Inference on the equilibrium flow problem

Research retreat and mini-workshop held at the IASC Cargese, Corsica, France, April 21-26, 2025

Organizers: Alfred Galichon (New York University and Sciences Po) and Antoine Jacquet (Sciences Po)

Funded by the European Research Council grant ERC-CoG No. 866274 “EQUIPRICE”.

This event combines a research retreat and a mini-workshop both seeking to cross perspectives in the construction of a general framework for estimation of the “equilibrium flow problem”. The “equilibrium flow problem,” extends minimum cost flow problems in order to provide a unified network-based framework to analyze problems such as matching problems, multinomial choice problems, hedonic pricing problems, shortest path problems, dynamic programming problems, international trade flows. The equilibrium flow problem is a far-reaching extension of Optimal Transport, and thus, this project can be seen as an extension of the increasingly popular topic of “Inverse Optimal Transport”.
We have been studying the mathematical properties of the problem, with questions such as existence of equilibrium prices, investigating possible uniqueness, and lattice structure, and special attention paid to the Nontransferable Utility (NTU) limit of the problem, which, in the bipartite case, provides the Gale and Shapley stable marriage problem. An inferential theory is build using maximum likelihood estimation and minimax-regret estimators. Model selection is incorporated to estimation procedures; more precisely, the proximal mapping operator will be inserted in between two iterative phases of these procedures. Several applications are developed, one to the gravity equation in international trade, one to hedonic models, one to matching models.

Investigators and speakers:

• Guillaume Carlier (Dauphine)
• Alfred Galichon (NYU and Sciences Po)
• Pierre Jacob (ESSEC)
• Antoine Jacquet (Sciences Po)
• Jean-Bernard Lasserre (Toulouse School of Economics)
• Guillaume Pouliot (University of Chicago)
• Maxime Sylvestre (Dauphine)

Talks are accessible to the public (on zoom) but registration is mandatory by emailing antoine.jacquet@sciencespo.fr.

Monday April 21, 2025

Morning: arrival and welcome coffee

Afternoon:
230pm – 330pm: Opening session, “leveling the playing field: what are the main results we are after?” (led by Antoine Jacquet and Maxime Sylvestre)
330pm – 4pm: coffee break.
4pm – 6pm: group work session.

Tuesday April 22, 2025

Morning
930am – 1030am: Guillaume Carlier, TBA.
1030am – 11am: coffee break.
11am – 1230pm: group work session.

Afternoon
230pm – 330pm: Maxime Sylvestre, “Convergence of a hybrid scheme for the computation of weak inverse optimal transport”.
330pm – 4pm: coffee break.
4pm – 6pm: group work session.

Wednesday April 23, 2025

Morning
930am – 1030am: Jean-Bernard Lasserr.e, “Gaussian mixtures closest to a given measure via optimal transport”.
1030am – 11am: coffee break.
11am – 1230pm: group work session.

Afternoon: free time

Thursday April 24, 2025

Morning
930am – 1030am: Antoine Jacquet, “Computation of NTU aggregate equilibria with a finite-time Newton-Jacobi method.”
1030am – 11am: coffee break.
11am – 1230pm: group work session.

Afternoon
230pm – 330pm: Pierre Jacob, TBA
330pm – 4pm: coffee break.
4pm – 6pm: group work session.

Friday April 25, 2025

Morning
1030am – 1030am: Alfred Galichon, “Some partial results on the inference in ITU matching models using LCP theory.”
1030am – 11am: coffee break.
11am – 1230pm: group work session.

Afternoon
230pm -330pm: Guillaume Pouliot, TBA.
330pm – 4pm: coffee break.
4pm – 6pm: group work session.

Saturday April 26, 2025

Morning
Departure

Applied microeconometrics

PhD Course, NYU Economics

Spring 2025

Alfred Galichon

This course will revisit some classical topics in microeconometrics (such as random utility models, dynamic discrete choice, demand estimation, matching models, and bundle choice problems) though the lenses of advanced computational methods (large scale optimization and machine learning). An important part of the course is dedicated to gaining familiarity with computational libraries such as scikit-learn, pytorch, openAI gym, chatGPT, gurobi, and others.

Lectures are delivered under a mix of in-person and online format. The language used is Python. Students not familiar with Python should contact the instructor to be provided a crash course before the start of classes.

Part 1. Random utility models

Content:
Poisson regression and logistic regression as generalized Linear Models, Lasso and Elastic Net, Min-Max Regret. Computation using Scikit-learn and TensorFlow.

Lectures:

• L1
• L2
• L3
• L4

References:

• An Introduction to Statistical Learning with applications in Python with by James, Witten, Hastie, Tibshirani and Taylor
• The Elements of Statistical Learning by Hastie, Tibshirani, and Friedman.
• Generalized Linear Models by McCullagh and Nelder.

Applications:

• housing prices https://www.kaggle.com/datasets/yasserh/housing-prices-dataset

• choice of transportation mode https://www.kaggle.com/datasets/thedevastator/california-commuting-mode-choice-from-2000-2010

Part 2. Dynamic discrete choice models
Content:
Rust, Markov Decision Processes, Multi-armed bandits, Q-Learning. Computation using OpenAI Gym.

Lectures:

• L5
• L6
• L7
• L8

References:

• Rust, J. (1987). Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher. Econometrica.
• Dynamic Programming and Optimal Control by Dimitri P. Bertsekas.
• Reinforcement Learning: An Introduction by Richard S. Sutton and Andrew G. Barto.

Applications:

• optimal maintenance decisions:  http://qed.econ.queensu.ca/jae/datasets/blevins001/
• career decisions https://respy.readthedocs.io/en/latest/projects/estimating-keane-and-wolpin-1997-msm.html
• tbd, taken from http://individual.utoronto.ca/vaguirre/wpapers/program_code_survey_joe_2008.html

Part 3. Characteristics models
Content:
Pure characteristics model, random coefficient logit model, Power diagrams, matching models. Berry, Levinsohn, Pakes. Simulation (Probit, GHK), stochastic GD. Computation using pyopt package, pyBLP, pyTorch.

Lectures:

• L9
• L10
• L11
• L12

References:

• Deep Learning by Aaron Courville, Ian Goodfellow, and Yoshua Bengio.
• Train, K. (2009). Discrete Choice Methods with Simulation.
• Galichon, A. (2016). Optimal Transport Methods in Economics.

Applications:

* automotive pricing https://www.kaggle.com/code/rkamath1/exploratory-analysis-tests-regression/input

https://pyblp.readthedocs.io/en/stable/_notebooks/tutorial/blp.html

* marriage market: https://github.com/TraME-Project/TraME-Datasets/

Cemmap masterclass, June 3-4, 2024

These lectures will introduce the optimal transport (OT) toolbox, with two applications in econometrics. The first one will pertain to the estimation of matching models. We start by introducing the discrete OT problem and its entropic regularization, and inverse OT, as well as its estimation using generalized linear models. The second application will deal with quantile methods. The one-dimensional OT problem will be discussed as well as its connections with the notions of quantile and rank is then covered. Connection with quantile regression will be discussed and the ‘vector quantile regression’ problem will then be introduced.

Part I Introduction (3h)
S1. Monge-Kantorovich duality (1h30)

S2. Computational optimal transport (1h30)

https://www.math-econ-code.org/optimal-assignment

Part II OT and matching models (3h)
S3. Matching with Transferable Utility and random utility (1h30)

https://www.math-econ-code.org/regularized-optimal-transport

S4. Estimation of matching models (1h30)

https://www.math-econ-code.org/matching-estimation

Part III OT and quantiles (2h)
S5. 1D optimal transport and quantiles (1h)

https://www.math-econ-code.org/one-dimensional-assignment

S5. Connection with quantile regression (1h)

https://www.math-econ-code.org/quantile-regression

Applied microeconometrics

PhD Course, NYU Economics

Spring 2024

Alfred Galichon

This course will revisit some classical topics in microeconometrics (such as random utility models, dynamic discrete choice, demand estimation, matching models, and bundle choice problems) though the lenses of machine learning and state-of-the-art optimization methods. An important part of the course is dedicated to gaining familiarity with computational libraries such as scikit-learn, pytorch, openAI gym, chatGPT, gurobi, and others.

Lectures are delivered under a mix of in-person and online format. The language used is Python. Students not familiar with Python should contact the instructor to be provided a crash course before the start of classes.

Part 1. Random utility models meet Machine learning

Content:
Poisson regression and logistic regression as generalized Linear Models, Lasso and Elastic Net, Min-Max Regret. Computation using Scikit-learn and TensorFlow.

Lectures:

• L1: Tue 1/30, 1145am-145pm (19W4, 802 and zoom)
• L2: Thu 2/1, 1pm-3pm (19W4, 802 and zoom)
• L3: Tue 2/6, 1145am-145pm (zoom)
• L4: Thu 2/15, 1pm-3pm (zoom)

References:

• An Introduction to Statistical Learning with applications in Python with by James, Witten, Hastie, Tibshirani and Taylor
• The Elements of Statistical Learning by Hastie, Tibshirani, and Friedman.
• Generalized Linear Models by McCullagh and Nelder.

Applications:

• housing prices https://www.kaggle.com/datasets/yasserh/housing-prices-dataset

• choice of transportation mode https://www.kaggle.com/datasets/thedevastator/california-commuting-mode-choice-from-2000-2010

Part 2. Dynamic discrete choice models meet Reinforcement Learning
Content:
Rust, Markov Decision Processes, Multi-armed bandits, Q-Learning. Computation using OpenAI Gym and Stable Baselines.

Lectures:

• L5: Thu 2/29, 1pm-3pm (zoom)
• L6: Thu 3/7, 1pm-3pm (zoom)
• L7: Wed 3/13, 330pm-530pm (19W4, 802 and zoom)
• L8: Thu Mar 3/14, 1pm-3pm (19W4, 802 and zoom)

References:

• Rust, J. (1987). Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher. Econometrica.
• Dynamic Programming and Optimal Control by Dimitri P. Bertsekas.
• Reinforcement Learning: An Introduction by Richard S. Sutton and Andrew G. Barto.

Applications:

• optimal maintenance decisions:  http://qed.econ.queensu.ca/jae/datasets/blevins001/
• career decisions https://respy.readthedocs.io/en/latest/projects/estimating-keane-and-wolpin-1997-msm.html
• tbd, taken from http://individual.utoronto.ca/vaguirre/wpapers/program_code_survey_joe_2008.html

Part 3. Characteristics models meet Deep Learning and Optimal Transport
Content:
Pure characteristics model, random coefficient logit model, Power diagrams, matching models.
Simulation (Probit, GHK), stochastic GD. Computation using pyopt package, pyBLP, pyTorch.

Lectures:

• L9: Tue 4/2, 1145am-145pm (zoom)
• L10: Tue 4/9, 1145am-145pm (zoom)
• L11: Tue 4/16, 1145am-145pm (19W4, 802 and zoom)
• L12: Thu 4/18, 1pm-3pm (19W4, 802 and zoom)

References:

• Deep Learning by Aaron Courville, Ian Goodfellow, and Yoshua Bengio.
• Train, K. (2009). Discrete Choice Methods with Simulation.
• Galichon, A. (2016). Optimal Transport Methods in Economics.

Applications:

* automotive pricing https://www.kaggle.com/code/rkamath1/exploratory-analysis-tests-regression/input

https://pyblp.readthedocs.io/en/stable/_notebooks/tutorial/blp.html

* marriage market: https://github.com/TraME-Project/TraME-Datasets/

Part 4. Recent advances on Bundle choice
Content:
Bundle choice, assortment problem, one-to-many matching, gross substitutes, greedy algorithm 

Lectures:

• L13: Tue 4/23, 1145am-145pm (zoom)
• L14: Thu 4/25, 1pm-3pm (zoom) 
• L15: Thu 5/2, 1pm-3pm (zoom)

Application:

• bundling in multichannel television markets, https://www.jstor.org/stable/23245430, Dataset: https://warwick.ac.uk/fac/soc/economics/staff/academic/crawford/research/bundling_welfare/

New Perspectives on Substitutes

Paris, June 2-3, 2023

Organizers: Alfred Galichon (New York University and Sciences Po) and Larry Samuelson (Yale University)

Substitutability has found itself at the core of modern economic modeling for two reasons.  First, it arises naturally in classes of models such as one-to-one matching, discrete choice, hedonic models, and many others.  Second, it leads to successful numerical methods, such as the greedy algorithm in the discrete case, and the Jacobi algorithm related ones in the continuous case.  Leading experts in the field will gather to make a link between discrete and continuous models and to explore recent theoretical developments and economic applications.

Abstracts can be found here.

Papers can be found here.

Slides can be found here.

The workshop is accessible to the public but registration is mandatory by emailing ag133@nyu.edu.

Friday June 2, 2023

Location: Sciences Po (27 rue Saint Guillaume, room AS)

1230pm-130pm: luncheon buffet (room AS)

Session 1

130pm-215pm: Renato Paes Leme: “Gross Substitutes: Representation and Approximation”

215pm-300pm: Xin Chen / Menglong Li: “S-Convexity and Gross Substitutability”

300pm-330pm: break

Session 2

330pm-415pm: Alexander Teytelboym: TBA

415pm-500pm: Jonathan Weinstein: “Direct Complementarity”

5pm-530pm: break

Session 3

530pm-615pm: Lucas Vernet: “Monotone Comparative Statics for Equilibrium Problems”

615-700pm: Ravi Jagadeesan: “Understanding Demand Types and Discrete Convexity”

700pm-730pm: dinner (invitation only): Chez Francoise, Aérogare des Invalides, 75007 Paris
https://goo.gl/maps/9HEZ9nMkRmcJrWkA6

Saturday June 3, 2023

Location: NYU Paris (57 boulevard Saint-Germain, 75005 Paris)

Session 4

0915am-10am: Faruk Gul: “Efficient Allocation of Indivisible Goods in Pseudo Markets with Constraints”

1000am-1045am: Pawel Dziewulski: “It’s all about parallelograms: comparative statics in quasilinear settings”

1045am-1115am: break

Session 5

1115am-1200pm: Elizabeth Baldwin: “Implementing Walrasian Equilibrium: The Language of Product-Mix Auctions”

1200pm-1245pm: Maxime Sylvestre: “Monotone comparative statics for submodular functions, with an application to aggregated deferred acceptance”

1245pm-130pm: Wolfgang Pesendorfer: “Lindahl Equilibrium as a Collective Choice Rule”

130pm lunch (invitation only): Les Papilles, 30 rue Gay Lussac, 75005 Paris
https://goo.gl/maps/73TVUbfvkG4cieXp7

Lecture series

Two lectures on matching models for family economics

Invited lectures given at the University of Toronto, April 1 and April 8, 2022

Content

The first lecture will cover transferable utility (TU) matching models with logit heterogeneity following the seminal paper by Choo and Siow (JPE 2006), its extension and parametric estimation following Galichon and Salanié (Restud 2022).
The second lecture will discuss the interplay between models of matching and collective models as depicted in Browning and Chiappori’s (Cambridge, 2014) monograph, and how models of imperfectly transferable utility (ITU) and logit or more general heterogeneity are needed to address this new class of problems, as is done in Galichon, Kominers and Weber (JPE, 2019).

Schedule

April 1 and April 8, 10am-12pm Eastern time, online.

Course material

The lecture slides are available from the following github repository.

References

These lectures are based on:
[GS] Galichon and Salanie (2022). Cupids invisible hands: Social Surplus and Identification in Matching Models. Review of Economic Studies.
[GKW] Galichon, Kominers and Weber (2019). Costly Concessions: An Empirical Framework for Matching with Imperfectly Transferable Utility (2019). Journal of Political Economy.
[MEC-OPTIM] Galichon (2022). `math+econ+code’ masterclass on optimal transport and economic applications. https://www.math-econ-code.org/mec-optim
[MEC-EQUIL] Galichon (2022). `math+econ+code’ masterclass on equilibrium transport and matching models in economics. https://www.math-econ-code.org/mec-equil

Outline

Lecture 1: Matching models with transferable utility:
Matching models as an optimization problem / regularized optimal transport / generalized linear models
Lecture 2: Matching models with imperfectly transferable / nontransferable utility:
Matching models as an equilibrium problem with substitutes

Lecture series

Gross substitutes, optimal transport and matching models

Optimal Transport Summer School, the University of Washington, Seattle, June 19-July 1, 2022

Content

Gross substitutes is a fundamental property in mathematics, economics and computation, almost as important as convexity. It is at the heart of optimal transport theory — although this is often underrecognized — and understanding the connection key to understanding the extension of optimal transport to other models of matching.

Schedule

TBD

Course material

The lecture slides are available before each lecture from the following github repository.

References

These lectures will be loosely based on my math+econ+code lectures:
A. Galichon, ‘math+econ+code’ masterclass on equilibrium transport and matching models in economics. June 2021. Available here.

Outline

Lecture 1. Introduction to gross substitutes
M-matrices and M-maps, nonlinear Perron-Froebenius theory, convergence of Jacobi algorithm. A toy hedonic model.

Lecture 2. Models of matching with transfers
Problem formulation, regularized and unregularized case. IPFP and its convergence. Existence and uniqueness of an equilibrium. Lattice structure.
Lecture 3. Models of matching without transfers
Gale and Shapley’s stable matchings. Adachi’s formulation. Kelso-Craford. Hatfield-Milgrom.

Kidney transplant hackaton

a math+econ+code event

When? TBA

Who? participation to this math+econ+code event is open to the public upon request.

What? Kidney transplant problems provide a very interesting real-life examples of dynamic matching problems. A large academic literature exists on the topic, both in economics and in operations research. Because the problem is a difficult problem computationally speaking, a number of algorithms exist out there to try to approximate the best solution. This hackaton will put you in the shoes of the transplant agency: at each period, you will receive the state of your population, which is made of your existing population of patients, plus new patients and minus deceased ones (simulated by our platform). Your role will be write the algorithm that matches donors and receivers, given the constraints on the way transplants can be done (state of the patient, compatibility between donor and receiver, length of transplant chains, etc.)

Each player plays in parallel and is evaluated by a score, which will reflect the state of their population of patients. The game is played over a very large number of periods. The player with the highest final score has come up with the best algorithm and wins the hackaton.

What is required? In order to participate, you need to be familiar with Python programming. Your only input is the matching function that assigns donors to receivers — a function whose length can be less than a hundred lines but can be more depending on your design. All the interfacing is done by the platform.

Interested? contact us at math.econ.code@gmail.com.