an introduction to mean eld games and their applications · academic positions at paris 7, then...
TRANSCRIPT
![Page 1: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/1.jpg)
An introduction to mean field games and their
applications
Pr. Olivier Gueant (Universite Paris 1 Pantheon-Sorbonne)
Mathematical Coffees – Huawei/FSMP joint seminar
September 2018
![Page 2: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/2.jpg)
Table of contents
1. Introduction
2. Static mean field games
3. MFG in continuous time (with continuous state space)
4. Numerics and examples
5. Special for Huawei: MFG on graphs
6. Conclusion
1
![Page 3: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/3.jpg)
Introduction
![Page 4: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/4.jpg)
The speaker
• PhD thesis on mean field games under the supervision of
Pierre-Louis Lions.
• Academic positions at Paris 7, then ENSAE, and now Paris 1.
• Main research field: optimal control and applications (incl.
mean field games, stochastic optimal control in finance,
reinforcement learning, etc.).
• Start-up (MFG Labs) with Lasry and Lions (created in 2009 –
acquired in 2013 by Havas).2
![Page 5: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/5.jpg)
Mean field games – In the beginning were...
Pierre-Louis Lions and Jean-Michel Lasry, who introduced mean
field games (MFG) in 2006.
→ Similar ideas arose in electrical engineering (Caines, Huang,
Malhame, 2006)
3
![Page 6: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/6.jpg)
Introduction - Mean field games
Game theory
• The study of strategic interactions.
• Central concept of Nash equilibrium.
• In MFG: the number of players is large.
Mean field
• Approximation as in physics, here to model strategic
interactions, not interactions between particles.
• Philosophical difference: freedom... however, as Spinoza said:This is that human freedom, which all boast that they possess, and which
consists solely in the fact, that men are conscious of their own desire, but
are ignorant of the causes whereby that desire has been determined.
• Difference for the maths: humans anticipate, particles do not!
Main consequence: the equations are not simply forward
in time.
4
![Page 7: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/7.jpg)
Introduction - Mean field games
Game theory
• The study of strategic interactions.
• Central concept of Nash equilibrium.
• In MFG: the number of players is large.
Mean field
• Approximation as in physics, here to model strategic
interactions, not interactions between particles.
• Philosophical difference: freedom... however, as Spinoza said:This is that human freedom, which all boast that they possess, and which
consists solely in the fact, that men are conscious of their own desire, but
are ignorant of the causes whereby that desire has been determined.
• Difference for the maths: humans anticipate, particles do not!
Main consequence: the equations are not simply forward
in time. 4
![Page 8: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/8.jpg)
Numerous applications
Economics
• Economic growth and
inequality.
• Oil extraction.
• Mining industries.
• Labor market.
• etc.
Population dynamics
• Waves in stadiums (ola).
• Structure of cities.
• Traffic jam and other forms
of congestion.
• etc.
Finance
• Competition between asset
managers.
• Optimal execution of several
brokers.
• etc.
5
![Page 9: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/9.jpg)
Numerous applications
Economics
• Economic growth and
inequality.
• Oil extraction.
• Mining industries.
• Labor market.
• etc.
Population dynamics
• Waves in stadiums (ola).
• Structure of cities.
• Traffic jam and other forms
of congestion.
• etc.
Finance
• Competition between asset
managers.
• Optimal execution of several
brokers.
• etc.
5
![Page 10: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/10.jpg)
Numerous applications
Economics
• Economic growth and
inequality.
• Oil extraction.
• Mining industries.
• Labor market.
• etc.
Population dynamics
• Waves in stadiums (ola).
• Structure of cities.
• Traffic jam and other forms
of congestion.
• etc.
Finance
• Competition between asset
managers.
• Optimal execution of several
brokers.
• etc.
5
![Page 11: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/11.jpg)
Numerous applications
Economics
• Economic growth and
inequality.
• Oil extraction.
• Mining industries.
• Labor market.
• etc.
Population dynamics
• Waves in stadiums (ola).
• Structure of cities.
• Traffic jam and other forms
of congestion.
• etc.
Finance
• Competition between asset
managers.
• Optimal execution of several
brokers.
• etc.
5
![Page 12: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/12.jpg)
Many forms but two main characteristics
Different forms of mean field games
• Static games / games in discrete time / differential games
(continuous time).
• Discrete / continuous state space.
Two main characteristics
• Continuum of anonymous players.
• All players maximize the same objective function (possible to
generalize to several populations of players).
A fixed-point equilibrium approach
• Each infinitesimal player takes the distribution of players as
given.
• The distribution of players proceeds from all individual choices.
6
![Page 13: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/13.jpg)
Many forms but two main characteristics
Different forms of mean field games
• Static games / games in discrete time / differential games
(continuous time).
• Discrete / continuous state space.
Two main characteristics
• Continuum of anonymous players.
• All players maximize the same objective function (possible to
generalize to several populations of players).
A fixed-point equilibrium approach
• Each infinitesimal player takes the distribution of players as
given.
• The distribution of players proceeds from all individual choices.
6
![Page 14: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/14.jpg)
Many forms but two main characteristics
Different forms of mean field games
• Static games / games in discrete time / differential games
(continuous time).
• Discrete / continuous state space.
Two main characteristics
• Continuum of anonymous players.
• All players maximize the same objective function (possible to
generalize to several populations of players).
A fixed-point equilibrium approach
• Each infinitesimal player takes the distribution of players as
given.
• The distribution of players proceeds from all individual choices.
6
![Page 15: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/15.jpg)
Many forms but two main characteristics
Different forms of mean field games
• Static games / games in discrete time / differential games
(continuous time).
• Discrete / continuous state space.
Two main characteristics
• Continuum of anonymous players.
• All players maximize the same objective function (possible to
generalize to several populations of players).
A fixed-point equilibrium approach
• Each infinitesimal player takes the distribution of players as
given.
• The distribution of players proceeds from all individual choices.
6
![Page 16: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/16.jpg)
A large and worldwide community (not exhaustive and slightly
outdated)
• College de France: P.-L. Lions + J.-M. Lasry
• Dauphine: P. Cardaliaguet, J. Salomon, G. Turinici
• Paris-Diderot: Y. Achdou + Ph.D. students
• Nice: F. Delarue
• Italy (Roma + Padua): I. Capuzzo-Dolcetta, F. Camilli, M.
Bardi
• Princeton: R. Carmona (+ Ph.D. students), B. Moll (econ)
• Columbia: Daniel Lacker
• Chicago: R. Lucas (econ)
• McGill: P. Caines + collaborators around the world
• KAUST: D. Gomes (+ Ph.D. students), P. Markowich
• Hong Kong + Dallas: A. Bensoussan
7
![Page 17: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/17.jpg)
Some references
Initial papers
• J.-M. Lasry and P.-L. Lions. Jeux a champ moyen i. le cas
stationnaire. C. R. Acad. Sci. Paris, 343(9), 2006.
• J.-M. Lasry and P.-L. Lions. Jeux a champ moyen ii. horizon fini
et controle optimal. C. R. Acad. Sci. Paris, 343(10), 2006.
• J.-M. Lasry and P.-L. Lions. Mean field games. Japanese
Journal of Mathematics, 2(1), Mar. 2007.
Courses and notes
• 5 years of PLL’s lectures about MFG available on the website
of the College de France (in French).
• Notes by P. Cardaliaguet
8
![Page 18: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/18.jpg)
Some references
Initial papers
• J.-M. Lasry and P.-L. Lions. Jeux a champ moyen i. le cas
stationnaire. C. R. Acad. Sci. Paris, 343(9), 2006.
• J.-M. Lasry and P.-L. Lions. Jeux a champ moyen ii. horizon fini
et controle optimal. C. R. Acad. Sci. Paris, 343(10), 2006.
• J.-M. Lasry and P.-L. Lions. Mean field games. Japanese
Journal of Mathematics, 2(1), Mar. 2007.
Courses and notes
• 5 years of PLL’s lectures about MFG available on the website
of the College de France (in French).
• Notes by P. Cardaliaguet
8
![Page 19: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/19.jpg)
Some references
Initial papers
• J.-M. Lasry and P.-L. Lions. Jeux a champ moyen i. le cas
stationnaire. C. R. Acad. Sci. Paris, 343(9), 2006.
• J.-M. Lasry and P.-L. Lions. Jeux a champ moyen ii. horizon fini
et controle optimal. C. R. Acad. Sci. Paris, 343(10), 2006.
• J.-M. Lasry and P.-L. Lions. Mean field games. Japanese
Journal of Mathematics, 2(1), Mar. 2007.
Courses and notes
• 5 years of PLL’s lectures about MFG available on the website
of the College de France (in French).
• Notes by P. Cardaliaguet
8
![Page 20: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/20.jpg)
Some references (cont’d)
Some applications: O. Gueant, J.-M. Lasry and P.-L. Lions. Mean
field games and applications, in Paris-Princeton Lectures on
Mathematical Finance, 2010
9
![Page 21: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/21.jpg)
Static mean field games
![Page 22: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/22.jpg)
Reminder about game theory
• Game theory studies strategic interactions.
• N players. Strategies (x1, x2, . . . , xN) ∈ EN (E compact set).
• Player i has utility (or score) ui (xi , x−i ).
• Key notion: Nash equilibrium
Nash equilibrium
(x∗1 , . . . , x∗N) is a Nash equilibrium ⇐⇒ for any player i , x∗i is the
best strategy when others play x∗−i .
i.e.:
∀i , x∗i maximizes xi 7→ ui (xi , x∗−i ).
10
![Page 23: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/23.jpg)
Reminder about game theory
• Game theory studies strategic interactions.
• N players. Strategies (x1, x2, . . . , xN) ∈ EN (E compact set).
• Player i has utility (or score) ui (xi , x−i ).
• Key notion: Nash equilibrium
Nash equilibrium
(x∗1 , . . . , x∗N) is a Nash equilibrium ⇐⇒ for any player i , x∗i is the
best strategy when others play x∗−i .
i.e.:
∀i , x∗i maximizes xi 7→ ui (xi , x∗−i ).
10
![Page 24: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/24.jpg)
N → +∞
Mean field hypotheses
• Players have the same objective function ui = u.
• Players are anonymous: ∀xi , x−i 7→ u(xi , x−i ) is a
symmetrical function.
u(xi , x−i ) = u
xi ,1
N − 1
∑j 6=i
δxj
= u(xi ,m
N−1(x−i )).
Static MFGs
A static MFG is given by a function
U : (x ,m) ∈ E × P(E ) 7→ U(x ,m),
where m stands for the distribution of the players’ strategies.
Remark: P(E ) is the (compact) set of probability measures on E .
11
![Page 25: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/25.jpg)
N → +∞
Mean field hypotheses
• Players have the same objective function ui = u.
• Players are anonymous: ∀xi , x−i 7→ u(xi , x−i ) is a
symmetrical function.
u(xi , x−i ) = u
xi ,1
N − 1
∑j 6=i
δxj
= u(xi ,m
N−1(x−i )).
Static MFGs
A static MFG is given by a function
U : (x ,m) ∈ E × P(E ) 7→ U(x ,m),
where m stands for the distribution of the players’ strategies.
Remark: P(E ) is the (compact) set of probability measures on E .11
![Page 26: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/26.jpg)
Nash-MFG equilibrium
What is a Nash equilibrium when N tends to +∞?
• A Nash equilibrium with N players is a tuple (x∗1 , x∗2 , . . . , x
∗N).
When N → +∞, an equilibrium is a probability measure m.
Definition: Nash-MFG
m is a Nash-MFG equilibrium
⇐⇒ The support of m is included in the argmax of x 7→ U(x ,m)
⇐⇒ For any probability measure f ∈ P(E ) on the set of
strategies E , ∫EU(x ,m)m ≥
∫EU(x ,m)f
This definition shows that m solves a (rather uncommon)
fixed-point problem.
12
![Page 27: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/27.jpg)
Nash-MFG equilibrium
What is a Nash equilibrium when N tends to +∞?
• A Nash equilibrium with N players is a tuple (x∗1 , x∗2 , . . . , x
∗N).
When N → +∞, an equilibrium is a probability measure m.
Definition: Nash-MFG
m is a Nash-MFG equilibrium
⇐⇒ The support of m is included in the argmax of x 7→ U(x ,m)
⇐⇒ For any probability measure f ∈ P(E ) on the set of
strategies E , ∫EU(x ,m)m ≥
∫EU(x ,m)f
This definition shows that m solves a (rather uncommon)
fixed-point problem.
12
![Page 28: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/28.jpg)
Nash-MFG equilibrium
What is a Nash equilibrium when N tends to +∞?
• A Nash equilibrium with N players is a tuple (x∗1 , x∗2 , . . . , x
∗N).
When N → +∞, an equilibrium is a probability measure m.
Definition: Nash-MFG
m is a Nash-MFG equilibrium
⇐⇒ The support of m is included in the argmax of x 7→ U(x ,m)
⇐⇒ For any probability measure f ∈ P(E ) on the set of
strategies E , ∫EU(x ,m)m ≥
∫EU(x ,m)f
This definition shows that m solves a (rather uncommon)
fixed-point problem.
12
![Page 29: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/29.jpg)
Underlying mathematical result
Theorem
• Let us assume that U is continuous.
• Let us consider a sequence ((xN1 , . . . , xNN ))N where
∀N, (xN1 , . . . , xNN ) is a Nash equilibrium of the N-player game
corresponding to U|E×EN/SN .
Then, up to a subsequence, ∃m ∈ P(E ) such that:
1. mN(xN1 , . . . , xNN ) weakly converges towards m.
2. m is a Nash-MFG equilibrium.
Remark: this results can also be adapted to prove an existence
result (by using mixed strategies).
13
![Page 30: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/30.jpg)
Underlying mathematical result
Theorem
• Let us assume that U is continuous.
• Let us consider a sequence ((xN1 , . . . , xNN ))N where
∀N, (xN1 , . . . , xNN ) is a Nash equilibrium of the N-player game
corresponding to U|E×EN/SN .
Then, up to a subsequence, ∃m ∈ P(E ) such that:
1. mN(xN1 , . . . , xNN ) weakly converges towards m.
2. m is a Nash-MFG equilibrium.
Remark: this results can also be adapted to prove an existence
result (by using mixed strategies).
13
![Page 31: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/31.jpg)
What about uniqueness?
Uniqueness
If U is decreasing in the sense that
∀m1 6= m2,
∫(U(x ,m1)− U(x ,m2))(m1 −m2) < 0
then, an equilibrium is unique.
This type of monotonicity result is ubiquitous in the MFG
literature.
14
![Page 32: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/32.jpg)
What about uniqueness?
Uniqueness
If U is decreasing in the sense that
∀m1 6= m2,
∫(U(x ,m1)− U(x ,m2))(m1 −m2) < 0
then, an equilibrium is unique.
This type of monotonicity result is ubiquitous in the MFG
literature.
14
![Page 33: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/33.jpg)
MFG and planning
Variational characterization (planner’s problem)
If there exists a function m 7→ F (m) on P(E ) such that DF = U,
then any maximum of F is a Nash-MFG equilibrium.
Remark 1: Sometimes, there exists a global problem whose
solution corresponds to a MFG equilibrium.
Remark 2: Uniqueness is related to the strict concavity of F , hence
the monotonicity assumption on U.
Put your towel on the beach
• Objective function: U(x ,m) = −x2 − γm(x).
• Global problem: F (m) =∫−x2m(x)dx − γ
2m(x)2dx .
• Unique equilibrium, of the form m(x) = 1γ (λ− x2)+
15
![Page 34: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/34.jpg)
MFG and planning
Variational characterization (planner’s problem)
If there exists a function m 7→ F (m) on P(E ) such that DF = U,
then any maximum of F is a Nash-MFG equilibrium.
Remark 1: Sometimes, there exists a global problem whose
solution corresponds to a MFG equilibrium.
Remark 2: Uniqueness is related to the strict concavity of F , hence
the monotonicity assumption on U.
Put your towel on the beach
• Objective function: U(x ,m) = −x2 − γm(x).
• Global problem: F (m) =∫−x2m(x)dx − γ
2m(x)2dx .
• Unique equilibrium, of the form m(x) = 1γ (λ− x2)+
15
![Page 35: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/35.jpg)
MFG and planning
Variational characterization (planner’s problem)
If there exists a function m 7→ F (m) on P(E ) such that DF = U,
then any maximum of F is a Nash-MFG equilibrium.
Remark 1: Sometimes, there exists a global problem whose
solution corresponds to a MFG equilibrium.
Remark 2: Uniqueness is related to the strict concavity of F , hence
the monotonicity assumption on U.
Put your towel on the beach
• Objective function: U(x ,m) = −x2 − γm(x).
• Global problem: F (m) =∫−x2m(x)dx − γ
2m(x)2dx .
• Unique equilibrium, of the form m(x) = 1γ (λ− x2)+
15
![Page 36: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/36.jpg)
MFG and planning
Variational characterization (planner’s problem)
If there exists a function m 7→ F (m) on P(E ) such that DF = U,
then any maximum of F is a Nash-MFG equilibrium.
Remark 1: Sometimes, there exists a global problem whose
solution corresponds to a MFG equilibrium.
Remark 2: Uniqueness is related to the strict concavity of F , hence
the monotonicity assumption on U.
Put your towel on the beach
• Objective function: U(x ,m) = −x2 − γm(x).
• Global problem: F (m) =∫−x2m(x)dx − γ
2m(x)2dx .
• Unique equilibrium, of the form m(x) = 1γ (λ− x2)+
15
![Page 37: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/37.jpg)
MFG and planning
Variational characterization (planner’s problem)
If there exists a function m 7→ F (m) on P(E ) such that DF = U,
then any maximum of F is a Nash-MFG equilibrium.
Remark 1: Sometimes, there exists a global problem whose
solution corresponds to a MFG equilibrium.
Remark 2: Uniqueness is related to the strict concavity of F , hence
the monotonicity assumption on U.
Put your towel on the beach
• Objective function: U(x ,m) = −x2 − γm(x).
• Global problem: F (m) =∫−x2m(x)dx − γ
2m(x)2dx .
• Unique equilibrium, of the form m(x) = 1γ (λ− x2)+
15
![Page 38: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/38.jpg)
MFG and planning
Variational characterization (planner’s problem)
If there exists a function m 7→ F (m) on P(E ) such that DF = U,
then any maximum of F is a Nash-MFG equilibrium.
Remark 1: Sometimes, there exists a global problem whose
solution corresponds to a MFG equilibrium.
Remark 2: Uniqueness is related to the strict concavity of F , hence
the monotonicity assumption on U.
Put your towel on the beach
• Objective function: U(x ,m) = −x2 − γm(x).
• Global problem: F (m) =∫−x2m(x)dx − γ
2m(x)2dx .
• Unique equilibrium, of the form m(x) = 1γ (λ− x2)+
15
![Page 39: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/39.jpg)
MFG in continuous time (with
continuous state space)
![Page 40: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/40.jpg)
Differential games
Static games are interesting but MFGs are really powerful in
continuous time (differential games):
The real power of MFGs in continuous time
• Differential/stochastic calculus.
• Ordinary and partial differential equations.
• Numerical methods.
Also, very general results have been obtained with probabilistic
methods (see Carmona, Delarue).
16
![Page 41: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/41.jpg)
Differential games
Static games are interesting but MFGs are really powerful in
continuous time (differential games):
The real power of MFGs in continuous time
• Differential/stochastic calculus.
• Ordinary and partial differential equations.
• Numerical methods.
Also, very general results have been obtained with probabilistic
methods (see Carmona, Delarue).
16
![Page 42: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/42.jpg)
Reminder of (stochastic) optimal control
Agent’s dynamics
dXt = αtdt + σdWt , X0 = x
Objective function
sup(αs)s≥0
E[∫ T
0(f (Xs)− L(αs)) ds + g(XT )
]
Remarks:
• f and L can also include a time dependency (e.g. discount
rate).
• Stationary (infinite horizon)/Ergodic problems can also be
considered.
17
![Page 43: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/43.jpg)
Reminder of (stochastic) optimal control
Agent’s dynamics
dXt = αtdt + σdWt , X0 = x
Objective function
sup(αs)s≥0
E[∫ T
0(f (Xs)− L(αs)) ds + g(XT )
]
Remarks:
• f and L can also include a time dependency (e.g. discount
rate).
• Stationary (infinite horizon)/Ergodic problems can also be
considered.
17
![Page 44: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/44.jpg)
Reminder of (stochastic) optimal control
Agent’s dynamics
dXt = αtdt + σdWt , X0 = x
Objective function
sup(αs)s≥0
E[∫ T
0(f (Xs)− L(αs)) ds + g(XT )
]
Remarks:
• f and L can also include a time dependency (e.g. discount
rate).
• Stationary (infinite horizon)/Ergodic problems can also be
considered.
17
![Page 45: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/45.jpg)
Reminder of (stochastic) optimal control
Main tool: value function
The best “score” an agent can expect when he is in x at time t:
u(t, x) = sup(αs)s≥t
E[∫ T
t(f (Xs)− L(αs)) ds + g(XT )|Xt = x
]
PDE
u “solves” the Hamilton-Jacobi(-Bellman) equation:
∂tu +σ2
2∆u + H(∇u) = −f (x), u(T , x) = g(x),
where H(p) = supα α · p − L(α).
Optimal control
The optimal control is α∗(t, x) = ∇H(∇u(t, x)).
18
![Page 46: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/46.jpg)
Reminder of (stochastic) optimal control
Main tool: value function
The best “score” an agent can expect when he is in x at time t:
u(t, x) = sup(αs)s≥t
E[∫ T
t(f (Xs)− L(αs)) ds + g(XT )|Xt = x
]
PDE
u “solves” the Hamilton-Jacobi(-Bellman) equation:
∂tu +σ2
2∆u + H(∇u) = −f (x), u(T , x) = g(x),
where H(p) = supα α · p − L(α).
Optimal control
The optimal control is α∗(t, x) = ∇H(∇u(t, x)).
18
![Page 47: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/47.jpg)
Reminder of (stochastic) optimal control
Main tool: value function
The best “score” an agent can expect when he is in x at time t:
u(t, x) = sup(αs)s≥t
E[∫ T
t(f (Xs)− L(αs)) ds + g(XT )|Xt = x
]
PDE
u “solves” the Hamilton-Jacobi(-Bellman) equation:
∂tu +σ2
2∆u + H(∇u) = −f (x), u(T , x) = g(x),
where H(p) = supα α · p − L(α).
Optimal control
The optimal control is α∗(t, x) = ∇H(∇u(t, x)).
18
![Page 48: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/48.jpg)
From optimal control problems to mean field games
• Continuum of players.
• Each player has a position X i that evolves according to:
dX it = αi
tdt + σdW it , X i
0 = x i
Remark: only independent idiosyncratic risks (common noise
has also been studied but it is more complicated).
• Each player optimizes:
max(αi
s )s≥0
E[∫ T
0
(f (X i
s ,m(s, ·))− L(αis ,m(s, ·))
)ds +g(X i
T ,m(T , ·))]
• The Nash-equilibrium t ∈ [0,T ] 7→ m(t, ·) must be consistent
with the decisions of the agents.
19
![Page 49: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/49.jpg)
From optimal control problems to mean field games
• Continuum of players.
• Each player has a position X i that evolves according to:
dX it = αi
tdt + σdW it , X i
0 = x i
Remark: only independent idiosyncratic risks (common noise
has also been studied but it is more complicated).
• Each player optimizes:
max(αi
s )s≥0
E[∫ T
0
(f (X i
s ,m(s, ·))− L(αis ,m(s, ·))
)ds +g(X i
T ,m(T , ·))]
• The Nash-equilibrium t ∈ [0,T ] 7→ m(t, ·) must be consistent
with the decisions of the agents.
19
![Page 50: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/50.jpg)
From optimal control problems to mean field games
• Continuum of players.
• Each player has a position X i that evolves according to:
dX it = αi
tdt + σdW it , X i
0 = x i
Remark: only independent idiosyncratic risks (common noise
has also been studied but it is more complicated).
• Each player optimizes:
max(αi
s )s≥0
E[∫ T
0
(f (X i
s ,m(s, ·))− L(αis ,m(s, ·))
)ds +g(X i
T ,m(T , ·))]
• The Nash-equilibrium t ∈ [0,T ] 7→ m(t, ·) must be consistent
with the decisions of the agents.
19
![Page 51: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/51.jpg)
From optimal control problems to mean field games
• Continuum of players.
• Each player has a position X i that evolves according to:
dX it = αi
tdt + σdW it , X i
0 = x i
Remark: only independent idiosyncratic risks (common noise
has also been studied but it is more complicated).
• Each player optimizes:
max(αi
s )s≥0
E[∫ T
0
(f (X i
s ,m(s, ·))− L(αis ,m(s, ·))
)ds +g(X i
T ,m(T , ·))]
• The Nash-equilibrium t ∈ [0,T ] 7→ m(t, ·) must be consistent
with the decisions of the agents.
19
![Page 52: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/52.jpg)
Examples
Repulsion
• f (x ,m) = −m(t, x)− δx2 and g = 0.
→ Willingness to be close to 0 but far from other players.
• Quadratic cost: L(α) = α2
2 .
Congestion
Cost of the form L(α,m(t, x)) = α2
2 (1 + m(t, x)).
20
![Page 53: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/53.jpg)
Examples
Repulsion
• f (x ,m) = −m(t, x)− δx2 and g = 0.
→ Willingness to be close to 0 but far from other players.
• Quadratic cost: L(α) = α2
2 .
Congestion
Cost of the form L(α,m(t, x)) = α2
2 (1 + m(t, x)).
20
![Page 54: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/54.jpg)
Partial differential equations
• u value function of the control problem (with given m).
• m distribution of the players
MFG PDEs
(HJB) ∂tu + σ2
2 ∆u + H(∇u,m) = −f (x ,m)
(K ) ∂tm +∇ · (m∇pH(∇u,m)) = σ2
2 ∆m
where H(p,m) = supα α · p − L(α,m).
u(T , x) = g(x), m(0, x) = m0(x)
The optimal control is α∗(t, x) = ∇pH(∇u(t, x),m(t, ·)).
21
![Page 55: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/55.jpg)
Partial differential equations
• u value function of the control problem (with given m).
• m distribution of the players
MFG PDEs
(HJB) ∂tu + σ2
2 ∆u + H(∇u,m) = −f (x ,m)
(K ) ∂tm +∇ · (m∇pH(∇u,m)) = σ2
2 ∆m
where H(p,m) = supα α · p − L(α,m).
u(T , x) = g(x), m(0, x) = m0(x)
The optimal control is α∗(t, x) = ∇pH(∇u(t, x),m(t, ·)).
21
![Page 56: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/56.jpg)
Partial differential equations
• u value function of the control problem (with given m).
• m distribution of the players
MFG PDEs
(HJB) ∂tu + σ2
2 ∆u + H(∇u,m) = −f (x ,m)
(K ) ∂tm +∇ · (m∇pH(∇u,m)) = σ2
2 ∆m
where H(p,m) = supα α · p − L(α,m).
u(T , x) = g(x), m(0, x) = m0(x)
The optimal control is α∗(t, x) = ∇pH(∇u(t, x),m(t, ·)).
21
![Page 57: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/57.jpg)
Remarks and variants
Forward/Backward
The system of PDEs is a forward/backward problem:
• The HJB equation is backward in time (terminal condition)
because agents anticipate the future.
• The transport equation is forward in time because it
corresponds to the dynamics of the agents.
Other frameworks
• Stationary setting (infinite horizon)
• Ergodic setting
Related problem
Same equations with initial and final conditions on m and no
terminal condition on u: the problem is then that of finding the
right terminal payoff g so that agents go from m0 to mT .
22
![Page 58: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/58.jpg)
Remarks and variants
Forward/Backward
The system of PDEs is a forward/backward problem:
• The HJB equation is backward in time (terminal condition)
because agents anticipate the future.
• The transport equation is forward in time because it
corresponds to the dynamics of the agents.
Other frameworks
• Stationary setting (infinite horizon)
• Ergodic setting
Related problem
Same equations with initial and final conditions on m and no
terminal condition on u: the problem is then that of finding the
right terminal payoff g so that agents go from m0 to mT .
22
![Page 59: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/59.jpg)
Remarks and variants
Forward/Backward
The system of PDEs is a forward/backward problem:
• The HJB equation is backward in time (terminal condition)
because agents anticipate the future.
• The transport equation is forward in time because it
corresponds to the dynamics of the agents.
Other frameworks
• Stationary setting (infinite horizon)
• Ergodic setting
Related problem
Same equations with initial and final conditions on m and no
terminal condition on u: the problem is then that of finding the
right terminal payoff g so that agents go from m0 to mT . 22
![Page 60: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/60.jpg)
Some results
Existence
A wide variety of PDE results, depending on f , L, g and σ.
Uniqueness
If the cost function L does not depend on m and if f is
decreasing in the sense:
∀m1 6= m2,
∫(f (x ,m1)− f (x ,m2))(m1 −m2) < 0
then a solution of the PDEs system is unique.
Remarks:
• Same criterion as above.
• For more general cost functions L (e.g. congestion), there is a
more general criterion (see Lions, or see the result in graphs).
23
![Page 61: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/61.jpg)
Some results
Existence
A wide variety of PDE results, depending on f , L, g and σ.
Uniqueness
If the cost function L does not depend on m and if f is
decreasing in the sense:
∀m1 6= m2,
∫(f (x ,m1)− f (x ,m2))(m1 −m2) < 0
then a solution of the PDEs system is unique.
Remarks:
• Same criterion as above.
• For more general cost functions L (e.g. congestion), there is a
more general criterion (see Lions, or see the result in graphs).
23
![Page 62: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/62.jpg)
Some results
Existence
A wide variety of PDE results, depending on f , L, g and σ.
Uniqueness
If the cost function L does not depend on m and if f is
decreasing in the sense:
∀m1 6= m2,
∫(f (x ,m1)− f (x ,m2))(m1 −m2) < 0
then a solution of the PDEs system is unique.
Remarks:
• Same criterion as above.
• For more general cost functions L (e.g. congestion), there is a
more general criterion (see Lions, or see the result in graphs).
23
![Page 63: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/63.jpg)
Some results
Existence
A wide variety of PDE results, depending on f , L, g and σ.
Uniqueness
If the cost function L does not depend on m and if f is
decreasing in the sense:
∀m1 6= m2,
∫(f (x ,m1)− f (x ,m2))(m1 −m2) < 0
then a solution of the PDEs system is unique.
Remarks:
• Same criterion as above.
• For more general cost functions L (e.g. congestion), there is a
more general criterion (see Lions, or see the result in graphs).
23
![Page 64: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/64.jpg)
Some results
Existence
A wide variety of PDE results, depending on f , L, g and σ.
Uniqueness
If the cost function L does not depend on m and if f is
decreasing in the sense:
∀m1 6= m2,
∫(f (x ,m1)− f (x ,m2))(m1 −m2) < 0
then a solution of the PDEs system is unique.
Remarks:
• Same criterion as above.
• For more general cost functions L (e.g. congestion), there is a
more general criterion (see Lions, or see the result in graphs).
23
![Page 65: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/65.jpg)
Some results
Existence
A wide variety of PDE results, depending on f , L, g and σ.
Uniqueness
If the cost function L does not depend on m and if f is
decreasing in the sense:
∀m1 6= m2,
∫(f (x ,m1)− f (x ,m2))(m1 −m2) < 0
then a solution of the PDEs system is unique.
Remarks:
• Same criterion as above.
• For more general cost functions L (e.g. congestion), there is a
more general criterion (see Lions, or see the result in graphs).23
![Page 66: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/66.jpg)
MFG with quadratic cost/Hamiltonian
MFG equations with quadratic cost function L(α) = α2
2 on the
domain [0,T ]× Ω, Ω standing for (0, 1)d :
(HJB) ∂tu +σ2
2∆u +
1
2|∇u|2 = −f (x ,m)
(K) ∂tm +∇ · (m∇u) =σ2
2∆m
Examples of conditions
• Boundary conditions: ∂u∂n = ∂m
∂n = 0 on (0,T )× ∂Ω
• Terminal condition: u(T , x) = g(x).
• Initial condition: m(0, x) = m0(x) ≥ 0.
The optimal control is α∗(t, x) = ∇u(t, x).
24
![Page 67: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/67.jpg)
MFG with quadratic cost/Hamiltonian
MFG equations with quadratic cost function L(α) = α2
2 on the
domain [0,T ]× Ω, Ω standing for (0, 1)d :
(HJB) ∂tu +σ2
2∆u +
1
2|∇u|2 = −f (x ,m)
(K) ∂tm +∇ · (m∇u) =σ2
2∆m
Examples of conditions
• Boundary conditions: ∂u∂n = ∂m
∂n = 0 on (0,T )× ∂Ω
• Terminal condition: u(T , x) = g(x).
• Initial condition: m(0, x) = m0(x) ≥ 0.
The optimal control is α∗(t, x) = ∇u(t, x).
24
![Page 68: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/68.jpg)
MFG with quadratic cost/Hamiltonian
MFG equations with quadratic cost function L(α) = α2
2 on the
domain [0,T ]× Ω, Ω standing for (0, 1)d :
(HJB) ∂tu +σ2
2∆u +
1
2|∇u|2 = −f (x ,m)
(K) ∂tm +∇ · (m∇u) =σ2
2∆m
Examples of conditions
• Boundary conditions: ∂u∂n = ∂m
∂n = 0 on (0,T )× ∂Ω
• Terminal condition: u(T , x) = g(x).
• Initial condition: m(0, x) = m0(x) ≥ 0.
The optimal control is α∗(t, x) = ∇u(t, x).
24
![Page 69: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/69.jpg)
Change of variables
Theorem: u = σ2 log(φ), m = φψ
Let’s consider a smooth solution (φ, ψ) (with φ > 0) of:
∂tφ+σ2
2∆φ = − 1
σ2f (x , φψ)φ (Eφ)
∂tψ −σ2
2∆ψ =
1
σ2f (x , φψ)ψ (Eψ)
• Boundary conditions: ∂φ∂n = ∂ψ
∂n = 0 on (0,T )× ∂Ω
• Terminal condition: φ(T , ·) = exp(uT (·)σ2
).
• Initial condition: ψ(0, ·) = m0(·)φ(0,·)
Then (u,m) = (σ2 log(φ), φψ) is a solution of (MFG).
Nice existence results exist on this system (see some of my papers).
25
![Page 70: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/70.jpg)
Change of variables
Theorem: u = σ2 log(φ), m = φψ
Let’s consider a smooth solution (φ, ψ) (with φ > 0) of:
∂tφ+σ2
2∆φ = − 1
σ2f (x , φψ)φ (Eφ)
∂tψ −σ2
2∆ψ =
1
σ2f (x , φψ)ψ (Eψ)
• Boundary conditions: ∂φ∂n = ∂ψ
∂n = 0 on (0,T )× ∂Ω
• Terminal condition: φ(T , ·) = exp(uT (·)σ2
).
• Initial condition: ψ(0, ·) = m0(·)φ(0,·)
Then (u,m) = (σ2 log(φ), φψ) is a solution of (MFG).
Nice existence results exist on this system (see some of my papers). 25
![Page 71: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/71.jpg)
Numerics and examples
![Page 72: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/72.jpg)
Numerical methods
• Variational formulation: when a global maximization problem
exists, gradient-descent/ascent can be used (see Lachapelle,
Salomon, Turinici)
• Finite difference method (Achdou and Capuzzo-Dolcetta)
• Specific methods in the quadratic cost case (see Gueant).
26
![Page 73: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/73.jpg)
Examples with population dynamics
Toy problem in the quadratic case
• f (x , ξ) = −16(x − 1/2)2 − 0.1 max(0,min(5, ξ)), i.e. agents
want to live near x = 12 but they do not want to live together.
• T = 0.5
• g = 0
• σ = 1
• m0(x) = µ(x)∫ 10 µ(x ′)dx ′
, where
µ(x) = 1 + 0.2 cos
(π
(2x − 3
2
))2
.
27
![Page 74: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/74.jpg)
Toy problem in the quadratic case
The functions φ and ψ.
28
![Page 75: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/75.jpg)
Toy problem in the quadratic case
The dynamics of the distribution m.
29
![Page 76: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/76.jpg)
Examples with population dynamics (videos provided by Y.
Achdou)
Going out of a movie theater (1)
• We consider a movie theatre with 6 rows, and 2 doors in the
front to exit.
• Neumann conditions on walls.
• Homogenous Dirichlet conditions at the doors.
• Running penalty while staying in the room.
• Congestion effects.
30
![Page 77: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/77.jpg)
Examples with population dynamics (videos provided by Y.
Achdou)
Going out of a movie theater (2)
• The same movie theatre with 6 rows, and 2 doors in the front
to exit.
• One door only will be open at a pre-defined time, but nobody
knows which one.
31
![Page 78: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/78.jpg)
Numerous economic applications
Many models in economics and finance – for instance:
• Interaction between economic growth and inequalities (where
Pareto distributions play a central role).
→ See Gueant, Lasry, Lions (Paris-Princeton lectures).
→ Similar ideas developed by Lucas and Moll.
• Competition between asset managers.
→ Gueant (Risk and decision analysis, 2013)
• Oil extraction (a la Hotelling) with noise.
→ See Gueant, Lasry, Lions (Paris-Princeton lectures).
• A long-term model for the mining industries.
→ Joint work of Achdou, Giraud, Lasry, Lions, and
Scheinkman.
32
![Page 79: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/79.jpg)
Numerous economic applications
Many models in economics and finance – for instance:
• Interaction between economic growth and inequalities (where
Pareto distributions play a central role).
→ See Gueant, Lasry, Lions (Paris-Princeton lectures).
→ Similar ideas developed by Lucas and Moll.
• Competition between asset managers.
→ Gueant (Risk and decision analysis, 2013)
• Oil extraction (a la Hotelling) with noise.
→ See Gueant, Lasry, Lions (Paris-Princeton lectures).
• A long-term model for the mining industries.
→ Joint work of Achdou, Giraud, Lasry, Lions, and
Scheinkman.
32
![Page 80: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/80.jpg)
Numerous economic applications
Many models in economics and finance – for instance:
• Interaction between economic growth and inequalities (where
Pareto distributions play a central role).
→ See Gueant, Lasry, Lions (Paris-Princeton lectures).
→ Similar ideas developed by Lucas and Moll.
• Competition between asset managers.
→ Gueant (Risk and decision analysis, 2013)
• Oil extraction (a la Hotelling) with noise.
→ See Gueant, Lasry, Lions (Paris-Princeton lectures).
• A long-term model for the mining industries.
→ Joint work of Achdou, Giraud, Lasry, Lions, and
Scheinkman.
32
![Page 81: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/81.jpg)
Numerous economic applications
Many models in economics and finance – for instance:
• Interaction between economic growth and inequalities (where
Pareto distributions play a central role).
→ See Gueant, Lasry, Lions (Paris-Princeton lectures).
→ Similar ideas developed by Lucas and Moll.
• Competition between asset managers.
→ Gueant (Risk and decision analysis, 2013)
• Oil extraction (a la Hotelling) with noise.
→ See Gueant, Lasry, Lions (Paris-Princeton lectures).
• A long-term model for the mining industries.
→ Joint work of Achdou, Giraud, Lasry, Lions, and
Scheinkman.32
![Page 82: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/82.jpg)
Special for Huawei: MFG on graphs
![Page 83: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/83.jpg)
Framework
MFGs are often written on continuous state spaces, but what
about discrete structures?
Notations for graph
• Graph G. Nodes indexed by integers from 1 to N.
• ∀i ∈ N = 1, . . . ,N:
• V(i) ⊂ N \ i the set of nodes j for which a directed edge
exists from i to j (cardinal: di ).
• V−1(i) ⊂ N \ i the set of nodes j for which a directed edge
exists from j to i .
33
![Page 84: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/84.jpg)
Framework
MFGs are often written on continuous state spaces, but what
about discrete structures?
Notations for graph
• Graph G. Nodes indexed by integers from 1 to N.
• ∀i ∈ N = 1, . . . ,N:
• V(i) ⊂ N \ i the set of nodes j for which a directed edge
exists from i to j (cardinal: di ).
• V−1(i) ⊂ N \ i the set of nodes j for which a directed edge
exists from j to i .
33
![Page 85: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/85.jpg)
Framework
MFGs are often written on continuous state spaces, but what
about discrete structures?
Notations for graph
• Graph G. Nodes indexed by integers from 1 to N.
• ∀i ∈ N = 1, . . . ,N:
• V(i) ⊂ N \ i the set of nodes j for which a directed edge
exists from i to j (cardinal: di ).
• V−1(i) ⊂ N \ i the set of nodes j for which a directed edge
exists from j to i .
33
![Page 86: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/86.jpg)
Framework
MFGs are often written on continuous state spaces, but what
about discrete structures?
Notations for graph
• Graph G. Nodes indexed by integers from 1 to N.
• ∀i ∈ N = 1, . . . ,N:
• V(i) ⊂ N \ i the set of nodes j for which a directed edge
exists from i to j (cardinal: di ).
• V−1(i) ⊂ N \ i the set of nodes j for which a directed edge
exists from j to i .
33
![Page 87: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/87.jpg)
Framework (continued)
Players, strategies, and costs
• Each player’s position: Markov chain (Xt)t with values in G.
• Instantaneous transition probabilities at time t:
λt(i , ·) : V(i)→ R+ (∀i ∈ N )
• Instantaneous cost L(i , (λi ,j)j∈V(i)) to set the value of λ(i , j)
to λi ,j .
34
![Page 88: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/88.jpg)
Framework (continued)
Players, strategies, and costs
• Each player’s position: Markov chain (Xt)t with values in G.
• Instantaneous transition probabilities at time t:
λt(i , ·) : V(i)→ R+ (∀i ∈ N )
• Instantaneous cost L(i , (λi ,j)j∈V(i)) to set the value of λ(i , j)
to λi ,j .
34
![Page 89: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/89.jpg)
Framework (continued)
Players, strategies, and costs
• Each player’s position: Markov chain (Xt)t with values in G.
• Instantaneous transition probabilities at time t:
λt(i , ·) : V(i)→ R+ (∀i ∈ N )
• Instantaneous cost L(i , (λi ,j)j∈V(i)) to set the value of λ(i , j)
to λi ,j .
34
![Page 90: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/90.jpg)
Framework (continued)
Players, strategies, and costs
• Each player’s position: Markov chain (Xt)t with values in G.
• Instantaneous transition probabilities at time t:
λt(i , ·) : V(i)→ R+ (∀i ∈ N )
• Instantaneous cost L(i , (λi ,j)j∈V(i)) to set the value of λ(i , j)
to λi ,j .
34
![Page 91: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/91.jpg)
Hypotheses
Hypotheses on L
• Super-linearity hypothesis:
∀i ∈ N , limλ∈Rdi
+ ,|λ|→+∞
L(i , λ)
|λ|= +∞
• Convexity hypothesis:
∀i ∈ N , λ ∈ Rdi+ 7→ L(i , λ) is strictly convex.
Also, we define:
∀i ∈ N , p ∈ Rdi 7→ H(i , p) = supλ∈Rdi
+
λ · p − L(i , λ).
35
![Page 92: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/92.jpg)
Hypotheses
Hypotheses on L
• Super-linearity hypothesis:
∀i ∈ N , limλ∈Rdi
+ ,|λ|→+∞
L(i , λ)
|λ|= +∞
• Convexity hypothesis:
∀i ∈ N , λ ∈ Rdi+ 7→ L(i , λ) is strictly convex.
Also, we define:
∀i ∈ N , p ∈ Rdi 7→ H(i , p) = supλ∈Rdi
+
λ · p − L(i , λ).
35
![Page 93: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/93.jpg)
Mean field game - control problem
Control problem
• Admissible Markovian controls:
A =
(λt(i , j))t∈[0,T ],i∈N ,j∈V(i) |t 7→ λt(i , j) ∈ L∞(0,T )
• For λ ∈ A and a given function m : [0,T ] 7→ PN we define
the payoff function: Jm : [0,T ]×N ×A → R by:
Jm(t, i , λ) = E
[∫ T
t(−L(Xs , λs(Xs , ·)) + f (Xs ,m(s))) ds
+g (XT ,m(T ))
]for (Xs)s∈[t,T ] a Markov chain on G, starting from i at time t,
with instantaneous transition probabilities given by (λs)s∈[t,T ].
Remark: ∀i ∈ N , f (i , ·) and g(i , ·) are assumed to be continuous
on PN .
36
![Page 94: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/94.jpg)
Mean field game - control problem
Control problem
• Admissible Markovian controls:
A =
(λt(i , j))t∈[0,T ],i∈N ,j∈V(i) |t 7→ λt(i , j) ∈ L∞(0,T )
• For λ ∈ A and a given function m : [0,T ] 7→ PN we define
the payoff function: Jm : [0,T ]×N ×A → R by:
Jm(t, i , λ) = E
[∫ T
t(−L(Xs , λs(Xs , ·)) + f (Xs ,m(s))) ds
+g (XT ,m(T ))
]for (Xs)s∈[t,T ] a Markov chain on G, starting from i at time t,
with instantaneous transition probabilities given by (λs)s∈[t,T ].
Remark: ∀i ∈ N , f (i , ·) and g(i , ·) are assumed to be continuous
on PN .
36
![Page 95: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/95.jpg)
Mean field game - control problem
Control problem
• Admissible Markovian controls:
A =
(λt(i , j))t∈[0,T ],i∈N ,j∈V(i) |t 7→ λt(i , j) ∈ L∞(0,T )
• For λ ∈ A and a given function m : [0,T ] 7→ PN we define
the payoff function: Jm : [0,T ]×N ×A → R by:
Jm(t, i , λ) = E
[∫ T
t(−L(Xs , λs(Xs , ·)) + f (Xs ,m(s))) ds
+g (XT ,m(T ))
]for (Xs)s∈[t,T ] a Markov chain on G, starting from i at time t,
with instantaneous transition probabilities given by (λs)s∈[t,T ].
Remark: ∀i ∈ N , f (i , ·) and g(i , ·) are assumed to be continuous
on PN .
36
![Page 96: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/96.jpg)
Mean field game - control problem
Control problem
• Admissible Markovian controls:
A =
(λt(i , j))t∈[0,T ],i∈N ,j∈V(i) |t 7→ λt(i , j) ∈ L∞(0,T )
• For λ ∈ A and a given function m : [0,T ] 7→ PN we define
the payoff function: Jm : [0,T ]×N ×A → R by:
Jm(t, i , λ) = E
[∫ T
t(−L(Xs , λs(Xs , ·)) + f (Xs ,m(s))) ds
+g (XT ,m(T ))
]for (Xs)s∈[t,T ] a Markov chain on G, starting from i at time t,
with instantaneous transition probabilities given by (λs)s∈[t,T ].
Remark: ∀i ∈ N , f (i , ·) and g(i , ·) are assumed to be continuous
on PN .
36
![Page 97: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/97.jpg)
Nash equilibrium
Nash-MFG equilibrium
A differentiable function m : t ∈ [0,T ] 7→ (m(t, i))i ∈ PN is said
to be a Nash-MFG equilibrium, if there exists an admissible control
λ ∈ A such that:
∀λ ∈ A, ∀i ∈ N , Jm(0, i , λ) ≥ Jm(0, i , λ)
and
∀i ∈ N , ddt
m(t, i) =∑
j∈V−1(i)
λt(j , i)m(t, j)−∑
j∈V(i)
λt(i , j)m(t, i)
In that case, λ is called an optimal control.
37
![Page 98: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/98.jpg)
The G-MFG equations
Definition (The G-MFG equations)
The G-MFG equations consist in a system of 2N equations, the
unknown being t ∈ [0,T ] 7→ (u(t),m(t)):
∀i ∈ N , d
dtu(t, i)+H
(i , (u(t, j)− u(t, i))j∈V(i)
)+f (i ,m(t)) = 0,
∀i , ddt
m(t, i) =∑
j∈V−1(i)
∂H(j , ·)∂pi
((u(t, k)− u(t, j))k∈V(j)
)m(t, j)
−∑
j∈V(i)
∂H(i , ·)∂pj
((u(t, k)− u(t, i))k∈V(i)
)m(t, i)
with u(T , i) = g(i ,m(T )) and m(0) = m0 ∈ PN given.
38
![Page 99: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/99.jpg)
The G-MFG equations
Proposition (The G-MFG equations as a sufficient condition)
Let m0 ∈ PN and let us consider a C 1 solution (u(t),m(t)) of the
G-MFG equations with (m(0, 1), . . . ,m(0,N)) = m0.
Then:
• t 7→ m(t) = (m(t, 1), . . . ,m(t,N)) is a Nash-MFG equilibrium
• The relations λt(i , j) = ∂H(i ,·)∂pj
((u(t, k)− u(t, i))k∈V(i)
)define an optimal control.
39
![Page 100: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/100.jpg)
The G-MFG equations
Proposition (The G-MFG equations as a sufficient condition)
Let m0 ∈ PN and let us consider a C 1 solution (u(t),m(t)) of the
G-MFG equations with (m(0, 1), . . . ,m(0,N)) = m0.
Then:
• t 7→ m(t) = (m(t, 1), . . . ,m(t,N)) is a Nash-MFG equilibrium
• The relations λt(i , j) = ∂H(i ,·)∂pj
((u(t, k)− u(t, i))k∈V(i)
)define an optimal control.
39
![Page 101: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/101.jpg)
Existence of a solution
Proposition (Existence of a solution to the G-MFG equations)
Let m0 ∈ PN . Under the assumptions made above, there exists a
C 1 solution (u,m) of the G-MFG equations such that m(0) = m0.
Sketch of proof (Fixed point):
• Comparison principle leads a priori bounds on u
supi∈N‖u(·, i)‖∞ ≤ sup
i∈N‖g(i , ·)‖∞
+
(supi∈N‖f (i , ·)‖∞ + sup
i∈N|H(i , 0)|
)T .
• ⇒ bounds on dmdt .
• Ascoli + Schauder to conclude.
40
![Page 102: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/102.jpg)
Existence of a solution
Proposition (Existence of a solution to the G-MFG equations)
Let m0 ∈ PN . Under the assumptions made above, there exists a
C 1 solution (u,m) of the G-MFG equations such that m(0) = m0.
Sketch of proof (Fixed point):
• Comparison principle leads a priori bounds on u
supi∈N‖u(·, i)‖∞ ≤ sup
i∈N‖g(i , ·)‖∞
+
(supi∈N‖f (i , ·)‖∞ + sup
i∈N|H(i , 0)|
)T .
• ⇒ bounds on dmdt .
• Ascoli + Schauder to conclude.
40
![Page 103: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/103.jpg)
Existence of a solution
Proposition (Existence of a solution to the G-MFG equations)
Let m0 ∈ PN . Under the assumptions made above, there exists a
C 1 solution (u,m) of the G-MFG equations such that m(0) = m0.
Sketch of proof (Fixed point):
• Comparison principle leads a priori bounds on u
supi∈N‖u(·, i)‖∞ ≤ sup
i∈N‖g(i , ·)‖∞
+
(supi∈N‖f (i , ·)‖∞ + sup
i∈N|H(i , 0)|
)T .
• ⇒ bounds on dmdt .
• Ascoli + Schauder to conclude.
40
![Page 104: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/104.jpg)
Uniqueness of smooth solutions
Proposition (Uniqueness for the solution of the G-MFG
equations)
Assume that f and g are such that:
∀(m, µ) ∈ PN×PN ,N∑i=1
(f (i ,m)−f (i , µ))(mi−µi ) ≥ 0 =⇒ m = µ
and
∀(m, µ) ∈ PN×PN ,N∑i=1
(g(i ,m)−g(i , µ))(mi−µi ) ≥ 0 =⇒ m = µ
Then, if (u, m) and (u, m) are two C 1 solutions of the G-MFG
equations, we have m = m and u = u.
41
![Page 105: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/105.jpg)
Uniqueness of smooth solutions
Proposition (Uniqueness for the solution of the G-MFG
equations)
Assume that f and g are such that:
∀(m, µ) ∈ PN×PN ,N∑i=1
(f (i ,m)−f (i , µ))(mi−µi ) ≥ 0 =⇒ m = µ
and
∀(m, µ) ∈ PN×PN ,N∑i=1
(g(i ,m)−g(i , µ))(mi−µi ) ≥ 0 =⇒ m = µ
Then, if (u, m) and (u, m) are two C 1 solutions of the G-MFG
equations, we have m = m and u = u.
41
![Page 106: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/106.jpg)
Uniqueness of smooth solutions
Proposition (Uniqueness for the solution of the G-MFG
equations)
Assume that f and g are such that:
∀(m, µ) ∈ PN×PN ,N∑i=1
(f (i ,m)−f (i , µ))(mi−µi ) ≥ 0 =⇒ m = µ
and
∀(m, µ) ∈ PN×PN ,N∑i=1
(g(i ,m)−g(i , µ))(mi−µi ) ≥ 0 =⇒ m = µ
Then, if (u, m) and (u, m) are two C 1 solutions of the G-MFG
equations, we have m = m and u = u.
41
![Page 107: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/107.jpg)
The G-Master equations
Definition (The G-Master equations)
The G-Master equations consist in N equations, the unknown
being (t,m) ∈ [0,T ]× PN 7→ (U1(t,m), . . . ,UN(t,m)).
∀i ∈ N , ∂Ui
∂t(t,m) + H
(i , (Uj(t,m)− Ui (t,m))j∈V(i)
)+
N∑l=1
∂Ui
∂ml(t,m)
∑j∈V−1(l)
∂H(j , ·)∂pl
((Uk(t,m)− Uj(t,m))k∈V(j)
)mj
−∑
j∈V(l)
∂H(l , ·)∂pj
((Uk(t,m)− Ul(t,m))k∈V(l)
)ml
+ f (i ,m) = 0
with Ui (T ,m) = g(i ,m).
42
![Page 108: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/108.jpg)
The G-Master equations
Proposition (From G-Master equations to G-MFG equations)
If (t,m) ∈ [0,T ]× PN 7→ (U1(t,m), . . . ,UN(t,m)) is a C 1
solution to the G-Master equations.
If a function m is such that m(0) = m0 ∈ PN and ddtm(t, i) =
∑j∈V−1(i)
∂H(j , ·)∂pi
((Uk(t,m(t))− Uj(t,m(t)))k∈V(j)
)m(t, j)
−∑
j∈V(i)
∂H(i , ·)∂pj
((Uk(t,m(t))− Ui (t,m(t)))k∈V(i)
)m(t, i)
Then t ∈ [0,T ] 7→ (U1(t,m(t)), . . . ,UN(t,m(t)),m(t)) is a
solution of the G-MFG equations.
43
![Page 109: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/109.jpg)
The G-Master equations
Proposition (From G-Master equations to G-MFG equations)
If (t,m) ∈ [0,T ]× PN 7→ (U1(t,m), . . . ,UN(t,m)) is a C 1
solution to the G-Master equations.
If a function m is such that m(0) = m0 ∈ PN and ddtm(t, i) =
∑j∈V−1(i)
∂H(j , ·)∂pi
((Uk(t,m(t))− Uj(t,m(t)))k∈V(j)
)m(t, j)
−∑
j∈V(i)
∂H(i , ·)∂pj
((Uk(t,m(t))− Ui (t,m(t)))k∈V(i)
)m(t, i)
Then t ∈ [0,T ] 7→ (U1(t,m(t)), . . . ,UN(t,m(t)),m(t)) is a
solution of the G-MFG equations.
43
![Page 110: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/110.jpg)
The G-Master equations
Proposition (From G-Master equations to G-MFG equations)
If (t,m) ∈ [0,T ]× PN 7→ (U1(t,m), . . . ,UN(t,m)) is a C 1
solution to the G-Master equations.
If a function m is such that m(0) = m0 ∈ PN and ddtm(t, i) =
∑j∈V−1(i)
∂H(j , ·)∂pi
((Uk(t,m(t))− Uj(t,m(t)))k∈V(j)
)m(t, j)
−∑
j∈V(i)
∂H(i , ·)∂pj
((Uk(t,m(t))− Ui (t,m(t)))k∈V(i)
)m(t, i)
Then t ∈ [0,T ] 7→ (U1(t,m(t)), . . . ,UN(t,m(t)),m(t)) is a
solution of the G-MFG equations.
43
![Page 111: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/111.jpg)
The G-Master equations
Proposition (From G-Master equations to G-MFG equations)
If (t,m) ∈ [0,T ]× PN 7→ (U1(t,m), . . . ,UN(t,m)) is a C 1
solution to the G-Master equations.
If a function m is such that m(0) = m0 ∈ PN and ddtm(t, i) =
∑j∈V−1(i)
∂H(j , ·)∂pi
((Uk(t,m(t))− Uj(t,m(t)))k∈V(j)
)m(t, j)
−∑
j∈V(i)
∂H(i , ·)∂pj
((Uk(t,m(t))− Ui (t,m(t)))k∈V(i)
)m(t, i)
Then t ∈ [0,T ] 7→ (U1(t,m(t)), . . . ,UN(t,m(t)),m(t)) is a
solution of the G-MFG equations.
43
![Page 112: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/112.jpg)
Potential games
Assumptions
We suppose that there exist two C 1 functions:
F : (m1, . . . ,mN) ∈ PN 7→ F (m1, . . . ,mN)
G : (m1, . . . ,mN) ∈ PN 7→ G (m1, . . . ,mN)
such that ∀i ∈ N :∂F
∂mi= f (i , ·)
∂G
∂mi= g(i , ·)
44
![Page 113: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/113.jpg)
Planning problem
We introduce for t ∈ [0,T ], mt ∈ PN and a given admissible
control (function) λ ∈ A, the payoff function
J (t,mt , λ) =∫ T
t
(F (m(s))−
N∑i=1
L(i , (λs(i , j))j∈V(i))m(s, i)
)ds + G (m(T ))
where ∀i ∈ N ,m(t, i) = mti and ∀i ∈ N ,∀s ∈ [t,T ]:
d
dsm(s, i) =
∑j∈V−1(i)
λs(j , i)m(s, j)−∑
j∈V(i)
λs(i , j)m(s, i)
Optimization problem
The (deterministic) optimization problem we consider is, for a
given m0 ∈ PN :
supλ∈AJ (0,m0, λ).
45
![Page 114: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/114.jpg)
HJ equation
Definition (The G-planning equation)
The G-planning equation consists in one PDE in Φ(t,m):
∂Φ
∂t(t,m1, . . . ,mN) +H(m1, . . . ,mN ,∇Φ) + F (m1, . . . ,mN) = 0
with the terminal conditions Φ(T ,m) = G (m), where:
H(m, p) = sup(λi,j )i∈N ,j∈V(i)
N∑i=1
[ ∑j∈V−1(i)
λj ,imj −∑
j∈V(i)
λi ,jmi
pi
−L(i , (λi ,j)j∈V(i))mi
]=
N∑i=1
miH(i , (pj − pi )j∈V(i)
)
46
![Page 115: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/115.jpg)
Solving the planning problem
Proposition
Let us consider a C 1 function Φ solution of the G-planning
equation. Then, Φ restricted to [0,T ]× PN is the value function
of the above planning problem, i.e.:
∀(t,mt) ∈ [0,T ]× PN ,Φ(t,mt) = supλ∈AJ (t,mt , λ)
47
![Page 116: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/116.jpg)
Solving the planning problem
Proposition
Moreover, if we define ∀i ∈ N ,m(t, i) = mti and
∀i ∈ N ,∀s ∈ [t,T ]
d
dsm(s, i) =
∑j∈V−1(i)
λs(j , i)m(s, j)−∑
j∈V(i)
λs(i , j)m(s, i)
with
λs(i , j) =∂H(i , ·)∂pj
((∂Φ
∂mk(s,m(s))− ∂Φ
∂mi(s,m(s))
)k∈V(i)
)
then λ is an optimal control for the planning problem.
48
![Page 117: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/117.jpg)
Going back to MFG
Proposition
Let Φ be a C 2 solution of the G-planning equation.
Define ∀i ∈ N , ∀t ∈ [0,T ], ∀m ∈ PN , Ui (t,m) = ∂Φ∂mi
(t,m).
Then, ∇Φ = U = (U1, . . . ,UN) verifies the G-Master equations.
Consequently, if we define ∀i ∈ N ,m(0, i) = m0i for a given
m0 ∈ PN and ∀i ∈ N ,∀s ∈ [t,T ]:ddsm(s, i) =
∑j∈V−1(i) λs(j , i)m(s, j)−
∑j∈V(i) λs(i , j)m(s, i)
with λs(i , j) = ∂H(i ,·)∂pj
((∂Φ∂mk
(s,m(s))− ∂Φ∂mi
(s,m(s)))k∈V(i)
)then m is a Nash-MFG equilibrium and λ is an optimal control for
the initial mean field game (the decentralized problem).
49
![Page 118: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/118.jpg)
Going back to MFG
Proposition
Let Φ be a C 2 solution of the G-planning equation.
Define ∀i ∈ N , ∀t ∈ [0,T ], ∀m ∈ PN , Ui (t,m) = ∂Φ∂mi
(t,m).
Then, ∇Φ = U = (U1, . . . ,UN) verifies the G-Master equations.
Consequently, if we define ∀i ∈ N ,m(0, i) = m0i for a given
m0 ∈ PN and ∀i ∈ N ,∀s ∈ [t,T ]:ddsm(s, i) =
∑j∈V−1(i) λs(j , i)m(s, j)−
∑j∈V(i) λs(i , j)m(s, i)
with λs(i , j) = ∂H(i ,·)∂pj
((∂Φ∂mk
(s,m(s))− ∂Φ∂mi
(s,m(s)))k∈V(i)
)then m is a Nash-MFG equilibrium and λ is an optimal control for
the initial mean field game (the decentralized problem).
49
![Page 119: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/119.jpg)
Going back to MFG
Proposition
Let Φ be a C 2 solution of the G-planning equation.
Define ∀i ∈ N , ∀t ∈ [0,T ], ∀m ∈ PN , Ui (t,m) = ∂Φ∂mi
(t,m).
Then, ∇Φ = U = (U1, . . . ,UN) verifies the G-Master equations.
Consequently, if we define ∀i ∈ N ,m(0, i) = m0i for a given
m0 ∈ PN and ∀i ∈ N ,∀s ∈ [t,T ]:ddsm(s, i) =
∑j∈V−1(i) λs(j , i)m(s, j)−
∑j∈V(i) λs(i , j)m(s, i)
with λs(i , j) = ∂H(i ,·)∂pj
((∂Φ∂mk
(s,m(s))− ∂Φ∂mi
(s,m(s)))k∈V(i)
)then m is a Nash-MFG equilibrium and λ is an optimal control for
the initial mean field game (the decentralized problem).
49
![Page 120: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/120.jpg)
Going back to MFG
Proposition
Let Φ be a C 2 solution of the G-planning equation.
Define ∀i ∈ N , ∀t ∈ [0,T ], ∀m ∈ PN , Ui (t,m) = ∂Φ∂mi
(t,m).
Then, ∇Φ = U = (U1, . . . ,UN) verifies the G-Master equations.
Consequently, if we define ∀i ∈ N ,m(0, i) = m0i for a given
m0 ∈ PN and ∀i ∈ N , ∀s ∈ [t,T ]:ddsm(s, i) =
∑j∈V−1(i) λs(j , i)m(s, j)−
∑j∈V(i) λs(i , j)m(s, i)
with λs(i , j) = ∂H(i ,·)∂pj
((∂Φ∂mk
(s,m(s))− ∂Φ∂mi
(s,m(s)))k∈V(i)
)then m is a Nash-MFG equilibrium and λ is an optimal control for
the initial mean field game (the decentralized problem).
49
![Page 121: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/121.jpg)
Going back to MFG
Proposition
Let Φ be a C 2 solution of the G-planning equation.
Define ∀i ∈ N , ∀t ∈ [0,T ], ∀m ∈ PN , Ui (t,m) = ∂Φ∂mi
(t,m).
Then, ∇Φ = U = (U1, . . . ,UN) verifies the G-Master equations.
Consequently, if we define ∀i ∈ N ,m(0, i) = m0i for a given
m0 ∈ PN and ∀i ∈ N , ∀s ∈ [t,T ]:ddsm(s, i) =
∑j∈V−1(i) λs(j , i)m(s, j)−
∑j∈V(i) λs(i , j)m(s, i)
with λs(i , j) = ∂H(i ,·)∂pj
((∂Φ∂mk
(s,m(s))− ∂Φ∂mi
(s,m(s)))k∈V(i)
)then m is a Nash-MFG equilibrium and λ is an optimal control for
the initial mean field game (the decentralized problem).
49
![Page 122: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/122.jpg)
Extending to models with congestion
We can extend existence and uniqueness of solutions of the
G-MFG equations to more general Hamiltonians.
We are not limited to
L(i , (λi ,j)j∈V(i),m) = L(i , (λi ,j)j∈V(i))− f (i ,m)
Assumptions
• Continuity: ∀i ∈ N ,L(i , ·, ·) is a continuous function from
Rdi+ × PN to R
• Asymptotic super-linearity:
∀i ∈ N ,∀m ∈ PN , limλ∈Rdi
+ ,|λ|→+∞L(i ,λ,m)|λ| = +∞
50
![Page 123: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/123.jpg)
Extending to models with congestion
We can extend existence and uniqueness of solutions of the
G-MFG equations to more general Hamiltonians.
We are not limited to
L(i , (λi ,j)j∈V(i),m) = L(i , (λi ,j)j∈V(i))− f (i ,m)
Assumptions
• Continuity: ∀i ∈ N ,L(i , ·, ·) is a continuous function from
Rdi+ × PN to R
• Asymptotic super-linearity:
∀i ∈ N ,∀m ∈ PN , limλ∈Rdi
+ ,|λ|→+∞L(i ,λ,m)|λ| = +∞
50
![Page 124: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/124.jpg)
Extending to models with congestion
We can extend existence and uniqueness of solutions of the
G-MFG equations to more general Hamiltonians.
We are not limited to
L(i , (λi ,j)j∈V(i),m) = L(i , (λi ,j)j∈V(i))− f (i ,m)
Assumptions
• Continuity: ∀i ∈ N ,L(i , ·, ·) is a continuous function from
Rdi+ × PN to R
• Asymptotic super-linearity:
∀i ∈ N ,∀m ∈ PN , limλ∈Rdi
+ ,|λ|→+∞L(i ,λ,m)|λ| = +∞
50
![Page 125: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/125.jpg)
Extending to models with congestion
We can extend existence and uniqueness of solutions of the
G-MFG equations to more general Hamiltonians.
We are not limited to
L(i , (λi ,j)j∈V(i),m) = L(i , (λi ,j)j∈V(i))− f (i ,m)
Assumptions
• Continuity: ∀i ∈ N ,L(i , ·, ·) is a continuous function from
Rdi+ × PN to R
• Asymptotic super-linearity:
∀i ∈ N ,∀m ∈ PN , limλ∈Rdi
+ ,|λ|→+∞L(i ,λ,m)|λ| = +∞
50
![Page 126: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/126.jpg)
Extending to models with congestion
Hamiltonian functions:
∀i ∈ N , p ∈ Rdi ,m ∈ PN 7→ H(i , p,m) = supλ∈Rdi
+
λ · p − L(i , λ,m)
Hypotheses
• ∀i ∈ N , H(i , ·, ·) is a continuous function.
• ∀i ∈ N ,∀m ∈ PN , H(i , ·,m) is a C 1 function with:
∂H
∂p(i , p,m) = argmax
λ∈Rdi+λ · p − L(i , λ,m)
• ∀i ∈ N ,∀j ∈ V(i), ∂H∂pj
(i , ·, ·) is a continuous function.
51
![Page 127: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/127.jpg)
Extending to models with congestion
Hamiltonian functions:
∀i ∈ N , p ∈ Rdi ,m ∈ PN 7→ H(i , p,m) = supλ∈Rdi
+
λ · p − L(i , λ,m)
Hypotheses
• ∀i ∈ N , H(i , ·, ·) is a continuous function.
• ∀i ∈ N , ∀m ∈ PN , H(i , ·,m) is a C 1 function with:
∂H
∂p(i , p,m) = argmax
λ∈Rdi+λ · p − L(i , λ,m)
• ∀i ∈ N ,∀j ∈ V(i), ∂H∂pj
(i , ·, ·) is a continuous function.
51
![Page 128: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/128.jpg)
Extending to models with congestion - Existence
Using the same proof as above:
Proposition (Existence)
Under the assumptions made above, there exists a C 1 solution
(u,m) of the G-MFG equations.
52
![Page 129: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/129.jpg)
Extending to models with congestion - Uniqueness
Proposition (Uniqueness)
Assume that g is such that:
∀(m, µ) ∈ PN×PN ,N∑i=1
(g(i ,m)−g(i , µ))(mi−µi ) ≥ 0 =⇒ m = µ
Assume that the hamiltonian functions can be written as:
∀i ∈ N ,∀p ∈ Rdi , ∀m ∈ PN ,H(i , p,m) = Hc(i , p,m) + f (i ,m)
with ∀i ∈ N , f (i , ·) a continuous function satisfying
∀(m, µ) ∈ PN×PN ,N∑i=1
(f (i ,m)−f (i , µ))(mi−µi ) ≥ 0 =⇒ m = µ
53
![Page 130: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/130.jpg)
Extending to models with congestion - Uniqueness
Proposition (Uniqueness)
Assume that g is such that:
∀(m, µ) ∈ PN×PN ,N∑i=1
(g(i ,m)−g(i , µ))(mi−µi ) ≥ 0 =⇒ m = µ
Assume that the hamiltonian functions can be written as:
∀i ∈ N ,∀p ∈ Rdi , ∀m ∈ PN ,H(i , p,m) = Hc(i , p,m) + f (i ,m)
with ∀i ∈ N , f (i , ·) a continuous function satisfying
∀(m, µ) ∈ PN×PN ,N∑i=1
(f (i ,m)−f (i , µ))(mi−µi ) ≥ 0 =⇒ m = µ
53
![Page 131: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/131.jpg)
Extending to models with congestion - Uniqueness
Proposition (Uniqueness)
Assume that g is such that:
∀(m, µ) ∈ PN×PN ,N∑i=1
(g(i ,m)−g(i , µ))(mi−µi ) ≥ 0 =⇒ m = µ
Assume that the hamiltonian functions can be written as:
∀i ∈ N ,∀p ∈ Rdi , ∀m ∈ PN ,H(i , p,m) = Hc(i , p,m) + f (i ,m)
with ∀i ∈ N , f (i , ·) a continuous function satisfying
∀(m, µ) ∈ PN×PN ,N∑i=1
(f (i ,m)−f (i , µ))(mi−µi ) ≥ 0 =⇒ m = µ
53
![Page 132: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/132.jpg)
Extending to models with congestion - Uniqueness
Proposition (Uniqueness (continued))
and ∀i ∈ N , Hc(i , ·, ·) a C 1 function with:
∀j ∈ V(i), ∂Hc∂pj
(i , ·, ·) a C 1 function on Rn × PN
Now, let us define
A : (q1, . . . , qN ,m) ∈∏N
i=1 Rdi × PN 7→ (αij(qi ,m))i ,j ∈MN
defined by:
αij(qi ,m) = −∂Hc
∂mj(i , qi ,m)
Let us also define, ∀i ∈ N ,
B i : (qi ,m) ∈ Rdi × PN 7→(βijk(qi ,m)
)j ,k∈MN,di defined by:
βijk(qi ,m) = mi∂2Hc
∂mj∂qik(i , qi ,m)
54
![Page 133: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/133.jpg)
Extending to models with congestion - Uniqueness
Proposition (Uniqueness (continued))
and ∀i ∈ N , Hc(i , ·, ·) a C 1 function with:
∀j ∈ V(i), ∂Hc∂pj
(i , ·, ·) a C 1 function on Rn × PNNow, let us define
A : (q1, . . . , qN ,m) ∈∏N
i=1 Rdi × PN 7→ (αij(qi ,m))i ,j ∈MN
defined by:
αij(qi ,m) = −∂Hc
∂mj(i , qi ,m)
Let us also define, ∀i ∈ N ,
B i : (qi ,m) ∈ Rdi × PN 7→(βijk(qi ,m)
)j ,k∈MN,di defined by:
βijk(qi ,m) = mi∂2Hc
∂mj∂qik(i , qi ,m)
54
![Page 134: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/134.jpg)
Extending to models with congestion - Uniqueness
Proposition (Uniqueness (continued))
and ∀i ∈ N , Hc(i , ·, ·) a C 1 function with:
∀j ∈ V(i), ∂Hc∂pj
(i , ·, ·) a C 1 function on Rn × PNNow, let us define
A : (q1, . . . , qN ,m) ∈∏N
i=1 Rdi × PN 7→ (αij(qi ,m))i ,j ∈MN
defined by:
αij(qi ,m) = −∂Hc
∂mj(i , qi ,m)
Let us also define, ∀i ∈ N ,
B i : (qi ,m) ∈ Rdi × PN 7→(βijk(qi ,m)
)j ,k∈MN,di defined by:
βijk(qi ,m) = mi∂2Hc
∂mj∂qik(i , qi ,m)
54
![Page 135: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/135.jpg)
Extending to models with congestion - Uniqueness
Proposition (Uniqueness (continued))
Let us also define, ∀i ∈ N ,
C i : (qi ,m) ∈ Rdi × PN 7→(γ ijk(qi ,m)
)j ,k∈Mdi ,N defined by:
γ ijk(qi ,m) = mi∂2Hc
∂mk∂qij(i , qi ,m)
Let us finally define, ∀i ∈ N ,
D i : (qi ,m) ∈ Rdi × PN 7→(δijk(qi ,m)
)j ,k∈Mdi defined by:
δijk(qi ,m) = mi∂2Hc
∂qij∂qik(i , qi ,m)
55
![Page 136: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/136.jpg)
Extending to models with congestion - Uniqueness
Proposition (Uniqueness (continued))
Let us also define, ∀i ∈ N ,
C i : (qi ,m) ∈ Rdi × PN 7→(γ ijk(qi ,m)
)j ,k∈Mdi ,N defined by:
γ ijk(qi ,m) = mi∂2Hc
∂mk∂qij(i , qi ,m)
Let us finally define, ∀i ∈ N ,
D i : (qi ,m) ∈ Rdi × PN 7→(δijk(qi ,m)
)j ,k∈Mdi defined by:
δijk(qi ,m) = mi∂2Hc
∂qij∂qik(i , qi ,m)
55
![Page 137: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/137.jpg)
Extending to models with congestion - Uniqueness
Proposition (Uniqueness (continued))
Let us also define, ∀i ∈ N ,
C i : (qi ,m) ∈ Rdi × PN 7→(γ ijk(qi ,m)
)j ,k∈Mdi ,N defined by:
γ ijk(qi ,m) = mi∂2Hc
∂mk∂qij(i , qi ,m)
Let us finally define, ∀i ∈ N ,
D i : (qi ,m) ∈ Rdi × PN 7→(δijk(qi ,m)
)j ,k∈Mdi defined by:
δijk(qi ,m) = mi∂2Hc
∂qij∂qik(i , qi ,m)
55
![Page 138: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/138.jpg)
Extending to models with congestion - Uniqueness
Proposition (Uniqueness (continued))
Assume that ∀(q1, . . . , qN ,m) ∈∏N
i=1 Rdi × PN
A(q1, . . . , qN ,m) B1(q1,m) · · · · · · · · · BN(qN ,m)
C 1(q1,m) D1(q1,m) 0 · · · · · · 0... 0
. . .. . .
......
.... . .
. . .. . .
......
.... . .
. . . 0
CN(qN ,m) 0 · · · · · · 0 DN(qN ,m)
≥ 0
Then, if (u, m) and (u, m) are two C 1 solutions of the G-MFG
equations, we have m = m and u = u.
56
![Page 139: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/139.jpg)
Extending to models with congestion - Uniqueness
Proposition (Uniqueness (continued))
Assume that ∀(q1, . . . , qN ,m) ∈∏N
i=1 Rdi × PN
A(q1, . . . , qN ,m) B1(q1,m) · · · · · · · · · BN(qN ,m)
C 1(q1,m) D1(q1,m) 0 · · · · · · 0... 0
. . .. . .
......
.... . .
. . .. . .
......
.... . .
. . . 0
CN(qN ,m) 0 · · · · · · 0 DN(qN ,m)
≥ 0
Then, if (u, m) and (u, m) are two C 1 solutions of the G-MFG
equations, we have m = m and u = u.
56
![Page 140: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/140.jpg)
Extending to models with congestion - Uniqueness
Proposition (Uniqueness (continued))
Assume that ∀(q1, . . . , qN ,m) ∈∏N
i=1 Rdi × PN
A(q1, . . . , qN ,m) B1(q1,m) · · · · · · · · · BN(qN ,m)
C 1(q1,m) D1(q1,m) 0 · · · · · · 0... 0
. . .. . .
......
.... . .
. . .. . .
......
.... . .
. . . 0
CN(qN ,m) 0 · · · · · · 0 DN(qN ,m)
≥ 0
Then, if (u, m) and (u, m) are two C 1 solutions of the G-MFG
equations, we have m = m and u = u.
56
![Page 141: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/141.jpg)
Conclusion
![Page 142: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/142.jpg)
Advantages and limitations of MFGs
Advantages
• Large class of problems can be modeled.
• Benefit from centuries of differential calculus.
• Numerical methods to solve PDEs are available.
• Possibility to extend the framework to several continuums of
agents and to add big players.
• Possibility to have common noise, but equations are more
difficult (Master equation).
Limitations/Drawbacks
• Only rational/perfect expectations → models are not flexible
enough sometimes.
• Numerical methods on graphs have not been proposed...
maybe Reinforcement Learning.
57
![Page 143: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/143.jpg)
Advantages and limitations of MFGs
Advantages
• Large class of problems can be modeled.
• Benefit from centuries of differential calculus.
• Numerical methods to solve PDEs are available.
• Possibility to extend the framework to several continuums of
agents and to add big players.
• Possibility to have common noise, but equations are more
difficult (Master equation).
Limitations/Drawbacks
• Only rational/perfect expectations → models are not flexible
enough sometimes.
• Numerical methods on graphs have not been proposed...
maybe Reinforcement Learning.
57
![Page 144: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/144.jpg)
Advantages and limitations of MFGs
Advantages
• Large class of problems can be modeled.
• Benefit from centuries of differential calculus.
• Numerical methods to solve PDEs are available.
• Possibility to extend the framework to several continuums of
agents and to add big players.
• Possibility to have common noise, but equations are more
difficult (Master equation).
Limitations/Drawbacks
• Only rational/perfect expectations → models are not flexible
enough sometimes.
• Numerical methods on graphs have not been proposed...
maybe Reinforcement Learning.
57
![Page 145: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/145.jpg)
Advantages and limitations of MFGs
Advantages
• Large class of problems can be modeled.
• Benefit from centuries of differential calculus.
• Numerical methods to solve PDEs are available.
• Possibility to extend the framework to several continuums of
agents and to add big players.
• Possibility to have common noise, but equations are more
difficult (Master equation).
Limitations/Drawbacks
• Only rational/perfect expectations → models are not flexible
enough sometimes.
• Numerical methods on graphs have not been proposed...
maybe Reinforcement Learning.
57
![Page 146: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/146.jpg)
Advantages and limitations of MFGs
Advantages
• Large class of problems can be modeled.
• Benefit from centuries of differential calculus.
• Numerical methods to solve PDEs are available.
• Possibility to extend the framework to several continuums of
agents and to add big players.
• Possibility to have common noise, but equations are more
difficult (Master equation).
Limitations/Drawbacks
• Only rational/perfect expectations → models are not flexible
enough sometimes.
• Numerical methods on graphs have not been proposed...
maybe Reinforcement Learning.
57
![Page 147: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/147.jpg)
Advantages and limitations of MFGs
Advantages
• Large class of problems can be modeled.
• Benefit from centuries of differential calculus.
• Numerical methods to solve PDEs are available.
• Possibility to extend the framework to several continuums of
agents and to add big players.
• Possibility to have common noise, but equations are more
difficult (Master equation).
Limitations/Drawbacks
• Only rational/perfect expectations → models are not flexible
enough sometimes.
• Numerical methods on graphs have not been proposed...
maybe Reinforcement Learning.
57
![Page 148: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/148.jpg)
Advantages and limitations of MFGs
Advantages
• Large class of problems can be modeled.
• Benefit from centuries of differential calculus.
• Numerical methods to solve PDEs are available.
• Possibility to extend the framework to several continuums of
agents and to add big players.
• Possibility to have common noise, but equations are more
difficult (Master equation).
Limitations/Drawbacks
• Only rational/perfect expectations → models are not flexible
enough sometimes.
• Numerical methods on graphs have not been proposed...
maybe Reinforcement Learning.
57
![Page 149: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/149.jpg)
Advantages and limitations of MFGs
Advantages
• Large class of problems can be modeled.
• Benefit from centuries of differential calculus.
• Numerical methods to solve PDEs are available.
• Possibility to extend the framework to several continuums of
agents and to add big players.
• Possibility to have common noise, but equations are more
difficult (Master equation).
Limitations/Drawbacks
• Only rational/perfect expectations → models are not flexible
enough sometimes.
• Numerical methods on graphs have not been proposed...
maybe Reinforcement Learning.
57
![Page 150: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/150.jpg)
Advantages and limitations of MFGs
Advantages
• Large class of problems can be modeled.
• Benefit from centuries of differential calculus.
• Numerical methods to solve PDEs are available.
• Possibility to extend the framework to several continuums of
agents and to add big players.
• Possibility to have common noise, but equations are more
difficult (Master equation).
Limitations/Drawbacks
• Only rational/perfect expectations → models are not flexible
enough sometimes.
• Numerical methods on graphs have not been proposed...
maybe Reinforcement Learning.
57
![Page 151: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/151.jpg)
Advantages and limitations of MFGs
Advantages
• Large class of problems can be modeled.
• Benefit from centuries of differential calculus.
• Numerical methods to solve PDEs are available.
• Possibility to extend the framework to several continuums of
agents and to add big players.
• Possibility to have common noise, but equations are more
difficult (Master equation).
Limitations/Drawbacks
• Only rational/perfect expectations → models are not flexible
enough sometimes.
• Numerical methods on graphs have not been proposed...
maybe Reinforcement Learning. 57
![Page 152: An introduction to mean eld games and their applications · Academic positions at Paris 7, then ENSAE, and now Paris 1. Main research eld: optimal control and applications (incl](https://reader034.vdocument.in/reader034/viewer/2022043020/5f3c51c8fa78c2505b6bdc44/html5/thumbnails/152.jpg)
The End
Thank you. Questions?
58