Mean-field game theory is the study of strategic decision making by small interacting agents in very large populations.
The use of the term "mean field" is inspired by mean-field theory in physics, which considers the behavior of systems of large numbers of particles where individual particles have negligible impacts upon the system.
[1] In traditional game theory, the subject of study is usually a game with two players and discrete time space, and extends the results to more complex situations by induction.
On the other hand with MFGs we can handle large numbers of players through the mean representative agent and at the same time describe complex state dynamics.
This class of problems was considered in the economics literature by Boyan Jovanovic and Robert W. Rosenthal,[2] in the engineering literature by Minyi Huang, Roland Malhame, and Peter E. Caines[3][4][5] and independently and around the same time by mathematicians Jean-Michel Lasry [fr] and Pierre-Louis Lions.
[6][7] In continuous time a mean-field game is typically composed of a Hamilton–Jacobi–Bellman equation that describes the optimal control problem of an individual and a Fokker–Planck equation that describes the dynamics of the aggregate distribution of agents.
Under fairly general assumptions it can be proved that a class of mean-field games is the limit as
[8] A related concept to that of mean-field games is "mean-field-type control".
[9] The following system of equations[10] 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
By controlling their movement, the agent aims to minimize their overall expected cost
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).
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.
The individual agent's dynamics are modeled as a stochastic differential equation
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.
[12] 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.
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,[13] departure time choice of morning commuters,[14] and decision-making processes for autonomous vehicle.
[15] 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,[16] control individuals’ decisions to limit their social interactions,[17] and support the government’s nonpharmaceutical interventions.