Auctions with Online Supply

TL;DR

This paper compares two models of selling identical goods to unit-demand bidders with unknown supply, showing that truthful mechanisms perform poorly in adversarial settings but can achieve constant welfare approximation under stochastic supply with known distribution.

Contribution

It introduces a separation between adversarial and stochastic supply models, demonstrating the effectiveness of truthful mechanisms in the stochastic case under certain conditions.

Findings

Truthful mechanisms cannot guarantee better than Omega(loglog n) approximation in adversarial supply.

Under a monotone hazard-rate condition, a truthful mechanism achieves a constant approximation in stochastic supply.

Almost optimal lower bounds are established for online-envy-free mechanisms.

Abstract

We study the problem of selling identical goods to n unit-demand bidders in a setting in which the total supply of goods is unknown to the mechanism. Items arrive dynamically, and the seller must make the allocation and payment decisions online with the goal of maximizing social welfare. We consider two models of unknown supply: the adversarial supply model, in which the mechanism must produce a welfare guarantee for any arbitrary supply, and the stochastic supply model, in which supply is drawn from a distribution known to the mechanism, and the mechanism need only provide a welfare guarantee in expectation. Our main result is a separation between these two models. We show that all truthful mechanisms, even randomized, achieve a diminishing fraction of the optimal social welfare (namely, no better than a Omega(loglog n) approximation) in the adversarial setting. In sharp contrast, in the stochastic model, under a standard monotone hazard-rate condition, we present a truthful mechanism that achieves a constant approximation. We show that the monotone hazard rate condition is necessary, and also characterize a natural subclass of truthful mechanisms in our setting, the set of online-envy-free mechanisms. All of the mechanisms we present fall into this class, and we prove almost optimal lower bounds for such mechanisms. Since auctions with unknown supply are regularly run in many online-advertising settings, our main results emphasize the importance of considering distributional information in the design of auctions in such environments.