User:TianmingLiu/sandbox

Introduction
Mean field games (MFGs) lie at the intersection of game theory with stochastic analysis and control theory. A mean field game is a strategic game dynamic and symmetric between a large number of agents in which the interactions between agents are negligible but each agent’s actions affect the mean of the population. In other words, each agent acts according to his minimization or maximization problem taking into account other agents’ decisions and because their population is large we can assume the number of agents goes to infinity and a representative agent exists (precise definitions for everything will be given later).

In traditional game theory, we usually study a game with 2 players and using induction we extend to several, but with games in continuous time with continuous states (differential games or stochastic differential games) this strategy cannot be used because of the complexity that the dynamic interactions generate. 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.

=== General form of a Mean-field Game === The following system of equations can be used to model a typical Mean-field game:

$$\begin{cases} \partial_t u-\nu \Delta u+H(x,m,Du)=0 &(1)\\ \partial_t m-\nu \Delta m-div(D_p H(x,m,Du) m)=0 &(2)\\ m(0)=m_0 &(3)\\ u(x,T)=G(x,m(T)) &(4) \end{cases}$$

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 $$\alpha$$ to influence the population's overall location by:

$$d X_t=\alpha_t d_t+\sqrt{2\nu}B_t$$

where $$\nu$$ 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 $$[0,T]$$:

$$C=\mathbb{E}[\int_{0}^TL(X_s,\alpha_s,m(s))ds+G(X_T,m(T))]$$

where $$L(X_s,\alpha_s,m(s))$$ 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:

$$u(t,x)=\inf_{\alpha}\mathbb{E}[\int_{t}^TL(X_s,\alpha_s,m(s))ds+G(X_T,m(T))]$$

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 $$\alpha^*(x,t)$$ can be determined as $$\alpha^*(x,t)=D_p H(x,m,Du)$$. 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).