An example of multiple mean field limits in ergodic differential games
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Nonlinear Differential Equations and Applications NoDEA
An example of multiple mean field limits in ergodic differential games Pierre Cardaliaguet and Catherine Rainer Abstract. We present an example of symmetric ergodic N -players differential games, played in memory strategies on the position of the players, for which the limit set, as N → +∞, of Nash equilibrium payoffs is large, although the game has a single mean field game equilibrium. This example is in sharp contrast with a result by Lacker (On the convergence of closed-loop Nash equilibria to the mean field game limit, 2018. arXiv:1808.02745) for finite horizon problems. Mathematics Subject Classification. 91A13, 35Q91, 91A15. Keywords. Mean field games, Differential games, Ergodic control, Folk Theorem.
1. Introduction In this note we want to underline the role of information in mean field games. For this we study the limit of Nash equilibrium payoffs in ergodic N -player stochastic differential games as the number N of players tends to infinity. Since the pioneering works by Lasry and Lions [24] (see also [19]) differential games with many agents have attracted a lot of attention under the terminology of mean field games. We also refer the reader to the monographs [3,10]. Mean field games are nonatomic dynamic games, in which the agents interact through the population density. Here we investigate in what extent the mean field game problem is the limit of the N -person differential games. This question is surprisingly difficult in general and is not completely understood so far in full generality. When, in the N -player game, players play in open-loop (i.e., observe only their own position but not the position of the other players), the mean field limit is a mean field game. The first result in that direction goes back to [24] in the ergodic setting (see also [1,16] for statements in the same direction); extensions to the non Markovian setting can be found in Fischer [17] while Lacker gave a complete characterization of the limit [22] (see also [25] for an exit time 0123456789().: V,-vol
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problem). Note that these (often technically difficult) results are not entirely surprising since, in the N-player game as well as in the mean field game, the players do not observe the position of the other players: therefore there is no real change of nature between the N -player problem and the mean field game. We are interested here in the N -player games in which players observe each other, the so-called closed-loop regime. In this setting, the mean field limit is much less understood and one possesses only partial results. In general, one formalizes the closed-loop Nash equilibria in the N -person game by a PDE (the Nash system) which describes the fact that players react in function of the current position of all the other players. The first convergence result in this setting [6] states that, in the finite horizon problem and under a suitable monotonicity assumption, the solution of the Nash system converges to a MFG equilibrium. The conve
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