BB MFG Variational formulations Density constraints for MFGs From Benamou-Brenier to mean field games (with density constraints) Alp´ ar Rich´ ard M´ esz´ aros LMO, Universit´ e Paris-Sud (based on ongoing joint works with F. Santambrogio, P . Cardaliaguet and F. J. Silva) W ORKSHOP ON O PTIMAL T RANSPORT IN THE A PPLIED S CIENCES , RICAM, D EC . 8-12, 2014, L INZ 1 / 20
BB MFG Variational formulations Density constraints for MFGs The content of the talk Benamou-Brenier formulation of Optimal Transport 1 Basic models of Mean Field Games, after J.-M. Lasry and P .-L. 2 Lions Variational approaches for MFGs 3 Study first order evolutive MFG systems with density constraints 4 Second order stationary MFGs under density constraints 5 2 / 20
BB MFG Variational formulations Density constraints for MFGs The Benamou-Brenier formulation of optimal transportation For two (regular enough) probability measures µ, ν ∈ P (Ω) ( Ω ⊂ R d is either compact or we set it T d ) we have for 1 < q < + ∞ W q q ( µ, ν ) = �� 1 � 1 � q | α t | q d m t d t : ∂ t m t + ∇ · ( m t α t ) = 0 , m 0 = µ, m 1 = ν = min α 0 Ω �� 1 � | w t | q � 1 = min d x d t : ∂ t m t + ∇ · ( w t ) = 0 , m 0 = µ, m 1 = ν . m q − 1 q ( m , w ) 0 Ω t This dynamic formulation is due to J.-D. Benamou and Y. Brenier, in ’00 The optimal curve [ 0 , 1 ] �→ m t gives a geodesic in ( P (Ω) , W q ) connecting µ and ν . 3 / 20
BB MFG Variational formulations Density constraints for MFGs A further model We define ℓ q : R × R d → R , | b | q 1 a q − 1 , if a > 0 , q ℓ q ( a , b ) := 0 , if ( a , b ) = ( 0 , 0 ) , + ∞ , otherwise . � 1 � and B q : C ( 0 , 1 ; ( P (Ω) , W q )) × M ([ 0 , 1 ] × Ω) d , B q ( m , w ) := ℓ q ( m t , w t ) d x d t . 0 Ω Question: What model does � � � min B q ( m , w ) + F ( m ) + u 1 d m 1 : ∂ t m t + ∇ · ( w t ) = 0 , m ( 0 , · ) = m 0 give ? ( m , w ) Ω Here F : C ( 0 , 1 ; ( P (Ω) , W q )) → R is a convex, l.s.c. functional, u 1 : Ω → R a smooth function and m 0 ∈ P (Ω) . Answer: it characterizes an MFG model. 4 / 20
BB MFG Variational formulations Density constraints for MFGs A short history of Mean Field Games This theory was introduced recently (in 2006-2007) by J.-M. Lasry and P .-L. Lions in a series of papers and a series of lectures by P .-L. Lions at Coll` ege de France → in the community of mathematicians; 5 / 20
BB MFG Variational formulations Density constraints for MFGs A short history of Mean Field Games This theory was introduced recently (in 2006-2007) by J.-M. Lasry and P .-L. Lions in a series of papers and a series of lectures by P .-L. Lions at Coll` ege de France → in the community of mathematicians; Similar models were introduced (around 2006) by P . Caines, M. Huang, R. Malham´ e → in the community of engineers; Analysis of differential games with a very large number of “small” players (agents); 5 / 20
BB MFG Variational formulations Density constraints for MFGs A short history of Mean Field Games This theory was introduced recently (in 2006-2007) by J.-M. Lasry and P .-L. Lions in a series of papers and a series of lectures by P .-L. Lions at Coll` ege de France → in the community of mathematicians; Similar models were introduced (around 2006) by P . Caines, M. Huang, R. Malham´ e → in the community of engineers; Analysis of differential games with a very large number of “small” players (agents); The models are derived from a “continuum limit”, letting the number of agents go to infinity (similarly to mean field limit in Statistical Mechanics and Physics → Bolzmann or Vlasov equations) 5 / 20
BB MFG Variational formulations Density constraints for MFGs A short history of Mean Field Games This theory was introduced recently (in 2006-2007) by J.-M. Lasry and P .-L. Lions in a series of papers and a series of lectures by P .-L. Lions at Coll` ege de France → in the community of mathematicians; Similar models were introduced (around 2006) by P . Caines, M. Huang, R. Malham´ e → in the community of engineers; Analysis of differential games with a very large number of “small” players (agents); The models are derived from a “continuum limit”, letting the number of agents go to infinity (similarly to mean field limit in Statistical Mechanics and Physics → Bolzmann or Vlasov equations) Real life applications in Economy, Finance and Social Sciences 5 / 20
BB MFG Variational formulations Density constraints for MFGs A typical model for second order MFG in ( 0 , T ) × R d − ∂ t u + ν ∆ u + H ( x , m , ∇ u ) = f ( x , m ) in ( 0 , T ) × R d ∂ t m − ν ∆ m − ∇ · ( D p H ( x , m , ∇ u ) m ) = 0 (1) in R d . m ( 0 ) = m 0 , u ( T , x ) = u T ( x , m ( T )) 6 / 20
BB MFG Variational formulations Density constraints for MFGs A typical model for second order MFG in ( 0 , T ) × R d − ∂ t u + ν ∆ u + H ( x , m , ∇ u ) = f ( x , m ) in ( 0 , T ) × R d ∂ t m − ν ∆ m − ∇ · ( D p H ( x , m , ∇ u ) m ) = 0 (1) in R d . m ( 0 ) = m 0 , u ( T , x ) = u T ( x , m ( T )) Assumptions: ν ≥ 0 is a parameter; the Hamiltonian H is convex in its last variable; m 0 (and m ( t ) ) is the density of a probability measure; f is an increasing smooth function; u T is a smooth function; 6 / 20
BB MFG Variational formulations Density constraints for MFGs A typical model for second order MFG in ( 0 , T ) × R d − ∂ t u + ν ∆ u + H ( x , m , ∇ u ) = f ( x , m ) in ( 0 , T ) × R d ∂ t m − ν ∆ m − ∇ · ( D p H ( x , m , ∇ u ) m ) = 0 (1) in R d . m ( 0 ) = m 0 , u ( T , x ) = u T ( x , m ( T )) Assumptions: ν ≥ 0 is a parameter; the Hamiltonian H is convex in its last variable; m 0 (and m ( t ) ) is the density of a probability measure; f is an increasing smooth function; u T is a smooth function; u is the value function of an arbitrary agent, m is the distribution of the agents; 6 / 20
BB MFG Variational formulations Density constraints for MFGs A heustistic interpretation An arbitrary agent controls the stochastic differential equation √ d X t = α t d t + 2 ν d B t , where B t is a standard Brownian motion. 7 / 20
BB MFG Variational formulations Density constraints for MFGs A heustistic interpretation An arbitrary agent controls the stochastic differential equation √ d X t = α t d t + 2 ν d B t , where B t is a standard Brownian motion. He aims at minimizing the quantity �� T � E L ( X s , m ( s ) , α s ) + f ( X s , m ( s )) d s + u T ( X T , m ( T )) , 0 where L is the usual Legendre-Flenchel conjugate of H w.r.t. the p variable. 7 / 20
BB MFG Variational formulations Density constraints for MFGs A heustistic interpretation An arbitrary agent controls the stochastic differential equation √ d X t = α t d t + 2 ν d B t , where B t is a standard Brownian motion. He aims at minimizing the quantity �� T � E L ( X s , m ( s ) , α s ) + f ( X s , m ( s )) d s + u T ( X T , m ( T )) , 0 where L is the usual Legendre-Flenchel conjugate of H w.r.t. the p variable. His optimal control is (at least heuristically) given in feedback form by α ∗ ( t , x ) = − D p H ( x , m , ∇ u ) . 7 / 20
BB MFG Variational formulations Density constraints for MFGs A typical model of first order MFG system A typical model for a first order (deterministic) MFG system is the following: 8 / 20
BB MFG Variational formulations Density constraints for MFGs A typical model of first order MFG system A typical model for a first order (deterministic) MFG system is the following: 2 |∇ u ( t , x ) | 2 = f ( x , m ( t )) − ∂ t u ( t , x ) + 1 in ( 0 , T ) × Ω ∂ t m ( t , x ) − ∇ · ( ∇ u ( t , x ) m ( t , x )) = 0 in ( 0 , T ) × Ω (2) m ( 0 ) = m 0 , u ( T , x ) = u T ( x , m ( T )) in Ω . 8 / 20
BB MFG Variational formulations Density constraints for MFGs A typical model of first order MFG system A typical model for a first order (deterministic) MFG system is the following: 2 |∇ u ( t , x ) | 2 = f ( x , m ( t )) − ∂ t u ( t , x ) + 1 in ( 0 , T ) × Ω ∂ t m ( t , x ) − ∇ · ( ∇ u ( t , x ) m ( t , x )) = 0 in ( 0 , T ) × Ω (2) m ( 0 ) = m 0 , u ( T , x ) = u T ( x , m ( T )) in Ω . u corresponds to the value function of a typical agent who controls his velocity α ( t ) and has to minimize his cost � T � 1 � 2 | α ( t ) | 2 + f ( x ( t ) , m ( t )) d t + u T ( x ( T ) , m ( T )) , 0 where x ′ ( s ) = α ( s ) and x ( 0 ) = x 0 . 8 / 20
BB MFG Variational formulations Density constraints for MFGs A typical model of first order MFG system A typical model for a first order (deterministic) MFG system is the following: 2 |∇ u ( t , x ) | 2 = f ( x , m ( t )) − ∂ t u ( t , x ) + 1 in ( 0 , T ) × Ω ∂ t m ( t , x ) − ∇ · ( ∇ u ( t , x ) m ( t , x )) = 0 in ( 0 , T ) × Ω (2) m ( 0 ) = m 0 , u ( T , x ) = u T ( x , m ( T )) in Ω . u corresponds to the value function of a typical agent who controls his velocity α ( t ) and has to minimize his cost � T � 1 � 2 | α ( t ) | 2 + f ( x ( t ) , m ( t )) d t + u T ( x ( T ) , m ( T )) , 0 where x ′ ( s ) = α ( s ) and x ( 0 ) = x 0 . The distribution of the other agents is represented by the density m ( t ) . Then their “feedback strategy” is given by α ( t , x ) = −∇ u ( t , x ) . 8 / 20
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