Mean Field Games

TL;DR: In this paper, the authors present three examples of the mean-field approach to modelling in economics and finance (or other related subjects) and show that these nonlinear problems are essentially well-posed problems with unique solutions.

Abstract: We survey here some recent studies concerning what we call mean-field models by analogy with Statistical Mechanics and Physics. More precisely, we present three examples of our mean-field approach to modelling in Economics and Finance (or other related subjects...). Roughly speaking, we are concerned with situations that involve a very large number of “rational players” with a limited information (or visibility) on the “game”. Each player chooses his optimal strategy in view of the global (or macroscopic) informations that are available to him and that result from the actions of all players. In the three examples we mention here, we derive a mean-field problem which consists in nonlinear differential equations. These equations are of a new type and our main goal here is to study them and establish their links with various fields of Analysis. We show in particular that these nonlinear problems are essentially well-posed problems i.e., have unique solutions. In addition, we give various limiting cases, examples and possible extensions. And we mention many open problems.

Summary

1.1 General introduction

• The authors present here some recent modelling issues arising in Economics and Finance which lead to new classes of nonlinear equations that they also briefly analyse here.
• And the authors analyse mathematically the limit equations.
• And the authors indicate various directions that can be (or need to be) investigated together with several open problems.
• In the one-dimensional case, the authors state and solve this problem showing the existence and uniqueness of a smooth solution.

1.2 Mathematical models

• The authors discuss here the mathematical structure of the three examples mentioned in the previous section.
• In general, the coupling between the first and the second equation in both systems (with the additional feature in the case of (2) of the equation in u written forward in time while the equation for m is backward in time) make these systems novel ones for which no existing theory or approach seems to be applicable directly.
• The second example concerns the formation of prices.
• And in (5) “σ(Φ).” simply corresponds to the multiplication operator.

1.3 The economical and financial context

• In many economical and financial situations, it is natural to consider a very large number of rational agents which have limited information.
• Here, the word rational is taken from the theory of rational anticipations and, roughly speaking, means that each agent tries to maximize his strategy (utility maximization).
• Section 3 is devoted to a toy model for price formation in an idealized situation where two populations of, respectively, vendors and buyers of a single good typically agree to a certain price at which some transactions take place.
• A fundamental contribution to this issue has been given by R. Aumann [1], and, since then, many works have investigated it (see the recent work by G. Carmona [9] and the references therein).
• The consistency of this equilibrium, that the authors call a “mean field equilibrium”, in the sense of rational anticipations is insured by the fact that the dynamics of the density of players results from the individual optimal strategies.

2.3 Stationary problems : mathematical analysis

• This is why the authors begin an analysis with a general uniqueness that they state, for the sake of simplicity, for smooth solutions of (22)-(24).
• In that case, the authors use the classical uniqueness results for ergodic Hamilton-Jacobi-Bellman equations to deduce that v1 ≡ v2.
• In order to restrict the length of this article which only aims at a survey of the problems, the authors shall not pursue here the discussion in such a technical direction which however is very interesting from a purely mathematical stand-point.
• Let us however mention that in strong anti-monotone situations, i.e. c < 0, β large in the above example, existence may not hold for arbitrary data.

2.4 Dynamical problems : mathematical analysis

• The authors now present the analogue of the mean-field games equations in the context of finite horizon control problems.
• The authors have been able to make this type of arguments rigorous only in very particular situations.
• This is only the authors shall first present a very general uniqueness result and then give some samples of existence results of various types (smooth or weak solutions).
• Another natural case concerns local operators V and V0 i.e. operators given by (35).
• All the remarks made in the previous section on stationary can be adapted to (32)-(34).

2.5 Deterministic limits

• This amounts to let the “noise” disappear from the player’s dynamics.
• And the authors begin with the stationary problem.
• And the authors assume that H satisfies (42) (for example).
• In other words, the authors recover the classical compressible Euler equations of Fluid Mechanics in the so-called barotropic and potential regime (see P-L. Lions [23] for more details . . . ).
• Once more, the authors see that the conditions on the monotonicity of F and F0 are natural ones.

2.6 Links with optimal control

• And the latter condition is precisely equivalent to the fact that V and V0 are monotone operators !.
• The authors also wish to mention, without any further explanation, that, in the convex case, these optimal control interpretations allow to give a notion of Pareto’s optimality to the mean-field equilibria they introduced and studied.
• It is also possible to adapt the optimal control problems above to the stationary meanfield equations.
• And in the example mentioned above, the optimal control problem takes the following form inf β {Φ(β)− λ} whee λ is the first eigenvalue of the Schrödinger operator −ν∆ + (f0 + β).

2.7 Variants and extensions

• The authors already mentioned in the previous sections many possible variants and extensions corresponding to more general dynamics, more complex interactions between the players and more complex criteria to minimize for each player.
• The authors also mentioned the possibility of incorporating in the equations source terms or even additional differential operators that may correspond to the “death and birth” of players, or to drift-diffusion phenomena for the density of players.
• One can also consider situations where the equation for v is an obstacle problem in which case, at least formally, the equation for m is naturally set on the zone where the solution does not coincide with the obstacle.
• Even in the cases the authors mentioned and studied, much remains to be done as far as existence, uniqueness and regularity are concerned.
• And there are fundamental open problems in the derivation of these mean-field equations (i.e. the rigorous treatment of the limit as N goes to +∞).

3.1 The model

• The authors consider an idealized population of players (which however somehow reflects the nature or microstructure of financial markets) consisting of two groups namely one group of buyers of a certain good and one group of vendors of the same good.
• Postulating some exogenous randomness in price preferences, the authors describe this population by two densities fB, fV i.e. nonnegative functions of (x, t) where t stands for time and x stands for a possible value of the price (roughly speaking fB(x, t) represents the number of potential buyers at a price x at time t).
• In the next two sections, the authors shall review some known results on the system (64)-(66).
• Let us mention that there are again many possible (and relevant) directions of research concerning the derivation of this model from Nash points and utility maximization, extensions to more general dynamics, to situations with several possible goods or where transactions may involve more than a unit quantity of the good . . .
• It is quite clear that much remains to be done both from a modelling and from a mathematical standpoint.the authors.

3.2 Main results

• The authors first perform a reduction of the system (64)-(66) which is not really necessary for their argument but allows to simplify the presentation.
• And the authors assume that f0 is a smooth function on R with fast decay at infinity (in all that follows, fast decay means that, for example, on can bound the function by C 1+|x|2 for some positive constant C).
• Then, their main result is the following Theorem 3.1 :.
• Under the above conditions, there exists a unique smooth solution (f, p) of (3) such that f has fast decay for all t ≥ 0.
• Here, the authors take the classical Dirac mass δ to make the explicit computations below as simple as possible.

4.1 The model

• For a detailed presentation of the financial background, the authors refer the reader to J-M. Lasry and P-L. Lions [17, 18].
• More precisely, let us consider for example pay-off functions in X = C2,αb (R) for some α ∈ (0, 1) fixed (where C2,αb means the space of bounded C 2,α functions with bounded first and second derivatives).

Full text

Mean Field Games
Jean-Michel LASRY
(∗)1
Pierre-Louis LIONS
(∗)2
(∗)
work partially supported by the chair “Finance and sustainable development”
Abstract : We survey here some recent studies concerning what we call mean-field models by
analogy with Statistical Mechanics and Physics. More precisely, we present three examples of
our mean-field approach to modelling in Economics and Finance (or other related subjects. . . ).
Roughly speaking, we are concerned with situations that involve a very large number of
“rational players” with a limited information (or visibility) on the “game”. Each player chooses
his optimal strategy in view of the global (or macroscopic) informations that are available to
him and that result from the actions of all players. In the three examples we mention here, we
derive a mean-field problem which consists in nonlinear differential equations. These equations
are of a new type and our main goal here is to study them and establish their links with
various fields of Analysis. We show in particular that these nonlinear problems are essentially
well-posed problems i.e. have unique solutions. In addition, we give various limiting cases,
examples and possible extensions. And we mention many open problems.
1
Institut de Finance, Universit´e Paris-Dauphine, Place du Mar´echal de Lattre de Tassigny, 75775 Paris
Cedex 16
2
Coll`ege de France, 3 rue d’Ulm, 75005 Paris and Ceremade - UMR CNRS 7549, Universit´e Paris-Dauphine,
Place du Mar´echal de Lattre de Tassigny, 75775 Paris Cedex 16

Contents
1 Introduction 3
1.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 The economical and financial context . . . . . . . . . . . . . . . . . . . . . . . 5
2 Mean Field Games 6
2.1 Stationary problems : Nash points or N players . . . . . . . . . . . . . . . . . 6
2.2 Stationary problems : N → +∞ . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Stationary problems : mathematical analysis . . . . . . . . . . . . . . . . . . . 12
2.4 Dynamical problems : mathematical analysis . . . . . . . . . . . . . . . . . . . 13
2.5 Deterministic limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Links with optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7 Variants and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Price formation and dynamic equilibria 21
3.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Stationary problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Formation of volatility 25
4.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Local well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Global solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References 28
2

1 Introduction
1.1 General introduction
We present here some recent modelling issues arising in Economics and Finance which lead
to new classes of nonlinear equations that we also briefly analyse here.
In a recent series of papers (J-M. Lasry and P-L. Lions [16, 17, 18, 19, 20, 21]), we introduce
a general mathematical mo de lling approach to situations which involve a great number of
“agents”. Roughly speaking, we derive these models from a “continuum limit” (in other words
letting the number of agents go to infinity) which is somewhat reminiscent of the classical
mean field approaches in Statistical Mechanics and Physics (as for instance, the derivation of
Boltzmann or Vlasov equations in the kinetic theory of gases) or in Quantum Mechanics and
Quantum Chemistry (density functional models, Hartree or Hartree-Fock type models. . . ).
This general approach leads in various situations to new nonlinear equations which contain
as particular examples many classical problems and are linked to several research fields of
Analysis. We describe rapidly these equations in the next section. And we conclude this
introduction by a brief overview of the economical and/or financial issues that we address
through our “mean-field” approach.
We consider here three different illustrations of such an approach that are treated in sections
2-4. In section 2, we consider stochastic differential games and N players Nash points. Then,
we derive rigorously the mean field limit equations as N goes to infinity (in a stationary
setting). And we analyse mathematically the limit e quations. We also consider time-dependent
problems and deterministic limits. We next give an interpretation of such systems of equations
in term of the optimal control of (some) partial differential equations. And we indicate various
directions that can be (or need to be) investigated together with several open problems.
Section 3 is devoted to the second example which leads to a new class of free boundary
problems. In the one-dimensional case, we state and solve this problem showing the existence
and uniqueness of a smooth solution. We next discuss briefly and explicitly some stationary
problems. And we mention various directions of interest.
The third example concerns the formation of volatility in financial markets. I n this context,
our mean field approach leads to a nonlinear differential equation in infinite dimensions. And
we show that i) its local solvability is induced by a striking property of solutions of parabolic
partial differential equations, ii) the model possesses remarkable invariance properties, iii)
which allow us to solve the equation globally in a semi-explicit way.
1.2 Mathematical models
We discuss here the mathematical s tructure of the three examples mentioned in the previous
section. We begin with the example which will be mostly detailed here namely “mean-field
differential games”. A typical example of the models we derive is given by the following
system of equations
3














−ν∆u + H(x, ∇u ) + λ = V (x, m)
−ν∆m − div(
∂H
∂p
(x, ∇u)m = 0
m > 0,
R
mdx = 1
(1)
where ν > 0, u is a scalar function, H(x, p) is a given function (or nonlinearity) which is
assumed to be convex in p, λ ∈ R is unknown, and V (x, r) is another given function (or
nonlinearity). Precise assumptions on H and V are given in section 2 below. Various types
of boundary conditions are possible and we mention here the simplest possible case where the
equations in (1) are set in R
d
, H(x, p), V (x, r) and the unknowns u and m are required to be
periodic in x
i
(of a given period T
i
> 0) for each 1 ≤ i ≤ d(x ∈ R
d
), and
R
m dx obviously
means
R
Q
m dx where Q =
d
Y
i=1
(0, T
i
).
The above problem corresponds to stationary situations. The time-dependent analogue is
typically of the following form (where t ∈ [0, T ])



















∂u
∂t
− ∆u + H(x, ∇u) = V (x, m), u|
t=0
= V
0
(x, m(x, 0))
∂m
∂t
+ ∆m + div(
∂H
∂p
(x, ∇u)m) = 0, m|
t=T
= m
0
m > 0,
Z
m dx = 1 for all t ∈ [0, T ].
(2)
The nonlinearities H and V are as above, m
0
is a given “initial” condition while V
0
(x, r)
is a given function (or nonlinearity).
If m were not present in the first equations of (1) and (2) (assume for instance that V and
u
0
only depend on x), these equations would simply be a general class of Hamilton-Jacobi-
Bellman equations arising in stochastic control theory (see for instance W.H. Fleming and H.
M. Soner [12], M. Bardi and I. Capuzzo-Dolcetta [5] and the references therein . . . ). And
the equations would then be seen as the linearized problem, backward in time in the case
of (2). In general, the coupling between the first and the second equation in both systems
(with the additional fe ature in the case of (2) of the equation in u written forward in time
while the equation for m is backward in time) make these systems novel ones for which no
existing theory or approach seems to be applicable directly. In addition, (1) and (2) contain
several particular cases of interest such as the Hartree equations in Quantum Mechanics, or
the compressible Euler equations (in the barotropic or isentropic regime) when we let ν go to
0+.
For these novel systems, we prove general existence results together with uniqueness results
at least when V (and V
0
) are nondecreasing in r (for all x). We also investigate the limit as ν
goes to 0
+
(deterministic limit) and show the links with optimal control problems (in which
case u – or m– may be interpreted as the primal state while m –or u– is then the dual state).
The second example concerns the formation of prices. Again, we indicate here a typical
example of the models we derive namely
4
















∂f
∂t
−
σ
2
2
∂
2
f
∂x
2
= −
σ
2
2
∂f
∂x
(p(t), t)

δ(x − p(t) + a) − δ(x − p(t) − a)

f(x, t) > 0 if x < p(t), t ≥ 0; f(x, t) < 0 if x > p(t), t ≥ 0;
f|
t=0
= f
0
on R, p(0) = p
0
(3)
where σ > 0,f
0
is a given smooth function with fast decay at infinity, p
0
∈ R and the following
compatibility condition holds
f
0
(x) > 0 if x < p
0
, f
0
(x) < 0 if x > p
0
.(4)
Finally, a > 0 is given and δ is a “delta-like” function i.e. a smooth non negative function
compactly supported in (−a, +a) such that
R
δ = 1. The problem (3) is clearly a free boundary
problem (note that f (p(t), t) = 0 for all t ≥ 0) which appears to be new. Our main results
state the existence and uniqueness of a smooth solution (u, p) (with fast decay at infinity).
The final example we consider concerns the formation of volatility in financial markets.
Postulating a simple linear elastic law for the impact of trading on stock prices, our mean field
approach leads to the following nonlinear differential equation in an infinite dimensional space
(which can be taken to be, for example, C
2,α
b
(R) i.e. the space of bounded C
2
functions with
bounded, H¨older continuous of exponent α ∈ (0, 1), first and second derivatives) : we look for
a mapping σ from C
2,α
b
(R) into C
0,α
b
(R × [0, T ]) —the space of bounded, H¨older continuous
of exponent α ∈ (0, 1), functions on R × [0, T ], where T > 0 is fixed— which satisfies
σ
0
(Φ) = kσ(Φ) . Γ for all Φ ∈ C
2,α
b
(R)(5)
where k > 0, Γ is the operator defined on C
2,α
b
by ΓΨ =
∂
2
u
∂x
2
and u solves the following parabolic
equation (written backward in time).
∂u
∂t
+
σ
2
2
∂
2
u
∂x
2
= 0 on R × (0, T ), u|
t=T
= Ψ .(6)
Of course, σ in (6) stands for the function of x and t given by σ(Φ). And in (5) “σ(Φ).”
simply c orresponds to the multiplication operator.
Such nonlinear differential equations as (5) cannot be solved in general unless some very
specific compatibility condition (related to the symmetry of second differentials) is satisfied.
And we show that this “symmetry” condition is indeed satisfied in our model thanks to a
remarkable property of solutions of parabolic equations. This insures the fact that (5) is
locally well-posed (in a maximal neighborhood of any Φ
0
, provided we specify σ(Φ
0
)). We
also show a general invariance property which allows us to prove that (5) is globally well-
posed and to construct its solutions in a semi-explicit way (via the solution of some nonlinear
parabolic equation...).
1.3 The economical and financial context
In many economical and financial situations, it is natural to consider a very large number
of rational agents which have limited information. Here, the word rational is taken from
the theory of rational anticipations and, roughly speaking, means that each agent tries to
5

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Mean field games" ?

More precisely, the authors present three examples of their mean-field approach to modelling in Economics and Finance ( or other related subjects. These equations are of a new type and their main goal here is to study them and establish their links with various fields of Analysis. The authors show in particular that these nonlinear problems are essentially well-posed problems i. e. have unique solutions. 

Q2. What is the possibility of incorporating in the equations source terms or even additional differential operators?

The authors also mentioned the possibility of incorporating in the equations source terms or even additional differential operators that may correspond to the “death and birth” of players, or to drift-diffusion phenomena for the densityof players. 

Q3. What is the first interpretation of the optimal control problem?

The first interpretation in term of optimal control corresponds to the optimal control of the following (backward) Fokker-Planck equation∂m∂t + ν∆m + div(αm) = 0 in Q× (0, T ), m |t=T = m0 in Q(56)where α = α(x, t) is a (distributed) control. 

Q4. What is the advantage of dealing with a continuum of players?

Another advantage of dealing with a continuum of players is the possibility of modelling easily the “birth and death” of players (in which case the total number of players may vary . . . ) through “source” terms in the equation form. 

Q5. what is the uniqueness of the solution of (32)-(34)?

The authors assume that F, F0, H are continuous and satisfy for all their arguments the following conditions for some contents a > 1, b > 1, q > 1, δ > 0, C ≥ 0F (x, , λ)λ ≥ δ|F (x, λ)|a − C,(43)F0(x, , λ)λ ≥ δ|F0(x, λ)|b − C,(44) δ|p|q − C ≤ H(x, p) ≤ C|p|q + C∂H ∂p (x, p).p ≥ qH − C, | ∂H ∂p (x, p)| ≤ C|p|q−1 + C. (45)Theorem 2.7 : Under the above conditions, there exists a solution of (32)-(34) such that v ∈ Lq(0, T ; W 1,q(Q)), v is bounded from below, m|∇v|q ∈ L1(0, T ; L1(Q)), v ∈ C([0, T ]; Lr(Q))where r = min(b, ad d−2(a−1)) if a < 1 + d 2 , r = b if a ≥ 1 + d 2 and m ∈ C([0, T ]; L1(Q)). 

Q6. What is the case of operators V, V0 that are non-negative L1 functions?

And the authors first consider the case of operators V, V0 that are smoothing operators namely map non-negative L1 functions such that ∫ Q mdx = 1 into the set of Lipschitz functions in x such that D2v(x) ≤ 

Q7. What is the simplest way to determine the volatility of a fixed interval?

if one assumes that the volatility in fact depends upon a “macroscopic” pay-off (some kind of cumulative pay-off), and that there exist a large number of players which all have a small impact on the volatility as soon as they trade an option, one is led to a mean field nonlinear differential equation for the volatility seen as a functional on a space of pay-offs. 

Q8. What is the simplest way to deduce that vNi is independent of i?

the authors consider v̄Ni (x i, t) = ∫ QN−1 v N i (x 1, . . . , xN , t) ∏ j 6=i m(xj, t)dxj which by symmetry should be essentially (asymptotically) independent of i. 

Q9. What is the general class of problems that the authors introduced?

One general class of problems the authors thus introduced (although most of the extensions mentioned above are not even contained in that class) is given by∂v∂t − F (x, t, v,Dxv, D2xv; m) = 0 in Q× (0, T ), v |t=0= V0[m],(60)∂m∂t + D2x(∂F ∂A m) +