Solving classical differential games Games with imperfect information Differential games with imperfect observation Problèmes d’information dans les jeux en temps continu P . Cardaliaguet Paris-Dauphine Based on joint works with C. Rainer and A. Souquière (Brest) SMAI 2011. Guidel, 23-27 Mai 2011 P . Cardaliaguet Jeux en temps continu
Solving classical differential games Games with imperfect information Differential games with imperfect observation Description of the game We investigate a stochastic differential game defined by � dX t = f ( X t , u t , v t ) dt + σ ( X t , u t , v t ) dB t , t ∈ [ t 0 , T ] , X t 0 = x 0 , where B is a d -dimensional standard Brownian motion R N × U × V → I R N and σ : I R n × U × V → I R N × d are f : I Lipschitz continuous and bounded, the processes u (controlled by Player I) and v (controlled by Player II) take their values in some compact sets U and V . The solution to (*) is denoted by t → X t 0 , x 0 , u , v . t P . Cardaliaguet Jeux en temps continu
Solving classical differential games Games with imperfect information Differential games with imperfect observation The payoffs R N → I Let g : I R be the terminal payoff, Player I minimises the payoff E [ g ( X T )] Player II maximises the payoff E [ g ( X T )] . P . Cardaliaguet Jeux en temps continu
Solving classical differential games Games with imperfect information Differential games with imperfect observation P . Cardaliaguet Jeux en temps continu
Solving classical differential games Games with imperfect information Differential games with imperfect observation Problems Describe the fact that the players chose their controls simultaneously by observing each other Compute (or characterize) their best payoffs. Compute (or characterize) their optimal strategies. P . Cardaliaguet Jeux en temps continu
Solving classical differential games Games with imperfect information Differential games with imperfect observation Outline Solving classical differential games 1 Formalisation Existence of the value Games with imperfect information 2 Description of the game Existence and characterization of the value Illustration through a simple game Differential games with imperfect observation 3 P . Cardaliaguet Jeux en temps continu
Solving classical differential games Formalisation Games with imperfect information Existence of the value Differential games with imperfect observation Outline Solving classical differential games 1 Formalisation Existence of the value Games with imperfect information 2 Description of the game Existence and characterization of the value Illustration through a simple game Differential games with imperfect observation 3 P . Cardaliaguet Jeux en temps continu
Solving classical differential games Formalisation Games with imperfect information Existence of the value Differential games with imperfect observation Outline Solving classical differential games 1 Formalisation Existence of the value Games with imperfect information 2 Description of the game Existence and characterization of the value Illustration through a simple game Differential games with imperfect observation 3 P . Cardaliaguet Jeux en temps continu
Solving classical differential games Formalisation Games with imperfect information Existence of the value Differential games with imperfect observation Strategies A strategy for Player I is a Borel measurable map α : [ t 0 , T ] × L 0 ([ t 0 , T ] , V ) × C 0 ([ t 0 , T ] , I R N ) → U such that there is τ > 0 with v 1 = v 2 and f 1 = f 2 on [ t 0 , t ] ⇒ α ( s , v 1 , f 1 ) = α ( s , v 2 , f 2 ) for s ∈ [ t 0 , t 0 + τ ] The set of strategies for Player I is denoted by A ( t 0 ) . The set of strategies for Player II is defined symmetrically and denoted by B ( t 0 ) . P . Cardaliaguet Jeux en temps continu
Solving classical differential games Formalisation Games with imperfect information Existence of the value Differential games with imperfect observation Playing pure strategies together Lemma R N , for all ( α, β ) ∈ A ( t 0 ) × B ( t 0 ) , there For all ( t 0 , x 0 ) ∈ [ 0 , T ] × I exists a unique couple of controls ( u , v ) that satisfies ( ∗ ) ( u , v ) = ( α ( · , v , B · − B t 0 ) , β ( · , u , B · − B t 0 )) on [ t 0 , T ] . Notation : X t 0 , x 0 ,α,β = X t 0 , x 0 , u , v T T where ( u , v ) is given by ( ∗ ) . P . Cardaliaguet Jeux en temps continu
Solving classical differential games Formalisation Games with imperfect information Existence of the value Differential games with imperfect observation Upper and lower value functions The upper value function is � � V + ( t 0 , x 0 ) = g ( X t 0 , x 0 ,α,β α ∈A ( t 0 ) sup inf E ) T β ∈B ( t 0 ) while the lower value function is � � g ( X t 0 , x 0 ,α,β V − ( t 0 , x 0 ) = ) sup α ∈A ( t 0 ) E inf T β ∈B ( t 0 ) P . Cardaliaguet Jeux en temps continu
Solving classical differential games Formalisation Games with imperfect information Existence of the value Differential games with imperfect observation Outline Solving classical differential games 1 Formalisation Existence of the value Games with imperfect information 2 Description of the game Existence and characterization of the value Illustration through a simple game Differential games with imperfect observation 3 P . Cardaliaguet Jeux en temps continu
Solving classical differential games Formalisation Games with imperfect information Existence of the value Differential games with imperfect observation Isaacs’ condition We assume that Isaacs’condition holds : for all R N , ξ ∈ I R N , and all A ∈ S N : ( t , x ) ∈ [ 0 , T ] × I H ( x , ξ, A ) := {� f ( x , u , v ) , ξ � + 1 2 Tr ( A σ ( x , u , v ) σ ∗ ( x , u , v )) } inf u sup v u {� f ( x , u , v ) , ξ � + 1 2 Tr ( A σ ( x , u , v ) σ ∗ ( x , u , v )) } = sup inf v P . Cardaliaguet Jeux en temps continu
Solving classical differential games Formalisation Games with imperfect information Existence of the value Differential games with imperfect observation Existence of a value Theorem (Fleming-Souganidis, 1989) Under Isaacs’ condition, the game has a value : R N . V + ( t , x ) = V − ( t , x ) ∀ ( t , x ) ∈ [ 0 , T ] × I The value V := V + = V − is the unique viscosity solution to the (backward) Hamilton-Jacobi equation � ∂ t w + H ( x , Dw , D 2 w ) = 0 R N in ( 0 , T ) × I R N w = g in { T } × I P . Cardaliaguet Jeux en temps continu
Solving classical differential games Formalisation Games with imperfect information Existence of the value Differential games with imperfect observation Idea of proof (1) Assume for simplicity that V + and V − are smooth. Lemma (Dynamic programming) R N and h > 0, For ( t 0 , x 0 ) ∈ [ 0 , T ] × I � V + � �� t 0 + h , X t 0 , x 0 ,α,β V + ( t 0 , x 0 ) = α ∈A ( t 0 ) sup inf E t 0 + h β ∈B ( t 0 ) and � V − � �� t 0 + h , X t 0 , x 0 ,α,β V − ( t 0 , x 0 ) = sup α ∈A ( t 0 ) E inf t 0 + h β ∈B ( t 0 ) P . Cardaliaguet Jeux en temps continu
Solving classical differential games Formalisation Games with imperfect information Existence of the value Differential games with imperfect observation Idea of proof (2) From dynamic programming : � V + � � � t 0 + h , X t 0 , x 0 ,α,β − V + ( t 0 , x 0 ) 0 = α ∈A ( t 0 ) sup inf E t 0 + h β ∈B ( t 0 ) � t 0 + h � � � DV + , f ( α, β ) � + 1 h ∂ t V + + 2 Tr ( σσ ∗ ( α, β ) D 2 V + ) ds ≈ inf α sup E β t 0 Divide by h and let h → 0 : � � DV + , f ( x 0 , u , v ) � + 1 � 0 = ∂ t V + + inf 2 Tr ( σσ ∗ ( x 0 , u , v ) D 2 V + ) u ∈ U sup v ∈ V = ∂ t V + ( t 0 , x 0 ) + H ( x 0 , DV + ( t 0 , x 0 ) , D 2 V + ( t 0 , x 0 )) . P . Cardaliaguet Jeux en temps continu
Solving classical differential games Formalisation Games with imperfect information Existence of the value Differential games with imperfect observation Sketch of proof (3) So V + and V − are both solutions to the Hamilton-Jacobi equation � ∂ t w + H ( x , Dw , D 2 w ) = 0 R N in ( 0 , T ) × I R N w = g in { T } × I Uniqueness of the solution ⇒ V + = V − . P . Cardaliaguet Jeux en temps continu
Solving classical differential games Formalisation Games with imperfect information Existence of the value Differential games with imperfect observation Sketch of proof (3) So V + and V − are both solutions to the Hamilton-Jacobi equation � ∂ t w + H ( x , Dw , D 2 w ) = 0 R N in ( 0 , T ) × I R N w = g in { T } × I Uniqueness of the solution ⇒ V + = V − . P . Cardaliaguet Jeux en temps continu
Solving classical differential games Formalisation Games with imperfect information Existence of the value Differential games with imperfect observation Comments Differential games were first investigated by Pontryagin and Isaacs in the mid-50ies. First proof of existence of a value : Fleming, 1961 The Hamilton-Jacobi equation has to be understood in the viscosity sense (introduced by Crandall-Lions, 1981) The above proof was made rigorous in Evans-Souganidis, 1984 (deterministic D.G.) Fleming-Souganidis, 1989 (stochastic D.G.) P . Cardaliaguet Jeux en temps continu
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