An Existence Result for a Class of Mean Field Games of Controls - PowerPoint PPT Presentation

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation An Existence Result for a Class of Mean Field Games of Controls Laurent Pfeiffer Inria and Ecole Polytechnique, Institut Polytechnique de Paris Joint work with J. Fr´ ed´ eric Bonnans, Justina Gianatti, and Saeed Hadikhanloo Two-Days Online Workshop on Mean-Field Games, June 18, 2020

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation Introduction The Mean-Field Game (MFG) model: Coupling (` a la Cournot) via endogenous price variable P Price related to the distribution of (states,controls). − → MFG of controls = extended MFG, strongly coupled MFG... Topics: Existence (2nd order case) Duality Lagrangian approach (1st order case).

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation 1 Cournot equilibria 2 MFG model 3 Reduction of the system and potential formulation 4 Existence result 5 Duality 6 Lagrangian formulation for the first-order case

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation 1 Cournot equilibria 2 MFG model 3 Reduction of the system and potential formulation 4 Existence result 5 Duality 6 Lagrangian formulation for the first-order case

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation Cournot equilibria Consider N producers, buy some raw material on a market. Quantity bought by producer i : v i Benefit resulting from v i : − L i ( v i ) Unitary price of raw material: P = Ψ( � N i =1 v i ). v ∈ R N such that Nash equilibrium: a vector ¯ � �� N � � v i ∈ arg min ¯ L i ( v i ) + Ψ j =1 ¯ v j v i , v i ∈ R for i = 1 , ..., N . Remark The producers do not take into account their contribution to the equilibrium price P.

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation Potential formulation Assumptions: L 1 ,..., L N are strongly convex Ψ = ∇ Φ, with Φ convex Potential formulation: �� N � Let B : v ∈ R N �→ B ( v ) = � N i =1 L i ( v i ) + Φ i =1 v i . Then, v ∈ R N is a Nash equilibrium ¯ �� N � ⇐ ⇒ ∇ L i (¯ v i ) + Ψ j =1 ¯ v j = 0 = ∇ v i B (¯ v ) , ∀ i = 1 , ..., N ⇐ ⇒ ¯ v minimizes B . Implies existence and uniqueness ( B is strongly convex). Remark MFG model: a dynamic version with infinitely many agents.

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation 1 Cournot equilibria 2 MFG model 3 Reduction of the system and potential formulation 4 Existence result 5 Duality 6 Lagrangian formulation for the first-order case

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation MFG model Coupled system:  ( i ) − ∂ t u − σ ∆ u + H ( ∇ u + P ) = 0 ( x , t ) ∈ Q ,     ( ii ) ∂ t m − σ ∆ m + div ( vm ) = 0 ( x , t ) ∈ Q ,    �� � ( iii ) P ( t ) = Ψ T d v ( x , t ) m ( x , t ) d x t ∈ [0 , T ] , (MFGC)   ( iv ) v = −∇ H ( ∇ u + P ) ( x , t ) ∈ Q ,      x ∈ T d , ( v ) m ( x , 0) = m 0 ( x ) , u ( x , T ) = g ( x ) u ( x , t ) value function v ( x , t ) feedback Unknowns: m ( x , t ) distribution P ( t ) price H Hamiltonian Ψ price function Data: m 0 initial distrib. g terminal cost

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation MFG of controls Equation (i): Hamilton-Jacobi-Bellman (HJB) equation . Associated stochastic optimal control problem:  �� T �   inf L ( α s ) + � P ( s ) , α s � d s + g ( X T ) , α ∈ L 2 ( t , T ) E u ( x , t ) = t √   s.t.: d X s = α s d s + 2 σ d W s , X t = x . X s stock at time s α s bought/sold quantity P ( s ) unitary price . Equation (ii): Fokker-Planck equation.

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation MFG of controls Equation (iii): price relation. � � � P ( t ) = Ψ T d v ( x , t ) m ( x , t ) d x � �� � → Demand Equation (iv): optimal feedback law. v ( x , t ) = −∇ H ( ∇ u ( x , t ) + P ( t )) . Remark Given m and u, the feedback v cannot be recovered in an explicit fashion → MFG of controls 1 . Equilibrium problem for each time t (involving P and v). 1 Graber & Bensoussan ’15, Gomes & Voskanyan ’16, Cardaliaguet & Lehalle ’18, Kobeissi ’19, Graber, Ignazio & Neufeld ’20,...

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation An example An idealized model from electrical engineering 2 : Large population of storage devices State variable x ∈ (0 , 2): State-of-charge Control α : Relative loading speed Reference demand D ref ( t ) Price function: � 2 P ( t ) = β D rel ( t ) , D rel ( t ) = D ref ( t ) + 0 v ( x , t ) m ( x , t ) d x Cost: L ( α ) = 1 2 α 2 , g ( x ) = − β D ref x Deterministic dynamics: σ = 0 2 Couillet et al. ’12, De Paola et al. ’16, Alasseur, Ben Tahar & Matoussi ’20, Gomes & Sa´ ude ’20

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation Results 2 2 0 1 -2 0 0 0.5 1 0 0.5 1 (a) Reference and relative demands (b) Distribution Figure: Equilibrium results β = 0 . 25 (small coupling)

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation Results 2 2 0 1 -2 0 0 0.5 1 0 0.5 1 (a) Reference and relative demands (b) Distribution Figure: Equilibrium results β = 2 (strong coupling)

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation 1 Cournot equilibria 2 MFG model 3 Reduction of the system and potential formulation 4 Existence result 5 Duality 6 Lagrangian formulation for the first-order case

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation Functional framework Given α ∈ (0 , 1) and X = [0 , T ], X = T d , or X = Q , � C j + α ( X ) := u ∈ C j ( X ) | ∃ C > 0 , ∀ x , y ∈ X , � � D i u ( y ) − D i u ( x ) � ≤ C � y − x � α X , ∀ i ≤ j � C α,α/ 2 ( Q ) := u ∈ C ( Q ) | ∃ C > 0 , ∀ x , y ∈ X , � x 2 − x 1 � α + | t 2 − t 1 | α/ 2 �� � | u ( x 2 , t 2 ) − u ( x 1 , t 1 ) | ≤ C � C 2+ α, 1+ α/ 2 ( Q ) := u ∈ C α,α/ 2 ( Q ) | ∂ t u ∈ C α,α/ 2 ( Q ) , � ∇ u ∈ C α,α/ 2 ( Q ) , ∇ 2 u ∈ C α,α/ 2 ( Q ) . We fix p > d + 2 and define the Sobolev space W 2 , 1 , p ( Q ) := L p (0 , T ; W 2 , p ( Q )) ∩ W 1 , p ( Q ) . Embedding: � u � C α ( Q ) + �∇ u � C α ( Q ) ≤ C � u � W 2 , 1 , p ( Q ) .

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation Assumptions Monotonicity assumptions: Ψ = ∇ Φ, where Φ is convex L is strongly convex. Growth assumptions: L ( v ) ≤ C (1 + � v � 2 ) Ψ( z ) ≤ C (1 + � z � ). Regularity assumptions: H ∈ C 2 ( R d ), H , ∇ H , ∇ 2 H are locally H¨ older continuous Ψ is locally H¨ older continuous m 0 ∈ C 2+ α ( T d ), g ∈ C 2+ α ( T d ) � m 0 ∈ D 1 ( T d ) := { h ∈ L ∞ ( T d ) | h ≥ 0 , T d h ( x ) d x = 1 } .

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation Auxiliary mappings We analyse ( iii ) and ( iv ) to eliminate v and P from (MFGC). Lemma For all m ∈ D 1 ( T d ) , for all w ∈ L ∞ ( T d , R d ) , there exists a unique pair ( v , P ) = ( v ( m , w ) , P ( m , w )) ∈ L ∞ ( T d , R d ) × R d such that � ∀ x ∈ T d , v ( x ) = −∇ H ( w ( x ) + P ) , ( ∗ ) �� � P = Ψ T d v ( x ) m ( x ) dx . Elements of proof. If m > 0, then ( v , P ) satisfies ( ∗ ) if and only if v minimizes the following convex functional: � �� � � � J ( v ): v �→ Φ T d v ( x ) m ( x ) d x + L ( v ( x ))+ � w ( x ) , v ( x ) � m ( x ) d x , T d which possesses a unique minimizer.

Cournot equilibria Model Potential formulation Existence result Duality Lagrangian formulation Auxiliary mappings Reduced coupled system:  − ∂ t u − σ ∆ u + H ( ∇ u + P ( m ( · , t ) , ∇ u ( · , t ))) = 0 ,   ∂ t m − σ ∆ m + div ( v ( m ( · , t ) , ∇ u ( · , t )) m ) = 0 ,   u ( x , T ) = g ( x ) , m ( x , 0) = m 0 ( x ) . ( MFGC ′ ) Lemma (Stability lemma) Let R > 0 , let m 1 and m 2 ∈ D 1 ( T d ) , let w 1 and w 2 ∈ L ∞ ( T d , R d ) with � w i � L ∞ ( T d , R d ) ≤ R. There exists C > 0 and α ∈ (0 , 1) , depending on R only such that � P ( m 2 , w 2 ) − P ( m 1 , w 1 ) � � � � w 2 − w 1 � α L ∞ ( T d ) + � m 2 − m 1 � α ≤ C . L 1 ( T d ) Idea of proof: stability analysis for convex optimization problems.