Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 1623 - 1639, Volume 34, Issue 4, April 2014      doi:10.3934/dcds.2014.34.1623

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Brendan Pass - Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1, Canada (email)

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.

Received: October 2012;      Revised: February 2013;      Available Online: October 2013.