Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions

Brendan Pass

doi:10.3934/dcds.2014.34.1623

Article Contents

2014, Volume 34, Issue 4: 1623-1639. Doi: 10.3934/dcds.2014.34.1623

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Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions

Brendan Pass^1,

1.
Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1

Received: September 30, 2012

Revised: January 31, 2013

Published: March 2014

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Abstract

We prove uniqueness and Monge solution results for multi-marginal optimal transportation problems with a certain class of surplus functions; this class arises naturally in multi-agent matching problems in economics. This result generalizes a seminal result of Gangbo and Święch [17]. Of particular interest, we show that this also yields a partial generalization of the Gangbo-Święch result to manifolds; alternatively, we can think of this as a partial extension of McCann's theorem for quadratic costs on manifolds to the multi-marginal setting [23].
We also show that the class of surplus functions considered here neither contains, nor is contained in, the class of surpluses studied in [27], another generalization of Gangbo and Święch's result.

Keywords:

Multi-marginal Monge Kantorovich problems,
optimal transport,
matching,
Monge solutions,
uniqueness,
equilibrium,
purity.

Mathematics Subject Classification: Primary: 49K20; Secondary: 91B68, 49Q15.

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