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ON MEAN FIELD GAMES

Pierre-Louis LIONS

College de France, Paris

(joint project with Jean-Michel LASRY)

Mathematical and Algorithmic Sciences LabFrance Research Center

Huawei Technologies

Boulogne-Billancourt, March 27, 2018

Pierre-Louis LIONS ON MEAN FIELD GAMES

I INTRODUCTION

II A REALLY SIMPLE EXAMPLE

III GENERAL STRUCTURE

IV THREE PARTICULAR CASES

V OVERVIEW AND PERSPECTIVES

VI MEANINGFUL DATA


I. INTRODUCTION

• New class of models for the average (Mean Field) behavior of“small” agents (Games) started in the early 2000’s by J-M. Lasryand P-L. Lions.

• Requires new mathematical theories.

• Numerous applications: economics, finance, social networks,crowd motions. . .

• Independent introduction of a particular class of MFG models byM. Huang, P.E. Caines and R.P. Malhame in 2006.

• A research community in expansion: mathematics, economics,finance. Economics: anonymous games, Krusell Smith!, jointprojects with Ph. Aghion, J. Scheinkman, B. Moll, P-N. Giraud. . .

• Some written references but most of the existing mathematicalmaterial to be found in the College de France videotapes (4 ×18h) that can be downloaded. . . !


• Combination of Mean Field theories (classical in Physics andMechanics) and the notion of Nash equilibria in Games theory.

• Nash equilibria for continua of “small” players: a singleheterogeneous group of players (adaptations to several groups. . . ).

• Interpretation in particular cases (but already general enough!)like process control of McKean-Vlasov. . .

• Each generic player is “rational” i.e. tries to optimize (control) acriterion that depends on the others (the whole group) and theoptimal decision affects the behavior of the group (however, thisinterpretation is limited to some particular situations. . . ).

• Huge class of models: agents → particles, no dep. on the groupare two extreme particular cases.


II. A REALLY SIMPLE EXAMPLE

• Simple example, not new but gives an idea of the general class ofmodels (other “simple” exs later on).

• E metric space, N players (1 6 i 6 N) choose a position xi ∈ Eaccording to a criterion Fi (X ) where X = (x1, . . . , xN) ∈ EN .

• Nash equilibrium: X = (x1, . . . , xN) if for all 1 6 i 6 N xi minover E of Fi (x1, . . . , xi−1, xi , xi+1, . . . xN).

• Usual difficulties with the notion

• N →∞ ? simpler ?

• Indistinguishable players:

Fi (X ) = F (xi , xjj 6=i ),F sym . in (xj)j 6=i


• Part of the mathematical theories is about N →∞:

Fi = F (x ,m) x ∈ E , m ∈ P(E )

where x = xi , m =1

N − 1

∑j 6=i

δxj

• “Thm”: Nash equilibria converge, as N →∞, to solutions of

(MFG) ∀x ∈ Supp m,F (x ,m) = infy∈E

F (y ,m)

• Facts: i) general existence and stability results

ii) uniqueness if (m→ F (•,m)) monotone

iii) If F = Φ′(m), then (minP(E)

Φ) yields one solution of MFG.


Example: E = Rd ,Fi (X ) = f (xi ) + g

(#j/|xi − xj | < ε

(N − 1)|Bε|

)g ↑ aversion crowds, g ↓ like crowds

F (x ,m) = f (x) + g(m ∗ 1Bε(x)(|Bε|−1)

ε→ 0 F (x ,m) = f (x) + g(m(x))

(MFG) supp m ⊂ Arg min

(f (x) + g(m(x))

)– g ↑ uniqueness, g ↓ non uniqueness

min

∫fm +

∫G (m)/m ∈ P(E )

, G =

∫ Z

0f (s)ds

– explicit solution if g ↑: m = g−1(λ− f ), λ ∈ R s.t.

∫m = 1


III. GENERAL STRUCTURE

• Particular case: dynamical problem, horizon T , continuous timeand space, Brownian noises (both indep. and common), nointertemporal preference rate, control on drifts (Hamiltonian H),criterion dep. only on m

• U(x ,m, t) (x ∈ Rd ,m ∈ P(Rd) or M+(Rd), t ∈ [0,T ] andH(x , p,m) (convex in p ∈ Rd)

• MFG master equation∂U∂t − (ν + α)∆xU + H(x ,∇xU,m)+

+〈 ∂U∂m ,−(ν + α)∆m + div (∂H∂p m)〉+

−α ∂U∂m2 (∇m,∇m) + 2α〈 ∂

∂m∇xU,∇m〉 = 0

and U |t=0= U0(x ,m) (final cost)

• ν amount of ind. rand. , α amount of common rand.


• ∞ d problem !

• If ν = 0 (ind): Nash N special case

using x = xi ,m = 1N−1

∑j 6=i

δxj

• Aggregation/decentralization: IF H(x , p,m) = H(x , p) + F ′(m)and U0 = Φ′0(m), then U = ∂Φ

∂m solves MFG if Φ solves HJB onP(E ) for the optimal control of a SPDE

• Particular case: many extensions and variants . . .


IV. THREE PARTICULAR CASES

• ∞ d problem in general but reductions to finite d in two cases

1. Indep. noises (α = 0)

int. along caract. in m yields

(MFGi)

∂u∂t − ν∆u + H(x ,∇u,m) = 0

u |t=0= U0(x ,m(0)),m |t=T = m

∂m∂t + ν∆m + div (∂H∂p m) = 0

where m is given

FORWARD — BACKWARD system !

contains as particular cases: HJB, heat, porous media, FP,V ,B,Hartree, semilinear elliptic, barotropic Euler . . .


2. Finite state space (i 6 i 6 k)

(MFGf) ∂U∂t + (F (x ,U) . ∇) U = G (x ,U),U |t=0 = U0

(no common noise here to simplify . . . )

x ∈ Rk , U → Rk , F and G : R2k → Rk

non-conservative hyperbolic system

Example: If F = F (U) = H ′(U),G ≡ 0

and if U0 = ∇ϕ0 (ϕ0 → R) then

– solve HJ

∂ϕ

∂t+ H(∇ϕ) = 0 , ϕ |t=0 = ϕ0

– take U = ∇ϕ , “U solves” (MFGf) in this case


3. Another point of view

(Ω,F ,P) a “rich enough” proba . space H Hilbert space of L2

Random Variables

Φ(m) = Φ(X ) if L(X ) = m(X → Rd)

Then MFG may be written as

∂U

∂t+ (F(X ,U).D)U = G(X ,U) + α∆dU

(+ν D2U(G ,G ) G ⊥ FX )

∆dU = ∆Z U(.+ Z )|Z = 0(Z ∈ Rd) ,

where U : H → HRemarks: 1) MFG U(X ) ∈ FX ,L(U(X )) = L(U(Y ))

if L(X ) = L(Y )2) U(X ) = ∇xU(x ,L(X ))|x=X

Allows to prove that the problem is well-posed in the “small”.


V. OVERVIEW AND PERSPECTIVES

Lots of questions, partial results exist, many open problems

Existence/regularity:(MFGi) “simple” if H “smooth” in m (or if H almost linear. . . ), OK if monotone (Zoom 1)(MFGf) OK if (G ,F ) mon. on R2k or small time (Zoom 2)

Uniqueness: OK if “monotone” or T small . . .

Non existence, non uniqueness, non regularity (!)

Qualitative properties, stationary states and stability,comparison, cycles . . .

N →∞ (see above)

Numerical methods (currently, 3 “general” methods and someparticular cases)

Variants: other noises, several populations . . .

random heterogeneity, partial info . . .

applications (MFG Labs . . . )


optimal stopping, impulsive controls

intertemporal preference rates (+λ→∞ effective models)

macroscopic limits

? Beyond MFG ? (fluctuations, LD, transitions)

Two more S . examples:

at which time will the meeting start ?

the (mexican) wave


ZOOM 1

(MFGi)

∂u∂t − ν∆u + H(x ,∇u) = f (x ,m)

u |t=0= U0(x),m |t=T = m

∂m∂t + ν∆m + div (∂H∂p m) = 0

m 7→ f (•,m) smoothing operator∃ regular solution

uniqueness if operator monotone or if T small

f (m(x)) ↑: ∃ ! regular solution ν > 0

f (m(x)) ↑: if ν = 0 m = f −1(∂u∂t + H(x ,∇u))

equation in m becomes quasilinear elliptic equation of secondorder (x ∈ Q, t ∈ [0,T ]) with “elliptic” boundary conditions

u |t=0= U0(x),∂u

∂t+ H(∇u) = f (m) if t = T


ZOOM 2

(MFGf)

∂u∂t + (F (x ,U) . ∇) U = G (x ,U) x ∈ Rd

U → Rd , U |t=0 = U0(x)

shocks (discontinuities of U) in finite time in general

well-posed problem on [0,Tmax) (Tmax 6 +∞)

∃ !regular solution monotone in x if U0 monotone and (G ,F )monotone of R2,k in R2k(+ . . .)

+ change of unknown functions:

ex.: ∂U∂t + (F (U).∇)U = 0


then V = F (U) solves

∂V

∂t+ (V .∇)V = 0

max class of regularity

∀δ > 0, infx∈Rd

dist(Sp(DV0(x)), (−∞, δ]) > 0

(V0 = F (U0) gives the maximum class of regularity ≈ composed of2 monotone applications)

Remark: gives new results of regularity for Hamilton-Jacobiequations of the first order.


VI. MEANINGFUL DATA

• MFG Labs

• Practical expertise and models mainly for “big” data involving“people”

• New models that include classical clustering models in M.L.(K-mean, EM . . . ), then algorithms

• No need for euclidean structures or for “a priori” distances


• Why “PEOPLE”

Ex. 1: Taxis

Ex. 2: Movies and Fb

People that are “close” will say they like movies that are “close”

→ consistency distance - like on items/people


• Even for “pure data” models make sense: data points becomeagents . . . (in fact lots of terminology from Game Theory inM.L./Data Analysis)

• Clustering: classical K-Mean

set of points x1, . . . , xv in Rd(N >> 1)


Find K points y1, . . . , yk s.t. ∃ partition (A1, . . . ,AK ) of1, . . . ,N for which

i) |yi − xj | 6 |yi ′ − xj | , ∀j ∈ Ai , ∀i ′ 6= i

ii) yi = (#Ai )−1∑j∈Ai

×j


MFG INTERPRETATION :

INTRODUCE • A GLOBAL CRITERION

F (u1, . . . , uk)Ex: min(u1, . . . , uk)

• K value functions (u1, . . . , uk)

• K “densities” (m1, . . . ,mk)

f being the initial density of “data” (no need to restrict to“discrete” data)

SOLVE MFG: EXAMPLE

ρui − ν∆ui +1

2(∇ui )2 = Fi (x ;mi )

ρmi − ν∆mi − div(∇uimi ) = ρ∂F

∂uif

ex. 1(ui<minj 6=i

uj )


BACK TO K-Mean

Fi =1 + ρ

2|x −

∫xmi∫mi|2 − νd

then indeed : ui =1

2|x − yi |2, yi =

∫xmi∫mi

and

∫mi =

∫f 1(ui<minj 6=i uj ),

∫xmi =

∫xfi1(ui<minj 6=i uj )

Next, this allows to

• create lots of new models


• smoothe clustering if needed, clusters within clusters, overlaps. . .

• no need for distances, no need for euclidean structure (choosecriterion F , class criteria Fi → ui . . .)

• transposition to graphs easy (ODE’s, massively //)

Remark : −∆u + |∇u|2 = eu(+∆)e−u

eui∑j

(e−uj − e−ui ) =∑j

(eui−uj − 1)

• social networks equilibria: “distance on items” ←→“distance on users” ←→ preferences ←→ “distances on items”. . .

• interpretation of “deep learning”. . .


on mean field games - univ-lorraine.frpcfm-ma.iecl.univ-lorraine.fr/slides/lions.pdfapplications...

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