Analysis of a Finite State Many Player Game Using Its Master Equation

We consider an n-player symmetric stochastic game with weak interactions between the players. Time is continuous and the horizon and the number of states are finite. We show that the value function of each of the players can be approximated by the solution of a partial differential equation called the master equation. Moreover, we analyze the fluctuations of the empirical measure of the states of the players in the game and show that it is governed by a solution to a stochastic differential equation. Finally, we prove the regularity of the master equation, which is required for the above results.

[1]  Peter E. Caines,et al.  Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle , 2006, Commun. Inf. Syst..

[2]  M. Manhart,et al.  Markov Processes , 2018, Introduction to Stochastic Processes and Simulation.

[3]  P. Lions,et al.  The Master Equation and the Convergence Problem in Mean Field Games , 2015, 1509.02505.

[4]  Peter E. Caines,et al.  The Nash certainty equivalence principle and McKean-Vlasov systems: An invariance principle and entry adaptation , 2007, 2007 46th IEEE Conference on Decision and Control.

[5]  Markus Fischer.  On the connection between symmetric $N$-player games and mean field games , 2014, 1405.1345.

[6]  P. Protter.  Stochastic integration and differential equations , 1990 .

[7]  A. Budhiraja,et al.  Rate control under heavy traffic with strategic servers , 2016, The Annals of Applied Probability.

[8]  Vassili Kolokoltsov,et al.  Sensitivity analysis for HJB equations with an application to a coupled backward-forward system , 2013, 1303.6234.

[9]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[10]  Markus Fischer,et al.  Probabilistic Approach to Finite State Mean Field Games , 2017, Applied Mathematics & Optimization.

[11]  R. Carmona,et al.  Finite State Mean Field Games with Major and Minor Players , 2016 .

[12]  D. Lacker.  A general characterization of the mean field limit for stochastic differential games , 2014, 1408.2708.

[13]  Daniel Lacker,et al.  Limit Theory for Controlled McKean-Vlasov Dynamics , 2016, SIAM J. Control. Optim..

[14]  Marie-Therese Wolfram,et al.  Socio-economic applications of finite state mean field games , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[15]  Francois Delarue,et al.  The Master Equation for Large Population Equilibriums , 2014, 1404.4694.

[16]  Alain Bensoussan,et al.  The Master equation in mean field theory , 2014, 1404.4150.

[17]  P. Lions,et al.  Mean field games , 2007 .

[18]  P. Dupuis,et al.  Large Deviation Principle For Finite-State Mean Field Interacting Particle Systems , 2016, 1601.06219.

[19]  D. Gomes,et al.  Continuous Time Finite State Mean Field Games , 2012, 1203.3173.

[20]  P. Cardaliaguet,et al.  Mean Field Games , 2020, Lecture Notes in Mathematics.

[21]  Erhan Bayraktar,et al.  A Numerical Scheme for a Mean Field Game in Some Queueing Systems Based on Markov Chain Approximation Method , 2017, SIAM J. Control. Optim..

[22]  René Carmona,et al.  Probabilistic Analysis of Mean-field Games , 2013 .

[23]  Diogo Gomes,et al.  Dual two-state mean-field games , 2014, 53rd IEEE Conference on Decision and Control.

[24]  Alekos Cecchin,et al.  Convergence, fluctuations and large deviations for finite state mean field games via the Master Equation , 2017, Stochastic Processes and their Applications.