red red red red red red red red red red red red red red red red red red red red Uniqueness for a class of linear quadratic mean field games with common noise Foguen Tchuendom Rinel Laboratoire J-A Dieudonne University of Nice Sophia Antipolis Hamburg, September 1 Workshop on Industrial and Applied Mathematics 2016
Introduction Mean field games theory is concerned with the study of differential games with: Exchangeable players (in a statistical sense) players in mean field interaction ( a weak interaction) Infinitely many players (or a continuum of players) PDE approach: Lasry-Lions ( 2006) Caines-Malhame-Huang (2006) (Nash Certainty Equivalence) Cardaliaguet, Gueant, ... (great contributions) Probabilistic approach: Carmona-Delarue(2012) Bensoussan, Fischer, ... (great contributions) Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
Some applications Applications: Mean field games and systemic risk Volatility formation, price formation and dynamic equilibria Crowd motion: mexican waves, congestion large population wireless power control problem Mean field games for marriage Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
N -players differential game ( In R to fix ideas!) Consider the dynamics of the i th player: i ∈ { 1 , .., N } � t t = ψ i + X i B ( X i µ s , α i s ) ds + σ W i s , ¯ t , t ∈ [ 0 , T ] 0 Mean field interaction through N µ t = 1 � ¯ δ X j N t j = 1 Each player wants to minimize the cost � T � � J i ( α 1 , α 2 , ..., α i , ..., α N ) = E G ( X i F ( X i µ t , α i T , ¯ µ T )+ t , ¯ t ) dt 0 Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
Nash equilibrium We say that a collection of controls ( α 1 ∗ , ..., α i ∗ , ..., α N ∗ ) form a Nash equilibrum if for all i = 1 , ..., N we have J i ( α 1 ∗ , ..., α i ∗ , ..., α N ∗ ) ≤ J i ( α 1 ∗ , ..., α i , ..., α N ∗ ) i.e Once an equilibrium is in force, no player has unilateral incentive to leave the equilibrium !!! Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
Why consider MFG theory? Finding Nash equilibria is a very complex problem for games with large number of players: MFG theory allows to construct approximate Nash equilibria for such games, and error term goes to zero as N → ∞ . MFG theory provide a decentralized way to compute approximate Nash equilibria for games with large number of players. Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
Approximate Nash Equilibria We say that a collection of controls ( α 1 ∗ , ..., α i ∗ , ..., α N ∗ ) form an approximate Nash equilibrum if there exists ǫ N > 0 such that for all i = 1 , ..., N we have J i ( α 1 ∗ , ..., α i ∗ , ..., α N ∗ ) ≤ J i ( α 1 ∗ , ..., α i , ..., α N ∗ ) + ǫ N Mean field games approach allows ǫ N → 0 as N → ∞ Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
Mean Field Games( ‘ N = ∞ ′ ) Players are indistinguishable so that the dynamics of players can be seen as dynamics of a single representative player. Propagation of chaos: � N µ t = 1 For specified players dynamics ¯ j = 1 δ X j t → µ t N (Sznitman 1991) Consistency demands a similar behaviour for the players at equilibrium Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
MFG-solution scheme 1 (mean field input) Fix a flow of probability measures ( µ t ) t ∈ [ 0 , T ] (candidate for the mass profile at equilibrium) 2 (cost minimization) Find α ∗ such that � T � � J ( α ∗ ) = min α J ( α ) := E G ( X T , µ T ) + F ( X t , µ t , α t ) dt 0 subject to � t X t = ψ + B ( X s , µ s , α s ) ds + σ W t , t ∈ [ 0 , T ] 0 3 (Consistency condition) Find ( µ t ) t ∈ [ 0 , T ] such that for all t ∈ [ 0 , T ] µ t = L ( X α ∗ t ) → ( α ∗ t , µ t ) t ∈ [ 0 , T ] is called an MFG-solution Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
Probabilistic approach (stochastic Pontryagin principle): α ∗ solves cost minimization problem if there is a solution to dX t = ∂ y H ( X t , Y t , α ∗ t , µ t ) dt + σ dW t dY t = − ∂ x H ( X t , Y t , α ∗ t , µ t ) dt + Z t dW t X 0 = ψ, Y T = ∂ x G ( X T , µ T ) where H ( X t , Y t , α ∗ t , µ t ) = min α t H ( X t , Y t , α t , µ t ) for all t P − a . s . → Forward-Backward SDEs involved Find µ such that for all t , µ t = L ( X α ∗ t ) Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
Solvability results of MFG-solution scheme For T > 0 small, existence and uniqueness. Existence for T > 0 large via Schauder-type theorems. Uniqueness for T > 0 large via the Lasry-Lions monotonicity conditions: �� [ F ( x , m ) − F ( x , m ′ )]( m − m ′ )( x ) dx ≥ 0 � [ G ( x , m ) − G ( x , m ′ )]( m − m ′ )( x ) dx ≥ 0 Numerical methods available in PDE approach. Not much is known with common noise Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
Noise and uniqueness (Peano Example!) Consider the ODE dx t = b ( x t ) dt , x 0 = 0 → mutliple solutions when b ( x ) = sign ( x ) Consider the SDE dx t = b ( x t ) dt + ǫ dB t x 0 = 0 → unique strong solution when b ( x ) = sign ( x ) Can additional noise yield uniqueness to MFGs for T > 0 large ? Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
Linear Quadratic N -players game with common noise Controlled dynamics of the i th player: i ∈ 1 , .., N � t t = ψ i + X i ( − X i u s )+ α i s ) ds + σ W i s + b (¯ t + σ 0 B t , t ∈ [ 0 , T ] 0 Mean field interaction through N u t = 1 � X j ¯ t N j = 1 Each player wants to minimize the cost � � T 1 u t )) 2 + ( α i J i ( α 1 , α 2 , ..., α i , ..., α N ) = E 2 (( X i t ) 2 ) dt t + f (¯ 0 � + 1 u T ) 2 2 ( X T + ¯ Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
LQ-MFG-solution scheme with common noise (Mean field input) Consider a process u = ( u t ) t ∈ [ 0 , T ] adapted to the filtration generated by B only. (Cost minimization) Find α ∗ such that � T � 1 � 1 2 ( X T + g ( u T )) 2 + 2 (( X t + f ( u t )) 2 + α 2 J ( α ∗ ) = min t )) dt α E 0 under the dynamics : � t X t = ψ + ( − X s + b ( u s ) + α s ) ds + σ W t + σ 0 B t , t ∈ [ 0 , T ] 0 (Consistency condition) Find u such that for all t ∈ [ 0 , T ] t |F B u t = E ( X α ∗ T ) → we remark that for all t ∈ [ 0 , T ] E ( X α ∗ t |F B t ) = E ( X α ∗ t |F B T ) Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
Solving LQ-MFG-solution scheme 1 Let t �→ η t be the unique solution to the Riccati ODE � η t = η 2 ˙ t + 2 η t − 1 , η T = 1 Proposition 1: There exists a solution ( α ∗ , u ) to LQ-MFG-solution scheme with common noise if and only if there exists a solution to the FBSDEs ∀ t ∈ [ 0 , T ] du t = ( − ( 1 + η t ) u t − h t + b ( u t )) dt + σ 0 dB t (1) dh t = (( 1 + η t ) h t − f ( u t ) − η t b ( u t )) dt + Z 1 t dB t and h T = g ( u T ) , u 0 = E [ ψ ] Moreover, α ∗ t = − η t X t − h t . Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
Stochastic Pontryagin principle 1 The Hamiltonian is given by H ( t , a , x , y , u ) = y ( − x + a + b ( u )) + 1 2 a 2 + 1 2 ( x + f ( u )) 2 The cost minimization problem has a solution α ∗ if we can solve the FBSDEs dX t = ∂ y H ( t , α ∗ t , X t , Y t , u t ) dt + σ dW t + σ 0 dB t t , X t , Y t , u t ) dt + Z t dW t + Z 0 dY t = − ∂ x H ( t , α ∗ t dB t X 0 = ψ, Y T = X T + g ( u T ) , t ∈ [ 0 , T ] . Subject to H ( t , α ∗ t , X t , Y t , u t ) = min a ∈ R H ( t , a , X t , Y t , u t ) , ∀ t ∈ [ 0 , T ] , a . s Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
Stochastic Pontryagin principle 2 Thanks to the strict convexity of ( x , a ) �→ H ( t , a , x , y , u ) , the cost minimization problem has a solution α ∗ = − Y if we can solve the FBSDEs dX t = ( − X t − Y t + b ( u t )) t + σ dW t + σ 0 dB t dY t = ( − X t + Y t − f ( u t )) dt + Z t dW t + Z 0 (2) t dB t X 0 = ψ, Y T = X T + g ( u T ) , t ∈ [ 0 , T ] . To solve a Linear FBSDEs, we seek solutions satisying Y t = η t X t + h t , t ∈ [ 0 , T ] (3) h t an Ito process depending only on B Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
Solving the cost minimization 1 We suppose that we are given a mean field input u and solve the cost minimization There exist a solution to (2) satisfying (3) if and only if there exist a solution � dh t = (( 1 + η t ) h t − f ( u t ) − η t b ( u t )) dt + Z 1 t dB t (4) h T = g ( u T ) , t ∈ [ 0 , T ] The proof uses Ito’s formula and the ansatz. Foguen Tchuendom Rinel Uniqueness for a class of linear quadratic mean field games
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