Pairwise Multi-marginal Optimal Transport via Universal Poisson Coupling
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We investigate the problem of pairwise multi-marginal optimal transport, that is, given a collection of probability distributions P_1,...,P_m on a Polish space, to find a coupling X_1,...,X_m, X_i∼ P_i, such that E[c(X_i,X_j)]< r_X∼ P_i, Y∼ P_jE[c(X,Y)] for any i,j, where c is a cost function and r>1. In other words, every pair (X_i,X_j) has an expected cost at most a factor of r from its lowest possible value. This can be regarded as a multi-agent matching problem with a fairness requirement. For the cost function c(x,y)= x-y_2^q over R^n, where q>0, we show that a finite r is attainable when either n=1 or q<1, and not attainable when n>2 and q>1. It is unknown whether such r exists when n>2 and q=1. Also, we show that r grows at least as fast as Ω(n^1{q=1}+q/2) when n→∞. The case of the discrete metric cost c(x,y)=1{x≠ y}, and more general metric and ultrametric costs are also investigated.
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