Optimal Transport Methods in Economics

Galichon, Alfred

doi:10.1515/9781400883592

Cited by 149 publications

(201 citation statements)

References 5 publications

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“…We follow the optimal transportation literature (Villani (2009), Galichon (2016) and Gretsky, Ostroy, and Zame (1992) in using a measure λ on X × Y to describe who is matched with whom and who remains unmatched. Formally, a match for a matching model (X Y …”

Section: Matches and Outcomesmentioning

confidence: 99%

“…The fundamental duality of linear programming also plays a central role in the theory of matching models with quasilinear (transferable) utility, from the theory's inception in Shapley and Shubik (1972) to the more recent adoption of optimal transport methods (cf. Galichon (2016)) based on the Kantorovich duality for infinite-dimensional linear programs (Villani (2009)). …”

Section: Introductionmentioning

confidence: 99%

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The Implementation Duality

Nöldeke¹,

Samuelson²

2018

Econometrica

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Conjugate duality relationships are pervasive in matching and implementation problems and provide much of the structure essential for characterizing stable matches and implementable allocations in models with quasilinear (or transferable) utility. In the absence of quasilinearity, a more abstract duality relationship, known as a Galois connection, takes the role of (generalized) conjugate duality. While weaker, this duality relationship still induces substantial structure. We show that this structure can be used to extend existing results for, and gain new insights into, adverse-selection principalagent problems and two-sided matching problems without quasilinearity.

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Section: Matches and Outcomesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Implementation Duality

Nöldeke¹,

Samuelson²

2018

Econometrica

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“…Formulated as a linear programming problem as in (3.1), the optimal transport problem can be solved using standard linear programming toolboxes; see Section 3.4 of Galichon (2016) for details of how to perform computations efficiently using the sparse structure of the constraint matrix.…”

Section: Discrete Optimal Assignment Problemmentioning

confidence: 99%

“…A short tutorial on convex analysis from the point of view of optimal transport is provided in Section 6.1 of Galichon (2016), where the basic notions used in the following, such as Legendre transform and subdifferential, are recapitulated. A short tutorial on convex analysis from the point of view of optimal transport is provided in Section 6.1 of Galichon (2016), where the basic notions used in the following, such as Legendre transform and subdifferential, are recapitulated.…”

Section: Scalar Product Surplusmentioning

confidence: 99%

“…However, until now, the appearance of optimal transport in economics was only in connection with two-sided models of matching; see, e.g. These methods are exposed in a comprehensive way in my recent monograph, Optimal Transport Methods in Economics (Galichon, 2016), aimed at an audience of economists. Indeed, as shown by Chiappori et al (2010), equilibrium outcomes in two-sided matching problems with transferable utility coincide with the solutions of an optimal transport problem.…”

Section: Introductionmentioning

confidence: 99%

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A survey of some recent applications of optimal transport methods to econometrics

Galichon

2017

The Econometrics Journal

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Multi‐to One‐Dimensional Optimal Transport

Chiappori

McCann

Pass

2017

Comm Pure Appl Math

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We consider the Monge-Kantorovich problem of transporting a probability density on R m to another on the line, so as to optimize a given cost function. We introduce a nestedness criterion relating the cost to the densities, under which it becomes possible to solve this problem uniquely, by constructing an optimal map one level set at a time. This map is continuous if the target density has connected support. We use level-set dynamics to develop and quantify a local regularity theory for this map and the Kantorovich potentials solving the dual linear program. We identify obstructions to global regularity through examples.More specifically, fix probability densities f and g on open sets X ⊂ R m and Y ⊂ R n with m ≥ n ≥ 1. Consider transporting f onto g so as to minimize the cost −s(x, y). We give a non-degeneracy condition (a) on s ∈ C 1,1 which ensures the set of * The authors are grateful to Toronto's Fields' Institute for the Mathematical Sciences for its kind hospitality during part of this work. RJM acknowledges partial support of this research by Natural Sciences and Engineering 1x paired with [g-a.e.] y ∈ Y lie in a codimension n submanifold of X. Specializing to the case m > n = 1, we discover a nestedness criteria relating s to (f, g) which allows us to construct a unique optimal solution in the form of a map F : X −→ Y . When s ∈ C 2 ∩ W 3,1 and log f and log g are bounded, the Kantorovich dual potentials (u, v) satisfy v ∈ C 1,1 loc (Y ), and the normal velocity V of F −1 (y) with respect to changes in y is given by V (x) = v ′′ (F (x)) − s yy (x, F (x)). Positivity (b) of V locally implies a Lipschitz bound on F ; moreover, v ∈ C 2 if F −1 (y) intersects ∂X ∈ C 1 transversally (c). On subsets where (a)-(c) can be be quantified, for each integer r ≥ 1 the norms of u, v ∈ C r+1,1 and F ∈ C r,1 are controlled by these bounds, log f, log g, ∂X C r−1,1 , ∂X C 1,1 , s C r+1,1 , and the smallness of F −1 (y). We give examples showing regularity extends from X to part of X, but not from Y to Y . We also show that when s remains nested for all (f, g), the problem in R m × R reduces to a supermodular problem in R × R.

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Optimal Transport Methods in Economics

Cited by 149 publications

References 5 publications

The Implementation Duality

The Implementation Duality

A survey of some recent applications of optimal transport methods to econometrics

Multi‐to One‐Dimensional Optimal Transport

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