An approximate Nash equilibrium for pure jump Markov games of mean-field-type on continuous state space

Abstract

We investigate mean-field games from the point of view of a large number of indistinguishable players, which eventually converges to infinity. The players are weakly coupled via their empirical measure. The dynamics of the states of the individual players is governed by a non-autonomous pure jump type semi group in a Euclidean space, which is not necessarily smoothing. Investigations are conducted in the framework of non-linear Markov processes. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics. In the limit as the number N of players goes to infinity this leads to a jump-type analog of the well-known non-linear McKean-Vlasov dynamics. The case where one player has an individual preference different from the ones of the remaining players is also covered. The two results combined reveal an epsilon-Nash Equilibrium for the Nplayer games.Mathematics Subject Classification (2010): 91A22, 91A13, 91A15, 60J75, 60J25.

Regularity and Sensitivity for McKean-Vlasov Type SPDEs Generated by Stable-like Processes

“…By Lemma 2 these derivatives are expressed in terms of the derivatives of the nonstochastic equation (7) or (11). For nonstochastic equations of this kind the sensitivity was obtained in our previous paper [20], which implies the point-wise (for almost all trajectories W t ) sensitivity of (3).…”

“…For the case of constant correlations σ com we shall be able to get bounds that are deterministic. To this end, we have to look at the main stages of the proof of the sensitivity of (7) and (11) and to see exactly how the estimates depend on the coefficients. The key point to note is that, sinceb andã are obtained by the shifting of b and a, the assumptions (C1)-(C3) on b and a are equivalent to the same assumptions onb andã with the same bounds on the norms in all spaces involved.…”

“…There is an extensive literature on properties of McKean-Vlasov SPDEs (see e.g. [1][2][3][4][5][6][7][8][9][10][11][12] and references therein), based on the diffusion proccesses. The well-posedness of the McKean-Vlasov SPDE was shown in [1] in the class of L 2 -functions, and for measures in [2], though under an additional monotonicity assumption.…”

On stationary fractional mean field games

“…In an equilibrium regime, the corresponding density of the average player is stable as time goes to +∞, and coincides with the population density m. From a PDE point of view, this equilibrium configuration is characterized by a system of a fractional Hamilton-Jacobi equation with Hamiltonian H given by the Legendre transform of L, coupled with a fractional stationary Fokker-Planck equation describing the long-time distribution of all agents, moving according to the control which minimizes the long time average cost (see [23], [20]). We recall that MFG with jumps have been very recently considered in the literature by using a completely different approach based on probabilistic techniques in [6], where the theory of non-linear Markovian propagators is used, and in [7], where the players control the intensity of jumps.…”

On Mean Field Games with Common Noise based on Stable-Like Processes

“…Mean field games with common noise present a quickly developing part of the mean field game theory. The theory of mean field games was initiated by Lasry-Lions [28] and Huang-Malhame-Caines [19,17,18], see [5,6,14,15,7,8,3,4] for recent surveys, as well as [9,10,16,2] and references therein. Notice that there is quite an extensive literature on the mean field games with common noise (e.g.…”