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A glimpse into the differential topology and geometry of optimal transport
April  2014, 34(4): 1623-1639. doi: 10.3934/dcds.2014.34.1623

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

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

Received  October 2012 Revised  February 2013 Published  October 2013

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.
Citation: Brendan Pass. Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1623-1639. doi: 10.3934/dcds.2014.34.1623
##### References:
 [1] M. Agueh and G. Carlier, Barycenters in the Wasserstein space,, SIAM J. Math. Anal., 43 (2011), 904. doi: 10.1137/100805741. Google Scholar [2] M. Beiglbock, P. Henry-Labordere and F. Penkner, Model independent bounds for option prices: A mass transport approach,, Finance Stoch., 17 (2013), 477. doi: 10.1007%2Fs00780-013-0205-8. Google Scholar [3] M. Bernot, J. Delon, G. Peyre and J. Rabin, Wasserstein barycenter and its application to texture mixing,, in, (6667), 435. Google Scholar [4] Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs,, (French) [Polar decomposition and increasing rearrangement of vector fields] C. R. Acad. Sci. Pair. Ser. I Math., 305 (1987), 805. Google Scholar [5] Y. Brenier, Extended Monge-Kantorovich theory,, in, (2001), 91. doi: 10.1007/978-3-540-44857-0_4. Google Scholar [6] G. Buttazzo, L. De Pascale and P. Gori-Giorgi, Optimal-transport formulation of electronic density-functional theory,, Phys. Rev. A, 85 (2012). doi: 10.1103/PhysRevA.85.062502. Google Scholar [7] L. Caffarelli, Allocation maps with general cost functions,, in, (1996), 29. Google Scholar [8] G. Carlier, On a class of multidimensional optimal transportation problems,, J. Convex Anal., 10 (2003), 517. Google Scholar [9] G. Carlier and I. Ekeland, Matching for teams,, Econom. Theory, 42 (2010), 397. doi: 10.1007/s00199-008-0415-z. Google Scholar [10] G. Carlier and B.Nazaret, Optimal transportation for the determinant,, ESAIM Control, 14 (2008), 678. doi: 10.1051/cocv:2008006. Google Scholar [11] P-A. Chiapporri, R. McCann and L. Nesheim, Hedonic price equilibria, stable matching and optimal transport: Equivalence, topology and uniqueness,, Econom. Theory, 42 (2010), 317. doi: 10.1007/s00199-009-0455-z. Google Scholar [12] C. Cotar, G. Friesecke and C. Klüppelberg, Density functional theory and optimal transportation with Coulomb cost,, Comm. Pure Appl. Math., 66 (2013), 548. doi: 10.1002/cpa.21437. Google Scholar [13] I. Ekeland, An optimal matching problem,, ESAIM Control, 11 (2005), 57. doi: 10.1051/cocv:2004034. Google Scholar [14] A. Galichon and N. Ghoussoub, Variational representations for N-cyclically monotone vector fields,, preprint, (). Google Scholar [15] A. Galichon and P. Henry-Labordere and N. Touzi, A stochastic control approach to non-arbitrage bounds given marginals, with an application to Lookback options,, Preprint available at , (). Google Scholar [16] W. Gangbo and R. McCann, The geometry of optimal transportation,, Acta Math., 177 (1996), 113. doi: 10.1007/BF02392620. Google Scholar [17] W. Gangbo and A. Święch, Optimal maps for the multidimensional Monge-Kantorovich problem,, Comm. Pure Appl. Math., 51 (1998), 23. doi: 10.1002/(SICI)1097-0312(199801)51:1<23::AID-CPA2>3.0.CO;2-H. Google Scholar [18] N. Ghoussoub and A. Moameni, Symmetric Monge Kantorovich problems and polar decomposition of vector fields,, preprint, (). Google Scholar [19] H. Heinich, Problème de Monge pour $n$ probabilités,, (French) [Monge problem for $n$ probabilities] C. R. Math. Acad. Sci. Paris, 334 (2002), 793. doi: 10.1016/S1631-073X(02)02341-5. Google Scholar [20] M. Knott and C. Smith, On a generalization of cyclic monotonicity and distances among random vectors,, Linear Algebra Appl., 199 (1994), 363. doi: 10.1016/0024-3795(94)90359-X. Google Scholar [21] V. Levin, Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem,, Set-Valued Analysis, 7 (1999), 7. doi: 10.1023/A:1008753021652. Google Scholar [22] X-N. Ma, N. Trudinger and Wang, X-J., Regularity of potential functions of the optimal transportation problem,, Arch. Rational Mech. Anal., 177 (2005), 151. doi: 10.1007/s00205-005-0362-9. Google Scholar [23] R. McCann, Polar factorization of maps on Riemannian manifolds,, Geom. Funct. Anal., 11 (2001), 589. doi: 10.1007/PL00001679. Google Scholar [24] I. Olkin and S. T. Rachev, Maximum submatrix traces for positive definite matrices,, SIAM J. Matrix Ana. Appl., 14 (1993), 390. doi: 10.1137/0614027. Google Scholar [25] B. Pass, Regularity properties of optimal transportation problems arising in hedonic pricing models,, ESAIM Control, 19 (2013), 668. doi: 10.1051/cocv/2012027. Google Scholar [26] B. Pass, Remarks on the semi-classical Hohenberg-Kohn functional,, Nonlinearity, 26 (2013), 2731. doi: 10.1088/0951-7715/26/9/2731. Google Scholar [27] B. Pass, Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem,, SIAM Journal on Mathematical Analysis, 43 (2011), 2758. doi: 10.1137/100804917. Google Scholar [28] B. Pass, On the local structure of optimal measures in the multi-marginal optimal transportation problem,, Calculus of Variations and Partial Differential Equations, 43 (2012), 529. doi: 10.1007/s00526-011-0421-z. Google Scholar [29] B. Pass, Optimal transportation with infinitely many marginals,, Journal of Functional Analysis, 264 (2013), 947. doi: 10.1016/j.jfa.2012.12.002. Google Scholar [30] L. Rüschendorf and L. Uckelmann, On optimal multivariate couplings,, Distributions with given marginals and moment problems (Prague, (1996), 261. Google Scholar [31] L. Rüschendorf and L. Uckelmann, On the $n$-coupling problem,, J. Multivariate Anal., 81 (2002), 242. doi: 10.1006/jmva.2001.2005. Google Scholar [32] C. Villani, "Optimal Transport: Old and New,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar

show all references

##### References:
 [1] M. Agueh and G. Carlier, Barycenters in the Wasserstein space,, SIAM J. Math. Anal., 43 (2011), 904. doi: 10.1137/100805741. Google Scholar [2] M. Beiglbock, P. Henry-Labordere and F. Penkner, Model independent bounds for option prices: A mass transport approach,, Finance Stoch., 17 (2013), 477. doi: 10.1007%2Fs00780-013-0205-8. Google Scholar [3] M. Bernot, J. Delon, G. Peyre and J. Rabin, Wasserstein barycenter and its application to texture mixing,, in, (6667), 435. Google Scholar [4] Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs,, (French) [Polar decomposition and increasing rearrangement of vector fields] C. R. Acad. Sci. Pair. Ser. I Math., 305 (1987), 805. Google Scholar [5] Y. Brenier, Extended Monge-Kantorovich theory,, in, (2001), 91. doi: 10.1007/978-3-540-44857-0_4. Google Scholar [6] G. Buttazzo, L. De Pascale and P. Gori-Giorgi, Optimal-transport formulation of electronic density-functional theory,, Phys. Rev. A, 85 (2012). doi: 10.1103/PhysRevA.85.062502. Google Scholar [7] L. Caffarelli, Allocation maps with general cost functions,, in, (1996), 29. Google Scholar [8] G. Carlier, On a class of multidimensional optimal transportation problems,, J. Convex Anal., 10 (2003), 517. Google Scholar [9] G. Carlier and I. Ekeland, Matching for teams,, Econom. Theory, 42 (2010), 397. doi: 10.1007/s00199-008-0415-z. Google Scholar [10] G. Carlier and B.Nazaret, Optimal transportation for the determinant,, ESAIM Control, 14 (2008), 678. doi: 10.1051/cocv:2008006. Google Scholar [11] P-A. Chiapporri, R. McCann and L. Nesheim, Hedonic price equilibria, stable matching and optimal transport: Equivalence, topology and uniqueness,, Econom. Theory, 42 (2010), 317. doi: 10.1007/s00199-009-0455-z. Google Scholar [12] C. Cotar, G. Friesecke and C. Klüppelberg, Density functional theory and optimal transportation with Coulomb cost,, Comm. Pure Appl. Math., 66 (2013), 548. doi: 10.1002/cpa.21437. Google Scholar [13] I. Ekeland, An optimal matching problem,, ESAIM Control, 11 (2005), 57. doi: 10.1051/cocv:2004034. Google Scholar [14] A. Galichon and N. Ghoussoub, Variational representations for N-cyclically monotone vector fields,, preprint, (). Google Scholar [15] A. Galichon and P. Henry-Labordere and N. Touzi, A stochastic control approach to non-arbitrage bounds given marginals, with an application to Lookback options,, Preprint available at , (). Google Scholar [16] W. Gangbo and R. McCann, The geometry of optimal transportation,, Acta Math., 177 (1996), 113. doi: 10.1007/BF02392620. Google Scholar [17] W. Gangbo and A. Święch, Optimal maps for the multidimensional Monge-Kantorovich problem,, Comm. Pure Appl. Math., 51 (1998), 23. doi: 10.1002/(SICI)1097-0312(199801)51:1<23::AID-CPA2>3.0.CO;2-H. Google Scholar [18] N. Ghoussoub and A. Moameni, Symmetric Monge Kantorovich problems and polar decomposition of vector fields,, preprint, (). Google Scholar [19] H. Heinich, Problème de Monge pour $n$ probabilités,, (French) [Monge problem for $n$ probabilities] C. R. Math. Acad. Sci. Paris, 334 (2002), 793. doi: 10.1016/S1631-073X(02)02341-5. Google Scholar [20] M. Knott and C. Smith, On a generalization of cyclic monotonicity and distances among random vectors,, Linear Algebra Appl., 199 (1994), 363. doi: 10.1016/0024-3795(94)90359-X. Google Scholar [21] V. Levin, Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem,, Set-Valued Analysis, 7 (1999), 7. doi: 10.1023/A:1008753021652. Google Scholar [22] X-N. Ma, N. Trudinger and Wang, X-J., Regularity of potential functions of the optimal transportation problem,, Arch. Rational Mech. Anal., 177 (2005), 151. doi: 10.1007/s00205-005-0362-9. Google Scholar [23] R. McCann, Polar factorization of maps on Riemannian manifolds,, Geom. Funct. Anal., 11 (2001), 589. doi: 10.1007/PL00001679. Google Scholar [24] I. Olkin and S. T. Rachev, Maximum submatrix traces for positive definite matrices,, SIAM J. Matrix Ana. Appl., 14 (1993), 390. doi: 10.1137/0614027. Google Scholar [25] B. Pass, Regularity properties of optimal transportation problems arising in hedonic pricing models,, ESAIM Control, 19 (2013), 668. doi: 10.1051/cocv/2012027. Google Scholar [26] B. Pass, Remarks on the semi-classical Hohenberg-Kohn functional,, Nonlinearity, 26 (2013), 2731. doi: 10.1088/0951-7715/26/9/2731. Google Scholar [27] B. Pass, Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem,, SIAM Journal on Mathematical Analysis, 43 (2011), 2758. doi: 10.1137/100804917. Google Scholar [28] B. Pass, On the local structure of optimal measures in the multi-marginal optimal transportation problem,, Calculus of Variations and Partial Differential Equations, 43 (2012), 529. doi: 10.1007/s00526-011-0421-z. Google Scholar [29] B. Pass, Optimal transportation with infinitely many marginals,, Journal of Functional Analysis, 264 (2013), 947. doi: 10.1016/j.jfa.2012.12.002. Google Scholar [30] L. Rüschendorf and L. Uckelmann, On optimal multivariate couplings,, Distributions with given marginals and moment problems (Prague, (1996), 261. Google Scholar [31] L. Rüschendorf and L. Uckelmann, On the $n$-coupling problem,, J. Multivariate Anal., 81 (2002), 242. doi: 10.1006/jmva.2001.2005. Google Scholar [32] C. Villani, "Optimal Transport: Old and New,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar
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