4 Presentations

• 
  09:00 AM - 09:25 AM

  A Hybrid Optimal Control Approach to Mean Field Games with Switching and Stopping Strategies

  • Dena Firoozi, presenter, McGill University
  • Ali Pakniyat, University of Michigan
  • Peter Caines, McGill University

  A novel framework is presented that combines Mean Field Games (MFG) and Hybrid Optimal Control (HOC) theory to obtain a unique Epsilon-Nash equilibrium for a non-cooperative game with stopping times. We consider the case where there exists one major agent with a significant influence on the system together with a large number of minor agents within two subpopulations, each with individually asymptotically negligible effect on the whole system. Each agent has stochastic linear dynamics with quadratic costs, and the agents are coupled in their dynamics by the average state of minor agents (i.e. the empirical mean field). The hybrid feature enters via the indexing by discrete states: (i) the switching of the major agent or (ii) the cessation of the feedback control action of one or both subpopulations of minor agents. Optimal switchings and stopping time strategies together with best response control actions for, respectively, the major agent and all minor agents are established with respect to their individual cost criteria by an application of LQG HOC theory.

• 
  09:25 AM - 09:50 AM

  On Mean Field Games with myopic players

  • Charafeddine Mouzouni, presenter, Ecole Centrale Lyon

  In this talk we shall present a new decision making mechanism in Mean Field Games. At each instant, agents set a strategy which optimizes their expected future cost by assuming their environment as immutable. As the system evolves, the players observe the evolution of the system and adapt to their new environment without anticipating. We present the model and the resulting system of equations. We give the main ideas for proving well-posedness for the system of equations, and show how to derive this system from N-players stochastic differential games models. In addition, we show under certain assumptions, that the myopic population can self-organize, and converge exponentially fast to the ergodic mean field games equilibrium.

• 
  09:50 AM - 10:15 AM

  Mean-field game price models

  • Diogo Gomes, presenter, KAUST
  • Levon Nurbekyan, KAUST
  • Joao Saude, CMU

  Here, we introduce a price-formation model for electricity markets where a large number of small players can store and trade electricity.
  Our model is a constrained mean-field game (MFG) where the price is a Lagrange multiplier for the supply vs. demand balance condition.
  Under mild conditions, we prove the uniqueness of the solution. Moreover, we establish several estimates for the solutions.
  Next, we examine model with finitely many agents and linear-quadratic models that have explicit solutions.
  Finally, we develop numerical methods and illustrate the behavior of the system numerically.

• 
  10:15 AM - 10:40 AM

  Risk-sensitive mean field games via the stochastic maximum principle

  • Tamer Başar, presenter, University of Illinois at Urbana-Champaign
  • Jun Moon, Ulsan National Institute of Science and Technology

  In this paper, we address the Nash equilibria of general stochastic nonzero-sum games with risk-sensitive finite-horizon cost for each player in the high-population regime. The state dynamics of each player are described by a nonlinear Itô stochastic differential equation, driven by the control of the corresponding player, the mean field term (that is common to all players) which is the empirical distribution of states of all (say, N) players, in addition to the Brownian motion, independent across players. The objective function of each player is the expected value of exponentiated integral cost that involves the state and control of the corresponding player as well as the mean field term. Hence, the coupling of the players in both state dynamics and objective functions is through the mean field term. The information available to the players is adapted closed-loop, both centralized and decentralized.

  As opposed to earlier work on this class problems based on the Hamilton-Jacobi-Bellman theory, our approach here is based on the stochastic maximum principle, which involves two sequential steps: (i) solving the generic risk-sensitive stochastic optimal control problem for a fixed probability measure (representing the mean field term), and (ii) solving the associated fixed-point problem. For step (i), we use the risk-sensitive maximum principle to obtain the optimal solution, which is characterized in terms of an associated forward-backward stochastic differential equation (FBSDE). In step (ii), we solve for the probability law induced by the state process driven by the optimal control of step (i). In particular, we show that the conditions of Schauder's fixed point theorem are satisfied, and hence that step (ii) of the problem is solvable. Then, we prove that the set of N optimal decentralized (and distributed) controls obtained from steps (i) and (ii) constitutes an approximate Nash equilibrium (more precisely, -Nash equilibrium) for the N player risk-sensitive stochastic differential game, where converges to zero as N goes to infinity at the rate of O(1/N^{1/(n+4)}). The convergence result follows from the fact that the 2-Wasserstein distance between the empirical distribution (or the mean field) of the N players and the fixed point of the law in step (ii) converges to zero as N goes to infinity at that rate, in view of the propagation of chaos and the law of large numbers. Finally, we discuss extensions to heterogeneous (non-symmetric) risk-sensitive mean field games, that is games where the players may have different models (in terms of both dynamics and cost functions).