On the Complexity of Simple and Optimal Deterministic Mechanisms for an Additive Buyer

TL;DR

This paper proves that finding the optimal deterministic mechanism for a single additive buyer is #P-hard in general, but becomes tractable under i.i.d. support size 2 conditions, with simple pricing strategies sufficing.

Contribution

It establishes the computational hardness of optimal deterministic mechanisms and identifies conditions where simple, polynomial-time solutions are possible.

Findings

Optimal deterministic mechanism design is #P-hard in general.

Simple pricing strategies suffice under i.i.d. support size 2 conditions.

Optimal revenue can be computed in polynomial time for i.i.d. distributions and fixed number of items.

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 distributions are i.i.d. with support size 2, the optimal revenue 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 polynomial time too when the number of items is constant.