Mean-field game theory

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Mean-field game theory is the study of strategic decision making in very large populations of small interacting agents. This class of problems was considered in the economics literature by Boyan Jovanovic and Robert W. Rosenthal,[1] in the engineering literature by Peter E. Caines and his co-workers[2][3][4] and independently and around the same time by mathematicians Jean-Michel Lasry [fr] and Pierre-Louis Lions.[5][6]

Use of the term "mean field" is inspired by mean-field theory in physics, which considers the behaviour of systems of large numbers of particles where individual particles have negligible impact upon the system.

In continuous time a mean-field game is typically composed by 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 ${\displaystyle N\to \infty }$ of a N-player Nash equilibrium.[7]

A related concept to that of mean-field games is "mean-field-type control". In this case a social planner controls a distribution of states and chooses a control strategy. The solution to a mean-field-type control problem can typically be expressed as dual adjoint Hamilton–Jacobi–Bellman equation coupled with Kolmogorov equation. Mean-field-type game theory is the multi-agent generalization of the single-agent mean-field-type control.[8]

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

${\displaystyle dx_{i}=(a_{i}x_{i}+b_{i}u_{i})\,dt+\sigma _{i}\,dw_{i},\quad i=1,\dots ,N,}$

where ${\displaystyle x_{i}}$ is the state of the ${\displaystyle i}$-th agent, and ${\displaystyle u_{i}}$ is the control. The individual agent's cost is

${\displaystyle J_{i}(u_{i},\nu )=\mathbb {E} \left\{\int _{0}^{\infty }e^{-\rho t}\left[(x_{i}-\nu )^{2}+ru_{i}^{2}\right]\,dt\right\},\quad \nu =\Phi \left({\frac {1}{N}}\sum _{k\neq i}^{N}x_{k}+\eta \right).}$

The coupling between agents occurs in the cost function.