M EAN F IELD G AMES WITH M AJOR AND M INOR P LAYERS René Carmona Department of Operations Research & Financial Engineering PACM Princeton University CEMRACS - Luminy, July 17, 2017
MFG WITH M AJOR AND M INOR P LAYERS S ET -U P R.C. - G. Zhu, R.C. - P. Wang State equations � dX 0 = b 0 ( t , X 0 t , µ t , α 0 t ) dt + σ 0 ( t , X 0 t , µ t , α 0 t ) dW 0 t t = b ( t , X t , µ t , X 0 t , α t , α 0 t ) dt + σ ( t , X t , µ t , X 0 t , α t , α 0 dX t t dW t , Costs �� T � J 0 ( α 0 , α ) 0 f 0 ( t , X 0 t , µ t , α 0 t ) dt + g 0 ( X 0 � = E T , µ T ) �� T J ( α 0 , α ) 0 f ( t , X t , µ N t , X 0 t , α t , α 0 = E t ) dt + g ( X T , µ T ) � ,
O PEN L OOP V ERSION OF THE MFG P ROBLEM The controls used by the major player and the representative minor player are of the form: α 0 t = φ 0 ( t , W 0 α t = φ ( t , W 0 [ 0 , T ] ) , and [ 0 , T ] , W [ 0 , T ] ) , (1) for deterministic progressively measurable functions φ 0 : [ 0 , T ] × C ([ 0 , T ]; R d 0 ) �→ A 0 and φ : [ 0 , T ] × C ([ 0 , T ]; R d ) × C ([ 0 , T ]; R d ) �→ A
T HE M AJOR P LAYER B EST R ESPONSE Assume representative minor player uses the open loop control given by φ : ( t , w 0 , w ) �→ φ ( t , w 0 , w ) , Major player minimizes �� T � J φ, 0 ( α 0 ) = E f 0 ( t , X 0 t , µ t , α 0 t ) dt + g 0 ( X 0 T , µ T ) 0 under the dynamical constraints: dX 0 = b 0 ( t , X 0 t , µ t , α 0 t ) dt + σ 0 ( t , X 0 t , µ t , α 0 t ) dW 0 t t = b ( t , X t , µ t , X 0 t , φ ( t , W 0 [ 0 , T ] , W [ 0 , T ] ) , α 0 dX t t ) dt + σ ( t , X t , µ t , X 0 t , φ ( t , W 0 [ 0 , T ] , W [ 0 , T ] ) , α 0 t ) dW t , µ t = L ( X t | W 0 [ 0 , t ] ) conditional distribution of X t given W 0 [ 0 , t ] . Major player problem as the search for: φ 0 , ∗ ( φ ) = arg J φ, 0 ( α 0 ) inf (2) α 0 t = φ 0 ( t , W 0 [ 0 , T ] ) Optimal control of the conditional McKean-Vlasov type!
T HE R EP . M INOR P LAYER B EST R ESPONSE System against which best response is sought comprises ◮ a major player ◮ a field of minor players different from the representative minor player ◮ Major player uses strategy α 0 t = φ 0 ( t , W 0 [ 0 , T ] ) ◮ Representative of the field of minor players uses strategy α t = φ ( t , W 0 [ 0 , T ] , W [ 0 , T ] ) . State dynamics dX 0 t = b 0 ( t , X 0 t , µ t , φ 0 ( t , W 0 [ 0 , T ] )) dt + σ 0 ( t , X 0 t , µ t , φ 0 ( t , W 0 [ 0 , T ] )) dW 0 t dX t = b ( t , X t , µ t , X 0 t , φ ( t , W 0 [ 0 , T ] , W [ 0 , T ] ) , φ 0 ( t , W 0 [ 0 , T ] )) dt + σ ( t , X t , µ t , X 0 t , φ ( t , W 0 [ 0 , T ] , W [ 0 , T ] ) , φ 0 ( t , W 0 [ 0 , T ] )) dW t , where µ t = L ( X t | W 0 [ 0 , t ] ) is the conditional distribution of X t given W 0 [ 0 , t ] . Given φ 0 and φ , SDE of (conditional) McKean-Vlasov type
T HE R EP . M INOR P LAYER B EST R ESPONSE ( CONT .) Representative minor player chooses a strategy α t = φ ( t , W 0 [ 0 , T ] , W [ 0 , T ] ) to minimize �� T J φ 0 ,φ (¯ f ( t , X t , X 0 α t , φ 0 ( t , W 0 � α ) = E t , µ t , ¯ [ 0 , T ] )) dt + g ( X T , µ t ) , 0 where the dynamics of the virtual state X t are given by: dX t = b ( t , X t , µ t , X 0 t , ¯ φ ( t , W 0 [ 0 , T ] , W [ 0 , T ] ) , φ 0 ( t , W 0 [ 0 , T ] )) dt + σ ( t , X t , µ t , X 0 t , ¯ φ ( t , W 0 [ 0 , T ] , W [ 0 , T ] ) , φ 0 ( t , W 0 [ 0 , T ] )) dW t , for a Wiener process W = ( W t ) 0 ≤ t ≤ T independent of the other Wiener processes. ◮ Optimization problem NOT of McKean-Vlasov type. ◮ Classical optimal control problem with random coefficients ∗ ( φ 0 , φ ) = arg J φ 0 ,φ (¯ φ inf α ) α t = φ ( t , W 0 [ 0 , T ] , W [ 0 , T ] )
N ASH E QUILIBRIUM Search for Best Response Map Fixed Point (ˆ φ 0 , ˆ φ 0 , ∗ (ˆ φ ) , ¯ φ ∗ (ˆ φ 0 , ˆ � � φ ) = φ ) . Fixed point in a space of controls , not measures !!!
C LOSED L OOP V ERSIONS OF THE MFG P ROBLEM ◮ Closed Loop Version Controls of the major player and the representative minor player are of the form: α 0 t = φ 0 ( t , X 0 α t = φ ( t , X [ 0 , T ] , µ t , X 0 [ 0 , T ] , µ t ) , and [ 0 , T ] ) , for deterministic progressively measurable functions φ 0 : [ 0 , T ] × C ([ 0 , T ]; R d 0 ) × P 2 ( R d ) �→ A 0 and φ : [ 0 , T ] × C ([ 0 , T ]; R d ) × P 2 ( R d ) × C ([ 0 , T ]; R d ) �→ A . ◮ Markovian Version Controls of the major player and the representative minor player are of the form: α 0 t = φ 0 ( t , X 0 α t = φ ( t , X t , µ t , X 0 t , µ t ) , and t ) , for deterministic feedback functions φ 0 : [ 0 , T ] × R d 0 × P 2 ( R d ) �→ A 0 and φ : [ 0 , T ] × R d × P 2 ( R d ) × R d 0 �→ A .
N ASH E QUILIBRIUM Search for Best Response Map Fixed Point (ˆ φ 0 , ˆ φ 0 , ∗ (ˆ φ ) , ¯ φ ∗ (ˆ φ 0 , ˆ � � φ ) = φ ) .
C ONTRACT T HEORY : A S TACKELBERG V ERSION R.C. - D. Possamaï - N. Touzi State equation dX t = σ ( t , X t , ν t , α t )[ λ ( t , X t , ν t , α t ) dt + dW t ] , ◮ X t Agent output ◮ α t agent effort (control) ◮ ν t distribution of output and effort (control) of agent Rewards � J 0 ( ξ ) = E � U P ( X [ 0 , T ] , ν T , ξ ) � � T � � J ( ξ, α ) = E − 0 f ( t , X t , ν t , α t ) dt + U A ( ξ ) , ◮ Given the choice of a contract ξ by the Principal ◮ Each agent in the field of exchangeable agents ◮ chooses an effort level α t ◮ meets his/her reservation price ◮ get the field of agents in a (MF) Nash equilibrium ◮ Principal chooses the contract to maximize his/her expected utility
L INEAR Q UADRATIC M ODELS State dynamics dX 0 = ( L 0 X 0 t + B 0 α 0 t + F 0 ¯ X t ) dt + D 0 dW 0 � t t = ( LX t + B α t + F ¯ X t + GX 0 dX t t ) dt + DdW t where ¯ X t = E [ X t |F 0 t ] , ( F 0 t ) t ≥ 0 filtration generated by W 0 Costs �� T � J 0 ( α 0 , α ) = E [( X 0 t − H 0 ¯ X t − η 0 ) † Q 0 ( X 0 t − H 0 ¯ X t − η 0 ) + α 0 † t R 0 α 0 t ] dt 0 �� T � J ( α 0 , α ) = E [( X t − HX 0 t − H 1 ¯ X t − η ) † Q ( X t − HX 0 t − H 1 ¯ X t − η ) + α † t R α t ] dt 0 in which Q , Q 0 , R , R 0 are symmetric matrices, and R , R 0 are assumed to be positive definite.
E QUILIBRIA ◮ Open Loop Version ◮ Optimization problems + fixed point = ⇒ large FBSDE ◮ affine FBSDE solved by a large matrix Riccati equation ◮ Closed Loop Version ◮ Fixed point step more difficult ◮ Search limited to controls of the form t , ¯ 2 ( t )¯ α 0 t = φ 0 ( t , X 0 X t ) = φ 0 0 ( t ) + φ 0 1 ( t ) X 0 t + φ 0 X t φ ( t , X t , X 0 t , ¯ X t ) = φ 0 ( t ) + φ 1 ( t ) X t + φ 2 ( t ) X 0 t + φ 3 ( t )¯ α t = X t ◮ Optimization problems + fixed point = ⇒ large FBSDE ◮ affine FBSDE solved by a large matrix Riccati equation Solutions are not the same !!!!
A PPLICATION TO B EE S WARMING ◮ V 0 , N velocity of the (major player) streaker bee at time t t ◮ V i , N the velocity of the i -th worker bee, i = 1 , · · · , N at time t t ◮ Linear dynamics � dV 0 , N = α 0 t dt + Σ 0 dW 0 t t dV i , N = α i t dt + Σ dW i t t ◮ Minimization of Quadratic costs �� T J 0 = E − ν t � 2 + λ 1 � V 0 , N t � 2 + ( 1 − λ 0 − λ 1 ) � α 0 � λ 0 � V 0 , N − ¯ V N t � 2 � � dt t t 0 ◮ ¯ � N i = 1 V i , N V N := 1 the average velocity of the followers, t t N ◮ deterministic function [ 0 , T ] ∋ t → ν t ∈ R d (leader’s free will) ◮ λ 0 and λ 1 are positive real numbers satisfying λ 0 + λ 1 ≤ 1 �� T J i = E l 0 � V i , N − V 0 , N � 2 + l 1 � V i , N t � 2 + ( 1 − l 0 − l 1 ) � α i � − ¯ V N t � 2 � � dt t t t 0 l 0 ≥ 0 and l 1 ≥ 0, l 0 + l 1 ≤ 1.
S AMPLE T RAJECTORIES IN E QUILIBRIUM ν ( t ) := [ − 2 π sin ( 2 π t ) , 2 π cos ( 2 π t )] k0 = 0.80 k1 = 0.19 l0 = 0.19 l1 = 0.80 k0 = 0.80 k1 = 0.19 l0 = 0.80 l1 = 0.19 1.0 1.0 0.5 0.5 y y 0.0 0.0 −0.5 −1.0 −0.5 0.0 0.5 −1.0 −0.5 0.0 0.5 x x F IGURE : Optimal velocity and trajectory of follower and leaders
S AMPLE T RAJECTORIES IN E QUILIBRIUM ν ( t ) := [ − 2 π sin ( 2 π t ) , 2 π cos ( 2 π t )] k0 = 0.19 k1 = 0.80 l0 = 0.19 l1 = 0.80 k0 = 0.19 k1 = 0.80 l0 = 0.80 l1 = 0.19 2.0 2.0 1.5 1.5 1.0 1.0 y y 0.5 0.5 0.0 0.0 0.0 0.5 1.0 1.5 2.0 −0.5 0.0 0.5 1.0 1.5 x x F IGURE : Optimal velocity and trajectory of follower and leaders
C ONDITIONAL P ROPAGATION OF C HAOS N = 5 N = 10 N = 20 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −0.2 −0.2 −0.2 −0.4 −0.4 −0.4 −0.6 −0.6 −0.6 −0.8 −0.8 −0.8 −1 −1 −1 N = 50 N = 100 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −1 F IGURE : Conditional correlation of 5 followers’ velocities
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