Let $X$ be some appropriate space (metric measure, Polish, whatever...) and $X\times X$ the product space with $\pi^1$ and $\pi^2$ as projections onto the first and second factor, respectively. Let $\mu$ and $\nu$ be two probability measures (again some appropriate space).
The Kantorovich formulation of optimal transport of measures $\mu$ and $\nu$ seeks a minimizer of \begin{equation} \Pi\mapsto\int c(x,y)\;d\Pi(x,y) \end{equation} such that \begin{equation} \begin{array}{c} \pi^1_*\Pi&=\mu\\ \pi^2_*\Pi&=\nu \end{array}\qquad\qquad \qquad (\#) \end{equation} I thought about it for a bit and in my context a different constraint makes more sense, namely: \begin{equation} \pi^1_*\Pi-\pi^2_*\Pi=\mu-\nu\;.\qquad\qquad \qquad (\dagger) \end{equation} I remember having seen this constraint somewhere, too.
Question: Since ($\dagger$) is more general I wonder what the advantages are. Which properties does $(\mu,\nu)\mapsto \min_{\Pi\in(\dagger)}\int c(x,y)\;d\Pi(x,y)$ have that $(\mu,\nu)\mapsto \min_{\Pi\in(\#)}\int c(x,y)\;d\Pi(x,y)$ doesn't have. I also can't find the reference anymore. Any help would be appreciated.