On the complexity of simple and optimal deterministic mechanisms for an additive buyer

Abstract

We show that the Revenue-Optimal Deterministic Mechanism Design problem for a single additive buyer is #P-hard, even when the distributions have support size 2 for each item and, more importantly, even when the optimal solution is guaranteed to be of a very simple kind: the seller picks a price for each individual item and a price for the grand bundle of all the items; the buyer can purchase either the grand bundle at its given price or any subset of items at their total individual prices. The following problems are also #P-hard, as immediate corollaries of the proof: 1. determining if individual item pricing is optimal for a given instance, 2. determining if grand bundle pricing is optimal, and 3. computing the optimal (deterministic) revenue. On the positive side, we show that when the distri-butions are i.i.d. with support size 2, the optimal rev-enue obtainable by any mechanism, even a randomized one, can be achieved by a simple solution of the above kind (individual item pricing with a discounted price for the grand bundle) and furthermore, it can be computed in polynomial time. The problem can be solved in poly-nomial time too when the number of items is constant.