Asymptotically Efficient Multi-Unit Auctions via Posted Prices

TL;DR

This paper investigates the efficiency of static posted prices in multi-unit auctions with many agents, showing near-optimal welfare under certain distributional conditions and establishing conditions for maximal welfare in adversarial settings.

Contribution

It demonstrates asymptotic near-optimal welfare using posted prices for i.i.d. valuations and introduces a necessary and sufficient condition for maximal welfare in adversarial order arrivals.

Findings

Expected revenue approaches optimal welfare for upper mass distributions.

No asymptotic full efficiency for most distributions without the upper mass condition.

A 'tiefree' condition characterizes when maximal welfare is achievable in adversarial settings.

Abstract

We study the asymptotic average-case efficiency of static and anonymous posted prices for $n$ agents and $m(n)$ multiple identical items with $m(n)=o\left(\frac{n}{\log n}\right)$. When valuations are drawn i.i.d from some fixed continuous distribution (each valuation is a vector in $\Re_+^m$ and independence is assumed only across agents) we show: (a) for any "upper mass" distribution there exist posted prices such that the expected revenue and welfare of the auction approaches the optimal expected welfare as $n$ goes to infinity; specifically, the ratio between the expected revenue of our posted prices auction and the expected optimal social welfare is $1-O\left(\frac{m(n)\log n}{n}\right)$, and (b) there do not exist posted prices that asymptotically obtain full efficiency for most of the distributions that do not satisfy the upper mass condition. When valuations are complete-information and only the arrival order is adversarial, we provide a "tiefree" condition that is sufficient and necessary for the existence of posted prices that obtain the maximal welfare. This condition is generically satisfied, i.e., it is satisfied with probability $1$ if the valuations are i.i.d.~from some continuous distribution.