Fitted Q-Learning in Mean-field Games

In the literature, existence of equilibria for discrete-time mean field games has been in general established via Kakutani's Fixed Point Theorem. However, this fixed point theorem does not entail any iterative scheme for computing equilibria. In this paper, we first propose a Q-iteration algorithm to compute equilibria for mean-field games with known model using Banach Fixed Point Theorem. Then, we generalize this algorithm to model-free setting using fitted Q-iteration algorithm and establish the probabilistic convergence of the proposed iteration. Then, using the output of this learning algorithm, we construct an approximate Nash equilibrium for finite-agent stochastic game with mean-field interaction between agents.Comment: 22 page

Mean Field Markov Decision Processes

We consider mean-field control problems in discrete time with discounted reward, infinite time horizon and compact state and action space. The existence of optimal policies is shown and the limiting mean-field problem is derived when the number of individuals tends to infinity. Moreover, we consider the average reward problem and show that the optimal policy in this mean-field limit is $\varepsilon$-optimal for the discounted problem if the number of individuals is large and the discount factor close to one. This result is very helpful, because it turns out that in the special case when the reward does only depend on the distribution of the individuals, we obtain a very interesting subclass of problems where an average reward optimal policy can be obtained by first computing an optimal measure from a static optimization problem and then achieving it with Markov Chain Monte Carlo methods. We give two applications: Avoiding congestion an a graph and optimal positioning on a market place which we solve explicitly