arXiv subjects

Alfred Galichon

Publications and source records attributed to Alfred Galichon.

At least 19 recordsLinked to original sources

arXiv2026

Optimal Transport in Economics

Optimal transport provides a common language for allocation, equilibrium, computation, and inference. Its primal problem assigns mass or agents, while its dual variables admit economic interpretations as utilities, prices, and scarcity rents. This review explains why this combination has proved unusually effective in economics. We first present the core results, including Kantorovich duality, integrality, cyclical monotonicity, the canonical distance and quadratic costs, and entropic regularization. We then trace the field's development from planning and operations research to modern analysis, statistics, and computation. The economic literature is organized around two roles for transport: as a model of matching, trade, hedonic equilibrium, and aggregate assignment; and as a tool for coupling distributions, measuring discrepancies, constructing multivariate ranks, solving inverse problems, and certifying economic conclusions. We conclude by examining extensions beyond transferable utility, one-to-one matching, static allocation, and unconstrained transport, together with open questions involving learning and identification across multiple markets.

econ.GN↗

arXiv2026

Aggregate Stable Matching with Money Burning

We propose an aggregate notion of non-transferable utility (NTU) stability for decentralized matching markets with fixed prices, where market clearing is achieved through one-sided money burning, which can be interpreted as waiting. Agents are grouped into observable types and are indifferent among individuals within type; equilibrium is defined at the type level and delivers equal indirect utility within each type. We introduce money burning into two types of NTU models: In a deterministic model, we relate our notion to classical Gale--Shapley stability and show how money burning decentralizes stable outcomes under aggregation. We then introduce separable random utility, obtaining an NTU counterpart to Choo and Siow (2006). We prove the existence and uniqueness of equilibrium and provide a stationary queueing interpretation. Finally, we develop a generalized deferred acceptance algorithm based on alternating constrained discrete-choice problems and prove its convergence to the unique equilibrium.

econ.TH↗

arXiv2026

An econometrician's guide to optimal transport

We propose an overview of optimal transport theory and its applications to econometric methodology. This review is specifically designed for practitioners, be they econometric theorists or applied econometricians. The review of applications of optimal transport to econometrics is organized around the particular aspects of the mathematical theory of optimal transport they rely on.

econ.EM↗

arXiv2025

Transferable Utility Matching Beyond Logit: Computation and Estimation with General Heterogeneity

We present a general framework for matching with transferable utility (TU) that accommodates arbitrary heterogeneity without relying on the logit structure. The optimal assignment problem is characterized by tractable linear programming formulation, allowing flexible error distributions and correlation patterns. We introduce an iterative algorithm that solves large-scale assignment problems with guaranteed convergence and an intuitive economic interpretation, and we show how the same structure supports a simulated moment-matching estimator of the systematic surplus. Experiments using simulated data demonstrate the algorithm's scalability and the estimator's consistency under correct specification, as well as systematic bias arising from logit misspecification.

econ.EM↗

arXiv2025

Repeated Matching Games: An Empirical Framework

We introduce a model of dynamic matching with transferable utility, extending the static model of Shapley and Shubik (1971). Forward-looking agents have individual states that evolve with current matches. Each period, a matching market with market-clearing prices takes place. We prove the existence of an equilibrium with time-varying distributions of agent types and show it is the solution to a social planner's problem. We also prove that a stationary equilibrium exists. We introduce econometric shocks to account for unobserved heterogeneity in match formation. We propose two algorithms to compute a stationary equilibrium. We adapt both algorithms for estimation. We estimate a model of accumulation of job-specific human capital using data on Swedish engineers.

econ.EM↗

arXiv2024

Substitutability, equilibrium transport, and matching models

This chapter explores the role of substitutability in economic models, particularly in the context of optimal transport and matching models. In equilibrium models with substitutability, market-clearing prices can often be recovered using coordinate update methods such as Jacobi's algorithm. We provide a detailed mathematical analysis of models with substitutability through the lens of Z- and M-functions, in particular regarding their role in ensuring the convergence of Jacobi's algorithm. The chapter proceeds by studying matching models using substitutability, first focusing on models with (imperfectly) transferable utility, and then on models with non-transferable utility. In both cases, the text reviews theoretical implications as well as computational approaches (Sinkhorn, Gale--Shapley), and highlights a practical economic application.

econ.TH↗

arXiv2024

Matching under Imperfectly Transferable Utility

In this paper, we examine matching models with imperfectly transferable utility (ITU). We provide motivating examples, discuss the theoretical foundations of ITU matching models and present methods for estimating them. We also explore connected topics and provide an overview of the related literature. This paper has been submitted as a draft chapter for the Handbook of the Economics of Matching, edited by Che, Chiappori and Salani\'e.

econ.GN↗

arXiv2024

The matching problem with linear transfers is equivalent to a hide-and-seek game

Matching problems with linearly transferable utility (LTU) generalize the well-studied transferable utility (TU) case by relaxing the assumption that utility is transferred one-for-one within matched pairs. We show that LTU matching problems can be reframed as nonzero-sum games between two players, thus generalizing a result from von Neumann. The underlying linear programming structure of TU matching problems, however, is lost when moving to LTU. These results draw a new bridge between non-TU matching problems and the theory of bimatrix games, with consequences notably regarding the computation of stable outcomes.

econ.TH↗

arXiv2023

On Sinkhorn's Algorithm and Choice Modeling

For a broad class of models widely used in practice for choice and ranking data based on Luce's choice axiom, including the Bradley--Terry--Luce and Plackett--Luce models, we show that the associated maximum likelihood estimation problems are equivalent to a classic matrix balancing problem with target row and column sums. This perspective opens doors between two seemingly unrelated research areas, and allows us to unify existing algorithms in the choice modeling literature as special instances or analogs of Sinkhorn's celebrated algorithm for matrix balancing. We draw inspirations from these connections and resolve some open problems on the study of Sinkhorn's algorithm. We establish the global linear convergence of Sinkhorn's algorithm for non-negative matrices whenever finite scaling matrices exist, and characterize its linear convergence rate in terms of the algebraic connectivity of a weighted bipartite graph. We further derive the sharp asymptotic rate of linear convergence, which generalizes a classic result of Knight (2008). To our knowledge, these are the first quantitative linear convergence results for Sinkhorn's algorithm for general non-negative matrices and positive marginals. Our results highlight the importance of connectivity and orthogonality structures in matrix balancing and Sinkhorn's algorithm, which could be of independent interest. More broadly, the connections we establish in this paper between matrix balancing and choice modeling could also help motivate further transmission of ideas and lead to interesting results in both disciplines.

math.OC↗

arXiv2023

Matching Function Equilibria with Partial Assignment: Existence, Uniqueness and Estimation

We argue that models coming from a variety of fields, such as matching models and discrete choice models among others, share a common structure that we call matching function equilibria with partial assignment. This structure includes an aggregate matching function and a system of nonlinear equations. We provide a proof of existence and uniqueness of an equilibrium and propose an efficient algorithm to compute it. For a subclass of matching models, we also develop a new parameter-free approach for constructing the counterfactual matching equilibrium. It has the advantage of not requiring parametric estimation when computing counterfactuals. We use our procedure to analyze the impact of the elimination of the Social Security Student Benefit Program in 1982 on the marriage market in the United States. We estimate several candidate models from our general class of matching functions and select the best fitting model using information based criterion.

econ.GN↗

arXiv2023

Existence of a Competitive Equilibrium with Substitutes, with Applications to Matching and Discrete Choice Models

We propose new results for the existence and uniqueness of a general nonparametric and nonseparable competitive equilibrium with substitutes. These results ensure the invertibility of a general competitive system. The existing literature has focused on the uniqueness of a competitive equilibrium assuming that existence holds. We introduce three properties that our supply system must satisfy: weak substitutes, pivotal substitutes, and responsiveness. These properties are sufficient to ensure the existence of an equilibrium, thus providing the existence counterpart to Berry, Gandhi, and Haile (2013)'s uniqueness results. For two important classes of models, bipartite matching models with full assignment and discrete choice models, we show that both models can be reformulated as a competitive system such that our existence and uniqueness results can be readily applied. We also provide an algorithm to compute the unique competitive equilibrium. Furthermore, we argue that our results are particularly useful for studying imperfectly transferable utility matching models with full assignment and non-additive random utility models.

econ.GN↗

arXiv2023

Monotone comparative statics for submodular functions, with an application to aggregated deferred acceptance

We propose monotone comparative statics results for maximizers of submodular functions, as opposed to maximizers of supermodular functions as in the classical theory put forth by Veinott, Topkis, Milgrom, and Shannon among others. We introduce matrons, a natural structure that is dual to sublattices that generalizes existing structures such as matroids and polymatroids in combinatorial optimization and M-sets in discrete convex analysis. Our monotone comparative statics result is based on a natural order on matrons, which is dual in some sense to Veinott's strong set order on sublattices. As an application, we propose a deferred acceptance algorithm that operates in the case of divisible goods, and we study its convergence properties.

econ.TH↗

arXiv2023

Stable and extremely unequal

We highlight the tension between stability and equality in non transferable utility matching. We consider many to one matchings and refer to the two sides of the market as students and schools. The latter have aligned preferences, which in this context means that a school's utility is the sum of its students' utilities. We show that the unique stable allocation displays extreme inequality between matched pairs.

econ.TH↗

arXiv2023

Cupid's Invisible Hand: Social Surplus and Identification in Matching Models

We investigate a model of one-to-one matching with transferable utility and general unobserved heterogeneity. Under a separability assumption that generalizes Choo and Siow (2006), we first show that the equilibrium matching maximizes a social gain function that trades off exploiting complementarities in observable characteristics and matching on unobserved characteristics. We use this result to derive simple closed-form formulae that identify the joint matching surplus and the equilibrium utilities of all participants, given any known distribution of unobserved heterogeneity. We provide efficient algorithms to compute the stable matching and to estimate parametric versions of the model. Finally, we revisit Choo and Siow's empirical application to illustrate the potential of our more general approach.

econ.GN↗

arXiv2023

A Note on the Estimation of Job Amenities and Labor Productivity

This paper introduces a maximum likelihood estimator of the value of job amenities and labor productivity in a single matching market based on the observation of equilibrium matches and wages. The estimation procedure simultaneously fits both the matching patterns and the wage curve. While our estimator is suited for a wide range of assignment problems, we provide an application to the estimation of the Value of a Statistical Life using compensating wage differentials for the risk of fatal injury on the job. Using US data for 2017, we estimate the Value of Statistical Life at \$ 6.3 million (\$2017).

econ.EM↗

arXiv2023

On the representation of the nested logit model

We give a two-line proof of a long-standing conjecture of Ben-Akiva and Lerman (1985) regarding the random utility representation of the nested logit model, thus providing a renewed and straightforward textbook treatment of that model. As an application, we provide a closed-form formula for the correlation between two Fréchet random variables coupled by a Gumbel copula.

math.ST↗

arXiv2022

The Existence of Equilibrium Flows

Galichon, Samuelson and Vernet (2022) introduced a class of problems, equilibrium flow problems, that nests several classical economic models such as bipartite matching models, minimum-cost flow problems and hedonic pricing models. We establish conditions for the existence of equilibrium prices in the equilibrium flow problem, in the process generalizing Hall's theorem.

econ.TH↗

arXiv2022

Monotone Comparative Statics for Equilibrium Problems

We introduce a notion of substitutability for correspondences and establish a monotone comparative static result, unifying results such as the inverse isotonicity of M-matrices, Berry, Gandhi and Haile's identification of demand systems, monotone comparative statics, and results on the structure of the core of matching games without transfers (Gale and Shapley) and with transfers (Demange and Gale). More specifically, we introduce the notions of 'unified gross substitutes' and 'nonreversingness' and show that if Q is a supply correspondence defined on a set of prices P which is a sublattice of R^N, and Q satisfies these two properties, then the set of prices yielding supply vector q is increasing (in the strong set order) in q; and it is a sublattice of P.

econ.TH↗