Asymptotically Efficient Multi-Unit Auctions via Posted Prices.

We study the asymptotic average-case efficiency of static and anonymous posted prices for $n$ agents and $m(n)$ multiple identical items with $m(n)=oleft(frac{n}{log n}right)$. 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 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 $1-Oleft(frac{m(n)log n}{n}right)$, 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.