scholarly journals Socio-economic applications of finite state mean field games

2014 ◽  
Vol 372 (2028) ◽  
pp. 20130405 ◽  
Author(s):  
Diogo Gomes ◽  
Roberto M. Velho ◽  
Marie-Therese Wolfram
Keyword(s):  
Consumer Choice ◽  
Numerical Experiments ◽  
Free Market ◽  
Hyperbolic Systems ◽  
Mean Field ◽  
Mean Field Games ◽  
Paradigm Shifts ◽  
Choice Behaviour ◽  
Mean Field Game ◽  
Finite State

In this paper, we present different applications of finite state mean field games to socio-economic sciences. Examples include paradigm shifts in the scientific community or consumer choice behaviour in the free market. The corresponding finite state mean field game models are hyperbolic systems of partial differential equations, for which we present and validate different numerical methods. We illustrate the behaviour of solutions with various numerical experiments, which show interesting phenomena such as shock formation. Hence, we conclude with an investigation of the shock structure in the case of two-state problems.

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.

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 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 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 Minimal-time mean field games

2019 ◽  
Vol 29 (08) ◽  
pp. 1413-1464 ◽  
Author(s):  
Guilherme Mazanti ◽  
Filippo Santambrogio
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Value Function ◽  
Mean Field ◽  
Mean Field Games ◽  
Mean Field Game ◽  
Minimal Time ◽  
Regularity Properties ◽  
The Value Function

This paper considers a mean field game model inspired by crowd motion where agents want to leave a given bounded domain through a part of its boundary in minimal time. Each agent is free to move in any direction, but their maximal speed is bounded in terms of the average density of agents around their position in order to take into account congestion phenomena. After a preliminary study of the corresponding minimal-time optimal control problem, we formulate the mean field game in a Lagrangian setting and prove existence of Lagrangian equilibria using a fixed point strategy. We provide a further study of equilibria under the assumption that agents may leave the domain through the whole boundary, in which case equilibria are described through a system of a continuity equation on the distribution of agents coupled with a Hamilton–Jacobi equation on the value function of the optimal control problem solved by each agent. This is possible thanks to the semiconcavity of the value function, which follows from some further regularity properties of optimal trajectories obtained through Pontryagin Maximum Principle. Simulations illustrate the behavior of equilibria in some particular situations.

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