On the Computational Complexity of Optimal Simple Mechanisms

TL;DR

This paper investigates the complexity of designing simple, revenue-maximizing partition mechanisms for a monopolist seller with additive, independent item valuations, providing a PTAS and proving NP-hardness.

Contribution

It introduces a polynomial-time approximation scheme for optimal partition mechanisms and establishes their computational hardness, along with structural insights into near-optimal solutions.

Findings

Existence of a PTAS for revenue-maximizing partition mechanisms.

Proven strong NP-hardness of the problem.

Structural property: near-optimal mechanisms use few non-trivial bundles.

Abstract

We consider a monopolist seller facing a single buyer with additive valuations over n heterogeneous, independent items. It is known that in this important setting optimal mechanisms may require randomization [HR12], use menus of infinite size [DDT15], and may be computationally intractable [DDT14]. This has sparked recent interest in finding simple mechanisms that obtain reasonable approximations to the optimal revenue [HN12, LY13, BILW14]. In this work we attempt to find the optimal simple mechanism. There are many ways to define simple mechanisms. Here we restrict our search to partition mechanisms, where the seller partitions the items into disjoint bundles and posts a price for each bundle; the buyer is allowed to buy any number of bundles. We give a PTAS for the problem of finding a revenue-maximizing partition mechanism, and prove that the problem is strongly NP-hard. En route, we prove structural properties of near-optimal partition mechanisms which may be of independent interest: for example, there always exists a near-optimal partition mechanism that uses only a constant number of non-trivial bundles (i.e. bundles with more than one item).