scholarly journals Learning in mean field games: The fictitious play

2017 ◽  
Vol 23 (2) ◽  
pp. 569-591 ◽  
Author(s):  
Pierre Cardaliaguet ◽  
Saeed Hadikhanloo
Keyword(s):  
Differential Games ◽  
Mean Field ◽  
Mean Field Games ◽  
Fictitious Play ◽  
Mean Field Game ◽  
Interacting Agents ◽  
Learning Procedure ◽  
Equilibrium Configurations ◽  
The Mean

Mean Field Game systems describe equilibrium configurations in differential games with infinitely many infinitesimal interacting agents. We introduce a learning procedure (similar to the Fictitious Play) for these games and show its convergence when the Mean Field Game is potential.

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 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.

scholarly journals A Probabilistic Approach to Extended Finite State Mean Field Games

2021 ◽  
Author(s):  
René Carmona ◽  
Peiqi Wang
Keyword(s):  
Nash Equilibrium ◽  
Continuous Time ◽  
Probabilistic Approach ◽  
Mean Field ◽  
Mean Field Games ◽  
Weak Formulation ◽  
Mean Field Game ◽  
Alternative Description ◽  
Finite State ◽  
The Mean

We develop a probabilistic approach to continuous-time finite state mean field games. Based on an alternative description of continuous-time Markov chains by means of semimartingales and the weak formulation of stochastic optimal control, our approach not only allows us to tackle the mean field of states and the mean field of control at the same time, but also extends the strategy set of players from Markov strategies to closed-loop strategies. We show the existence and uniqueness of Nash equilibrium for the mean field game as well as how the equilibrium of a mean field game consists of an approximative Nash equilibrium for the game with a finite number of players under different assumptions of structure and regularity on the cost functions and transition rate between states.

The Mean Field Games

2013 ◽  
pp. 11-14
Author(s):  
Alain Bensoussan ◽  
Jens Frehse ◽  
Phillip Yam
Keyword(s):  
Mean Field ◽  
Mean Field Games ◽  
The Mean

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 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.

scholarly journals COMPUTATION OF MEAN FIELD EQUILIBRIA IN ECONOMICS

2010 ◽  
Vol 20 (04) ◽  
pp. 567-588 ◽  
Author(s):  
AIME LACHAPELLE ◽  
JULIEN SALOMON ◽  
GABRIEL TURINICI
Keyword(s):  
Theoretical Result ◽  
Optimization Problem ◽  
Existence Result ◽  
Mean Field ◽  
Numerical Results ◽  
Mean Field Games ◽  
Economy Of Scale ◽  
Technological Transition ◽  
The Mean

Motivated by a mean field games stylized model for the choice of technologies (with externalities and economy of scale), we consider the associated optimization problem and prove an existence result. To complement the theoretical result, we introduce a monotonic algorithm to find the mean field equilibria. We close with some numerical results, including the multiplicity of equilibria describing the possibility of a technological transition.

scholarly journals Selective model-predictive control for flocking systems

2018 ◽  
Vol 9 (2) ◽  
pp. 4-21 ◽  
Author(s):  
Giacomo Albi ◽  
Lorenzo Pareschi
Keyword(s):  
Model Predictive Control ◽  
Predictive Control ◽  
Computational Cost ◽  
Mean Field ◽  
Stochastic Algorithm ◽  
Field Limit ◽  
Mean Field Limit ◽  
Constrained Dynamics ◽  
Interacting Agents ◽  
The Mean

Abstract In this paper the optimal control of alignment models composed by a large number of agents is investigated in presence of a selective action of a controller, acting in order to enhance consensus. Two types of selective controls have been presented: an homogeneous control filtered by a selective function and a distributed control active only on a selective set. As a first step toward a reduction of computational cost, we introduce a model predictive control (MPC) approximation by deriving a numerical scheme with a feedback selective constrained dynamics. Next, in order to cope with the numerical solution of a large number of interacting agents, we derive the mean-field limit of the feedback selective constrained dynamics, which eventually will be solved numerically by means of a stochastic algorithm, able to simulate effciently the selective constrained dynamics. Finally, several numerical simulations are reported to show the effciency of the proposed techniques.

scholarly journals Convergence to the mean field game limit: A case study

2020 ◽  
Vol 30 (1) ◽  
pp. 259-286 ◽  
Author(s):  
Marcel Nutz ◽  
Jaime San Martin ◽  
Xiaowei Tan
Keyword(s):  
Mean Field ◽  
Mean Field Game ◽  
The Mean
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