We consider a revenue-maximizing seller with a single item facing a single buyer with a private budget. The (value, budget) pair is drawn from an arbitrary and possibly correlated distribution. We characterize the optimal mechanism in such cases, and quantify the amount of price discrimination that might be present. For example, there could be up to 3·2k-1 -1 distinct non-trivial menu options in the optimal mechanism for such a buyer with k distinct possible budgets (compared to k if the marginal distribution of values conditioned on each budget has decreasing marginal revenue [CG00], or 2 if there is an arbitrary distribution and one possible budget [CMM11]).
Our approach makes use of the duality framework of [CDW16], and duality techniques related to the “FedEx Problem” of [FGKK16]. In contrast to [FGKK16] and other prior work, we characterize the optimal primal/dual without nailing down an explicit closed form.