scholarly journals Numerical Methods for Finite-State Mean-Field Games Satisfying a Monotonicity Condition

2018 ◽  
Author(s):  
Diogo A. Gomes ◽  
João Saúde
Keyword(s):  
Numerical Methods ◽  
Mean Field ◽  
Mean Field Games ◽  
Monotonicity Condition ◽  
Finite State

scholarly journals Mean field games models of segregation

2017 ◽  
Vol 27 (01) ◽  
pp. 75-113 ◽  
Author(s):  
Yves Achdou ◽  
Martino Bardi ◽  
Marco Cirant
Keyword(s):  
Numerical Methods ◽  
Existence Of Solutions ◽  
Large Population ◽  
Mean Field ◽  
Mean Field Games ◽  
Residential Choice ◽  
Urban Settlements ◽  
Thomas Schelling ◽  
Two Populations ◽  
Large Population Limit

This paper introduces and analyzes some models in the framework of mean field games (MFGs) describing interactions between two populations motivated by the studies on urban settlements and residential choice by Thomas Schelling. For static games, a large population limit is proved. For differential games with noise, the existence of solutions is established for the systems of partial differential equations of MFG theory, in the stationary and in the evolutive case. Numerical methods are proposed with several simulations. In the examples and in the numerical results, particular emphasis is put on the phenomenon of segregation between the populations.

scholarly journals Mean Field Games: Numerical Methods for the Planning Problem

2012 ◽  
Vol 50 (1) ◽  
pp. 77-109 ◽  
Author(s):  
Yves Achdou ◽  
Fabio Camilli ◽  
Italo Capuzzo-Dolcetta
Keyword(s):  
Numerical Methods ◽  
Mean Field ◽  
Mean Field Games ◽  
Planning Problem

scholarly journals Linear quadratic mean field game with control input constraint

2018 ◽  
Vol 24 (2) ◽  
pp. 901-919 ◽  
Author(s):  
Ying Hu ◽  
Jianhui Huang ◽  
Xun Li
Keyword(s):  
Backward Stochastic Differential Equation ◽  
Consistency Condition ◽  
Mean Field ◽  
Control Process ◽  
Mean Field Games ◽  
Monotonicity Condition ◽  
Projection Operators ◽  
Control Input ◽  
Linear Quadratic ◽  
The Individual

In this paper, we study a class of linear-quadratic (LQ) mean-field games in which the individual control process is constrained in a closed convex subset Γ of full space ℝm. The decentralized strategies and consistency condition are represented by a class of mean-field forward-backward stochastic differential equation (MF-FBSDE) with projection operators on Γ. The wellposedness of consistency condition system is obtained using the monotonicity condition method. The related ϵ-Nash equilibrium property is also verified.

scholarly journals A machine learning framework for solving high-dimensional mean field game and mean field control problems

2020 ◽  
Vol 117 (17) ◽  
pp. 9183-9193
Author(s):  
Lars Ruthotto ◽  
Stanley J. Osher ◽  
Wuchen Li ◽  
Levon Nurbekyan ◽  
Samy Wu Fung
Keyword(s):  
Machine Learning ◽  
Numerical Methods ◽  
Mean Field ◽  
Two Dimensions ◽  
High Dimensional ◽  
Mean Field Games ◽  
Spatial Discretization ◽  
Learning Framework ◽  
Field Control ◽  
Crowd Motion

Mean field games (MFG) and mean field control (MFC) are critical classes of multiagent models for the efficient analysis of massive populations of interacting agents. Their areas of application span topics in economics, finance, game theory, industrial engineering, crowd motion, and more. In this paper, we provide a flexible machine learning framework for the numerical solution of potential MFG and MFC models. State-of-the-art numerical methods for solving such problems utilize spatial discretization that leads to a curse of dimensionality. We approximately solve high-dimensional problems by combining Lagrangian and Eulerian viewpoints and leveraging recent advances from machine learning. More precisely, we work with a Lagrangian formulation of the problem and enforce the underlying Hamilton–Jacobi–Bellman (HJB) equation that is derived from the Eulerian formulation. Finally, a tailored neural network parameterization of the MFG/MFC solution helps us avoid any spatial discretization. Our numerical results include the approximate solution of 100-dimensional instances of optimal transport and crowd motion problems on a standard work station and a validation using a Eulerian solver in two dimensions. These results open the door to much-anticipated applications of MFG and MFC models that are beyond reach with existing numerical methods.

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 Mean Field Games: Numerical Methods

2010 ◽  
Vol 48 (3) ◽  
pp. 1136-1162 ◽  
Author(s):  
Yves Achdou ◽  
Italo Capuzzo-Dolcetta
Keyword(s):  
Numerical Methods ◽  
Mean Field ◽  
Mean Field Games
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