Introduction Variational formulations Density constraints and penalizations for MFGs Summary and perspectives Variational approach 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) C ONFERENCE ON O PTIMIZATION , T RANSPORTATION AND E QUILIBRIUM IN E CONOMICS , S EPT . 15-19, 2014, T ORONTO 1 / 18
Introduction Variational formulations Density constraints and penalizations for MFGs Summary and perspectives The content of the talk Basic models of Mean Field Games, after J.-M. Lasry and P .-L. 1 Lions Variational approaches for MFGs 2 A link with Optimal Transport and its Benamou-Brenier 3 formulation Second order stationary MFGs 4 Treating density constraints and penalizations for second order 5 stationary MFGs Summary, works in progress and open questions 6 2 / 18
Introduction Variational formulations Density constraints and penalizations for MFGs Summary and perspectives A short history of Mean Field Games This theory was introduced recently (in 2006-2007) by J.-M. .-L. Lions 12 3 in a series of papers and a series of Lasry and P .-L. Lions at Coll` lectures by P ege de France; 1 Lasry, J.-M., Lions, P .-L. Jeux ` a champ moyen I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris , (2006). 2 Lasry, J.-M., Lions, P .-L. Jeux ` a champ moyen II. Horizon fini et contrˆ ole optimal, C. R. Math. Acad. Sci. Paris , (2006). 3 Lasry, J.-M., Lions, P .-L. Mean field games, Jpn. J. Math. , (2007). 3 / 18
Introduction Variational formulations Density constraints and penalizations for MFGs Summary and perspectives A short history of Mean Field Games This theory was introduced recently (in 2006-2007) by J.-M. .-L. Lions 12 3 in a series of papers and a series of Lasry and P .-L. Lions at Coll` lectures by P ege de France; Analysis of differential games with a very large number of “small” players (agents); 1 Lasry, J.-M., Lions, P .-L. Jeux ` a champ moyen I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris , (2006). 2 Lasry, J.-M., Lions, P .-L. Jeux ` a champ moyen II. Horizon fini et contrˆ ole optimal, C. R. Math. Acad. Sci. Paris , (2006). 3 Lasry, J.-M., Lions, P .-L. Mean field games, Jpn. J. Math. , (2007). 3 / 18
Introduction Variational formulations Density constraints and penalizations for MFGs Summary and perspectives A short history of Mean Field Games This theory was introduced recently (in 2006-2007) by J.-M. .-L. Lions 12 3 in a series of papers and a series of Lasry and P .-L. Lions at Coll` lectures by P ege de France; 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) 1 Lasry, J.-M., Lions, P .-L. Jeux ` a champ moyen I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris , (2006). 2 Lasry, J.-M., Lions, P .-L. Jeux ` a champ moyen II. Horizon fini et contrˆ ole optimal, C. R. Math. Acad. Sci. Paris , (2006). 3 Lasry, J.-M., Lions, P .-L. Mean field games, Jpn. J. Math. , (2007). 3 / 18
Introduction Variational formulations Density constraints and penalizations for MFGs Summary and perspectives A short history of Mean Field Games This theory was introduced recently (in 2006-2007) by J.-M. .-L. Lions 12 3 in a series of papers and a series of Lasry and P .-L. Lions at Coll` lectures by P ege de France; 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 1 Lasry, J.-M., Lions, P .-L. Jeux ` a champ moyen I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris , (2006). 2 Lasry, J.-M., Lions, P .-L. Jeux ` a champ moyen II. Horizon fini et contrˆ ole optimal, C. R. Math. Acad. Sci. Paris , (2006). 3 Lasry, J.-M., Lions, P .-L. Mean field games, Jpn. J. Math. , (2007). 3 / 18
Introduction Variational formulations Density constraints and penalizations for MFGs Summary and perspectives A typical model for second order MFG in ( 0 , T ) × R d ( i ) − ∂ t u + ν ∆ u + H ( x , m , ∇ u ) = f ( x , m ) in ( 0 , T ) × R d ∂ t m − ν ∆ m − ∇ · ( ∇ p H ( x , m , ∇ u ) m ) = 0 ( ii ) in R d . ( iii ) m ( 0 ) = m 0 , u ( T , x ) = G ( x , m ( T )) (1) 4 / 18
Introduction Variational formulations Density constraints and penalizations for MFGs Summary and perspectives A typical model for second order MFG in ( 0 , T ) × R d ( i ) − ∂ t u + ν ∆ u + H ( x , m , ∇ u ) = f ( x , m ) in ( 0 , T ) × R d ∂ t m − ν ∆ m − ∇ · ( ∇ p H ( x , m , ∇ u ) m ) = 0 ( ii ) in R d . ( iii ) m ( 0 ) = m 0 , u ( T , x ) = G ( x , m ( T )) (1) 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; 4 / 18
Introduction Variational formulations Density constraints and penalizations for MFGs Summary and perspectives A typical model for second order MFG in ( 0 , T ) × R d ( i ) − ∂ t u + ν ∆ u + H ( x , m , ∇ u ) = f ( x , m ) in ( 0 , T ) × R d ∂ t m − ν ∆ m − ∇ · ( ∇ p H ( x , m , ∇ u ) m ) = 0 ( ii ) in R d . ( iii ) m ( 0 ) = m 0 , u ( T , x ) = G ( x , m ( T )) (1) 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; u is the value function of an arbitrary agent, m is the distribution of the agents; 4 / 18
Introduction Variational formulations Density constraints and penalizations for MFGs Summary and perspectives A heustistical 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. 5 / 18
Introduction Variational formulations Density constraints and penalizations for MFGs Summary and perspectives A heustistical 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 + G ( X T , m ( T )) , 0 where L is the usual Legendre-Flenchel conjugate of H w.r.t. the p variable. 5 / 18
Introduction Variational formulations Density constraints and penalizations for MFGs Summary and perspectives A heustistical 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 + G ( 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 ) = −∇ p H ( x , m , ∇ u ) . 5 / 18
Introduction Variational formulations Density constraints and penalizations for MFGs Summary and perspectives A typical model of first order MFG system A typical model for a first order (deterministic) MFG system is the following: 6 / 18
Introduction Variational formulations Density constraints and penalizations for MFGs Summary and perspectives 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 ) × R d ( i ) in ( 0 , T ) × R d ( ii ) ∂ t m ( t , x ) − ∇ · ( ∇ u ( t , x ) m ( t , x )) = 0 in R d . ( iii ) m ( 0 ) = m 0 , u ( T , x ) = G ( x , m ( T )) (2) 6 / 18
Introduction Variational formulations Density constraints and penalizations for MFGs Summary and perspectives 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 ) × R d ( i ) in ( 0 , T ) × R d ( ii ) ∂ t m ( t , x ) − ∇ · ( ∇ u ( t , x ) m ( t , x )) = 0 in R d . ( iii ) m ( 0 ) = m 0 , u ( T , x ) = G ( x , m ( T )) (2) 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 + G ( x ( T ) , m ( T )) , 0 where x ′ ( s ) = α ( s ) and x ( 0 ) = x 0 . 6 / 18
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