A Note on Selling Optimally Two Uniformly Distributed Goods

We provide a new, much simplified and straightforward proof to a result of Pavlov [2011] regarding the revenue maximizing mechanism for selling two items with uniformly i.i.d. valuations over intervals [c, c + 1], to an additive bidder. This is done by explicitly defining optimal dual solutions to a relaxed version of the problem, where the convexity requirement for the bidder’s utility has been dropped. Their optimality comes directly from their structure, through the use of exact complementarity. For c = 0 and c ≥ 0.092 it turns out that the corresponding optimal primal solution is a feasible auction, thus the initial relaxation comes without a loss, and revenue maximality follows. However, for 0 < c < 0.092 that’s not the case, providing the first clear example where relaxing convexity provably does not come for free, even in a two-item regularly i.i.d. setting.