## Tutorial : Mean Field Stochastic Games

**A short version of the tutorial is now available.**

**Content of the tutorial**

# I-Introduction

*Short history, motivations**Connection to mathematical physics, population games, random matrix theory**Replica method, fluid approximation, stochastic approximation**Dynamical system approach to large-scale systems**What is mean field?**Why mean field stochastic games?**Relevance in large-scale systems**Internet of things with 2 billions of nodes**Network of sensors deployed along a volcano, collecting large quantities of data to monitor seismic activities where transmissions are from relay-node to relay-node until finally delivered to a base station**Disruption-tolerant networks with opportunistic meeting in a large population of 20.000.000 nodes*

*Advantages and Limitations of Mean Field Approaches*

# II-Preliminaries

*Introduction to dynamic games (discrete time, continous time)**Evolutionary games**Stochastic games**Difference games**Differential games**Dynamic cooperative games*

# III-Mean field stochastic game models

- Discrete time models
- Probabilistic transitions between states
*State evolution described by difference equation or inclusion*

- Continuous time models
- Deterministic state dynamics
- Stochastic differential equation
*Imperfect state observation*

# IV-Asymptotic analysis

- Mean field convergence
- Deterministic mean field limit
- Stochastic mean field limit / noisy mean field with diffusion
*Reading: (1) Bruno de Finetti (1931), (2) Edwin Hewitt and**Leonard Jimmie Savage (1955), (3) David Aldous (1983), (4) Olav Kallenberg (1989), (5) Alain-Sol Sznitman (1989), (6) Carl Graham*

- Propagation of chaos and asymptotic indistinguishability per class/type
- Non-commutativity in stationary regime, double limit
*How to stabilize a controlled mean field game dynamics?*

- Mean field optimality :Backward-Forward optimality equation
- Mean field Hamilton-Jacobi-Bellman-Fleming equation
- Fokker-Planck-Kolmogorov equation
- Link with optimal control
- Bellman-Shapley optimality coupled with Kolmogorov forward equation
- McKean-Vlasov equations
*Backward-forward FPK-McV: existence, uniqueness conditions*

- Mean field equilibria, epsilon-equilibria, population-size dependent approximation, mean field response
*Sufficiency conditions for Existence of solution**Sufficiency conditions for uniqueness of regular solutions (HJBF/**FPK-McV)*

*Mean field learning**Hybrid learning in large populations**Q-learning for coupled BS-K equations**H-learning for coupled HJBF-K equations*

# V-Applications

*Information dissemination in opportunistic networks**Access control in stochastic networks*- Evolutionary biology: resource competition and aggregative interaction
*Power allocation under log-normal channels**Mean field predictor in noisy environment**Controlled McKean-Vlasov equations**How mobility influences the energy consumption in heterogeneous ad hoc networks?**Mobility and power saving in wireless networks (link with dynamic stochastic geometry)**Mean field dynamics in chemical reaction networks*

# V-Extensions

*Risk-sensitive mean field stochastic games: Markov games**Multiplicative Poisson equation, Multiplicative dynamic programming*

*Risk-sensitive mean field stochastic games: difference games**Risk-sensitive mean field stochastic games: differential games**Hybrid mean field stochastic games**Robust mean field stochastic games*

## V-Related references on MEAN FIELD STOCHASTIC GAMES

Probably very imcomplete.If you think something should be includes, please write me an e-mail.

*Selten, R. (1970), Preispolitik der Mehrprodktenunternehmung in der statischen Theorie, Springer-Verlag.**B. Jovanovic and R. W. Rosenthal. Anonymous sequential games. Journal of Mathematical Economics, 17:77-87, 1988**Corchon, L. (1994): ``Comparative Statics for Aggregative Games. The Strong Concavity Case'', Mathematical Social Sciences 28, 151-165**Minyi Huang; Caines PE; Malhame, R.P.; Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions, Proceedings of 42nd IEEE Conference on Decision and Control, 2003.**Jean-Michel Lasry and Pierre-Louis Lions, Mean field games, Japanese Journal of Mathematics, Volume 2, Number 1, 229-260, DOI: 10.1007/s11537-007-0657-8, Special Feature: The 1st Takagi Lecture**Michel Benaïm, J. Y. LeBoudec, A class of mean field interaction models for computer and communication systems,**Olivier Gueant, A reference case for mean field games models, Journal de Mathématiques Pures et Appliques, Volume 92, Issue 3, September 2009, Pages 276-294**Y. Achdou, I.C. Dolcetta, Mean field games: numerical methods, SIAM Journal on Numerical Anal., 48(3), 1136-1162, 2010**Diogo A. Gomes Joana Mohr and Rafael Rigao Souza, Discrete time, finite state space mean field games,Journal de Mathematiques Pures et Appliques***,**Volume 93, Issue 3, March 2010, Pages 308-328*C. Dogbe, Modeling crowds by the mean-field limit approach. Mathematical and Computer Modelling, 52, (2010), 1506–1520**A. Lachapelle, J. Salomon and G. Turinici, Computation of Mean Field Equilibria in Economics, Math. Models, Methods in Applied Sciences, Vol 20, Issue 4, 2010, DOI: 10.1142/S0218202510004349**Gabriel Y. Weintraub, Lanier Benkard, and Benjamin Van Roy, "Markov Perfect Industry Dynamics with Many Firms." Econometrica (Nov 2008): 1375-1411.**Tao Lia, Ji-Feng Zhang, Decentralized tracking-type games for multi-agent systems with coupled**S. Adlakha and R. Johari . Mean field equilibrium in dynamic games with complementarities, IEEE Conference on Decision and Control (CDC), 2010,**Huibing Yin, Mehta, P.G., Meyn, S.P., Shanbhag, U.V., Synchronization of coupled oscillators is a game, American Control Conference (ACC), 2010**Bingchang Wang and Ji-Feng Zhang. Mean field games for large population stochastic multi-agent systems with**etc*