Mean Field Games
Introduction
Suppose we have a large number of agents who are all competing against each other in a game over time, that is they are selecting strategy processes in an attempt to optimize their objective functionals. The solutions to these games are Nash equlibria (NE) which are combinations of strategies such that no agent can improve their outcome given no other agent changes their strategy. In stochastic differential games, agent states and strategies are modeled by continuous time stochastic processes with game dynamics decribed by coupled stochastic differential equations (SDEs), one for each agent. The complexity of these games grows exponentially as the number of agents increases, meaning that determining solutions quickly becomes intractable. However, if we assume that the number of agents is very large, and that agents are sufficiently homogenous, then we can consider the limit as the number of agents goes to infinity. In this limit, the game dynamics are described by a single representative agent interacts with the distribution of the population without impacting the distribution itself. This limiting case is called a Mean Field Game (MFG).
First introuced by Lasry and Lions in 2006, Mean Field Game (MFG) theory stems primarily from mean field models in mathematical physics which analyze the behavior of many idential particals. Another branch of study is Mean Field Control (MFC) theory, which is concerned with the optimal control of a population of agents.
MFG General Definition
The typical model for Mean Field Games (MFGs) is the following system
\[ \begin{cases} -\partial_t u - \nu \delta u + H(x, m, Du) = 0 \quad \text{in } \mathbb{R}^d\times (0,T) \\ \partial_t m - \nu \delta m - \text{div}(D_p H(x, m, Du)m) = 0 \quad \text{in } \mathbb{R}^d\times (0,T) \\ m(0) = m_0(x), \quad u(x,T) = G(x, m(T)), \end{cases} \]
where \(\nu\) is a nonnegative parameter.
The first equation is a Hamilton-Jacobi-Bellman Equation and is understood in backward time. The second equation is a Fokker-Planck Equation and is understood in forward time. There are two crucial structural conditions for this system:
Convexity \(H=H(x, m, p)\) with respect to the last variable \(p\), which implies that the HJB equation is associated with an optimal control problem. Specifically, this is the value function associated with a representative agent.
\(m_0\) (and therefore \(m_t\)) is (the density of) a probability measure, associated with the evolution of a population (continuum) of agents.