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We introduce graphs associated to transport problems between discretemarginals, that allow to characterize the set of all optimizers given oneprimal optimizer. In particular, we establish that connectivity of those graphsis a necessary and sufficient condition for uniqueness of the dual optimizers.Moreover, we provide an algorithm that can efficiently compute the dualoptimizer that is the limit, as the regularization parameter goes to zero, ofthe dual entropic optimizers. Our results find an application in aStackelberg-Cournot-Nash game, for which we obtain existence andcharacterization of the equilibria.

Characterization of transport optimizers via graphs and applications to Stackelberg-Cournot-Nash equilibria