Mean-field game theory

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-\operatorname{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 dt+\sqrt{2\nu} dB_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}\left[\int_{0}^TL(X_s,\alpha_s,m(s))ds+G(X_T,m(T))\right] 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}\left[\int_{t}^TL(X_s,\alpha_s,m(s))ds+G(X_T,m(T))\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 \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). == Finite State Games ==

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 \mathcal{G}=(\mathcal{E}, \mathcal{A}, \{ Q_a \}, {\bf m }_0, \{ c_a \}, \beta), where \mathcal{E} is the state space, \mathcal{A} the action set, Q_{a} the transition rate matrices, {\bf m }_0 the initial state, \{c_a\} the cost functions and \beta \in \mathbb{R} a discount factor. Furthermore, a mixed strategy is a measurable function \pi: \mathbb{E} \times \mathbb{R}^+ \xrightarrow[]{} \mathcal{P(A)}, that associates to each state i \in \mathcal{E} and each time t \geq 0 a probability measure \pi_i(t) \in \mathcal{P(A)} on the set of possible actions. Thus \pi_{i,a}(t) is the probability that, at time t a player in state i takes action a, under strategy \pi. Additionally, rate matrices \{ Q_a ({\bf m }^{\pi}(t)) \}_{a \in \mathcal{A}} define the evolution over the time of population distribution, where {\bf m }^{\pi}(t) \in \mathcal{P(\mathcal{E})} is the population distribution at time t. == Linear-quadratic Gaussian game problem ==

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) \,dt + \sigma_i \,dW_i, \quad i = 1, \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, \dots, N. The individual agent's cost is 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. == General and applied use ==

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 Rene Carmona reviews applications in financial engineering and economics that can be cast and tackled within the framework of the MFG paradigm. 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, departure time choice of morning commuters, and decision-making processes for autonomous vehicle. 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, control individuals’ decisions to limit their social interactions, and support the government’s nonpharmaceutical interventions. ==See also==