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 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 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 A probabilistic weak formulation of mean field games and applications

2015 ◽  
Vol 25 (3) ◽  
pp. 1189-1231 ◽  
Author(s):  
René Carmona ◽  
Daniel Lacker
Keyword(s):  
Mean Field ◽  
Mean Field Games ◽  
Weak Formulation

scholarly journals Deep Learning and Mean-Field Games: A Stochastic Optimal Control Perspective

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 14
Author(s):  
Luca Di Persio ◽  
Matteo Garbelli
Keyword(s):  
Optimal Control ◽  
Deep Learning ◽  
Stochastic Optimal Control ◽  
Mathematical Formulation ◽  
Mean Field ◽  
Mean Field Games ◽  
Theoretical Frameworks ◽  
Depth Analysis ◽  
The Mean ◽  
Hamilton Jacobi Bellman

We provide a rigorous mathematical formulation of Deep Learning (DL) methodologies through an in-depth analysis of the learning procedures characterizing Neural Network (NN) models within the theoretical frameworks of Stochastic Optimal Control (SOC) and Mean-Field Games (MFGs). In particular, we show how the supervised learning approach can be translated in terms of a (stochastic) mean-field optimal control problem by applying the Hamilton–Jacobi–Bellman (HJB) approach and the mean-field Pontryagin maximum principle. Our contribution sheds new light on a possible theoretical connection between mean-field problems and DL, melting heterogeneous approaches and reporting the state-of-the-art within such fields to show how the latter different perspectives can be indeed fruitfully unified.

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 The Master Equation and the Convergence Problem in Mean Field Games

2019 ◽  
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Optimal Control ◽  
Master Equation ◽  
Differential Games ◽  
Mean Field ◽  
Mathematical Finance ◽  
Mean Field Games ◽  
Mean Field Limit ◽  
Continuum Of Players ◽  
Developing Area ◽  
Suitable Notion

This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity. Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players as the number of players tends to infinity. The book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit. The book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.

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