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Transition to nestedness in multi- to one-dimensional optimal transport

Part of: Mathematical economics

Published online by Cambridge University Press:  01 October 2018

PIERRE-ANDRÉ CHIAPPORI ,
ROBERT MCCANN  and
BRENDAN PASS
PIERRE-ANDRÉ CHIAPPORI
Affiliation:
Department of Economics, Columbia University, New York, NY, USA email: [email protected]
ROBERT MCCANN
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada email: [email protected]
BRENDAN PASS*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada email: [email protected]
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Abstract

We study a one-parameter class of examples of optimal transport problems between a two-dimensional source and a one-dimensional target. Our earlier work identified a nestedness condition on the surplus function and marginals, under which it is possible to solve the problem semi-explicitly. In the family of examples we consider, we classify the values of parameters which lead to nestedness. In those cases, we derive an almost explicit characterisation of the solution.

Keywords

Monge-Kantorovich problemoptimal transportunequal dimensionsmatching

MSC classification

Secondary: 91B68: Matching models
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 6: Applied Optimal Transport , December 2019 , pp. 1220 - 1228
Copyright
© Cambridge University Press 2018 

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Footnotes

†

The authors are grateful to Toronto’s Fields’ Institute for the Mathematical Sciences for its kind hospitality during part of this work, and to Boyang Wu for independently checking the simulations. They acknowledge partial support of RJM’s research by Natural Sciences and Engineering Research Council of Canada Grants No. 217006-08 and No. 217006-15. Chiappori gratefully acknowledges financial support from the NSF (Award No. 1124277). Pass is pleased to acknowledge support from Natural Sciences and Engineering Research Council of Canada Grant No. 412779-2012 and a University of Alberta start-up grant.

References

Chiappori, P.-A., McCann, R. J. & Pass, B. (submitted, 2016) Multidimensional matching. Preprint at arXiv:1604.05771.Google Scholar
Chiappori, P.-A., McCann, R. J. & Pass, B. (2017) Multi- to one-dimensional optimal transport. Commun. Pure Appl. Math. 70(12), 2405–2444.CrossRefGoogle Scholar
Gangbo, W. & McCann, R. J. (2000) Shape recognition via Wasserstein distance. Quart. Appl. Math. 58(4), 705–737.CrossRefGoogle Scholar
McCann, R. J. & Pass, B. (submitted, 2018) Optimal transportation between unequal dimensions. Preprint at https://arxiv.org/abs/1805.11187.Google Scholar
Pass, B. (2012) Regularity of optimal transportation between spaces with different dimensions. Math. Res. Lett. 19(2), 291–307.CrossRefGoogle Scholar
Santambrogio, F. (2015) Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling. Progress in Nonlinear Differential Equations and their Applications, Vol. 87, Birkhäuser/Springer, Cham.CrossRefGoogle Scholar
Villani, C. (2003) Topics in Optimal Transportation. Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence.Google Scholar
Villani, C. (2009) Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften, Vol. 338, Springer, New York.CrossRefGoogle Scholar