This program provides mathematical background for the long program in Fall 2021 on Distributed Solutions to Complex Societal Problems. Examples of the problems to be addressed in the long program include modeling phenomena such as the macroeconomy, conflict, financial regulation, crowd movement, big data, and advertising, as well as engineering problems involving decentralized intelligence, machine learning, and telecommunications.

An important mathematical development contributing to the understanding of such problems is the theory of Mean Field Games. This is a mathematical framework well-suited to the study of models in which a large number of agents interact strategically in a stochastically evolving environment, all responding to a range of incentives, and all trying to simultaneously forecast the decisions of other agents.

The tutorial lectures will be complemented by discussions and short talks on related topics.

The intended audience consists of advanced graduate students, postdocs, and researchers interested in the general topic who have some knowledge of probability, stochastic analysis, and partial differential equations. If necessary, some of the more advanced background material will be presented in additional lectures.

Material from the first two weeks is a prerequisite for the rest of the program. Attendees who are not familiar with that material are encouraged to participate in the entire program.

The program will consist of the following six tutorials.

DatesTopicOrganizer
June 1-4 Mean Field Games: The Analytic Approach Daniela Tonon
University of Padova
June 7-11 Mean Field Games: The Probabilistic approach Daniel Lacker
IEOR, Columbia University
June 14-18 Mean Field Games and Applications: Numerical Methods Yves Achdou
Mathematics, Université Paris-Diderot
Economic Models and MFG theory Fernando Alvarez
Economics, University of Chicago
June 21-25 Network Games René Carmona
ORFE, Princeton University
Crowd and Social Dynamics Benedetto Piccoli
Mathematics, Rutgers University

Week 1: June 1-4 

Mean Field Games: The Analytic Approach 

  • Daniela Tonon (University of Padova)
  • Marco Cirant (University of Padova)

Abstract: Mean Field Games systems have been introduced simultaneously in 2006 by Lasry-Lions and Huang-Caines-Malhamé to describe Nash equilibria in differential games with infinitely many players.  After introducing the model we show how the existence and uniqueness of classical solutions can be proved using a fixed point argument in some simple cases. Then, we present a variational framework that allows to construct weak solutions for general first order and second order degenerate Mean Field Games. For non-degenerate second order problems, we finally explore some methods in parabolic regularity to obtain smoothness of solutions.

Daily schedule:

8:45-9:00 (Tuesday, June 1 only) Introductory remarks by Takis Souganidis on fall long program
9:00-11:00 Lectures by Marco Cirant
11-11:30 Break
11:30-1:30 Lectures by Daniela Tonon
3:30-4:30 Office hours by Peter Morfe and Sebastian Munoz (see the participant information page for Zoom links)
4:30-5:30 (June 2-4) Office hours by Nikiforos Mimikos (see the participant information page for Zoom link)

Week 2: June 7-11 

Mean Field Games: The Probabilistic  Approach

  • Daniel Lacker (Industrial Engineering and Operations Research, Columbia University)
  • Francois Delarue (University of Nice)
  • Ludovic Tangpi (Princeton University)

Abstract: This week’s course is devoted to techniques from probability and stochastic analysis which have played a central role in the development of mean field game theory in the past decade. This includes (1) the basics of McKean-Vlasov equations and propagation of chaos, (2) analysis of the mean field game fixed point problem via forward-backward stochastic differential equations and probabilistic compactification techniques, and (3) construction of approximate equilibria for finite-population games. As time permits, the course will cover also (4) the convergence problem and (5) aspects of the master equation.

Daily schedule:

9:00-10:15: Lectures by Francois Delarue
10:45-12:00: Lectures by Daniel Lacker
2:00-3:15 Lectures by Ludovic Tangpi
3:30-4:30 Problem Session and Office Hours by Mark Cerenzia

Week 3:  June 14-18  

1.     Mean Field Games and Applications: Numerical Methods (June 14-16)

  • Yves Achdou (Université de Paris, Laboratoire Jacques-Louis Lions (LJLL))
  • Mathieu Laurière (ORFE, Princeton)

Abstract: The theory of  mean field games  aims at studying deterministic or stochastic differential games (Nash equilibria) as the number of agents tends to infinity. Since only very few mean field games have explicit or semi-explicit solutions, numerical simulations  play a crucial role in obtaining quantitative information from this class of models.  When there is no common noise, mean field games  lead to  systems of  evolutive partial differential equations coupling a backward Bellman equation and a forward Fokker-Planck equation. In this mini-course,  several aspects of a finite difference method used to approximate the previously mentioned system of PDEs are discussed, including convergence, variational aspects and algorithms for solving the resulting  systems of nonlinear equations.  Stochastic methods based on neural networks will also be addressed.  Examples of applications  to  macroeconomics and to  models  of crowd motion will be discussed.

Schedule:

Monday and Tuesday, June 14-15
9:00-11:00 Lectures by Yves Achdou
12:00-2:00 Lectures by Mathieu Laurière

Wednesday, June 16
9:00-11:00 Lecture by Yves Achdou

2.     Economic Models and MFG theory (June16-18)

  • Fernando Alvarez (Economics, University of Chicago) 
  • Adrien Bilal (Economics, Harvard University)

Abstract: Applications of MFG to price setting (i.e. equilibrium models where firms face cost of adjustment nominal prices) and sectoral reallocation models (equilibrium models where firms and or households face cost of moving across locations, industries or occupations). MFG and perturbation to characterize the dynamic response to “shocks” in equilibrium models.  Combination of MFG master equation and  local perturbation methods as a tool to characterize equilibrium processes.

Schedule:

Wednesday, June 16
12:00-2:00 Lecture

Thursday and Friday, June 17-18
9:00-11:00 Lectures
12:00-2:00 Lectures

Week 4: June 21-25

1.  June 21-23 Network Games  

  • Rene Carmona (ORFE, Princeton University) 
  • Alex Aurell (ORFE, Princeton University)
  • Yichen Feng (UCSB)
  • Francesca Parise (Electrical and Computer Engineering, Cornell University)
  • Agathe Sorel (Industrial Engineering and Operations Research, Columbia University)

Abstract: The goal of the tutorial is to present game models, both static and dynamic, both deterministic and stochastic, for which the interactions between the players are underpinned by a graph, possibly random. The foundations will be laid down for the participants to benefit from the current research on stochastic network games for large dense graphs and graphons, and limits of sparse graphs.

Detailed schedule:

Monday, June 21, 2021 

Lecture #1 9:00 am – 10:20 am: A Crash Course on Static Games

Lecture #2 10:40 am – 12:00 pm: Introduction to Graphs and Network Games

Invited Talk #1 2:00 pm – 2:40 pm: Y. Feng, Linear-Quadratic Stochastic Differential Games on Directed Chain Networks

Invited Talk #1 2:40 pm – 3:20 pm: A. Sorel, A case Study on Stochastic Games on Large Graphs in Mean Field and Sparse Regimes

Coffee Break 3:20 pm – 3:40 pm

Lightning Talks: 3:40 pm – 4:00 pm 

Tuesday, June 22, 2021 

Lecture #3 9:00 am – 10:20 am: Games with Incomplete Information and Auctions

Lecture #4 10:40 am – 12:00 pm: Graphon Games

Invited Talk #1 2:00 pm – 2:40 pm: F. Parise, Analysis and interventions in large networks: graphon games and graphon contagion

Invited Talk #1 2:40 pm – 3:20 pm: A. Aurell, Stochastic Dynamic Graphon Games

Coffee Break 3:20 pm – 3:40 pm

Lightning Talks: 3:40 pm – 4:00 pm

Wednesday, June 23, 2021 

Lecture #1 9:00 am – 10:20 am: Signals and Correlated Equilibria

Lecture #2 10:40 am – 12:00 pm: Large Bayesian Games and Cournot Competition

The “lightning talks” will be short slots of 10 minutes for participants who want to present something they did, ask a question, etc. 

2.  June 23-25 Crowd and Social Dynamics  8 Lectures 

  • Benedetto Piccoli (Mathematics, Rutgers)
  • Felisia Angela Chiarello (Politecnico di Torino)
  • Xiaoqian Gong (Mathematics, Arizona State University)
  • Andrea Tosin  (Mathematics,  Polytecnico di Torino)

Abstract: Applications of mean-field games and mean-field controlled models will be introduced, focusing on opinion dynamics, crowd dynamics and vehicular traffic.  The mean-field controlled models will be introduced along the Neitzert-Dobrushin approach and Wasserstein spaces. The mean-field limit for controlled dynamics is then addressed with its limitations.   Control by leaders and other techniques are proposed are remedies.   Finally, measure differential equations are presented.

Schedule:

Wednesday, June 23
2:00 Benedetto Piccoli, “Introduction to the course and first examples from social dynamics, crowd dynamics and vehicular traffic”
3:00 Xiaoqian Gong, “Basics of optimal transport, Wasserstein distance and mean-field limits”
4:00 Xiaoqian Gong, office hours

Thursday, June 24
9:00 Andrea Tosin, “Introduction to kinetic model for vehicular traffic”
10:00 Felisia Angela Chiarello, “Chapman-Enskog expansions, kinetic and macroscopic models for vehicular traffic”
11:00 Benedetto Piccoli, “Crowd dynamics: from phenomenology to multiscale models”
3:00 Xiaoqian Gong, “Generalized Wasserstein distance and models for multilane traffic”
4:00 Xiaoqian Gong, office hours

Friday, June 25
9:00 Andrea Tosin, “Autonomous driving, controlled microscopic models and their kinetic and fluid dynamic limits”
10:00 Felisia Angela Chiarello, office hours
11:00 Benedetto Piccoli, “Multipopulation models and smoothing of traffic with AVs on highways”