ON MEAN FIELD GAMES Pierre-Louis LIONS Coll` ege de France, Paris - - PowerPoint PPT Presentation

ON MEAN FIELD GAMES

Pierre-Louis LIONS

Coll` ege de France, Paris (joint project with Jean-Michel LASRY) Mathematical and Algorithmic Sciences Lab France 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. Lasry and P-L. Lions.

• Requires new mathematical theories.
• Numerous applications: economics, finance, social networks,

crowd motions. . .

• Independent introduction of a particular class of MFG models by
• M. Huang, P.E. Caines and R.P. Malham´

e in 2006.

• A research community in expansion: mathematics, economics,
• finance. Economics: anonymous games, Krusell Smith!, joint

projects with Ph. Aghion, J. Scheinkman, B. Moll, P-N. Giraud. . .

• Some written references but most of the existing mathematical

material to be found in the Coll` ege de France videotapes (4 × 18h) that can be downloaded. . . !

Pierre-Louis LIONS ON MEAN FIELD GAMES

• Combination of Mean Field theories (classical in Physics and

Mechanics) and the notion of Nash equilibria in Games theory.

• Nash equilibria for continua of “small” players: a single

heterogeneous 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) a

criterion that depends on the others (the whole group) and the

• ptimal decision affects the behavior of the group (however, this

interpretation is limited to some particular situations. . . ).

• Huge class of models: agents → particles, no dep. on the group

are 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 of

models (other “simple” exs later on).

• E metric space, N players (1 i N) choose a position xi ∈ E

according to a criterion Fi(X) where X = (x1, . . . , xN) ∈ E N.

• Nash equilibrium: ¯

X = (¯ x1, . . . , ¯ xN) if for all 1 i N ¯ xi min

• ver E of Fi(¯

x1, . . . , ¯ xi−1, xi, ¯ xi+1, . . . ¯ xN).

• Usual difficulties with the notion
• N → ∞ ? simpler ?
• Indistinguishable players:

Fi(X) = F(xi, {xj}j=i), F sym . in (xj)j=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=i

δxj

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

(MFG) ∀x ∈ Supp m, F(x, m) = inf

y∈E F(y, m)

• Facts: i) general existence and stability results

ii) uniqueness if (m → F(•, m)) monotone iii) If F = Φ′(m), then (min

P(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 f (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 time

and space, Brownian noises (both indep. and common), no intertemporal 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] and

H(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 =

1 N−1

• j=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 on

P(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 i 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 → H Remarks: 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 some particular cases) Variants: other noises, several populations . . . random heterogeneity, partial info . . . applications (MFG Labs . . . )

Pierre-Louis LIONS ON MEAN FIELD GAMES

• ptimal 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 → 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 second

• rder (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 +∞) ∃ !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, inf

x∈Rd dist(Sp(DV0(x)), (−∞, δ]) > 0

(V0 = F(U0) gives the maximum class of regularity ≈ composed of 2 monotone applications) Remark: gives new results of regularity for Hamilton-Jacobi equations 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 become

agents . . . (in fact lots of terminology from Game Theory in M.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) of {1, . . . , N} for which i) |yi − xj| |yi′ − xj| , ∀j ∈ Ai , ∀i′ = 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 ∂ui f ex. 1(ui<min

j=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=i uj),
• xmi =
• xfi1(ui<minj=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 (choose

criterion 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