On the Complexity of Bundle-Pricing and Simple Mechanisms.

We show that the problem of finding an optimal bundle-pricing for a single additive buyer is #P-hard, even when the distributions have support size 2 for each item and the optimal solution is guaranteed to be a simple one: the seller picks a price for the grand bundle and a price for each individual item; the buyer can purchase either the grand bundle at the given price or any bundle of items at their total individual prices. We refer to this simple and natural family of pricing schemes as discounted item-pricings. In addition to the hardness result, we show that when the distributions are i.i.d. with support size 2, a discounted item-pricing can achieve the optimal revenue obtainable by lottery-pricings and it can be found in polynomial time.