# A variational mean field game of controls with free final time and pairwise interactions

Guilherme Mazanti, Laurent Pfeiffer, Saeed Sadeghi Arjmand

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Guilherme Mazanti, Laurent Pfeiffer, Saeed Sadeghi Arjmand. A variational mean field game of controls with free final time and pairwise interactions. 2026. hal-05518729

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# A variational mean field game of controls with free final time and pairwise interactions

Guilherme Mazanti*†

Laurent Pfeiffer*†

Saeed Sadeghi Arjmand

February 19, 2026

## Abstract

This article considers a mean field game model inspired by crowd motion models in which agents aim at reaching a given target set and wish to minimize a cost consisting of an individual running cost, an individual cost depending on the arrival time at the target set, and an interaction running cost, which takes the form of pairwise interactions with other agents through both positions and velocities. We subsume this game under a more general class of games on abstract Polish spaces with pairwise interactions, and prove that the latter games have a variational structure (in the sense that their equilibria can be characterized as critical points of some potential functional) and admit equilibria. We also discuss two a priori distinct notions of equilibria, providing a sufficient condition under which both notions coincide. The results for the games in abstract Polish spaces are applied to our mean field game model, and a numerical illustration concludes the paper.

**Keywords.** Mean field games, Lagrangian equilibria, congestion games, variational games, potential games, pairwise interactions, existence of equilibria.

**Mathematics Subject Classification 2020.** 49N80, 49J27, 49K27, 91A07, 91A14, 91A16.

## Contents

<table><tr><td>1</td><td>Introduction</td><td>1</td></tr><tr><td>2</td><td>Mean field game model with pairwise interactions</td><td>4</td></tr><tr><td>3</td><td>A general non-atomic game with pairwise interactions</td><td>6</td></tr><tr><td>3.1</td><td>First properties</td><td>8</td></tr><tr><td>3.2</td><td>Potential game</td><td>9</td></tr><tr><td>3.3</td><td>Existence of equilibria</td><td>13</td></tr><tr><td>3.4</td><td>Strong equilibria</td><td>14</td></tr><tr><td>4</td><td>Application to the mean field game model with pairwise interactions</td><td>15</td></tr><tr><td>5</td><td>Numerical illustration</td><td>23</td></tr></table>

## 1 Introduction

Since the introduction of mean field games (MFGs) in 2006 by Jean-Michel Lasry and Pierre-Louis Lions [63–65] and the simultaneous independent works by Peter E. Caines, Minyi Huang, and Roland P. Malhamé [55–57], many authors have proposed MFG models for (or inspired by) the analysis of crowd motion [8, 10, 20, 38, 62, 68, 78]. While the mathematical description of the

<footnote>*Université Paris-Saclay, CNRS, CentraleSupélec, Inria, Laboratoire des signaux et systèmes, 91190 Gif-sur-Yvette, France.</footnote>

<footnote>†Fédération de Mathématiques de CentraleSupélec, 91190, Gif-sur-Yvette, France.</footnote>
movement of crowds of people has attracted the interest of several researchers along the years (see, e.g., [7, 53, 54, 58, 66, 71]), the MFG approach to crowd motion has the particularity that, by considering agents of the crowd to be rational, one may consider that each agent chooses their trajectory based not only on the present or past states of other agents, but also on a rational anticipation of their future behavior. Our aim in this paper is to propose and study an MFG model inspired by crowd motion in which agents interact through a Cucker-Smale-type interaction, in the spirit of [78].

To the authors' knowledge, the first work to be fully dedicated to a mean field game model for crowd motion is [62], which proposes an MFG model for a two-population crowd with trajectories perturbed by additive Brownian motion and considers both their stationary distributions and their evolution on a prescribed time interval. Other works have later proposed MFG models for crowd motion taking into account different characteristics. To cite a few examples, let us mention [20], which considers the fast exit of a crowd and proposes a mean field game model which is studied numerically; [30], which is not originally motivated by the modeling of crowd motion but considers an MFG model with a density constraint, which is a natural assumption in some crowd motion models (see also [69] for the case of second-order mean field games with density constraints); [1, 33], which study a two-population MFG model motivated by urban settlements; [10], which presents numerical simulations for some variational mean field games related to crowd motion; [8], which, inspired by cities crowded by tourists such as Venice, Italy, proposes a mean field game model for the movement of pedestrian tourists; [38], which provides a generalized MFG model for pedestrians with limited predictive abilities; or also [43, 44, 67, 68, 75, 76], which consider MFG models in which the goal of each agent is to reach a given target set in minimal time, interactions between agents being modeled through a congestion-dependent maximal velocity. Other works studying mean field games motivated by or related to crowd motion include [2, 6, 9, 32, 41].

A common feature of several MFG models for crowd motion is that, at a given time $t$, an agent at a space position $x$ will interact with other agents only through the distribution $m_t$ of their positions in the state space at time $t$. While this already provides models with interesting features, a position-only interaction fails to capture important aspects of crowd motion. Indeed, in [69], given two agents at a certain distance from one another, it is natural to expect that their interaction will be qualitatively different depending on whether they are moving towards or away from one another, since in the first case they are expected to deviate in order to avoid a collision, while, in the second case, their movements are not expected to have a mutual interference. This motivates the study of MFG models for crowd motion in which the interaction through agents is based not only on their positions, but also on their velocities, or, equivalently for many models, on their controls.

Mean field game models in which the dependence on other agents takes into account not only their positions but also their controls are called *MFGs of controls*. Their analysis, started in [48, 49], has been much developed in recent years (see, e.g., [16, 22, 29, 42, 51, 52, 60, 61, 78], and, in particular, the PhD thesis [59]). In the context of crowd motion, [78] has proposed a model<fnref n="1" /> which assumes that agents evolve in a given set $\Omega \subset \mathbb{R}^d$, that the trajectory $\gamma$ of an agent choosing a control $u$ is described by the control system $\dot{\gamma}(t) = u(t)$, and that an agent chooses their control $u$ in order to minimize a cost of the form

$$ \int_0^T \left( \frac{\delta}{2} |\dot{\gamma}(t)|^2 + \frac{\lambda}{2} \int_{C([0,T], \Omega)} \eta(\gamma(t) - \tilde{\gamma}(t)) |\dot{\gamma}(t) - \dot{\tilde{\gamma}}(t)|^2 dQ(\tilde{\gamma}) \right) dt + \Psi(\gamma(T)), \quad (1) $$

where $\delta$ and $\lambda$ are positive constants, $\eta: \mathbb{R}^d \to \mathbb{R}_+$ is an interaction kernel, $T > 0$ is a fixed time horizon, and $Q$ is a probability measure on the set of all continuous trajectories $C([0, T], \Omega)$ describing the distribution of trajectories of all agents.

Given an initial distribution of agents $m_0$, described mathematically as a probability measure on $\Omega$, an equilibrium of this MFG model with initial condition $m_0$ is then a probability measure $Q$ on $C([0, T], \Omega)$ whose evaluation at time $0$ is$^2 m_0$ and which is concentrated on trajectories $\gamma$ that are optimal for (1) with a fixed initial condition. It is proved in [78, Proposition 3.1] that this

<footnote n="1">¹We point out that the authors of [78] argue that whether the model they analyze is indeed an MFG of controls is up to debate, and that “classifying it or not as an MFG of controls is a matter of taste”.</footnote>

<footnote n="2">²That is, $e_0 \neq Q = m_0$, using the notation described in page 4 below.</footnote>
MFG model is variational (also called potential), i.e., that there exists a function J, defined over the space of probability measures on C([0, T], Ω) whose evaluation at time 0 is m₀, such that (local) minimizers of J are equilibria of the MFG with initial condition m₀. In particular, this variational structure allows one to prove the existence of an equilibrium for the MFG by proving the existence of a (global) minimizer of J, which is done in [78, Theorem 3.1]. We also mention that [78] provides additional properties of equilibria, such as the fact that, under suitable additional assumptions, optimal trajectories are C¹,¹ and unique.

Note that this way of describing equilibria as probability measures on the space of trajectories corresponds to the Lagrangian formulation of mean field games. The Lagrangian formulation is a classical approach in optimal transport problems (see, e.g., [4, 77, 79]), which has been used for instance in [18] to study incompressible flows, in [31] for Wardrop equilibria in traffic flow, or in [11] for branched transport problems. The use of the Lagrangian approach in mean field games dates back at least to [27, 30], and since then it has been used in several works, such as [10, 24–26, 44, 45, 67, 68, 74–76].

The model of [78] is inspired by the Cucker–Smale model. Introduced in [39] to model the evolution of a flock of birds, the Cucker–Smale model assumes that a finite number of agents (which represent birds in the original model) evolve according to a prescribed law taking into account the fact that agents wish to align their velocities with others, and that this alignment effect is stronger when the birds are closer. In (1), this is modeled by the term $η(γ(t) - \tilde{γ}(t))|\dot{γ}(t) - \tilde{γ}(t)|^2$, which penalizes large differences in velocities for agents that are close if one assumes that $η$ is a Gaussian-like kernel (for instance, nonnegative, depending only on the modulus of its argument, and nonincreasing with it). Note also that the interactions between agents in (1) are pairwise: for a given agent with a trajectory $γ$, its total interaction is the integral of the interactions of the pairs of trajectories $(γ, \tilde{γ})$ for all trajectories $\tilde{γ}$ distributed according to $Q$.

Most mean field games considered in the literature, including the one from [78] but also those from [1, 2, 6, 8, 10, 20, 30, 32, 33, 41, 57, 62, 64, 65], assume that the time interval $[0, T]$ of the game is fixed, with all agents starting at time 0 and ending their movement at time $T$. However, in many practical applications, including crowd motion but also in economics, for instance, it is interesting to study models in which the final time for the movement of an agent is free, and is actually part of their minimization criterion. In particular, agents may leave the game before its end. MFGs with a free final time have been considered, for instance, in [21, 35], which consider mean field games with applications to bank run, i.e., situations in which clients of a bank, believing that the bank is about to fail, withdraw all their money, and try to choose the time to withdraw the money in an optimal way. These models belong to a more general class of mean field game problems known as mean field games of optimal stopping, in which the main choice of an agent is when to stop the game [12, 13, 17, 47, 72, 73]. Other works, such as [37, 50], consider MFGs with free final time for the production of exhaustible resources, in which firms, who wish to maximize their profit, produce goods based on exhaustible resources, and they leave the game when they deplete their capacities. These games can be seen as mean field games with an absorbing boundary, i.e., in which agents who reach a certain part of the boundary of the domain immediately leave the game [23, 40]. In the context of crowd motion, MFG models with free final time were developed in [38, 43, 44, 67, 68, 75, 76].

In this paper, we generalize the model from [78] by considering that (i) one has a more general individual running cost than $\delta/2|\dot{γ}(t)|^2$ from (1); (ii) one has a more general interaction cost than $η(γ(t) - \tilde{γ}(t))|\dot{γ}(t) - \tilde{γ}(t)|^2$ from (1); (iii) the final time of the movement of an agent is not a prescribed time $T$, but is part of the optimization criterion of each agent; and (iv) the aim of each agent is to reach a given common target set $\Xi$. More precisely, we shall consider that each agent of the game minimizes a cost of the form

$$ \int_{0}^{+\infty} \ell(t, \gamma(t), \dot{\gamma}(t)) dt + \Psi(\tau(\gamma)) + \int_{\Gamma} \int_{0}^{\tau(\gamma) \wedge \tau(\tilde{\gamma})} h(t, \gamma(t), \tilde{\gamma}(t), \dot{\gamma}(t), \tilde{\dot{\gamma}}(t)) dt dQ(\tilde{\gamma}), \quad (2) $$

where $ℓ, h$, and $Ψ$ are given functions and, for a given trajectory $γ, τ(γ)$ denotes the exit time of $γ$, i.e., the first time at which $γ$ reaches the target set $\Xi$ (or $τ(γ) = +\infty$ if $γ$ does not reach $\Xi$). While (i) and (ii) do not bring many technical difficulties with respect to [78], novel ideas are
required to treat the free final-time setting from (iii) and (iv) and address the difficulties coming
from a noncompact time interval and the lack of continuity of the exit-time function $\tau(\cdot)$.

It turns out that the potential structure highlighted in [78] is not specific to the MFG considered
in that reference nor to the MFG described by (2) and comes instead mostly from the pairwise
nature of the interaction between agents. Hence, instead of working directly with (2), we consider
in most of the paper a more general non-atomic game model in abstract Polish spaces (which are
not necessarily spaces of trajectories) in which agents minimize the sum of an individual cost and
a pairwise interaction cost. The analysis of more general non-atomic games and their potential
structure has also been the subject of other works in the literature, such as [14], which considers
a non-atomic game which may contain pairwise interaction terms as a particular case. We point
out, however, that the model we consider cannot be subsumed under that of [14] (see Remark 3.3).

In addition to showing, as in [78], that the non-atomic game we consider has a potential
structure in the sense that local minimizers of a potential $\mathcal{J}$ are equilibria of the game, we go
further in the potential formulation by showing (Theorem 3.20) that the set of equilibria of the
game is exactly the set of critical points of its potential function $\mathcal{J}$. We then deduce (Theorem 3.22)
existence of an equilibrium by proving the existence of a minimizer for $\mathcal{J}$. We also discuss two a
priori distinct notions of equilibria, strong and weak equilibria, and prove (Theorem 3.24) that,
under suitable assumptions, both notions coincide for our model. Finally, we apply the results
obtained for the general non-atomic game to the MFG consisting in each agent minimizing (2),
proving existence of equilibria (Theorem 4.9) and equivalence between strong and weak equilibria
(Theorem 4.12).

The sequel of the paper is organized as follows. Section 2 presents the MFG model consisting
on each agent minimizing (2), providing our standing assumptions on the functions $\ell$, $h$, and
$\Psi$ from (2) as well as the precise definition of the exit time function $\tau(\cdot)$ and the definition of
equilibrium of the game. Section 3 moves to the analysis of a more general non-atomic game model
in Polish spaces, presenting the model and its assumptions, providing its elementary properties
in Section 3.1, proving its potential structure in Section 3.2, deducing existence of equilibria in
Section 3.3, and discussing the equivalence of the a priori distinct notions of strong and weak
equilibria in Section 3.4. These general results are applied to the MFG model from Section 2 in
Section 4. The paper is concluded by a numerical simulation in Section 5 illustrating behavior of
the equilibria in an $N$-player setting.

**Notation.** In this paper, $d$ is a fixed positive integer, the set of nonnegative real numbers is denoted by $\mathbb{R}_+$, the set of nonnegative rational numbers is denoted by $\mathbb{Q}_+$, and we define $\bar{\mathbb{R}}_+$ as $\bar{\mathbb{R}}_+ = \mathbb{R}_+ \cup \{+\infty\}$. For a given $\Omega \subset \mathbb{R}^d$ nonempty, open, and bounded, we let $\mathcal{C}(\mathbb{R}_+; \bar{\Omega})$ be the spaces of continuous curves from $\mathbb{R}_+$ to $\bar{\Omega}$ endowed with the topology of uniform convergence on compact time intervals, with which it is a Polish space. For simplicity, we denote $\mathcal{C}(\mathbb{R}_+; \bar{\Omega})$ by $\Gamma$.

Given a Polish space $X$, we denote the space of Borel probability measures on $X$ by $\mathcal{P}(X)$ and endow it with the topology of weak convergence of measures. The support of $\mu \in \mathcal{P}(X)$ is denoted by $\text{spt}(\mu)$, and is defined as the set of all points $x \in X$ such that $\mu(N_x) > 0$ for every neighborhood $N_x$ of $x$. We denote the Dirac measure centered at a point $x_0 \in X$ by $\delta_{x_0}$.

For two metric spaces $X$ and $Y$ endowed with their Borel $\sigma$-algebras and a Borel-measurable
map $f: X \to Y$, the pushforward of a measure $\mu$ on $X$ through $f$ is the measure $f_{\#}\mu$ on $Y$ defined
by $f_{\#}\mu(B) = \mu(f^{-1}(B))$ for every Borel subset $B$ of $Y$. For $t \in \mathbb{R}_+$, we denote by $e_t: \Gamma \to \bar{\Omega}$ the
evaluation map at time $t$, defined by $e_t(\gamma) = \gamma(t)$ for every $\gamma \in \Gamma$.

## 2 Mean field game model with pairwise interactions

We consider here that agents move in $\bar{\Omega}$, where $\Omega$ is a nonempty open bounded subset of $\mathbb{R}^d$, and
that their goal is to reach a certain target $\Xi \subset \bar{\Omega}$, which is a nonempty closed set. The initial
distribution of agents $m_0 \in \mathcal{P}(\bar{\Omega})$ at time $t=0$ is known, agents are submitted to the trivial
control system $\dot{\gamma}(t) = u(t)$, and the goal of an agent starting at position $x_0$ is to choose a control
$u: \mathbb{R}_+ \to \mathbb{R}^d$ such that $\gamma(t) \in \bar{\Omega}$ for every $t \in \mathbb{R}_+$ and the cost $F(\gamma, Q)$ defined by

$$F(\gamma, Q) = \int_{0}^{+\infty} \ell(t, \gamma(t), \dot{\gamma}(t)) \, dt + \Psi(\tau(\gamma)) + \int_{\Gamma} \int_{0}^{\tau(\gamma) \wedge \tau(\tilde{\gamma})} h(t, \gamma(t), \tilde{\gamma}(t), \dot{\gamma}(t), \dot{\tilde{\gamma}}(t)) \, dt \, dQ(\tilde{\gamma}) \quad (3)$$
is minimized among all trajectories $\gamma$ starting at the same position $x_0$. In (3), $\ell: \mathbb{R}_+ \times \bar{\Omega} \times \mathbb{R}^d \to \mathbb{R}_+$, $h: \mathbb{R}_+ \times \bar{\Omega} \times \bar{\Omega} \times \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}_+$, and $\Psi: \mathbb{R}_+ \to \mathbb{R}_+$ are given functions, $Q \in \mathcal{P}(\Gamma)$ represents the distribution of the trajectories of all agents, and $\tau: \Gamma \to \mathbb{R}_+$ is the first exit time function, defined, for $\gamma \in \Gamma$, by

$$ \tau(\gamma) = \inf\{t \in \mathbb{R}_+ | \gamma(t) \in \Xi\}, \quad (4) $$

with the convention $\inf \emptyset = +\infty$. We also set, by convention, $F(\gamma, Q) = +\infty$ if $\gamma$ is not absolutely continuous, if $Q$ gives positive measure to a set of functions which are not absolutely continuous, or if $\tilde{\gamma} \mapsto \int_0^{\tau(\gamma)\wedge\tau(\tilde{\gamma})} h(t, \gamma(t), \tilde{\gamma}(t), \dot{\gamma}(t), \tilde{\dot{\gamma}}(t)) dt$ is not $Q$-integrable.
Let us provide the definition of equilibrium of this mean field game.

**Definition 2.1.** Let $m_0 \in \mathcal{P}(\bar{\Omega})$. A measure $Q \in \mathcal{P}(\Gamma)$ is said to be an equilibrium with initial condition $m_0$ if $e_{0\#}Q = m_0$, $\int_{\Gamma} F(\gamma, Q) dQ(\gamma) < +\infty$, and $Q$-almost every $\gamma$ satisfies

$$ F(\gamma, Q) = \inf_{\substack{\omega \in \Gamma \\ \omega(0) = \gamma(0)}} F(\omega, Q). $$

Let us now introduce the main assumptions on $\Omega$, $\Xi$, $\ell$, $h$, and $\Psi$ that we will need for the analysis of this mean field game.

(H1) $\Omega \subset \mathbb{R}^d$ is nonempty, open, and bounded.

(H2) $\Xi \subset \bar{\Omega}$ is nonempty and closed.

(H3) The function $\ell: \mathbb{R}_+ \times \bar{\Omega} \times \mathbb{R}^d \to \mathbb{R}_+$ is such that

(a) $t \mapsto \ell(t, x, p)$ is measurable for every $(x, p) \in \bar{\Omega} \times \mathbb{R}^d$;

(b) $(x, p) \mapsto \ell(t, x, p)$ is continuous for almost every $t \in \mathbb{R}_+$;

(c) $p \mapsto \ell(t, x, p)$ is convex for almost every $t \in \mathbb{R}_+$ and every $x \in \bar{\Omega}$; and

(d) there exist constants $\alpha > 0$ and $\theta > 1$ such that

$$ \ell(t, x, p) \ge \alpha |p|^{\theta}, \quad \text{for all } (t, x, p) \in \mathbb{R}_+ \times \bar{\Omega} \times \mathbb{R}^d. $$

(H4) The function $h: \mathbb{R}_+ \times \bar{\Omega} \times \bar{\Omega} \times \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}_+$ is such that

(a) $t \mapsto h(t, x, \tilde{x}, p, \tilde{p})$ is measurable for every $(x, \tilde{x}, p, \tilde{p}) \in \bar{\Omega} \times \bar{\Omega} \times \mathbb{R}^d \times \mathbb{R}^d$;

(b) $(x, \tilde{x}, p, \tilde{p}) \mapsto h(t, x, \tilde{x}, p, \tilde{p})$ is continuous for almost every $t \in \mathbb{R}_+$;

(c) $(p, \tilde{p}) \mapsto h(t, x, \tilde{x}, p, \tilde{p})$ is convex for almost every $t \in \mathbb{R}_+$ and every $(x, \tilde{x}) \in \bar{\Omega} \times \bar{\Omega}$;

(d) for all $(t, x, \tilde{x}, p, \tilde{p}) \in \mathbb{R}_+ \times \bar{\Omega} \times \bar{\Omega} \times \mathbb{R}^d \times \mathbb{R}^d$, we have the symmetry property

$$ h(t, x, \tilde{x}, p, \tilde{p}) = h(t, \tilde{x}, x, \tilde{p}, p); \quad \text{and} $$

(e) there exist constants $C > 0$ and $\beta \in (0, \theta]$, where $\theta$ is the constant from (H3)(d), such that

$$ h(t, x, \tilde{x}, p, \tilde{p}) \le C(|p|^{\beta} + |\tilde{p}|^{\beta}), \quad \text{for all } (t, x, \tilde{x}, p, \tilde{p}) \in \mathbb{R}_+ \times \bar{\Omega} \times \bar{\Omega} \times \mathbb{R}^d \times \mathbb{R}^d. $$

(H5) The function $\Psi: \mathbb{R}_+ \to \mathbb{R}_+$ is lower semicontinuous, nondecreasing, and there exist positive constants $a$ and $b$ such that

$$ \Psi(t) \ge at - b, \quad \text{for all } t \in \mathbb{R}_+. $$

(H6) There exists a constant $\kappa > 0$ such that, for every $x_0 \in \bar{\Omega}$, there exists an absolutely continuous curve $\gamma \in \Gamma$ with $\gamma(0) = x_0$ and

$$ \int_0^{+\infty} \ell(t, \gamma(t), \dot{\gamma}(t)) dt + \Psi(\tau(\gamma)) \le \kappa. $$
Note that (H6) can be seen as an accessibility assumption: wherever in $\bar{\Omega}$ is the starting point $x_0$, there always exists an absolutely continuous trajectory allowing an agent to reach the target set $\Xi$ within a time that is uniformly bounded, and with a running cost that is also uniformly bounded. It can be ensured, for instance, if $\Omega$ is connected and has smooth boundary.

In the sequel, instead of directly studying this mean field game, we will consider in Section 3 a more general non-atomic game containing this MFG as a particular case. We will prove that this non-atomic game has a potential structure and use this fact to deduce the existence of equilibria. Finally, we will apply the results obtained for the general non-atomic game to the above MFG in Section 4.

### 3 A general non-atomic game with pairwise interactions

As a preliminary step towards the study of the mean field game model we are interested in this paper, we consider here a more general non-atomic game with pairwise interactions. Let $X$ and $Y$ be Polish spaces and $\pi: X \to Y$, $L: X \to \bar{\mathbb{R}}_+$, and $H: X \times X \to \bar{\mathbb{R}}_+$ be Borel-measurable functions. Each agent of the game is assumed to be associated with an element $y \in Y$ (but this association is not necessarily injective: different agents can be associated with the same element of $Y$), and we assume that the distribution of agents according to the elements associated with them in $Y$ is described by a known probability measure $m_0 \in \mathcal{P}(Y)$. An agent of the game associated with some $y \in Y$ wishes to minimize the cost $F(x, Q)$ given by

$$F(x, Q) = L(x) + \int_X H(x, \tilde{x}) \, dQ(\tilde{x}) \quad (5)$$

with the constraint $\pi(x) = y$, i.e., the agent wants to choose $x \in X$ solving the minimization problem

$$\text{Min}(y, Q): \begin{cases} \text{minimize } F(x, Q) \\ x \in X \\ \text{subject to } \pi(x) = y, \end{cases} \quad (6)$$

where $Q \in \mathcal{P}(X)$ denotes the distribution of the choices of all agents. Note that the cost $F(x, Q)$ can be interpreted as containing an individual cost $L(x)$ and an average pairwise interaction cost $\int_X H(x, \tilde{x}) \, dQ(\tilde{x})$. In the sequel, we refer to this non-atomic game as NAG($X, Y, \pi, L, H, m_0$) and we look for equilibria of this game, according to the following definition.

**Definition 3.1.** Let $X$ and $Y$ be Polish spaces, $\pi: X \to Y$, $L: X \to \bar{\mathbb{R}}_+$, and $H: X \times X \to \bar{\mathbb{R}}_+$ be Borel-measurable functions, and $m_0 \in \mathcal{P}(Y)$. We say that $Q \in \mathcal{P}(X)$ is an equilibrium of NAG($X, Y, \pi, L, H, m_0$) if $\pi_{\#}Q = m_0$, $\int_X F(x, Q) \, dQ(x) < +\infty$, and $Q$-almost every $x \in X$ solves $\text{Min}(\pi(x), Q)$ with $F$ given by (5).

**Remark 3.2.** The mean field game from Section 2 is a particular case of NAG($X, Y, \pi, L, H, m_0$). To see that, it suffices to set $Y = \bar{\Omega}$, $X = \Gamma$ with the topology of uniform convergence on compact sets, $\pi = e_0$, the evaluation map at time zero, and define the functions $L: \Gamma \to \bar{\mathbb{R}}_+$ and $H: \Gamma \times \Gamma \to \bar{\mathbb{R}}_+$ by

$$L(\gamma) = \begin{cases} \int_{0}^{+\infty} \ell(t, \gamma(t), \dot{\gamma}(t)) \, dt + \Psi(\tau(\gamma)) & \text{if } \gamma \text{ is absolutely continuous,} \\ +\infty & \text{otherwise} \end{cases}$$

and

$$H(\gamma, \tilde{\gamma}) = \begin{cases} \int_{0}^{\tau(\gamma) \wedge \tau(\tilde{\gamma})} h(t, \gamma(t), \tilde{\gamma}(t), \dot{\gamma}(t), \tilde{\dot{\gamma}}(t)) \, dt & \text{if } L(\gamma) < +\infty \text{ and } L(\tilde{\gamma}) < +\infty, \\ +\infty & \text{otherwise.} \end{cases}$$

We will justify later (see Proposition 4.8) that, with these definitions, the notions of equilibria from Definitions 2.1 and 3.1 coincide.
**Remark 3.3.** For an individual agent associated with a given element $y \in Y$, their cost (5) and their minimization problem (6) are particular cases of the more general problem of, given $y \in Y$,

$$ \underset{x \in X}{\text{minimize}} \ c(x, y) + \mathcal{V}[Q](x), \qquad (7) $$

without the constraint $\pi(x) = y$. Indeed, (7) reduces to (6) with the choice $c(x, y) = L(x) + \chi_{\text{graph}(\pi)}(x, y)$ and $\mathcal{V}[Q](x) = \int_X H(x, \tilde{x}) dQ(\tilde{x})$, where $\chi_{\text{graph}(\pi)}$ is the characteristic function of the graph of $\pi$, i.e., $\chi_{\text{graph}(\pi)}(x, y) = 0$ if $y = \pi(x)$ and $\chi_{\text{graph}(\pi)}(x, y) = +\infty$ otherwise.

The non-atomic game in which each agent minimizes (7) was previously considered in the literature, for instance, in [14], where the authors study equilibria of the game, named in that context as Cournot-Nash equilibria, through a variational approach, exploiting in particular links between such equilibria and optimal transport problems. These ideas, however, cannot be directly applied to the non-atomic game NAG($X, Y, \pi, L, H, m_0$) that we consider in this article, as some key assumptions of [14] cannot be satisfied in our context. In particular, [14] requires $c$ to be continuous in order to obtain good properties for the associated Wasserstein distance $W_c$, but such a continuity assumption can never be satisfied in our model due to the presence of the characteristic function $\chi_{\text{graph}(\pi)}$ in the expression of $c$.

Let us now state the main assumptions under which we will consider the abstract non-atomic game NAG($X, Y, \pi, L, H, m_0$).

(A1) $X$ and $Y$ are Polish spaces.

(A2) $\pi: X \to Y$ is continuous.

(A3) $L: X \to \overline{\mathbb{R}}_+$ is lower semicontinuous and $H: X \times X \to \overline{\mathbb{R}}_+$ is Borel measurable.

(A4) $H$ is symmetric, i.e., $H(x, \tilde{x}) = H(\tilde{x}, x)$ for every $(x, \tilde{x}) \in X \times X$.

(A5) The function $X \times X \ni (x, \tilde{x}) \mapsto L(x) + L(\tilde{x}) + H(x, \tilde{x}) \in \overline{\mathbb{R}}_+$ is lower semicontinuous.

(A6) There exists $\kappa > 0$ such that, for every $y \in Y$, there exists $x \in X$ with $\pi(x) = y$ and $L(x) \le \kappa$.

(A7) For every $\kappa > 0$, the set $\{x \in X \mid L(x) \le \kappa\}$ is compact.

(A8) There exists $C > 0$ such that $H(x, \tilde{x}) \le C(L(x) + L(\tilde{x}) + 1)$ for every $(x, \tilde{x}) \in X \times X$.

Our main goals concerning NAG($X, Y, \pi, L, H, m_0$) are to show that (i) it is a potential game, i.e., its equilibria can be obtained as minimizers of a certain potential $\mathcal{J}$; and (ii) the potential $\mathcal{J}$ admits a minimizer, and hence NAG($X, Y, \pi, L, H, m_0$) admits an equilibrium.

Let us introduce some notation to be used in the sequel. We define $J: X \times X \to \overline{\mathbb{R}}_+$ by

$$ J(x, \tilde{x}) = L(x) + L(\tilde{x}) + H(x, \tilde{x}), \qquad (8) $$

and we define the functions $\mathcal{L}: \mathcal{P}(X) \to \overline{\mathbb{R}}_+$, $\mathcal{H}: \mathcal{P}(X) \times \mathcal{P}(X) \to \overline{\mathbb{R}}_+$, and $\mathcal{J}: \mathcal{P}(X) \to \overline{\mathbb{R}}_+$ by

$$ \mathcal{L}(Q) = \int_X L(x) \, dQ(x), \qquad (9a) $$

$$ \mathcal{H}(Q, \tilde{Q}) = \int_{X \times X} H(x, \tilde{x}) \, d(Q \otimes \tilde{Q})(x, \tilde{x}), \qquad (9b) $$

$$ \mathcal{J}(Q) = \int_{X \times X} J(x, \tilde{x}) \, d(Q \otimes \tilde{Q})(x, \tilde{x}). \qquad (9c) $$

Given $m_0 \in \mathcal{P}(Y)$, we also set

$$ \mathcal{P}_{m_0}(X) = \{Q \in \mathcal{P}(X) \mid \pi_{\#} Q = m_0\}. \qquad (10) $$

**Remark 3.4.** Under (A2), the set $\mathcal{P}_{m_0}(X)$ is a closed subset of $\mathcal{P}(X)$.

**Remark 3.5.** According to the definitions of $F$, $\mathcal{L}$, $\mathcal{H}$, and $\mathcal{J}$, from (5) and (9), one can easily check that $\mathcal{J}(Q) = 2\mathcal{L}(Q) + \mathcal{H}(Q, Q)$ and $\int_X F(x, Q) dQ(x) = \mathcal{L}(Q) + \mathcal{H}(Q, Q)$, for every $Q \in \mathcal{P}(X)$. In particular, $\mathcal{J}(Q) < +\infty$ if and only if $\int_X F(x, Q) dQ(x) < +\infty$.
## 3.1 First properties

Let us now establish some elementary properties of the functions introduced in (9). We start with the following consequence of (A8), whose proof is immediate.

**Lemma 3.6.** Assume that (A1), (A3), and (A8) are satisfied. Let $C > 0$ be the constant from (A8) and $\mathcal{L}$ and $\mathcal{H}$ be defined as in (9a) and (9b). Then, for every $(Q, \tilde{Q}) \in \mathcal{P}(X) \times \mathcal{P}(X)$, we have

$$ \mathcal{H}(Q, \tilde{Q}) \leq C(\mathcal{L}(Q) + \mathcal{L}(\tilde{Q}) + 1). $$

As a consequence of Remark 3.5 and Lemma 3.6, we also immediately obtain the following result.

**Corollary 3.7.** Assume that (A1), (A3), and (A8) are satisfied. Let $J$, $\mathcal{L}$, and $\mathcal{J}$ be defined as in (8), (9a) and (9c). Then $\text{dom} \mathcal{L} = \text{dom} \mathcal{J}$.

**Remark 3.8.** From Lemma 3.6, we deduce that the function $\mathcal{H}: \text{dom} \mathcal{L} \times \text{dom} \mathcal{L} \to \mathbb{R}_+$ can be extended to bilinearity to a unique bilinear function, still denoted by $\mathcal{H}$, defined in $\text{Span}(\text{dom} \mathcal{L}) \times \text{Span}(\text{dom} \mathcal{L})$ and taking values in $\mathbb{R}$, where the linear span is taken in the space of signed measures on $X$.

In order to provide additional properties of $\text{dom} \mathcal{J}$, we first establish the following preliminary result, which states that, in (A6), the element $x \in X$ can be selected as a Borel-measurable function of $y \in Y$.

**Lemma 3.9.** Assume that (A1)-(A3), (A6), and (A7) are satisfied and let $\kappa > 0$ be as in (A6). Then there exists a Borel measurable function $\Phi: Y \to X$ such that, for every $y \in Y$, we have $\pi(\Phi(y)) = y$ and $L(\Phi(y)) \leq \kappa$.

*Proof.* Let us consider the set valued map $\mathcal{G}: Y \rightrightarrows X$ defined by

$$ \mathcal{G}(y) = \{x \in X \mid \pi(x) = y \text{ and } L(x) \leq \kappa\}. $$

Note that, by (A6), we have $\mathcal{G}(y) \neq \emptyset$ for every $y \in Y$. Since $\pi$ is continuous and $L$ is lower semicontinuous, one immediately deduces that the graph of $\mathcal{G}$ is closed and, in particular, $\mathcal{G}(y)$ is closed for every $y \in Y$. In addition, $\mathcal{G}$ takes values in the set $\{x \in X \mid L(x) \leq \kappa\}$, which is compact thanks to (A7). Hence, by [5, Proposition 1.4.8], we get that $\mathcal{G}$ is upper semicontinuous and, by [5, Proposition 1.4.4], the set $\mathcal{G}^{-1}(A) = \{y \in Y \mid \mathcal{G}(y) \cap A \neq \emptyset\}$ is closed for every closed set $A \subset X$. Hence, by [36, Proposition III.11], $\mathcal{G}^{-1}(B)$ is open for every open set $B \subset X$, and thus $\mathcal{G}$ is measurable. Now, by applying [5, Theorem 8.1.3], we conclude that the set-valued map $\mathcal{G}$ admits a Borel-measurable selection, which is our desired function $\Phi$. $\square$

**Remark 3.10.** Notice that it is not possible to apply [5, Theorem 8.1.4] in order to extract directly a Borel measurable selection from the fact that $\mathcal{G}$ has a closed graph since the measure $m_0$ is not complete in the measurable space $\overline{\Omega}$ endowed with its Borel $\sigma$-algebra.

As a consequence of Lemma 3.9, we deduce the following result on $\text{dom} \mathcal{J}$.

**Corollary 3.11.** Assume that (A1)-(A3) and (A6)-(A8) are satisfied, and let $J$ and $\mathcal{J}$ be given by (8) and (9c). Then, for every $m_0 \in \mathcal{P}(Y)$, the set $\text{dom} \mathcal{J} \cap \mathcal{P}_{m_0}(X)$ is nonempty and convex.

*Proof.* By Corollary 3.7, it suffices to show that $\text{dom} \mathcal{L} \cap \mathcal{P}_{m_0}(X)$ is nonempty and convex, where $\mathcal{L}$ is given by (9a). Let $\Phi: Y \to X$ be the map from Lemma 3.9 and define $Q = \Phi_{\#}m_0$. Then $\pi_{\#}Q = m_0$ and an immediate computation shows that $\mathcal{L}(Q) \leq \kappa < +\infty$, where $\kappa$ is the constant from (A6). Hence $Q \in \text{dom} \mathcal{L} \cap \mathcal{P}_{m_0}(X)$.

Note that $\mathcal{P}_{m_0}(X)$ is clearly convex from (10), and $\mathcal{L}$ is linear, implying that $\mathcal{L}(tQ + (1-t)\tilde{Q}) = t\mathcal{L}(Q) + (1-t)\mathcal{L}(\tilde{Q})$ for every $Q, \tilde{Q} \in \mathcal{P}(X)$ and $t \in [0, 1]$. Hence $\text{dom} \mathcal{L}$ is convex, yielding that $\text{dom} \mathcal{L} \cap \mathcal{P}_{m_0}(X)$ is convex. $\square$

We will also need in the sequel the following property of the function $F$ from (5).
**Lemma 3.12.** Assume that (A1), (A3), and (A5) are satisfied, and let $\mathcal{L}$ be the function defined in (9a). For every $Q \in \text{dom} \mathcal{L}$, the function $x \mapsto F(x, Q)$ is lower semicontinuous.

*Proof.* Note that, defining $J$ as in (8), we have

$$F(x, Q) + \mathcal{L}(Q) = \int_X J(x, \tilde{x}) \, dQ(\tilde{x}),$$

and thus, since $\mathcal{L}(Q) < +\infty$, it suffices to prove the lower semicontinuity of the function $x \mapsto \int_X J(x, \tilde{x}) \, dQ(\tilde{x})$.

Let $(x_n)_{n \in \mathbb{N}}$ be a sequence in $X$ converging to some $x \in X$. Since $J$ is lower semicontinuous thanks to (A5), we have, for every $\tilde{x} \in X$,

$$J(x, \tilde{x}) \leq \liminf_{n \to +\infty} J(x_n, \tilde{x}).$$

Then, Fatou's lemma shows that

$$\int_X J(x, \tilde{x}) \, dQ(\tilde{x}) \leq \int_X \liminf_{n \to +\infty} J(x_n, \tilde{x}) \, dQ(\tilde{x}) \leq \liminf_{n \to +\infty} \int_X J(x_n, \tilde{x}) \, dQ(\tilde{x}),$$

yielding the conclusion.

□

Note that Lemma 3.12 also yields the following result on the existence of optimizers for $F(\cdot, Q)$.

**Corollary 3.13.** Assume that (A1)-(A3) and (A5)-(A8) are satisfied and let $\mathcal{L}$ be the function defined in (9a). For every $Q \in \text{dom} \mathcal{L}$ and $y \in Y$, there exists $x \in X$ with $\pi(x) = y$ and such that

$$F(x, Q) = \inf_{\substack{z \in X \\ \pi(z)=y}} F(z, Q).$$

*Proof.* Let $(x_n)_{n \in \mathbb{N}}$ be a minimizing sequence for $F(\cdot, Q)$, i.e., $(x_n)_{n \in \mathbb{N}}$ is a sequence in $X$ with $\pi(x_n) = y$ for every $n \in \mathbb{N}$ and

$$\lim_{n \to +\infty} F(x_n, Q) = \inf_{\substack{z \in X \\ \pi(z)=y}} F(z, Q). \qquad (11)$$

By (A6), there exists $z \in X$ with $\pi(z) = y$ such that $L(z) \leq \kappa$, where $\kappa$ is the constant from (A6). Then, by (A8), we have $F(z, Q) \leq (C+1)L(z) + \mathcal{C}\mathcal{L}(Q) + C \leq (C+1)\kappa + \mathcal{C}\mathcal{L}(Q) + C$, where $C$ is the constant from (A8). Hence, the limit in the left-hand side in (11) is finite, and we assume, up to removing finitely many elements from the sequence, that $F(x_n, Q) < +\infty$ for every $n \in \mathbb{N}$, and thus the sequence $(F(x_n, Q))_{n \in \mathbb{N}}$ is bounded. Thus, the sequence $(L(x_n))_{n \in \mathbb{N}}$ is also bounded and, by (A7), this implies that, up to extracting a subsequence, there exists $x \in X$ such that $x_n \to x$ as $n \to +\infty$. By (A2), we obtain that $\pi(x) = y$ and, by (11) and Lemma 3.12, we obtain that

$$F(x, Q) \leq \lim_{n \to +\infty} F(x_n, Q) = \inf_{\substack{z \in X \\ \pi(z)=y}} F(z, Q) \leq F(x, Q),$$

yielding the conclusion.

□

## 3.2 Potential game

Our next goal is to prove that NAG($X, Y, \pi, L, H, m_0$) is a potential game with potential given by the function $\mathcal{J}$ from (9c). For games with finitely many players, a potential is a function such that, when a single player changes their strategy, the variation in the cost of that player is equal to (or proportional to) the variation of the potential. In particular, in potential games, equilibria can be found as minimizers of the potential. To adapt these ideas to our setting with a continuum of players, one needs to provide a suitable notion of “variation” of the potential $\mathcal{J}$ as a player changes their trajectory. It turns out that the good notion is that of differentiability of $\mathcal{J}$, in the sense of the following definition.
**Definition 3.14.** Assume that (A3) is satisfied and let $\mathcal{J}$ be defined by (9c). Given $Q_0 \in \text{dom} \mathcal{J}$, we say that $\mathcal{J}$ is differentiable at $Q_0$ if there exists a Borel-measurable function $G: X \to \mathbb{R}$ such that, for every $Q \in \text{dom} \mathcal{J}$, $G$ is $(Q - Q_0)$-integrable and

$$ \lim_{t \to 0^+} \frac{\mathcal{J}(Q_0 + t(Q - Q_0)) - \mathcal{J}(Q_0)}{t} = \int_X G(x) \, d(Q - Q_0)(x). \quad (12) $$

In this case, we denote $G$ by $\frac{\delta \mathcal{J}}{\delta Q}(Q_0)$ and we write the right-hand side of (12) as $\langle \frac{\delta \mathcal{J}}{\delta Q}(Q_0), Q - Q_0 \rangle$.

**Remark 3.15.** There are several notions of derivation for functions defined in the space of probability measures, and the one presented in Definition 3.14 resembles the one from [28, Definition 2.2.1] and [34, Definition 5.43], which is widely used in the analysis of mean field games. The main difference is that the notion we provide above requires less regularity of $(Q_0, x) \mapsto \frac{\delta \mathcal{J}}{\delta Q}(Q_0)(x)$, which is not necessarily continuous, and might even fail to be defined for some $Q_0 \in \text{dom} \mathcal{J}$. This weaker notion suitably fits our purposes in the sequel. We refer the interested reader to [4, Chapter 10], [28, Section 2.2.3 and Appendices A.1 and A.2], and [34, Chapter 5] for other notions of derivation for functions defined in the space of probability measures and further discussion on the relations among those notions.

Note that, if $G$ is a Borel-measurable function satisfying (12), then $G + \lambda$ also satisfies (12) for any constant $\lambda \in \mathbb{R}$, since the integral of a constant with respect to the measure $Q - Q_0$ is zero. Hence, $\frac{\delta \mathcal{J}}{\delta Q}$ is not uniquely defined. We prove, however, in our next result, that, similarly to the analogous notion of derivative from [28, Definition 2.2.1] and [34, Definition 5.43], $\frac{\delta \mathcal{J}}{\delta Q}$ is indeed unique up to an additive constant.

**Lemma 3.16.** Assume that (A1)-(A3), (A6), and (A7) are satisfied and let $\mathcal{J}$ be defined by (9c). Let $Q_0 \in \text{dom} \mathcal{J}$ and assume that $G_1$ and $G_2$ are two Borel measurable functions such that, for every $Q \in \text{dom} \mathcal{J}$, $G_1$ and $G_2$ are $(Q - Q_0)$-integrable and

$$ \int_X G_1(x) \, d(Q - Q_0)(x) = \int_X G_2(x) \, d(Q - Q_0)(x). $$

Then there exists $\lambda \in \mathbb{R}$ such that $G_1(x) = G_2(x) + \lambda$ for $Q_0$-almost every $x \in X$.

*Proof.* Let $\mathcal{L}$ be the function defined in (9a). Note that, for $Q_0$-almost every $x \in X$, we have $\delta_x \in \text{dom} \mathcal{J}$. Indeed, since $Q_0 \in \text{dom} \mathcal{J} = \text{dom} \mathcal{L}$ by Corollary 3.7, we deduce that $L(x) < +\infty$ for $Q_0$-almost every $x \in X$, and thus $\mathcal{L}(\delta_x) = L(x) < +\infty$, showing that $\delta_x \in \text{dom} \mathcal{L} = \text{dom} \mathcal{J}$.

We now set $\lambda = \int_X (G_1 - G_2)(x) \, dQ_0(x)$ and observe that, by assumption, for every $Q \in \text{dom} \mathcal{J}$, we have

$$ \int_X G_1(x) \, dQ(x) = \int_X G_2(x) \, dQ(x) + \lambda, $$

and hence, for $Q_0$-almost every $x \in X$, we can take $Q = \delta_x$ in the above identity to deduce that $G_1(x) = G_2(x) + \lambda$, as required. $\square$

Our next result is an important step in formulating NAG($X, Y, \pi, L, H, m_0$) as a potential non-atomic game with potential $\mathcal{J}$, since, under the additional assumption (A4), it relates the variations of $\mathcal{J}$, in the sense of Definition 3.14, with the individual cost $F(x, Q)$ of an agent from (5).

**Theorem 3.17.** Assume that (A1)-(A4), (A6), and (A7) are satisfied and let $\mathcal{J}$ be defined by (9c). Then, for every $Q_0 \in \text{dom} \mathcal{J}$, the function $\mathcal{J}$ is differentiable at $Q_0$, with $\frac{\delta \mathcal{J}}{\delta Q}(Q_0)(x) = 2F(x, Q_0)$ for $Q_0$-almost every $x \in X$.

*Proof.* Fix $Q_0 \in \text{dom} \mathcal{J}$ and let $\mathcal{L}$ and $\mathcal{H}$ be defined as in (9a) and (9b). For every $Q \in \text{dom} \mathcal{J}$ and $t \in (0, 1]$, using Remark 3.5, we have that

$$ \frac{\mathcal{J}(Q_0 + t(Q - Q_0)) - \mathcal{J}(Q_0)}{t} = 2(\mathcal{L}(Q) + \mathcal{H}(Q_0, Q)) - 2(\mathcal{L}(Q_0) + \mathcal{H}(Q_0, Q_0)) \\ + t\mathcal{H}(Q - Q_0, Q - Q_0), $$
where $\mathcal{H}(Q - Q_0, Q - Q_0) \in \mathbb{R}$ is well-defined by Remark 3.8. Hence, letting $t \to 0^+$, we obtain from Remark 3.5 that

$$ 
\begin{aligned}
\lim_{t \to 0} \frac{\mathcal{J}(Q_0 + t(Q - Q_0)) - \mathcal{J}(Q_0)}{t} &= 2(\mathcal{L}(Q) + \mathcal{H}(Q_0, Q)) - 2(\mathcal{L}(Q_0) + \mathcal{H}(Q_0, Q_0)) \\
&= \int_X 2F(x, Q_0) \, dQ(x) - \int_X 2F(x, Q_0) \, dQ_0(x) \\
&= \int_X 2F(x, Q_0) \, d(Q - Q_0)(x),
\end{aligned}
$$

yielding the conclusion.

□

Now that we have established a link between variations of $\mathcal{J}$ and the individual cost $F(x, Q)$ in Theorem 3.17, our next step to study the potential structure of NAG($X, Y, \pi, L, H, m_0$) is to relate equilibria of this game with minimizers of $\mathcal{J}$. For that purpose, we start with the following definition.

**Definition 3.18.** Assume that (A3) is satisfied and let $\mathcal{J}$ be defined by (9c). We say that $Q_0 \in \text{dom} \mathcal{J} \cap \mathcal{P}_{m_0}(X)$ is a critical point of $\mathcal{J}$ in $\mathcal{P}_{m_0}(X)$ if $\mathcal{J}$ is differentiable at $Q_0$ and

$$ \left\langle \frac{\delta \mathcal{J}}{\delta Q}(Q_0), Q - Q_0 \right\rangle \ge 0 \quad \text{for every } Q \in \text{dom} \mathcal{J} \cap \mathcal{P}_{m_0}(X). \qquad (13) $$

Inequalities of the form (13) are sometimes known as *Euler inequalities*, and they are usually necessary conditions for minimization of differentiable functions (see, e.g., [3, Theorem 10.2.1 and Remark 10.2.2] for the case of functions defined on Hilbert spaces). Our next result states that (13) is indeed a necessary condition for the minimization of $\mathcal{J}$ on $\mathcal{P}_{m_0}(X)$.

**Proposition 3.19.** Assume that (A3) satisfied and let $Q_0 \in \text{dom} \mathcal{J}$ be a local minimizer of $\mathcal{J}$ in $\mathcal{P}_{m_0}(X)$, i.e., there exists a neighborhood $V$ of $Q_0$ in $\mathcal{P}_{m_0}(X)$, with the topology of weak convergence of probability measures, such that $\mathcal{J}(Q) \ge \mathcal{J}(Q_0)$ for every $Q \in V$. If $\mathcal{J}$ is differentiable at $Q_0$, then $Q_0$ is a critical point of $\mathcal{J}$ in $\mathcal{P}_{m_0}(X)$.

*Proof.* Let $Q_0 \in \text{dom} \mathcal{J}$ be a local minimizer of $\mathcal{J}$ in $\mathcal{P}_{m_0}(X)$ and assume that $\mathcal{J}$ is differentiable at $Q_0$. Then, for every $Q \in \text{dom} \mathcal{J} \cap \mathcal{P}_{m_0}(X)$, we deduce from the facts that $Q_0$ is a local minimizer of $\mathcal{J}$ in $\mathcal{P}_{m_0}(X)$ and $Q_0 + t(Q - Q_0) \to Q_0$ as $t \to 0^+$ in the topology of weak convergence in $\mathcal{P}_{m_0}(X)$ that

$$ \frac{\mathcal{J}(Q_0 + t(Q - Q_0)) - \mathcal{J}(Q_0)}{t} \ge 0 $$

for small enough $t > 0$. Taking the limit as $t \to 0^+$ immediately yields (13).

□

We next establish one of our main results on the potential structure of NAG($X, Y, \pi, L, H, m_0$), stating that equilibria of this game coincide with critical points of $\mathcal{J}$ in $\mathcal{P}_{m_0}(X)$. The proof we present here is closely based on that of [78, Lemma 3.3].

**Theorem 3.20.** Assume that (A1)-(A8) are satisfied and let $\mathcal{J}$ be given by (9c). A measure $Q_0 \in \mathcal{P}_{m_0}(X)$ is an equilibrium for NAG($X, Y, \pi, L, H, m_0$) if and only if it is a critical point of $\mathcal{J}$ in $\mathcal{P}_{m_0}(X)$.

As a consequence, any local minimizer of $\mathcal{J}$ in $\mathcal{P}_{m_0}(X)$ is an equilibrium for NAG($X, Y, \pi, L, H, m_0$).

*Proof.* Recall that, according to Remark 3.5, $Q_0 \in \text{dom} \mathcal{J}$ is equivalent to having $\int_X F(x, Q_0) \, dQ_0(x) < +\infty$.

Assume first that $Q_0 \in \mathcal{P}_{m_0}(X)$ is a critical point of $\mathcal{J}$ in $\mathcal{P}_{m_0}(X)$ and note that, by definition, we have $Q_0 \in \text{dom} \mathcal{J}$, and hence $\int_X F(x, Q_0) \, dQ_0(x) < +\infty$. To obtain a contradiction, suppose that the set

$$ \{x \in X \mid \exists z \in X \text{ such that } \pi(z) = \pi(x) \text{ and } F(z, Q_0) < F(x, Q_0)\} $$
is not $Q_0$-negligible. Note that the above set is equal to

$$ \bigcup_{(q,r) \in \mathbb{Q}_+^2 \atop 0 < q < r} \{x \in X \mid F(x, Q_0) > r \text{ and } \{z \in X \mid \pi(z) = \pi(x) \text{ and } F(z, Q_0) \le q\} \ne \emptyset\} $$

and thus there exists a pair of positive rational numbers $(q, r)$ with $q < r$ such that the set $A$ defined by

$$ A = \{x \in X \mid F(x, Q_0) > r \text{ and } \{z \in X \mid \pi(z) = \pi(x) \text{ and } F(z, Q_0) \le q\} \ne \emptyset\} \quad (14) $$

is not $Q_0$-negligible.

For any $p \in \mathbb{R}$, define $A^p$ as

$$ A^p = \{x \in X \mid F(x, Q_0) \le p\}. $$

Note that $A^p$ is closed since, by Lemma 3.12, $x \mapsto F(x, Q_0)$ is lower semicontinuous. In addition, $A^p \subset \{x \in X \mid L(x) \le p\}$ and the latter set is compact thanks to (A7). Hence, $A^p$ is compact.

Let us now define the multifunction $\mathcal{S}$ as

$$ \mathcal{S}: \begin{cases} X \rightleftharpoons A^q, \\ x \mapsto \mathcal{S}(x) = \{z \in A^q \mid \pi(z) = \pi(x)\}. \end{cases} $$

Using (A2), it is immediate to verify that $\mathcal{S}$ has closed graph and, using the fact that $A^q$ is compact, one can also easily verify that the domain $\text{dom} \, \mathcal{S}$, defined as $\text{dom} \, \mathcal{S} = \{x \in X \mid \mathcal{S}(x) \ne \emptyset\}$, is closed. It is also clearly nonempty, since $A^q \subset \text{dom} \, \mathcal{S}$. By [5, Proposition 1.4.8], we deduce that $\mathcal{S}$ is upper semicontinuous. Hence, by [5, Proposition 1.4.4], $\mathcal{S}^{-1}(A) = \{x \in X \mid \mathcal{S}(x) \cap A \ne \emptyset\}$ is closed for every closed set $A$. Now by definition of measurability and [36, Proposition III.11], we obtain that $\mathcal{S}$ is Borel-measurable, and, by [5, Theorem 8.1.3], there exists a Borel-measurable function $s: \text{dom} \, \mathcal{S} \to A^q$ satisfying

$$ s(x) \in \mathcal{S}(x), \quad \text{for all } x \in \text{dom} \, \mathcal{S}. $$

Note that the set $A$ defined in (14) is Borel measurable, since it can be written as

$$ A = \{x \in X \mid F(x, Q_0) > r\} \cap \text{dom} \, \mathcal{S}, $$

$\text{dom} \, \mathcal{S}$ is closed, and $\{x \in X \mid F(x, Q_0) > r\}$ is the complement of $A^r$, and hence it is open. We can thus define a measure $\tilde{Q} \in \mathcal{P}(X)$ by $\tilde{Q} = s_{\#}(Q_0|_A) + Q_0|_{(X \setminus A)}$, which is well-defined since $\tilde{Q}(X) = Q_0(A \cap s^{-1}(X)) + Q_0(X \setminus A) = Q_0(A) + Q_0(X \setminus A) = 1$, and in addition $\tilde{Q} \in \mathcal{P}_{m_0}(X)$, since

$$ \begin{aligned} \pi_{\#} \tilde{Q} &= \pi_{\#} (s_{\#}(Q_0|_A) + Q_0|_{(X \setminus A)}) = \pi_{\#} (s_{\#}(Q_0|_A)) + \pi_{\#}(Q_0|_{(X \setminus A)}) \\ &= (\pi \circ s)_{\#} (Q_0|_A) + \pi_{\#}(Q_0|_{(X \setminus A)}) = \pi_{\#}(Q_0|_A) + \pi_{\#}(Q_0|_{(X \setminus A)}) \\ &= \pi_{\#}(Q_0|_A + Q_0|_{(X \setminus A)}) = \pi_{\#} Q_0 = m_0, \end{aligned} $$

where we use that $\pi \circ s(x) = \pi(x)$ for every $x \in \text{dom} \, \mathcal{S}$. Using that $Q_0(A) > 0$, we also compute

$$ \begin{aligned} \int_X F(x, Q_0) \, d\tilde{Q}(x) &= \int_{X \setminus A} F(x, Q_0) \, dQ_0(x) + \int_A F(s(x), Q_0) \, dQ_0(x) \\ &\le \int_{X \setminus A} F(x, Q_0) \, dQ_0(x) + qQ_0(A) \\ &< \int_{X \setminus A} F(x, Q_0) \, dQ_0(x) + rQ_0(A) \\ &< \int_{X \setminus A} F(x, Q_0) \, dQ_0(x) + \int_A F(x, Q_0) \, dQ_0(x) \\ &= \int_X F(x, Q_0) \, dQ_0(x), \end{aligned} $$

which, using Theorem 3.17, contradicts the fact that $Q_0$ is a critical point of $\mathcal{J}$ in $\mathcal{P}_{m_0}(X)$.
For the converse implication, assume that $Q_0 \in \mathcal{P}_{m_0}(X)$ is an equilibrium of NAG$(X, Y, \pi, L, H, m_0)$. In particular, $\int_X F(x, Q_0) dQ_0(x) < +\infty$, and thus $Q_0 \in \text{dom} \mathcal{J}$. Define $\nu: Y \to \mathbb{R}_+$ by

$$ \nu(y) = \inf_{\substack{z \in X \\ \pi(z)=y}} F(z, Q_0) \qquad (15) $$

and note that $\nu$ is Borel-measurable. Indeed, for every $\rho \ge 0$, we have

$$ \begin{aligned} \{y \in Y \mid \nu(y) \le \rho\} &= \bigcap_{n \in \mathbb{N}^*} \left\{ y \in Y \mid \exists z \in X \text{ such that } \pi(z) = y \text{ and } F(z, Q_0) \le \rho + \frac{1}{n} \right\} \\ &= \bigcap_{n \in \mathbb{N}^*} \pi \left( \left\{ z \in X \mid F(z, Q_0) \le \rho + \frac{1}{n} \right\} \right). \end{aligned} $$

For every $n \in \mathbb{N}^*$, by Lemma 3.12 and (A7), we have that $\{z \in X \mid F(z, Q_0) \le \rho + \frac{1}{n}\}$ is a closed subset of the compact set $\{z \in X \mid L(z) \le \rho + \frac{1}{n}\}$, and hence it is compact. Thus, by (A2), the set $\pi(\{z \in X \mid F(z, Q_0) \le \rho + \frac{1}{n}\})$ is compact, and thus it is closed, allowing one to conclude that $\{y \in Y \mid \nu(y) \le \rho\}$ is a Borel set. This yields the Borel-measurability of $\nu$, as required.

By definition of equilibrium, we have $F(x, Q_0) = \nu(\pi(x))$ for $Q_0$-almost every $x \in X$. Hence

$$ \int_X F(x, Q_0) dQ_0(x) = \int_X \nu(\pi(x)) dQ_0(x). $$

On the other hand, if one takes $Q \in \text{dom} \mathcal{J} \cap \mathcal{P}_{m_0}(X)$, then

$$ \begin{aligned} \int_X F(z, Q_0) dQ(z) &\ge \int_X \nu(\pi(z)) dQ(z) = \int_Y \nu(y) d(\pi_{\#} Q)(y) = \int_Y \nu(y) dm_0(y) \\ &= \int_Y \nu(y) d(\pi_{\#} Q_0)(y) = \int_X \nu(\pi(x)) dQ_0(x) = \int_X F(x, Q_0) dQ_0(x), \end{aligned} $$

and thus, using Theorem 3.17, we deduce that $Q_0$ is a critical point of $\mathcal{J}$ in $\mathcal{P}_{m_0}(X)$.

Finally, the last assertion of the statement follows as an immediate consequence of Proposition 3.19. $\square$

## 3.3 Existence of equilibria

We now turn to the question of existence of an equilibrium for NAG$(X, Y, \pi, L, H, m_0)$ which, thanks to the potential structure of this game highlighted in Section 3.2, can be addressed by studying the existence of minimizers for the function $\mathcal{J}$ from (9c). We start with the following preliminary result on $\mathcal{J}$.

**Lemma 3.21.** Assume that (A1) and (A5) are satisfied and let $\mathcal{J}$ be the function defined in (9c). Then $\mathcal{J}$ is lower semicontinuous on $\mathcal{P}(X)$.

*Proof.* Let $(Q_n)_{n \in \mathbb{N}}$ be a sequence in $\mathcal{P}(X)$ converging to some $Q \in \mathcal{P}(X)$. Since $X$ is a Polish space, it follows from [78, Lemma A.1] that the sequence of the product measures $(Q_n \otimes Q_n)_{n \in \mathbb{N}}$ converges to $Q \otimes Q$. For $C > 0$, let $J_C: X \times X \to \mathbb{R}_+$ be defined by $J_C(x, \tilde{x}) = \min\{J(x, \tilde{x}), C\}$ and notice that, by (A5), the function $J_C$ is lower semicontinuous and bounded. Then, by [15, Corollary 8.2.5], we have

$$ \begin{aligned} \int_{X \times X} J_C(x, \tilde{x}) \, d(Q \otimes Q)(x, \tilde{x}) &\le \liminf_{n \to +\infty} \int_{X \times X} J_C(x, \tilde{x}) \, d(Q_n \otimes Q_n)(x, \tilde{x}) \\ &\le \liminf_{n \to +\infty} \int_{X \times X} J(x, \tilde{x}) \, d(Q_n \otimes Q_n)(x, \tilde{x}) = \liminf_{n \to +\infty} \mathcal{J}(Q_n). \end{aligned} $$

Since the above holds for every $C > 0$, by Lebesgue's monotone convergence theorem, we deduce that

$$ \mathcal{J}(Q) = \int_{X \times X} J(x, \tilde{x}) \, d(Q \otimes Q)(x, \tilde{x}) = \lim_{C \to +\infty} \int_{X \times X} J_C(x, \tilde{x}) \, d(Q \otimes Q)(x, \tilde{x}) \le \liminf_{n \to +\infty} \mathcal{J}(Q_n), $$

yielding the conclusion. $\square$
Our main result on the existence of minimizers for $\mathcal{J}$ and of equilibria for NAG($X, Y, \pi, L, H, m_0$) is the following

**Theorem 3.22.** Assume that (A1)–(A3) and (A5)–(A8) are satisfied and let $\mathcal{J}$ be the function from (9c). Then, for every $m_0 \in \mathcal{P}(Y)$, $\mathcal{J}$ admits a minimizer in $\mathcal{P}_{m_0}(X)$.

As a consequence, if in addition (A4) is also satisfied, then there exists an equilibrium $Q$ for NAG($X, Y, \pi, L, H, m_0$).

*Proof.* Let $m_0 \in \mathcal{P}(Y)$ and note that, by Corollary 3.11, we have $\text{dom} \mathcal{J} \cap \mathcal{P}_{m_0}(X) \neq \emptyset$. Let $(Q_n)_{n \in \mathbb{N}}$ be a minimizing sequence for $\mathcal{J}$ in $\mathcal{P}_{m_0}(X)$, i.e., $(Q_n)_{n \in \mathbb{N}}$ is a sequence in $\text{dom} \mathcal{J} \cap \mathcal{P}_{m_0}(X)$ and

$$ \lim_{n \to +\infty} \mathcal{J}(Q_n) = \inf_{Q \in \mathcal{P}_{m_0}(X)} \mathcal{J}(Q). $$

We claim that the sequence $(Q_n)_{n \in \mathbb{N}}$ is tight. Indeed, by Markov's inequality, for every $M > 0$, we have

$$ Q_n(\{x \in X \mid L(x) > M\}) \leq \frac{1}{M} \mathcal{L}(Q_n) \leq \frac{1}{M} \mathcal{J}(Q_n), $$

and the conclusion follows since $(\mathcal{J}(Q_n))_{n \in \mathbb{N}}$ is a bounded sequence and, by (A7), $\{x \in X \mid L(x) \leq M\}$ is compact for every $M > 0$. Therefore, by Prokhorov's theorem (see, e.g., [4, Theorem 5.1.3]), up to extracting a subsequence (which we still denote by $(Q_n)_{n \in \mathbb{N}}$ for simplicity), there exists $Q \in \mathcal{P}(X)$ such that $Q_n \to Q$ as $n \to +\infty$. In addition, by Remark 3.4, we have $Q \in \mathcal{P}_{m_0}(X)$. By Lemma 3.21, we deduce that $\mathcal{J}(Q) \leq \lim_{n \to +\infty} \mathcal{J}(Q_n) = \inf_{\tilde{Q} \in \mathcal{P}_{m_0}(X)} \mathcal{J}(\tilde{Q})$, and thus $Q$ is a minimizer of $\mathcal{J}$ in $\mathcal{P}_{m_0}(X)$.

The last part of the statement is an immediate consequence of Theorem 3.20. $\square$

## 3.4 Strong equilibria

Definitions 2.1 and 3.1 of equilibria of the mean field game from Section 2 and of NAG($X, Y, \pi, L, H, m_0$) require an optimization problem to be solved by $Q$-almost every element of the space, similarly to other Lagrangian approaches in the literature, such as those from [68, 75, 76]. An alternative definition consists in requiring the optimization problem to be solved for every element in the support of $Q$. This stronger notion of equilibrium was used, for instance, in [24–26, 44, 78], and, in many situations, as shown in [44, Remark 4.6] and [76, Proposition 3.7], both notions of equilibria coincide. We aim at proving that this is also the case in our setting, at the cost of an additional assumption on NAG($X, Y, \pi, L, H, m_0$). To do that, let us start by defining the notion of strong equilibrium.

**Definition 3.23.** Let $X$ and $Y$ be Polish spaces, $\pi: X \to Y$, $L: X \to \overline{\mathbb{R}}_+$, and $H: X \times X \to \overline{\mathbb{R}}_+$ be Borel-measurable functions, and $m_0 \in \mathcal{P}(Y)$. We say that $Q \in \mathcal{P}(X)$ is a strong equilibrium of NAG($X, Y, \pi, L, H, m_0$) if $\pi_{\#}Q = m_0$, $\int_X F(x, Q) dQ(x) < +\infty$, and every $x \in \text{spt}(Q)$ solves $\text{Min}(\pi(x), F)$ with $F$ given by (5).

To prove that the notions of equilibrium and strong equilibrium are equivalent, we will make use of an additional technical assumption, which we now state.

(A9) For every $Q \in \text{dom} \mathcal{L}$, the set $\text{Opt}(Q)$ defined by

$$ \text{Opt}(Q) = \left\{ x \in X \mid F(x, Q) = \inf_{\substack{\tilde{x} \in X \\ \pi(\tilde{x}) = \pi(x)}} F(\tilde{x}, Q) \right\} \quad (16) $$

is closed.

As a consequence, we obtain the following result on the equivalence between equilibrium and strong equilibrium for NAG($X, Y, \pi, L, H, m_0$).

**Theorem 3.24.** Assume that (A1)–(A3), (A5), and (A9) are satisfied. Then $Q \in \mathcal{P}_{m_0}(X)$ is an equilibrium of the non-atomic game NAG($X, Y, \pi, L, H, m_0$) if and only if it is a strong equilibrium of this game.
Proof. It is immediate from Definitions 3.1 and 3.23 that every strong equilibrium is also an equilibrium of NAG(X, Y, π, L, H, m₀). To prove the converse statement, let Q ∈ Pm₀(X) be an equilibrium of NAG(X, Y, π, L, H, m₀) and note that, by Remark 3.5, we have Q ∈ dom J, implying that Q ∈ dom L. Note also that, since Q is an equilibrium of NAG(X, Y, π, L, H, m₀), we have Q(Opt(Q)) = 1, where Opt(Q) is the set defined in (16).

Let $x \in \text{spt}(Q)$. Then, by definition of support and using the fact that $Q(\text{Opt}(Q)) = 1$, there exists a sequence $(x_n)_{n \in \mathbb{N}}$ in $\text{Opt}(Q)$ such that $x_n \to x$ as $n \to +\infty$. Since $\text{Opt}(Q)$ is closed by (A9), we deduce that $x \in \text{Opt}(Q)$. Thus $\text{spt}(Q) \subset \text{Opt}(Q)$, concluding the proof. $\square$

**Remark 3.25.** If one also assumes (A9) in Theorem 3.20, then one can provide a simpler argument for the first implication in its proof, i.e., the fact that critical points of J in Pm₀(X) are equilibria of NAG(X, Y, π, L, H, m₀).

Indeed, notice first that, under the assumptions of Theorem 3.20 and (A9), for every $Q_0 \in \text{dom } \mathcal{L}$, we have

$$ \text{Opt}(Q_0) \subset \{x \in X \mid L(x) \leq (C+1)\kappa + C\mathcal{L}(Q_0) + C\}, \quad (17) $$

where $\kappa$ and $C$ are the constants from (A6) and (A8) and Opt($Q_0$) is the set defined in (16). This is the case since, given $x \in \text{Opt}(Q_0)$, we have $L(x) \leq F(x, Q_0) \leq F(z, Q_0)$ for every $z \in X$ with $\pi(z) = \pi(x)$. By (A8), we have $F(z, Q_0) \leq (C+1)L(z) + C\mathcal{L}(Q_0) + C$ for every $z \in X$, and, combining with (A6), we deduce that there exists $z \in X$ such that $\pi(z) = \pi(x)$ and $F(z, Q_0) \leq (C+1)\kappa + C\mathcal{L}(Q_0) + C$, yielding the inclusion (17). Now, by (A7) and (A9), we deduce from (17) that Opt($Q_0$) is compact.

Assume that $Q_0 \in Pm_0(X)$ is a critical point of J in $Pm_0(X)$. Let Opt: $Y \rightrightarrows X$ be the set-valued map defined for $y \in Y$ by

$$ \text{Opt}(y) = \left\{ x \in X \mid \pi(x) = y \text{ and } F(x, Q_0) = \inf_{\substack{z \in X \\ \pi(z)=y}} F(z, Q_0) \right\}. $$

Clearly, Opt($y$) $\subset$ Opt($Q_0$) for every $y \in Y$, and it follows from Corollary 3.13 that Opt($y$) $\neq \emptyset$ for every $y \in Y$. Using the closedness of Opt($Q_0$) and the continuity of $\pi$, it is easy to check that the graph of Opt is closed. Since Opt($Q_0$) is compact, [5, Proposition 1.4.8] shows that Opt is upper semicontinuous and, combining [5, Proposition 1.4.4], [36, Proposition III.11], and [5, Theorem 8.1.3], we conclude that the set valued map Opt admits a Borel-measurable selection, which we denote by $\phi: Y \to X$.

Define $\hat{Q} = \phi_{\#}m_0$ and notice that $\pi_{\#}\hat{Q} = m_0$ and

$$ \int_Y \nu(y) \, dm_0(y) = \int_X F(x, Q_0) \, d\hat{Q}(x), $$

where $\nu$ is the function defined in (15). Since $Q_0$ is a critical point of J in $Pm_0(X)$, we obtain, using Theorem 3.17, that

$$ \int_X F(x, Q_0) \, dQ_0(x) \le \int_X F(x, Q_0) \, d\hat{Q}(x) = \int_Y \nu(y) \, dm_0(y) = \int_X \nu(\pi(x)) \, dQ_0(x). \quad (18) $$

Let $\Theta: X \to \mathbb{R}$ be the function defined by $\Theta(x) = F(x, Q_0) - \nu(\pi(x))$. By definition, $\Theta(x) \ge 0$ for any $x \in X$. It follows from (18) that

$$ \int_X \Theta(x) \, dQ_0(x) \le 0, $$

therefore, $\Theta(x) = 0$ for $Q_0$-a.e. $x$, which implies that $Q_0$ is an equilibrium of NAG(X, Y, π, L, H, m₀).

# 4 Application to the mean field game model with pairwise interactions

We now come back to the mean field game model of Section 2, and our aim is to apply the results from Section 3 for the non-atomic game NAG(X, Y, π, L, H, m₀) to the former MFG. To do so, we
let X, Y, π, L, and H be defined from the MFG from Section 2 as in Remark 3.2 and we aim at proving that (A1)–(A8) are satisfied if we assume (H1)–(H6), and that the notions of equilibria from Definitions 2.1 and 3.1 coincide. Clearly, (A1) and (A2) are satisfied as soon as one assumes (H1) and (H2), (A4) follows from (H4)(d), and (A6) follows from (H6). We are thus left to show (A3), (A5), (A7), and (A8). We start by the following result.

**Lemma 4.1.** Assume that (H1), (H2), and (H5) are satisfied and let $\tau$ be the function defined in (4). Then the functions $\tau$ and $\Psi \circ \tau$ are lower semicontinuous.

*Proof.* Let $(\gamma_n)_{n \in \mathbb{N}}$ be a sequence in $\Gamma$ converging to some $\gamma \in \Gamma$ uniformly on compact time intervals. We want to prove that

$$ \liminf_{n \to \infty} \tau(\gamma_n) \geq \tau(\gamma). $$

Let $\tau_*$ denote the left-hand side of the above inequality. If $\tau_* = +\infty$, there is nothing to prove, and thus we only consider the case $\tau_* < +\infty$. Let $(\gamma_{n_k})_{k \in \mathbb{N}}$ be a subsequence such that $\lim_{k \to \infty} \tau(\gamma_{n_k}) = \tau_*$. Since $\Xi$ is closed, we have, from the definition of $\tau$, that $\gamma_{n_k}(\tau(\gamma_{n_k})) \in \Xi$ for every $k \in \mathbb{N}$, and thus $\gamma(\tau_*) \in \Xi$. Hence, by the definition of $\tau$, we deduce that $\tau(\gamma) \leq \tau_*$, which is the desired inequality.

Regarding the function $\Psi \circ \tau$, since $\Psi$ is lower semicontinuous and nondecreasing, we obtain that, for any sequence $(\gamma_n)_{n \in \mathbb{N}}$ in $\Gamma$ converging to some $\gamma \in \Gamma$, we have

$$ \liminf_{n \to \infty} \Psi(\tau(\gamma_n)) \geq \Psi(\liminf_{n \to \infty} \tau(\gamma_n)) \geq \Psi(\tau(\gamma)), $$

as required.

□

We now provide the following technical result, which is a preliminary step towards showing the assertion on $L$ from (A3) as well as (A7).

**Lemma 4.2.** Assume that (H1)–(H3) and (H5) are satisfied and let $\theta > 1$ be the constant from (H3)(d) and $L$ be the function defined in Remark 3.2. Let $(\gamma_n)_{n \in \mathbb{N}}$ be a sequence in $\Gamma$ such that the sequence $(L(\gamma_n))_{n \in \mathbb{N}}$ is bounded. Then, for every $T > 0$, $(\gamma_n)_{n \in \mathbb{N}}$ is a bounded sequence in $W^{1,\theta}([0, T], \bar{\Omega})$ and there exists a subsequence $(\gamma_{n_k})_{k \in \mathbb{N}}$ of $(\gamma_n)_{n \in \mathbb{N}}$ and an element $\gamma \in W^{1,\theta}(\mathbb{R}_+, \bar{\Omega})$ such that $\gamma_{n_k} \to \gamma$ as $k \to +\infty$ in the topology of $\Gamma$ and, for every $T > 0$, we have the weak convergence $\dot{\gamma}_{n_k} \to \dot{\gamma}$ as $k \to +\infty$ in $L^\theta([0, T], \mathbb{R}^d)$.

*Proof.* Let $\kappa > 0$ and $(\gamma_n)_{n \in \mathbb{N}}$ be a sequence in $\Gamma$ such that $L(\gamma_n) \leq \kappa$ for every $n \in \mathbb{N}$. Hence, by (H3), we deduce that, for every $n \in \mathbb{N}$,

$$ \int_{0}^{+\infty} |\dot{\gamma}_n(t)|^\theta dt \leq \kappa. $$

Thus, for every $T > 0$, the sequence $(\gamma_n)_{n \in \mathbb{N}}$ is bounded in $W^{1,\theta}([0, T], \bar{\Omega})$. Recall also that the injection of $W^{1,\theta}([0, T], \bar{\Omega})$ into $\mathcal{C}([0, T], \bar{\Omega})$ is compact and that bounded sets of $W^{1,\theta}([0, T], \bar{\Omega})$ are relatively compact for the weak convergence.

We extract the required subsequence by a standard diagonal argument as follows. Since $(\gamma_n)_{n \in \mathbb{N}}$ is bounded in $W^{1,\theta}([0, 1], \bar{\Omega})$, we extract a subsequence $(\gamma_n^1)_{n \in \mathbb{N}}$ of $(\gamma_n)_{n \in \mathbb{N}}$ which converges strongly in $\mathcal{C}([0, 1], \bar{\Omega})$ and weakly in $W^{1,\theta}([0, 1], \bar{\Omega})$ to some $\gamma^1 \in W^{1,\theta}([0, 1], \bar{\Omega})$. Now, assuming that $k \in \mathbb{N}^*$ is such that we have constructed a subsequence $(\gamma_n^k)_{n \in \mathbb{N}}$ of $(\gamma_n)_{n \in \mathbb{N}}$ converging strongly in $\mathcal{C}([0, k], \bar{\Omega})$ and weakly in $W^{1,\theta}([0, k], \bar{\Omega})$ to some element $\gamma^k \in W^{1,\theta}([0, k], \bar{\Omega})$, using the fact that $(\gamma_n^k)_{n \in \mathbb{N}}$ is bounded in $W^{1,\theta}([0, k+1], \bar{\Omega})$, we extract a subsequence $(\gamma_n^{k+1})_{n \in \mathbb{N}}$ of $(\gamma_n^k)_{n \in \mathbb{N}}$ converging strongly in $\mathcal{C}([0, k+1], \bar{\Omega})$ and weakly in $W^{1,\theta}([0, k+1], \bar{\Omega})$ to some $\gamma^{k+1} \in W^{1,\theta}([0, k+1], \bar{\Omega})$, and clearly, by uniqueness of the limit, $\gamma^{k+1}$ coincides with $\gamma^k$ in their common domain $[0, k]$.

Define $\gamma: \mathbb{R}_+ \to \bar{\Omega}$ by setting $\gamma(t) = \gamma^{[t]}(t)$, and note that, by construction, we have $\gamma \in W^{1,\theta}(\mathbb{R}_+, \bar{\Omega})$, since, for every $k \in \mathbb{N}^*$, $\gamma$ coincides with $\gamma^k$ in $[0, k]$. The diagonal sequence $(\gamma_n^k)_{n \in \mathbb{N}}$ is a subsequence of $(\gamma_n)_{n \in \mathbb{N}}$ which, by construction, converges strongly to $\gamma$ in $\Gamma$, i.e., uniformly in any compact subset of $\mathbb{R}_+$, and, for every $T > 0$, it also converges weakly to $\gamma$ in $W^{1,\theta}([0, T], \bar{\Omega})$, yielding the conclusion.

□

A first consequence of Lemma 4.2 is the following.
**Corollary 4.3.** Assume that (H1)-(H3) and (H5) are satisfied and let L be the function defined in Remark 3.2. Then L is lower semicontinuous.

*Proof.* Let $(\gamma_n)_{n \in \mathbb{N}}$ be a sequence in $\Gamma$ such that $\gamma_n \to \gamma$ as $n \to +\infty$ for some $\gamma \in \Gamma$. We need to prove that

$$L(\gamma) \leq \liminf_{n \to +\infty} L(\gamma_n). \qquad (19)$$

If the right-hand side of (19) is equal to $+\infty$, there is nothing to prove, so we only consider the case where it is finite. In this case, we extract from $(\gamma_n)_{n \in \mathbb{N}}$ a subsequence $(\gamma_{n_k})_{k \in \mathbb{N}}$ such that $(L(\gamma_{n_k}))_{k \in \mathbb{N}}$ converges to the right-hand side of (19). For simplicity, we will denote $(\gamma_{n_k})_{k \in \mathbb{N}}$ by $(\gamma_n)_{n \in \mathbb{N}}$ in the sequel. Up to removing at most finitely many elements of the sequence, we have $L(\gamma_n) < +\infty$ for every $n \in \mathbb{N}$, and thus $(L(\gamma_n))_{n \in \mathbb{N}}$ is bounded. Hence, by Lemma 4.2, up to extracting once again a subsequence, which we still denote by the same notation, we deduce that $\gamma \in W_{\text{loc}}^{1,\theta}(\mathbb{R}_+, \bar{\Omega})$ and that, for every $T > 0$, we have the weak convergence $\dot{\gamma}_n \to \dot{\gamma}$ in $L^\theta([0, T], \mathbb{R}^d)$.

By Lemma 4.1, we have $\Psi(\tau(\gamma)) \leq \liminf_{n \to +\infty} \Psi(\tau(\gamma_n))$. In addition, using [46, Theorem 4.5], we obtain that, for every $T > 0$,

$$\int_0^T \ell(t, \gamma(t), \dot{\gamma}(t)) dt \leq \liminf_{n \to +\infty} \int_0^T \ell(t, \gamma_n(t), \dot{\gamma}_n(t)) dt \leq \liminf_{n \to +\infty} \int_0^{+\infty} \ell(t, \gamma_n(t), \dot{\gamma}_n(t)) dt,$$

and, since this holds for every $T > 0$, we deduce that

$$\int_0^{+\infty} \ell(t, \gamma(t), \dot{\gamma}(t)) dt \leq \liminf_{n \to +\infty} \int_0^{+\infty} \ell(t, \gamma_n(t), \dot{\gamma}_n(t)) dt,$$

yielding the conclusion.

□

We also deduce from Lemma 4.2 and Corollary 4.3 the following result.

**Corollary 4.4.** Assume that (H1)-(H3) and (H5) are satisfied and let L be the function defined in Remark 3.2. Then (A7) is satisfied.

*Proof.* Let $\kappa > 0$ and consider a sequence $(\gamma_n)_{n \in \mathbb{N}}$ in $\Gamma$ such that $L(\gamma_n) \leq \kappa$ for every $n \in \mathbb{N}$. Then, by Lemma 4.2, $(\gamma_n)_{n \in \mathbb{N}}$ admits a subsequence, still denoted by $(\gamma_n)_{n \in \mathbb{N}}$, converging to some $\gamma \in W_{\text{loc}}^{1,\theta}(\mathbb{R}_+, \bar{\Omega})$, both in the topology of $\Gamma$ and with $\dot{\gamma}_n \to \dot{\gamma}$ in $L^\theta([0, T], \mathbb{R}^d)$ for every $T > 0$. Since L is lower semicontinuous by Corollary 4.3 and $\gamma_n \to \gamma$ in $\Gamma$ as $n \to +\infty$, we deduce that $L(\gamma) \leq \liminf_{n \to +\infty} L(\gamma_n) \leq \kappa$, yielding the conclusion. □

Let us now show that J is also lower semicontinuous and H is Borel measurable.

**Lemma 4.5.** Assume that (H1)-(H5) are satisfied, let L and H be the functions defined in Remark 3.2, and J be the function defined in (8). Then J is lower semicontinuous and H is Borel measurable.

*Proof.* We note first that it suffices to prove the assertion on J. Indeed, since L is lower semi-continuous by Corollary 4.3, it is Borel measurable, and thus the set $\{(\gamma, \tilde{\gamma}) \in \Gamma \times \Gamma \mid L(\gamma) = +\infty \text{ or } L(\tilde{\gamma}) = +\infty\}$ is Borel measurable. Since H is constant and equal to $+\infty$ in this set, it suffices to show that H is Borel measurable in the complementary of this set, that is, on $\{(\gamma, \tilde{\gamma}) \in \Gamma \times \Gamma \mid L(\gamma) < +\infty \text{ and } L(\tilde{\gamma}) < +\infty\}$. In this set, we have $H(\gamma, \tilde{\gamma}) = J(\gamma, \tilde{\gamma}) - L(\gamma) - L(\tilde{\gamma})$, and thus, if J is shown to be lower semicontinuous, it will also be Borel measurable, and hence H will be Borel measurable as the difference of Borel measurable functions.

Let us then prove that J is lower semicontinuous. Let $(\gamma_n, \tilde{\gamma}_n)_{n \in \mathbb{N}}$ be a sequence in $\Gamma \times \Gamma$ converging to some $(\gamma, \tilde{\gamma}) \in \Gamma \times \Gamma$. We want to prove that

$$J(\gamma, \tilde{\gamma}) \leq \liminf_{n \to +\infty} J(\gamma_n, \tilde{\gamma}_n). \qquad (20)$$

As in the proof of Corollary 4.3, we consider only the case where the right-hand side of the above inequality is finite and, up to extracting a subsequence, we also have that $J(\gamma_n, \tilde{\gamma}_n)$ converges as $n \to +\infty$ to $\liminf_{n \to +\infty} J(\gamma_n, \tilde{\gamma}_n)$ and that the sequence $(J(\gamma_n, \tilde{\gamma}_n))_{n \in \mathbb{N}}$ is bounded. In particular, $(L(\gamma_n))_{n \in \mathbb{N}}$ and $(L(\tilde{\gamma}_n))_{n \in \mathbb{N}}$ are bounded sequences, so, by Lemma 4.2, we deduce that $\gamma$ and $\tilde{\gamma}$
belong to $W_{\text{loc}}^{1,\theta}(\mathbb{R}_+, \bar{\Omega})$ and that, up to a further subsequence extraction, $\dot{\gamma}_n \to \dot{\gamma}$ and $\dot{\tilde{\gamma}}_n \to \dot{\tilde{\gamma}}$ as $n \to +\infty$ in $L^\theta([0, T], \mathbb{R}^d)$, for every $T > 0$.

Note that, by Corollary 4.3, we already have

$$L(\gamma) \le \liminf_{n \to +\infty} L(\gamma_n), \quad L(\tilde{\gamma}) \le \liminf_{n \to +\infty} L(\tilde{\gamma}_n),$$

so we are left to show that

$$H(\gamma, \tilde{\gamma}) \le \liminf_{n \to +\infty} H(\gamma_n, \tilde{\gamma}_n). \qquad (21)$$

Let us denote $\sigma_n = \tau(\gamma_n) \wedge \tau(\tilde{\gamma}_n)$ and $\sigma = \tau(\gamma) \wedge \tau(\tilde{\gamma})$ and recall that, by Lemma 4.1, we have $\sigma \le \liminf_{n \to +\infty} \sigma_n$. We have nothing to prove in the case $\sigma = 0$ since, in this case, the left-hand side of (21) is zero. We thus only consider the case $\sigma > 0$ from now on. Fix an increasing sequence $(T_k)_{k \in \mathbb{N}}$ in $\mathbb{R}_+$ with $\lim_{k \to +\infty} T_k = \sigma$. For each $k \in \mathbb{N}$, applying [46, Theorem 4.5] to the interval $[0, T_k]$, we obtain that

$$\int_0^{T_k} h(t, \gamma(t), \tilde{\gamma}(t), \dot{\gamma}(t), \dot{\tilde{\gamma}}(t)) \, dt \le \liminf_{n \to +\infty} \int_0^{T_k} h(t, \gamma_n(t), \tilde{\gamma}_n(t), \dot{\gamma}_n(t), \dot{\tilde{\gamma}}_n(t)) \, dt. \quad (22)$$

Since $T_k < \sigma \le \liminf_{n \to +\infty} \sigma_n$, we have $\sigma_n > T_k$ for $n$ large enough (depending on $k$), and thus, as $h$ is nonnegative, we get

$$\int_0^{T_k} h(t, \gamma_n(t), \tilde{\gamma}_n(t), \dot{\gamma}_n(t), \dot{\tilde{\gamma}}_n(t)) \, dt \le \int_0^{\sigma_n} h(t, \gamma_n(t), \tilde{\gamma}_n(t), \dot{\gamma}_n(t), \dot{\tilde{\gamma}}_n(t)) \, dt$$

for $n$ large enough. Hence, taking the $\liminf$ as $n \to +\infty$, we obtain, combining with (22), that

$$\int_0^{T_k} h(t, \gamma(t), \tilde{\gamma}(t), \dot{\gamma}(t), \dot{\tilde{\gamma}}(t)) \, dt \le \liminf_{n \to +\infty} \int_0^{\sigma_n} h(t, \gamma_n(t), \tilde{\gamma}_n(t), \dot{\gamma}_n(t), \dot{\tilde{\gamma}}_n(t)) \, dt.$$

Since this holds true for every $k \in \mathbb{N}$, we then obtain that

$$\int_0^{\sigma} h(t, \gamma(t), \tilde{\gamma}(t), \dot{\gamma}(t), \dot{\tilde{\gamma}}(t)) \, dt \le \liminf_{n \to +\infty} \int_0^{\sigma_n} h(t, \gamma_n(t), \tilde{\gamma}_n(t), \dot{\gamma}_n(t), \dot{\tilde{\gamma}}_n(t)) \, dt,$$

as required. $\square$

<figure><img src="image_3.png" /></figure>

**Remark 4.6.** The proof of Lemma 4.5 does not imply that $H$ is lower semicontinuous. Indeed, although we prove the inequality (21), this is not done for any sequence $(\gamma_n, \tilde{\gamma}_n)_{n \in \mathbb{N}}$ converging in $\Gamma \times \Gamma$: as we want to prove (20), we restrict our attention to sequences for which the right-hand side of (20) is finite. Hence, our proof of the inequality (21) does not take into account sequences $(\gamma_n, \tilde{\gamma}_n)_{n \in \mathbb{N}}$ for which the right-hand side of (21) is finite, but that of (20) is infinite. One should also take these sequences into account in order to obtain lower semicontinuity of $H$, and the main issue is that we cannot apply the compactness result from Lemma 4.2 to such sequences.

Our next result establishes (A8) as a consequence of (H1)–(H5).

**Lemma 4.7.** Assume that (H1)–(H5) are satisfied and let $L$ and $H$ be the functions defined in *Remark 3.2*. Then (A8) is satisfied.

*Proof.* Consider the constants $\alpha, \theta, C, \beta, a$, and $b$ from (H3)–(H5). For every $\gamma \in \Gamma$ with $L(\gamma) < +\infty$, we have, using Young’s inequality, that

$$\begin{aligned} \int_0^{\tau(\gamma)} |\dot{\gamma}(t)|^\beta \, dt &\le \frac{\beta}{\theta} \int_0^{+\infty} |\dot{\gamma}(t)|^\theta \, dt + \frac{\theta - \beta}{\theta} \tau(\gamma) \\ &\le \frac{\beta}{\alpha\theta} \int_0^{+\infty} \ell(t, \gamma(t), \dot{\gamma}(t)) \, dt + \frac{(\theta - \beta)(\Psi(\tau(\gamma)) + b)}{a\theta} \\ &\le M \left( L(\gamma) + \frac{1}{2} \right), \end{aligned}$$
where $M = \max \left\{ \frac{\beta}{a\theta}, \frac{\theta-\beta}{a\theta}, \frac{2b(\theta-\beta)}{a\theta} \right\}$. On the other hand, using (H4), for every $(\gamma, \tilde{\gamma}) \in \Gamma \times \Gamma$ with $L(\gamma) < +\infty$ and $L(\tilde{\gamma}) < +\infty$, we have

$$ 
\begin{aligned}
H(\gamma, \tilde{\gamma}) & \le \int_0^{\tau(\gamma) \wedge \tau(\tilde{\gamma})} C(|\dot{\gamma}(t)|^\beta + |\dot{\tilde{\gamma}}(t)|^\beta) \, dt \\
& \le C \left( \int_0^{\tau(\gamma)} |\dot{\gamma}(t)|^\beta \, dt \right) + C \left( \int_0^{\tau(\tilde{\gamma})} |\dot{\tilde{\gamma}}(t)|^\beta \, dt \right) \\
& \le CM(L(\gamma) + L(\tilde{\gamma}) + 1),
\end{aligned}
$$

yielding the conclusion.

□

We are thus finally in position to conclude that the mean field game from Section 2 can be studied through the non-atomic game NAG($X, Y, \pi, L, H, m_0$).

**Proposition 4.8.** Assume that (H1)–(H6) are satisfied and consider the non-atomic game NAG($X, Y, \pi, L, H, m_0$) with $X, Y, \pi, L$, and $H$ defined as in Remark 3.2. Then (A1)–(A8) are satisfied for NAG($X, Y, \pi, L, H, m_0$). In addition, $Q$ is an equilibrium of the mean field game from Section 2 with initial condition $m_0 \in \mathcal{P}(\bar{\Omega})$ if and only if it is an equilibrium of NAG($X, Y, \pi, L, H, m_0$).

*Proof.* The fact that (A1)–(A8) are satisfied was already established in the previous results. As for the second part of the statement, note that there is a subtlety to be addressed: the term $H(\gamma, \tilde{\gamma})$ does not necessarily coincide with the integral $\int_0^{\tau(\gamma) \wedge \tau(\tilde{\gamma})} h(t, \gamma(t), \tilde{\gamma}(t), \dot{\gamma}(t), \dot{\tilde{\gamma}}(t)) \, dt$, since the first one is defined to be $+\infty$ whenever $L(\gamma) = +\infty$ or $L(\tilde{\gamma}) = +\infty$, but the second one can be finite even in some cases where $L(\gamma) = +\infty$ or $L(\tilde{\gamma}) = +\infty$. Hence, the functions defined in (3) and (5) do not necessarily coincide for all $(\gamma, Q)$. In this proof, we denote by $F_1(\gamma, Q)$ the function defined in (3) and by $F_2(\gamma, Q)$ the function defined in (5).

Notice first that, if $Q \in \mathcal{P}(\Gamma)$ is such that $Q(\text{dom } L) = 1$, then $F_1(\gamma, Q) = F_2(\gamma, Q)$ for every $\gamma \in \Gamma$. Indeed, recall that, for every $(\gamma, \tilde{\gamma}) \in \text{dom } L \times \text{dom } L$, we have

$$ H(\gamma, \tilde{\gamma}) = \int_0^{\tau(\gamma) \wedge \tau(\tilde{\gamma})} h(t, \gamma(t), \tilde{\gamma}(t), \dot{\gamma}(t), \dot{\tilde{\gamma}}(t)) \, dt. $$

Since $Q(\text{dom } L) = 1$, we can integrate the above equality in $\tilde{\gamma}$ with respect to $Q$ to deduce that

$$ \int_{\Gamma} H(\gamma, \tilde{\gamma}) \, dQ(\tilde{\gamma}) = \int_{\Gamma} \int_{0}^{\tau(\gamma) \wedge \tau(\tilde{\gamma})} h(t, \gamma(t), \tilde{\gamma}(t), \dot{\gamma}(t), \dot{\tilde{\gamma}}(t)) \, dt \, dQ(\tilde{\gamma}) $$

for every $\gamma \in \text{dom } L$. Thus, $F_1(\gamma, Q) = F_2(\gamma, Q)$ for every $\gamma \in \text{dom } L$. On the other hand, if $\gamma \notin \text{dom } L$, we have $L(\gamma) = +\infty$, hence $F_1(\gamma, Q) = F_2(\gamma, Q) = +\infty$. Hence, $F_1(\gamma, Q) = F_2(\gamma, Q)$ for every $\gamma \in \Gamma$.

If $Q$ is an equilibrium of the mean field game from Section 2 with initial condition $m_0 \in \mathcal{P}(\bar{\Omega})$, then $\int_{\Gamma} F_1(\gamma, Q) \, dQ(\gamma) < +\infty$, showing that $F_1(\gamma, Q) < +\infty$ for $Q$-almost every $\gamma \in \Gamma$. In particular, $L(\gamma) < +\infty$ for $Q$-almost every $\gamma$, i.e., $Q(\text{dom } L) = 1$, and thus, by the above argument, $F_1(\gamma, Q) = F_2(\gamma, Q)$ for every $\gamma \in \Gamma$, proving that $Q$ is also an equilibrium of NAG($X, Y, \pi, L, H, m_0$). Conversely, if $Q$ is an equilibrium of NAG($X, Y, \pi, L, H, m_0$), then $\int_{\Gamma} F_2(\gamma, Q) \, dQ(\gamma) < +\infty$, showing that $F_2(\gamma, Q) < +\infty$ for $Q$-almost every $\gamma \in \Gamma$. In particular, $L(\gamma) < +\infty$ for $Q$-almost every $\gamma$, i.e., $Q(\text{dom } L) = 1$, and thus, by the above argument, $F_1(\gamma, Q) = F_2(\gamma, Q)$ for every $\gamma \in \Gamma$, proving that $Q$ is also an equilibrium of the mean field game from Section 2 with initial condition $m_0 \in \mathcal{P}(\bar{\Omega})$. □

As an immediate consequence of Theorem 3.22 and Proposition 4.8, we obtain our main result on the mean field game model from Section 2.

**Theorem 4.9.** Assume that (H1)–(H6) are satisfied and let $m_0 \in \mathcal{P}(\bar{\Omega})$. Then there exists an equilibrium $Q \in \mathcal{P}(\Gamma)$ of the mean field game from Section 2 with initial condition $m_0$.

We conclude our discussion on the model from Section 2 by addressing strong equilibria. Similarly to Definition 3.23, we shall say that $Q \in \mathcal{P}(\Gamma)$ is a strong equilibrium of the mean field game from Section 2 if it satisfies Definition 2.1 with “$Q$-almost every $\gamma$” replaced by “every $\gamma \in \text{spt}(Q)$”.

Let us introduce two additional assumptions on the model from Section 2.
(H7) For every $x \in \bar{\Omega}$ and every sequence $(x_n)_{n \in \mathbb{N}}$ with $x_n \to x$ as $n \to +\infty$, we have that $d_{\text{geo}}(x_n, x) \to 0$ as $n \to +\infty$, where $d_{\text{geo}}$ denotes the geodesic distance in $\bar{\Omega}$.

(H8) There exists $\alpha^* > 0$ such that $\ell(t, x, p) \le \alpha^*|p|^\theta$ for every $(t, x, p) \in \mathbb{R}_+ \times \bar{\Omega} \times \mathbb{R}^d$, where $\theta$ is the same constant as in (H3).

One of the consequences of (H8) is that trajectories minimizing $F(\gamma, Q)$ starting at some $x_0$ already in the target set $\Xi$ must necessarily remain constant at all times.

**Lemma 4.10.** Assume that (H1)-(H5) and (H8) are satisfied for the mean field game of Section 2 and let $x_0 \in \Xi$ and $Q \in \mathcal{P}(\Gamma)$. Then the unique trajectory $\gamma_0$ minimizing (3) with the constraint $\gamma_0(0) = x_0$ is the constant trajectory $\gamma_0(t) = x_0$ for every $t \in \mathbb{R}_+$.

*Proof.* Clearly, $F(\gamma, Q) \ge \Psi(0)$ for every $\gamma \in \Gamma$, and the value $\Psi(0)$ is attained for the constant trajectory $\gamma_0$ since $\tau(\gamma_0) = 0$ and $0 \le \ell(t, \gamma_0(t), \dot{\gamma}_0(t)) \le \alpha^*|\dot{\gamma}_0(t)|^\theta = 0$. To prove uniqueness, note that $F(\gamma, Q) = \Psi(0)$ implies $\int_0^{+\infty} \ell(t, \gamma(t), \dot{\gamma}(t)) dt = 0$, showing that $\gamma$ is absolutely continuous and $\ell(t, \gamma(t), \dot{\gamma}(t)) = 0$ for almost every $t \in \mathbb{R}_+$. By (H3)(d), this implies that $\dot{\gamma}(t) = 0$ for almost every $t \in \mathbb{R}_+$, and hence $\gamma$ is constant. The constraint $\gamma(0) = x_0$ then implies that $\gamma = \gamma_0$. $\square$

We now use (H7) and (H8) to prove that (A9) is satisfied.

**Lemma 4.11.** Assume that (H1)-(H6), (H7), and (H8) are satisfied for the mean field game of Section 2 and consider the non-atomic game NAG($X, Y, \pi, L, H, m_0$) defined as in Remark 3.2. Then (A9) is satisfied.

*Proof.* Let $Q \in \text{dom} \mathcal{L}$ and $\text{Opt}(Q)$ be the set defined in (16). Recall that we have the inclusion

$$ \text{Opt}(Q_0) \subset \{\gamma \in \Gamma \mid L(\gamma) \le (C+1)\kappa + C\mathcal{L}(Q_0) + C\}, $$

where $\kappa$ and $C$ are the constants from (A6) and (A8): this was proved in Remark 3.25 under (A6) and (A8), and these two assumptions are satisfied here thanks to (H6) and Lemma 4.7. Let $(\gamma_n)_{n \in \mathbb{N}}$ be a sequence in $\text{Opt}(Q)$ and $\gamma \in \Gamma$ with $\gamma_n \to \gamma$ as $n \to +\infty$. In particular, by Lemma 4.2, we deduce that $\gamma$ and $\gamma_n$ belong to $W_{\text{loc}}^{1,\theta}(\mathbb{R}_+, \bar{\Omega})$ for every $n \in \mathbb{N}$ and that $\dot{\gamma}_n \to \dot{\gamma}$ in $L^\theta([0, T], \mathbb{R}^d)$ as $n \to +\infty$, for every $T > 0$.

Let $x_0 = \gamma(0)$ and, for $n \in \mathbb{N}$, set $x_n = \gamma_n(0)$. We split the proof in three cases.

*Case 1:* $x_0 \in \Xi$ and there exists a subsequence $(x_{n_k})_{k \in \mathbb{N}}$ of $(x_n)_{n \in \mathbb{N}}$ such that $x_{n_k} \in \Xi$ for every $k \in \mathbb{N}$.

In this case, it follows from Lemma 4.10 that $\gamma_{n_k}$ is constant and, since $\gamma_{n_k} \to \gamma$ as $k \to +\infty$, it follows that $\gamma$ is also constant. Hence, applying once again Lemma 4.10, we deduce that $\gamma$ minimizes (3) with the constraint $\gamma(0) = x_0$, yielding that $\gamma \in \text{Opt}(Q)$.

*Case 2:* $x_0 \in \Xi$ and $x_n \notin \Xi$ for every $n \in \mathbb{N}$.

Let $\varepsilon_n = d_{\text{geo}}(x_n, x_0)$ and $\sigma_n: [0, \varepsilon_n] \to \bar{\Omega}$ be a geodesic curve such that $\sigma_n(0) = x_n$, $\sigma_n(\varepsilon_n) = x_0$, and $|\dot{\sigma}_n(t)| = 1$ for almost every $t \in (0, \varepsilon_n)$. Notice that the existence of such a geodesic curve can be ensured by assumption (H7) and [19, Proposition 2.5.19] and, in addition, by (H7), we have $\varepsilon_n \to 0$ as $n \to +\infty$. One can easily extend the domain of $\sigma_n$ to $\mathbb{R}_+$ by setting $\sigma_n(t) = x_0$ for all $t \ge \varepsilon_n$. We have $0 < \tau(\sigma_n) \le \varepsilon_n \to 0$ as $n \to +\infty$, and thus $\Psi(\tau(\sigma_n)) \to \Psi(0^+)$ as $n \to +\infty$, where $\Psi(0^+) = \lim_{t \to 0^+} \Psi(t)$, which exists since $\Psi$ is nondecreasing. We also compute

$$ \int_0^{+\infty} \ell(t, \sigma_n(t), \dot{\sigma}_n(t)) dt \le \alpha^* \int_0^{+\infty} |\dot{\sigma}_n(t)|^\theta dt \le \alpha^* \varepsilon_n \xrightarrow{n \to +\infty} 0 \quad (23) $$

and, for every $\omega \in \text{dom} L$,

$$ \int_0^{\tau(\sigma_n) \wedge \tau(\omega)} h(t, \sigma_n(t), \omega(t), \dot{\sigma}_n(t), \dot{\omega}(t)) dt \le C\varepsilon_n + C \int_0^{\varepsilon_n} |\dot{\omega}(t)|^\beta dt \xrightarrow{n \to +\infty} 0, \quad (24) $$
where we use the fact that $\omega \in \text{dom } L$ to deduce from (H3)(d) that $\dot{\omega} \in L^\theta(\mathbb{R}_+, \mathbb{R}^d)$, and thus $|\dot{\omega}|^\beta \in L^1([0, T], \mathbb{R}^d)$ for every $T > 0$ since $\beta \in (0, \theta]$. In addition, for all $n$ large enough so that $\varepsilon_n \le 1$, we have

$$ \int_0^{\tau(\sigma_n) \wedge \tau(\omega)} h(t, \sigma_n(t), \omega(t), \dot{\sigma}_n(t), \dot{\omega}(t)) \, dt \le C + C \int_0^1 |\dot{\omega}(t)|^\beta \, dt, \quad (25) $$

and the right-hand side of the above inequality is $Q$-integrable, since, by Hölder's inequality and (H3)(d), we have

$$ \begin{aligned} \int_{\Gamma} \int_{0}^{1} |\dot{\omega}(t)|^{\beta} dt \, dQ(\omega) &\le \left( \int_{\Gamma} \int_{0}^{1} |\dot{\omega}(t)|^{\theta} dt \, dQ(\omega) \right)^{\frac{\beta}{\theta}} \\ &\le \left( \frac{1}{\alpha} \int_{\Gamma} \int_{0}^{+\infty} \ell(t, \omega(t), \dot{\omega}(t)) dt \, dQ(\omega) \right)^{\frac{\beta}{\theta}} \le \left( \frac{\mathcal{L}(Q)}{\alpha} \right)^{\frac{\beta}{\theta}} < +\infty. \end{aligned} \quad (26) $$

Hence, it follows from (24), (25), and Lebesgue's dominated convergence theorem that

$$ \lim_{n \to +\infty} \int_{\Gamma} \int_{0}^{\tau(\sigma_n) \wedge \tau(\omega)} h(t, \sigma_n(t), \omega(t), \dot{\sigma}_n(t), \dot{\omega}(t)) \, dt \, dQ(\omega) = 0. $$

Combining this with (23) and the fact that $\Psi(\tau(\sigma_n)) \to \Psi(0^+)$ as $n \to +\infty$, we deduce that $\lim_{n \to +\infty} F(\sigma_n, Q) = \Psi(0^+)$. Since $\gamma_n$ is a minimizer of (3) with fixed initial condition $x_n$ and $\sigma_n \in \Gamma$ with $\sigma_n(0) = x_n$, we have $F(\gamma_n, Q) \le F(\sigma_n, Q)$, and, on the other hand, since $\tau(\gamma_n) > 0$, we have $\Psi(0^+) \le \Psi(\tau(\gamma_n)) \le F(\gamma_n, Q)$. Hence, $\lim_{n \to +\infty} F(\gamma_n, Q) = \Psi(0^+)$, and (H3)(d) yields $\int_0^{+\infty} |\dot{\gamma}_n(t)|^\theta dt \to 0$ as $n \to +\infty$. Recalling that, for every $T > 0$, we have $\dot{\gamma}_n \to \dot{\gamma}$ in $L^\theta([0, T], \mathbb{R}^d)$ as $n \to +\infty$, we deduce, for the lower semicontinuity of the $L^\theta$ norm with respect to weak convergence, that

$$ \|\dot{\gamma}\|_{L^\theta([0,T], \mathbb{R}^d)} \le \liminf_{n \to +\infty} \|\dot{\gamma}_n\|_{L^\theta([0,T], \mathbb{R}^d)} = 0, $$

and, since $T > 0$ is arbitrary, we deduce that $\gamma$ is constant. Hence, by Lemma 4.10, we deduce that $\gamma$ minimizes (3) with the constraint $\gamma(0) = x_0$, yielding that $\gamma \in \mathbf{Opt}(Q)$.

Case 3: $x_0 \notin \Xi$.

By (H2), we have in this case $x_n \notin \Xi$ for $n$ large enough. To prove that $\gamma \in \mathbf{Opt}(Q)$, we will prove that $F(\gamma, Q) \le F(\tilde{\gamma}, Q)$ for every $\tilde{\gamma} \in \Gamma$ with $\tilde{\gamma}(0) = x_0$. For that purpose, we will construct a sequence of trajectories $(\tilde{\gamma}_n)_{n \in \mathbb{N}}$ in $\Gamma$ with $\tilde{\gamma}_n(0) = x_n$ for every $n \in \mathbb{N}$ and such that

$$ \liminf_{n \to +\infty} F(\tilde{\gamma}_n, Q) \le F(\tilde{\gamma}, Q). \quad (27) $$

This will allow us to conclude, since, by Lemma 3.12 and by the fact that $\gamma_n \in \mathbf{Opt}(Q)$, we will then have

$$ F(\gamma, Q) \le \liminf_{n \to +\infty} F(\gamma_n, Q) \le \liminf_{n \to +\infty} F(\tilde{\gamma}_n, Q) \le F(\tilde{\gamma}, Q), $$

yielding that $\gamma \in \mathbf{Opt}(Q)$. We then focus on showing (27).

Let $\tilde{\gamma} \in \Gamma$ with $\tilde{\gamma}(0) = x_0$. Note that there is nothing to be proved in (27) if $F(\tilde{\gamma}, Q) = +\infty$, so we assume in the sequel that $F(\tilde{\gamma}, Q) < +\infty$. In particular, $\tilde{\gamma} \in \text{dom } L$, which implies by (H3) and (H5) that $\dot{\tilde{\gamma}} \in L^\theta(\mathbb{R}_+, \mathbb{R}^d)$ and $\tau(\tilde{\gamma}) < +\infty$.

Let $\varepsilon_n = \text{d}_{\text{geo}}(x_n, x_0)$ and $\zeta_n: [0, \varepsilon_n] \to \bar{\Omega}$ be a geodesic curve such that $\zeta_n(0) = x_n$, $\zeta_n(\varepsilon_n) = \tilde{\gamma}(\varepsilon_n)$, and $|\zeta_n(t)| = \frac{\text{d}_{\text{geo}}(x_n, \tilde{\gamma}(\varepsilon_n))}{\varepsilon_n}$ for every $t \in [0, \varepsilon_n]$ (with the convention that, in the case $\varepsilon_n = 0$, we have that $\zeta_n: \{0\} \to \bar{\Omega}$ is given by $\zeta_n(0) = x_n = x_0$). Notice that, as in Case 2, the existence of such a geodesic curve can be ensured by assumption (H7) and [19, Proposition 2.5.19] and, in addition, by (H7), we have $\varepsilon_n \to 0$ as $n \to +\infty$. Let us define $\tilde{\gamma}_n \in \Gamma$ by

$$ \tilde{\gamma}_n(t) = \begin{cases} \zeta_n(t) & \text{if } t \in [0, \varepsilon_n], \\ \tilde{\gamma}(t) & \text{if } t \ge \varepsilon_n. \end{cases} $$
In particular, we also have $\tilde{\gamma}_n \in \text{dom } L$, $\dot{\tilde{\gamma}}_n \in L^\theta(\mathbb{R}_+, \mathbb{R}^d)$, and $0 < \tau(\tilde{\gamma}_n) \le \tau(\tilde{\gamma}) < +\infty$ for every $n \in \mathbb{N}$ large enough.

Note that, since $\tilde{\gamma}$ and $\tilde{\gamma}_n$ belong to dom L, we have

$$ \begin{aligned} & \left| \int_{0}^{+\infty} \ell(t, \tilde{\gamma}(t), \dot{\tilde{\gamma}}(t)) dt - \int_{0}^{+\infty} \ell(t, \tilde{\gamma}_n(t), \dot{\tilde{\gamma}}_n(t)) dt \right| \\ & \le \int_{0}^{\varepsilon_n} \ell(t, \tilde{\gamma}(t), \dot{\tilde{\gamma}}(t)) dt + \int_{0}^{\varepsilon_n} \ell(t, \varsigma_n(t), \dot{\varsigma}_n(t)) dt \quad (28) \\ & \le \int_{0}^{\varepsilon_n} \ell(t, \tilde{\gamma}(t), \dot{\tilde{\gamma}}(t)) dt + \alpha^* \frac{d_{\text{geo}}(x_n, \tilde{\gamma}(\varepsilon_n))^\theta}{\varepsilon_n^{\theta-1}}. \end{aligned} $$

Since $\tilde{\gamma} \in \text{dom } L$, we have

$$ \int_{0}^{\varepsilon_n} \ell(t, \tilde{\gamma}(t), \dot{\tilde{\gamma}}(t)) dt \xrightarrow{n \to +\infty} 0. \quad (29) $$

Notice also that

$$ d_{\text{geo}}(x_n, \tilde{\gamma}(\varepsilon_n))^\theta \le (d_{\text{geo}}(x_n, x_0) + d_{\text{geo}}(\tilde{\gamma}(0), \tilde{\gamma}(\varepsilon_n)))^\theta \le 2^{\theta-1} \left( d_{\text{geo}}(x_n, x_0)^\theta + \left( \int_0^{\varepsilon_n} |\dot{\tilde{\gamma}}(t)| dt \right)^\theta \right) $$

and, by Hölder's inequality,

$$ \left( \int_{0}^{\varepsilon_n} |\dot{\tilde{\gamma}}(t)| dt \right)^{\theta} \le \varepsilon_n^{\theta-1} \int_{0}^{\varepsilon_n} |\dot{\tilde{\gamma}}(t)|^{\theta} dt. $$

Hence

$$ \frac{d_{\text{geo}}(x_n, \tilde{\gamma}(\varepsilon_n))^\theta}{\varepsilon_n^{\theta-1}} \le 2^{\theta-1} \left( \varepsilon_n + \int_0^{\varepsilon_n} |\dot{\tilde{\gamma}}(t)|^\theta dt \right) \xrightarrow{n \to +\infty} 0, \quad (30) $$

since $\dot{\tilde{\gamma}} \in L^\theta(\mathbb{R}_+, \bar{\Omega})$. Combining (28), (29), and (30), we deduce that

$$ \lim_{n \to +\infty} \int_{0}^{+\infty} \ell(t, \tilde{\gamma}_n(t), \dot{\tilde{\gamma}}_n(t)) dt = \int_{0}^{+\infty} \ell(t, \tilde{\gamma}(t), \dot{\tilde{\gamma}}(t)) dt. \quad (31) $$

Note now that, for every $t \in [0, \varepsilon_n]$, we have

$$ \begin{aligned} |\varsigma_n(t) - x_0| &= \left| \varsigma_n(0) + \int_0^t \dot{\varsigma}_n(s) ds - x_0 \right| \le |x_n - x_0| + t \frac{d_{\text{geo}}(x_n, \tilde{\gamma}(\varepsilon_n))}{\varepsilon_n} \\ &\le |x_n - x_0| + d_{\text{geo}}(x_n, \tilde{\gamma}(\varepsilon_n)) \xrightarrow{n \to +\infty} 0, \end{aligned} $$

where we use (30) in the last step. Hence, since $x_0 \notin \Xi$, we deduce that, for $n$ large enough (independently of $t$), $\varsigma_n(t) \notin \Xi$ for every $t \in [0, \varepsilon_n]$. In addition, for $n$ large enough (independently of $t$), we clearly have $\tilde{\gamma}(t) \notin \Xi$ for every $t \in [0, \varepsilon_n]$. Hence $\tau(\tilde{\gamma}_n) = \tau(\tilde{\gamma})$ for $n$ large enough, and thus $\Psi(\tau(\tilde{\gamma}_n)) = \Psi(\tau(\tilde{\gamma}))$ for $n$ large enough. Combining this with (31), we deduce that

$$ \lim_{n \to +\infty} L(\tilde{\gamma}_n) = L(\tilde{\gamma}). \quad (32) $$

Let us now study whether $\int_{\Gamma} H(\tilde{\gamma}_n, \omega) dQ(\omega)$ converges to $\int_{\Gamma} H(\tilde{\gamma}, \omega) dQ(\omega)$ as $n \to +\infty$. For every $\omega \in \text{dom } L$, recalling that $\tau(\tilde{\gamma}_n) = \tau(\tilde{\gamma})$ for $n$ large enough, we have

$$ |H(\tilde{\gamma}_n, \omega) - H(\tilde{\gamma}, \omega)| \le \int_0^{\varepsilon_n} h(t, \tilde{\gamma}(t), \omega(t), \dot{\tilde{\gamma}}(t), \dot{\omega}(t)) dt + \int_0^{\varepsilon_n} h(t, \varsigma_n(t), \omega(t), \dot{\varsigma}_n(t), \dot{\omega}(t)) dt. \quad (33) $$

We have

$$ \int_{0}^{\varepsilon_n} h(t, \tilde{\gamma}(t), \omega(t), \dot{\tilde{\gamma}}(t), \dot{\omega}(t)) dt \le C \left( \int_{0}^{\varepsilon_n} |\dot{\tilde{\gamma}}(t)|^{\beta} dt + \int_{0}^{\varepsilon_n} |\dot{\omega}(t)|^{\beta} dt \right) \xrightarrow{n \to +\infty} 0 \quad (34) $$

since $\tilde{\gamma}$ and $\omega$ belong to dom L and, as such, their time derivatives belong to $L^\theta(\mathbb{R}_+, \mathbb{R}^d)$, which implies that $|\dot{\tilde{\gamma}}|^\beta$ and $|\dot{\omega}|^\beta$ belong to $L^1([0, T], \mathbb{R}^d)$ for every $T > 0$, since $\beta \in (0, \theta]$. On the other hand,

$$ \int_{0}^{\varepsilon_n} h(t, \varsigma_n(t), \omega(t), \dot{\varsigma}_n(t), \dot{\omega}(t)) dt \le C \left( \frac{d_{\text{geo}}(x_n, \tilde{\gamma}(\varepsilon_n))^\beta}{\varepsilon_n^{\beta-1}} + \int_{0}^{\varepsilon_n} |\dot{\omega}(t)|^\beta dt \right). \quad (35) $$
As before, we have $\int_{0}^{\epsilon_n} |\dot{\omega}(t)|^\beta dt \to 0$ as $n \to +\infty$, and, using (30), we have

$$ \frac{\mathrm{d}_{\mathrm{geo}}(x_n, \tilde{\gamma}(\epsilon_n))^\beta}{\epsilon_n^{\beta-1}} = \epsilon_n^{\frac{\theta-\beta}{\theta}} \left( \frac{\mathrm{d}_{\mathrm{geo}}(x_n, \tilde{\gamma}(\epsilon_n))^\theta}{\epsilon_n^{\theta-1}} \right)^{\frac{\beta}{\theta}} \underset{n \to +\infty}{\longrightarrow} 0 $$

since $\beta \in (0, \theta]$. Combining the above with (33), (34), and (35), we deduce that

$$ \lim_{n \to +\infty} H(\tilde{\gamma}_n, \omega) = H(\tilde{\gamma}, \omega) \qquad (36) $$

for every $\omega \in \mathrm{dom}\,L$. Notice also that we can estimate, for $n$ large enough (independently of $\omega$) in order to have in particular $\epsilon_n \le 1$,

$$ \begin{aligned} H(\tilde{\gamma}_n, \omega) &\le H(\tilde{\gamma}, \omega) + \int_0^{\epsilon_n} h(t, \varsigma_n(t), \omega(t), \dot{\varsigma}_n(t), \dot{\omega}(t)) \, dt \\ &\le H(\tilde{\gamma}, \omega) + C \frac{\mathrm{d}_{\mathrm{geo}}(x_n, \tilde{\gamma}(\epsilon_n))^\beta}{\epsilon_n^{\beta-1}} + C \int_0^1 |\dot{\omega}(t)|^\beta \, dt \qquad (37) \\ &\le H(\tilde{\gamma}, \omega) + 1 + C \int_0^1 |\dot{\omega}(t)|^\beta \, dt, \end{aligned} $$

where we have chosen $n$ large enough to have $C \frac{\mathrm{d}_{\mathrm{geo}}(x_n, \tilde{\gamma}(\epsilon_n))^\beta}{\epsilon_n^{\beta-1}} \le 1$. The right-hand side of (37) is $Q$-integrable with respect to $\omega$ thanks to Lemma 4.7, the facts that $\tilde{\gamma} \in \mathrm{dom}\,L$ and $Q \in \mathrm{dom}\,\mathcal{L}$, and (26). Hence, by (36), (37), and Lebesgue's dominated convergence theorem, we deduce that

$$ \lim_{n \to +\infty} \int_{\Gamma} H(\tilde{\gamma}_n, \omega) \, dQ(\omega) = \int_{\Gamma} H(\tilde{\gamma}, \omega) \, dQ(\omega). \qquad (38) $$

Combining (32) and (38), we deduce that

$$ \lim_{n \to +\infty} F(\tilde{\gamma}_n, Q) = F(\tilde{\gamma}, Q) $$

which clearly implies (27).

<figure><img src="image_4.png" /></figure>

Finally, as a consequence of Theorem 3.24 and Lemma 4.11, we immediately obtain the following result.

**Theorem 4.12.** Assume that (H1)–(H6), (H7), and (H8) are satisfied and let $m_0 \in \mathbb{P}(\bar{\Omega})$. Then $Q \in \mathbb{P}(\Gamma)$ is an equilibrium of the mean field game from Section 2 with initial condition $m_0$ if and only if it is a strong equilibrium of the same mean field game with the same initial condition.

# 5 Numerical illustration

We now provide a numerical illustration for the mean field game from Section 2. In addition to (H1)–(H6), we work here under the additional assumption that

$$ h(t, x, \tilde{x}, p, \tilde{p}) = 0 \quad \text{if } (x, p) = (\tilde{x}, \tilde{p}), $$

which implies that the interaction cost between an agent and themself is zero.

Note that the corresponding $N$-player game is also a potential game. More precisely, consider the game with $N$ agents where the aim of agent $i \in \{1, \dots, N\}$ starting at the position $x_{0,i}$ is to minimize with respect to $\gamma_i \in \Gamma$ the cost

$$ F_N(\gamma_i, \gamma_{-i}) = L(\gamma_i) + \frac{1}{N} \sum_{j=1}^{N} H(\gamma_i, \gamma_j) $$

with the constraint $\gamma_i(0) = x_{0,i}$, where $L$ and $H$ are given as in Remark 3.2 and $\gamma_{-i} = (\gamma_1, \dots, \gamma_{i-1}, \gamma_{i+1}, \dots, \gamma_N) \in \Gamma^{N-1}$ represents the choices of trajectories of other agents. Note that, since
$H(\gamma_i, \gamma_i) = 0$, the above sum can be equivalently taken over $j \in \{1, \dots, N\} \setminus \{i\}$. It is easy to verify that this $N$-player game is a potential game, with potential

$$ \mathcal{J}_N(\gamma_1, \dots, \gamma_N) = \frac{2}{N} \sum_{i=1}^{N} L(\gamma_i) + \frac{1}{N^2} \sum_{i=1}^{N} \sum_{j=1}^{N} H(\gamma_i, \gamma_j). \quad (39) $$

In addition, $F_N(\gamma_i, \gamma_{-i})$ is nothing but the cost $F(\gamma_i, Q)$ from (3) with $Q = \frac{1}{N} \sum_{j=1}^{N} \delta_{\gamma_j}$, and $\mathcal{J}_N(\gamma_1, \dots, \gamma_N)$ is nothing but $\mathcal{J}(Q)$ with the same $Q$ and $\mathcal{J}$ given by (9c).

Numerical approximations of equilibria of this $N$-player game can be obtained by minimizing the function $\mathcal{J}_N$ from (39) with the constraints $\gamma_i(0) = x_{0,i}$ for every $i \in \{1, \dots, N\}$. In order to avoid direct minimization over the whole space $\Gamma^N$, one can minimize $\mathcal{J}_N$ one variable at each time through a classical coordinate descent algorithm (see, e.g., [81] for an overview of coordinate descent algorithms in finite dimension), as described in Algorithm 1.

**Algorithm 1** Coordinate descent algorithm for minimizing the function $\mathcal{J}_N$ from (39) with constraints $\gamma_i(0) = x_{0,i}$ for every $i \in \{1, \dots, N\}$

**Require:** Positive integer $N$, initial conditions $x_{0,1}, \dots, x_{0,N}$ in $\bar{\Omega}$.

Initialize $\gamma_1, \dots, \gamma_N$ in $\Gamma$ with $\gamma_i(0) = x_{0,i}$ for every $i \in \{1, \dots, N\}$.

**repeat**

**for** $i \in \{1, \dots, N\}$ **do**

Select a new $\gamma_i \in \arg\min_{\tilde{\gamma}_i \in \Gamma} \mathcal{J}_N(\gamma_1, \dots, \gamma_{i-1}, \tilde{\gamma}_i, \gamma_{i+1}, \dots, \gamma_N)$
$\tilde{\gamma}_i(0) = x_{0,i}$

**end for**

**until** some convergence criterion is met.

Note that, even though $\mathcal{J}_N$ is a sum over $N + N^2$ terms, at each iteration of the algorithm, when minimizing for some agent $i$, $\mathcal{J}_N(\gamma_1, \dots, \gamma_{i-1}, \tilde{\gamma}_i, \gamma_{i+1}, \dots, \gamma_N)$ can be replaced by the individual cost $F_N(\tilde{\gamma}_i, \gamma_{-i})$, which is a sum of only $1 + N$ terms, as all other terms appearing in $\mathcal{J}_N(\gamma_1, \dots, \gamma_{i-1}, \tilde{\gamma}_i, \gamma_{i+1}, \dots, \gamma_N)$ are independent of $\tilde{\gamma}_i$. Hence, the coordinate descent algorithm coincides with a best response algorithm from game theory, in which, at each step, one player changes their strategy to their best response, while all other players remain with the same strategy (see, e.g., [70], where it is shown that the best response algorithm converges to a Nash equilibrium for potential games with finitely many actions for each player).

For our numerical illustration, we have applied Algorithm 1 to a slight modification of the mean field game from Section 2 in which there are two different populations. More precisely, we consider that there are two populations evolving in $\bar{\Omega}$, the distributions of their trajectories being described by two probability measures $Q_1, Q_2 \in \mathcal{P}(\Gamma)$. The populations are identical except for their initial distributions and target sets: for $i \in \{1, 2\}$, we assume that population $i$ is distributed at time $t=0$ according to a measure $m_0^i \in \mathcal{P}(\bar{\Omega})$ and that the goal of each agent of population $i$ is to reach a given target set $\Xi_i$ satisfying (H2). The cost optimized by an agent of population $i$ is of the form (3), with $\Psi(\tau(\gamma))$ replaced by $\Psi(\tau_i(\gamma))$, where $\tau_i(\gamma)$ is defined as in (4) with $\Xi_i$ replaced by $\Xi_i$, and the integral of $h$ in (3) is replaced by the sum of two integrals, one with respect to $Q_1$, in which $\tau(\gamma) \wedge \tau(\tilde{\gamma})$ is replaced by $\tau_i(\gamma) \wedge \tau_1(\tilde{\gamma})$, and another with respect to $Q_2$, in which $\tau(\gamma) \wedge \tau(\tilde{\gamma})$ is replaced by $\tau_i(\gamma) \wedge \tau_2(\tilde{\gamma})$. The analysis carried out in Sections 3 and 4 can be easily adapted to this two-population setting.

In our illustration, we consider that agents move in the square $\Omega = (0, 1) \times (0, 1) \subset \mathbb{R}^2$. Agents of population 1 are initially distributed according to the uniform measure $m_0^1$ in the left side of the square, $\{0\} \times [0, 1]$, and their goal is to reach the target set $\Xi_1 = \{1\} \times [0, 1]$. Agents of population 2 are initially distributed according to the uniform measure $m_0^2$ in the right side of the square, $\{1\} \times [0, 1]$, and their goal is to reach the target set $\Xi_2 = \{0\} \times [0, 1]$. For the illustration, we have chosen the functions $\ell$ and $\Psi$ as

$$ \ell(t, x, p) = \frac{1}{2} |p|^2, \quad \Psi(t) = t. $$
As for the function h, our aim is to select a function such that, when computed along two trajectories γ and $\tilde{\gamma}$, gives

$$h(t, \gamma(t), \tilde{\gamma}(t), \dot{\gamma}(t), \tilde{\dot{\gamma}}(t)) = 8 \exp \left( -\frac{|\gamma(t) - \tilde{\gamma}(t)|^2}{2\sigma^2} \right) \max \left[ 0, -\frac{d}{dt}|\gamma(t) - \tilde{\gamma}(t)| \right],$$

where $\sigma > 0$, since the above expression penalizes two agents that are close to each other, but only when the distance $|\gamma(t) - \tilde{\gamma}(t)|$ between them is decreasing. As

$$\frac{d}{dt}|\gamma(t) - \tilde{\gamma}(t)| = (\dot{\gamma}(t) - \tilde{\dot{\gamma}}(t)) \cdot \frac{\gamma(t) - \tilde{\gamma}(t)}{|\gamma(t) - \tilde{\gamma}(t)|},$$

we then wish to choose

$$h(t, x, \tilde{x}, p, \tilde{p}) = 8 \exp \left( -\frac{|x - \tilde{x}|^2}{2\sigma^2} \right) \max \left[ 0, -(p - \tilde{p}) \cdot \frac{x - \tilde{x}}{|x - \tilde{x}|} \right].$$

However, the above expression is not defined for $x = \tilde{x}$, and it is not possible to provide a continuous extension of it. We then choose a parameter $\delta > 0$ and replace the term $-(p - \tilde{p}) \cdot \frac{x - \tilde{x}}{|x - \tilde{x}|}$ by a convex combination of itself and the continuous expression $|p - \tilde{p}|$, with weights that are respectively 1 and 0 when $|x - \tilde{x}| \ge \delta$, and that have an affine dependence on $|x - \tilde{x}|$ when this term is less than $\delta$. Hence, the final expression of the function h chosen for our simulation is

$$h(t, x, \tilde{x}, p, \tilde{p}) = 8 \exp \left( -\frac{|x - \tilde{x}|^2}{2\sigma^2} \right) \max \left[ 0, -\min \left( 1, \frac{|x - \tilde{x}|}{\delta} \right) (p - \tilde{p}) \cdot \frac{x - \tilde{x}}{|x - \tilde{x}|} \right. \\ \left. + \max \left( 0, 1 - \frac{|x - \tilde{x}|}{\delta} \right) |p - \tilde{p}| \right], \quad (40)$$

and we selected $\sigma = \frac{1}{4}$ and $\delta = \frac{1}{5}$.

We have simulated the N-player game with 20 players in each population, and the corresponding results are provided in Figure 5.1. For the simulation, we have replaced the space $\Gamma$ by $\mathcal{C}([0, T], \bar{\Omega})$ with $T = 3$ (this choice was validated a posteriori by noticing that all agents in the simulation reach their desired target set much before the final time $T$). The time interval $[0, T]$ was discretized in $N_t = 100$ equally spaced points and trajectories where thus represented as elements of $\bar{\Omega}^{N_t}$. The initial distribution $m_0^1$ was replaced by a discrete distribution concentrated on 20 equally spaced points in the segment $\{0\} \times [0, 1]$, with a similar discretization for $m_0^2$. For a given trajectory $\gamma$ of population $i$ represented as a sequence of $N_t$ points in $\bar{\Omega}$, its velocity was computed as the sequence of $N_t - 1$ points obtained by dividing the difference between two successive positions in the trajectory by the time step $\Delta t = \frac{T}{N_t - 1}$, while its exit time $\tau(\gamma)$ was approximated by $\int_0^T \chi(\gamma(t)) dt$, where $\chi$ is a smooth function approximating the indicator function $1_{\bar{\Omega} \setminus \Xi_i}$. The nonsmooth functions $\max(0, \cdot)$ and $\min(1, \cdot)$ appearing in (40) were replaced by smooth approximations, while the integrals from (3) were approximated using the rectangle method. Algorithm 1 was initialized with trajectories going in a straight line with constant velocity from their initial position to the point in the target set with the same ordinate as the initial position, while, for the minimization step in Algorithm 1, we used the function `minimize` from Python's `scipy{ optimize` toolbox [80], alternating between Nelder-Mead and L-BFGS-B methods for optimization.

We observe, in Figure 5.1, a behavior close to the expected qualitative behavior for a crowd motion model. Agents reach the target set while minimizing the cost (3), and one can see the effect of the interaction term represented by $h$ by noticing that agents seem to anticipate the arrival of agents in the opposite direction, leaving them some space to pass in order to avoid a large congestion term due to $h$.

## References

[1] Y. Achdou, M. Bardi, and M. Cirant. Mean field games models of segregation. *Math. Models Methods Appl. Sci.*, 27(1):75–113, 2017.
<figure><img src="image_5.png" /><figcaption>Figure 5.1: Trajectories of the agents of the game described in Section 5 at times (a) $t = 0$, (b) $t = 0.24$, (c) $t = 0.36$, and (d) $t = 0.76$. Each population is represented by a different color, trajectories are represented by solid lines, and circles represent the current positions of the agents.</figcaption></figure>

[2] Y. Achdou and J.-M. Lasry. Mean field games for modeling crowd motion. In *Contributions to partial differential equations and applications*, volume 47 of *Comput. Methods Appl. Sci.*, pages 17–42. Springer, Cham, 2019.

[3] G. Allaire. *Numerical analysis and optimization*. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, 2007. Translated from the French by Alan Craig.

[4] L. Ambrosio, N. Gigli, and G. Savaré. *Gradient flows in metric spaces and in the space of probability measures*. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005.

[5] J.-P. Aubin and H. Frankowska. *Set-valued analysis*. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2009. Reprint of the 1990 edition [MR1048347].
[6] A. Aurell and B. Djehiche. Behavior near walls in the mean-field approach to crowd dynamics. *SIAM J. Appl. Math.*, 80(3):1153–1174, 2020.

[7] B. Aylaj, N. Bellomo, L. Gibelli, and D. Knopoff. *Crowd dynamics by kinetic theory modeling—complexity, modeling, simulations, and safety*, volume 36 of *Synthesis Lectures on Mathematics and Statistics*. Springer, Cham, 2022. Reprint of the 2020 original.

[8] F. Bagagiolo, S. Faggian, R. Maggistro, and R. Pesenti. Optimal control of the mean field equilibrium for a pedestrian tourists' flow model. *Netw. Spat. Econ.*, 22:243–266, 2022.

[9] M. Bardi and P. Cardaliaguet. Convergence of some mean field games systems to aggregation and flocking models. *Nonlinear Anal.*, 204:Paper No. 112199, 24, 2021.

[10] J.-D. Benamou, G. Carlier, and F. Santambrogio. Variational mean field games. In *Active particles. Vol. 1. Advances in theory, models, and applications*, Model. Simul. Sci. Eng. Technol., pages 141–171. Birkhäuser/Springer, Cham, 2017.

[11] M. Bernot, V. Caselles, and J.-M. Morel. *Optimal transportation networks*, volume 1955 of *Lecture Notes in Mathematics*. Springer-Verlag, Berlin, 2009. Models and theory.

[12] C. Bertucci. Optimal stopping in mean field games, an obstacle problem approach. *J. Math. Pures Appl. (9)*, 120:165–194, 2018.

[13] C. Bertucci. Monotone solutions for mean field games master equations: finite state space and optimal stopping. *J. Éc. polytech. Math.*, 8:1099–1132, 2021.

[14] A. Blanchet and G. Carlier. Optimal transport and Cournot-Nash equilibria. *Math. Oper. Res.*, 41(1):125–145, 2016.

[15] V. I. Bogachev. *Measure theory. Vol. I, II*. Springer-Verlag, Berlin, 2007.

[16] J. F. Bonnans, J. Gianatti, and L. Pfeiffer. A Lagrangian approach for aggregative mean field games of controls with mixed and final constraints. *SIAM J. Control Optim.*, 61(1):105–134, 2023.

[17] G. Bouveret, R. Dumitrescu, and P. Tankov. Mean-field games of optimal stopping: a relaxed solution approach. *SIAM J. Control Optim.*, 58(4):1795–1821, 2020.

[18] Y. Brenier. The least action principle and the related concept of generalized flows for incompressible perfect fluids. *J. Amer. Math. Soc.*, 2(2):225–255, 1989.

[19] D. Burago, Y. Burago, and S. Ivanov. *A course in metric geometry*, volume 33 of *Graduate Studies in Mathematics*. American Mathematical Society, Providence, RI, 2001.

[20] M. Burger, M. Di Francesco, P. A. Markowich, and M.-T. Wolfram. On a mean field game optimal control approach modeling fast exit scenarios in human crowds. In *52nd IEEE Conference on Decision and Control*. IEEE, dec 2013.

[21] M. Burzoni and L. Campi. Mean field games with absorption and common noise with a model of bank run. *Stochastic Process. Appl.*, 164:206–241, 2023.

[22] F. Camilli and C. Marchi. On quasi-stationary mean field games of controls. *Appl. Math. Optim.*, 87(3):Paper No. 47, 31, 2023.

[23] L. Campi and M. Fischer. N-player games and mean-field games with absorption. *Ann. Appl. Probab.*, 28(4):2188–2242, 2018.

[24] P. Cannarsa and R. Capuani. Existence and uniqueness for mean field games with state constraints. In *PDE models for multi-agent phenomena*, volume 28 of Springer INdAM Ser., pages 49–71. Springer, Cham, 2018.
[25] P. Cannarsa, R. Capuani, and P. Cardaliaguet. $C^{1,1}$-smoothness of constrained solutions in the calculus of variations with application to mean field games. *Math. Eng.*, 1(1):174–203, 2019.

[26] P. Cannarsa, R. Capuani, and P. Cardaliaguet. Mean field games with state constraints: from mild to pointwise solutions of the PDE system. *Calc. Var. Partial Differential Equations*, 60(3):Paper No. 108, 33, 2021.

[27] P. Cardaliaguet. Weak solutions for first order mean field games with local coupling. In *Analysis and geometry in control theory and its applications*, volume 11 of Springer INdAM Ser., pages 111–158. Springer, Cham, 2015.

[28] P. Cardaliaguet, F. Delarue, J.-M. Lasry, and P.-L. Lions. *The master equation and the convergence problem in mean field games*, volume 201 of *Annals of Mathematics Studies*. Princeton University Press, Princeton, NJ, 2019.

[29] P. Cardaliaguet and C.-A. Lehalle. Mean field game of controls and an application to trade crowding. *Math. Financ. Econ.*, 12(3):335–363, 2018.

[30] P. Cardaliaguet, A. R. Mészáros, and F. Santambrogio. First order mean field games with density constraints: pressure equals price. *SIAM J. Control Optim.*, 54(5):2672–2709, 2016.

[31] G. Carlier, C. Jimenez, and F. Santambrogio. Optimal transportation with traffic congestion and Wardrop equilibria. *SIAM J. Control Optim.*, 47(3):1330–1350, 2008.

[32] E. Carlini, A. Festa, F. J. Silva, and M.-T. Wolfram. A semi-Lagrangian scheme for a modified version of the Hughes' model for pedestrian flow. *Dyn. Games Appl.*, 7(4):683–705, 2017.

[33] E. Carlini and F. J. Silva. A fully-discrete scheme for systems of nonlinear Fokker-Planck-Kolmogorov equations. In *PDE models for multi-agent phenomena*, volume 28 of Springer INdAM Ser., pages 195–218. Springer, Cham, 2018.

[34] R. Carmona and F. Delarue. *Probabilistic theory of mean field games with applications. I*, volume 83 of *Probability Theory and Stochastic Modelling*. Springer, Cham, 2018. Mean field FBSDEs, control, and games.

[35] R. Carmona, F. Delarue, and D. Lacker. Mean field games of timing and models for bank runs. *Appl. Math. Optim.*, 76(1):217–260, 2017.

[36] C. Castaing and M. Valadier. *Convex analysis and measurable multifunctions*. Lecture Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977.

[37] P. Chan and R. Sircar. Fracking, renewables, and mean field games. *SIAM Rev.*, 59(3):588–615, 2017.

[38] E. Cristiani, A. de Santo, and M. Menci. A generalized mean-field game model for the dynamics of pedestrians with limited predictive abilities. *Commun. Math. Sci.*, 21(1):65–82, 2023.

[39] F. Cucker and S. Smale. Emergent behavior in flocks. *IEEE Trans. Automat. Control*, 52(5):852–862, 2007.

[40] L. Di Persio, M. Garbelli, and M. Ricciardi. The master equation in a bounded domain with absorption. Preprint arXiv:2203.15583.

[41] B. Djehiche, A. Tcheukam, and H. Tembine. A mean-field game of evacuation in mutlilevel building. *IEEE Trans. Automat. Control*, 62(10):5154–5169, 2017.

[42] M. F. Djete. Large population games with interactions through controls and common noise: convergence results and equivalence between open-loop and closed-loop controls. *ESAIM Control Optim. Calc. Var.*, 29:Paper No. 39, 42, 2023.

[43] R. Ducasse, G. Mazanti, and F. Santambrogio. Second order local minimal-time mean field games. *NoDEA Nonlinear Differential Equations Appl.*, 29:Paper No. 37, 32 pp., 2022.
[44] S. Dweik and G. Mazanti. Sharp semi-concavity in a non-autonomous control problem and $L^p$ estimates in an optimal-exit MFG. *NoDEA Nonlinear Differential Equations Appl.*, 27(2):Paper No. 11, 59 pp., 2020.

[45] M. Fischer and F. J. Silva. On the asymptotic nature of first order mean field games. *Appl. Math. Optim.*, 84(2):2327-2357, 2021.

[46] E. Giusti. *Direct methods in the calculus of variations*. World Scientific Publishing Co., Inc., River Edge, NJ, 2003.

[47] D. A. Gomes and S. Patrizi. Obstacle mean-field game problem. *Interfaces Free Bound.*, 17(1):55-68, 2015.

[48] D. A. Gomes, S. Patrizi, and V. K. Voskanyan. On the existence of classical solutions for stationary extended mean field games. *Nonlinear Anal.*, 99:49-79, 2014.

[49] D. A. Gomes and V. K. Voskanyan. Extended deterministic mean-field games. *SIAM J. Control Optim.*, 54(2):1030-1055, 2016.

[50] P. J. Graber and C. Mouzouni. On mean field games models for exhaustible commodities trade. *ESAIM Control Optim. Calc. Var.*, 26:Paper No. 11, 38, 2020.

[51] P. J. Graber, A. Mullenix, and L. Pfeiffer. Weak solutions for potential mean field games of controls. *NoDEA Nonlinear Differential Equations Appl.*, 28(5):Paper No. 50, 34, 2021.

[52] P. J. Graber and R. Sircar. Master equation for Cournot mean field games of control with absorption. *J. Differential Equations*, 343:816-909, 2023.

[53] D. Helbing and P. Molnár. Social force model for pedestrian dynamics. *Physical Review E*, 51(5):4282-4286, 1995.

[54] L. F. Henderson. The statistics of crowd fluids. *Nature*, 229(5284):381-383, 1971.

[55] M. Huang, P. E. Caines, and R. P. Malhamé. Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions. In *42nd IEEE Conference on Decision and Control, 2003. Proceedings*, volume 1, pages 98-103. IEEE, 2003.

[56] M. Huang, P. E. Caines, and R. P. Malhamé. Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized $\epsilon$-Nash equilibria. *IEEE Trans. Automat. Control*, 52(9):1560-1571, 2007.

[57] M. Huang, R. P. Malhamé, and P. E. Caines. Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. *Commun. Inf. Syst.*, 6(3):221-251, 2006.

[58] R. L. Hughes. A continuum theory for the flow of pedestrians. *Transport. Res. B-Meth*, 36(6):507-535, 2002.

[59] Z. Kobeissi. *Contributions to the theory of Mean Field Games*. PhD thesis, Université de Paris, October 2020.

[60] Z. Kobeissi. Mean field games with monotonous interactions through the law of states and controls of the agents. *NoDEA Nonlinear Differential Equations Appl.*, 29(5):Paper No. 52, 33, 2022.

[61] Z. Kobeissi. On classical solutions to the mean field game system of controls. *Comm. Partial Differential Equations*, 47(3):453-488, 2022.

[62] A. Lachapelle and M.-T. Wolfram. On a mean field game approach modeling congestion and aversion in pedestrian crowds. *Transport. Res. B-Meth*, 45(10):1572-1589, 2011.
[63] J.-M. Lasry and P.-L. Lions. Jeux à champ moyen. I. Le cas stationnaire. *C. R. Math. Acad. Sci. Paris*, 343(9):619-625, 2006.

[64] J.-M. Lasry and P.-L. Lions. Jeux à champ moyen. II. Horizon fini et contrôle optimal. *C. R. Math. Acad. Sci. Paris*, 343(10):679-684, 2006.

[65] J.-M. Lasry and P.-L. Lions. Mean field games. *Jpn. J. Math.*, 2(1):229-260, 2007.

[66] B. Maury and S. Faure. *Crowds in equations. An introduction to the microscopic modeling of crowds*. Advanced Textbooks in Mathematics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019.

[67] G. Mazanti. A note on existence and asymptotic behavior of Lagrangian equilibria for first-order optimal-exit mean field games. *Math. Control Relat. Fields*, 14(4):1776-1813, 2024.

[68] G. Mazanti and F. Santambrogio. Minimal-time mean field games. *Math. Models Methods Appl. Sci.*, 29(8):1413-1464, 2019.

[69] A. R. Mészáros and F. J. Silva. A variational approach to second order mean field games with density constraints: the stationary case. *J. Math. Pures Appl. (9)*, 104(6):1135-1159, 2015.

[70] D. Monderer and L. S. Shapley. Potential games. *Games Econom. Behav.*, 14(1):124-143, 1996.

[71] A. Muntean and F. Toschi, editors. *Collective dynamics from bacteria to crowds*, volume 553 of *CISM International Centre for Mechanical Sciences. Courses and Lectures*. Springer, Vienna, 2014. An excursion through modeling, analysis and simulation.

[72] M. Nutz. A mean field game of optimal stopping. *SIAM J. Control Optim.*, 56(2):1206-1221, 2018.

[73] M. Nutz, J. San Martin, and X. Tan. Convergence to the mean field game limit: a case study. *Ann. Appl. Probab.*, 30(1):259-286, 2020.

[74] S. Sadeghi Arjmand and G. Mazanti. On the characterization of equilibria of nonsmooth minimal-time mean field games with state constraints. In *2021 60th IEEE Conference on Decision and Control (CDC)*, pages 5300-5305, 2021.

[75] S. Sadeghi Arjmand and G. Mazanti. Multipopulation minimal-time mean field games. *SIAM J. Control Optim.*, 60(4):1942-1969, 2022.

[76] S. Sadeghi Arjmand and G. Mazanti. Nonsmooth mean field games with state constraints. *ESAIM Control Optim. Calc. Var.*, 28:Paper No. 74, 42 pp., 2022.

[77] F. Santambrogio. *Optimal transport for applied mathematicians*, volume 87 of *Progress in Nonlinear Differential Equations and their Applications*. Birkhäuser/Springer, Cham, 2015. Calculus of variations, PDEs, and modeling.

[78] F. Santambrogio and W. Shim. A Cucker-Smale inspired deterministic mean field game with velocity interactions. *SIAM J. Control Optim.*, 59(6):4155-4187, 2021.

[79] C. Villani et al. *Optimal transport: old and new*, volume 338. Springer, 2009.

[80] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and SciPy 1.0 Contributors. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. *Nature Methods*, 17:261-272, 2020.

[81] S. J. Wright. Coordinate descent algorithms. *Math. Program.*, 151(1):3-34, 2015.