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Abstract: In this paper we study a mean field model for discrete time, finite number ofstates, dynamic games. These models arise in situations that involve a verylarge number of agents moving from state to state according to certainoptimality criteria. The mean field approach for optimal control anddifferential games continuous state and time was introduced by Lasry andLions. Here we consider a discrete version of the problem. Our setting is thefollowing: we assume that there is a very large number of identical agentswhich can be in a finite number of states. Because the number of agents is verylarge, we assume the mean field hypothesis, that is, that the only relevantinformation for the global evolution is the fraction $\pi^n i$ of players ineach state $i$ at time $n$. The agents look for minimizing a running cost,which depends on $\pi$, plus a terminal cost $V^N$. In contrast with optimalcontrol, where usually only the terminal cost $V^N$ is necessary to solve theproblem, in mean-field games both the initial distribution of agents $\pi^0$and the terminal cost $V^N$ are necessary to determine the solutions, that is,the distribution of players $\pi^n$ and value function $V^n$, for $0\leq n\leqN$. Because both initial and terminal data needs to be specified, we call thisproblem the initial-terminal value problem. Existence of solutions isnon-trivial. Nevertheless, following the ideas of Lasry and Lions, we were ableto establish existence and uniqueness, both for the stationary and for theinitial-terminal value problems. In the last part of the paper we prove themain result of the paper, namely the exponential convergence to a stationarysolution of $\pi^0, V^0$, as $N\to \infty$, for the initial-terminal valueproblem with fixed data $\pi^{-N}$ and $V^N$.



Autor: Diogo A. Gomes, Joana Mohr, Rafael R. Souza

Fuente: https://arxiv.org/







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