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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
Pierre-Louis LIONS ON MEAN FIELD GAMES
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. . . !
Pierre-Louis LIONS ON MEAN FIELD GAMES
• 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.
Pierre-Louis LIONS ON MEAN FIELD GAMES
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
Pierre-Louis LIONS ON MEAN FIELD GAMES
• 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.
Pierre-Louis LIONS ON MEAN FIELD GAMES
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
Pierre-Louis LIONS ON MEAN FIELD GAMES
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.
Pierre-Louis LIONS ON MEAN FIELD GAMES
• ∞ 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 . . .
Pierre-Louis LIONS ON MEAN FIELD GAMES
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 . . .
Pierre-Louis LIONS ON MEAN FIELD GAMES
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
Pierre-Louis LIONS ON MEAN FIELD GAMES
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”.
Pierre-Louis LIONS ON MEAN FIELD GAMES
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 . . . )
Pierre-Louis LIONS ON MEAN FIELD GAMES
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
Pierre-Louis LIONS ON MEAN FIELD GAMES
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
Pierre-Louis LIONS ON MEAN FIELD GAMES
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
Pierre-Louis LIONS ON MEAN FIELD GAMES
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.
Pierre-Louis LIONS ON MEAN FIELD GAMES
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
Pierre-Louis LIONS ON MEAN FIELD GAMES
• 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
Pierre-Louis LIONS ON MEAN FIELD GAMES
• 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)
Pierre-Louis LIONS ON MEAN FIELD GAMES
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
Pierre-Louis LIONS ON MEAN FIELD GAMES
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 )
Pierre-Louis LIONS ON MEAN FIELD GAMES
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
Pierre-Louis LIONS ON MEAN FIELD GAMES
• 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”. . .
Pierre-Louis LIONS ON MEAN FIELD GAMES
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