Extended mean-field games

Pierre-Louis Lions, Panagiotis E Souganidis

► To cite this version:

Pierre-Louis Lions, Panagiotis E Souganidis. Extended mean-field games. Rendiconti Lincei. Matematica e Applicazioni, 2020, 31 (3), pp.611-625. 10.4171/RLM/907. hal-04999174

HAL Id: hal-04999174

https://college-de-france.hal.science/hal-04999174v1

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**Partial Differential Equations** — *Extended mean-field games*, by PIERRE-LOUIS LIONS and PANAGIOTIS E. SOUGANIDIS, communicated on 28 April 2020.

**ABSTRACT.** — We introduce a new class of coupled forward-backward in time systems consisting of a forward Hamilton–Jacobi and a backward quasilinear transport equation, which we call extended mean-field games system. This new class of equations strictly contains the classical mean-field games system with no common noise and its homogenization limit, and optimal transportation-type control problems. We also identify a new and meaningful “monotonicity”-type condition that yields well-posedness. The same condition yields uniqueness in the Hilbertian setting for the master equation without common noise as well as the hyperbolic system describing finite-state mean-field games.

**KEY WORDS:** Mean-field games, back-forward systems

**MATHEMATICS SUBJECT CLASSIFICATION:** 35B27, 35B40, 35K40, 35K59, 91A13

# 1. INTRODUCTION

We introduce a new forward-backward in time system partial differential (PDE for short) which we call extended mean-field games (MFG for short) system. It consists of a forward (in time) Hamilton–Jacobi equation and a backward (in time) quasilinear transport equation. This new class of systems strictly contains the classical MFG-system without common noise introduced by Lions and Lasry [4] and, in a special case, by Huang, Malhamé, and Caines [3], the homogenization limit of MFG-systems with no common noise and local interactions that was studied by the authors in [6] and general optimal transportation-type control problems. We note that the terminology “extended MFG” has been used previously but somehow incorrectly. Indeed, the models called previously extended turned out to be classical MFG.

We also bring forward a new “monotonicity-type” condition generalizing the classical Lasry–Lions monotonicity condition (see [4]) which yields the well-posedness of the extended MFG system. We verify that this criterion is meaningful, and, hence, the correct one, for all the three general classes of problems mentioned above. Finally, we point out the equivalence between this new condition and the ellipticity of an associated second-oder PDE and its linerization as well as the monotonicity condition needed for the well-posedness in the Hilbertian setting for the master equation without common noise as well as the hyperbolic system describing finite-state mean-field games.

In full generality we consider the system
$$ (1.1) \qquad \left\{ \begin{array}{l} \partial_t u + H(Du, x, m) = 0 \quad \text{in } \mathbb{R}^d \times (0, T], \\ \partial_t m + \text{div } B(Du, x, m) = 0 \quad \text{in } \mathbb{R}^d \times [0, T), \\ m(\cdot, T) = m_T \quad \text{and} \quad u(\cdot, 0) = G(\cdot, m(\cdot, 0)) \quad \text{on } \mathbb{R}^d. \end{array} \right. $$

To avoid any difficulties due to unboundedness of the domain we assume that

$$ (1.2) \qquad u, m : \mathbb{R}^d \times [0, T] \to \mathbb{R} \text{ are periodic in the unit cube } \mathbb{T}^d. $$

We also assume that

$$ (1.3) \qquad \left\{ \begin{array}{l} H : \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R} \to \mathbb{R} \text{ is Lipschitz continuous in all arguments,} \\ \text{convex in the first, and } \mathbb{T}^d\text{-periodic in the second,} \end{array} \right. $$

$$ (1.4) \qquad \left\{ \begin{array}{l} G : \mathbb{R}^d \times \mathbb{R} \to \mathbb{R} \text{ is Lipschitz continuous in both arguments,} \\ \text{and } \mathbb{T}^d\text{-periodic in the first,} \end{array} \right. $$

and

$$ (1.5) \qquad \left\{ \begin{array}{l} B : \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R} \to \mathbb{R}^d \text{ is Lipschitz continuous in all arguments,} \\ \text{and } \mathbb{T}^d\text{-periodic in the second.} \end{array} \right. $$

We note that we have made no effort to optimize the assumptions on $H$ and $B$, and, routinely, throughout the paper we work as they were smooth. In particular, the Lipschitz assumption is indeed very strong and can be easily replaced by locally Lipschitz.

We are interested in identifying a general condition guaranteeing the uniqueness of solutions of (1.1). We also discuss briefly the existence for short time of smooth solutions of (1.1).

Throughout the paper and unless otherwise said, we work with functions that have all the necessary regularity to perform the calculations.

For the uniqueness, in addition the classical in the MFG theory assumptions that $H$ is nonincreasing in $m$, that is,

$$ (1.6) \qquad H_m \le 0 \quad \text{a.e. in } (p, x, m) \in \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R}, $$

and $G$ is nondecreasing in $m$, that is,

$$ (1.7) \qquad G_m \ge 0 \quad \text{a.e. in } (x, m) \in \mathbb{R}^d \times \mathbb{R}, $$

we introduce a new hypothesis, that is, we assume that, in the sense of symmetric $d \times d$-matrices, and a.e. in $(p, x, m) \in \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R}$,

$$ (1.8) \qquad (B_m - D_p H) \otimes (B_m - D_p H) \le 4 D_p B(-H_m). $$
Note that, at least when $H_m < 0$, (1.8) implies that, a.e. in $(p, x, m) \in \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R}$ and in the sense of symmetric $d \times d$-matrices,

$$ (1.9) \qquad D_p B(p, x, m) \geq 0. $$

The result we prove is stated next.

**THEOREM 1.1.** Assume (1.3), (1.4), (1.5), (1.6), (1.7) and (1.8). Then (1.1) has at most one smooth solution, which exists if the horizon $T > 0$ is sufficiently small.

We remark that Theorem 1.1 also holds true, if we assume the presence of idiosyncratic noise of strength $\sqrt{2\nu}$, that is, for the system

$$ (1.10) \qquad \begin{cases} \partial_t u - \nu \Delta u + H(Du, x, m) = 0 & \text{in } \mathbb{R}^d \times (0, T], \\ \partial_t m + \nu \Delta m + \text{div} \, B(Du, x, m) = 0 & \text{in } \mathbb{R}^d \times [0, T), \\ m(\cdot, T) = m_T & \text{and } u(\cdot, 0) = G(\cdot, m(\cdot, 0)) \text{ on } \mathbb{R}^d. \end{cases} $$

For simplicity, here we assume that $\nu = 0$, that is, we only discuss (1.1).

We mention next briefly some examples of systems that are strictly covered by it. All of them are discussed in detail in subsequent sections.

The classical MFG-system is

$$ (1.11) \qquad \begin{cases} \partial_t u - \nu \Delta u + H(Du, x, m) = 0 & \text{in } \mathbb{R}^d \times (0, T], \\ \partial_t m + \nu \Delta m + \text{div} [D_p H(Du, x, m)m] = 0 & \text{in } \mathbb{R}^d \times [0, T), \\ m(\cdot, T) = m_T & \text{and } u(\cdot, 0) = G(\cdot, m(\cdot, 0)) \text{ on } \mathbb{R}^d. \end{cases} $$

Obviously (1.11) is a special case of (1.10) with $B(p, x, m) = D_p H(p, x, m)m$. In the case of local interactions, it was shown in [5] that, if $H$ satisfies (1.3) and, in the sense of symmetric matrices and for a.e. all $(p, x, m) \in \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R}$,

$$ (1.12) \qquad (D_{pm}^2 H(p, x, m) \otimes D_{pm}^2 H(p, x, m)) \leq 4D_{pp}^2 H(p, x, m)m(-H_m), $$

then (1.11) has a unique solution. It is immediate that, when $B(p, x, m) = D_p H(p, x, m)m$, (1.8) reduces to (1.12).

A second example is the system obtained in the homogenization limit $\epsilon \to 0$ of an oscillatory MFG-system with local interactions like

$$ (1.13) \qquad \begin{cases} \partial_t u^\epsilon - \epsilon \Delta u^\epsilon + H(Du^\epsilon, \frac{x}{\epsilon}, m^\epsilon) = 0 & \text{in } \mathbb{R}^d \times (0, T], \\ \partial_t m^\epsilon + \epsilon \Delta m^\epsilon + \text{div} \left[ D_p H(Du^\epsilon, \frac{x}{\epsilon}, m^\epsilon) m^\epsilon \right] = 0 & \text{in } \mathbb{R}^d \times [0, T), \\ u^\epsilon(\cdot, 0) = u_0 & \text{and } m(\cdot, T) = m_T \text{ on } \mathbb{R}^d, \end{cases} $$

which was studied in [6] and leads to the system
$$ (1.14) \qquad \begin{cases} \partial_t \bar{u} + \bar{H}(D\bar{u}, \bar{m}) = 0 & \text{in } \mathbb{R}^d \times (0, T], \\ \partial_t \bar{m} + \text{div}[\bar{A}(D\bar{u}, \bar{m})\bar{m}] = 0 & \text{in } \mathbb{R}^d \times [0, T), \\ \bar{u}(\cdot, 0) = u_0 & \text{and } \bar{m}(\cdot, T) = m_T \text{ on } \mathbb{R}^d, \end{cases} $$

where are $\bar{H}$ and $\bar{A}$ are respectively the effective Hamiltonian and density flow. It was shown in [6] that, for a general class of MFG,

$$ (1.15) \qquad \bar{A}(p,m) \neq D_p \bar{H}(p,m), $$

that is the limiting system is not of MFG-type. It is, however, a special case of (1.1) with $B(p,x,m) = \bar{A}(p,m)m$. This raised the obvious question whether the limit of well-posed problems may not be itself well-posed. Details about the above are described in Section 3.

The last example of problems giving rise to an extended MFG-system arises in the optimal transportation-type control problems which are described in detail in Section 4.

*Organization of the paper.* In Section 2 we prove Theorem 1.1 and discuss other contexts in which (1.8) appears and makes sense. In Section 3 we discuss the homogenization example. The optimal transportation problem is explained in Section 4.

## 2. THE WELL-POSEDNESS OF THE EXTENDED MFG SYSTEM

We begin with the proof of Theorem 1.1. The uniqueness proof follows the general Lasry-Lions uniqueness argument, see, for example, [4]. At the heart of the argument is the elementary calculus observation that, given $\phi: \mathbb{R}^m \times \mathbb{R}^m \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ with $\phi(a, a, b, b) = 0$, then a sufficient condition to have $\phi(a_1, a_2, b_1, b_2) \ge 0$ for all $a_1, a_2 \in \mathbb{R}^m$ and all $b_1, b_2 \in \mathbb{R}^n$ is that, for all $a, p \in \mathbb{R}^m$ and $b, q \in \mathbb{R}^n$,

$$ (2.1) \qquad \frac{d}{d\epsilon}_{|\epsilon=0} \phi(a + \epsilon p, a, b + \epsilon q, b) \ge 0. $$

**THE PROOF OF THEOREM 1.1.** We concentrate here on the uniqueness of the solutions. The short time existence of smooth solutions follows as in the classical MFG-system (1.11). We refer to this at the end of the current section.

For the uniqueness, we assume that $(u_1, m_1)$ and $(u_2, m_2)$ are solutions of (1.1) with $m_1(\cdot, T) = m_2(\cdot, T)$ and $u_1(\cdot, 0) = G(\cdot, m_1(\cdot, 0))$ and $u_2(\cdot, 0) = G(\cdot, m_2(\cdot, 0))$. Recall that we have assumed that we work on $\mathbb{T}^d$ and that the solutions are sufficiently regular to perform the calculations that follow.

Following [4] we multiply the equations satisfied by $u_1 - u_2$ and $m_1 - m_2$ by $m_1 - m_2$ and $u_1 - u_2$ respectively and then we integrate over $\mathbb{T}^d$.

It follows that
$$ (2.2) \qquad \left\{ \begin{aligned} & \frac{d}{dt} \int (u_1 - u_2)(m_1 - m_2) \, dx \\ & = \int [(B(Du_1, x, m_1) - B(Du_2, x, m_2)) \cdot D(u_1 - u_2) \\ & \phantom{=} - (H(Du_1, x, m_1) - H(Du_2, x, m_2))(m_1 - m_2)] \, dx. \end{aligned} \right. $$

After integrating (2.2) in time over $[0, T]$ and using the initial and terminal conditions, we get

$$ (2.3) \qquad I + II = 0, $$

where

$$ (2.4) \qquad \left\{ \begin{aligned} I &= \int_0^T \int [(B(Du_1, x, m_1) - B(Du_2, x, m_2)) \cdot D(u_1 - u_2) \\ & \phantom{=} - (H(Du_1, x, m_1) - H(Du_2, x, m_2))(m_1 - m_2)] \, dx \, dt, \end{aligned} \right. $$

and

$$ (2.5) \qquad II = \int (G(x, m_1(x, 0)) - G(x, m_2(x, 0)))(m_1(x, 0) - m_2(x, 0)) \, dx \ge 0, $$

the last inequality following from (1.4).

To obtain a contradiction, it suffices to have

$$ I \ge 0, $$

which follows, if, for all $p_1, p_2 \in \mathbb{R}^d, x \in \mathbb{T}^d$ and $m_1, m_2 \in \mathbb{R}$,

$$ (2.6) \qquad \left\{ \begin{aligned} \Phi(p_1, p_2, m_1, m_2) &= (B(p_1, x, m_1) - B(p_2, x, m_2)) \cdot (p_1 - p_2) \\ & \phantom{=} - (H(p_1, x, m_1) - H(p_2, x, m_2))(m_1 - m_2) \ge 0. \end{aligned} \right. $$

Actually to get a contradiction we need to have either $II > 0$, which means that $G$ must be strictly increasing in $m$ or $I > 0$, which requires (1.8) to hold with a strict inequality.

In what follows, we ignore this point and proceed to identify a local condition that implies (2.6). Moreover, since the space variable $x$ can be thought as parameter, from now on we ignore it.

It is immediate from (2.6) that $\Phi(p_2, p_2, m_1, m_2) = 0$. Hence, following the observation before the beginning of the ongoing proof, for $\varepsilon > 0$, $p, q \in \mathbb{R}^d$ and $m, n \in \mathbb{R}$, we set $p_2 = p, p_1 = p + \varepsilon q, m_2 = m$ and $m_1 = m + \varepsilon n$ in (2.6), we differentiate with respect to $\varepsilon$, let $\varepsilon = 0$ and require that the derivative is nonnegative. It follows that, for all $p, q \in \mathbb{R}^d, x \in \mathbb{R}^d$ and $m, n \in \mathbb{R}$,

$$ (2.7) \qquad D_p B(p, x, m) q \cdot q + B_m(p, x, m) \cdot qn - D_p H(p, x, m) \cdot qn \\ \qquad \phantom{(2.7) \qquad} - H_m(p, x, m) n^2 \ge 0. $$
Since $-H_m \ge 0$, for all $p, q \in \mathbb{R}^d$, $x \in \mathbb{R}^d$ and $m, n \in \mathbb{R}$, we must have

$$ (2.8) \quad |(B_m(p, x, m) - D_p H(p, x, m)) \cdot q|^2 \le 4(-H_m(p, x, m)) D_p B(p, x, m) q \cdot q. $$

The claim now follows. $\square$

We discuss next an interpretation of (1.8) in terms of the ellipticity of certain second-order pde.

Following an observation of the first author in the context of (1.11) (see [5]), when $H$ is separated and strictly decreasing in $m$, it is possible to rewrite (1.1) as a single equation. For simplicity, in what follows, we also assume that there is no explicit dependence on $x$.

We consider the system

$$ (2.9) \quad \left\{ \begin{array}{ll} \partial_t u + H(Du, x) = f(m, x) & \text{in } \mathbb{R}^d \times (0, T], \\ \partial_t m + \operatorname{div} B(Du, x, m) = 0 & \text{in } \mathbb{R}^d \times [0, T), \\ m(\cdot, T) = m_T & \text{and } u(\cdot, 0) = G(\cdot, m_T) \text{ on } \mathbb{R}^d, \end{array} \right. $$

we assume that $m \to f(m, x)$ is strictly decreasing for all $x$ and denote by $F(\cdot, x)$ its inverse.

Then

$$ m = F(\partial_t u + H(Du), x) $$

and the second equation in (2.9) gives rise to the second-order PDE

$$ \partial_t F(\partial_t u + H(Du), x) + \operatorname{div}[B(Du, x, F(\partial_t u + H(Du), x))] = 0 $$

which, after some calculations, reduces to

$$ (2.10) \quad u_{tt} + (D_p H + B_m) \cdot Du_t + \operatorname{tr}[(f' D_p B + D_p H \otimes B_m) D^2 u] \\ \qquad + f' \operatorname{div}_x B + B_m \cdot (D_x H + D_x f) = 0, $$

where $\operatorname{div} B = \sum_{i=1}^d B_{x_i}^i$ and, to simplify the expression, we omitted the arguments of $f$, $D_p H$ and $B_m$, $D_p B$ which are respectively $F(\partial_t u + H(Du), x)$, $(Du, x)$ and $(Du, x, F(\partial_t u + H(Du))$.

It is immediate to check that (2.10) is degenerate elliptic if and only if (1.8) holds.

The interpretation of (1.8) as an “ellipticity property” can be seen more easily by considering the linearization of (1.1) around a smooth solution $(u, m)$.

The linearized system is

$$ (2.11) \quad \left\{ \begin{array}{ll} \partial_t \phi + D_p H(Du, x, m) \cdot D\phi + H_m(Du, x, m)\psi = 0, \\ \partial_t \psi + \operatorname{div}[D_p B(Du, x, m) \cdot D\phi + B_m \psi] = 0. \end{array} \right. $$
Hence, if $H_m < 0$, from the first equation in (2.11) we find

$$ \psi = - \frac{1}{H_m(Du, x, m)} (\hat{\partial}_t \phi + D_p H(Du, x, m) \cdot D\phi). $$

Inserting the information above in the second equation of (2.11) and omitting, for notational simplicity, the dependence of $H_m$, $D_p H$, $D_p B$ and $B_m$ on $(Du, x, m)$ we get

$$ (2.12) \qquad \partial_t \left[ -\frac{1}{H_m} (\partial_t \phi + D_p H \cdot D\phi) \right] \\ \qquad \qquad + \operatorname{div} \left[ D_p B \cdot D\phi - \frac{1}{H_m} B_m (\partial_t \phi + D_p H \cdot D\phi) = 0 \right]. $$

Given that $H_m < 0$, (2.12) is elliptic if and only if

$$ (-H_m)DB_p + B_m \otimes D_p H \geq \frac{1}{4}(D_p H + B_m) \otimes (D_p H + B_m). $$

The last inequality is clearly equivalent to (1.8).

We also point out that, employing the implicit function theorem, we can use the ellipticity of the linearized system to prove the existence and uniqueness of smooth solutions of (1.1) for small $T > 0$.

We conclude this section with a discussion the importance of (1.8) for the uniqueness of smooth solutions in the Hilbertian approach (see [5]) for the master equation for MFG without common noise or the general nonsymmetric hyperbolic systems arising in the study of discrete in space MFG.

In the latter case, it is necessary to study the uniqueness (and existence) of solutions of the system

$$ (2.13) \qquad \frac{dU}{dt} + (B(U, X) \cdot D)U + A(U, X) = 0 $$

which is set in a Hilbert space $H$ which may be $\mathbb{R}^d$ or $L^2((\Omega, \mathcal{F}, \mathbb{P}); \mathbb{R}^d)$ for some fixed probability $(\Omega, \mathcal{F}, \mathbb{P})$.

The classical structural condition for the well-posedness of (2.13) is the monotonicity in $(X, U)$ of the map $(X, U) \to (A(U, X), B(U, X))$, which is equivalent to

$$ (2.14) \qquad \left[ \begin{array}{cc} -A_X & \frac{B_U - A_X}{2} \\ \frac{B_U - A_X}{2} & B_U \end{array} \right] \geq 0. $$

It is immediate to check that (2.14), with $A$ in place of $H$, and (1.8) are the same.
In the Hilbertian approach for MFG without common noise, the problem reduces to studying (2.13) with A and B replaced respectively by H and $D_pH$.

### 3. THE HOMOGENIZATION EXAMPLE

We review now the homogenization problem studied in [6]. The aim is to show that, although the limit system (1.14) is, in general, not of MFG-type, it does satisfy, for all $p \in \mathbb{R}^d$ and $m \in \mathbb{R}$,

$$ (3.1) \quad \begin{cases} (\bar{A}(p,m) - D_p \bar{H}(p,m) + m\bar{A}_m) \otimes (\bar{A}(p,m) - D_p \bar{H}(p,m) + m\bar{A}_m(p,m)) \\ \le 4D_p \bar{A}(p,m)m(-\bar{H}_m(p,m)), \end{cases} $$

which is the condition following from (1.8) when $B(p,m) = \bar{A}(p,m)m$ and is the same as (1.12) when $A = D_p \bar{H}$, that is when the limit system is of MFG-type.

We argue formally and we refer to [6] for the details. In what follows $\langle \phi \rangle = \int_{\mathbb{T}^d} \phi(x) dx$ and $|\mathbb{T}^d| = 1$.

We assume that the solution $(u^e, m^e)$ of (1.13) has the formal expansion

$$ (3.2) \quad \begin{aligned} u^e(x,t) &\approx \bar{u}(x,t) + \epsilon v \left(x, \frac{x}{\epsilon}, t\right) + \dots \quad \text{and} \\ m^e(x,t) &\approx \bar{m}(x,t) \mu \left(x, \frac{x}{\epsilon}, t\right) + \dots, \end{aligned} $$

with $\mu \ge 0$ and $\int_{\mathbb{T}^d} \mu(x, y, t) dy = 1$, and $v, \mu \in \mathbb{T}^d$-periodic in the second argument that we denote by $y$. In what follows we omit the dependence of $v$ and $\mu$ in $x$ and $t$.

After substituting (3.2) in (1.13) we find that there must exist $\bar{H}(p, \lambda)$, where $p = D\bar{u}(x, t)$ and $\lambda = \bar{m}(x, t)$, such that $v$ and $\mu$ in (3.2) satisfy the system

$$ (3.3) \quad \left\{ \begin{array}{ll} -\Delta_y v + H(p + D_y v, y, \lambda \mu) = \bar{H}(p, \lambda) & \text{in } \mathbb{T}^d, \\ -\Delta_y \mu + \operatorname{div}(D_p H(p + D_y v, y, \lambda \mu)) = 0 & \text{in } \mathbb{T}^d, \\ \mu \ge 0 \quad \text{and} \quad \langle \mu \rangle = 1. \end{array} \right. $$

Once such $\bar{H}$, $v$ and $\mu$ are found, it follows that $\bar{u}$ and $\bar{\mu}$ solve the homogenized system (1.14) with

$$ (3.4) \quad \bar{A}(p,m) = \langle D_p H(p + D_y v, y, \lambda \mu) \mu \rangle. $$

The solvability of the cell-system (3.3) was established in [4]. The homogenization of (1.13), under the assumption that (1.14) has a smooth solution was proved in [6].

The same reference also identifies a class of MFG (potential games) which homogenize to a system that is not of MFG-type, that is, (1.15) holds. A question left unanswered in [6] was whether (1.14) has a unique solution.

Before we address the last point, we revisit briefly the issue of whether (1.14) is of MFG-type or not.
The class of MFG identified in [6] are potential games with “separated” dependence on $Du$ and $m$, that is, Hamiltonians of the form

$$H(p, y) - f(y, m)$$

and

$$f(y, m) = \frac{\delta F}{\delta m}(y, m),$$

for some $\mathbb{T}^d$-periodic in the first and convex in the second argument $F: \mathbb{R}^d \times \mathbb{R} \to \mathbb{R}$.

Potential MFG admit a variational representation which is amenable to $\Gamma$-convergence methods, and, hence, have a limit functional. In general, this functional does not have separated dependence in $p$ and $m$, which is a necessary and sufficient condition for an MFG to have a variational characterization.

Next we derive the property needed for (1.14) to be of MFG-type.

Differentiating the first equation of (3.3) in $p_i$, for each $i = 1, \dots, d$, we find

$$-\Delta \frac{\partial v}{\partial p_i} + H_{p_i} + D_p H \cdot D \frac{\partial v}{\partial p_i} + \lambda H_m \frac{\partial \mu}{\partial p_i} = \bar{H}_{p_i}.$$

Multiplying the identity above by $\mu$, integrating over $\mathbb{T}^d$ and using the equation satisfied by $\mu$ we find

$$(3.5) \quad \bar{A}(p, \lambda) = \langle D_p H(p, y, \lambda\mu)\mu \rangle = D_p \bar{H}(p, \lambda) - \lambda \langle H_m(p, y, \lambda\mu)\mu D_p \mu \rangle.$$

Hence, (1.14) is of MFG-type if and only if

$$(3.6) \qquad \langle H_m(p, y, \lambda\mu) D_p(\mu)^2 \rangle = 0.$$

For the separated MFG with Hamiltonian $H(p, y) - f(m)$, (3.6) takes the form

$$(3.7) \qquad \langle f'(\lambda\mu) D_p(\mu^2) \rangle = 0.$$

In the particular case that $f(0) = 0$, $f'(0) \neq 0$ and $\lambda = 0$, (3.7) implies that $D_p\langle\mu^2\rangle = 0$, that is,

$$(3.8) \qquad \langle \mu^2 \rangle \text{ is independent of } p.$$

We discuss next (3.8) in the context of a family of MFG systems with $d = 1$,

$$H(p, y, m) = \frac{1}{2} |p|^2 - V(y) - f(m),$$

and

$$f(0) = 0, \quad f'(0) \neq 0 \quad \text{and} \quad V \text{ periodic in } \mathbb{T} \text{ and } \min V = 0.$$
For $\delta > 0$, $p \in \mathbb{R}$ and $\lambda = 0$ the associated cell-system is of the form

$$ (3.9) \qquad \begin{cases} -\delta v_{yy}^{\delta} + \frac{1}{2} |p + v^{\delta}|^2 = V(y) + \bar{H}^{\delta}(p) & \text{in } \mathbb{T}, \\ \delta \mu^{\delta} + ((p + v_y^{\delta})\mu^{\delta})_y = 0 & \text{in } \mathbb{T}, \\ \mu^{\delta} \ge 0 \quad \text{and} \quad \langle \mu^{\delta} \rangle = 1. \end{cases} $$

In view of the previous discussion, for the corresponding homogenized systems to be of MFG-type, we must have that

$$ (3.10) \qquad \langle (\mu^{\delta})^2 \rangle \text{ is independent of } p. $$

The stability properties of (3.9) imply (3.10) must persist the limit $\delta \to 0$ at least as long as the limit problem has a smooth solution.

Consider the cell-system

$$ (3.11) \qquad \begin{cases} \frac{1}{2} |p + v_y|^2 = V(y) + \bar{H}^0(p) & \text{in } \mathbb{T}, \\ ((p + v_y)\mu)_y = 0 & \text{in } \mathbb{T}, \\ \mu^0 \ge 0 \quad \text{and} \quad \langle \mu^0 \rangle = 1. \end{cases} $$

It is known that, for each $p \in \mathbb{R}$, there exists a unique $\bar{H}^0(p)$ such that the first equation of (3.11) has a solution.

If $p > \langle V^{1/2} \rangle$, then $H^0(p)$ is defined by $|p| = \langle (V + \bar{H}^0(p))^{1/2} \rangle$, and (3.11) has a smooth solution $(v^0, \mu^0)$ given by $v_y^0 + p = (V + \bar{H}^0(p))^{1/2}$ and

$$ \mu^0 = \frac{c(p)}{(V + \bar{H}^0(p))^{1/2}} $$

with $c(p)$ chosen so that $\langle \mu^0 \rangle = 1$, that is,

$$ c(p) = \frac{1}{\int \frac{1}{(V + \bar{H}^0(p))^{1/2}}} $$

It is known that, as $\delta \to 0$, $\bar{H}^\delta(p)$ and $v^\delta \to v^0$ uniformly in $\mathbb{T}$, and, for $|p| > \langle V^{1/2} \rangle$, $\mu^\delta \to \mu^0$ in $L^1(\mathbb{T}) \cap L^2(\mathbb{T})$. Then $\langle \mu^{0,2} \rangle$ must be independent of $p$.

Thus, there must exist $\theta > 0$, which is independent of $p$ and such that, for $p > \langle V^{1/2} \rangle$,

$$ c^2(p) \int \frac{1}{V + \bar{H}^0(p)} = \theta, $$

that is,

$$ (3.12) \qquad \int \frac{1}{V + \bar{H}^0(p)} = \theta \left( \int \frac{1}{(V + \bar{H}^0(p))^{1/2}} \right)^2. $$
Note that, since $|p| = \langle (V + \bar{H}^0(p))^{1/2} \rangle$, it follows that

$$ \lim_{|p| \to \infty} \frac{\bar{H}^0(p)}{|p|^2} = 1. $$

Using this last observation in (3.12) yields that we must have $\theta = 1$, and, hence,

$$ \int \frac{1}{V + \bar{H}^0(p)} = \left( \int \frac{1}{(V + \bar{H}^0(p))^{1/2}} \right)^2, $$

which is not possible unless $V$ is a constant.

We return now to the question of the wellposedness of (1.14). Instead of verifying that (3.1) holds, which requires considerable amount of calculations involving differentiating the cell-system, we provide here a more direct proof.

For this we assume that we have two solutions of $(\bar{u}_1, \bar{m}_1)$ and $(\bar{u}_2, \bar{m}_2)$ of (1.14) which are associated with the solutions $(v_1, \mu_1)$ and $(v_2, \mu_2)$ of the cell system (3.3) with $p_1 = D\bar{u}_1(x, t)$, $p_2 = D\bar{u}_2(x, t)$, $\lambda_1 = \bar{m}_1(x, t)$ and $\lambda_2 = \bar{m}_2(x, t)$ and we repeat the general uniqueness argument described in the previous section.

Uniqueness follows, if we show that

$$ (3.13) \qquad \begin{cases} (\bar{A}(D\bar{u}_1, \bar{m}_1)\bar{m}_1 - \bar{A}(D\bar{u}_2, \bar{m}_2)\bar{m}_2) \cdot (D\bar{u}_1 - D\bar{u}_2) \\ - (\bar{H}(D\bar{u}_1, \bar{m}_1) - \bar{H}(D\bar{u}_2, \bar{m}_2))(\bar{m}_1 - \bar{m}_2) \ge 0. \end{cases} $$

We claim that (3.13) follows from the analogous property of the original problem. Since $x$ and $t$ are constant below, we omit them.

Subtracting the first two equations of the cell systems, multiplying the difference by $(\bar{m}_1\mu_1 - \bar{m}_2\mu_2)$ and integrating over $\mathbb{T}^d$ we get

$$ \begin{aligned} & (\bar{H}(p_1, \lambda_1) - \bar{H}(p_2, \lambda_2))(\lambda_1 - \lambda_2) \\ &= \langle (\bar{H}(p_1, \lambda_1) - \bar{H}(p_2, \lambda_2))(\lambda_1 \mu_1 - \lambda_2 \mu_2) \rangle \\ &= \langle (H(p_1 + Dv_1, \lambda_1 \mu_1) - H(p_2 + Dv_2, \lambda_2 \mu_2))(\lambda_1 \mu_1 - \lambda_2 \mu_2) \rangle \\ &\quad - \langle (v_1 - v_2)\Delta(\lambda_1 \mu_1 - \lambda_2 \mu_2) \rangle. \end{aligned} $$

Multiplying the second equation of each cell system by $\lambda_1$ and $\lambda_2$ respectively, subtracting the two equations, multiplying the difference by $(v_1 - v_2)$ and integrating over $\mathbb{T}^d$ gives

$$ \begin{aligned} 0 &= \langle (v_1 - v_2)\Delta(\lambda_1 \mu_1 - \lambda_2 \mu_2) \rangle \\ &\quad + \langle D(v_1 - v_2) \cdot (D_p H(p_1 + Dv_1, \lambda_1 \mu_1)\lambda_1 \mu_1 - D_p H(p_2 + Dv_2, \lambda_2 \mu_2)\lambda_2 \mu_2) \rangle \\ &= \langle (v_1 - v_2)\Delta(\lambda_1 \mu_1 - \lambda_2 \mu_2) \rangle \\ &\quad + \langle ((Dv_1 + p_1) - (Dv_2 + p_2)) \cdot (D_p H(p_1 + Dv_1, \lambda_1 \mu_1)\lambda_1 \mu_1 \\ &\quad - D_p H(p_2 + Dv_2, \lambda_2 \mu_2)\lambda_2 \mu_2) \rangle \\ &\quad - \langle (p_1 - p_2) \cdot (D_p H(p_1 + Dv_1, \lambda_1 \mu_1)\lambda_1 \mu_1 - D_p H(p_2 + Dv_2, \lambda_2 \mu_2)\lambda_2 \mu_2) \rangle \end{aligned} $$
$$
\begin{align*}
&= \langle (v_1 - v_2)\Delta(\lambda_1\mu_1 - \lambda_2\mu_2) \rangle \\
&\quad + \langle (Dv_1 + p_1) - (Dv_2 + p_2) \rangle \cdot (D_p H(p_1 + Dv_1, \lambda_1\mu_1)\lambda_1\mu_1 \\
&\quad - D_p H(p_2 + Dv_2, \lambda_2\mu_2)\lambda_2\mu_2) \rangle \\
&\quad - (p_1 - p_2) \cdot (\bar{A}(p_1, \lambda_1)\lambda_1 - \bar{A}(p_2, \lambda_2)\lambda_2).
\end{align*}
$$

Combining the outcomes of the last two sets of equalities we find that

$$
\left\{
\begin{aligned}
& (\bar{A}(D\bar{u}_1, \bar{m}_1)\bar{m}_1 - \bar{A}(D\bar{u}_2, \bar{m}_2)\bar{m}_2) \cdot (D\bar{u}_1 - D\bar{u}_2) \\
& \quad - (\bar{H}(D\bar{u}_1, \bar{m}_1) - \bar{H}(D\bar{u}_2, \bar{m}_2))(\bar{m}_1 - \bar{m}_2) \\
& = \langle (Dv_1 + p_1) - (Dv_2 + p_2) \rangle \cdot (D_p H(p_1 + Dv_1, \lambda_1\mu_1)\lambda_1\mu_1 \\
& \quad - D_p H(p_2 + Dv_2, \lambda_2\mu_2)\lambda_2\mu_2) \rangle \\
& \quad - \langle (H(p_1 + Dv_1, \lambda_1\mu_1) - H(p_2 + Dv_2, \lambda_2\mu_2))(\lambda_1\mu_1 - \lambda_2\mu_2) \rangle \ge 0,
\end{aligned}
\right.
$$

the last inequality following from the well-posedness condition of the original
system.

4. OPTIMAL TRANSPORTATION CONTROL PROBLEMS

We introduce general optimal transportation control problems and show that
they are contained in the class of extended MFG.

The goal is to optimize, for definiteness here minimize, for a given $u_T$ and a
convex $L: \mathbb{R}^d \times \mathbb{R} \to \mathbb{R}$, the functional

$$
(4.1) \quad \int_0^T \int m(x,t)L(\alpha(x,t),m(x,t)) \,dx\,dt + \int u_T(x)m(x,T) \,dx
$$

over all controls $\alpha \in \mathcal{A}$ and $m \ge 0$ solving the continuity equation

$$
(4.2) \quad \partial_t m + \operatorname{div}(\alpha m) = 0 \quad \text{in } \mathbb{R}^d \quad \text{and} \quad m(\cdot, 0) = m_0 \ge 0 \quad \text{in } \mathbb{R}^d;
$$

note that, although $L$ is convex, the functional (4.1) is may not be convex in $\alpha$
and $m$.

The first example of the type of minimization problems discussed here was
introduced by Brenier [1] in the context of optimal transportation/mass transport
and has $L(a) = \frac{1}{2}|\alpha|^2$, (4.2) been the mass continuity equation.

Another example is the functional

$$
(4.3) \quad \int_0^T \int m L(\alpha) + G(m) \,dx\,dt + \int u_T(x) m(x, T) \,dx
$$

with $L: \mathbb{R}^d \to \mathbb{R}$ and $G: \mathbb{R} \to \mathbb{R}$ convex, which (see [5]) is equivalent to a MFG-system.
To turn (4.1) into a convex functional, we write $\beta = m\alpha$. Then (4.1) is rewritten as

$$ (4.4) \qquad \int_0^T \int m(x,t) \mathcal{L}(\beta(x,t), m(x,t)) \, dx \, dt + \int u_T(x) m(x, T) \, dx, $$

with

$$ (4.5) \qquad \mathcal{L}(\beta, m) = mL\left(\frac{\beta}{m}, m\right), $$

and $m$ solving

$$ (4.6) \qquad \partial_t m + \text{div} \beta = 0 \quad \text{in } \mathbb{R}^d \quad \text{and} \quad m(\cdot, 0) = m_0 \ge 0 \quad \text{in } \mathbb{R}^d. $$

It is now a simple calculation to check $\mathcal{L}$ and, hence (4.4), is convex.

Another important example is the so called congestion model with

$$ (4.7) \qquad L(\alpha, m) = \frac{1}{2} m^{\gamma} |\alpha|^{2}, $$

which yields, for $\gamma \in [1, 2]$, the convex functional

$$ (4.8) \qquad \mathcal{L}(\beta, m) = \frac{1}{2} m^{\gamma-1} |\beta|^2. $$

The existence of minimizers of the congestion problem has been established in Cardaliaguet, Graber, Porretta and Tonon [2] were we refer for more discussion.

Here we assume that there exist minimizers. Classical arguments yield the Euler-Lagrange system

$$ (4.9) \qquad \left\{ \begin{array}{l} \partial_t u + H(Du, m) - mH_m(Du, m) = 0 \quad \text{in } \mathbb{R}^d \times (0, T), \\ \partial_t m + \text{div}(D_p H(Du, m)m) = 0 \quad \text{in } \mathbb{R}^d \times [0, T), \\ u(\cdot, T) = u_T \quad \text{and} \quad m(\cdot, 0) = m_0 \quad \text{in } \mathbb{R}^d, \end{array} \right. $$

with $H(p, m) = \sup_{\alpha}\{p \cdot \alpha - L(\alpha, m)\}$.

It is, of course, clear that (4.9) is an MFG-system if and only if $H_m = 0$.

Next we check whether (4.9) satisfies (1.8). Letting $\mathcal{H}(p, m) = H(p, m) - mH_m(p, m)$ and $A(p, m) = D_p H(p, m)$ we rewrite (4.9) as

$$ (4.10) \qquad \left\{ \begin{array}{l} \partial_t u + \mathcal{H}(Du, m) = 0 \quad \text{in } \mathbb{R}^d \times (0, T), \\ \partial_t m + \text{div}(A(Du, m)m) = 0 \quad \text{in } \mathbb{R}^d \times [0, T), \\ u(\cdot, T) = u_T \quad \text{and} \quad m(\cdot, 0) = m_0 \quad \text{in } \mathbb{R}^d, \end{array} \right. $$

and we claim that (4.10) satisfies (3.1) and, thus, is well posed.
It is immediate that $A + mA_m - D_p\mathcal{H} = 0$, while, the convexity of $H$ gives $D_pA = D^2H \ge 0$.

Moreover, we observe that $-\mathcal{H}_m \ge 0$, is equivalent to the concavity of $m \to mH(p, m)$ in $m$.

Indeed, since

$$mL(\frac{\beta}{m}, m) = \sup_{\beta}\{\beta \cdot p - mH(p, m)\},$$

the convexity of $\mathcal{L}(\beta, m)$ in $\beta$ and $m$ is equivalent to the concavity of the map $m \to mH_m(p, m)$.

**ACKNOWLEDGMENTS.** The first author was partially supported by the Air Force Office for Scientific Research grant FA9550-18-1-0494. The second author was partially supported by the National Science Foundation grants DMS-1600129 and DMS-1900599, the Office for Naval Research grant N000141712095 and the Air Force Office for Scientific Research grant FA9550-18-1-0494.

## REFERENCES

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MR 1658919

[2] P. CARDALIAGUET - J. GRABER - A. PORRETTA - D. TONON, Second order mean field games with degenerate diffusion and local coupling, Nonlinear Differential Equations and Applications NoDEA 22, 1287–1317 (2015). MR 3791252

[3] M. Y. HUANG - R. P. MALHAMÉ - P. E. CAINES, *Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle*, Commun. Inf. Syst. 6 (2006), no. 3, 221–251. MR 2346927

[4] J.-M. LASRY - P.-L. LIONS, *Mean field games*, Jpn. J. Math. 2 (2007), no. 1, 229–260.
MR 2295621

[5] P.-L. LIONS, *Mean field games*, Cours de Collège de France.

[6] P.-L. LIONS - P. E. SOUGANIDIS, *Homogenization of the backward-forward system in mean-field games in periodic environments*, Rendiconti Lincei: Matematica.

Received 23 April 2020,
and in revised form 27 April 2020

Pierre-Louis Lions
College de France
11 Place Marcelin Berthelot
75005 Paris, France
and
CEREMADE
Université de Paris-Dauphine
Place du Maréchal de Lattre de Tassigny
75016 Paris, France
lions@ceremade.dauphine.fr
Panagiotis E. Souganidis
Department of Mathematics
University of Chicago
5734 S. University Ave.
Chicago, IL 60637, USA
souganidis@math.uchicago.edu
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