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 optimal 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 x i ∈ E according to a criterion F i ( X ) where X = ( x 1 , . . . , x N ) ∈ E N . • Nash equilibrium: ¯ X = (¯ x 1 , . . . , ¯ x N ) if for all 1 � i � N ¯ x i min over E of F i (¯ x 1 , . . . , ¯ x i − 1 , x i , ¯ x i +1 , . . . ¯ x N ). • Usual difficulties with the notion • N → ∞ ? simpler ? • Indistinguishable players: F i ( X ) = F ( x i , { x j } j � = i ) , F sym . in ( x j ) j � = i Pierre-Louis LIONS ON MEAN FIELD GAMES
• Part of the mathematical theories is about N → ∞ : F i = F ( x , m ) x ∈ E , m ∈ P ( E ) 1 � where x = x i , m = δ x j N − 1 j � = i • “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
� # { j / | x i − x j | < ε } � Example: E = R d , F i ( X ) = f ( x i ) + g ( N − 1) | B ε | g ↑ aversion crowds, g ↓ like crowds F ( x , m ) = f ( x ) + g ( m ∗ 1 B ε ( x )( | B ε | − 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 � Z �� � � min fm + G ( m ) / m ∈ P ( E ) , G = f ( s ) ds 0 � – 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 ∈ R d , m ∈ P ( R d ) or M + ( R d ) , t ∈ [0 , T ] and H ( x , p , m ) (convex in p ∈ R d ) • MFG master equation ∂ U ∂ t − ( ν + α )∆ x U + H ( x , ∇ x U , m )+ + � ∂ U ∂ m , − ( ν + α )∆ m + div ( ∂ H ∂ p m ) � + − α ∂ U ∂ m 2 ( ∇ m , ∇ m ) + 2 α � ∂ ∂ m ∇ x U , ∇ m � = 0 and U | t =0 = U 0 ( 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 � 1 using x = x i , m = δ x j N − 1 j � = i • Aggregation/decentralization: IF H ( x , p , m ) = H ( x , p ) + F ′ ( m ) 0 ( m ), then U = ∂ Φ and U 0 = 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 ∂ u ∂ t − ν ∆ u + H ( x , ∇ u , m ) = 0 u | t =0 = U 0 ( x , m (0)) , m | t = T = ¯ ( MFGi ) 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 ) ∂ U (MFGf) ∂ t + ( F ( x , U ) . ∇ ) U = G ( x , U ) , U | t =0 = U 0 (no common noise here to simplify . . . ) x ∈ R k , U → R k , F and G : R 2 k → R k non-conservative hyperbolic system Example: If F = F ( U ) = H ′ ( U ) , G ≡ 0 and if U 0 = ∇ ϕ 0 ( ϕ 0 → R ) then – solve HJ ∂ϕ ∂ t + H ( ∇ ϕ ) = 0 , ϕ | t =0 = ϕ 0 – 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 L 2 Random Variables Φ( m ) = Φ( X ) if L ( X ) = m ( X → R d ) Then MFG may be written as ∂ U ∂ t + ( F ( X , U ) . D ) U = G ( X , U ) + α ∆ d U (+ ν D 2 U ( G , G ) G ⊥ F X ) ∆ d U = ∆ Z U ( . + Z ) | Z = 0( Z ∈ R d ) , where U : H → H Remarks: 1) MFG U ( X ) ∈ F X , L ( U ( X )) = L ( U ( Y )) if L ( X ) = L ( Y ) 2) U ( X ) = ∇ x U ( 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 R 2 k 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
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 ∂ u ∂ t − ν ∆ u + H ( x , ∇ u ) = f ( x , m ) ( MFGi ) u | t =0 = U 0 ( 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 order ( x ∈ Q , t ∈ [0 , T ]) with “elliptic” boundary conditions u | t =0 = U 0 ( x ) , ∂ u ∂ t + H ( ∇ u ) = f ( ¯ m ) if t = T Pierre-Louis LIONS ON MEAN FIELD GAMES
ZOOM 2 � ∂ u ∂ t + ( F ( x , U ) . ∇ ) U = G ( x , U ) x ∈ R d ( MFGf ) U → R d , U | t =0 = U 0 ( x ) shocks (discontinuities of U ) in finite time in general well-posed problem on [0 , T max ) ( T max � + ∞ ) ∃ !regular solution monotone in x if U 0 monotone and ( G , F ) monotone of R 2 , k in R 2 k (+ . . . ) + 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 , x ∈ R d dist ( Sp ( DV 0 ( x )) , ( −∞ , δ ]) > 0 inf ( V 0 = F ( U 0 ) 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
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