Some remarks on mean field games

Abstract

We study in this paper three aspects of Mean Field Games. The first one is the case when the dynamics of each player depend on the strategies of the other players. The second one concerns the modeling of "noise" in discrete space models and the formulation of the Master Equation in this case. Finally, we show how Mean Field Games reduce to agent based models when the intertemporal preference rate goes to infinity, i.e. when the anticipation of the players vanishes. 2

Strategic advantages in mean field games with a major player

“…the limit λ → ∞. In [4], we studied the same type of limit without a major player. In the present case, the same type of limits occurs : U → 0 as λ → ∞ and the system reduces to…”

Convergence of some Mean Field Games systems to aggregation and flocking models

“…Our results are inspired on one hand by the last part of [8], in which the authors show how to derive a McKean-Vlasov equation from a mean field game system and, on the other hand, by [21] (see also [4]) which discusses how multi-agent control problems in which the players have limiting anticipation converge to aggregation models. Let us briefly recall the content of both papers.…”

“…Let us briefly recall the content of both papers. In [8], the authors study MFG systems of the form $ & %´B t u λ´ν ∆u λ`H px, Du λ , m λ ptqq`λu " 0 in R dˆp 0, T q, B t m λ´ν ∆m λ´d ivpm λ D p Hpx, Du λ , m λ ptqq " 0 in R dˆp 0, T q, u λ pT, xq " u T pxq, m λ p0q " m 0 in R d . (1) Here ν ą 0 is fixed, λ ą 0 is a large parameter which describes the impatience of the players and H " Hpx, p, mq is the Hamiltonian of the problem which includes interaction terms between the players.…”

Markov Distributional Equilibrium Dynamics in Games with Complementarities and No Aggregate Risk

“…A second approach (often used in conjunction with aggregation and stationarity) is to simplify interaction in the equilibrium by imposing some behavioral features in its definition. Such approaches include notions of oblivious equilibria (as in Lasry and Lions, 2007, Achdou et al, 2014, Bertucci et al, 2018, Light and Weintraub, 2019, Achdou et al, 2020, mean-field equilibria (as in Weintraub et al, 2008, Adlakha et al, 2015, and Ifrach and Weintraub, 2016, or imagined-continuum equilibria (as in Kalai and Shmaya, 2018), among others. 3 In this paper, we argue that such simplifications need not play a crucial role if one wants to analyze the equilibrium dynamics in a class of games with strategic complementarities we consider.…”