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Tutorial : Mean Field Stochastic Games

A short version of the tutorial is now available.

Content of the tutorial

I-Introduction

  • puceblackShort history, motivations
  • puceblackConnection to mathematical physics, population games, random matrix theory
  • puceblackReplica method, fluid approximation, stochastic approximation
  • puceblackDynamical system approach to large-scale systems
  • puceblackWhat is mean field?
  • puceblackWhy mean field stochastic games?
  • puceblackRelevance in large-scale systems
    • puceblackInternet of things with 2 billions of nodes
    • puceblackNetwork 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
    • puceblackDisruption-tolerant networks with opportunistic meeting in a large population of 20.000.000 nodes
  • puceblackAdvantages and Limitations of Mean Field Approaches

II-Preliminaries

  • puceblackIntroduction to dynamic games (discrete time, continous time)
    • puceblackEvolutionary games
    • puceblackStochastic games
    • puceblackDifference games
    • puceblackDifferential games
    • puceblackDynamic cooperative games

III-Mean field stochastic game models

  • puceblackDiscrete time models
    • puceblackProbabilistic transitions between states
    • puceblackState evolution described by difference equation or inclusion
  • puceblackContinuous time models
    • puceblackDeterministic state dynamics
    • puceblackStochastic differential equation
    • puceblackImperfect state observation

IV-Asymptotic analysis

    • puceblackMean field convergence
    • puceblackPropagation of chaos and asymptotic indistinguishability per class/type
      • puceblackNon-commutativity in stationary regime, double limit
      • puceblackHow to stabilize a controlled mean field game dynamics?
    • puceblackMean field optimality :Backward-Forward optimality equation
      • puceblackMean field Hamilton-Jacobi-Bellman-Fleming equation
      • puceblackFokker-Planck-Kolmogorov equation
      • puceblackLink with optimal control
      • puceblackBellman-Shapley optimality coupled with Kolmogorov  forward equation
      • puceblackMcKean-Vlasov equations
      • puceblackBackward-forward FPK-McV: existence, uniqueness conditions
    • puceblackMean field equilibria, epsilon-equilibria, population-size dependent approximation, mean field response
      • puceblackSufficiency conditions for Existence of solution
      • puceblackSufficiency conditions for uniqueness of regular solutions (HJBF/FPK-McV)
    • puceblackMean field learning
      • puceblackHybrid learning in large populations
      • puceblackQ-learning for coupled BS-K equations
      • puceblackH-learning for coupled HJBF-K equations

V-Applications

      • puceblackInformation dissemination in opportunistic networks
      • puceblackAccess control in stochastic networks
      • puceblackEvolutionary biology: resource competition and aggregative interaction
      • puceblackPower allocation under log-normal channels
      • puceblackMean field predictor in noisy environment
      • puceblackControlled McKean-Vlasov equations
      • puceblackHow mobility influences the energy consumption in heterogeneous ad hoc networks?
      • puceblackMobility and power saving in wireless networks (link with dynamic stochastic geometry)
      • puceblackMean field dynamics in chemical reaction networks

V-Extensions

      • puceblackRisk-sensitive mean field stochastic games: Markov games
        • puceblackMultiplicative Poisson equation, Multiplicative dynamic programming
      • puceblackRisk-sensitive mean field stochastic games: difference games
      • puceblackRisk-sensitive mean field stochastic games: differential games
      • puceblackHybrid mean field stochastic games
      • puceblackRobust 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.

    • puceblackSelten, 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
    • puceblackCorchon, L. (1994): ``Comparative Statics for Aggregative Games. The Strong Concavity Case'', Mathematical Social Sciences 28, 151-165
    • puceblackMinyi 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. 
    • puceblackJean-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
    • puceblackMichel Benaïm, J. Y. LeBoudec, A class of mean field interaction models for computer and communication systems, Performance Evaluation, Volume 65 Issue 11-12, November, 2008 
    • puceblackOlivier Gueant, A reference case for mean field games models, Journal de Mathématiques Pures et Appliques, Volume 92, Issue 3, September 2009, Pages 276-294
    • puceblackY. Achdou, I.C. Dolcetta, Mean field games: numerical methods, SIAM Journal on Numerical Anal., 48(3), 1136-1162, 2010
    • puceblackDiogo A. Gomes Joana Mohr and Rafael Rigao Souza, Discrete time, finite state space mean field games,Journal de Mathematiques Pures et AppliquesVolume 93, Issue 3, March 2010, Pages 308-328 
    • puceblackC. Dogbe, Modeling crowds by the mean-field limit approach.   Mathematical and Computer Modelling, 52, (2010), 1506–1520
    • puceblackA. 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
    • puceblackGabriel Y. Weintraub, Lanier Benkard, and Benjamin Van Roy, "Markov Perfect Industry Dynamics with Many Firms." Econometrica (Nov 2008): 1375-1411.
    • puceblackTao Lia, Ji-Feng Zhang, Decentralized tracking-type games for multi-agent systems with coupled ARXmodels: Asymptotic Nash equilibria, Automatica 44 (2008) 713-725
    • puceblackS. Adlakha and R. Johari . Mean field equilibrium in dynamic games with complementarities, IEEE Conference on Decision and Control (CDC), 2010,
    • puceblackHuibing Yin,   Mehta, P.G.,   Meyn, S.P.,   Shanbhag, U.V., Synchronization of coupled oscillators is a game, American Control Conference (ACC), 2010 
    • puceblackBingchang Wang and Ji-Feng Zhang. Mean field games for large population stochastic multi-agent systems with markov jumps. Proceedings of the 29th Chinese ControlConference, 4572-4577, Beijing, China, July 29-31 2010
    • etc