Descending Price Auctions with Bounded Number of Price Levels and Batched Prophet Inequality

TL;DR

This paper introduces a descending price auction model with a limited number of price levels, connecting it to a new variant of prophet inequality, and demonstrates near-optimal revenue performance with few price levels.

Contribution

It formulates a batched prophet inequality problem and derives revenue guarantees for descending auctions with bounded price levels, extending to multiple units.

Findings

For single-unit sales, 4 price levels achieve over 98% of optimal revenue.

The auction's competitive ratio improves with more price levels and units.

A closed-form bound relates auction performance to parameters m and k.

Abstract

We consider descending price auctions for selling $m$ units of a good to unit demand i.i.d. buyers where there is an exogenous bound of $k$ on the number of price levels the auction clock can take. The auctioneer's problem is to choose price levels $p_1 > p_2 > \cdots > p_{k}$ for the auction clock such that auction expected revenue is maximized. The prices levels are announced prior to the auction. We reduce this problem to a new variant of prophet inequality, which we call \emph{batched prophet inequality}, where a decision-maker chooses $k$ (decreasing) thresholds and then sequentially collects rewards (up to $m$) that are above the thresholds with ties broken uniformly at random. For the special case of $m=1$ (i.e., selling a single item), we show that the resulting descending auction with $k$ price levels achieves $1- 1/e^k$ of the unrestricted (without the bound of $k$) optimal revenue. That means a descending auction with just 4 price levels can achieve more than 98\% of the optimal revenue. We then extend our results for $m>1$ and provide a closed-form bound on the competitive ratio of our auction as a function of the number of units $m$ and the number of price levels $k$.