Mean Field Games in the context of crowd motion Y. Achdou (LJLL, - - PowerPoint PPT Presentation

Mean Field Games in the context of crowd motion

• Y. Achdou

(LJLL, Universit´ e Paris-Diderot) June, 2019 — CIRM Marseille-Luminy

• Y. Achdou

Mean field games

Models of congestion

Outline

1 Models of congestion 2 Numerical simulations 3 Common noise 4 Several populations, segregation 5 Mean field games of control 6 MFG versus mean field type control

• Y. Achdou

Mean field games

Models of congestion

Mean field games

MFG: Nash equilibria with a continuum of identical agents, interacting via some global information. The dynamics of a representative agent is dXt = √ 2νdWt + γtdt where (Wt) is a d-dimensional Brownian motion (idiosynchratic noise) (γt) is the control of the agent.

1

Individual optimal control problem: the representative agent minimizes Et,x T

t

L(Xs, γs; ms)ds + G(XT ; mT )

• ,

where ms is the distribution of states (a single agent is assumed to have no influence on ms). Dynamic programming yields an optimal feedback γ∗

t and an optimal

trajectory X∗

t . 2

MFG equilibrium: mt = law of X∗

t .

• Y. Achdou

Mean field games

Models of congestion

Congestion

The cost of motion at x depends on m(x) in an increasing manner. A typical example was introduced by P-L. Lions (lectures at Coll` ege de France): L(x, γ; m) ∼ (µ+m(x))σ|γ|q′ + F(x, m(x)) where µ ≥ 0, σ > 0 and q′ > 1. The corresponding Hamiltonian is of the form H(x, p; m) = |p|q q(µ + m(x))α − F(x, m(x)), with α = σ(q − 1). Remarks Degeneracy of the Hamiltonian H as m → +∞ Difficulty in empty regions if µ = 0 This model is named “Soft Congestion” by Santambrogio and his coauthors. Their “Hard Congestion” models include inequality contraints on m: m ≤ ¯ m

• Y. Achdou

Mean field games

Models of congestion

The system of PDEs

                 −∂tu − ν∆u + 1

q |Du|q (m+µ)α = F(m) ,

(t, x) ∈ (0, T) × Ω ∂tm − ν∆m − div

• m |Du|q−2Du

(m+µ)α

• = 0 ,

(t, x) ∈ (0, T) × Ω m(0, x) = m0(x) , u(T, x) = G(x, m(T)) , x ∈ Ω (1) + boundary conditions on ∂Ω... Main assumption: Either µ > 0 (non singular case) or µ = 0 (singular case) Natural growth w.r.t. |Du|: 1 < q ≤ 2 0 < α ≤ 4 q − 1 q = 4 q′

• Y. Achdou

Mean field games

Models of congestion

The condition α ≤ 4(q − 1)/q is related to uniqueness and stability

General MFG with local coupling lead to systems of the form              −∂tu − ν∆u + H(x, p, m) = F(m) (t, x) ∈ (0, T) × Ω, ∂tm − ν∆m − div(mHp(x, p, m)) = 0 , (t, x) ∈ (0, T) × Ω, m(0, x) = m0(x) , u(T, x) = G(m(T, x)) , x ∈ Ω . P-L. Lions proved that a sufficient condition for the uniqueness of classical solutions is that G be non decreasing, F be strictly increasing and that −Hm (x, p, m)

1 2 mHT m,p(x, p, m) 1 2 mHm,p(x, p, m)

mHp,p(x, p, m)

• > 0,

for all x ∈ Ω, m > 0 and p ∈ Rd. In the present congestion model, this condition ⇔ α ≤ 4(q − 1)/q.

• Y. Achdou

Mean field games

Models of congestion

Proof of uniqueness (in the periodic case) (1/2)

Consider two solutions of (⋆): (u1, m1) and (u2, m2): multiply HJB1 − HJB2 by m1 − m2: T

• Td(u1 − u2)(∂tm1 − ∂tm2) + ν∇(u1 − u2) · ∇(m1 − m2)

+ T

• Td
• H(x, ∇u1, m1) − H(x, ∇u2, m2)
• (m1 − m2)

= T

• Td(F(·; m1) − F(·; m2))(m1 − m2)

+

• Td
• G(·; m1|t=T ) − G(·; m2|t=T )(m1|t=T − m2|t=T
• .

multiply FP1 − FP2 by u1 − u2: 0 = T

• Td(u1 − u2)(∂tm1 − ∂tm2) + ν∇(u1 − u2) · ∇(m1 − m2)

+ T

• Td
• m1

∂H ∂p (x, ∇u1, m1) − m2 ∂H ∂p (x, ∇u2, m2)

• · ∇(u1 − u2).
• Y. Achdou

Mean field games

Models of congestion

Proof of uniqueness (in the periodic case) (2/2)

subtract: 0 =        T

• Td E(m1, ∇u1, m2, ∇u2) +

T

• Td(F(·; m1) − F(·; m2))(m1 − m2)

+

• Td G(·; m1|t=T ) − G(·; m2|t=T ))(m1|t=T − m2|t=T )

where E(m1, p1, m2, p2) = m2 − m1 p2 − p1

• ·
• H(p1, m1) − H(p2, m2)

m2 ∂H

∂p (p2, m2) − m1 ∂H ∂p (p1, m1)

• =

m2 − m1 p2 − p1

• ·
• Θ

m2 p2

• − Θ

m1 p1

• with

Θ m p

• =

−H(p, m) m ∂H

∂p (p, m)

• .

E positive ⇔ Θ monotone ⇔

• −Hm (p, m)

mHT

m,p(p, m)/2

mHm,p(p, m)/2 mHp,p(p, m)

• > 0,

∀m > 0. If Θ, F and G are monotone, then the 3 terms vanish. If for example F is strictly increasing and G is nondecreasing, we get that u1(t = T) = u2(t = T) and m1 = m2. Then, u1 = u2 is obtained from uniqueness results for the Bellman equation.

• Y. Achdou

Mean field games

Models of congestion

Some references

P-L. Lions [∼ 2011]: lectures at Coll` ege de France. In particular, the condition for uniqueness of classical solutions. Gomes-Mitake [2015]: existence of classical solutions in a specific stationary case: purely quadratic Hamiltonian, i.e. H(x, p, m) = |p|2

mα , with a special

trick Gomes-Voskanyan[2015] and Graber[2015]: short-time existence results of classical solutions for evolutive MFG with congestion More generally, for the existence of classical solutions, restrictive assumptions (e.g.

• n the growth of F and G) are needed.

Moreover, if H(x, p, m) = |p|q

mα , one needs to prove that m does not vanish.

It seems more feasible to work with weak solutions: (Y.A-Porretta [2018])

• Y. Achdou

Mean field games

Models of congestion

References on weak solutions of the MFG systems

Weak solutions = distributional sol. with suitable integrability properties First introduced by Lasry and Lions in 2007 For Hamiltonians with separate dependencies: H(x, p, m) = H(x, p) − F(m), Porretta, [ARMA 2015], showed that weak sol. allow to build a very general well-posed setting: stability results for weak sol. of Fokker-Planck equations uniqueness of weak sol. of the MFG system (Ok but much harder than for classical sol.) Allow to prove general convergence results for numerical schemes [Y.A-Porretta 2016] When the MFG system can be seen as the optimality conditions of an

• ptimal control problem driven by a PDE, weak solutions are the minima of

a relaxed primal-dual pb [Cardaliaguet-Graber-Porretta-Tonon 2015] Allow to handle degenerate diffusion [C-G-P-T 2015] Difficulty with the present congestion model: it is not possible to use a variational approach. Yet, the proof of existence relies on the monotonicity assumption.

• Y. Achdou

Mean field games

Models of congestion

Weak solutions in the non singular case: µ > 0

Consider the model problem:        −∂tu − ν∆u + 1

q |Du|q (m+µ)α = F(m) ,

(t, x) ∈ (0, T) × Td, ∂tm − ν∆m − div

• m |Du|q−2Du

(m+µ)α

• = 0 ,

(t, x) ∈ (0, T) × Td, m(0, x) = m0(x) , u(T, x) = G(x, m(T, x)) , x ∈ Td. Additional assumptions: m0 ∈ C(Td), F, G bounded from below, and λ f(m) − κ ≤ F(t, x, m) ≤ 1 λ f(m) + κ, ∀m ≥ 0, for some nondecreasing function f such that s → sf(s) is convex. Similar assumption on G. We set H(x, p, m) = 1 q |p|q (m + µ)α .

• Y. Achdou

Mean field games

Models of congestion

Weak solutions in the non singular case: µ > 0

Definition A weak solution (u, m) is a distributional solution of the boundary value problem s.t. mF(m) ∈ L1, mT G(mT ) ∈ L1(Td), m

|Du|q (µ+m)α ∈ L1, |Du|q (µ+m)α ∈ L1,

and the energy identity holds

• Td m0 u(0) dx=
• Td G(x, m(T)) m(T) dx +

T

• Td F(t, x, m)m dxdt

+ T

• Td m [Hp(t, x, m, Du) · Du − H(t, x, m, Du)] dxdt.

Theorem [Y.A- A.Porretta 2018] Under the previous assumptions, (mainly α ≤ 4/q′ ), there exists a weak solution. Uniqueness under further monotonicity assumptions on F and G.

• Y. Achdou

Mean field games

Models of congestion

Important steps in the proof

Use renormalized solutions for the F-P. equation (Porretta 2014) Theorem : Crossed energy inequalities (also used in the proof of existence) Consider (u, m) such that mF(m) ∈ L1, m|t=T G(m|t=T ) ∈ L1(Td), m

|Du|q (µ+m)α ∈ L1, |Du|q (µ+m)α ∈ L1

m is a weak sol. of

• F.P. equation + m|t=0 = m0
• u is a distrib. subsol. of
• the Bellman equation + u|t=T ≤ G(mt=T )
• For any pair (˜

u, ˜ m) with the same properties as (u, m), we have the crossed-integrability: ˜ m |Du|q (µ + m)α ∈ L1, m |D˜ u|q (µ + ˜ m)α ∈ L1, . . . and the energy inequality: ˜ m0 , u(0) ≤

• Td G(x, m(T)) ˜

m(T) dx + T

• Td F(t, x, m) ˜

m dxdt + T

• Td [ ˜

m Hp(t, x, ˜ m, D˜ u) · Du − ˜ m H(t, x, m, Du)] dxdt

• Y. Achdou

Mean field games

Models of congestion

Weak solutions in the singular case: µ = 0 (1/3)

               −∂tu − ν∆u + 1

q |Du|q mα

= F(m) , (t, x) ∈ (0, T) × Td ∂tm − ν∆m − div(m |Du|q−2Du

mα

) = 0 , (t, x) ∈ (0, T) × Td m(0, x) = m0(x) , u(T, x) = G(x, m(T)) , x ∈ Td.

• Y. Achdou

Mean field games

Models of congestion

Weak solutions in the singular case: µ = 0 (2/3)

Definition The pair (u, m) is a weak solution if

1

mF(m) ∈ L1, mT G(mT ) ∈ L1(Td), m1{m>0}

|Du|q mα

∈ L1, 1{m>0}

|Du|q mα

∈ L1.

2

u is a subsolution of the Bellman equation: for any 0 ≤ ϕ ∈ C∞

c ((0, T] × Td),

T

• Td u ϕt dxdt − ν

T

• Td u ∆ϕ dxdt +

T

• Td H(t, x, m, Du)1{m>0}ϕ dxdt

≤ T

• Td F(t, x, m)ϕ dxdt +
• Td G(x, m(T))ϕ(T) dx

3

m is a weak solution of ∂tm − ν∆m − div

• m1{m>0}

|Du| mα

• =

m(t = 0) = m0

4

The energy identity holds:

• Td m0 u(0) dx =
• Td G(x, m(T)) m(T) dx +

T

• Td F(t, x, m)m dxdt

+ T

• Td m [Hp(t, x, m, Du) · Du − H(t, x, m, Du)] 1{m>0}dxdt.

Note that the left hand side term is meaningful.

• Y. Achdou

Mean field games

Models of congestion

Weak solutions in the singular case: µ = 0 (3/3)

Theorem Y.A-Porretta [2018] If either

• q < 2 and α ≤

4 q′

• r
• q = 2 and α < 4/q′ = 2
• , and if F and G are

nondecreasing, then there exists a unique weak solution. Remark We only miss the limit case q = 2 and α = 2.

• Y. Achdou

Mean field games

Models of congestion

Extension

Existence and uniqueness of weak solutions holds for              −∂tu − ν∆u + H(t, x, p, m) = F(m) (t, x) ∈ (0, T) × Td ∂tm − ν∆m − div(mHp(t, x, p, m)) = 0 , (t, x) ∈ (0, T) × Td m(0, x) = m0(x) , u(T, x) = G(m(T, x)) , x ∈ Td , under the structure conditions H(t, x, 0, m) ≤ 0 , H(t, x, p, m) ≥ c0 |p|q (m + µ)α − c1 (1 + m

α q−1 ) ,

|Hp(t, x, p, m)| ≤ c2 (1 + |p|q−1 (m + µ)α ) , Hp(t, x, p, m) · p ≥ (1 + σ) H(t, x, p, m) − c3 (1 + m

α q−1 ) ,

for a.e. (t, x) ∈ QT and every p ∈ RN, where σ, c0, . . . , c3 are positive constants, and the same assumptions on F, G α and q.

• Y. Achdou

Mean field games

Numerical simulations

Outline

1 Models of congestion 2 Numerical simulations 3 Common noise 4 Several populations, segregation 5 Mean field games of control 6 MFG versus mean field type control

• Y. Achdou

Mean field games

Numerical simulations

Numerical method

Finite difference schemes with monotone discrete versions of the Hamiltonian The discrete Fokker-Planck operator is the adjoint of the linearized Bellman

• perator (with respect to u)

This yields a monotone scheme for the Fokker-Planck equation: preserves the total mass and the positivity We use implicit time schemes (to avoid restrictions on the time step). This allows to tackle infinite horizon problems The discrete PDE system has exactly the same stucture as the continuous

• ne

Existence by Brouwer theorem The proof of uniqueness can be transposed to the discrete case, (in the congestion models in particular) Resolution of the system of nonlinear equations obtained from the finite difference scheme: Newton method

• Y. Achdou

Mean field games

Numerical simulations

A Prototypical Case: Exit from a Hall with Obstacles

The geometry 50 50

exit exit

• Y. Achdou

Mean field games

Numerical simulations

The Data

The initial density m0 is piecewise constant and takes two values 0 and 4 people/m2. There are 3300 people in the hall. ν = 0.012 H(x, p, m) =

8 (1+m)

3 4 |p|2 −

1 3200

F(m) ∼ 0

density at t=0 seconds 10 20 30 40 50 x 10 20 30 40 50 y 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5

which leads to the following HJB equation ∂u ∂t + 6 500 ∆u − 8 (1 + m)

3 4

|∇u|2 = − 1 3200 Boundary conditions: ∂u ∂n =

∂m ∂n

= at walls, u = m = at exits.

• Y. Achdou

Mean field games

Numerical simulations

The Results

The horizon is T = 40 min. The two doors stay open from t = 0 to t = T.

500 1000 1500 2000 2500 3000 3500 5 10 15 20 25 30 35 40

The number of people in the room vs. time

• Y. Achdou

Mean field games

Numerical simulations

Evolution of the Distribution

density at t=10 seconds 10 20 30 40 50 x 10 20 30 40 50 y 0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4 density at t=2 minutes 10 20 30 40 50 x 10 20 30 40 50 y 1 2 3 4 5 6 1 2 3 4 5 6 density at t=5 minutes 10 20 30 40 50 x 10 20 30 40 50 y 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 density at t=15 minutes 10 20 30 40 50 x 10 20 30 40 50 y 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(the scale varies w.r.t. t)

• Y. Achdou

Mean field games

Numerical simulations

(Loading m2doors.mov) Figure : The evolution of the density

• Y. Achdou

Mean field games

Common noise

Outline

1 Models of congestion 2 Numerical simulations 3 Common noise 4 Several populations, segregation 5 Mean field games of control 6 MFG versus mean field type control

• Y. Achdou

Mean field games

Common noise

MFG with some common noise

When there is an infinity of admissible states, MFG models with common noise generally lead to an infinite dimensional PDE, named the master equation, whose solutions are functions depending on the state variables and

• n the distribution of states (a measure)

If the state space is finite, then the master equation takes the form of a system of first or second order PDEs The state space may be infinite but the common noise may consist of a random variable that may jump between a finite number of values at a finite number of dates. Then the MFG can be described by a collection of several systems of of forward/backward PDEs, with suitable couplings.

• Y. Achdou

Mean field games

Common noise

Exit from a hall with a common uncertainty

Same geometry. The horizon is T. Before t = T/2, the two doors are closed.

• Y. Achdou

Mean field games

Common noise

Exit from a hall with a common uncertainty

Same geometry. The horizon is T. Before t = T/2, the two doors are closed. People know that one of the two doors will be opened at t = T/2 and will stay open until t = T, but they do not know which.

• Y. Achdou

Mean field games

Common noise

Exit from a hall with a common uncertainty

Same geometry. The horizon is T. Before t = T/2, the two doors are closed. People know that one of the two doors will be opened at t = T/2 and will stay open until t = T, but they do not know which. At T/2, the probability that a given door be opened is 1/2.

• Y. Achdou

Mean field games

Common noise

Exit from a hall with a common uncertainty

Same geometry. The horizon is T. Before t = T/2, the two doors are closed. People know that one of the two doors will be opened at t = T/2 and will stay open until t = T, but they do not know which. At T/2, the probability that a given door be opened is 1/2. Hence the model involves three pairs of unknown functions (uC, mC) is defined on (0, T/2) × Ω and corresponds to the situation when the room is closed. (uL, mL) and (uR, mR) are defined on (T/2, T) × Ω and resp. correspond to the case when the left (resp. right) door is open.

• Y. Achdou

Mean field games

Common noise

The boundary value problem

The systems of PDEs: for j = C, L, R, ∂tuj + ν∆uj − H(mj, ∇uj) = −F(mj), ∂tmj − ν∆mj − div

• mj∂pH(mj, ∇uj)
• = 0,

in (0, T/2) × Ω for j = C and in (T/2, T) × Ω for j = L, R. Boundary conditions

∂uC ∂n = ∂mC ∂n

= 0

• n (0, T

2 ) × ∂Ω,

and for j = L, R,    ∂uj ∂n = ∂mj ∂n =

• n ( T

2 , T) × Γj N,

uj = mj =

• n ( T

2 , T) × Γj D

Transmission conditions at t = T/2 mL( T 2 , x) = mR( T 2 , x) = mC( T 2 , x) in Ω, uC( T 2 , x) = 1 2

• uL( T

2 , x) + uR( T 2 , x)

• in Ω.
• Y. Achdou

Mean field games

Common noise

Results

T = 40 min.

500 1000 1500 2000 2500 3000 5 10 15 20 25 30 35 40 number of individuals time (minutes)

The number of people in the room vs. time

• Y. Achdou

Mean field games

Common noise

(Loading densitynuonethird.mov)

• Y. Achdou

Mean field games

Several populations, segregation

Outline

1 Models of congestion 2 Numerical simulations 3 Common noise 4 Several populations, segregation 5 Mean field games of control 6 MFG versus mean field type control

• Y. Achdou

Mean field games

Several populations, segregation

2 populations : the system of PDEs

∂u1 ∂t + ν∆u1 − H1(x, ∇u1; m1, m2) = −Φ1(m1, m2) ∂m1 ∂t − ν∆m1 − div

• m1

∂H1 ∂p (x, ∇u1; m1, m2)

• = 0

∂u2 ∂t + ν∆u2 − H2(x, ∇u2; m2, m1) = −Φ2(m2, m1) ∂m2 ∂t − ν∆m2 − div

• m2

∂H2 ∂p (x; ∇u2; m2, m1)

• = 0

+ boundary conditions... The Hamiltonian for the population labeled i is for example Hi(x, p; mi, mj) = |p|2 1 + ciimi + cijmj The coupling cost for the population labeled i is for example Φi(x, mi, mj) = fi(x) + αi

• mi

mi + mj + ǫ − βi

• −

+ γi(mi + mj − ti)+ Existence of solutions: Ok in general (Cirant 2017, Y.A-Bardi-Cirant 2018) Uniqueness: no in general

• Y. Achdou

Mean field games

Several populations, segregation

An example of a MFG with 2 populations and congestion/segregation effects

Purpose: try a MFG model for a crossroad with two main flows of vehicles.

• Y. Achdou

Mean field games

Several populations, segregation

The system of PDEs

∂u1 ∂t + ν∆u1 − H1(x, ∇u1; m1, m2) = −Φ1(m1, m2) ∂m1 ∂t − ν∆m1 − div

• m1

∂H1 ∂p (x, ∇u1; m1, m2)

• = 0

∂u2 ∂t + ν∆u2 − H2(x, ∇u2; m2, m1) = −Φ2(m2, m1) ∂m2 ∂t − ν∆m2 − div

• m2

∂H2 ∂p (x; ∇u2; m2, m1)

• = 0

The Hamiltonian for the population labeled i is Hi(x, p; mi, mj) = |p|2 1 + mi + 5mj The coupling cost for the population labeled i is Φi(x, mi, mj) = 0.5 + 0.5

• mi

mi + mj + ǫ − 0.5

• −

+ (mi + mj − 4)+

• Y. Achdou

Mean field games

Several populations, segregation

Boundary conditions

Exit costs for the population labeled 0:

1

North-West and South-East exits: 0

2

South-West exit : −8.5

3

North-East exit : −4

4

South exit : −7 Exit costs for the population labeled 1:

1

North-West and South-East exits: 0

2

South-West exit : −7

3

North-East and South exit : −4 Entry fluxes

1

Population 0: at the North-West exit, the entry flux is 1

2

Population 1: at the South-East exit, the entry flux is 1

• Y. Achdou

Mean field games

Several populations, segregation

Stationary equilibrium for ν ∼ 0.3 ((a)) Distributions of

the two populations

((b)) Value functions

• f the two populations

((c)) Fluxes for

population 0

((d)) Fluxes for

population 1

Figure : Stationary Equilibrium ν ∼ 0.31

• Y. Achdou

Mean field games

Several populations, segregation

Stationary equilibrium for ν ∼ 0.15 ((a)) Distributions of

the two populations

((b)) Value functions

• f the two populations

((c)) Fluxes for

population 0

((d)) Fluxes for

population 1

Figure : Stationary Equilibrium ν ∼ 0.16

• Y. Achdou

Mean field games

Mean field games of control

Outline

1 Models of congestion 2 Numerical simulations 3 Common noise 4 Several populations, segregation 5 Mean field games of control 6 MFG versus mean field type control

• Y. Achdou

Mean field games

Mean field games of control

MFGs of control (with Z. Kobeissi)

The agents interact via the distribution of states and controls : µ(t) = L(Xt, γt) Let V (t, x) be an average drift at x ∈ Ω and t ∈ [0, T]: V (t, x) = 1 Z(t, x)

• Ω×Rd γ K(x, y) dµ(t, y, γ), with µ(t) = L(Xt, γt),

where K is a kernel and Z(t, x) is a normalization factor. The cost to be minimized by a given agent is J(γ) = E

• G (XT ; m(T)) +

T α 2 |γt − λV (t, Xt)|2 + 1 − α 2 |γt|2 + F (Xt; m(t)) dt

• ,

with λ < 1, 0 ≤ α ≤ 1, m(t) = L(Xt). λ > 0 means that the agents aim at having locally the same strategy The Hamiltonian associated to this control problem is H(x, p, V ) = 1 2 |p − λαV |2 − λ2α 2 |V |2.

• Y. Achdou

Mean field games

Mean field games of control

The forward-backward system

The system of PDEs is                          − ∂tu(t, x) − ν∆u(t, x) + 1 2

• ∇xu(t, x) − λαV (t, x)
• 2

− λ2α 2

• V (t, x)
• 2

= F(x; m(t)), ∂tmt(t, x) − ν∆m(t, x) − div

• ∇xu − λαV (t, x)
• m(t, x)
• = 0,

V (t, x) =

• Ω
• −∇xu(t, y) + λαV (t, y)

K(x, y) Z(t, x) dm(t, y), Z(t, x) =

• Ω

K(x, y)dm(t, y), with boundary conditions    u(T, x) = G(x; m(T)), m(0) = m◦, ∂u ∂n − λαV · n = 0, ∂m ∂n = 0, on ∂Ω. Existence of classical solutions (Z. Kobeissi) Uniqueness: no, in general

• Y. Achdou

Mean field games

Mean field games of control

Existence results (Z.Kobeissi)

J(γ) = E

• G (XT ; m(T)) +

T α a |γt − λV (t, Xt)|a + 1 − α b |γt|b + F (Xt; m(t))

• dt
• ,

with a, b ≥ 2, 0 ≤ α ≤ 1, λ < 1. case Existence Uniqueness α = 1 Yes No K = 1 ⇒ V (t, x) = V (t) Yes No a = b = 2, λ < 0 Yes Yes a = b, λ ≥ 0 Yes No a = b ≥ 2, λ ≥ 0 Yes if λ small enough No a = b ≥ 2, λ ≥ 0 Yes if α(1 − α) small enough No Main step in the existence proof: bounds on Du∞ knowing u∞ Difficulty : in some cases, it is not easy to get a bound on u∞

• Y. Achdou

Mean field games

Mean field games of control

Parameters of the simulations

Ω = [−0.5, 0.5]2. V is a piecewise linear interpolation of averages of the control in 3 × 3 identical subdomains. T = 1, λ = 0.9, α = 1 and ν = 10−4. No coupling cost. The initial mass is distributed in two symmetric corners of the domain and the terminal cost pushes the agents to go towards the other two corners of the domains.

Initial distribution Terminal cost

2

5 5

10−4

• Y. Achdou

Mean field games

Mean field games of control

A first symmetrical solution

• Y. Achdou

Mean field games

Mean field games of control

Non-symmetrical solutions

An evanescent part is added to the initial distribution, and a continuation method is used

ε →

• 10−4+

5 5 ε →

• 10−4+

10−4

Initial Condition 0 < t < T

• Y. Achdou

Mean field games

Mean field games of control

Non-symmetrical solutions

• Y. Achdou

Mean field games

Mean field games of control

With another cost which models crowd aversion and a more local formula for V

• Y. Achdou

Mean field games

MFG versus mean field type control

Outline

1 Models of congestion 2 Numerical simulations 3 Common noise 4 Several populations, segregation 5 Mean field games of control 6 MFG versus mean field type control

• Y. Achdou

Mean field games

MFG versus mean field type control

MFG vs. Mean field type control (MFTC) (1/3)

MFG: look for Nash equilibria with N identical agents, then let N → ∞ Carmona and Delarue / Bensoussan et al have studied the control of McKean-Vlasov dynamics: Assume that the all N agents use the same feedback law γ The perturbations of γ impact the empirical distribution Pass to the limit as N → ∞ first, then minimize the asymptotic cost

• Y. Achdou

Mean field games

MFG versus mean field type control

MFG vs. Mean field type control (MFTC) (1/3)

MFG: look for Nash equilibria with N identical agents, then let N → ∞ Carmona and Delarue / Bensoussan et al have studied the control of McKean-Vlasov dynamics: Assume that the all N agents use the same feedback law γ The perturbations of γ impact the empirical distribution Pass to the limit as N → ∞ first, then minimize the asymptotic cost MFTC models consists of an optimal control problem driven by a Fokker-Planck equation: Find a feedback γs = γ(s, Xs; ms) which minimizes J(t) = E T

t

L (Xs, γs; ms) ds + G (XT ; mT )

• subject to

dXt = √ 2νdWt + γtdt mt is the law of Xt, therefore ∂m ∂t − ν∆m + div (mγ) = 0 m|t=0 = m0.

• Y. Achdou

Mean field games

MFG versus mean field type control

MFG vs. MFTC (2/3)

Assume local dependency: L (x, γ; m) = L(x, γ, m(x)) and G(x; m) = G(x, m(x)): the cost can be expressed as J(t) = T

t

• Ω

L (x, γ(s, x, ms)m(s, x)) m(s, x)dsdx +

• Ω

G (x; m(T, x)) m(T, x)dx and the optimality conditions read ∂u ∂t + ν∆u − H(x, ∇u, m(t, x))−m(t, x) ∂H ∂m (x, ∇u(x, t), m(t, x)) = 0 ∂m ∂t − ν∆m − div

• m ∂H

∂p (x, ∇u; m)

• = 0

with the terminal and initial conditions u(t = T, x) = G(x, m(T, x)) +m(T, x) ∂G ∂m (x, m(T, x)) m(0, x) = m0(x)

• Y. Achdou

Mean field games

MFG versus mean field type control

MFG vs. MFTC (3/3)

The latter PDE system enjoys uniqueness if −(mH)m,m mHp,p

• > 0

for all x ∈ Ω, m > 0 and p ∈ Rd. If H(x, p, m) =

|p|q (µ+m)α , then the latter condition holds if α ≤ 1, (while the

condition is α ≤

4q q−1 for the MFG system with the same Hamiltonian)

Existence and uniqueness for weak solutions of the mean field type control problem with congestion and possibly degenerate diffusion were proved in [Y.A.-Lauri` ere 2015], using a variational approach. Numerical methods using the variational approach were studied in [Y.A.-Lauri` ere 2016].

• Y. Achdou

Mean field games