Generic Properties of First Order Mean Field Games Alberto Bressanand Khai T. Nguyen

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Generic Properties of First Order

Mean Field Games

Alberto Bressan(∗)and Khai T. Nguyen(∗∗)

(∗)Department of Mathematics, Penn State University,

(∗∗)Department of Mathematics, North Carolina State University.

E-mails: axb62@psu.edu, khai@math.ncsu.edu.

October 27, 2022

Abstract

We consider a class of deterministic mean ﬁeld games, where the state associated with

each player evolves according to an ODE which is linear w.r.t. the control. Existence,

uniqueness, and stability of solutions are studied from the point of view of generic theory.

Within a suitable topological space of dynamics and cost functionals, we prove that, for

“nearly all” mean ﬁeld games (in the Baire category sense) the best reply map is single

valued for a.e. player. As a consequence, the mean ﬁeld game admits a strong (not ran-

domized) solution. Examples are given of open sets of games admitting a single solution,

and other open sets admitting multiple solutions. Further examples show the existence

of an open set of MFG having a unique solution which is asymptotically stable w.r.t. the

best reply map, and another open set of MFG having a unique solution which is unstable.

We conclude with an example of a MFG with terminal constraints which does not have

any solution, not even in the mild sense with randomized strategies.

1 Introduction

This paper deals with a class of mean ﬁeld games with a continuum of players, where the state

associated with each player evolves according to a controlled ODE. We study the existence,

uniqueness, and stability of solutions from the point of view of generic theory. Namely, we

seek properties of solutions that are satisﬁed either on some open set of MFG, or for “nearly

all” MFG in the topological sense [12, 20]; i.e., for all MFG in the intersection of countably

many open dense sets.

Let (Ω,B, µ) be a probability space. More precisely, we assume that Ω is a metric space

with Borel σ-algebra B, while µis an atomless probability measure on Ω. Without loss of

generality, throughout the following we assume Ω = [0,1] with Lebesgue measure. We regard

ξ∈Ω as a Lagrangian variable, labelling one particular player. Accordingly, we shall denote

by t7→ x(t, ξ) a trajectory for player ξ. By selecting one trajectory x(·, ξ)∈ C[0, T ]; IRnfor

arXiv:2210.14643v1 [math.OC] 26 Oct 2022

each player (depending measurably on ξ), one obtains an element Xin the space

L1Ω ; C[0, T ]; IRn.(1.1)

The space (1.1) is naturally endowed with the Banach norm

kXk.

=ZΩ sup

t∈[0,T ]x(t, ξ)!dξ. (1.2)

To deﬁne a (deterministic) mean ﬁeld game, for each player ξ∈Ω we consider an optimal

control problem where the dynamics and the cost functions also depend on the cumulative

distribution Xof all other players. To express this dependence, we consider a ﬁnite number

of smooth scalar functions φ1, . . . , φN∈ C2[0, T ]×IRn, and deﬁne η(t) = (η1, . . . , ηN)(t) to

be the vector of “moments”

ηi(t) = ZΩ

φit, x(t, ξ)dξ, i = 1, . . . , N. (1.3)

The control problem for player ξtakes the form

minimize: ZT

Lt, x(t), u(t), η(t)dt +ψx(T),(1.4)

subject to the dynamics

˙x(t) = ft, x(t), u(t), η(t)t∈[0, T ],(1.5)

and with initial datum

x(ξ, 0) = ¯x(ξ).(1.6)

Deﬁnition 1.1 In the above setting, by a strong solution to the mean ﬁeld game we mean a

family of control functions t7→ u(t, ξ)∈IRmand corresponding trajectories t7→ x(t, ξ)∈IRn,

deﬁned for ξ∈Ωand t∈[0, T ], such that the following holds.

For a.e. ξ∈Ω, the control u(·, ξ)and the trajectory x(·, ξ)provide an optimal solution to the

optimal control problem (1.4)–(1.6) for player ξ, where η(t) = (η1, . . . , ηN)(t)is the vector of

moments deﬁned at (1.3).

A mean ﬁeld game thus yields a (possibly multivalued) map η7→ Φ(η) from C([0, T ]; IRN)

into itself. Namely, given η(·), for each ξ∈Ω consider an optimal trajectory xη(·, ξ) of the

corresponding optimal control problem (1.4)–(1.6). We then set

Φ(η).

=eη= (eη1,...,eηN),eηi(t).

=ZΩ

φit, xη(t, ξ)dξ, (1.7)

under suitable assumptions that will ensure that the integral in (1.7) is well deﬁned. By

deﬁnition, a ﬁxed point of this composed map

η(·)7→ xη(·, ξ) ; ξ∈Ω7→ eη= Φ(η)

[moments] 7→ [optimal trajectories] 7→ [moments] (1.8)

yields a strong solution to the mean ﬁeld game.

Remark 1.1 In general, the map Φ can be multivalued. Indeed, for some η(·), there can be

a subset V⊆Ω with positive measure, such that each player ξ∈Vhas two or more optimal

trajectories. For this reason, a mean ﬁeld game may not have a solution in the strong sense

considered in Deﬁnition 1.1. In order to achieve a general existence theorem one needs to relax

the concept of solution, allowing the possibility of randomized strategies [1, 6, 9]. This leads to

the problem of ﬁnding a ﬁxed point of an upper semicontinuous convex-valued multifunction,

which exists by Kakutani’s theorem [10, 17].

Following the standard literature on ﬁxed points of continuous or multivalued maps, we in-

troduce

Deﬁnition 1.2 A solution x=x∗(t, ξ)to the above mean ﬁeld game is stable if the corre-

sponding function η∗∈ C0[0, T ]; IRNat (1.3) is a stable ﬁxed point of the multifunction Φ

at (1.7). Namely, for every ε > 0there exists δ > 0such that the following holds. For every

sequence η(k)k≥0such that

kη(0) −η∗kC0< δ, η(k)∈Φη(k−1)for all k≥1,(1.9)

one has kη(k)−η∗kC0< ε for all k≥1.

If, in addition, every such sequence η(k)converges to η∗, then we say that the solution is

asymptotically stable.

If the solution is not stable, we say that it is unstable.

Next, we say that a solution of the mean ﬁeld game is structurally stable if it persists under

small perturbations of the dynamics and the cost functionals. More precisely:

Deﬁnition 1.3 We say that a solution x=x(t, ξ)to the above mean ﬁeld game (1.3–(1.6) is

structurally stable (or equivalently: essential) if, given ε > 0, there exists δ > 0such that

the following holds. For any perturbations (f†, L†, ψ†, φ†,¯x†)satisfying

max nkf†−fkC2,kL†−LkC2,kψ†−ψkC2,kφ†−φkC2o< δ, k¯x†−¯xkL∞< δ, (1.10)

the corresponding perturbed game has a solution x†=x†(t, ξ)such that

sup

t∈[0,T ]ZΩx†(t, ξ)−x(t, ξ)dξ < ε. (1.11)

Throughout the following, we shall assume that the dynamics is aﬃne w.r.t. the control vari-

able:

f(x, u, η) = f0(x, η) +

i=1

fi(x, η)ui,(1.12)

and all functions f, ψ, L have at least C2regularity.

Since our MFG at (1.3)–(1.6) is characterized by the 5-tuple of functions (f, L, ψ, φ, ¯x), we are

interested in properties which are satisﬁed either (i) for all games where (f, L, ψ, φ, ¯x) ranges

inside an open set (in a suitable Banach space), or (ii) for generic games, i.e., for all games

where (f, L, ψ, φ, ¯x) ranges over the intersection of countably many open dense sets. Roughly

speaking, the main results of the paper can be summarized as follows.

(i) Given a triple (f, L, φ)∈ C3×C3×C3, for a generic pair (ψ, ¯x)∈ C3×L∞, the best reply

map η7→ Φ(η)in (1.8) is single valued. As a consequence, the MFG (1.3)–(1.6) admits

a strong solution.

(ii) There is an open set of mean ﬁeld games with a unique solution, which is stable and

essential.

(iii) There is an open set of mean ﬁeld games with a unique solution, which is unstable, and

essential.

(iv) There is an open set of mean ﬁeld games with two solutions, both essential.

More precise statements of these results will be given in the following sections. The remainder

of the paper is organized as follows.

As a warm-up, in Section 2 we review the basic tools for proving generic properties. Here we

consider a family of optimal control problems where the dynamics is linear w.r.t. the control

functions. We show that, for generic dynamics f, running cost Land terminal cost ψ, for

a.e. initial datum x(0) = xthe optimal control is unique.

Section 3 provides a simple way to construct mean ﬁeld games with multiple solutions. Given

an optimal control problem and a pair (x∗, u∗) (not necessarily optimal) which satisﬁes the

Pontryagin necessary conditions, we show the existence of a mean ﬁeld game where u∗is the

optimal control for every player. As a consequence, for any control problem where the Pon-

tryagin equations have multiple solutions, one can construct a MFG with multiple solutions.

Under generic assumptions, all of these solutions are structurally stable.

Section 4 contains the main result of the paper. Namely, for a generic MFG of the form

(1.3)–(1.6), the best reply map η7→ Φ(η) is single valued. Hence the MFG admits a strong

solution. Here the analysis is far more delicate than in the proof of the generic uniqueness for

the optimal control problem in Section 2. Indeed, we need to show that the statement

•The set of initial points x, for which the problem (1.4)–(1.6) has multiple solutions, has

measure zero

is true not just for one function η(·), but simultaneously for all functions η= (η1, . . . , ηN), in

a suitable domain.

Finally, Section 5 collects a variety of examples, where the MFG have multiple strong solutions,

Some of these are stable, in the sense of Deﬁnition 1.2, while others are unstable.

We conclude with two examples of MFG without solution. The ﬁrst one is a well known case

where nonexistence is due to the fact that the best reply of each player is not unique. No

strong solution exists, but one can construct a mild solution where each player adopts a

randomized strategy. In the second example, the presence of a terminal constraint lacking a

transversality condition prevents the existence of any solution, even in the mild (randomized)

sense.

Some concluding remarks, pointing to future research directions, are given in Section 6.

Mean ﬁeld games with stochastic dynamics have been introduced by Lasry and Lions [18] and

by Huang, Malham´e and Caine [16], to model the behavior of a large number of interacting

agents. Their solution leads to a well known system of forward-backward parabolic equations.

Solutions to ﬁrst order MFG (with deterministic dynamics) can be obtained as a vanishing

viscosity limit of these parabolic PDEs, i.e., as viscosity solutions to a corresponding Hamilton-

Jacobi equation [6, 7, 8, 9]. Equivalently, one can take a Lagrangian approach, describing the

optimal control and the optimal trajectory of each single agent. This is the approach followed

in the present paper. Some examples of MFG with unique or with multiple solutions can be

found in [1]. A concept of structural stability for solutions to ﬁrst order MFG was proposed

in [5].

2 Generic uniqueness for optimal control problems

Consider an optimal control problem of the form

minimize: J[u].

=ZT

Lx(t), u(t)dt +ψx(T),(2.1)

with dynamics which is aﬃne in the control:

˙x(t) = fx(t), u(t)=f0(x(t)) +

i=1

fi(x(t)) ui(t), x(0) = ¯x. (2.2)

Here u(t)∈IRmwhile x(t)∈IRn. To ﬁx ideas, we shall consider the couple (f, L) satisfying

the following assumptions.

(A1) The functions fi:IRn7→ IRn,i= 0, . . . , m, are twice continuously diﬀerentiable. More-

over the vector ﬁelds fisatisfy the sublinear growth condition

fi(x)≤c1|x|+ 1(2.3)

for some constant c1>0and all x∈IRn.

(A2) The running cost L:IRn×IRm7→ IR is twice continuously diﬀerentiable and satisﬁes

(L(x, u)≥c2|u|2−1,

|Lx(x, u)| ≤ `(|x|)·(1 + |u|2),(2.4)

for some constant c2>0and some continuous function `. Moreover, Lis uniformly

convex w.r.t. u. Namely, for some δL>0, the m×mmatrix of second derivatives

w.r.t. usatisﬁes

Luu(x, u)> δL·Imfor all x, u. (2.5)

Here Imdenotes the m×midentity matrix

Throughout the following, the open ball centered at the origin with radius ris denoted by Br=

B(0, r), while Brdenotes its closure. Under the previous assumptions, optimal controls and

optimal trajectories of the optimization problem (2.1)-(2.2) satisfy uniform a priori bounds:

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GenericPropertiesofFirstOrderMeanFieldGamesAlbertoBressan()andKhaiT.Nguyen()()DepartmentofMathematics,PennStateUniversity,()DepartmentofMathematics,NorthCarolinaStateUniversity.E-mails:axb62@psu.edu,khai@math.ncsu.edu.October27,2022AbstractWeconsideraclassofdeterministicmeaneldgames,wherethes...

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