In this paper, zerosum meanfield type games (ZSMFTG) with linear dynamics and quadratic cost are studied under infinitehorizon discounted utility function. ZSMFTG are a class of games in which two decision makers whose utilities sum to zero, compete to influence a large population of indistinguishable agents. In particular, the case in which the transition and utility functions depend on the state, the action of the controllers, and the mean of the state and the actions, is investigated. The optimality conditions of the game are analysed for both openloop and closedloop controls, and explicit expressions for the Nash equilibrium strategies are derived. Moreover, two policy optimization methods that rely on policy gradient are proposed for both modelbased and samplebased frameworks. In the modelbased case, the gradients are computed exactly using the model, whereas they are estimated using MonteCarlo simulations in the samplebased case. Numerical experiments are conducted to show the convergence of the utility function as well as the two players' controls.
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Carmona, Rene ; Lauriere, Mathieu ; Tan, Zongjun ( , ArXivorg)We investigate reinforcement learning for mean field control problems in discrete time, which can be viewed as Markov decision processes for a large number of exchangeable agents interacting in a mean field manner. Such problems arise, for instance when a large number of robots communicate through a central unit dispatching the optimal policy computed by minimizing the overall social cost. An approximate solution is obtained by learning the optimal policy of a generic agent interacting with the statistical distribution of the states of the other agents. We prove rigorously the convergence of exact and modelfree policy gradient methods in a meanfield linearquadratic setting. We also provide graphical evidence of the convergence based on implementations of our algorithms.more » « less

Carmona, Rene ; Graves, Christy V. ; Tan, Zongjun ( , ESAIM. Proceedings and surveys)The price of anarchy, originally introduced to quantify the inefficiency of selfish behavior in routing games, is extended to mean field games. The price of anarchy is defined as the ratio of a worst case social cost computed for a mean field game equilibrium to the optimal social cost as computed by a central planner. We illustrate properties of such a price of anarchy on linear quadratic extended mean field games, for which explicit computations are possible. A sufficient and necessary condition to have no price of anarchy is presented . Various asymptotic behaviors of the price of anarchy are proved for limiting behaviors of the coefficients in the model and numerics are presented .more » « less