October 7th, 2021, 14h00, room It will be online. Subscribe to the mailing list [here](https://listes.gipsa-lab.grenoble-inp.fr/sympa/subscribe/gaia-seminaire) to receive the connection details..
Bruno Gaujal (INRIA)
Title: Discrete Mean Field Games: Existence of Equilibria and Convergence (joint for with Nicolas Gast and Josu Doncel)
Abstract: Mean field games have been introduced by Lasry and Lions as well as Huang, Caines and Malhame in 2006 to model interactions between a large number of strategic agents (players) and have had a large success ever since. Most of the literature concerns continuous state spaces and describes a mean field game as a coupling between a Hamilton-Jacobi-Bellman equation with a Fokker- Planck equation. Here, we are interested in presenting mean field games with a finite number of states and finite number of actions per player. In this case, the analog of the Hamilton-Jacobi- Bellman equation is a Bellman equation and the discrete version of the Fokker- Planck equation is a Kolmogorov equation. The models we present in this seminar, both in the synchronous and asynchronous cases includes non-linear dynamics with explicit interactions between players. This covers several natural phenomena such as information/infection propagation or resource congestion. We show that the only requirement needed to guarantee the existence of a Mean Field Equilibrium in mixed strategies is that the cost is continuous with respect to the population distribution (convexity is not needed). This result nicely mimics the conditions for existence of a Nash equilibrium in the simpler case of static population games. The second part of the seminar concerns convergence of finite games to mean field limits.We show that a mean field equilibrium is always an ε-approximation of an equilibrium of a corresponding game with a finite number N of players, where ε goes to 0 when N goes to infinity. This is the discrete version of similar results in continuous games. However, we show also that not all equilibria for the finite version converge to a Nash equilibrium of the mean field limit of the game. We provide several counter- examples to illustrate this fact. They are all based on the following idea: The “tit for tat” principle allows one to define many equilibria in repeated games with N players. However, when the number of players is infinite, the deviation of a single player is not visible by the population that cannot punish him in retaliation for her deviation. This implies that while the games with N players may have many equilibria, as stated by the folk theorem, this may not be the case for the limit game. This fact is well-known for large repeated games (the Anti-folk Theorem). However, up to our knowledge, these results had not yet been investigated in mean field games.