Approximation Schemes for Sequential Posted Pricing in Multi-Unit Auctions

TL;DR

This paper develops polynomial-time approximation schemes for revenue-maximizing sequential posted-pricing mechanisms in multi-unit auctions, showing near-optimal solutions and the diminishing value of adaptivity as the number of units increases.

Contribution

The paper introduces the first PTAS for both adaptive and non-adaptive SPMs in multi-unit auctions, with algorithms that perform near-optimally and analyze the adaptivity gap.

Findings

A non-adaptive SPM achieves near-optimal revenue compared to adaptive SPMs.

The approximation ratio improves as the number of units increases, approaching optimality.

The adaptivity gap diminishes with larger K, simplifying mechanism design.

Abstract

We design algorithms for computing approximately revenue-maximizing {\em sequential posted-pricing mechanisms (SPM)} in $K$-unit auctions, in a standard Bayesian model. A seller has $K$ copies of an item to sell, and there are $n$ buyers, each interested in only one copy, who have some value for the item. The seller must post a price for each buyer, the buyers arrive in a sequence enforced by the seller, and a buyer buys the item if its value exceeds the price posted to it. The seller does not know the values of the buyers, but have Bayesian information about them. An SPM specifies the ordering of buyers and the posted prices, and may be {\em adaptive} or {\em non-adaptive} in its behavior. The goal is to design SPM in polynomial time to maximize expected revenue. We compare against the expected revenue of optimal SPM, and provide a polynomial time approximation scheme (PTAS) for both non-adaptive and adaptive SPMs. This is achieved by two algorithms: an efficient algorithm that gives a $(1-\frac{1}{\sqrt{2\pi K}})$-approximation (and hence a PTAS for sufficiently large $K$), and another that is a PTAS for constant $K$. The first algorithm yields a non-adaptive SPM that yields its approximation guarantees against an optimal adaptive SPM -- this implies that the {\em adaptivity gap} in SPMs vanishes as $K$ becomes larger.