Mean-field game theory

The following system of equations{{Cite book |last=Achdou |first=Yves |url=https://www.worldcat.org/oclc/1238206187 |title=Mean field games : Cetraro, Italy 2019 |date=2020 |others=Pierre Cardaliaguet, F. Delarue, Alessio Porretta, Filippo Santambrogio |isbn=978-3-030-59837-2 |location=Cham |oclc=1238206187}} can be used to model a typical Mean-field game:

The basic dynamics of this set of Equations can be explained by an average agent's optimal control problem. In a mean-field game, an average agent can control their movement $\backslash alpha$ to influence the population's overall location by:

$d\; X\_t=\backslash alpha\_t\; dt+\backslash sqrt\{2\backslash nu\}\; dB\_t$

where $\backslash nu$ is a parameter and $B\_t$ is a standard Brownian motion. By controlling their movement, the agent aims to minimize their overall expected cost $C$ throughout the time period $[0,T]$:

$C=\backslash mathbb\{E\}\backslash left[\backslash int\_\{0\}^TL(X\_s,\backslash alpha\_s,m(s))ds+G(X\_T,m(T))\backslash right]$

where $L(X\_s,\backslash alpha\_s,m(s))$ is the running cost at time $s$ and $G(X\_T,m(T))$

is the terminal cost at time $T$. By this definition, at time $t$ and position $x$, the value function $u(t,x)$ can be determined as:

$u(t,x)=\backslash inf\_\{\backslash alpha\}\backslash mathbb\{E\}\backslash left[\backslash int\_\{t\}^TL(X\_s,\backslash alpha\_s,m(s))ds+G(X\_T,m(T))\backslash right]$

Given the definition of the value function $u(t,x)$, it can be tracked by the Hamilton-Jacobi equation (1). The optimal action of the average players $\backslash alpha^*(x,t)$ can be determined as $\backslash alpha^*(x,t)=D\_p\; H(x,m,Du)$. As all agents are relatively small and cannot single-handedly change the dynamics of the population, they will individually adapt the optimal control and the population would move in that way. This is similar to a Nash Equilibrium, in which all agents act in response to a specific set of others' strategies. The optimal control solution then leads to the Kolmogorov-Fokker-Planck equation (2).

A prominent category of mean field is games with a finite number of states and a finite number of actions per player. For those games, the analog of the Hamilton-Jacobi-Bellman equation is the Bellman equation, and the discrete version of the Fokker-Planck equation is the Kolmogorov equation. Specifically, for discrete-time models, the players' strategy is the Kolmogorov equation's probability matrix. In continuous time models, players have the ability to control the transition rate matrix.

A discrete mean field game can be defined by a tuple $\backslash mathcal\{G\}=(\backslash mathcal\{E\},\; \backslash mathcal\{A\},\; \backslash \{\; Q\_a\; \backslash \},\; \{\backslash bf\; m\; \}\_0,\; \backslash \{\; c\_a\; \backslash \},\; \backslash beta)$, where $\backslash mathcal\{E\}$ is the state space, $\backslash mathcal\{A\}$ the action set, $Q\_\{a\}$ the transition rate matrices, $\{\backslash bf\; m\; \}\_0$ the initial state, $\backslash \{c\_a\backslash \}$ the cost functions and $\backslash beta$ $\backslash in\; \backslash mathbb\{R\}$ a discount factor. Furthermore, a mixed strategy is a measurable function $\backslash pi:\; \backslash mathbb\{E\}\; \backslash times\; \backslash mathbb\{R\}^+\; \backslash xrightarrow[]\{\}\; \backslash mathcal\{P(A)\}$, that associates to each state $i\; \backslash in\; \backslash mathcal\{E\}$ and each time $t\; \backslash geq\; 0$ a probability measure $\backslash pi\_i(t)\; \backslash in\; \backslash mathcal\{P(A)\}$ on the set of possible actions. Thus $\backslash pi\_\{i,a\}(t)$ is the probability that, at time $t$ a player in state $i$ takes action $a$, under strategy $\backslash pi$. Additionally, rate matrices $\backslash \{\; Q\_a\; (\{\backslash bf\; m\; \}^\{\backslash pi\}(t))\; \backslash \}\_\{a\; \backslash in\; \backslash mathcal\{A\}\}$ define the evolution over the time of population distribution, where $\{\backslash bf\; m\; \}^\{\backslash pi\}(t)\; \backslash in\; \backslash mathcal\{P(\backslash mathcal\{E\})\}$ is the population distribution at time $t$.{{cite journal | arxiv=1909.01209 | doi=10.3934/jdg.2019016 | title=Discrete mean field games: Existence of equilibria and convergence | year=2019 | last1=Doncel | first1=Josu | last2=Gast | first2=Nicolas | last3=Gaujal | first3=Bruno | journal=Journal of Dynamics & Games | pages=1–19 | s2cid=197507580 }}

From Caines (2009), a relatively simple model of large-scale games is the linear-quadratic Gaussian model. The individual agent's dynamics are modeled as a stochastic differential equation

$$dX\_i\; =\; (a\_i\; X\_i\; +\; b\_i\; u\_i)\; \backslash ,dt\; +\; \backslash sigma\_i\; \backslash ,dW\_i,\; \backslash quad\; i\; =\; 1,\; \backslash dots,\; N,$$

where $X\_i$ is the state of the $i$-th agent, $u\_i$ is the control of the $i$-th agent, and $W\_i$ are independent Wiener processes for all $i\; =\; 1,\; \backslash dots,\; N$. The individual agent's cost is

$$J\_i(u\_i,\; \backslash nu)\; =\; \backslash mathbb\{E\}\backslash left\backslash \{\; \backslash int\_0^\backslash infty\; e^\{-\backslash rho\; t\}\; \backslash left[(X\_i\; -\; \backslash nu)^2\; +\; ru\_i^2\backslash right]\; \backslash ,dt\backslash right\backslash \},\; \backslash quad\; \backslash nu\; =\; \backslash Phi\backslash left(\backslash frac\{1\}\{N\}\; \backslash sum\_\{k\; \backslash neq\; i\}^N\; X\_k\; +\; \backslash eta\backslash right).$$

The coupling between agents occurs in the cost function.

The paradigm of Mean Field Games has become a major connection between distributed decision-making and stochastic modeling. Starting out in the stochastic control literature, it is gaining rapid adoption across a range of applications, including:

a. Financial market

Carmona reviews applications in financial engineering and economics that can be cast and tackled within the framework of the MFG paradigm.{{cite arXiv |last1=Carmona |first1=Rene |title=Applications of mean field games in financial engineering and economic theory |eprint=2012.05237 |date=2020|class=q-fin.GN }} Carmona argues that models in macroeconomics, contract theory, finance, …, greatly benefit from the switch to continuous time from the more traditional discrete-time models. He considers only continuous time models in his review chapter, including systemic risk, price impact, optimal execution, models for bank runs, high-frequency trading, and cryptocurrencies.

b. Crowd motions

MFG assumes that individuals are smart players which try to optimize their strategy and path with respect to certain costs (equilibrium with rational expectations approach). MFG models are useful to describe the anticipation phenomenon: the forward part describes the crowd evolution while the backward gives the process of how the anticipations are built. Additionally, compared to multi-agent microscopic model computations, MFG only requires lower computational costs for the macroscopic simulations. Some researchers have turned to MFG in order to model the interaction between populations and study the decision-making process of intelligent agents, including aversion and congestion behavior between two groups of pedestrians,{{cite journal |last1=Lachapelle |first1=Aimé |last2=Wolfram |first2=Marie-Therese|author2-link=Marie-Therese Wolfram |title=On a mean field game approach modeling congestion and aversion in pedestrian crowds |journal=Transportation Research Part B: Methodological |date=2011 |volume=45 |issue=10 |pages=1572–1589|doi=10.1016/j.trb.2011.07.011 |s2cid=55991774 |url=https://basepub.dauphine.fr/handle/123456789/5946 }} departure time choice of morning commuters,{{cite arXiv |last1=Feinstein |first1=Zachary |last2=Sojmark |first2=Andreas |title=A dynamic default contagion model: From Eisenberg-Noe to the mean field |eprint=1912.08695 |date=2019|class=q-fin.MF }} and decision-making processes for autonomous vehicle.{{cite journal |last1=Huang |first1=Kuang |last2=Chen |first2=Xu |last3=Di |first3=Xuan |last4=Du |first4=Qiang |title=Dynamic driving and routing games for autonomous vehicles on networks: A mean field game approach |journal=Transportation Research Part C: Emerging Technologies |date=2021 |volume=128 |page=103189|doi=10.1016/j.trc.2021.103189 |s2cid=235436377 |arxiv=2012.08388 }}

c. Control and mitigation of Epidemics

Since the epidemic has affected society and individuals significantly, MFG and mean-field controls (MFCs) provide a perspective to study and understand the underlying population dynamics, especially in the context of the Covid-19 pandemic response. MFG has been used to extend the SIR-type dynamics with spatial effects or allowing for individuals to choose their behaviors and control their contributions to the spread of the disease. MFC is applied to design the optimal strategy to control the virus spreading within a spatial domain,{{cite journal |last1=Lee |first1=Wonjun |last2=Liu |first2=Siting |last3=Tembine |first3=Hamidou |last4=Li |first4=Wuchen |last5=Osher |first5=Stanley |title=Controlling propagation of epidemics via mean-field control |journal=SIAM Journal on Applied Mathematics |date=2021 |volume=81 |issue=1 |pages=190–207|doi=10.1137/20M1342690 |arxiv=2006.01249 |s2cid=226299517 }} control individuals’ decisions to limit their social interactions,{{cite journal |last1=Aurell |first1=Alexander |last2=Carmona |first2=Rene |last3=Dayanikli |first3=Gokce |last4=Lauriere |first4=Mathieu |title=Optimal incentives to mitigate epidemics: a Stackelberg mean field game approach |journal=SIAM Journal on Control and Optimization |date=2022 |volume=60 |issue=2 |page=S294–S322|doi=10.1137/20M1377862 |arxiv=2011.03105 |s2cid=226278147 }} and support the government’s nonpharmaceutical interventions.{{cite journal |last1=Elie |first1=Romuald |last2=Hubert |first2=Emma |last3=Turinici |first3=Gabriel |title=Contact rate epidemic control of COVID-19: an equilibrium view |journal=Mathematical Modelling of Natural Phenomena |date=2020 |volume=15 |page=35|doi=10.1051/mmnp/2020022 |s2cid=215814201 |doi-access=free |arxiv=2004.08221 }}

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• Aggregative game
• Complex adaptive system
• Differential game
• Evolutionary game theory
• Quantal response equilibrium
• Potential game

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{{Reflist|30em}}

• {{usurped|1=[https://web.archive.org/web/20120415020409/http://www.ieeecss-oll.org/lectures/2009/mean-field-stochastic-control Mean Field Stochastic Control]}} ({{usurped|1=[https://web.archive.org/web/20150924032856/http://www.ieeecss-oll.org/sites/default/files/Caines.pdf Slides]}}), 2009 IEEE Control Systems Society Bode Prize Lecture by Peter E. Caines
• {{cite book |doi=10.1007/978-1-4471-5102-9_30-1 |chapter=Mean Field Games |title=Encyclopedia of Systems and Control |pages=1–6 |year=2013 |last1=Caines |first1=Peter E. |isbn=978-1-4471-5102-9 |s2cid=33954904 }}
• [https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf Notes on Mean Field Games], from Pierre-Louis Lions' lectures at Collège de France
• {{in lang|fr}} [http://www.college-de-france.fr/site/pierre-louis-lions/index.htm#|m=course|q=/site/pierre-louis-lions/course-2011-2012.htm Video lectures] by Pierre-Louis Lions
• [https://www.oliviergueant.com/uploads/4/3/0/9/4309511/paris-princeton.pdf Mean field games and applications] by Olivier Guéant, Jean-Michel Lasry, and Pierre-Louis Lions

{{Game theory}}

Category:Game theory

Category:Mathematical economics