Selected topics in mean field games · 2022-econ-12

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Question 1

Let $d\geq3$, $E=\{1,\ldots,d\}$, and

$$\Delta=\left\{m\in[0,1]^d:\sum_{i=1}^d m_i=1\right\}.$$

Let $F=(F^i)_{i\in E}$ and $G=(G^i)_{i\in E}$ be smooth potential couplings: there are smooth scalar functions $\mathcal F,\mathcal G:\Delta\to\mathbb R$ such that, for every $m$ in the interior of $\Delta$ and every $h\in\mathbb R^d$ with $\sum_i h_i=0$,

$$D\mathcal F(m)[h]=\sum_{i=1}^dF^i(m)h_i, \qquad D\mathcal G(m)[h]=\sum_{i=1}^dG^i(m)h_i.$$

Fix $T>0$ and an interior initial distribution $m_0$. The associated deterministic mean field control problem is to minimize

$$\int_0^T\left[ \frac12\sum_{\substack{i,j\in E\\i\ne j}}p_t^i|\beta_t^{i,j}|^2 +\mathcal F(p_t)\right]dt+\mathcal G(p_T) \tag{1}$$

over bounded measurable transition rates $\beta_t^{i,j}\geq0$ for $i\ne j$, with $\beta_t^{i,i}=-\sum_{j\ne i}\beta_t^{i,j}$, subject to

$$\dot p_t^i=\sum_{j\in E}p_t^j\beta_t^{j,i}, \qquad p_0=m_0. \tag{2}$$

For each $N\geq2$, consider the symmetric $N$-player continuous-time Markov game in which player $n$ has state $X_t^n\in E$, controls the transition rate $\alpha_t^{n,i,j}$ from $i$ to $j$, and minimizes

$$\mathbb E\left[ \int_0^T\left( \frac12\sum_{j\ne X_t^n}|\alpha_t^{n,X_t^n,j}|^2 +F^{X_t^n}(m_t^{N,n}) \right)dt +G^{X_T^n}(m_T^{N,n}) \right], \tag{3}$$

where

$$m_t^{N,n}=\frac1{N-1}\sum_{\ell\ne n}\delta_{X_t^\ell}.$$

Let $m_t^N=N^{-1}\sum_{n=1}^N\delta_{X_t^n}$ be the empirical distribution under a symmetric Markovian Nash equilibrium, with initial empirical distributions converging to $m_0$.

Determine whether, for every sequence of such finite-player equilibria, every weak limit as $N\to\infty$ of the laws of $m^N$ is supported on the population trajectories $p$ generated by minimizers of the deterministic control problem (1)--(2), thereby excluding potential mean field game equilibria that are merely nonminimizing critical points of (1).

Source reference: Pierre Cardaliaguet and François Delarue, “Selected topics in mean field games,” Proceedings of the International Congress of Mathematicians 2022, vol. 5, EMS Press, 2022, pp. 3660–3703, Section 4.3 (“Selection of equilibria,” unnumbered open question), https://doi.org/10.4171/ICM2022/17.

Question 2

Let $d\geq3$, $E=\{1,\ldots,d\}$, and

$$\Delta=\left\{m\in[0,1]^d:\sum_{i=1}^d m_i=1\right\}.$$

Let $F=(F^i)_{i\in E}$ and $G=(G^i)_{i\in E}$ be smooth potential couplings: there are smooth scalar functions $\mathcal F,\mathcal G:\Delta\to\mathbb R$ such that, for every $m$ in the interior of $\Delta$ and every $h\in\mathbb R^d$ with $\sum_i h_i=0$,

$$D\mathcal F(m)[h]=\sum_{i=1}^dF^i(m)h_i, \qquad D\mathcal G(m)[h]=\sum_{i=1}^dG^i(m)h_i.$$

Fix $T>0$ and an interior initial distribution $m_0$. Define the deterministic mean field control problem by

$$\inf_{\beta}\left\{ \int_0^T\left[ \frac12\sum_{\substack{i,j\in E\\i\ne j}}p_t^i|\beta_t^{i,j}|^2 +\mathcal F(p_t)\right]dt+\mathcal G(p_T) \right\}, \tag{1}$$

where $\beta_t^{i,j}\geq0$ for $i\ne j$, $\beta_t^{i,i}=-\sum_{j\ne i}\beta_t^{i,j}$, and

$$\dot p_t^i=\sum_{j\in E}p_t^j\beta_t^{j,i}, \qquad p_0=m_0. \tag{2}$$

For every $\varepsilon>0$, let $(W^{i,j})_{i,j\in E}$ be independent Brownian motions and choose a nonnegative smooth function $\varphi^\varepsilon:[0,\infty)\to[0,\infty)$, positive near $0$, whose inward drift is strong enough to give uniqueness and such that $\varphi^\varepsilon\to0$ locally uniformly on $(0,\infty)$. Let $(u^\varepsilon,m^\varepsilon,v^\varepsilon)$ be the unique solution of

$$\begin{aligned} du_t^{\varepsilon,i}={}&\Bigg[ \frac12\sum_{j\in E}(u_t^{\varepsilon,i}-u_t^{\varepsilon,j})_+^2 -F^i(m_t^\varepsilon) -\sqrt\varepsilon\sum_{j\in E}\sqrt{m_t^{\varepsilon,i}m_t^{\varepsilon,j}} \bigl(v_t^{\varepsilon,i,i,j}-v_t^{\varepsilon,i,j,i}\bigr)\\ &\qquad -\sum_{j\in E}\varphi^\varepsilon(m_t^{\varepsilon,j}) (u_t^{\varepsilon,j}-u_t^{\varepsilon,i}) \Bigg]dt +\sum_{j,k\in E}v_t^{\varepsilon,i,j,k}\,dW_t^{j,k},\\ dm_t^{\varepsilon,i}={}& \sum_{j\in E}\Bigl[ m_t^{\varepsilon,j}\bigl(\varphi^\varepsilon(m_t^{\varepsilon,i}) +(u_t^{\varepsilon,j}-u_t^{\varepsilon,i})_+\bigr) -m_t^{\varepsilon,i}\bigl(\varphi^\varepsilon(m_t^{\varepsilon,j}) +(u_t^{\varepsilon,i}-u_t^{\varepsilon,j})_+\bigr) \Bigr]dt\\ &\qquad +\sqrt\varepsilon\sum_{j\in E} \sqrt{m_t^{\varepsilon,i}m_t^{\varepsilon,j}}\, d(W_t^{i,j}-W_t^{j,i}),\\ m_0^\varepsilon={}&m_0,\qquad u_T^{\varepsilon,i}=G^i(m_T^\varepsilon), \qquad i\in E. \end{aligned} \tag{3}$$

The running and terminal couplings in (3) are the original $F$ and $G$; no additional cost perturbation designed to make the positive-noise game potential is allowed.

Determine whether every weak limit, as $\varepsilon\downarrow0$, of the laws of $m^\varepsilon$ on $C([0,T];\Delta)$ is supported on the population trajectories $p$ generated by minimizers of the deterministic mean field control problem (1)--(2).

Source reference: Pierre Cardaliaguet and François Delarue, “Selected topics in mean field games,” Proceedings of the International Congress of Mathematicians 2022, vol. 5, EMS Press, 2022, pp. 3660–3703, Section 4.3 (“Selection of equilibria,” unnumbered open question), https://doi.org/10.4171/ICM2022/17.