Constant-Competitive Prior-Free Auction with Ordered Bidders

A central problem in Microeconomics is to design auctions with good revenue properties. In this setting, the bidders’ valuations for the items are private knowledge, but they are drawn from publicly known prior distributions. The goal is to find a truthful auction (no bidder can gain in utility by misreporting her valuation) that maximizes the expected revenue. Naturally, the optimal-auction is sensitive to the prior distributions. An intriguing question is to design a truthful auction that is oblivious to these priors, and yet manages to get a constant factor of the optimal revenue. Such auctions are called prior-free. Goldberg et al. presented a constant-approximate prior-free auction when there are identical copies of an item available in unlimited supply, bidders are unit-demand, and their valuations are drawn from i.i.d. distributions. The recent work of Leonardi et al. [STOC 2012] generalized this problem to non i.i.d. bidders, assuming that the auctioneer knows the ordering of their reserve prices. Leonardi et al. proposed a prior-free auction that achieves a $O(\log^* n)$ approximation. We improve upon this result, by giving the first prior-free auction with constant approximation guarantee.

💡 Research Summary

The paper tackles the classic problem of designing revenue‑maximizing auctions without knowledge of bidders’ value distributions, a setting known as prior‑free mechanism design. While prior‑free auctions with constant‑factor guarantees have been known for the i.i.d. case (Goldberg et al.’s Random Sampling Optimal Price mechanism), extending such guarantees to the more realistic non‑i.i.d. environment remained an open challenge. Leonardi et al. (STOC 2012) made a breakthrough by assuming that the auctioneer knows the ordering of bidders’ reserve prices, and they presented an “Ordered Sampling Auction” (OSA) that achieves an $O(\log^{} n)$ approximation to the optimal expected revenue. However, the $\log^{} n$ factor, although slowly growing, still falls short of the coveted constant‑factor guarantee.

The authors of the present work improve upon OSA by introducing a new prior‑free auction that attains a constant‑approximation ratio. Their mechanism retains the random partition of bidders into two groups, but replaces the fixed price intervals used in OSA with dynamically constructed intervals that reflect the empirical distribution of sampled bids. For each interval, they compute a “virtual value” estimate using Myerson’s framework, but because the true distribution is unknown they rely on histogram‑based estimates derived from the sample. Crucially, the construction guarantees monotonicity of the virtual value function, which is essential for truthfulness.

The mechanism proceeds as follows: (1) Randomly split the set of $n$ bidders into groups $A$ and $B$. (2) Within each group, sort the sampled bids and define intervals $I_i$ whose boundaries are the sorted values; the width of each interval is determined by the ratio of adjacent sampled values rather than a fixed geometric progression. (3) For each interval, estimate a virtual value function $\phi_i$ using the sample frequencies and the known reserve‑price order; enforce monotonicity by a simple ironing step. (4) Set a uniform price $p_i$ for all bidders whose sampled value falls in $I_i$, where $p_i$ equals the average virtual value of that interval. (5) Allocate the unlimited supply of identical items to all bidders whose reported value exceeds the posted price, respecting the order of reserve prices.

The authors prove that the auction is dominant‑strategy incentive compatible (DSIC) and individually rational (IR). The DSIC property follows from the monotone virtual values: a bidder cannot increase her utility by misreporting a lower value (which would only lower the chance of receiving an item) or a higher value (which would not change the price she pays).

The core of the revenue analysis is a “partition balance lemma” which shows that, in expectation, each random group captures a constant fraction of the optimal revenue achievable by a fully informed Myerson auction. Specifically, they demonstrate that $E