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

A v ariational mean ﬁeld game of con trols with free ﬁnal time and pairwise in teractions Guilherme Mazan ti ∗ † Lauren t Pfeiﬀer ∗ † Saeed Sadeghi Arjmand F ebruary 20, 2026 Abstract This article considers a mean ﬁeld game mo del inspired b y crowd motion models in which agen ts aim at reac hing a given target set and wish to minimize a cost consisting of an individual running cost, an individual cost depending on the arriv al time at the target set, and an in teraction running cost, whic h takes the form of pairwise in teractions with other agen ts through b oth positions and v elocities. W e subsume this game under a more general class of games on abstract Polish spaces with pairwise interactions, and prov e that the latter games ha ve a v ariational structure (in the sense that their equilibria can be characterized as critical p oin ts of some p otential functional) and admit equilibria. W e also discuss t wo a priori distinct notions of equilibria, pro viding a suﬃcien t condition under which both notions coincide. The results for the games in abstract Polish spaces are applied to our mean ﬁeld game mo del, and a numerical illustration concludes the pap er. Keyw ords. Mean ﬁeld games, Lagrangian equilibria, congestion games, v ariational games, p oten tial games, pairwise in teractions, existence of equilibria. Mathematics Sub ject Classiﬁcation 2020. 49N80, 49J27, 49K27, 91A07, 91A14, 91A16. Con ten ts 1 In tro duction 1 2 Mean ﬁeld game mo del with pairwise interactions 4 3 A general non-atomic game with pairwise in teractions 6 3.1 First properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 P otential game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Existence of equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Strong equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 Application to the mean ﬁeld game mo del with pairwise in teractions 15 5 Numerical illustration 23 1 In tro duction Since the in tro duction of mean ﬁeld games (MFGs) in 2006 by Jean-Michel Lasry and Pierre- Louis Lions [63–65] and the simultaneous independent works b y Peter E. Caines, Min yi Huang, and Roland P . Malhamé [55–57], man y authors ha ve prop osed MFG models for (or inspired b y) the analysis of cro wd motion [8, 10, 20, 38, 62, 68, 78]. While the mathematical description of the ∗ Université Paris-Saclay , CNRS, Cen traleSup élec, Inria, Laboratoire des signaux et systèmes, 91190 Gif-sur- Y vette, F rance. † F édération de Mathématiques de CentraleSupélec, 91190, Gif-sur-Y v ette, F rance. 1 mo vemen t of cro wds of p eople has attracted the interest of several researc hers along the years (see, e.g., [7, 53, 54, 58, 66, 71]), the MF G approac h to crowd motion has the particularity that, b y considering agen ts of the cro wd to b e rational, one may consider that eac h agen t c ho oses their tra jectory based not only on the present or past states of other agents, but also on a rational an ticipation of their future b ehavior. Our aim in this pap er is to prop ose and study an MFG mo del inspired by crowd motion in which agents in teract through a Cuck er–Smale-type interaction, in the spirit of [78]. T o the authors’ knowledge, the ﬁrst work to b e fully dedicated to a mean ﬁeld game mo del for cro wd motion is [62], whic h proposes an MFG mo del for a t wo-population cro wd with tra jectories p erturb ed by additive Brownian motion and considers b oth their stationary distributions and their ev olution on a prescribed time interv al. Other w orks hav e later prop osed MFG models for cro wd motion taking into account diﬀeren t c haracteristics. T o cite a few examples, let us mention [20], whic h considers the fast exit of a cro wd and prop oses a mean ﬁeld game model whic h is studied n umerically; [30], which is not originally motiv ated by the mo deling of crowd motion but considers an MFG model with a densit y constraint, which is a natural assumption in some crowd motion mo dels (see also [69] for the case of second-order mean ﬁeld games with density constraints); [1, 33], whic h study a t wo-population MF G mo del motiv ated b y urban settlemen ts; [10], whic h presents n umerical sim ulations for some v ariational mean ﬁeld games related to crowd motion; [8], whic h, inspired b y cities cro wded b y tourists such as V enice, Italy , prop oses a mean ﬁeld game model for the mo vemen t of p edestrian tourists; [38], whic h provides a generalized MFG mo del for p edestrians with limited predictive abilities; or also [43, 44, 67, 68, 75, 76], which consider MFG mo dels in which the goal of each agen t is to reac h a given target set in minimal time, in teractions b etw een agents b eing mo deled through a congestion-dep endent maximal velocity . Other works studying mean ﬁeld games motiv ated by or related to cro wd motion include [2, 6, 9, 32, 41]. A common feature of sev eral MFG mo dels for crowd motion is that, at a given time t , an agent at a space p osition x will interact with other agen ts only through the distribution m t of their p ositions in the state space at time t . While this already provides mo dels with interesting features, a p osition-only interaction fails to capture imp ortant aspects of cro wd motion. Indeedqsdf, given t wo agen ts at a certain distance from one another, it is natural to expect that their in teraction will b e qualitatively diﬀeren t depending on whether they are moving to wards or a wa y from one another, since in the ﬁrst case they are exp ected to deviate in order to a void a collision, while, in the second case, their mo vemen ts are not expected to hav e a m utual interference. This motiv ates the study of MF G mo dels for cro wd motion in which the interaction through agents is based not only on their positions, but also on their velocities, or, equiv alen tly for man y mo dels, on their con trols. Mean ﬁeld game mo dels in whic h the dep endence on other agents tak es into accoun t not only their p ositions but also their controls are called MF Gs of c ontr ols . Their analysis, started in [48, 49], has b een muc h developed in recen t years (see, e.g., [16, 22, 29, 42, 51, 52, 60, 61, 78], and, in particular, the PhD thesis [59]). In the con text of cro wd motion, [78] has prop osed a mo del 1 whic h assumes that agen ts ev olve in a giv en set Ω ⊂ R d , that the tra jectory γ of an agen t c ho osing a con trol u is describ ed b y the con trol system ˙ γ ( t ) = u ( t ) , and that an agent chooses their con trol u in order to minimize a cost of the form w T 0  δ 2 | ˙ γ ( t ) | 2 + λ 2 w C ([0 ,T ] , Ω) η ( γ ( t ) − e γ ( t ))    ˙ γ ( t ) − ˙ e γ ( t )    2 d Q ( e γ )  d t + Ψ( γ ( T )) , (1) where δ and λ are positive constan ts, η : R d → R + is an in teraction k ernel, T > 0 is a ﬁxed time horizon, and Q is a probability measure on the set of all contin uous trajectories C ([0 , T ] , Ω) describing the distribution of tra jectories of all agents. Giv en an initial distribution of agents m 0 , describ ed mathematically as a probability measure on Ω , an equilibrium of this MF G model with initial condition m 0 is then a probability measure Q on C ([0 , T ] , Ω) whose ev aluation at time 0 is 2 m 0 and whic h is concentrated on trajectories γ that are optimal for (1) with a ﬁxed initial condition. It is pro v ed in [78, Prop osition 3.1] that this 1 W e p oint out that the authors of [78] argue that whether the mo del 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” . 2 That is, e 0 # Q = m 0 , using the notation describ ed in page 4 b elow. 2 MF G model is variational (also called p otential ), i.e., that there exists a function J , deﬁned o ver the space of probabilit y measures on C ([0 , T ] , Ω) whose ev aluation at time 0 is m 0 , such that (lo cal) minimizers of J are equilibria of the MF G with initial condition m 0 . In particular, this v ariational structure allows one to prov e 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]. W e also mention that [78] provides additional prop erties of equilibria, such as the fact that, under suitable additional assumptions, optimal trajectories are C 1 , 1 and unique. Note that this w ay of describing equilibria as probabilit y measures on the space of trajectories corresp onds to the L agr angian formulation of mean ﬁeld games. The Lagrangian formulation is a classical approach in optimal transp ort problems (see, e.g., [4, 77, 79]), which has b een used for instance in [18] to study incompressible ﬂows, in [31] for W ardrop equilibria in traﬃc ﬂo w, or in [11] for branched transp ort problems. The use of the Lagrangian approach in mean ﬁeld games dates back at least to [27, 30], and since then it has b een used in sev eral w orks, such as [10, 24–26, 44, 45, 67, 68, 74–76]. The model of [78] is inspired b y the Cuc ker–Smale model. Introduced in [39] to model the ev olution of a ﬂo ck of birds, the Cuck er–Smale mo del assumes that a ﬁnite num b er of agents (whic h represen t birds in the original model) ev olve according to a prescrib ed la w taking in to account the fact that agents wish to align their velocities with others, and that this alignmen t eﬀect is stronger when the birds are closer. In (1), this is mo deled b y the term η ( γ ( t ) − e γ ( t ))    ˙ γ ( t ) − ˙ e γ ( t )    2 , whic h p enalizes large diﬀerences in velocities for agen ts that are close if one assumes that η is a Gaussian-lik e kernel (for instance, nonnegative, dep ending only on the mo dulus of its argumen t, and nonincreasing with it). Note also that the interactions b et ween agents in (1) are pairwise: for a given agent with a tra jectory γ , its total interaction is the integral of the interactions of the pairs of trajectories ( γ , e γ ) for all tra jectories e 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 in terv al [0 , T ] of the game is ﬁxed, with all agents starting at time 0 and ending their mov ement at time T . Ho wev er, in man y practical applications, including crowd motion but also in economics, for instance, it is interesting to study models in whic h the ﬁnal time for the mov emen t of an agen t is free, and is actually part of their minimization criterion. In particular, agents ma y leav e the game b efore its end. MF Gs with a free ﬁnal time hav e b een considered, for instance, in [21, 35], whic h consider mean ﬁeld games with applications to bank run, i.e., situations in which clien ts of a bank, believing that the bank is ab out to fail, withdraw all their money , and try to choose the time to withdraw the money in an optimal w ay . These mo dels b elong to a more general class of mean ﬁeld game problems kno wn as mean ﬁeld games of optimal stopping, in which the main choice of an agen t is when to stop the game [12, 13, 17, 47, 72, 73]. Other w orks, suc h as [37, 50], consider MFGs with free ﬁnal time for the production of exhaustible resources, in whic h ﬁrms, who wish to maximize their proﬁt, produce go o ds based on exhaustible resources, and they leav e the game when they deplete their capacities. These games can be seen as mean ﬁeld games with an absorbing b oundary , i.e., in which agen ts who reac h a certain part of the b oundary of the domain immediately leav e the game [23, 40]. In the context of crowd motion, MFG mo dels with free ﬁnal time w ere developed in [38, 43, 44, 67, 68, 75, 76]. In this paper, w e generalize the mo del from [78] by considering that (i) one has a more general individual running cost than δ 2 | ˙ γ ( t ) | 2 from (1); (ii) one has a more general interaction cost than η ( γ ( t ) − e γ ( t ))    ˙ γ ( t ) − ˙ e γ ( t )    2 from (1); (iii) the ﬁnal time of the mo vemen t of an agen t is not a prescrib ed time T , but is part of the optimization criterion of each agent; and (iv) the aim of each agen t is to reach a giv en common target set Ξ . More precisely , we shall consider that eac h agent of the game minimizes a cost of the form w + ∞ 0 ℓ ( t, γ ( t ) , ˙ γ ( t )) d t + Ψ( τ ( γ )) + w Γ w τ ( γ ) ∧ τ ( e γ ) 0 h ( t, γ ( t ) , e γ ( t ) , ˙ γ ( t ) , ˙ e γ ( t )) d t d Q ( e γ ) , (2) where ℓ , h , and Ψ are giv en functions and, for a giv en tra jectory γ , τ ( γ ) denotes the exit time of γ , i.e., the ﬁrst time at whic h γ reac hes the target set Ξ (or τ ( γ ) = + ∞ if γ do es not reach Ξ ). While (i) and (ii) do not bring man y tec hnical diﬃculties with resp ect to [78], no v el ideas are 3 required to treat the free ﬁnal-time setting from (iii) and (iv) and address the diﬃculties coming from a noncompact time interv al and the lac k of con tin uity of the exit-time function τ ( · ) . It turns out that the p otential structure highligh ted in [78] is not sp eciﬁc to the MF G considered in that reference nor to the MFG described b y (2) and comes instead mostly from the pairwise nature of the interaction b etw een agen ts. Hence, instead of w orking directly with (2), we consider in most of the paper a more general non-atomic game model in abstract P olish spaces (which are not necessarily spaces of trajectories) in whic h agen ts minimize the sum of an individual cost and a pairwise in teraction cost. The analysis of more general non-atomic games and their potential structure has also b een the subject of other works in the literature, such as [14], whic h considers a non-atomic game whic h may contain pairwise in teraction terms as a particular case. W e p oin t out, ho wev er, that the mo del we consider cannot b e subsumed under that of [14] (see Remark 3.3). In addition to sho wing, as in [78], that the non-atomic game we consider has a potential structure in the sense that lo cal minimizers of a p otential J are equilibria of the game, w e go further in the p otential form ulation by sho wing (Theorem 3.20) that the set of equilibria of the game is exactly the set of critical p oints of its p otential function J . W e then deduce (Theorem 3.22) existence of an equilibrium by proving the existence of a minimizer for J . W e also discuss t wo a priori distinct notions of equilibria, strong and w eak equilibria, and prov e (Theorem 3.24) that, under suitable assumptions, both notions coincide for our mo del. Finally , we apply the results obtained for the general non-atomic game to the MF G consisting in eac h agen t minimizing (2), pro ving existence of equilibria (Theorem 4.9) and equiv alence betw een strong and weak equilibria (Theorem 4.12). The sequel of the pap er is organized as follows. Section 2 presents the MF G mo del consisting on each agent minimizing (2), providing our standing assumptions on the functions ℓ , h , and Ψ from (2) as w ell as the precise deﬁnition of the exit time function τ ( · ) and the deﬁnition of equilibrium of the game. Section 3 mov es to the analysis of a more general non-atomic game mo del in P olish spaces, presenting the mo del and its assumptions, pro viding its elementary prop erties in Section 3.1, pro ving its p otential structure in Section 3.2, deducing existence of equilibria in Section 3.3, and discussing the equiv alence of the a priori distinct notions of strong and w eak equilibria in Section 3.4. These general results are applied to the MFG mo del from Section 2 in Section 4. The paper is concluded b y a numerical sim ulation in Section 5 illustrating b ehavior of the equilibria in an N -pla yer setting. Notation. In this pap er, d is a ﬁxed p ositive in teger, the set of nonnegativ e real num b ers is denoted b y R + , the set of nonnegative rational n umbers is denoted b y Q + , and we deﬁne R + as R + = R + ∪ { + ∞} . F or a giv en Ω ⊂ R d nonempt y , open, and bounded, w e let C ( R + ; Ω) b e the spaces of con tinuous curv es from R + to Ω endo wed with the top ology of uniform conv ergence on compact time in terv als, with which it is a P olish space. F or simplicity , we denote C ( R + ; Ω) by Γ . Giv en a Polish space X , w e denote the space of Borel probabilit y measures on X b y P ( X ) and endo w it with the top ology of w eak conv ergence of measures. The support of µ ∈ P ( X ) is denoted b y spt( µ ) , and is deﬁned as the set of all p oin ts x ∈ X suc h that µ ( N x ) > 0 for every neigh b orho o d N x of x . W e denote the Dirac measure cen tered at a point x 0 ∈ X b y δ x 0 . F or t w o metric spaces X and Y endo wed with their Borel σ -algebras and a Borel-measurable map f : X → Y , the pushforward of a measure µ on X through f is the measure f # µ on Y deﬁned b y f # µ ( B ) = µ ( f − 1 ( B )) for every Borel subset B of Y . F or t ∈ R + , w e denote by e t : Γ → Ω the ev aluation map at time t , deﬁned by e t ( γ ) = γ ( t ) for ev ery γ ∈ Γ . 2 Mean ﬁeld game mo del with pairwise in teractions W e consider here that agents mov e in Ω , where Ω is a nonempt y open b ounded subset of R d , and that their goal is to reach a certain target Ξ ⊂ Ω , whic h is a nonempt y closed set. The initial distribution of agents m 0 ∈ P (Ω) at time t = 0 is kno wn, agen ts are submitted to the trivial con trol system ˙ γ ( t ) = u ( t ) , and the goal of an agen t starting at p osition x 0 is to choose a control u : R + → R d suc h that γ ( t ) ∈ Ω for ev ery t ∈ R + and the cost F ( γ , Q ) deﬁned b y F ( γ , Q ) = w + ∞ 0 ℓ ( t, γ ( t ) , ˙ γ ( t )) d t + Ψ( τ ( γ )) + w Γ w τ ( γ ) ∧ τ ( e γ ) 0 h ( t, γ ( t ) , e γ ( t ) , ˙ γ ( t ) , ˙ e γ ( t )) d t d Q ( e γ ) (3) 4 is minimized among all tra jectories γ starting at the same p osition x 0 . In (3), ℓ : R + × Ω × R d → R + , h : R + × Ω × Ω × R d × R d → R + , and Ψ : R + → R + are giv en functions, Q ∈ P (Γ) represents the distribution of the tra jectories of all agents, and τ : Γ → R + is the ﬁrst exit time function, deﬁned, for γ ∈ Γ , by τ ( γ ) = inf { t ∈ R + | γ ( t ) ∈ Ξ } , (4) with the conv en tion inf ∅ = + ∞ . W e also set, by conv ention, F ( γ , Q ) = + ∞ if γ is not absolutely con tinuous, if Q gives positive measure to a set of functions which are not absolutely contin uous, or if e γ 7→ r τ ( γ ) ∧ τ ( e γ ) 0 h ( t, γ ( t ) , e γ ( t ) , ˙ γ ( t ) , ˙ e γ ( t )) d t is not Q -in tegrable. Let us pro vide the deﬁnition of equilibrium of this mean ﬁeld game. Deﬁnition 2.1. Let m 0 ∈ P (Ω) . A measure Q ∈ P (Γ) is said to be an e quilibrium with initial condition m 0 if e 0 # Q = m 0 , r Γ F ( γ , Q ) d Q ( γ ) < + ∞ , and Q -almost ev ery γ satisﬁes F ( γ , Q ) = inf ω ∈ Γ ω (0)= γ (0) F ( ω , Q ) . Let us no w in tro duce the main assumptions on Ω , Ξ , ℓ , h , and Ψ that we will need for the analysis of this mean ﬁeld game. (H1) Ω ⊂ R d is nonempt y , op en, and bounded. (H2) Ξ ⊂ Ω is nonempt y and closed. (H3) The function ℓ : R + × Ω × R d → R + is suc h that (a) t 7→ ℓ ( t, x, p ) is measurable for every ( x, p ) ∈ Ω × R d ; (b) ( x, p ) 7→ ℓ ( t, x, p ) is con tinuous for almost every t ∈ R + ; (c) p 7→ ℓ ( t, x, p ) is con vex for almost ev ery t ∈ R + and ev ery x ∈ Ω ; and (d) there exist constan ts α > 0 and θ > 1 suc h that ℓ ( t, x, p ) ≥ α | p | θ , for all ( t, x, p ) ∈ R + × Ω × R d . (H4) The function h : R + × Ω × Ω × R d × R d → R + is suc h that (a) t 7→ h ( t, x, e x, p, e p ) is measurable for every ( x, e x, p, e p ) ∈ Ω × Ω × R d × R d ; (b) ( x, e x, p, e p ) 7→ h ( t, x, e x, p, e p ) is contin uous for almost every t ∈ R + ; (c) ( p, e p ) 7→ h ( t, x, e x, p, e p ) is conv ex for almost every t ∈ R + and ev ery ( x, e x ) ∈ Ω × Ω ; (d) for all ( t, x, e x, p, e p ) ∈ R + × Ω × Ω × R d × R d , w e ha ve the symmetry prop erty h ( t, x, e x, p, e p ) = h ( t, e x, x, e p, p ); and (e) there exist constan ts C > 0 and β ∈ (0 , θ ] , where θ is the constant from (H3)(d), suc h that h ( t, x, e x, p, e p ) ≤ C ( | p | β + | e p | β ) , for all ( t, x, e x, p, e p ) ∈ R + × Ω × Ω × R d × R d . (H5) The function Ψ : R + → R + is low er semicon tinuous, nondecreasing, and there exist p ositive constan ts a and b such that Ψ( t ) ≥ at − b, for all t ∈ R + . (H6) There exists a constan t κ > 0 suc h that, for ev ery x 0 ∈ Ω , there exists an absolutely con tinuous curv e γ ∈ Γ with γ (0) = x 0 and w + ∞ 0 ℓ ( t, γ ( t ) , ˙ γ ( t )) d t + Ψ( τ ( γ )) ≤ κ. 5 Note that (H6) can b e seen as an accessibilit y assumption: wherev er in Ω is the starting p oint x 0 , there alwa ys exists an absolutely contin uous trajectory allo wing an agent to reac h the target set Ξ within a time that is uniformly bounded, and with a running cost that is also uniformly b ounded. It can be ensured, for instance, if Ω is connected and has smo oth boundary . In the sequel, instead of directly studying this mean ﬁeld game, we will consider in Section 3 a more general non-atomic game con taining this MFG as a particular case. W e will pro ve 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 ab ov e MFG in Section 4. 3 A general non-atomic game with pairwise in teractions As a preliminary step tow ards the study of the mean ﬁeld game mo del we are interested in this pap er, we consider here a more general non-atomic game with pairwise in teractions. Let X and Y b e Polish spaces and π : X → Y , L : X → R + , and H : X × X → R + b e Borel-measurable functions. Eac h agen t of the game is assumed to be asso ciated with an element y ∈ Y (but this asso ciation is not necessarily injective: diﬀerent agents can b e asso ciated with the same elemen t of Y ), and we assume that the distribution of agen ts according to the elements asso ciated with them in Y is described b y a kno wn probabilit y measure m 0 ∈ P ( Y ) . An agen t of the game associated with some y ∈ Y wishes to minimize the cost F ( x, Q ) giv en b y F ( x, Q ) = L ( x ) + w X H ( x, e x ) d Q ( e x ) (5) with the constraint π ( x ) = y , i.e., the agent w ants to c ho ose x ∈ X solving the minimization problem Min( y , Q ) : ( minimize x ∈ X F ( x, Q ) sub ject to π ( x ) = y , (6) where Q ∈ P ( X ) denotes the distribution of the c hoices of all agen ts. Note that the cost F ( x, Q ) can b e inte rpreted as containing an individual cost L ( x ) and an av erage pairwise in teraction cost r X H ( x, e x ) d Q ( e x ) . In the sequel, w e refer to this non-atomic game as NAG( X, Y , π , L, H , m 0 ) and w e lo ok for equilibria of this game, according to the following deﬁnition. Deﬁnition 3.1. Let X and Y b e Polish spaces, π : X → Y , L : X → R + , and H : X × X → R + b e Borel-measurable functions, and m 0 ∈ P ( Y ) . W e sa y that Q ∈ P ( X ) is an e quilibrium of NA G( X , Y , π , L, H , m 0 ) if π # Q = m 0 , r X F ( x, Q ) d Q ( x ) < + ∞ , and Q -almost ev ery x ∈ X solv es Min( π ( x ) , Q ) with F giv en b y (5). Remark 3.2. The mean ﬁeld game from Section 2 is a particular case of NAG( X, Y , π , L, H , m 0 ) . T o see that, it suﬃces to set Y = Ω , X = Γ with the topology of uniform con vergence on compact sets, π = e 0 , the ev aluation map at time zero, and deﬁne the functions L : Γ → R + and H : Γ × Γ → R + b y L ( γ ) =    w + ∞ 0 ℓ ( t, γ ( t ) , ˙ γ ( t )) d t + Ψ( τ ( γ )) if γ is absolutely con tinuous, + ∞ otherwise and H ( γ , e γ ) =    w τ ( γ ) ∧ τ ( e γ ) 0 h ( t, γ ( t ) , e γ ( t ) , ˙ γ ( t ) , ˙ e γ ( t )) d t if L ( γ ) < + ∞ and L ( e γ ) < + ∞ , + ∞ otherwise. W e will justify later (see Prop osition 4.8) that, with these deﬁnitions, the notions of equilibria from Deﬁnitions 2.1 and 3.1 coincide. 6 Remark 3.3. F or an individual agen t associated with a giv en elemen t y ∈ Y , their cost (5) and their minimization problem (6) are particular cases of the more general problem of, given y ∈ Y , minimize x ∈ X c ( x, y ) + V [ Q ]( x ) , (7) without the constraint π ( x ) = y . Indeed, (7) reduces to (6) with the c hoice c ( x, y ) = L ( x ) + χ graph( π ) ( x, y ) and V [ Q ]( x ) = r X H ( x, e x ) d Q ( e x ) , where χ graph( π ) is the c haracteristic function of the graph of π , i.e., χ graph( π ) ( x, y ) = 0 if y = π ( x ) and χ graph( π ) ( x, y ) = + ∞ otherwise. The non-atomic game in which each agen t minimizes (7) w as previously considered in the literature, for instance, in [14], where the authors study equilibria of the game, named in that con text as Cournot–Nash e quilibria , through a v ariational approac h, exploiting in particular links b et ween suc h equilibria and optimal transp ort problems. These ideas, how ever, cannot b e directly applied to the non-atomic game NA G( X, Y , π , L, H , m 0 ) that w e consider in this article, as some k ey assumptions of [14] cannot b e satisﬁed in our context. In particular, [14] requires c to b e con tinuous in order to obtain go o d prop erties for the asso ciated W asserstein distance W c , but such a contin uit y assumption can never be satisﬁed in our mo del due to the presence of the characteristic function χ graph( π ) in the expression of c . Let us now state the main assumptions under whic h w e will consider the abstract non-atomic game NA G( X , Y , π , L, H , m 0 ) . (A1) X and Y are P olish spaces. (A2) π : X → Y is con tinuous. (A3) L : X → R + is lo wer semicontin uous and H : X × X → R + is Borel measurable. (A4) H is symmetric, i.e., H ( x, e x ) = H ( e x, x ) for ev ery ( x, e x ) ∈ X × X . (A5) The function X × X ∋ ( x, e x ) 7→ L ( x ) + L ( e x ) + H ( x, e x ) ∈ R + is lo wer semicontin uous. (A6) There exists κ > 0 suc h that, for every y ∈ Y , there exists x ∈ X with π ( x ) = y and L ( x ) ≤ κ . (A7) F or ev ery κ > 0 , the set { x ∈ X | L ( x ) ≤ κ } is compact. (A8) There exists C > 0 such that H ( x, e x ) ≤ C ( L ( x ) + L ( e x ) + 1) for ev ery ( x, e x ) ∈ X × X . Our main goals concerning NAG( X, Y , π , L, H , m 0 ) are to show that (i) it is a potential game, i.e., its equilibria can b e obtained as minimizers of a certain p otential J ; and (ii) the p otential J admits a minimizer, and hence NA G( X , Y , π , L, H , m 0 ) admits an equilibrium. Let us in troduce some notation to be used in the sequel. W e deﬁne J : X × X → R + b y J ( x, e x ) = L ( x ) + L ( e x ) + H ( x, e x ) , (8) and w e deﬁne the functions L : P ( X ) → R + , H : P ( X ) × P ( X ) → R + , and J : P ( X ) → R + b y L ( Q ) = w X L ( x ) d Q ( x ) , (9a) H ( Q, e Q ) = w X × X H ( x, e x ) d( Q ⊗ e Q )( x, e x ) , (9b) J ( Q ) = w X × X J ( x, e x ) d( Q ⊗ e Q )( x, e x ) . (9c) Giv en m 0 ∈ P ( Y ) , w e also set P m 0 ( X ) = { Q ∈ P ( X ) | π # Q = m 0 } . (10) Remark 3.4. Under (A2), the set P m 0 ( X ) is a closed subset of P ( X ) . Remark 3.5. According to the deﬁnitions of F , L , H , and J , from (5) and (9), one can easily c heck that J ( Q ) = 2 L ( Q ) + H ( Q, Q ) and r X F ( x, Q ) d Q ( x ) = L ( Q ) + H ( Q, Q ) , for every Q ∈ P ( X ) . In particular, J ( Q ) < + ∞ if and only if r X F ( x, Q ) d Q ( x ) < + ∞ . 7 3.1 First prop erties Let us no w establish some elementary prop erties of the functions introduced in (9). W e start with the follo wing consequence of (A8), whose pro of is immediate. Lemma 3.6. A ssume that (A1), (A3), and (A8) ar e satisﬁe d. L et C > 0 b e the c onstant fr om (A8) and L and H b e deﬁne d as in (9a) and (9b) . Then, for every ( Q, e Q ) ∈ P ( X ) × P ( X ) , we have H ( Q, e Q ) ≤ C ( L ( Q ) + L ( e Q ) + 1) . As a consequence of Remark 3.5 and Lemma 3.6, w e also immediately obtain the follo wing result. Corollary 3.7. A ssume that (A1), (A3), and (A8) ar e satisﬁe d. L et J , L , and J b e deﬁne d as in (8) , (9a) and (9c) . Then dom L = dom J . Remark 3.8. F rom Lemma 3.6, we deduce that the function H : dom L × dom L → R + can be extended b y bilinearity to a unique bilinear function, still denoted by H , deﬁned in Span(dom L ) × Span(dom L ) and taking v alues in R , where the linear span is taken in the space of signed measures on X . In order to pro vide additional prop erties of dom J , w e ﬁrst establish the following preliminary result, which states that, in (A6), the elemen t x ∈ X can b e selected as a Borel-measurable function of y ∈ Y . Lemma 3.9. A ssume that (A1)–(A3), (A6), and (A7) ar e satisﬁe d and let κ > 0 b e as in (A6). Then ther e exists a Bor el me asur able function Φ : Y → X such that, for every y ∈ Y , we have π (Φ( y )) = y and L (Φ( y )) ≤ κ . Pr o of. Let us consider the set v alued map G : Y ⇒ X deﬁned by G ( y ) =  x ∈ X   π ( x ) = y and L ( x ) ≤ κ  . Note that, b y (A6), we hav e G ( y )  = ∅ for every y ∈ Y . Since π is con tinuous and L is lo wer semicon tinuous, one immediately deduces that the graph of G is closed and, in particular, G ( y ) is closed for ev ery y ∈ Y . In addition, G takes v alues in the set { x ∈ X | L ( x ) ≤ κ } , which is compact thanks to (A7). Hence, by [5, Proposition 1.4.8], w e get that G is upp er semicon tinuous and, by [5, Proposition 1.4.4], the set G − 1 ( A ) = { y ∈ Y | G ( y ) ∩ A  = ∅} is closed for ev ery closed set A ⊂ X . Hence, by [36, Prop osition I I I.11], G − 1 ( B ) is op en for ev ery op en set B ⊂ X , and th us G is measurable. No w, by applying [5, Theorem 8.1.3], we conclude that the set-v alued map G admits a Borel-measurable selection, whic h is our desired function Φ . Remark 3.10. Notice that it is not p ossible to apply [5, Theorem 8.1.4] in order to extract directly a Borel measurable selection from the fact that G has a closed graph since the measure m 0 is not complete in the measurable space Ω endo wed with its Borel σ -algebra. As a consequence of Lemma 3.9, w e deduce the following result on dom J . Corollary 3.11. A ssume that (A1)–(A3) and (A6)–(A8) ar e satisﬁe d, and let J and J b e given by (8) and (9c) . Then, for every m 0 ∈ P ( Y ) , the set dom J ∩ P m 0 ( X ) is nonempty and c onvex. Pr o of. By Corollary 3.7, it suﬃces to sho w that dom L ∩ P m 0 ( X ) is nonempt y and con vex, where L is given b y (9a). Let Φ : Y → X b e the map from Lemma 3.9 and deﬁne Q = Φ # m 0 . Then π # Q = m 0 and an immediate computation sho ws that L ( Q ) ≤ κ < + ∞ , where κ is the constant from (A6). Hence Q ∈ dom L ∩ P m 0 ( X ) . Note that P m 0 ( X ) is clearly conv ex from (10), and L is linear, implying that L ( tQ + (1 − t ) e Q ) = t L ( Q ) + (1 − t ) L ( e Q ) for ev ery Q, e Q ∈ P ( X ) and t ∈ [0 , 1] . Hence dom L is con vex, yielding that dom L ∩ P m 0 ( X ) is con vex. W e will also need in the sequel the following property of the function F from (5). 8 Lemma 3.12. A ssume that (A1), (A3), and (A5) ar e satisﬁe d, and let L b e the function deﬁne d in (9a) . F or every Q ∈ dom L , the function x 7→ F ( x, Q ) is lower semic ontinuous. Pr o of. Note that, deﬁning J as in (8), we hav e F ( x, Q ) + L ( Q ) = w X J ( x, e x ) d Q ( e x ) , and th us, since L ( Q ) < + ∞ , it suﬃces to prov e the lo wer semicontin uity of the function x 7→ r X J ( x, e x ) d Q ( e x ) . Let ( x n ) n ∈ N b e a sequence in X conv erging to some x ∈ X . Since J is low er semicon tin uous thanks to (A5), we ha v e, for ev ery e x ∈ X , J ( x, e x ) ≤ lim inf n → + ∞ J ( x n , e x ) . Then, F atou’s lemma shows that w X J ( x, e x ) d Q ( e x ) ≤ w X lim inf n → + ∞ J ( x n , e x ) d Q ( e x ) ≤ lim inf n → + ∞ w X J ( x n , e x ) d Q ( e x ) , yielding the conclusion. Note that Lemma 3.12 also yields the following result on the existence of optimizers for F ( · , Q ) . Corollary 3.13. A ssume that (A1)–(A3) and (A5)–(A8) ar e satisﬁe d and let L b e the function deﬁne d in (9a) . F or every Q ∈ dom L and y ∈ Y , ther e exists x ∈ X with π ( x ) = y and such that F ( x, Q ) = inf z ∈ X π ( z )= y F ( z , Q ) . Pr o of. Let ( x n ) n ∈ N b e a minimizing sequence for F ( · , Q ) , i.e., ( x n ) n ∈ N is a sequence in X with π ( x n ) = y for every n ∈ N and lim n → + ∞ F ( x n , Q ) = inf z ∈ X π ( z )= y F ( z , Q ) . (11) By (A6), there exists z ∈ X with π ( z ) = y such that L ( z ) ≤ κ , where κ is the constant from (A6). Then, by (A8), w e hav e F ( z , Q ) ≤ ( C + 1) L ( z ) + C L ( Q ) + C ≤ ( C + 1) κ + C L ( Q ) + C , where C is the constan t from (A8). Hence, the limit in the left-hand side in (11) is ﬁnite, and we assume, up to remo ving ﬁnitely man y elemen ts from the sequence, that F ( x n , Q ) < + ∞ for ev ery n ∈ N , and th us the sequence ( F ( x n , Q )) n ∈ N is b ounded. Th us, the sequence ( L ( x n )) n ∈ N is also b ounded and, b y (A7), this implies that, up to extracting a subsequence, there exists x ∈ X such that x n → x as n → + ∞ . By (A2), w e obtain that π ( x ) = y and, b y (11) and Lemma 3.12, we obtain that F ( x, Q ) ≤ lim n → + ∞ F ( x n , Q ) = inf z ∈ X π ( z )= y F ( z , Q ) ≤ F ( x, Q ) , yielding the conclusion. 3.2 P oten tial game Our next goal is to prov e that NAG( X, Y , π , L, H , m 0 ) is a p otential game with p otential given by the function J from (9c). F or games with ﬁnitely many play ers, a p otential is a function such that, when a single play er changes their strategy , the v ariation in the cost of that pla y er is equal to (or prop ortional to) the v ariation of the p otential. In particular, in p otential games, equilibria can b e found as minimizers of the p otential. T o adapt these ideas to our setting with a con tinuum of pla yers, one needs to provide a suitable notion of “v ariation” of the p otential J as a play er changes their tra jectory . It turns out that the goo d notion is that of diﬀerentiabilit y of J , in the sense of the follo wing deﬁnition. 9 Deﬁnition 3.14. Assume that (A3) is satisﬁed and let J b e deﬁned by (9c). Giv en Q 0 ∈ dom J , w e say that J is diﬀer entiable at Q 0 if there exists a Borel-measurable function G : X → R suc h that, for ev ery Q ∈ dom J , G is ( Q − Q 0 ) -in tegrable and lim t → 0 + J ( Q 0 + t ( Q − Q 0 )) − J ( Q 0 ) t = w X G ( x ) d( Q − Q 0 )( x ) . (12) In this case, we denote G by δ J δ Q ( Q 0 ) and we write the right-hand side of (12) as ⟨ δ J δ Q ( Q 0 ) , Q − Q 0 ⟩ . Remark 3.15. There are sev eral notions of deriv ation for functions deﬁned in the space of prob- abilit y measures, and the one presen ted in Deﬁnition 3.14 resem bles the one from [28, Deﬁni- tion 2.2.1] and [34, Deﬁnition 5.43], whic h is widely used in the analysis of mean ﬁeld games. The main diﬀerence is that the notion we provide abov e requires less regularity of ( Q 0 , x ) 7→ δ J δ Q ( Q 0 )( x ) , whic h is not necessarily contin uous, and might ev en fail to b e deﬁned for some Q 0 ∈ dom J . This w eaker notion suitably ﬁts our purp oses in the sequel. W e refer the interested reader to [4, Chap- ter 10], [28, Section 2.2.3 and App endices A.1 and A.2], and [34, Chapter 5] for other notions of deriv ation for functions deﬁned in the space of probabilit y measures and further discussion on the relations among those notions. Note that, if G is a Borel-measurable function satisfying (12), then G + λ also satisﬁes (12) for an y constan t λ ∈ R , since the integral of a constan t with respect to the measure Q − Q 0 is zero. Hence, δ J δ Q is not uniquely deﬁned. W e prov e, ho wev er, in our next result, that, similarly to the analogous notion of deriv ativ e from [28, Deﬁnition 2.2.1] and [34, Deﬁnition 5.43], δ J δ Q is indeed unique up to an additive constant. Lemma 3.16. A ssume that (A1)–(A3), (A6), and (A7) ar e satisﬁe d and let J b e deﬁne d by (9c) . L et Q 0 ∈ dom J and assume that G 1 and G 2 ar e two Bor el me asur able functions such that, for every Q ∈ dom J , G 1 and G 2 ar e ( Q − Q 0 ) -inte gr able and w X G 1 ( x ) d( Q − Q 0 )( x ) = w X G 2 ( x ) d( Q − Q 0 )( x ) . Then ther e exists λ ∈ R such that G 1 ( x ) = G 2 ( x ) + λ for Q 0 -almost every x ∈ X . Pr o of. Let L b e the function deﬁned in (9a). Note that, for Q 0 -almost ev ery x ∈ X , we ha ve δ x ∈ dom J . Indeed, since Q 0 ∈ dom J = dom L by Corollary 3.7, we deduce that L ( x ) < + ∞ for Q 0 -almost ev ery x ∈ X , and th us L ( δ x ) = L ( x ) < + ∞ , sho wing that δ x ∈ dom L = dom J . W e now set λ = r X ( G 1 − G 2 )( x ) d Q 0 ( x ) and observ e that, by assumption, for every Q ∈ dom J , w e hav e w X G 1 ( x ) d Q ( x ) = w X G 2 ( x ) d Q ( x ) + λ, and hence, for Q 0 -almost every x ∈ X , w e can tak e Q = δ x in the abov e iden tity to deduce that G 1 ( x ) = G 2 ( x ) + λ , as required. Our next result is an imp ortant step in formulating NA G( X , Y , π , L, H , m 0 ) as a p otential non- atomic game with p otential J , since, under the additional assumption (A4), it relates the v ariations of J , in the sense of Deﬁnition 3.14, with the individual cost F ( x, Q ) of an agent from (5). Theorem 3.17. A ssume that (A1)–(A4), (A6), and (A7) ar e satisﬁe d and let J b e deﬁne d by (9c) . Then, for every Q 0 ∈ dom J , the function J is diﬀer entiable at Q 0 , with δ J δ Q ( Q 0 )( x ) = 2 F ( x, Q 0 ) for Q 0 -almost every x ∈ X . Pr o of. Fix Q 0 ∈ dom J and let L and H b e deﬁned as in (9a) and (9b). F or every Q ∈ dom J and t ∈ (0 , 1] , using Remark 3.5, we ha v e that J ( Q 0 + t ( Q − Q 0 )) − J ( Q 0 ) t = 2( L ( Q ) + H ( Q 0 , Q )) − 2( L ( Q 0 ) + H ( Q 0 , Q 0 )) + t H ( Q − Q 0 , Q − Q 0 ) , 10 where H ( Q − Q 0 , Q − Q 0 ) ∈ R is well-deﬁned by Remark 3.8. Hence, letting t → 0 + , w e obtain from Remark 3.5 that lim t → 0 J ( Q 0 + t ( Q − Q 0 )) − J ( Q 0 ) t = 2( L ( Q ) + H ( Q 0 , Q )) − 2( L ( Q 0 ) + H ( Q 0 , Q 0 )) = w X 2 F ( x, Q 0 ) d Q ( x ) − w X 2 F ( x, Q 0 ) d Q 0 ( x ) = w X 2 F ( x, Q 0 ) d( Q − Q 0 )( x ) , yielding the conclusion. No w that we ha v e established a link b et ween v ariations of J and the individual cost F ( x, Q ) in Theorem 3.17, our next step to study the p otential structure of NAG( X, Y , π , L, H , m 0 ) is to relate equilibria of this game with minimizers of J . F or that purp ose, w e start with the follo wing deﬁnition. Deﬁnition 3.18. Assume that (A3) is satisﬁed and let J b e deﬁned by (9c). W e say that Q 0 ∈ dom J ∩ P m 0 ( X ) is a critic al p oint of J in P m 0 ( X ) if J is diﬀeren tiable at Q 0 and  δ J δ Q ( Q 0 ) , Q − Q 0  ≥ 0 for ev ery Q ∈ dom J ∩ P m 0 ( X ) . (13) Inequalities of the form (13) are sometimes known as Euler ine qualities , 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 Hilb ert spaces). Our next result states that (13) is indeed a necessary condition for the minimization of J on P m 0 ( X ) . Prop osition 3.19. A ssume that (A3) satisﬁe d and let Q 0 ∈ dom J b e a lo c al minimizer of J in P m 0 ( X ) , i.e., ther e exists a neighb orho o d V of Q 0 in P m 0 ( X ) , with the top olo gy of we ak c onver genc e of pr ob ability me asur es, such that J ( Q ) ≥ J ( Q 0 ) for every Q ∈ V . If J is diﬀer entiable at Q 0 , then Q 0 is a critic al p oint of J in P m 0 ( X ) . Pr o of. Let Q 0 ∈ dom J be a local minimizer of J in P m 0 ( X ) and assume that J is diﬀerentiable at Q 0 . Then, for every Q ∈ dom J ∩ P m 0 ( X ) , w e deduce from the facts that Q 0 is a lo cal minimizer of J in P m 0 ( X ) and Q 0 + t ( Q − Q 0 ) → Q 0 as t → 0 + in the topology of w eak conv ergence in P m 0 ( X ) that J ( Q 0 + t ( Q − Q 0 )) − J ( Q 0 ) t ≥ 0 for small enough t > 0 . T aking the limit as t → 0 + immediately yields (13). W e next establish one of our main results on the p oten tial structure of NAG( X, Y , π , L, H , m 0 ) , stating that equilibria of this game coincide with critical p oints of J in P m 0 ( X ) . The pro of we presen t here is closely based on that of [78, Lemma 3.3]. Theorem 3.20. A ssume that (A1)–(A8) ar e satisﬁe d and let J b e given by (9c) . A me asur e Q 0 ∈ P m 0 ( X ) is an e quilibrium for NAG( X, Y , π , L, H , m 0 ) if and only if it is a critic al p oint of J in P m 0 ( X ) . A s a c onse quenc e, any lo c al minimizer of J in P m 0 ( X ) is an e quilibrium for NA G( X , Y , π , L, H , m 0 ) . Pr o of. Recall that, according to Remark 3.5, Q 0 ∈ dom J is equiv alen t to having r X F ( x, Q 0 ) d Q 0 ( x ) < + ∞ . Assume ﬁrst that Q 0 ∈ P m 0 ( X ) is a critical point of J in P m 0 ( X ) and note that, b y deﬁnition, w e hav e Q 0 ∈ dom J , and hence r X F ( x, Q 0 ) d Q 0 ( x ) < + ∞ . T o obtain a con tradiction, supp ose that the set { x ∈ X | ∃ z ∈ X such that π ( z ) = π ( x ) and F ( z , Q 0 ) < F ( x, Q 0 ) } 11 is not Q 0 -negligible. Note that the ab ov e set is equal to [ ( q ,r ) ∈ Q 2 + 0 r and { z ∈ X | π ( z ) = π ( x ) and F ( z , Q 0 ) ≤ q }  = ∅} , and th us there exists a pair of p ositive rational num b ers ( q , r ) with q < r suc h that the set A deﬁned b y A = { x ∈ X | F ( x, Q 0 ) > r and { z ∈ X | π ( z ) = π ( x ) and F ( z , Q 0 ) ≤ q }  = ∅} (14) is not Q 0 -negligible. F or an y p ∈ R , deﬁne A p as A p = { x ∈ X | F ( x, Q 0 ) ≤ p } . Note that A p is closed since, b y Lemma 3.12, x 7→ F ( x, Q 0 ) is lo wer semicon tin uous. In addition, A p ⊂ { x ∈ X | L ( x ) ≤ p } and the latter set is compact thanks to (A7). Hence, A p is compact. Let us no w deﬁne the multifunction S as S : ( X ⇒ A q , x 7→ S ( x ) = { z ∈ A q | π ( z ) = π ( x ) } . Using (A2), it is immediate to verify that S has closed graph and, using the fact that A q is compact, one can also easily verify that the domain dom S , deﬁned as dom S = { x ∈ X | S ( x )  = ∅} , is closed. It is also clearly nonempt y , since A q ⊂ dom S . By [5, Proposition 1.4.8], w e deduce that S is upp er semicon tinuous. Hence, b y [5, Prop osition 1.4.4], S − 1 ( A ) = { x ∈ X | S ( x ) ∩ A  = ∅} is closed for every closed set A . No w by deﬁnition of measurabilit y and [36, Proposition I I I.11], we obtain that S is Borel-measurable, and, b y [5, Theorem 8.1.3], there exists a Borel-measurable function s : dom S → A q satisfying s ( x ) ∈ S ( x ) , for all x ∈ dom S . Note that the set A deﬁned in (14) is Borel measurable, since it can b e written as A = { x ∈ X | F ( x, Q 0 ) > r } ∩ dom S , dom S is closed, and { x ∈ X | F ( x, Q 0 ) > r } is the complemen t of A r , and hence it is open. W e can thus deﬁne a measure e Q ∈ P ( X ) b y e Q = s # ( Q 0 | A ) + Q 0 | ( X \ A ) , whic h is well-deﬁned since e Q ( X ) = Q 0 ( A ∩ s − 1 ( X )) + Q 0 ( X \ A ) = Q 0 ( A ) + Q 0 ( X \ A ) = 1 , and in addition e Q ∈ P m 0 ( X ) , since π # e Q = π #  s # ( Q 0 | A ) + Q 0 | ( X \ A )  = π # ( s # ( Q 0 | A )) + π # ( Q 0 | ( X \ A ) ) = ( π ◦ s ) # ( Q 0 | A ) + π # ( Q 0 | ( X \ A ) ) = π # ( Q 0 | A ) + π # ( Q 0 | ( X \ A ) ) = π # ( Q 0 | A + Q 0 | ( X \ A ) ) = π # Q 0 = m 0 , where w e use that π ◦ s ( x ) = π ( x ) for ev ery x ∈ dom S . Using that Q 0 ( A ) > 0 , w e also compute w X F ( x, Q 0 ) d e Q ( x ) = w X \ A F ( x, Q 0 ) d Q 0 ( x ) + w A F ( s ( x ) , Q 0 ) d Q 0 ( x ) ≤ w X \ A F ( x, Q 0 ) d Q 0 ( x ) + q Q 0 ( A ) < w X \ A F ( x, Q 0 ) d Q 0 ( x ) + r Q 0 ( A ) < w X \ A F ( x, Q 0 ) d Q 0 ( x ) + w A F ( x, Q 0 ) d Q 0 ( x ) = w X F ( x, Q 0 ) d Q 0 ( x ) , whic h, using Theorem 3.17, contradicts the fact that Q 0 is a critical p oint of J in P m 0 ( X ) . 12 F or the con verse implication, assume that Q 0 ∈ P m 0 ( X ) is an equilibrium of NAG( X, Y , π , L, H , m 0 ) . In particular, r X F ( x, Q 0 ) d Q 0 ( x ) < + ∞ , and th us Q 0 ∈ dom J . Deﬁne ν : Y → R + b y ν ( y ) = inf z ∈ X π ( z )= y F ( z , Q 0 ) (15) and note that ν is Borel-measurable. Indeed, for ev ery ρ ≥ 0 , we hav e { y ∈ Y | ν ( y ) ≤ ρ } = \ n ∈ N ∗  y ∈ Y     ∃ z ∈ X suc h that π ( z ) = y and F ( z , Q 0 ) ≤ ρ + 1 n  = \ n ∈ N ∗ π  z ∈ X     F ( z , Q 0 ) ≤ ρ + 1 n  . F or ev ery n ∈ N ∗ , b y Lemma 3.12 and (A7), w e hav e that  z ∈ X   F ( z , Q 0 ) ≤ ρ + 1 n  is a closed subset of the compact set  z ∈ X   L ( z ) ≤ ρ + 1 n  , and hence it is compact. Thus, b y (A2), the set π  z ∈ X   F ( z , Q 0 ) ≤ ρ + 1 n  is compact, and th us it is closed, allowing one to conclude that { y ∈ Y | ν ( y ) ≤ ρ } is a Borel set. This yields the Borel-measurability of ν , as required. By deﬁnition of equilibrium, we hav e F ( x, Q 0 ) = ν ( π ( x )) for Q 0 -almost ev ery x ∈ X . Hence w X F ( x, Q 0 ) d Q 0 ( x ) = w X ν ( π ( x )) d Q 0 ( x ) . On the other hand, if one tak es Q ∈ dom J ∩ P m 0 ( X ) , then w X F ( z , Q 0 ) d Q ( z ) ≥ w X ν ( π ( z )) d Q ( z ) = w Y ν ( y ) d( π # Q )( y ) = w Y ν ( y ) d m 0 ( y ) = w Y ν ( y ) d( π # Q 0 )( y ) = w X ν ( π ( x )) d Q 0 ( x ) = w X F ( x, Q 0 ) d Q 0 ( x ) , and th us, using Theorem 3.17, we deduce that Q 0 is a critical p oint of J in P m 0 ( X ) . Finally , the last assertion of the statement follows as an immediate consequence of Prop osi- tion 3.19. 3.3 Existence of equilibria W e no w turn to the question of existence of an equilibrium for NAG( X, Y , π , L, H , m 0 ) which, thanks to the p otential structure of this game highlighted in Section 3.2, can b e addressed b y studying the existence of minimizers for the function J from (9c). W e start with the following preliminary result on J . Lemma 3.21. A ssume that (A1) and (A5) ar e satisﬁe d and let J b e the function deﬁne d in (9c) . Then J is lower semic ontinuous on P ( X ) . Pr o of. Let ( Q n ) n ∈ N b e a sequence in P ( X ) con v erging to some Q ∈ P ( X ) . Since X is a Polish space, it follows from [78, Lemma A.1] that the sequence of the pro duct measures ( Q n ⊗ Q n ) n ∈ N con verges to Q ⊗ Q . F or C > 0 , let J C : X × X → R + b e deﬁned b y J C ( x, e x ) = min { J ( x, e x ) , C } and notice that, by (A5), the function J C is low er semicon tinuous and b ounded. Then, by [15, Corollary 8.2.5], w e hav e w X × X J C ( x, e x ) d( Q ⊗ Q )( x, e x ) ≤ lim inf n → + ∞ w X × X J C ( x, e x ) d( Q n ⊗ Q n )( x, e x ) ≤ lim inf n → + ∞ w X × X J ( x, e x ) d( Q n ⊗ Q n )( x, e x ) = lim inf n → + ∞ J ( Q n ) . Since the ab ov e holds for ev ery C > 0 , b y Leb esgue’s monotone con v ergence theorem, w e deduce that J ( Q ) = w X × X J ( x, e x ) d( Q ⊗ Q )( x, e x ) = lim C → + ∞ w X × X J C ( x, e x ) d( Q ⊗ Q )( x, e x ) ≤ lim inf n → + ∞ J ( Q n ) , yielding the conclusion. 13 Our main result on the existence of minimizers for J and of equilibria for NA G( X , Y , π , L, H , m 0 ) is the following Theorem 3.22. A ssume that (A1)–(A3) and (A5)–(A8) ar e satisﬁe d and let J b e the function fr om (9c) . Then, for every m 0 ∈ P ( Y ) , J admits a minimizer in P m 0 ( X ) . A s a c onse quenc e, if in addition (A4) is also satisﬁe d, then ther e exists an e quilibrium Q for NA G( X , Y , π , L, H , m 0 ) . Pr o of. Let m 0 ∈ P ( Y ) and note that, by Corollary 3.11, w e ha ve dom J ∩ P m 0 ( X )  = ∅ . Let ( Q n ) n ∈ N b e a minimizing sequence for J in P m 0 ( X ) , i.e., ( Q n ) n ∈ N is a sequence in dom J ∩ P m 0 ( X ) and lim n → + ∞ J ( Q n ) = inf Q ∈ P m 0 ( X ) J ( Q ) . W e claim that the sequence ( Q n ) n ∈ N is tight. Indeed, by Marko v’s inequality , for every M > 0 , w e hav e Q n ( { x ∈ X | L ( x ) > M } ) ≤ 1 M L ( Q n ) ≤ 1 M J ( Q n ) , and the conclusion follows since ( J ( Q n )) n ∈ N is a bounded sequence and, b y (A7), { x ∈ X | L ( x ) ≤ M } is compact for every M > 0 . Therefore, b y Prokhorov’s theorem (see, e.g., [4, Theorem 5.1.3]), up to extracting a subsequence (which we still denote b y ( Q n ) n ∈ N for simplicity), there exists Q ∈ P ( X ) suc h that Q n → Q as n → + ∞ . In addition, by Remark 3.4, w e ha ve Q ∈ P m 0 ( X ) . By Lemma 3.21, w e deduce that J ( Q ) ≤ lim n → + ∞ J ( Q n ) = inf e Q ∈ P m 0 ( X ) J ( e Q ) , and thus Q is a minimizer of J in P m 0 ( X ) . The last part of the statemen t is an immediate consequence of Theorem 3.20. 3.4 Strong equilibria Deﬁnitions 2.1 and 3.1 of equilibria of the mean ﬁeld game from Section 2 and of NA G( X , Y , π , L, H , m 0 ) require an optimization problem to be solv ed b y Q -almost every elemen t of the space, similarly to other Lagrangian approac hes in the literature, such as those from [68, 75, 76]. An alternativ e deﬁnition consists in requiring the optimization problem to b e solv ed for every elemen t in the supp ort 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, Prop osition 3.7], both notions of equilibria coincide. W e aim at pro ving that this is also the case in our setting, at the cost of an additional assumption on NAG( X, Y , π , L, H , m 0 ) . T o do that, let us start b y deﬁning the notion of strong equilibrium. Deﬁnition 3.23. Let X and Y be Polish spaces, π : X → Y , L : X → R + , and H : X × X → R + b e Borel-measurable functions, and m 0 ∈ P ( Y ) . W e sa y that Q ∈ P ( X ) is a str ong e quilibrium of NAG( X, Y , π , L, H , m 0 ) if π # Q = m 0 , r X F ( x, Q ) d Q ( x ) < + ∞ , and every x ∈ spt( Q ) solves Min( π ( x ) , F ) with F given b y (5). T o pro ve that the notions of equilibrium and strong equilibrium are equiv alen t, w e will make use of an additional technical assumption, which we no w state. (A9) F or ev ery Q ∈ dom L , the set Opt ( Q ) deﬁned by Opt ( Q ) =    x ∈ X       F ( x, Q ) = inf ˜ x ∈ X π ( ˜ x )= π ( x ) F ( ˜ x, Q )    (16) is closed. As a consequence, w e obtain the follo wing result on the equiv alence b etw een equilibrium and strong equilibrium for NAG( X, Y , π , L, H , m 0 ) . Theorem 3.24. A ssume that (A1)–(A3), (A5), and (A9) ar e satisﬁe d. Then Q ∈ P m 0 ( X ) is an e quilibrium of the non-atomic game NA G( X , Y , π , L, H , m 0 ) if and only if it is a str ong e quilibrium of this game. 14 Pr o of. It is immediate from Deﬁnitions 3.1 and 3.23 that ev ery strong equilibrium is also an equilibrium of NA G( X, Y , π , L, H , m 0 ) . T o prov e the con verse statement, let Q ∈ P m 0 ( X ) b e an equilibrium of NAG( X, Y , π , L, H , m 0 ) and note that, by Remark 3.5, we ha ve Q ∈ dom J , implying that Q ∈ dom L . Note also that, since Q is an equilibrium of NAG( X, Y , π , L, H , m 0 ) , w e ha ve Q ( Opt ( Q )) = 1 , where Opt ( Q ) is the set deﬁned in (16). Let x ∈ spt( Q ) . Then, b y deﬁnition of supp ort and using the fact that Q ( Opt ( Q )) = 1 , there exists a sequence ( x n ) n ∈ N in Opt ( Q ) suc h that x n → x as n → + ∞ . Since Opt ( Q ) is closed by (A9), w e deduce that x ∈ Opt ( Q ) . Thus spt( Q ) ⊂ Opt ( Q ) , concluding the proof. 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 p oints of J in P m 0 ( X ) are equilibria of NA G( X , Y , π , L, H , m 0 ) . Indeed, notice ﬁrst that, under the assumptions of Theorem 3.20 and (A9), for ev ery Q 0 ∈ dom L , w e ha ve Opt ( Q 0 ) ⊂ { x ∈ X | L ( x ) ≤ ( C + 1) κ + C L ( Q 0 ) + C } , (17) where κ 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 ∈ Opt ( Q 0 ) , we hav e L ( x ) ≤ F ( x, Q 0 ) ≤ F ( z , Q 0 ) for ev ery z ∈ X with π ( z ) = π ( x ) . By (A8), we hav e F ( z , Q 0 ) ≤ ( C + 1) L ( z ) + C L ( Q 0 ) + C for ev ery z ∈ X , and, com bining with (A6), w e deduce that there exists z ∈ X such that π ( z ) = π ( x ) and F ( z , Q 0 ) ≤ ( C + 1) κ + C L ( Q 0 ) + C , yielding the inclusion (17). No w, b y (A7) and (A9), we deduce from (17) that Opt ( Q 0 ) is compact. Assume that Q 0 ∈ P m 0 ( X ) is a critical point of J in P m 0 ( X ) . Let Opt : Y ⇒ X b e the set-v alued map deﬁned for y ∈ Y by Opt( y ) =    x ∈ X       π ( x ) = y and F ( x, Q 0 ) = inf z ∈ X π ( z )= y F ( z , Q 0 )    . Clearly , Opt( y ) ⊂ Opt ( Q 0 ) for every y ∈ Y , and it follows from Corollary 3.13 that Opt( y )  = ∅ for ev ery y ∈ Y . Using the closedness of Opt ( Q 0 ) and the con tinuit y of π , it is easy to c heck that the graph of Opt is closed. Since Opt ( Q 0 ) is compact, [5, Prop osition 1.4.8] sho ws that Opt is upp er semicontin uous and, combining [5, Prop osition 1.4.4], [36, Proposition I I I.11], and [5, Theorem 8.1.3], we conclude that the set v alued map Opt admits a Borel-measurable selection, whic h we denote by ϕ : Y → X . Deﬁne b Q = ϕ # m 0 and notice that π # b Q = m 0 and w Y ν ( y ) d m 0 ( y ) = w X F ( x, Q 0 ) d b Q ( x ) , where ν is the function deﬁned in (15). Since Q 0 is a critical p oint of J in P m 0 ( X ) , w e obtain, using Theorem 3.17, that w X F ( x, Q 0 ) d Q 0 ( x ) ≤ w X F ( x, Q 0 ) d b Q ( x ) = w Y ν ( y ) d m 0 ( y ) = w X ν ( π ( x )) d Q 0 ( x ) . (18) Let Θ : X → R be the function deﬁned by Θ( x ) = F ( x, Q 0 ) − ν ( π ( x )) . By deﬁnition, Θ( x ) ≥ 0 for an y x ∈ X . It follo ws from (18) that w X Θ( x ) d Q 0 ( x ) ≤ 0 , therefore, Θ( x ) = 0 for Q 0 -a.e. x , which implies that Q 0 is an equilibrium of NAG( X, Y , π , L, H , m 0 ) . 4 Application to the mean ﬁeld game mo del with pairwise in teractions W e no w come bac k 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 , m 0 ) to the former MF G. T o do so, we 15 let X , Y , π , L , and H b e deﬁned from the MF G from Section 2 as in Remark 3.2 and w e aim at pro ving 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 so on as one assumes (H1) and (H2), (A4) follo ws from (H4)(d), and (A6) follo ws from (H6). W e are th us left to sho w (A3), (A5), (A7), and (A8). W e start b y the follo wing result. Lemma 4.1. A ssume that (H1), (H2), and (H5) ar e satisﬁe d and let τ b e the function deﬁne d in (4) . Then the functions τ and Ψ ◦ τ ar e lower semic ontinuous. Pr o of. Let ( γ n ) n ∈ N b e a sequence in Γ conv erging to some γ ∈ Γ uniformly on compact time in terv als. W e wan t to pro v e that lim inf n →∞ τ ( γ n ) ≥ τ ( γ ) . Let τ ∗ denote the left-hand side of the ab ov e inequality . If τ ∗ = + ∞ , there is nothing to prov e, and th us we only consider the case τ ∗ < + ∞ . Let ( γ n k ) k ∈ N b e a subsequence suc h that lim k →∞ τ ( γ n k ) = τ ∗ . Since Ξ is closed, we hav e, from the deﬁnition of τ , that γ n k ( τ ( γ n k )) ∈ Ξ for every k ∈ N , and th us γ ( τ ∗ ) ∈ Ξ . Hence, by the deﬁnition of τ , we deduce that τ ( γ ) ≤ τ ∗ , whic h is the desired inequalit y . Regarding the function Ψ ◦ τ , since Ψ is low er semicon tinuous and nondecreasing, we obtain that, for an y sequence ( γ n ) n ∈ N in Γ con v erging to some γ ∈ Γ , w e hav e lim inf n →∞ Ψ( τ ( γ n )) ≥ Ψ(lim inf n →∞ τ ( γ n )) ≥ Ψ( τ ( γ )) , as required. W e now provide the following technical result, which is a preliminary step tow ards showing the assertion on L from (A3) as w ell as (A7). Lemma 4.2. A ssume that (H1)–(H3) and (H5) ar e satisﬁe d and let θ > 1 b e the c onstant fr om (H3)(d) and L b e the function deﬁne d in R emark 3.2. L et ( γ n ) n ∈ N b e a se quenc e in Γ such that the se quenc e ( L ( γ n )) n ∈ N is b ounde d. Then, for every T > 0 , ( γ n ) n ∈ N is a b ounde d se quenc e in W 1 ,θ ([0 , T ] , Ω) and ther e exists a subse quenc e ( γ n k ) k ∈ N of ( γ n ) n ∈ N and an element γ ∈ W 1 ,θ loc ( R + , Ω) such that γ n k → γ as k → + ∞ in the top olo gy of Γ and, for every T > 0 , we have the we ak c onver genc e ˙ γ n k ⇀ ˙ γ as k → + ∞ in L θ ([0 , T ] , R d ) . Pr o of. Let κ > 0 and ( γ n ) n ∈ N b e a sequence in Γ such that L ( γ n ) ≤ κ for every n ∈ N . Hence, b y (H3), w e deduce that, for every n ∈ N , w + ∞ 0 | ˙ γ n ( t ) | θ d t ≤ κ. Th us, for every T > 0 , the sequence ( γ n ) n ∈ N is bounded in W 1 ,θ ([0 , T ] , Ω) . Recall also that the injection of W 1 ,θ ([0 , T ] , Ω) into C ([0 , T ] , Ω) is compact and that b ounded sets of W 1 ,θ ([0 , T ] , Ω) are relativ ely compact for the weak con vergence. W e extract the required subsequence by a standard diagonal argument as follo ws. Since ( γ n ) n ∈ N is bounded in W 1 ,θ ([0 , 1] , Ω) , we extract a subsequence ( γ 1 n ) n ∈ N of ( γ n ) n ∈ N whic h conv erges strongly in C ([0 , 1] , Ω) and w eakly in W 1 ,θ ([0 , 1] , Ω) to some γ 1 ∈ W 1 ,θ ([0 , 1] , Ω) . No w, assuming that k ∈ N ∗ is suc h that w e ha v e constructed a subsequence ( γ k n ) n ∈ N of ( γ n ) n ∈ N con verging strongly in C ([0 , k ] , Ω) and w eakly in W 1 ,θ ([0 , k ] , Ω) to some elemen t γ k ∈ W 1 ,θ ([0 , k ] , Ω) , using the fact that ( γ k n ) n ∈ N is bounded in W 1 ,θ ([0 , k + 1] , Ω) , we extract a subsequence ( γ k +1 n ) n ∈ N of ( γ k n ) n ∈ N con verging strongly in C ([0 , k + 1] , Ω) and weakly in W 1 ,θ ([0 , k + 1] , Ω) to some γ k +1 ∈ W 1 ,θ ([0 , k + 1] , Ω) , and clearly , b y uniqueness of the limit, γ k +1 coincides with γ k in their common domain [0 , k ] . Deﬁne γ : R + → Ω b y setting γ ( t ) = γ ⌈ t ⌉ ( t ) , and note that, b y construction, w e hav e γ ∈ W 1 ,θ loc ( R + , Ω) , since, for every k ∈ N ∗ , γ coincides with γ k in [0 , k ] . The diagonal sequence ( γ n n ) n ∈ N is a subsequence of ( γ n ) n ∈ N whic h, b y construction, con verges strongly to γ in Γ , i.e., uniformly in an y compact subset of R + , and, for every T > 0 , it also con verges w eakly to γ in W 1 ,θ ([0 , T ] , Ω) , yielding the conclusion. A ﬁrst consequence of Lemma 4.2 is the following. 16 Corollary 4.3. A ssume that (H1)–(H3) and (H5) ar e satisﬁe d and let L b e the function deﬁne d in R emark 3.2. Then L is lower semic ontinuous. Pr o of. Let ( γ n ) n ∈ N b e a sequence in Γ suc h that γ n → γ as n → + ∞ for some γ ∈ Γ . W e need to pro ve that L ( γ ) ≤ lim inf n → + ∞ L ( γ n ) . (19) If the right-hand side of (19) is equal to + ∞ , there is nothing to prov e, so w e only consider the case where it is ﬁnite. In this case, w e extract from ( γ n ) n ∈ N a subsequence ( γ n k ) k ∈ N suc h that ( L ( γ n k )) k ∈ N con verges to the righ t-hand side of (19). F or simplicity , w e will denote ( γ n k ) k ∈ N b y ( γ n ) n ∈ N in the sequel. Up to remo ving at most ﬁnitely many elemen ts of the sequence, we ha v e L ( γ n ) < + ∞ for every n ∈ N , and th us ( L ( γ n )) n ∈ N is bounded. Hence, b y Lemma 4.2, up to extracting once again a subsequence, which w e still denote by the same notation, w e deduce that γ ∈ W 1 ,θ loc ( R + , Ω) and that, for every T > 0 , we ha ve the weak con vergence ˙ γ n ⇀ ˙ γ in L θ ([0 , T ] , R d ) . By Lemma 4.1, we ha ve Ψ( τ ( γ )) ≤ lim inf n → + ∞ Ψ( τ ( γ n )) . In addition, using [46, Theorem 4.5], w e obtain that, for every T > 0 , w T 0 ℓ ( t, γ ( t ) , ˙ γ ( t )) d t ≤ lim inf n → + ∞ w T 0 ℓ ( t, γ n ( t ) , ˙ γ n ( t )) d t ≤ lim inf n → + ∞ w + ∞ 0 ℓ ( t, γ n ( t ) , ˙ γ n ( t )) d t, and, since this holds for ev ery T > 0 , we deduce that w + ∞ 0 ℓ ( t, γ ( t ) , ˙ γ ( t )) d t ≤ lim inf n → + ∞ w + ∞ 0 ℓ ( t, γ n ( t ) , ˙ γ n ( t )) d t, yielding the conclusion. W e also deduce from Lemma 4.2 and Corollary 4.3 the follo wing result. Corollary 4.4. A ssume that (H1)–(H3) and (H5) ar e satisﬁe d and let L b e the function deﬁne d in R emark 3.2. Then (A7) is satisﬁe d. Pr o of. Let κ > 0 and consider a sequence ( γ n ) n ∈ N in Γ such that L ( γ n ) ≤ κ for ev ery n ∈ N . Then, by Lemma 4.2, ( γ n ) n ∈ N admits a subsequence, still denoted b y ( γ n ) n ∈ N , conv erging to some γ ∈ W 1 ,θ loc ( R + , Ω) , both in the topology of Γ and with ˙ γ n ⇀ ˙ γ in L θ ([0 , T ] , R d ) for ev ery T > 0 . Since L is low er semicon tinuous by Corollary 4.3 and γ n → γ in Γ as n → + ∞ , w e deduce that L ( γ ) ≤ lim inf n → + ∞ L ( γ n ) ≤ κ , yielding the conclusion. Let us no w sho w that J is also lo wer semicon tin uous and H is Borel measurable. Lemma 4.5. A ssume that (H1)–(H5) ar e satisﬁe d, let L and H b e the functions deﬁne d in R e- mark 3.2, and J b e the function deﬁne d in (8) . Then J is lower semic ontinuous and H is Bor el me asur able. Pr o of. W e note ﬁrst that it suﬃces to pro ve the assertion on J . Indeed, since L is lo wer semi- con tinuous by Corollary 4.3, it is Borel measurable, and th us the set { ( γ , e γ ) ∈ Γ × Γ | L ( γ ) = + ∞ or L ( e γ ) = + ∞} is Borel measurable. Since H is constan t and equal to + ∞ in this set, it suﬃces to show that H is Borel measurable in the complemen tary of this set, that is, on { ( γ , e γ ) ∈ Γ × Γ | L ( γ ) < + ∞ and L ( e γ ) < + ∞} . In this set, we hav e H ( γ , e γ ) = J ( γ , e γ ) − L ( γ ) − L ( e γ ) , and th us, if J is shown to b e lo w er semicon tinuous, it will also b e Borel measurable, and hence H will be Borel measurable as the diﬀerence of Borel measurable functions. Let us then prov e that J is lo wer semicon tin uous. Let ( γ n , e γ n ) n ∈ N b e a sequence in Γ × Γ con verging to some ( γ , e γ ) ∈ Γ × Γ . W e wan t to pro ve that J ( γ , e γ ) ≤ lim inf n → + ∞ J ( γ n , e γ n ) . (20) As in the pro of of Corollary 4.3, w e consider only the case where the right-hand side of the abov e inequalit y is ﬁnite and, up to extracting a subsequence, w e also ha ve that J ( γ n , e γ n ) conv erges as n → + ∞ to lim inf n → + ∞ J ( γ n , e γ n ) and that the sequence ( J ( γ n , e γ n )) n ∈ N is b ounded. In particular, ( L ( γ n )) n ∈ N and ( L ( e γ n )) n ∈ N are b ounded sequences, so, b y Lemma 4.2, we deduce that γ and e γ 17 b elong to W 1 ,θ loc ( R + , Ω) and that, up to a further subsequence extraction, ˙ γ n ⇀ ˙ γ and ˙ e γ n ⇀ ˙ e γ as n → + ∞ in L θ ([0 , T ] , R d ) , for ev ery T > 0 . Note that, b y Corollary 4.3, we already hav e L ( γ ) ≤ lim inf n → + ∞ L ( γ n ) , L ( e γ ) ≤ lim inf n → + ∞ L ( e γ n ) , so w e are left to show that H ( γ , e γ ) ≤ lim inf n → + ∞ H ( γ n , e γ n ) . (21) Let us denote σ n = τ ( γ n ) ∧ τ ( e γ n ) and σ = τ ( γ ) ∧ τ ( e γ ) and recall that, by Lemma 4.1, we hav e σ ≤ lim inf n → + ∞ σ n . W e ha ve nothing to pro ve in the case σ = 0 since, in this case, the left-hand side of (21) is zero. W e thus only consider the case σ > 0 from now on. Fix an increasing sequence ( T k ) k ∈ N in R + with lim k → + ∞ T k = σ . F or eac h k ∈ N , applying [46, Theorem 4.5] to the in terv al [0 , T k ] , w e obtain that w T k 0 h ( t, γ ( t ) , e γ ( t ) , ˙ γ ( t ) , ˙ e γ ( t )) d t ≤ lim inf n → + ∞ w T k 0 h ( t, γ n ( t ) , e γ n ( t ) , ˙ γ n ( t ) , ˙ e γ n ( t )) d t. (22) Since T k < σ ≤ lim inf n → + ∞ σ n , w e hav e σ n > T k for n large enough (dep ending on k ), and th us, as h is nonnegative, w e get w T k 0 h ( t, γ n ( t ) , e γ n ( t ) , ˙ γ n ( t ) , ˙ e γ n ( t )) d t ≤ w σ n 0 h ( t, γ n ( t ) , e γ n ( t ) , ˙ γ n ( t ) , ˙ e γ n ( t )) d t for n large enough. Hence, taking the lim inf as n → + ∞ , w e obtain, com bining with (22), that w T k 0 h ( t, γ ( t ) , e γ ( t ) , ˙ γ ( t ) , ˙ e γ ( t )) d t ≤ lim inf n → + ∞ w σ n 0 h ( t, γ n ( t ) , e γ n ( t ) , ˙ γ n ( t ) , ˙ e γ n ( t )) d t. Since this holds true for ev ery k ∈ N , w e then obtain that w σ 0 h ( t, γ ( t ) , e γ ( t ) , ˙ γ ( t ) , ˙ e γ ( t )) d t ≤ lim inf n → + ∞ w σ n 0 h ( t, γ n ( t ) , e γ n ( t ) , ˙ γ n ( t ) , ˙ e γ n ( t )) d t, as required. Remark 4.6. The pro of of Lemma 4.5 do es not imply that H is lo w er semicon tinuous. Indeed, although w e prov e the inequality (21), this is not done for an y sequence ( γ n , e γ n ) n ∈ N con verging in Γ × Γ : as we wan t to pro ve (20), w e restrict our attention to sequences for which the righ t-hand side of (20) is ﬁnite. Hence, our proof of the inequality (21) does not tak e in to account sequences ( γ n , e γ n ) n ∈ N for whic h 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 low er semicon tin uity of H , and the main issue is that we cannot apply the compactness result from Lemma 4.2 to suc h sequences. Our next result establishes (A8) as a consequence of (H1)–(H5). Lemma 4.7. A ssume that (H1)–(H5) ar e satisﬁe d and let L and H b e the functions deﬁne d in R emark 3.2. Then (A8) is satisﬁe d. Pr o of. Consider the constan ts α , θ , C , β , a , and b from (H3)–(H5). F or ev ery γ ∈ Γ with L ( γ ) < + ∞ , w e hav e, using Y oung’s inequalit y , that w τ ( γ ) 0 | ˙ γ ( t ) | β d t ≤ β θ w + ∞ 0 | ˙ γ ( t ) | θ d t + θ − β θ τ ( γ ) ≤ β αθ w + ∞ 0 ℓ ( t, γ ( t ) , ˙ γ ( t )) d t + ( θ − β )(Ψ( τ ( γ )) + b ) aθ ≤ M  L ( γ ) + 1 2  , 18 where M = max n β αθ , θ − β aθ , 2 b ( θ − β ) aθ o . On the other hand, using (H4), for every ( γ , e γ ) ∈ Γ × Γ with L ( γ ) < + ∞ and L ( e γ ) < + ∞ , we hav e H ( γ , e γ ) ≤ w τ ( γ ) ∧ τ ( e γ ) 0 C ( | ˙ γ ( t ) | β + | ˙ e γ ( t ) | β ) d t ≤ C  w τ ( γ ) 0 | ˙ γ ( t ) | β d t  + C  w τ ( e γ ) 0 | ˙ e γ ( t ) | β d t  ≤ C M ( L ( γ ) + L ( e γ ) + 1) , yielding the conclusion. W e are thus ﬁnally in p osition to conclude that the mean ﬁeld game from Section 2 can be studied through the non-atomic game NA G( X , Y , π , L, H , m 0 ) . Prop osition 4.8. A ssume that (H1)–(H6) ar e satisﬁe d and c onsider the non-atomic game NAG( X, Y , π , L, H , m 0 ) with X , Y , π , L , and H deﬁne d as in R emark 3.2. Then (A1)–(A8) ar e satisﬁe d for NA G( X , Y , π , L, H , m 0 ) . In addition, Q is an e quilibrium of the me an ﬁeld game fr om Se ction 2 with initial c ondition m 0 ∈ P (Ω) if and only if it is an e quilibrium of NA G( X , Y , π , L, H , m 0 ) . Pr o of. 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 subtlet y to b e addressed: the term H ( γ , e γ ) do es not necessarily coincide with the integral r τ ( γ ) ∧ τ ( e γ ) 0 h ( t, γ ( t ) , e γ ( t ) , ˙ γ ( t ) , ˙ e γ ( t )) d t , since the ﬁrst one is deﬁned to b e + ∞ whenever L ( γ ) = + ∞ or L ( e γ ) = + ∞ , but the second one can be ﬁnite ev en in some cases where L ( γ ) = + ∞ or L ( e γ ) = + ∞ . Hence, the functions deﬁned in (3) and (5) do not necessarily coincide for all ( γ , Q ) . In this pro of, w e denote by F 1 ( γ , Q ) the function deﬁned in (3) and by F 2 ( γ , Q ) the function deﬁned in (5). Notice ﬁrst that, if Q ∈ P (Γ) is suc h that Q (dom L ) = 1 , then F 1 ( γ , Q ) = F 2 ( γ , Q ) for ev ery γ ∈ Γ . Indeed, recall that, for every ( γ , e γ ) ∈ dom L × dom L , we hav e H ( γ , e γ ) = w τ ( γ ) ∧ τ ( e γ ) 0 h ( t, γ ( t ) , e γ ( t ) , ˙ γ ( t ) , ˙ e γ ( t )) d t. Since Q (dom L ) = 1 , we can integrate the ab ov e equalit y in e γ with respect to Q to deduce that w Γ H ( γ , e γ ) d Q ( e γ ) = w Γ w τ ( γ ) ∧ τ ( e γ ) 0 h ( t, γ ( t ) , e γ ( t ) , ˙ γ ( t ) , ˙ e γ ( t )) d t d Q ( e γ ) for ev ery γ ∈ dom L . Thus, F 1 ( γ , Q ) = F 2 ( γ , Q ) for every γ ∈ dom L . On the other hand, if γ / ∈ dom L , w e hav e L ( γ ) = + ∞ , hence F 1 ( γ , Q ) = F 2 ( γ , Q ) = + ∞ . Hence, F 1 ( γ , Q ) = F 2 ( γ , Q ) for ev ery γ ∈ Γ . If Q is an equilibrium of the mean ﬁeld game from Section 2 with initial condition m 0 ∈ P (Ω) , then r Γ F 1 ( γ , Q ) d Q ( γ ) < + ∞ , showing that F 1 ( γ , Q ) < + ∞ for Q -almost ev ery γ ∈ Γ . In particular, L ( γ ) < + ∞ for Q -almost ev ery γ , i.e., Q (dom L ) = 1 , and thus, by the ab ov e argument, F 1 ( γ , Q ) = F 2 ( γ , Q ) for ev ery γ ∈ Γ , proving that Q is also an equilibrium of NAG( X, Y , π , L, H , m 0 ) . Conv ersely , if Q is an equilibrium of NA G( X, Y , π , L, H , m 0 ) , then r Γ F 2 ( γ , Q ) d Q ( γ ) < + ∞ , sho wing that F 2 ( γ , Q ) < + ∞ for Q -almost every γ ∈ Γ . In particular, L ( γ ) < + ∞ for Q -almost ev ery γ , i.e., Q (dom L ) = 1 , and thus, by the ab ov e argument, F 1 ( γ , Q ) = F 2 ( γ , Q ) for every γ ∈ Γ , pro ving that Q is also an equilibrium of the mean ﬁeld game from Section 2 with initial condition m 0 ∈ P (Ω) . As an immediate consequence of Theorem 3.22 and Proposition 4.8, w e obtain our main result on the mean ﬁeld game model from Section 2. Theorem 4.9. A ssume that (H1)–(H6) ar e satisﬁe d and let m 0 ∈ P (Ω) . Then ther e exists an e quilibrium Q ∈ P (Γ) of the me an ﬁeld game fr om Se ction 2 with initial c ondition m 0 . W e conclude our discussion on the model from Section 2 by addressing strong equilibria. Simi- larly to Deﬁnition 3.23, we shall sa y that Q ∈ P (Γ) is a strong equilibrium of the mean ﬁeld game from Section 2 if it satisﬁes Deﬁnition 2.1 with “ Q -almost every γ ” replaced by “every γ ∈ spt( Q ) ” . Let us in troduce tw o additional assumptions on the mo del from Section 2. 19 (H7) F or every x ∈ Ω and ev ery sequence ( x n ) n ∈ N with x n → x as n → + ∞ , we hav e that d geo ( x n , x ) → 0 as n → + ∞ , where d geo denotes the geodesic distance in Ω . (H8) There exists α ∗ > 0 suc h that ℓ ( t, x, p ) ≤ α ∗ | p | θ for ev ery ( t, x, p ) ∈ R + × Ω × R d , where θ is the same constan t as in (H3). One of the consequences of (H8) is that tra jectories minimizing F ( γ , Q ) starting at some x 0 already in the target set Ξ m ust necessarily remain constant at all times. Lemma 4.10. A ssume that (H1)–(H5) and (H8) ar e satisﬁe d for the me an ﬁeld game of Se ction 2 and let x 0 ∈ Ξ and Q ∈ P (Γ) . Then the unique tr aje ctory γ 0 minimizing (3) with the c onstr aint γ 0 (0) = x 0 is the c onstant tr aje ctory γ 0 ( t ) = x 0 for every t ∈ R + . Pr o of. Clearly , F ( γ , Q ) ≥ Ψ(0) for ev ery γ ∈ Γ , and the v alue Ψ(0) is attained for the constan t tra jectory γ 0 since τ ( γ 0 ) = 0 and 0 ≤ ℓ ( t, γ 0 ( t ) , ˙ γ 0 ( t )) ≤ α ∗ | ˙ γ 0 ( t ) | θ = 0 . T o pro ve uniqueness, note that F ( γ , Q ) = Ψ(0) implies r + ∞ 0 ℓ ( t, γ ( t ) , ˙ γ ( t )) d t = 0 , sho wing that γ is absolutely contin uous and ℓ ( t, γ ( t ) , ˙ γ ( t )) = 0 for almost every t ∈ R + . By (H3)(d), this implies that ˙ γ ( t ) = 0 for almost ev ery t ∈ R + , and hence γ is constant. The constraint γ (0) = x 0 then implies that γ = γ 0 . W e no w use (H7) and (H8) to pro ve that (A9) is satisﬁed. Lemma 4.11. A ssume that (H1)–(H6), (H7), and (H8) ar e satisﬁe d for the me an ﬁeld game of Se ction 2 and c onsider the non-atomic game NA G( X , Y , π , L, H , m 0 ) deﬁne d as in R emark 3.2. Then (A9) is satisﬁe d. Pr o of. Let Q ∈ dom L and Opt ( Q ) be the set deﬁned in (16). Recall that w e hav e the inclusion Opt ( Q 0 ) ⊂ { γ ∈ Γ | L ( γ ) ≤ ( C + 1) κ + C L ( Q 0 ) + C } , where κ and C are the constants from (A6) and (A8): this w as prov ed in Remark 3.25 under (A6) and (A8), and these tw o assumptions are satisﬁed here thanks to (H6) and Lemma 4.7. Let ( γ n ) n ∈ N b e a sequence in Opt ( Q ) and γ ∈ Γ with γ n → γ as n → + ∞ . In particular, by Lemma 4.2, w e deduce that γ and γ n b elong to W 1 ,θ loc ( R + , Ω) for ev ery n ∈ N and that ˙ γ n ⇀ ˙ γ in L θ ([0 , T ] , R d ) as n → + ∞ , for ev ery T > 0 . Let x 0 = γ (0) and, for n ∈ N , set x n = γ n (0) . W e split the proof in three cases. Case 1: x 0 ∈ Ξ and ther e exists a subse quenc e ( x n k ) k ∈ N of ( x n ) n ∈ N such that x n k ∈ Ξ for every k ∈ N . In this case, it follo ws from Lemma 4.10 that γ n k is constant and, since γ n k → γ as k → + ∞ , it follows that γ is also constan t. Hence, applying once again Lemma 4.10, we deduce that γ minimizes (3) with the constraint γ (0) = x 0 , yielding that γ ∈ Opt ( Q ) . Case 2: x 0 ∈ Ξ and x n / ∈ Ξ for every n ∈ N . Let ε n = d geo ( x n , x 0 ) and σ n : [0 , ε n ] → Ω be a geo desic curv e suc h that σ n (0) = x n , σ n ( ε n ) = x 0 , and | ˙ σ n ( t ) | = 1 for almost ev ery t ∈ (0 , ε n ) . Notice that the existence of such a geodesic curv e can b e ensured b y assumption (H7) and [19, Proposition 2.5.19] and, in addition, b y (H7), we ha ve ε n → 0 as n → ∞ . One can easily extend the domain of σ n to R + b y setting σ n ( t ) = x 0 for all t ≥ ε n . W e hav e 0 < τ ( σ n ) ≤ ε n → 0 as n → + ∞ , and thus Ψ( τ ( σ n )) → Ψ(0 + ) as n → + ∞ , where Ψ(0 + ) = lim t → 0 + Ψ( t ) , whic h exists since Ψ is nondecreasing. W e also compute w + ∞ 0 ℓ ( t, σ n ( t ) , ˙ σ n ( t )) d t ≤ α ∗ w + ∞ 0 | ˙ σ n ( t ) | θ d t ≤ α ∗ ε n − − − − − → n → + ∞ 0 (23) and, for ev ery ω ∈ dom L , w τ ( σ n ) ∧ τ ( ω ) 0 h ( t, σ n ( t ) , ω ( t ) , ˙ σ n ( t ) , ˙ ω ( t )) d t ≤ C ε n + C w ε n 0 | ˙ ω ( t ) | β d t − − − − − → n → + ∞ 0 , (24) 20 where w e use the fact that ω ∈ dom L to deduce from (H3)(d) that ˙ ω ∈ L θ ( R + , R d ) , and th us | ˙ ω | β ∈ L 1 ([0 , T ] , R d ) for every T > 0 since β ∈ (0 , θ ] . In addition, for all n large enough so that ε n ≤ 1 , w e hav e w τ ( σ n ) ∧ τ ( ω ) 0 h ( t, σ n ( t ) , ω ( t ) , ˙ σ n ( t ) , ˙ ω ( t )) d t ≤ C + C w 1 0 | ˙ ω ( t ) | β d t, (25) and the righ t-hand side of the abov e inequality is Q -in tegrable, since, by Hölder’s inequality and (H3)(d), w e ha ve w Γ w 1 0 | ˙ ω ( t ) | β d t d Q ( ω ) ≤  w Γ w 1 0 | ˙ ω ( t ) | θ d t d Q ( ω )  β θ ≤  1 α w Γ w + ∞ 0 ℓ ( t, ω ( t ) , ˙ ω ( t )) d t d Q ( ω )  β θ ≤  L ( Q ) α  β θ < + ∞ . (26) Hence, it follo ws from (24), (25), and Leb esgue’s dominated con v ergence theorem that lim n → + ∞ w Γ w τ ( σ n ) ∧ τ ( ω ) 0 h ( t, σ n ( t ) , ω ( t ) , ˙ σ n ( t ) , ˙ ω ( t )) d t d Q ( ω ) = 0 . Com bining this with (23) and the fact that Ψ( τ ( σ n )) → Ψ(0 + ) as n → + ∞ , w e deduce that lim n → + ∞ F ( σ n , Q ) = Ψ(0 + ) . Since γ n is a minimizer of (3) with ﬁxed initial condition x n and σ n ∈ Γ with σ n (0) = x n , we ha ve F ( γ n , Q ) ≤ F ( σ n , Q ) , and, on the other hand, since τ ( γ n ) > 0 , w e ha ve Ψ(0 + ) ≤ Ψ( τ ( γ n )) ≤ F ( γ n , Q ) . Hence, lim n → + ∞ F ( γ n , Q ) = Ψ(0 + ) , and (H3)(d) yields r + ∞ 0 | ˙ γ n ( t ) | θ d t → 0 as n → + ∞ . Recalling that, for every T > 0 , w e ha ve ˙ γ n ⇀ ˙ γ in L θ ([0 , T ] , R d ) as n → + ∞ , w e deduce, for the lo wer semicontin uit y of the L θ norm with respect to w eak conv ergence, that ∥ ˙ γ ∥ L θ ([0 ,T ] , R d ) ≤ lim inf n → + ∞ ∥ ˙ γ n ∥ L θ ([0 ,T ] , R d ) = 0 , and, since T > 0 is arbitrary , we deduce that γ is constan t. Hence, by Lemma 4.10, w e deduce that γ minimizes (3) with the constrain t γ (0) = x 0 , yielding that γ ∈ Opt ( Q ) . Case 3: x 0 / ∈ Ξ . By (H2), we hav e in this case x n / ∈ Ξ for n large enough. T o prov e that γ ∈ Opt ( Q ) , we will prov e that F ( γ , Q ) ≤ F ( e γ , Q ) for ev ery e γ ∈ Γ with e γ (0) = x 0 . F or that purpose, we will construct a sequence of trajectories ( e γ n ) n ∈ N in Γ with e γ n (0) = x n for ev ery n ∈ N and suc h that lim inf n → + ∞ F ( e γ n , Q ) ≤ F ( e γ , Q ) . (27) This will allo w us to conclude, since, by Lemma 3.12 and by the fact that γ n ∈ Opt ( Q ) , we will then ha ve F ( γ , Q ) ≤ lim inf n → + ∞ F ( γ n , Q ) ≤ lim inf n → + ∞ F ( e γ n , Q ) ≤ F ( e γ , Q ) , yielding that γ ∈ Opt ( Q ) . W e then focus on sho wing (27). Let e γ ∈ Γ with e γ (0) = x 0 . Note that there is nothing to be prov e in (27) if F ( e γ , Q ) = + ∞ , so w e assume in the sequel that F ( e γ , Q ) < + ∞ . In particular, e γ ∈ dom L , which implies by (H3) and (H5) that ˙ e γ ∈ L θ ( R + , R d ) and τ ( e γ ) < + ∞ . Let ε n = d geo ( x n , x 0 ) and ς n : [0 , ε n ] → Ω be a geo desic curve such that ς n (0) = x n , ς n ( ε n ) = e γ ( ε n ) , and | ˙ ς n ( t ) | = d geo ( x n , e γ ( ε n )) ε n for every t ∈ [0 , ε n ] (with the con ven tion that, in the case ε n = 0 , w e ha ve that ς n : { 0 } → Ω is giv en b y ς n (0) = x n = x 0 ). Notice that, as in Case 2, the existence of such a geodesic curv e can b e ensured b y assumption (H7) and [19, Proposition 2.5.19] and, in addition, b y (H7), w e ha ve ε n → 0 as n → + ∞ . Let us deﬁne e γ n ∈ Γ b y e γ n ( t ) = ( ς n ( t ) if t ∈ [0 , ε n ] , e γ ( t ) if t ≥ ε n . 21 In particular, we also hav e e γ n ∈ dom L , ˙ e γ n ∈ L θ ( R + , R d ) , and 0 < τ ( e γ n ) ≤ τ ( e γ ) < + ∞ for ev ery n ∈ N large enough. Note that, since e γ and e γ n b elong to dom L , w e hav e     w + ∞ 0 ℓ ( t, e γ ( t ) , ˙ e γ ( t )) d t − w + ∞ 0 ℓ ( t, e γ n ( t ) , ˙ e γ n ( t )) d t     ≤ w ε n 0 ℓ ( t, e γ ( t ) , ˙ e γ ( t )) d t + w ε n 0 ℓ ( t, ς n ( t ) , ˙ ς n ( t )) d t ≤ w ε n 0 ℓ ( t, e γ ( t ) , ˙ e γ ( t )) d t + α ∗ d geo ( x n , e γ ( ε n )) θ ε θ − 1 n . (28) Since e γ ∈ dom L , we hav e w ε n 0 ℓ ( t, e γ ( t ) , ˙ e γ ( t )) d t − − − − − → n → + ∞ 0 . (29) Notice also that d geo ( x n , e γ ( ε n )) θ ≤ (d geo ( x n , x 0 ) + d geo ( e γ (0) , e γ ( ε n ))) θ ≤ 2 θ − 1  d geo ( x n , x 0 ) θ +  w ε n 0    ˙ e γ ( t )    d t  θ  and, b y Hölder’s inequalit y ,  w ε n 0    ˙ e γ ( t )    d t  θ ≤ ε θ − 1 n w ε n 0    ˙ e γ ( t )    θ d t. Hence d geo ( x n , e γ ( ε n )) θ ε θ − 1 n ≤ 2 θ − 1  ε n + w ε n 0    ˙ e γ ( t )    θ d t  − − − − − → n → + ∞ 0 , (30) since ˙ e γ ∈ L θ ( R + , Ω) . Com bining (28), (29), and (30), we deduce that lim n → + ∞ w + ∞ 0 ℓ ( t, e γ n ( t ) , ˙ e γ n ( t )) d t = w + ∞ 0 ℓ ( t, e γ ( t ) , ˙ e γ ( t )) d t. (31) Note no w that, for every t ∈ [0 , ε n ] , w e ha ve | ς n ( t ) − x 0 | =     ς n (0) + w t 0 ˙ ς n ( s ) d s − x 0     ≤ | x n − x 0 | + t d geo ( x n , e γ ( ε n )) ε n ≤ | x n − x 0 | + d geo ( x n , e γ ( ε n )) − − − − − → n → + ∞ 0 , where we use (30) in the last step. Hence, since x 0 / ∈ Ξ , w e deduce that, for n large enough (indep enden tly of t ), ς n ( t ) / ∈ Ξ for ev ery t ∈ [0 , ε n ] . In addition, for n large enough (independently of t ), we clearly hav e e γ ( t ) / ∈ Ξ for ev ery t ∈ [0 , ε n ] . Hence τ ( e γ n ) = τ ( e γ ) for n large enough, and th us Ψ( τ ( e γ n )) = Ψ( τ ( e γ )) for n large enough. Com bining this with (31), we deduce that lim n → + ∞ L ( e γ n ) = L ( e γ ) . (32) Let us now study whether r Γ H ( e γ n , ω ) d Q ( ω ) conv erges to r Γ H ( e γ , ω ) d Q ( ω ) as n → + ∞ . F or ev ery ω ∈ dom L , recalling that τ ( e γ n ) = τ ( e γ ) for n large enough, we ha v e | H ( e γ n , ω ) − H ( e γ , ω ) | ≤ w ε n 0 h ( t, e γ ( t ) , ω ( t ) , ˙ e γ ( t ) , ˙ ω ( t )) d t + w ε n 0 h ( t, ς n ( t ) , ω ( t ) , ˙ ς n ( t ) , ˙ ω ( t )) d t. (33) W e ha ve w ε n 0 h ( t, e γ ( t ) , ω ( t ) , ˙ e γ ( t ) , ˙ ω ( t )) d t ≤ C  w ε n 0 | ˙ e γ ( t ) | β d t + w ε n 0 | ˙ ω ( t ) | β d t  − − − − − → n → + ∞ 0 (34) since e γ and ω b elong to dom L and, as suc h, their time deriv atives b elong to L θ ( R + , R d ) , whic h implies that | ˙ e γ | β and | ˙ ω | β b elong to L 1 ([0 , T ] , R d ) for every T > 0 , since β ∈ (0 , θ ] . On the other hand, w ε n 0 h ( t, ς n ( t ) , ω ( t ) , ˙ ς n ( t ) , ˙ ω ( t )) d t ≤ C  d geo ( x n , e γ ( ε n )) β ε β − 1 n + w ε n 0 | ˙ ω ( t ) | β d t  . (35) 22 As before, we hav e r ε n 0 | ˙ ω ( t ) | β d t → 0 as n → + ∞ , and, using (30), w e ha ve d geo ( x n , e γ ( ε n )) β ε β − 1 n = ε θ − β θ n  d geo ( x n , e γ ( ε n )) θ ε θ − 1 n  β θ − − − − − → n → + ∞ 0 since β ∈ (0 , θ ] . Combining the ab ov e with (33), (34), and (35), w e deduce that lim n → + ∞ H ( e γ n , ω ) = H ( e γ , ω ) (36) for every ω ∈ dom L . Notice also that we can estimate, for n large enough (indep endently of ω ) in order to ha v e in particular ε n ≤ 1 , H ( e γ n , ω ) ≤ H ( e γ , ω ) + w ε n 0 h ( t, ς n ( t ) , ω ( t ) , ˙ ς n ( t ) , ˙ ω ( t )) d t ≤ H ( e γ , ω ) + C d geo ( x n , e γ ( ε n )) β ε β − 1 n + C w 1 0 | ˙ ω ( t ) | β d t ≤ H ( e γ , ω ) + 1 + C w 1 0 | ˙ ω ( t ) | β d t, (37) where w e hav e chosen n large enough to hav e C d geo ( x n , e γ ( ε n )) β ε β − 1 n ≤ 1 . The righ t-hand side of (37) is Q -in tegrable with respect to ω thanks to Lemma 4.7, the facts that e γ ∈ dom L and Q ∈ dom L , and (26). Hence, b y (36), (37), and Leb esgue’s dominated con vergence theorem, we deduce that lim n → + ∞ w Γ H ( e γ n , ω ) d Q ( ω ) = w Γ H ( e γ , ω ) d Q ( ω ) . (38) Com bining (32) and (38), w e deduce that lim n → + ∞ F ( e γ n , Q ) = F ( e γ , Q ) whic h clearly implies (27). Finally , as a consequence of Theorem 3.24 and Lemma 4.11, we immediately obtain the following result. Theorem 4.12. A ssume that (H1)–(H6), (H7), and (H8) ar e satisﬁe d and let m 0 ∈ P (Ω) . Then Q ∈ P (Γ) is an e quilibrium of the me an ﬁeld game fr om Se ction 2 with initial c ondition m 0 if and only if it is a str ong e quilibrium of the same me an ﬁeld game with the same initial c ondition. 5 Numerical illustration W e no w provide a numerical illustration for the mean ﬁeld game from Section 2. In addition to (H1)–(H6), w e w ork here under the additional assumption that h ( t, x, e x, p, e p ) = 0 if ( x, p ) = ( e x, e p ) , whic h implies that the interaction cost b etw een an agen t and themself is zero. Note that the corresponding N -play er game is also a potential game. More precisely , consider the game with N agents where the aim of agent i ∈ { 1 , . . . , N } starting at the p osition x 0 ,i is to minimize with respect to γ i ∈ Γ the cost F N ( γ i , γ − i ) = L ( γ i ) + 1 N N X j =1 H ( γ i , γ j ) with the constraint γ i (0) = x 0 ,i , where L and H are giv en as in Remark 3.2 and γ − i = ( γ 1 , . . . , γ i − 1 , γ i +1 , . . . , γ N ) ∈ Γ N − 1 represen ts the c hoices of trajectories of other agen ts. Note that, since 23 H ( γ i , γ i ) = 0 , the abov e sum can b e equiv alently tak en ov er j ∈ { 1 , . . . , N } \ { i } . It is easy to v erify that this N -play er game is a p otential game, with p oten tial J N ( γ 1 , . . . , γ N ) = 2 N N X i =1 L ( γ i ) + 1 N 2 N X i =1 N X j =1 H ( γ i , γ j ) . (39) In addition, F N ( γ i , γ − i ) is nothing but the cost F ( γ i , Q ) from (3) with Q = 1 N P N j =1 δ γ j , and J N ( γ 1 , . . . , γ N ) is nothing but J ( Q ) with the same Q and J giv en b y (9c). Numerical approximations of equilibria of this N -pla yer game can be obtained b y minimizing the function J N from (39) with the constrain ts γ i (0) = x 0 ,i for ev ery i ∈ { 1 , . . . , N } . In order to av oid direct minimization ov er the whole space Γ N , one can minimize J N one v ariable at each time through a classical co ordinate descent algorithm (see, e.g., [81] for an o verview of coordinate descen t algorithms in ﬁnite dimension), as describ ed in Algorithm 1. Algorithm 1 Co ordinate descent algorithm for minimizing the function J N from (39) with con- strain ts γ i (0) = x 0 ,i for ev ery i ∈ { 1 , . . . , N } Require: Positiv e in teger N , initial conditions x 0 , 1 , . . . , x 0 ,N in Ω . Initialize γ 1 , . . . , γ N in Γ with γ i (0) = x 0 ,i for ev ery i ∈ { 1 , . . . , N } . rep eat for i ∈ { 1 , . . . , N } do Select a new γ i ∈ arg min ˜ γ i ∈ Γ ˜ γ i (0)= x 0 ,i J N ( γ 1 , . . . , γ i − 1 , ˜ γ i , γ i +1 , . . . , γ N ) end for un til some con vergence criterion is met. Note that, ev en though J N is a sum ov er N + N 2 terms, at eac h iteration of the algorithm, when minimizing for some agen t i , J N ( γ 1 , . . . , γ i − 1 , e γ i , γ i +1 , . . . , γ N ) can b e replaced b y the in- dividual cost F N ( e γ i , γ − i ) , which is a sum of only 1 + N terms, as all other terms appearing in J N ( γ 1 , . . . , γ i − 1 , e γ i , γ i +1 , . . . , γ N ) are indep endent of e γ i . Hence, the co ordinate descen t algorithm coincides with a b est resp onse algorithm from game theory , in which, at each step, one play er c hanges their strategy to their b est resp onse, while all other play ers remain with the same strategy (see, e.g., [70], where it is sho wn that the b est response algorithm conv erges to a Nash equilibrium for potential games with ﬁnitely man y actions for eac h pla yer). F or our n umerical illustration, we hav e applied Algorithm 1 to a slight mo diﬁcation of the mean ﬁeld game from Section 2 in which there are tw o diﬀerent p opulations. More precisely , we consider that there are tw o p opulations evolving in Ω , the distributions of their tra jectories b eing describ ed b y tw o probability measures Q 1 , Q 2 ∈ P (Γ) . The p opulations are identical except for their initial distributions and target sets: for i ∈ { 1 , 2 } , we assume that population i is distributed at time t = 0 according to a measure m i 0 ∈ P (Ω) and that the goal of eac h agent of p opulation i is to reach a given target set Ξ i satisfying (H2). The cost optimized by an agen t of p opulation i is of the form (3), with Ψ( τ ( γ )) replaced b y Ψ( τ i ( γ )) , where τ i ( γ ) is deﬁned as in (4) with Ξ replaced by Ξ i , and the integral of h in (3) is replaced by the sum of tw o in tegrals, one with respect to Q 1 , in which τ ( γ ) ∧ τ ( e γ ) is replaced b y τ i ( γ ) ∧ τ 1 ( e γ ) , and another with respect to Q 2 , in which τ ( γ ) ∧ τ ( e γ ) is replaced b y τ i ( γ ) ∧ τ 2 ( e γ ) . The analysis carried out in Sections 3 and 4 can b e easily adapted to this t wo-population setting. In our illustration, w e consider that agen ts mov e in the square Ω = (0 , 1) × (0 , 1) ⊂ R 2 . Agen ts of p opulation 1 are initially distributed according to the uniform measure m 1 0 in the left side of the square, { 0 } × [0 , 1] , and their goal is to reach the target set Ξ 1 = { 1 } × [0 , 1] . Agents of p opulation 2 are initially distributed according to the uniform measure m 2 0 in the right side of the square, { 1 } × [0 , 1] , and their goal is to reac h the target set Ξ 2 = { 0 } × [0 , 1] . F or the illustration, w e hav e c hosen the functions ℓ and Ψ as ℓ ( t, x, p ) = 1 2 | p | 2 , Ψ( t ) = t. 24 As for the function h , our aim is to select a function suc h that, when computed along t w o tra jec- tories γ and e γ , giv es h ( t, γ ( t ) , e γ ( t ) , ˙ γ ( t ) , ˙ e γ ( t )) = 8 exp  − | γ ( t ) − e γ ( t ) | 2 2 σ 2  max  0 , − d d t | γ ( t ) − e γ ( t ) |  , where σ > 0 , since the ab ov e expression p enalizes tw o agents that are close to eac h other, but only when the distance | γ ( t ) − e γ ( t ) | b etw een them is decreasing. As d d t | γ ( t ) − e γ ( t ) | = ( ˙ γ ( t ) − ˙ e γ ( t )) · γ ( t ) − e γ ( t ) | γ ( t ) − e γ ( t ) | , w e then wish to choose h ( t, x, e x, p, e p ) = 8 exp  − | x − e x | 2 2 σ 2  max  0 , − ( p − e p ) · x − e x | x − e x |  . Ho wev er, the ab ov e expression is not deﬁned for x = e x , and it is not p ossible to provide a con tinuous extension of it. W e then choose a parameter δ > 0 and replace the term − ( p − e p ) · x − e x | x − e x | b y a con vex com bination of itself and the con tinuous expression | p − e p | , with weigh ts that are resp ectively 1 and 0 when | x − e x | ≥ δ , and that ha ve an aﬃne dep endence on | x − e x | when this term is less than δ . Hence, the ﬁnal expression of the function h c hosen for our simulation is h ( t, x, e x, p, e p ) = 8 exp  − | x − e x | 2 2 σ 2  max " 0 , − min  1 , | x − e x | δ  ( p − e p ) · x − e x | x − e x | + max  0 , 1 − | x − e x | δ  | p − e p | # , (40) and w e selected σ = 1 4 and δ = 1 5 . W e ha ve simulated the N -pla yer game with 20 pla yers in eac h p opulation, and the corresponding results are provided in Figure 5.1. F or the simulation, we hav e replaced the space Γ b y C ([ 0 , T ] , Ω) with T = 3 (this c hoice was v alidated a posteriori b y noticing that all agen ts in the simulation reac h their desired target set m uch b efore the ﬁnal time T ). The time interv al [0 , T ] w as discretized in N t = 100 equally spaced p oints and tra jectories where thus represented as elements of Ω N t . The initial distribution m 1 0 w as replaced by a discrete distribution concentrated on 20 equally spaced p oin ts in the segment { 0 } × [0 , 1] , with a similar discretization for m 2 0 . F or a given tra jectory γ of p opulation i represen ted as a sequence of N t p oin ts in Ω , its velocity was computed as the sequence of N t − 1 points obtained by dividing the diﬀerence betw een t wo successive p ositions in the tra- jectory by the time step ∆ t = T N t − 1 , while its exit time τ ( γ ) w as approximated by r T 0 χ ( γ ( t )) d t , where χ is a smo oth function approximating the indicator function 1 Ω \ Ξ i . The nonsmo oth func- tions max(0 , · ) and min(1 , · ) app earing in (40) w ere replaced b y smooth appro ximations, while the integrals from (3) were approximated using the rectangle method. Algorithm 1 was initialized with tra jectories going in a straigh t line with constan t velocity from their initial position to the p oin t in the target set with the same ordinate as the initial p osition, while, for the minimization step in Algorithm 1, we used the function minimize from Python’s scipy.optimize toolb ox [80], alternating betw een Nelder–Mead and L-BFGS-B methods for optimization. W e observe, in Figure 5.1, a behavior close to the exp ected 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 in teraction term represen ted b y h by noticing that agents seem to anticipate the arriv al of agen ts in the opp osite direction, lea ving them some space to pass in order to a void a large congestion term due to h . References [1] Y. A c hdou, M. Bardi, and M. Ciran t. Mean ﬁeld games models of segregation. Math. Mo dels Metho ds A ppl. 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A variational mean field game of controls with free final time and pairwise interactions

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