# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1812.05870