On Quasi-stationary Mean Field Games Models

Abstract

We explore a mechanism of decision-making in mean field games with myopic players. At each instant, agents set a strategy which optimizes their expected future cost by assuming their environment as immutable. As the system evolves, the players observe the evolution of the system and adapt to their new environment without anticipating. With a specific cost structures, these models give rise to coupled systems of partial differential equations of quasi-stationary nature. We provide sufficient conditions for the existence and uniqueness of classical solutions for these systems, and give a rigorous derivation of these systems from N-players stochastic differential games models. Finally, we show that the population can self-organize and converge exponentially fast to the ergodic mean field games equilibrium, if the initial distribution is sufficiently close to it and the Hamiltonian is quadratic.

Appendix A. Elementary Facts on the Fokker–Planck Equation

Let $V:[0,T]\times Q\rightarrow {\mathbb {R}}$ be a given bounded vector field, which is continuous in time and Hölder continuous in space, and we consider the following Fokker–Planck equation:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}m-\sigma \Delta m-\mathrm{div}(m V)=0 \quad \text{ in } (0,T)\times Q,\\&m(0)=m_{0} \quad \text{ in } Q; \end{aligned} \right. \end{aligned}$$

(A.1)

and the following stochastic differential equation:

$$\begin{aligned} \,\mathrm{d}\mathbb {X}_{t}= V(t,\mathbb {X}_{t})\,\mathrm{d}t+ \sqrt{2\sigma }\,\mathrm{d}B_{t}\quad t\in (0,T], \quad \mathbb {X}_{0}= Z_{0}, \end{aligned}$$

(A.2)

where $(B_{t})$ is a standard d-dimensional Brownian motion over some probability space $(\Omega , \mathcal {F},\mathbb {P})$ and $Z_{0}\in L^{1}(\Omega )$ is random and independent of $(B_{t})$. Under these assumptions, there is a unique solution to (A.2) and the following hold:

Lemma A.1

If $m_{0}=\mathcal {L}(Z_{0})$ then, $m(t)=\mathcal {L}(\mathbb {X}_{t})$ is a weak solution to (A.1) and there exists a constant $C_{T}>0$ such that, for any $t,s \in [0,T],$

$$\begin{aligned} \,{\mathbf{d}}_{1}(m(t),m(s)) \le C_{T}(1+\Vert V\Vert _{\infty })|t-s |^{1/2}. \end{aligned}$$

Proof

The first assertion is a straightforward consequence of Itô’s formula. On the other hand, for any 1-Lipschitz continuous function $\phi$ and any $t\ge s$, one has

$$\begin{aligned} \int _{{\mathbb {T}}^{d}}\phi (x) \,\mathrm{d}(m(t)-m(s))(x)\le & {} {\mathbb {E}}\vert \phi (\mathbb {X}_{t})-\phi (\mathbb {X}_{s}) \vert \le {\mathbb {E}}\vert \mathbb {X}_{t}-\mathbb {X}_{s} \vert \\\le & {} {\mathbb {E}}\left[ \int _{s}^{t}\vert V(u,\mathbb {X}_{u}) \vert \,\mathrm{d}u + \sqrt{2\sigma } \vert B_{t}-B_{s} \vert \right] \\\le & {} \Vert V \Vert _{\infty }(t-s)+ \sqrt{2\sigma (t-s)}. \end{aligned}$$

$\square$