scholarly journals Adaptive Mean Field Games for Large Population Coupled ARX Systems with Unknown Coupling Strength

2013 ◽  
Vol 3 (4) ◽  
pp. 489-507 ◽  
Author(s):  
Tao Li ◽  
Ji-Feng Zhang
Keyword(s):  
Coupling Strength ◽  
Large Population ◽  
Mean Field ◽  
Mean Field Games

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 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 On the Convergence of Model Free Learning in Mean Field Games

2020 ◽  
Vol 34 (05) ◽  
pp. 7143-7150
Author(s):  
Romuald Elie ◽  
Julien Pérolat ◽  
Mathieu Laurière ◽  
Matthieu Geist ◽  
Olivier Pietquin
Keyword(s):  
Learning Algorithms ◽  
Large Population ◽  
Single Agent ◽  
Mean Field ◽  
Space Environment ◽  
Iteration Step ◽  
Mean Field Games ◽  
Scalable Algorithms ◽  
Model Free ◽  
Approximate Nash Equilibrium

Learning by experience in Multi-Agent Systems (MAS) is a difficult and exciting task, due to the lack of stationarity of the environment, whose dynamics evolves as the population learns. In order to design scalable algorithms for systems with a large population of interacting agents (e.g., swarms), this paper focuses on Mean Field MAS, where the number of agents is asymptotically infinite. Recently, a very active burgeoning field studies the effects of diverse reinforcement learning algorithms for agents with no prior information on a stationary Mean Field Game (MFG) and learn their policy through repeated experience. We adopt a high perspective on this problem and analyze in full generality the convergence of a fictitious iterative scheme using any single agent learning algorithm at each step. We quantify the quality of the computed approximate Nash equilibrium, in terms of the accumulated errors arising at each learning iteration step. Notably, we show for the first time convergence of model free learning algorithms towards non-stationary MFG equilibria, relying only on classical assumptions on the MFG dynamics. We illustrate our theoretical results with a numerical experiment in a continuous action-space environment, where the approximate best response of the iterative fictitious play scheme is computed with a deep RL algorithm.

scholarly journals Cemracs 2017: numerical probabilistic approach to MFG

2019 ◽  
Vol 65 ◽  
pp. 84-113
Author(s):  
Andrea Angiuli ◽  
Christy V. Graves ◽  
Houzhi Li ◽  
Jean-François Chassagneux ◽  
François Delarue ◽  
...  
Keyword(s):  
Optimization Problems ◽  
Continuation Method ◽  
Large Population ◽  
Probabilistic Approach ◽  
Mean Field ◽  
Picard Iteration ◽  
Mean Field Games ◽  
Benchmark Problems ◽  
Control Problems ◽  
Great Success

This project investigates numerical methods for solving fully coupled forward-backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. Having numerical solvers for such mean field FBSDEs is of interest because of the potential application of these equations to optimization problems over a large population, say for instance mean field games (MFG) and optimal mean field control problems. Theory for this kind of problems has met with great success since the early works on mean field games by Lasry and Lions, see [29], and by Huang, Caines, and Malhamé, see [26]. Generally speaking, the purpose is to understand the continuum limit of optimizers or of equilibria (say in Nash sense) as the number of underlying players tends to infinity. When approached from the probabilistic viewpoint, solutions to these control problems (or games) can be described by coupled mean field FBSDEs, meaning that the coefficients depend upon the own marginal laws of the solution. In this note, we detail two methods for solving such FBSDEs which we implement and apply to five benchmark problems. The first method uses a tree structure to represent the pathwise laws of the solution, whereas the second method uses a grid discretization to represent the time marginal laws of the solutions. Both are based on a Picard scheme; importantly, we combine each of them with a generic continuation method that permits to extend the time horizon (or equivalently the coupling strength between the two equations) for which the Picard iteration converges.

scholarly journals Finite-State Contract Theory with a Principal and a Field of Agents

2020 ◽  
Author(s):  
René Carmona ◽  
Peiqi Wang
Keyword(s):  
Optimal Control ◽  
Stochastic Models ◽  
Contract Theory ◽  
Large Population ◽  
Mean Field ◽  
Theory Model ◽  
Mean Field Games ◽  
Weak Formulation ◽  
Finite State ◽  
Set Up

We use the recently developed probabilistic analysis of mean field games with finitely many states in the weak formulation to set up a principal/agent contract theory model where the principal faces a large population of agents interacting in a mean field manner. We reduce the problem to the optimal control of dynamics of the McKean-Vlasov type, and we solve this problem explicitly for a class of models with concave rewards. The paper concludes with a numerical example demonstrating the power of the results when applied to an example of epidemic containment. This paper was accepted by Baris Ata, stochastic models and simulation.

scholarly journals Backward-forward linear-quadratic mean-field Stackelberg games

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kehan Si ◽  
Zhen Wu
Keyword(s):  
Large Population ◽  
Consistency Condition ◽  
Market Price ◽  
Mean Field ◽  
Mapping Method ◽  
Open Loop ◽  
Stackelberg Games ◽  
Linear Quadratic ◽  
Population System ◽  
The Cost

AbstractThis paper studies a controlled backward-forward linear-quadratic-Gaussian (LQG) large population system in Stackelberg games. The leader agent is of backward state and follower agents are of forward state. The leader agent is dominating as its state enters those of follower agents. On the other hand, the state-average of all follower agents affects the cost functional of the leader agent. In reality, the leader and the followers may represent two typical types of participants involved in market price formation: the supplier and producers. This differs from standard MFG literature and is mainly due to the Stackelberg structure here. By variational analysis, the consistency condition system can be represented by some fully-coupled backward-forward stochastic differential equations (BFSDEs) with high dimensional block structure in an open-loop sense. Next, we discuss the well-posedness of such a BFSDE system by virtue of the contraction mapping method. Consequently, we obtain the decentralized strategies for the leader and follower agents which are proved to satisfy the ε-Nash equilibrium property.

scholarly journals Lax connection and conserved quantities of quadratic mean field games

2021 ◽  
Vol 62 (8) ◽  
pp. 083302
Author(s):  
Thibault Bonnemain ◽  
Thierry Gobron ◽  
Denis Ullmo
Keyword(s):  
Mean Field ◽  
Mean Field Games ◽  
Conserved Quantities ◽  
Quadratic Mean
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