Submodular mean field games: Existence and approximation of solutions

We study mean field games with scalar Ito-type dynamics and costs that are submodular with respect to a suitable order relation on the state and measure space. The submodularity assumption has a number of interesting consequences. Firstly, it allows us to prove existence of solutions via an application of Tarski's fixed point theorem, covering cases with discontinuous dependence on the measure variable. Secondly, it ensures that the set of solutions enjoys a lattice structure: in particular, there exist a minimal and a maximal solution. Thirdly, it guarantees that those two solutions can be obtained through a simple learning procedure based on the iterations of the best-response-map. The mean field game is first defined over ordinary stochastic controls, then extended to relaxed controls. Our approach allows also to treat a class of submodular mean field games with common noise in which the representative player at equilibrium interacts with the (conditional) mean of its state's distribution.

[1]  Brigitte Maier,et al.  Supermodularity And Complementarity , 2016 .

[2]  A note on stochastic dominance and compactness , 2019, 1909.03740.

[3]  Martino Bardi,et al.  On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games , 2017, ESAIM: Control, Optimisation and Calculus of Variations.

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

[5]  Marco Fuhrman,et al.  Existence of Optimal Stochastic Controls and Global Solutions of Forward-Backward Stochastic Differential Equations , 2004, SIAM J. Control. Optim..

[6]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[7]  Pierre Cardaliaguet,et al.  Learning in mean field games: The fictitious play , 2015, 1507.06280.

[8]  ALFRED MÜLLER,et al.  Stochastic Order Relations and Lattices of Probability Measures , 2006, SIAM J. Optim..

[9]  M. Nendel.  A note on stochastic dominance, uniform integrability and lattice properties , 2020, Bulletin of the London Mathematical Society.

[10]  D. M. Topkis.  Equilibrium Points in Nonzero-Sum n-Person Submodular Games , 1979 .

[11]  R. Amir.  Supermodularity and Complementarity in Economics: An Elementary Survey , 2003 .

[12]  T. K. Chandra.  de La Vallée Poussin’s theorem, uniform integrability, tightness and moments , 2015 .

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

[14]  Daniel Lacker,et al.  Mean field games via controlled martingale problems: Existence of Markovian equilibria , 2014, 1404.2642.

[15]  Rinel Foguen Tchuendom,et al.  Uniqueness for Linear-Quadratic Mean Field Games with Common Noise , 2018, Dyn. Games Appl..

[16]  Piotr Wiecek.  Total Reward Semi-Markov Mean-Field Games with Complementarity Properties , 2017, Dyn. Games Appl..

[17]  Rene Carmona,et al.  Mean Field Games of Timing and Models for Bank Runs , 2016, 1606.03709.

[18]  X. Vives.  Oligopoly Pricing: Old Ideas and New Tools , 1999 .

[19]  Stochastic Orders , 2008 .

[20]  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..

[21]  Arthur Stoddart,et al.  Existence of optimal controls , 1967 .

[22]  T. Kamae,et al.  Stochastic Inequalities on Partially Ordered Spaces , 1977 .

[23]  R. Carmona,et al.  Control of McKean–Vlasov dynamics versus mean field games , 2012, 1210.5771.

[24]  R. Amir.  Cournot Oligopoly and the Theory of Supermodular Games , 1996 .

[25]  Alain Bensoussan,et al.  Linear-Quadratic Mean Field Games , 2014, Journal of Optimization Theory and Applications.

[26]  F. Delarue,et al.  Selection of equilibria in a linear quadratic mean-field game , 2018, Stochastic Processes and their Applications.

[27]  Ramesh Johari,et al.  Mean Field Equilibrium in Dynamic Games with Strategic Complementarities , 2013, Oper. Res..

[28]  Matti Vihola,et al.  Stochastic order characterization of uniform integrability and tightness , 2011 .

[29]  R. Carmona,et al.  Mean field games with common noise , 2014, 1407.6181.

[30]  Markus Fischer,et al.  On the Convergence Problem in Mean Field Games: A Two State Model without Uniqueness , 2019, SIAM J. Control. Optim..

[31]  A. Tarski.  A LATTICE-THEORETICAL FIXPOINT THEOREM AND ITS APPLICATIONS , 1955 .

[32]  François Delarue,et al.  Probabilistic Theory of Mean Field Games with Applications I: Mean Field FBSDEs, Control, and Games , 2018 .

[33]  R. Carmona.  Lectures on Bsdes, Stochastic Control, and Stochastic Differential Games with Financial Applications , 2016 .

[34]  KarouiNicole El,et al.  Compactification methods in the control of degenerate diffusions: existence of an optimal control , 1987 .

[35]  Paul R. Milgrom,et al.  Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities , 1990 .

[36]  William H. Sandholm,et al.  ON THE GLOBAL CONVERGENCE OF STOCHASTIC FICTITIOUS PLAY , 2002 .

[37]  X. Vives.  Nash equilibrium with strategic complementarities , 1990 .