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 Quantum Mean-Field Games with the Observations of Counting Type

Games ◽  
2021 ◽  
Vol 12 (1) ◽  
pp. 7
Author(s):  
Vassili N. Kolokoltsov
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Open Problem ◽  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Mean Field ◽  
Mean Field Games ◽  
Quantum Game ◽  
Quantum Games ◽  
Interesting Open Problem

Quantum games and mean-field games (MFG) represent two important new branches of game theory. In a recent paper the author developed quantum MFGs merging these two branches. These quantum MFGs were based on the theory of continuous quantum observations and filtering of diffusive type. In the present paper we develop the analogous quantum MFG theory based on continuous quantum observations and filtering of counting type. However, proving existence and uniqueness of the solutions for resulting limiting forward-backward system based on jump-type processes on manifolds seems to be more complicated than for diffusions. In this paper we only prove that if a solution exists, then it gives an ϵ-Nash equilibrium for the corresponding N-player quantum game. The existence of solutions is suggested as an interesting open problem.

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 A Novel Mean Field Game-Based Strategy for Charging Electric Vehicles in Solar Powered Parking Lots

Energies ◽  
2021 ◽  
Vol 14 (24) ◽  
pp. 8517
Author(s):  
Samuel M. Muhindo ◽  
Roland P. Malhamé ◽  
Geza Joos
Keyword(s):  
Nash Equilibrium ◽  
Solar Energy ◽  
Electric Vehicles ◽  
Large Population ◽  
Mean Field ◽  
Mean Field Games ◽  
Parking Lot ◽  
Equal Sharing ◽  
Solar Powered ◽  
Energy Forecast

We develop a strategy, with concepts from Mean Field Games (MFG), to coordinate the charging of a large population of battery electric vehicles (BEVs) in a parking lot powered by solar energy and managed by an aggregator. A yearly parking fee is charged for each BEV irrespective of the amount of energy extracted. The goal is to share the energy available so as to minimize the standard deviation (STD) of the state of charge (SOC) of batteries when the BEVs are leaving the parking lot, while maintaining some fairness and decentralization criteria. The MFG charging laws correspond to the Nash equilibrium induced by quadratic cost functions based on an inverse Nash equilibrium concept and designed to favor the batteries with the lower SOCs upon arrival. While the MFG charging laws are strictly decentralized, they guarantee that a mean of instantaneous charging powers to the BEVs follows a trajectory based on the solar energy forecast for the day. That day ahead forecast is broadcasted to the BEVs which then gauge the necessary SOC upon leaving their home. We illustrate the advantages of the MFG strategy for the case of a typical sunny day and a typical cloudy day when compared to more straightforward strategies: first come first full/serve and equal sharing. The behavior of the charging strategies is contrasted under conditions of random arrivals and random departures of the BEVs in the parking lot.

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 Existence and non-existence for time-dependent mean field games with strong aggregation

2021 ◽  
Author(s):  
Marco Cirant ◽  
Daria Ghilli
Keyword(s):  
Population Density ◽  
State Space ◽  
Time Horizon ◽  
Existence Of Solutions ◽  
Mean Field ◽  
The State ◽  
Time Dependent ◽  
Classical Solutions ◽  
Mean Field Games ◽  
Quadratic Mean

AbstractWe investigate the existence of classical solutions to second-order quadratic Mean-Field Games systems with local and strongly decreasing couplings of the form $$-\sigma m^\alpha $$ - σ m α ,$$\alpha \ge 2/N$$ α ≥ 2 / N , where m is the population density and N is the dimension of the state space. We prove the existence of solutions under the assumption that $$\sigma $$ σ is small enough. For large $$\sigma $$ σ , we show that existence may fail whenever the time horizon T is large.

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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