Asymptotically Optimal Welfare of Posted Pricing for Multiple Items with MHR Distributions

TL;DR

This paper demonstrates that posted pricing mechanisms can asymptotically achieve near-optimal social welfare for multiple items with unit-demand buyers drawn from MHR distributions, using dynamic or static prices.

Contribution

It introduces mechanisms with dynamic and static prices that attain near-optimal welfare guarantees, extending prior single-item results to multiple items with MHR distributions.

Findings

Dynamic prices achieve a $1 - rac{ ext{constant}}{ ext{log} n}$ welfare approximation.

Static prices achieve a $1 - rac{ ext{log log log n}}{ ext{log} n}$ welfare approximation.

Both guarantees are proven to be asymptotically optimal even for single-item cases.

Abstract

We consider the problem of posting prices for unit-demand buyers if all $n$ buyers have identically distributed valuations drawn from a distribution with monotone hazard rate. We show that even with multiple items asymptotically optimal welfare can be guaranteed. Our main results apply to the case that either a buyer's value for different items are independent or that they are perfectly correlated. We give mechanisms using dynamic prices that obtain a $1 - \Theta \left( \frac{1}{\log n}\right)$-fraction of the optimal social welfare in expectation. Furthermore, we devise mechanisms that only use static item prices and are $1 - \Theta \left( \frac{\log\log\log n}{\log n}\right)$-competitive compared to the optimal social welfare. As we show, both guarantees are asymptotically optimal, even for a single item and exponential distributions.