Rep. 31 79–144.
• [3] Bensoussan, A. and Frehse, J. (1983). Nonlinear Elliptic Systems in Stochastic Game Theory, Universität Bonn. SFB 72. Approximation und Optimierung, 1983.
• [4] Bensoussan, A., Frehse, J. and Yam, S. C. P. (2015). The master equation in mean field theory. J. Math. Pures Appl. (9) 103 1441–1474.
• [5] Bensoussan, A., Frehse, J. and Yam, S. C. P. (2017). On the interpretation of the Master Equation. Stochastic Process. Appl. 127 2093–2137.
• [6] Bobkov, S. G. and Götze, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 1–28.
• [7] Bolley, F., Guillin, A. and Malrieu, F. (2010). Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov–Fokker–Planck equation. ESAIM Math. Model. Numer. Anal. 44 867–884.
• [8] Bolley, F., Guillin, A. and Villani, C. (2007). Quantitative concentration inequalities for empirical measures on non-compact spaces. Probab. Theory Related Fields 137 541–593.
• [9] Budhiraja, A., Dupuis, P. and Fischer, M. (2012). Large deviation properties of weakly interacting processes via weak convergence methods. Ann. Probab. 40 74–102.
• [10] Cardaliaguet, P., Delarue, F., Lasry, J.-M. and Lions, P.-L. (2019) The Master Equation and the Convergence Problem in Mean Field Games. Annals of Mathematics Studies 12 Princeton Univ. Press.
• [11] Carmona, R. and Delarue, F. (2014). The master equation for large population equilibriums. In Stochastic Analysis and Applications 2014. Springer Proc. Math. Stat. 100 77–128. Springer, Cham.
• [12] Carmona, R. and Delarue, F. (2017). Probabilistic Theory of Mean Field Games: Vol. I, Mean Field FBSDEs, Control, and Games. Stochastic Analysis and Applications, Springer, Berlin.
• [13] Carmona, R. and Delarue, F. (2017). Probabilistic Theory of Mean Field Games: Vol. II, Mean Field Games with Common Noise and Master Equations. Stochastic Analysis and Applications, Springer, Berlin.
• [14] Carmona, R., Fouque, J.-P. and Sun, L.-H. (2015). Mean field games and systemic risk. Commun. Math. Sci. 13 911–933.
• [15] Carmona, R. and Zhu, X. (2016). A probabilistic approach to mean field games with major and minor players. Ann. Appl. Probab. 26 1535–1580.
• [16] Cecchin, A. and Fischer, M. (2017). Probabilistic approach to finite state mean field games. Preprint. Available at arXiv:1704.00984.
• [17] Cecchin, A. and Pelino, G. (2019). Convergence, fluctuations and large deviations for finite state mean field games via the master equation. Stochastic Process. Appl. 129 4510–4555.
• [18] Chassagneux, J.-F., Crisan, D. and Delarue, F. (2014). A probabilistic approach to classical solutions of the master equation for large population equilibria. Preprint. Available at arXiv:1411.3009.
• [19] Dawson, D. A. and Gärtner, J. (1987). Large deviations from the McKean–Vlasov limit for weakly interacting diffusions. Stochastics 20 247–308.
• [20] Delarue, F., Lacker, D. and Ramanan, K. (2018). From the master equation to mean field game limit theory: A central limit theorem. Electron. J. Probab. 24 1–54.
• [21] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Springer, Berlin.
• [22] Djellout, H., Guillin, A. and Wu, L. (2004). Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 32 2702–2732.
• [23] Fischer, M. (2014). On the form of the large deviation rate function for the empirical measures of weakly interacting systems. Bernoulli 20 1765–1801.
• [24] Fischer, M. (2017). On the connection between symmetric $N$-player games and mean field games. Ann. Appl. Probab. 27 757–810.
• [25] Fournier, N. and Guillin, A. (2015). On the rate of convergence in Wasserstein distance of the empirical measure. Probab. Theory Related Fields 162 707–738.
• [26] Gangbo, W. and Świȩch, A. (2015). Existence of a solution to an equation arising from the theory of mean field games. J. Differential Equations 259 6573–6643.
• [27] Gozlan, N. (2009). A characterization of dimension free concentration in terms of transportation inequalities. Ann. Probab. 37 2480–2498.
• [28] Gozlan, N. and Léonard, C. (2010). Transport inequalities. A survey. Markov Process. Related Fields 16 635–736.
• [29] Horowitz, J. and Karandikar, R. L. (1994). Mean rates of convergence of empirical measures in the Wasserstein metric. J. Comput. Appl. Math. 55 261–273.
• [30] Lacker, D. (2016). A general characterization of the mean field limit for stochastic differential games. Probab. Theory Related Fields 165 581–648.
• [31] Lacker, D. and Ramanan, K. (2019). Rare Nash equilibria and the price of anarchy in large static games. Math. Oper. Res. 44 400–422.
• [32] Lacker, D. and Zariphopoulou, T. (2017). Mean field and n-agent games for optimal investment under relative performance criteria. Preprint. Available at arXiv:1703.07685.
• [33] Lasry, J.-M. and Lions, P.-L. (2007). Mean field games. Jpn. J. Math. 2 229–260.
• [34] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI.
• [35] Strömberg, T. (2009). A note on the differentiability of conjugate functions. Arch. Math. (Basel) 93 481–485.
• [36] Stroock, D. W. and Varadhan, S. R. S. (1972). On the support of diffusion processes with applications to the strong maximum principle. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory Univ. California Press, Berkeley, CA. 333–359.
• [37] Sznitman, A.-S. (1991). Topics in propagation of chaos. In École D’Été de Probabilités de Saint-Flour XIX—1989. Lecture Notes in Math. 1464 165–251. Springer, Berlin.
• [38] Üstünel, A. S. (2012). Transportation cost inequalities for diffusions under uniform distance. In Stochastic Analysis and Related Topics. Springer Proc. Math. Stat. 22 203–214. Springer, Heidelberg.
• [39] Wang, R., Wang, X. and Wu, L. (2010). Sanov’s theorem in the Wasserstein distance: A necessary and sufficient condition. Statist. Probab. Lett. 80 505–512.