Mean field games and applications

Guilherme Mazanti

IFAC 2017 World Congress

iCODE WorkshopControl and Decision at Paris-Saclay

Challenge Energy

Toulouse — July 9th, 2017

LMO, Université Paris-SudUniversité Paris-Saclay

Mean field games Crowd motion MFGs with velocity constraint

Outline

1 Mean field gamesFrameworkA few examplesMain goalsA simple modelReferences

2 Crowd motionFrameworkMean field games with congestion penalizationMean field games with congestion constraint

3 Mean field games with velocity constraintThe modelThe Lagrangian approachExistence of a Lagrangian equilibriumCharacterization of equilibriaSimulations

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesFramework

Mean field games (MFGs) are differential games with a continuumof players / agents, assumed to be rational, indistinguishable, andinfluenced only by the average behavior of other players.

Ω: set of possible states for a player (Ω ⊂ Rd , a smoothmanifold, a network, etc.).ρt ∈ P(Ω): (Borel) probability measure (“=” probability density);ρt (x) = ρ(t , x) is the density of players on x ∈ Ω at time t .Dynamics of a player: γ(t) = f (t ,γ(t), u(t)), where f is thesame for all players, γ is the trajectory of the player in Ω, andu is a control each player can choose, with u(t) ∈ U.Choice of u: each player wants to minimize some costw T

0L(t ,γ(t), u(t), ρt ) dt + G(γ(T ), ρT ),

where L : [0, T ]×Ω×U ×P(Ω) → R and G : Ω×P(Ω) → Rare the same for all players and T ∈ (0, +∞].

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesFramework

Mean field games (MFGs) are differential games with a continuumof players / agents, assumed to be rational, indistinguishable, andinfluenced only by the average behavior of other players.

Ω: set of possible states for a player (Ω ⊂ Rd , a smoothmanifold, a network, etc.).

ρt ∈ P(Ω): (Borel) probability measure (“=” probability density);ρt (x) = ρ(t , x) is the density of players on x ∈ Ω at time t .Dynamics of a player: γ(t) = f (t ,γ(t), u(t)), where f is thesame for all players, γ is the trajectory of the player in Ω, andu is a control each player can choose, with u(t) ∈ U.Choice of u: each player wants to minimize some costw T

0L(t ,γ(t), u(t), ρt ) dt + G(γ(T ), ρT ),

where L : [0, T ]×Ω×U ×P(Ω) → R and G : Ω×P(Ω) → Rare the same for all players and T ∈ (0, +∞].

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesFramework

Mean field games (MFGs) are differential games with a continuumof players / agents, assumed to be rational, indistinguishable, andinfluenced only by the average behavior of other players.

Ω: set of possible states for a player (Ω ⊂ Rd , a smoothmanifold, a network, etc.).ρt ∈ P(Ω): (Borel) probability measure (“=” probability density);ρt (x) = ρ(t , x) is the density of players on x ∈ Ω at time t .

Dynamics of a player: γ(t) = f (t ,γ(t), u(t)), where f is thesame for all players, γ is the trajectory of the player in Ω, andu is a control each player can choose, with u(t) ∈ U.Choice of u: each player wants to minimize some costw T

0L(t ,γ(t), u(t), ρt ) dt + G(γ(T ), ρT ),

where L : [0, T ]×Ω×U ×P(Ω) → R and G : Ω×P(Ω) → Rare the same for all players and T ∈ (0, +∞].

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesFramework

Mean field games (MFGs) are differential games with a continuumof players / agents, assumed to be rational, indistinguishable, andinfluenced only by the average behavior of other players.

Ω: set of possible states for a player (Ω ⊂ Rd , a smoothmanifold, a network, etc.).ρt ∈ P(Ω): (Borel) probability measure (“=” probability density);ρt (x) = ρ(t , x) is the density of players on x ∈ Ω at time t .

Dynamics of a player: γ(t) = f (t ,γ(t), u(t)), where f is thesame for all players, γ is the trajectory of the player in Ω, andu is a control each player can choose, with u(t) ∈ U.Choice of u: each player wants to minimize some costw T

0L(t ,γ(t), u(t), ρt ) dt + G(γ(T ), ρT ),

where L : [0, T ]×Ω×U ×P(Ω) → R and G : Ω×P(Ω) → Rare the same for all players and T ∈ (0, +∞].

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesFramework

Mean field games (MFGs) are differential games with a continuumof players / agents, assumed to be rational, indistinguishable, andinfluenced only by the average behavior of other players.

Ω: set of possible states for a player (Ω ⊂ Rd , a smoothmanifold, a network, etc.).ρt ∈ P(Ω): (Borel) probability measure (“=” probability density);ρt (x) = ρ(t , x) is the density of players on x ∈ Ω at time t .Dynamics of a player: γ(t) = f (t ,γ(t), u(t)), where f is thesame for all players, γ is the trajectory of the player in Ω, andu is a control each player can choose, with u(t) ∈ U.

Choice of u: each player wants to minimize some costw T

0L(t ,γ(t), u(t), ρt ) dt + G(γ(T ), ρT ),

where L : [0, T ]×Ω×U ×P(Ω) → R and G : Ω×P(Ω) → Rare the same for all players and T ∈ (0, +∞].

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesFramework

Mean field games (MFGs) are differential games with a continuumof players / agents, assumed to be rational, indistinguishable, andinfluenced only by the average behavior of other players.

Ω: set of possible states for a player (Ω ⊂ Rd , a smoothmanifold, a network, etc.).ρt ∈ P(Ω): (Borel) probability measure (“=” probability density);ρt (x) = ρ(t , x) is the density of players on x ∈ Ω at time t .Dynamics of a player: γ(t) = f (t ,γ(t), u(t)), where f is thesame for all players, γ is the trajectory of the player in Ω, andu is a control each player can choose, with u(t) ∈ U.

Choice of u: each player wants to minimize some costw T

0L(t ,γ(t), u(t), ρt ) dt + G(γ(T ), ρT ),

where L : [0, T ]×Ω×U ×P(Ω) → R and G : Ω×P(Ω) → Rare the same for all players and T ∈ (0, +∞].

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesFramework

Mean field games (MFGs) are differential games with a continuumof players / agents, assumed to be rational, indistinguishable, andinfluenced only by the average behavior of other players.

Ω: set of possible states for a player (Ω ⊂ Rd , a smoothmanifold, a network, etc.).ρt ∈ P(Ω): (Borel) probability measure (“=” probability density);ρt (x) = ρ(t , x) is the density of players on x ∈ Ω at time t .Dynamics of a player: γ(t) = f (t ,γ(t), u(t)), where f is thesame for all players, γ is the trajectory of the player in Ω, andu is a control each player can choose, with u(t) ∈ U.Choice of u: each player wants to minimize some costw T

0L(t ,γ(t), u(t), ρt ) dt + G(γ(T ), ρT ),

where L : [0, T ]×Ω×U ×P(Ω) → R and G : Ω×P(Ω) → Rare the same for all players and T ∈ (0, +∞].

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesFramework

Mean field games (MFGs) are differential games with a continuumof players / agents, assumed to be rational, indistinguishable, andinfluenced only by the average behavior of other players.

Ω: set of possible states for a player (Ω ⊂ Rd , a smoothmanifold, a network, etc.).ρt ∈ P(Ω): (Borel) probability measure (“=” probability density);ρt (x) = ρ(t , x) is the density of players on x ∈ Ω at time t .Dynamics of a player: γ(t) = f (t ,γ(t), u(t)), where f is thesame for all players, γ is the trajectory of the player in Ω, andu is a control each player can choose, with u(t) ∈ U.Choice of u: each player wants to minimize some costw T

0L(t ,γ(t), u(t), ρt ) dt + G(γ(T ), ρT ),

where L : [0, T ]×Ω×U ×P(Ω) → R and G : Ω×P(Ω) → Rare the same for all players and T ∈ (0, +∞].

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesA few examples

Example 1: Mexican wave (adapted from [Guéant, Lasry, Lions;2011]).Setting: people in a stadium doing a Mexican wave.

Ω = S1 × [0, 1]: θ ∈ S1 represents position at the stadium,z ∈ [0, 1] represents whether a person is seated (0), standing(1), or at an intermediate position.ρ0 ∈ P(Ω): initial distribution of people.(θ(t)z(t)

)=

(0

u(t)

), where (θ(t), z(t)) ∈ Ω and u(t) ∈ R.

Each person wants to minimizew T

0

[|u(t)|2

2︸ ︷︷ ︸cost ofmoving

+α z(t)p(1 – z(t))q︸ ︷︷ ︸cost of intermediate

positions

+βwΩ

(z(t) – ζ)2χ(ϑ – θ(t)) dρt (ζ, ϑ)︸ ︷︷ ︸wish to behave as neighbors

]dt ,

where α,β > 0, p, q ≥ 1, χ ≥ 0 on S1,rS1 χ(ϑ) dϑ = 1, and χ

is concentrated around zero.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesA few examples

Example 1: Mexican wave (adapted from [Guéant, Lasry, Lions;2011]).Setting: people in a stadium doing a Mexican wave.

Ω = S1 × [0, 1]: θ ∈ S1 represents position at the stadium,z ∈ [0, 1] represents whether a person is seated (0), standing(1), or at an intermediate position.ρ0 ∈ P(Ω): initial distribution of people.(θ(t)z(t)

)=

(0

u(t)

), where (θ(t), z(t)) ∈ Ω and u(t) ∈ R.

Each person wants to minimizew T

0

[|u(t)|2

2︸ ︷︷ ︸cost ofmoving

+α z(t)p(1 – z(t))q︸ ︷︷ ︸cost of intermediate

positions

+βwΩ

(z(t) – ζ)2χ(ϑ – θ(t)) dρt (ζ, ϑ)︸ ︷︷ ︸wish to behave as neighbors

]dt ,

where α,β > 0, p, q ≥ 1, χ ≥ 0 on S1,rS1 χ(ϑ) dϑ = 1, and χ

is concentrated around zero.Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesA few examples

Example 2: Jet lag (adapted from the talk [Carmona; 2017]).Setting: ≈ 104 neuronal cells responsible for circadian rhythm.Each cell behaves like an oscillator.

Ω = S1: θ ∈ S1 represents oscillatory phase of a cell.ρ0 ∈ P(Ω): initial distribution of phases.θ(t) = ω0 + u(t), where θ(t) ∈ Ω, u(t) ∈ R, and ω0 ≈ 2π

24.5 .Cost:

lim supT→+∞

1T

w T

0

[|u(t)|2

2︸ ︷︷ ︸cost of the

control

–αw 2π

0cos (θ(t) – ϑ) dρt (ϑ)︸ ︷︷ ︸

synchronization withother cells

– β cos (θ(t) –ωSt –φ(t))︸ ︷︷ ︸synchronization with

the Sun

]dt ,

where α,β > 0, ωS = 2π24 , and φ(t) corresponds to the phase

of the current time zone.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesA few examples

Example 2: Jet lag (adapted from the talk [Carmona; 2017]).Setting: ≈ 104 neuronal cells responsible for circadian rhythm.Each cell behaves like an oscillator.

Ω = S1: θ ∈ S1 represents oscillatory phase of a cell.ρ0 ∈ P(Ω): initial distribution of phases.θ(t) = ω0 + u(t), where θ(t) ∈ Ω, u(t) ∈ R, and ω0 ≈ 2π

24.5 .Cost:

lim supT→+∞

1T

w T

0

[|u(t)|2

2︸ ︷︷ ︸cost of the

control

–αw 2π

0cos (θ(t) – ϑ) dρt (ϑ)︸ ︷︷ ︸

synchronization withother cells

– β cos (θ(t) –ωSt –φ(t))︸ ︷︷ ︸synchronization with

the Sun

]dt ,

where α,β > 0, ωS = 2π24 , and φ(t) corresponds to the phase

of the current time zone.Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesMain goals

Given a MFG model (i.e., Ω, ρ0 ∈ P(Ω), a control systemγ(t) = f (t ,γ(t), u(t)), and an optimization criterion), the goals are to

characterize equilibrium states, i.e., states where almost allagents satisfy their optimization criterion;

obtain existence (and possibly uniqueness) results for theequations characterizing equilibria;

obtain further properties of equilibria states (e.g. regularity);

study convergence to equilibria;

study convergence of games with N players to mean fieldgames as N → ∞;

numerically approximate equilibria; etc.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesA simple model

[Cardaliaguet, Notes on Mean Field Games; 2013], based on P.-L.Lions’ lectures at Collège de France.

Ω = Rd ; ρ0 ∈ P(Rd ); γ(t) = u(t), u(t) ∈ Rd ;

Minimizew T

0

[ |u(t)|2

2+ F (γ(t), ρt )

]dt + G(γ(T ), ρT ).

Equilibrium:

Given ρ : [0, T ] → P(Ω), compute optimal trajectories.

Define an optimal flow: t 7→ Φ(t , x) is an optimal trajectorystarting from x.

ρ is an equilibrium if Φ is well-defined and Φ(t , ·)#ρ0 = ρt forevery t ∈ [0, T ].

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesA simple model

Theorem

If F and G are continuous and satisfy some C2 bounds, and if ρ0 isabsolutely continuous with bounded and compactly supporteddensity, then there exists an equilibrium ρ for this MFG.

Moreover,there exists ψ : [0, T ] × Rd → R such that (ρ,ψ) solves the meanfield equations

∂tρ(t , x) – div (∇xψ(t , x)ρ(t , x)) = 0, (t , x) ∈ (0, T ) × Rd ,

–∂tψ(t , x) +12|∇xψ(t , x)|2 = F (x, ρt ), (t , x) ∈ (0, T ) × Rd ,

ρ(0, x) = ρ0(x), x ∈ Rd ,

ψ(T , x) = G(x, ρT ), x ∈ Rd .

Coupled system of nonlinear PDEs: a continuity equation forwardin time (in the sense of distributions) and a Hamilton–Jacobiequation backward in time (in the viscosity sense).

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesA simple model

Theorem

If F and G are continuous and satisfy some C2 bounds, and if ρ0 isabsolutely continuous with bounded and compactly supporteddensity, then there exists an equilibrium ρ for this MFG. Moreover,there exists ψ : [0, T ] × Rd → R such that (ρ,ψ) solves the meanfield equations

∂tρ(t , x) – div (∇xψ(t , x)ρ(t , x)) = 0, (t , x) ∈ (0, T ) × Rd ,

–∂tψ(t , x) +12|∇xψ(t , x)|2 = F (x, ρt ), (t , x) ∈ (0, T ) × Rd ,

ρ(0, x) = ρ0(x), x ∈ Rd ,

ψ(T , x) = G(x, ρT ), x ∈ Rd .

Coupled system of nonlinear PDEs: a continuity equation forwardin time (in the sense of distributions) and a Hamilton–Jacobiequation backward in time (in the viscosity sense).

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesA simple model

Theorem

If F and G are continuous and satisfy some C2 bounds, and if ρ0 isabsolutely continuous with bounded and compactly supporteddensity, then there exists an equilibrium ρ for this MFG. Moreover,there exists ψ : [0, T ] × Rd → R such that (ρ,ψ) solves the meanfield equations

∂tρ(t , x) – div (∇xψ(t , x)ρ(t , x)) = 0, (t , x) ∈ (0, T ) × Rd ,

–∂tψ(t , x) +12|∇xψ(t , x)|2 = F (x, ρt ), (t , x) ∈ (0, T ) × Rd ,

ρ(0, x) = ρ0(x), x ∈ Rd ,

ψ(T , x) = G(x, ρT ), x ∈ Rd .

Coupled system of nonlinear PDEs: a continuity equation forwardin time (in the sense of distributions) and a Hamilton–Jacobiequation backward in time (in the viscosity sense).

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesReferences

Origins of MFGs: [Huang, Malhamé, Caines; 2006], [Lasry, Lions; 2006], [Lasry,Lions; 2006], [Huang, Caines, Malhamé; 2007], [Lasry, Lions; 2007].

A tutorial on MFGs: [Cardaliaguet, Notes on Mean Field Games; 2013].

More general models: [Guéant; 2012], [Feleqi; 2013], [Fornasier, Piccoli, Rossi;2014], [Gomes, Saúde; 2014], [Cirant; 2015], [Carmona, Delarue, Lacker; 2016],[Moon, Basar; 2017].

Limits of games with N players: [Lasry, Lions; 2006], [Huang, Caines, Malhamé;2007], [Bardi, Priuli; 2013], [Kolokoltsov, Troeva, Yang; 2014], [Fischer; 2017].

Numerical approximations for MFGs: [Achdou, Camilli, Capuzzo-Dolcetta; 2012],[Guéant; 2012], [Achdou; 2013], [Carlini, Silva; 2014].

Variational MFGs: [Mészáros, Silva; 2015], [Cardaliaguet, Mészáros, Santambrogio;2016], [Prosinski, Santambrogio; 2016], [Benamou, Carlier, Santambrogio; 2017].

Limit of large time horizon T → ∞: [Cardaliaguet, Lasry, Lions, Porretta; 2013].

MFGs on graphs and networks: [Gomes, Mohr, Souza; 2013], [Camilli, Carlini,Marchi; 2015], [Guéant; 2015], [Cacace, Camilli, Marchi; 2017].

Master equation: [Bensoussan, Frehse, Yam; 2015], [Cardaliaguet, Delarue, Lasry,Lions; 2015].

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field gamesReferences

Origins of MFGs: [Huang, Malhamé, Caines; 2006], [Lasry, Lions; 2006], [Lasry,Lions; 2006], [Huang, Caines, Malhamé; 2007], [Lasry, Lions; 2007].

A tutorial on MFGs: [Cardaliaguet, Notes on Mean Field Games; 2013].

More general models: [Guéant; 2012], [Feleqi; 2013], [Fornasier, Piccoli, Rossi;2014], [Gomes, Saúde; 2014], [Cirant; 2015], [Carmona, Delarue, Lacker; 2016],[Moon, Basar; 2017].

Limits of games with N players: [Lasry, Lions; 2006], [Huang, Caines, Malhamé;2007], [Bardi, Priuli; 2013], [Kolokoltsov, Troeva, Yang; 2014], [Fischer; 2017].

Numerical approximations for MFGs: [Achdou, Camilli, Capuzzo-Dolcetta; 2012],[Guéant; 2012], [Achdou; 2013], [Carlini, Silva; 2014].

Variational MFGs: [Mészáros, Silva; 2015], [Cardaliaguet, Mészáros, Santambrogio;2016], [Prosinski, Santambrogio; 2016], [Benamou, Carlier, Santambrogio; 2017].

Limit of large time horizon T → ∞: [Cardaliaguet, Lasry, Lions, Porretta; 2013].

MFGs on graphs and networks: [Gomes, Mohr, Souza; 2013], [Camilli, Carlini,Marchi; 2015], [Guéant; 2015], [Cacace, Camilli, Marchi; 2017].

Master equation: [Bensoussan, Frehse, Yam; 2015], [Cardaliaguet, Delarue, Lasry,Lions; 2015].

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionFramework

Goal: Provide mathematical models for the motion of a largenumber of people.

Shibuya Crossing, Tokyo, 2014.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionFramework

A vast literature, with several different models: [Henderson; 1971],[Gipps, Marksjö; 1985], [Borgers, Timmermans; 1986], [Yuhaski, Smith;1989], [Løvås; 1994], [Helbing, Molnár; 1995], [Helbing, Farkas, Vicsek;2000], [Blue, Adler; 2001], [Burstedde, Klauck, Schadschneider, Zittartz;2001], [Hoogendoorn, Bovy; 2004], [Maury, Venel; 2009], [Maury,Roudneff-Chupin, Santambrogio, Venel; 2011], [Cristiani, Priuli, Tosin;2015], ...

Microscopic vs. macroscopic.

Passive vs. active.

Deterministic vs. stochastic.

How to handle congestion.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionFramework

A vast literature, with several different models: [Henderson; 1971],[Gipps, Marksjö; 1985], [Borgers, Timmermans; 1986], [Yuhaski, Smith;1989], [Løvås; 1994], [Helbing, Molnár; 1995], [Helbing, Farkas, Vicsek;2000], [Blue, Adler; 2001], [Burstedde, Klauck, Schadschneider, Zittartz;2001], [Hoogendoorn, Bovy; 2004], [Maury, Venel; 2009], [Maury,Roudneff-Chupin, Santambrogio, Venel; 2011], [Cristiani, Priuli, Tosin;2015], ...

Microscopic vs. macroscopic.

Passive vs. active.

Deterministic vs. stochastic.

How to handle congestion.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionMean field games with congestion penalization

Based on [Benamou, Carlier, Santambrogio; 2017]; parametersα > 0, m > 0.

Ω = Td = Rd /Zd ; ρ0 ∈ P(Td ); γ(t) = u(t), u(t) ∈ Rd ;

Minimizew T

0

[|u(t)|2

2︸ ︷︷ ︸speed

penalization

+α(ρ(t ,γ(t))

)m︸ ︷︷ ︸congestionpenalization

]dt + G(γ(T ))︸ ︷︷ ︸

target

.

As before, equilibria characterized by the mean field equations

∂tρ(t , x) – div (∇xψ(t , x)ρ(t , x)) = 0, (t , x) ∈ (0, T ) × Td ,

– ∂tψ(t , x) +12|∇xψ(t , x)|2 = αρ(t , x)m, (t , x) ∈ (0, T ) × Td ,

ρ(0, x) = ρ0(x), x ∈ Td ,

ψ(T , x) = G(x), x ∈ Td .

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionMean field games with congestion penalization

Based on [Benamou, Carlier, Santambrogio; 2017]; parametersα > 0, m > 0.

Ω = Td = Rd /Zd ; ρ0 ∈ P(Td ); γ(t) = u(t), u(t) ∈ Rd ;

Minimizew T

0

[|u(t)|2

2︸ ︷︷ ︸speed

penalization

+α(ρ(t ,γ(t))

)m︸ ︷︷ ︸congestionpenalization

]dt + G(γ(T ))︸ ︷︷ ︸

target

.

As before, equilibria characterized by the mean field equations

∂tρ(t , x) – div (∇xψ(t , x)ρ(t , x)) = 0, (t , x) ∈ (0, T ) × Td ,

– ∂tψ(t , x) +12|∇xψ(t , x)|2 = αρ(t , x)m, (t , x) ∈ (0, T ) × Td ,

ρ(0, x) = ρ0(x), x ∈ Td ,

ψ(T , x) = G(x), x ∈ Td .

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionMean field games with congestion penalization

Equilibria also minimize a global energy: (ρ, v = –∇ψ) minimizew T

0

wTd

[ρ(t , x)

|v(t , x)|2

2+

α

m + 1ρ(t , x)m+1

]dx dt +

wTd

G(x)ρ(T , x) dx

among all (ρ, v) such that ∂tρ + div(vρ) = 0.

cost:

w T

0

[||2

2+ αρ(t , )m

]dt + G().

Global energy and total cost are not the same!Total cost at MFG equilibrium – Optimal total cost =: Cost of anarchy.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionMean field games with congestion penalization

Equilibria also minimize a global energy: (ρ, v = –∇ψ) minimizew T

0

wTd

[ρ(t , x)

|v(t , x)|2

2+

α

m + 1ρ(t , x)m+1

]dx dt +

wTd

G(x)ρ(T , x) dx

among all (ρ, v) such that ∂tρ + div(vρ) = 0.

Individual cost:

w T

0

[|γ(t)|2

2+ αρ(t ,γ(t))m

]dt + G(γ(T )).

Global energy and total cost are not the same!Total cost at MFG equilibrium – Optimal total cost =: Cost of anarchy.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionMean field games with congestion penalization

Equilibria also minimize a global energy: (ρ, v = –∇ψ) minimizew T

0

wTd

[ρ(t , x)

|v(t , x)|2

2+

α

m + 1ρ(t , x)m+1

]dx dt +

wTd

G(x)ρ(T , x) dx

among all (ρ, v) such that ∂tρ + div(vρ) = 0.

Total cost:

w T

0

wTd

[ρ(t , x)

|v(t , x)|2

2+ αρ(t , x)m+1

]dx dt +

wTd

G(x)ρ(t , x) dx.

Global energy and total cost are not the same!Total cost at MFG equilibrium – Optimal total cost =: Cost of anarchy.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionMean field games with congestion penalization

Equilibria also minimize a global energy: (ρ, v = –∇ψ) minimizew T

0

wTd

[ρ(t , x)

|v(t , x)|2

2+

α

m + 1ρ(t , x)m+1

]dx dt +

wTd

G(x)ρ(T , x) dx

among all (ρ, v) such that ∂tρ + div(vρ) = 0.

Total cost:

w T

0

wTd

[ρ(t , x)

|v(t , x)|2

2+ αρ(t , x)m+1

]dx dt +

wTd

G(x)ρ(t , x) dx.

Global energy and total cost are not the same!Total cost at MFG equilibrium – Optimal total cost =: Cost of anarchy.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionMean field games with congestion penalization

Thanks to the fact that (ρ, v) minimizes a global energy, one canobtain more regularity on ρ.

Theorem (Benamou, Carlier, Santambrogio; 2017)

The equilibrium density ρ satisfies ρm+1

2 ∈ H1loc((0, T ) × Td ).

Proof based on duality arguments. See also [Brenier; 1999],[Cardaliaguet, Mészáros, Santambrogio; 2016].

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionMean field games with congestion penalization

Simulations by [Benamou, Carlier, Santambrogio; 2017], α = 1,m = 5.

Function G

Density ρ

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionMean field games with congestion constraint

Based on [Santambrogio; 2012], [Cardaliaguet, Mészáros,Santambrogio; 2016], and [Benamou, Carlier, Santambrogio;2017].

Main idea: replace the congestion penalization (ρ(t , x))m in thecost by a constraint ρ(t , x) ≤ ρmax.

Naive approach: every agent minimizes the non-penalized costw T

0

|u(t)|2

2dt + G(γ(T ))

while respecting the constraint ρ(t ,γ(t)) ≤ ρmax. But one extraagent does not violate the constraint...

Better idea: consider the minimization of the global energyw T

0

wTdρ(t , x)

|v(t , x)|2

2dx dt +

wTd

G(x)ρ(T , x) dx

among all (ρ, v) such that ∂tρ + div(vρ) = 0 and ρ ≤ ρmax.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionMean field games with congestion constraint

Based on [Santambrogio; 2012], [Cardaliaguet, Mészáros,Santambrogio; 2016], and [Benamou, Carlier, Santambrogio;2017].

Main idea: replace the congestion penalization (ρ(t , x))m in thecost by a constraint ρ(t , x) ≤ ρmax.

Naive approach: every agent minimizes the non-penalized costw T

0

|u(t)|2

2dt + G(γ(T ))

while respecting the constraint ρ(t ,γ(t)) ≤ ρmax. But one extraagent does not violate the constraint...

Better idea: consider the minimization of the global energyw T

0

wTdρ(t , x)

|v(t , x)|2

2dx dt +

wTd

G(x)ρ(T , x) dx

among all (ρ, v) such that ∂tρ + div(vρ) = 0 and ρ ≤ ρmax.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionMean field games with congestion constraint

This corresponds to taking α = 1ρm+1

maxin the global energy

w T

0

wTd

[ρ(t , x)

|v(t , x)|2

2+

α

m + 1ρ(t , x)m+1

]dx dt +

wTd

G(x)ρ(T , x) dx

and letting m → +∞ formally.

Previous mean field equations:∂tρ(t , x) – div (∇xψ(t , x)ρ(t , x)) = 0, (t , x) ∈ (0, T ) × Td ,

–∂tψ(t , x) +12|∇xψ(t , x)|2 =

(ρ(t , x)ρmax

)m

, (t , x) ∈ (0, T ) × Td ,

ρ(0, x) = ρ0(x), x ∈ Td ,ψ(T , x) = G(x), x ∈ Td .p(t , x) ≥ 0, ρ(t , x) ≤ ρmax,

(t , x) ∈ (0, T ) × Td .p(t , x) [ρmax – ρ(t , x)] = 0,

p: pressure due to the density constraint. Similar to fluidmechanics.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionMean field games with congestion constraint

This corresponds to taking α = 1ρm+1

maxin the global energy

w T

0

wTd

[ρ(t , x)

|v(t , x)|2

2+

α

m + 1ρ(t , x)m+1

]dx dt +

wTd

G(x)ρ(T , x) dx

and letting m → +∞ formally.

New mean field equations:

∂tρ(t , x) – div (∇xψ(t , x)ρ(t , x)) = 0, (t , x) ∈ (0, T ) × Td ,

–∂tψ(t , x) +12|∇xψ(t , x)|2 = p(t , x), (t , x) ∈ (0, T ) × Td ,

ρ(0, x) = ρ0(x), x ∈ Td ,ψ(T , x) = G(x) + p(T , x), x ∈ Td ,

p(t , x) ≥ 0, ρ(t , x) ≤ ρmax,(t , x) ∈ (0, T ) × Td .p(t , x) [ρmax – ρ(t , x)] = 0,

p: pressure due to the density constraint. Similar to fluidmechanics.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionMean field games with congestion constraint

Each agent now minimizesw T

0

[|u(t)|2

2+ p(t ,γ(t))

]dt + G(γ(T )) + p(T ,γ(T )).

p: price to discourage agents to go through saturated areas.

Results proved in [Cardaliaguet, Mészáros, Santambrogio; 2016]:

existence of solutions to the mean field equations;

convergence of solutions as m → +∞;

regularity results.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Crowd motionMean field games with congestion constraint

Simulations by [Benamou, Carlier, Santambrogio; 2017].

Penalizationm = 5

Constraint

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintThe model

Based on ongoing works with F. Santambrogio and S. Dweik.

Main ideas:Propose a model where the final time T is not fixed.Maximal speed of agents depends on local congestion.

Model:Ω ⊂ Rd non-empty, open, and bounded; ρ0 ∈ P(Ω).Γ ⊂ ∂Ω non-empty and closed.Maximal speed of the agents is bounded:γ(t) = f (ρt ,γ(t))u(t), u(t) ∈ B(0; 1) = closed unit ball.

Optimization criterion: agents want to leave Ω through Γ inminimal time.

infT ≥ 0 | γ(t) = f (ρt ,γ(t))u(t), u : R+ → B(0, 1),

γ(0) = x, γ(T ) ∈ Γ , γ(t) ∈ Ω for t ∈ [0, T ],

γ(t) = 0 for t > T .

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintThe model

Based on ongoing works with F. Santambrogio and S. Dweik.

Main ideas:Propose a model where the final time T is not fixed.Maximal speed of agents depends on local congestion.

Model:Ω ⊂ Rd non-empty, open, and bounded; ρ0 ∈ P(Ω).Γ ⊂ ∂Ω non-empty and closed.Maximal speed of the agents is bounded:γ(t) = f (ρt ,γ(t))u(t), u(t) ∈ B(0; 1) = closed unit ball.

Optimization criterion: agents want to leave Ω through Γ inminimal time.

infT ≥ 0 | γ(t) = f (ρt ,γ(t))u(t), u : R+ → B(0, 1),

γ(0) = x, γ(T ) ∈ Γ , γ(t) ∈ Ω for t ∈ [0, T ],

γ(t) = 0 for t > T .

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintThe model

Based on ongoing works with F. Santambrogio and S. Dweik.

Main ideas:Propose a model where the final time T is not fixed.Maximal speed of agents depends on local congestion.

Model:Ω ⊂ Rd non-empty, open, and bounded; ρ0 ∈ P(Ω).Γ ⊂ ∂Ω non-empty and closed.Maximal speed of the agents is bounded:γ(t) = f (ρt ,γ(t))u(t), u(t) ∈ B(0; 1) = closed unit ball.

Optimization criterion: agents want to leave Ω through Γ inminimal time.

infT ≥ 0 | γ(t) = f (ρt ,γ(t))u(t), u : R+ → B(0, 1),

γ(0) = x, γ(T ) ∈ Γ , γ(t) ∈ Ω for t ∈ [0, T ],

γ(t) = 0 for t > T .

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintThe Lagrangian approach

Eulerian approach: ρ : R+ → P(Ω) is a curve on the set ofmeasures. Motion is described by the density and the velocityfield of the population.

Lagrangian approach: Q ∈ P(C), where C = C(R+,Ω), is ameasure on the set of curves. Motion is described by thetrajectory of each agent.

Lagrangian framework for mean field games already used in theliterature, cf. e.g. the survey in [Benamou, Carlier, Santambrogio;2017].

Link between Eulerian and Lagrangian: ρt = et #Q, whereet : C → Ω is the evaluation at time t of a curve, et (γ) = γ(t).

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintThe Lagrangian approach

Eulerian approach: ρ : R+ → P(Ω) is a curve on the set ofmeasures. Motion is described by the density and the velocityfield of the population.

Lagrangian approach: Q ∈ P(C), where C = C(R+,Ω), is ameasure on the set of curves. Motion is described by thetrajectory of each agent.

Lagrangian framework for mean field games already used in theliterature, cf. e.g. the survey in [Benamou, Carlier, Santambrogio;2017].

Link between Eulerian and Lagrangian: ρt = et #Q, whereet : C → Ω is the evaluation at time t of a curve, et (γ) = γ(t).

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintThe Lagrangian approach

DefinitionA measure Q ∈ P(C) is a Lagrangian equilibrium of the mean fieldgame if e0#Q = ρ0 and Q-almost every γ ∈ C is optimal for

infT ≥ 0 | γ(t) = f (et #Q,γ(t))u(t), u : R+ → B(0, 1),

γ(0) = x, γ(T ) ∈ Γ , γ(t) ∈ Ω for t ∈ [0, T ],

γ(t) = 0 for t > T .

In the sequel, we consider

the existence of a Lagrangian equilibrium;

the characterization of equilibria by a system of mean fieldequations;

some simulations.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintExistence of a Lagrangian equilibrium

We use the following assumptions on f and Ω.

Hypotheses

f : P(Ω) ×Ω→ R+ is Lipschitz continuous andfmax = sup

µ∈P(Ω)x∈Ω

f (µ, x) < +∞, fmin = infµ∈P(Ω)

x∈Ω

f (µ, x) > 0.

∂Ω is C1,1 (can be generalized).

With no loss of generality (change in time scale): fmax = 1.

Distance in P(Ω): Wasserstein distance

W1(µ,ν) = minγ∈P(Ω×Ω)

π1#γ=µ, π2#γ=ν

wΩ×Ω

|x – y| dγ(x, y).

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintExistence of a Lagrangian equilibrium

We use the following assumptions on f and Ω.

Hypotheses

f : P(Ω) ×Ω→ R+ is Lipschitz continuous andfmax = sup

µ∈P(Ω)x∈Ω

f (µ, x) < +∞, fmin = infµ∈P(Ω)

x∈Ω

f (µ, x) > 0.

∂Ω is C1,1 (can be generalized).

With no loss of generality (change in time scale): fmax = 1.

Distance in P(Ω): Wasserstein distance

W1(µ,ν) = minγ∈P(Ω×Ω)

π1#γ=µ, π2#γ=ν

wΩ×Ω

|x – y| dγ(x, y).

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintExistence of a Lagrangian equilibrium

TheoremUnder the previous hypotheses, there exists a Lagrangianequilibrium Q ∈ P(C) for this game.

Main ideas:

For fixed Q ∈ P(C), compute the set ΓΓΓQ ⊂ C of all optimaltrajectories for the measure Q.To study the optimal control problem, we introduce the value function

τQ(t0, x0) = infT ≥ 0 | γ(t) = f (et #Q,γ(t))u(t), u : R+ → B(0, 1),

γ(t0) = x0, γ(t0 + T ) ∈ Γ , γ(t) ∈ Ω for t ∈ [t0, t0 + T ],

γ(t) = 0 for t > t0 + T .

Define F (Q) = Q | e0#Q = ρ0 and Q(ΓΓΓQ) = 1. Equilibrium⇐⇒ fixed point of the set-valued map F , i.e., Q ∈ F (Q).

Conclude by Kakutani fixed point theorem.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintExistence of a Lagrangian equilibrium

TheoremUnder the previous hypotheses, there exists a Lagrangianequilibrium Q ∈ P(C) for this game.

Main ideas:

For fixed Q ∈ P(C), compute the set ΓΓΓQ ⊂ C of all optimaltrajectories for the measure Q.To study the optimal control problem, we introduce the value function

τQ(t0, x0) = infT ≥ 0 | γ(t) = f (et #Q,γ(t))u(t), u : R+ → B(0, 1),

γ(t0) = x0, γ(t0 + T ) ∈ Γ , γ(t) ∈ Ω for t ∈ [t0, t0 + T ],

γ(t) = 0 for t > t0 + T .

Define F (Q) = Q | e0#Q = ρ0 and Q(ΓΓΓQ) = 1. Equilibrium⇐⇒ fixed point of the set-valued map F , i.e., Q ∈ F (Q).

Conclude by Kakutani fixed point theorem.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintCharacterization of equilibria

We have proved the existence of a Lagrangian equilibrium to theminimal time mean field game.

Advantage: easier than to prove than in the Eulerianapproach. Application of Kakutani fixed point theoremrequires fewer properties of the optimal trajectories.

Drawback: we have no information on ρt = et #Q.

Goal: characterize τQ and ρ as solutions of a system of PDEs.

We will treat in the sequel only the case Γ = ∂Ω.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintCharacterization of equilibria

We have proved the existence of a Lagrangian equilibrium to theminimal time mean field game.

Advantage: easier than to prove than in the Eulerianapproach. Application of Kakutani fixed point theoremrequires fewer properties of the optimal trajectories.

Drawback: we have no information on ρt = et #Q.

Goal: characterize τQ and ρ as solutions of a system of PDEs.

We will treat in the sequel only the case Γ = ∂Ω.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintCharacterization of equilibria

In addition to the previous hypotheses, we also assume:

Hypotheses

f : P(Ω) ×Ω→ R∗+ is given by f (µ, x) = K [E(µ, x)], with

E(µ, x) =wΩχ(x – y)η(y)dµ(y),

K ∈ C1,1(R+,R∗+) is bounded, χ ∈ C1,1(Rd ,R+), and

η ∈ C1,1(Rd ,R+) with η(x) = 0 and ∇η(x) = 0 for x ∈ ∂Ω.

An agent looks around (according to the convolution kernel χ) tosee the local concentration of agents, ignoring people who alreadyleft (η(y) = 0 on Γ , η(y) = 1 outside a neighborhood of Γ ).

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintCharacterization of equilibria

TheoremUnder the previous assumptions, τQ and ρ solve the mean fieldequations

∂tρ(t , x) – divx

[f (ρt , x)

∇xτQ(t , x)|∇xτQ(t , x)|

ρ(t , x)

]= 0, R+ ×Ω,

– ∂tτQ(t , x) + |∇xτQ(t , x)| f (ρt , x) – 1 = 0, R+ ×Ω,

ρ(0, x) = ρ0(x), Ω,

τQ(t , x) = 0, R+ × ∂Ω.

Continuity equation satisfied in the sense of distributions,Hamilton–Jacobi equation satisfied in the viscosity sense.

Hamilton–Jacobi equation can be obtained by standard techniqueson optimal control (and does not use the previous assumptions).But the situation is much harder for the continuity equation.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintCharacterization of equilibria

TheoremUnder the previous assumptions, τQ and ρ solve the mean fieldequations

∂tρ(t , x) – divx

[f (ρt , x)

∇xτQ(t , x)|∇xτQ(t , x)|

ρ(t , x)

]= 0, R+ ×Ω,

– ∂tτQ(t , x) + |∇xτQ(t , x)| f (ρt , x) – 1 = 0, R+ ×Ω,

ρ(0, x) = ρ0(x), Ω,

τQ(t , x) = 0, R+ × ∂Ω.

Continuity equation satisfied in the sense of distributions,Hamilton–Jacobi equation satisfied in the viscosity sense.

Hamilton–Jacobi equation can be obtained by standard techniqueson optimal control (and does not use the previous assumptions).But the situation is much harder for the continuity equation.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintCharacterization of equilibria

TheoremUnder the previous assumptions, τQ and ρ solve the mean fieldequations

∂tρ(t , x) – divx

[f (ρt , x)

∇xτQ(t , x)|∇xτQ(t , x)|

ρ(t , x)

]= 0, R+ ×Ω,

– ∂tτQ(t , x) + |∇xτQ(t , x)| f (ρt , x) – 1 = 0, R+ ×Ω,

ρ(0, x) = ρ0(x), Ω,

τQ(t , x) = 0, R+ × ∂Ω.

Continuity equation satisfied in the sense of distributions,Hamilton–Jacobi equation satisfied in the viscosity sense.

Hamilton–Jacobi equation can be obtained by standard techniqueson optimal control (and does not use the previous assumptions).

But the situation is much harder for the continuity equation.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintCharacterization of equilibria

TheoremUnder the previous assumptions, τQ and ρ solve the mean fieldequations

∂tρ(t , x) – divx

[f (ρt , x)

∇xτQ(t , x)|∇xτQ(t , x)|

ρ(t , x)

]= 0, R+ ×Ω,

– ∂tτQ(t , x) + |∇xτQ(t , x)| f (ρt , x) – 1 = 0, R+ ×Ω,

ρ(0, x) = ρ0(x), Ω,

τQ(t , x) = 0, R+ × ∂Ω.

Continuity equation satisfied in the sense of distributions,Hamilton–Jacobi equation satisfied in the viscosity sense.

Hamilton–Jacobi equation can be obtained by standard techniqueson optimal control (and does not use the previous assumptions).But the situation is much harder for the continuity equation.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintCharacterization of equilibria

The continuity equation:

∂tρ(t , x) – divx

[f (ρt , x)

∇xτQ(t , x)|∇xτQ(t , x)|

ρ(t , x)

]= 0.

If τQ is differentiable at a point (t ,γ(t)) of an optimal trajectory,then ∇xτQ(t ,γ(t)) = 0 and the optimal control at this point is

u(t) = – ∇xτQ(t ,γ(t))|∇xτQ(t ,γ(t))| .

τQ is Lipschitz, hence differentiable a.e., but it may benowhere differentiable along a particular trajectory...We need more properties of τQ and the optimal trajectories

obtained byPontryagin Maximum Principle =⇒ optimal γ and u are C1,1;

⇓proving semiconcavity of τQ ⇐= (t , x) 7→ f (ρt , x) is also C1,1 +[Cannarsa, Sinestrari; 2004];using properties of semiconcave functions.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintCharacterization of equilibria

The continuity equation:

∂tρ(t , x) – divx

[f (ρt , x)

∇xτQ(t , x)|∇xτQ(t , x)|

ρ(t , x)

]= 0.

If τQ is differentiable at a point (t ,γ(t)) of an optimal trajectory,then ∇xτQ(t ,γ(t)) = 0 and the optimal control at this point is

u(t) = – ∇xτQ(t ,γ(t))|∇xτQ(t ,γ(t))| .

τQ is Lipschitz, hence differentiable a.e., but it may benowhere differentiable along a particular trajectory...We need more properties of τQ and the optimal trajectoriesobtained by

Pontryagin Maximum Principle =⇒ optimal γ and u are C1,1;⇓

proving semiconcavity of τQ ⇐= (t , x) 7→ f (ρt , x) is also C1,1 +[Cannarsa, Sinestrari; 2004];using properties of semiconcave functions.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintSimulations

Simulation 1: going from A to B through two different paths.

A B

1 – ℓ

ℓ

Ω: the network;Γ = B;ρ0 = δA;ℓ ∈ (0, 1).

ρt = mδx(t) + (1 – m)δy(t);x(t): lower path;y(t): upper path;m ∈ [0, 1].

χ(x) =

1+cos(πx

ε )2ε , if |x| < ε,

0, if |x| ≥ ε,

η(x) =

1–cos(

πd(x,B)ε

)2 , if d(x, B) < ε,

1, if d(x, B) ≥ ε,

K (x) =1

1 +(

2x15

)4 ,

ε =110

.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintSimulations

Simulation 1: going from A to B through two different paths.

A B

1 – ℓ

ℓ

Ω: the network;Γ = B;ρ0 = δA;ℓ ∈ (0, 1).

ρt = mδx(t) + (1 – m)δy(t);x(t): lower path;y(t): upper path;m ∈ [0, 1].

χ(x) =

1+cos(πx

ε )2ε , if |x| < ε,

0, if |x| ≥ ε,

η(x) =

1–cos(

πd(x,B)ε

)2 , if d(x, B) < ε,

1, if d(x, B) ≥ ε,

K (x) =1

1 +(

2x15

)4 ,

ε =110

.

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintSimulations

Simulation 1: going from A to B through two different paths.

A B

1 – ℓ

ℓ

Ω: the network;Γ = B;ρ0 = δA;ℓ ∈ (0, 1).

ρt = mδx(t) + (1 – m)δy(t);x(t): lower path;y(t): upper path;m ∈ [0, 1].

χ(x) =

1+cos(πx

ε )2ε , if |x| < ε,

0, if |x| ≥ ε,

η(x) =

1–cos(

πd(x,B)ε

)2 , if d(x, B) < ε,

1, if d(x, B) ≥ ε,

K (x) =1

1 +(

2x15

)4 ,

ε =110

. ℓ

m

0 1

1

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintSimulations

Simulation 1: going from A to B through two different paths.

A B

1 – ℓ

ℓ

Ω: the network;Γ = B;ρ0 = δA;ℓ ∈ (0, 1).

ρt = mδx(t) + (1 – m)δy(t);x(t): lower path;y(t): upper path;m ∈ [0, 1].

χ(x) =

1+cos(πx

ε )2ε , if |x| < ε,

0, if |x| ≥ ε,

η(x) =

1–cos(

πd(x,B)ε

)2 , if d(x, B) < ε,

1, if d(x, B) ≥ ε,

K (x) =1

1 +(

2x15

)4 ,

ε =110

. ℓ

Exit time

0 1

1

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintSimulations

Simulation 2: going from A to B in a network.

ℓ

1

B

L

1

A

L

m1m

2

m1

1–

m1

1 – m1+ m1m2

m1 (1

–m2 )

Ω: the network;Γ = B;ρ0 = δA;L ∈ (1, +∞), ℓ ∈ (0, 1);χ, K , and ε as before;η(y) = 1 for all y ∈ Ω.

If 0 < t < T1:ρt = m1δx(t) + (1 – m1)δy(t).

If T1 < t < T2:ρt = m1(1 – m2)δx(t) + m1m2δz(t)

+ (1 – m1)δy(t).

If t > T2:ρt = m1(1 – m2)δx(t)

+ (1 – m1 + m1m2)δy(t).

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintSimulations

Simulation 2: going from A to B in a network.

ℓ

1

B

L

1

A

L

m1m

2

m1

1–

m1

1 – m1+ m1m2

m1 (1

–m2 )

Ω: the network;Γ = B;ρ0 = δA;L ∈ (1, +∞), ℓ ∈ (0, 1);χ, K , and ε as before;η(y) = 1 for all y ∈ Ω.

If 0 < t < T1:ρt = m1δx(t) + (1 – m1)δy(t).

If T1 < t < T2:ρt = m1(1 – m2)δx(t) + m1m2δz(t)

+ (1 – m1)δy(t).

If t > T2:ρt = m1(1 – m2)δx(t)

+ (1 – m1 + m1m2)δy(t).

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintSimulations

ℓ

1

B

L

1

A

L

m1m

2

m1

1–

m1

1 – m1+m1m2

m1 (1

–m2 )

By symmetry,

m1 = 1 – m1 + m1m2 and 1 – m1 = m1(1 – m2)

=⇒ m1 =1

2 – m2.

ℓ

m1, m2

0 0.35

0.6

ℓ

Exit time

0

2.70.35

3

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintSimulations

ℓ

1

B

L

1

A

L

m1m

2

m1

1–

m1

1 – m1+m1m2

m1 (1

–m2 )

By symmetry,

m1 = 1 – m1 + m1m2 and 1 – m1 = m1(1 – m2)

=⇒ m1 =1

2 – m2.

ℓ

m1, m2

0 0.35

0.6

ℓ

Exit time

0

2.70.35

3

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintSimulations

ℓ

1

B

L

1

A

L

m1m

2

m1

1–

m1

1 – m1+m1m2

m1 (1

–m2 )

By symmetry,

m1 = 1 – m1 + m1m2 and 1 – m1 = m1(1 – m2)

=⇒ m1 =1

2 – m2.

ℓ

m1, m2

0 0.35

0.6

ℓ

Exit time

0

2.70.35

3

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games with velocity constraintSimulations

Simulation 3: exiting a circle.Ω = B(0; 1)

Γ = ∂Ω

ρ0 = δp

χ(x) = max

(0,ε – |x|ε2

)η(x) = 1

K (x) =1

1 +(x

2

)2

p = (0.08, 0)

ε = 0.1

Mean field games and applications Guilherme Mazanti

Mean field games Crowd motion MFGs with velocity constraint

Mean field games and applications Guilherme Mazanti