Pairwise Multi-marginal Optimal Transport via Universal Poisson Coupling

We investigate the problem of pairwise multi-marginal optimal transport, that is, given a collection of probability distributions $P_{1},\ldots,P_{m}$ on a Polish space, to find a coupling $X_{1},\ldots,X_{m}$, $X_{i}\sim P_{i}$, such that $\mathbf{E}[c(X_{i},X_{j})]\le r\inf_{X\sim P_{i},\,Y\sim P_{j}}\mathbf{E}[c(X,Y)]$ for any $i,j$, where $c$ is a cost function and $r\ge1$. 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 $c(x,y)=\Vert x-y\Vert_{2}^{q}$ over $\mathbb{R}^{n}$, where $q>0$, we show that a finite $r$ is attainable when either $n=1$ or $q<1$, and not attainable when $n\ge2$ and $q>1$. It is unknown whether such $r$ exists when $n\ge2$ and $q=1$. Also, we show that $r$ grows at least as fast as $\Omega(n^{\mathbf{1}\{q=1\}+q/2})$ when $n\to\infty$. The case of the discrete metric cost $c(x,y)=\mathbf{1}\{x\neq y\}$, and more general metric and ultrametric costs are also investigated.