scholarly journals Viability analysis of the first-order mean field games

2020 ◽  
Vol 26 ◽  
pp. 33
Author(s):  
Yurii Averboukh
Keyword(s):  
Mean Field ◽  
Initial Distribution ◽  
Initial Time ◽  
Mean Field Games ◽  
Sufficient Condition ◽  
Mean Field Game ◽  
First Order ◽  
Viability Analysis ◽  
A Value ◽  
Object Of Study

In the paper, we examine the dependence of the solution of the deterministic mean field game on the initial distribution of players. The main object of study is the mapping which assigns to the initial time and the initial distribution of players the set of expected rewards of the representative player corresponding to solutions of mean field game. This mapping can be regarded as a value multifunction. We obtain the sufficient condition for a multifunction to be a value multifunction. It states that if a multifunction is viable with respect to the dynamics generated by the original mean field game, then it is a value multifunction. Furthermore, the infinitesimal variant of this condition is derived.

The Second-Order Master Equation

2019 ◽  
pp. 128-158
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Master Equation ◽  
Mean Field ◽  
Second Order ◽  
Well Posedness ◽  
Mean Field Game ◽  
First Order ◽  
Common Noise ◽  
System P ◽  
The Mean

This chapter investigates the second-order master equation with common noise, which requires the well-posedness of the mean field game (MFG) system. It also defines and analyzes the solution of the master equation. The chapter explains the forward component of the MFG system that is recognized as the characteristics of the master equation. The regularity of the solution of the master equation is explored through the tangent process that solves the linearized MFG system. It also analyzes first-order differentiability and second-order differentiability in the direction of the measure on the same model as for the first-order derivatives. This chapter concludes with further description of the derivation of the master equation and well-posedness of the stochastic MFG system.

scholarly journals Long-Time Behavior of First-Order Mean Field Games on Euclidean Space

2019 ◽  
Vol 10 (2) ◽  
pp. 361-390
Author(s):  
Piermarco Cannarsa ◽  
Wei Cheng ◽  
Cristian Mendico ◽  
Kaizhi Wang
Keyword(s):  
Euclidean Space ◽  
Mean Field ◽  
Mean Field Games ◽  
Time Behavior ◽  
Long Time Behavior ◽  
First Order ◽  
Long Time

scholarly journals First-order, stationary mean-field games with congestion

2018 ◽  
Vol 173 ◽  
pp. 37-74 ◽  
Author(s):  
David Evangelista ◽  
Rita Ferreira ◽  
Diogo A. Gomes ◽  
Levon Nurbekyan ◽  
Vardan Voskanyan
Keyword(s):  
Mean Field ◽  
Mean Field Games ◽  
First Order

scholarly journals Schrödinger approach to Mean Field Games with negative coordination

2020 ◽  
Vol 9 (4) ◽  
Author(s):  
Thibault Bonnemain ◽  
Thierry Gobron ◽  
Denis Ullmo
Keyword(s):  
Mean Field ◽  
Time Limit ◽  
External Potential ◽  
Mean Field Games ◽  
Relative Importance ◽  
Mean Field Game ◽  
Non Linear ◽  
The Mean ◽  
The Way

Mean Field Games provide a powerful framework to analyze the dynamics of a large number of controlled agents in interaction. Here we consider such systems when the interactions between agents result in a negative coordination and analyze the behavior of the associated system of coupled PDEs using the now well established correspondence with the non linear Schrödinger equation. We focus on the long optimization time limit and on configurations such that the game we consider goes through different regimes in which the relative importance of disorder, interactions between agents and external potential vary, which makes possible to get insights on the role of the forward-backward structure of the Mean Field Game equations in relation with the way these various regimes are connected.

scholarly journals The selection problem for some first-order stationary Mean-field games

2020 ◽  
Vol 15 (4) ◽  
pp. 681-710
Author(s):  
Diogo A. Gomes ◽  
◽  
Hiroyoshi Mitake ◽  
Kengo Terai ◽  
Keyword(s):  
Mean Field ◽  
Selection Problem ◽  
Mean Field Games ◽  
First Order

scholarly journals Mean Field Games Flock! The Reinforcement Learning Way

2021 ◽  
Author(s):  
Sarah Perrin ◽  
Mathieu Laurière ◽  
Julien Pérolat ◽  
Matthieu Geist ◽  
Romuald Élie ◽  
...  
Keyword(s):  
Reinforcement Learning ◽  
Nash Equilibrium ◽  
Population Distribution ◽  
Mean Field ◽  
High Dimensional ◽  
Mean Field Games ◽  
Fictitious Play ◽  
Mean Field Game ◽  
Best Response

We present a method enabling a large number of agents to learn how to flock. This problem has drawn a lot of interest but requires many structural assumptions and is tractable only in small dimensions. We phrase this problem as a Mean Field Game (MFG), where each individual chooses its own acceleration depending on the population behavior. Combining Deep Reinforcement Learning (RL) and Normalizing Flows (NF), we obtain a tractable solution requiring only very weak assumptions. Our algorithm finds a Nash Equilibrium and the agents adapt their velocity to match the neighboring flock’s average one. We use Fictitious Play and alternate: (1) computing an approximate best response with Deep RL, and (2) estimating the next population distribution with NF. We show numerically that our algorithm can learn multi-group or high-dimensional flocking with obstacles.

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