Prior-free Dynamic Mechanism Design With Limited Liability

TL;DR

This paper develops auction mechanisms for repeated sales with limited bidder rationality and adversarial valuations, achieving a constant fraction of the ideal benchmark and introducing a new optimal mechanism for a single bidder scenario.

Contribution

It introduces a novel mechanism for repeated auctions with adversarial valuations and limited bidder rationality, achieving tight bounds on revenue relative to the benchmark.

Findings

Mechanism attains a $1/e$ fraction of the 'sell the business' benchmark.

No mechanism can exceed $2/e$ of the benchmark under the constraints.

Optimal single-bidder mechanism achieves revenue $V/e$, which is proven tight.

Abstract

We study the problem of repeatedly auctioning off an item to one of $k$ bidders where: a) bidders have a per-round individual rationality constraint, b) bidders may leave the mechanism at any point, and c) the bidders' valuations are adversarially chosen (the prior-free setting). Without these constraints, the auctioneer can run a second-price auction to "sell the business" and receive the second highest total value for the entire stream of items. We show that under these constraints, the auctioneer can attain a constant fraction of the "sell the business" benchmark, but no more than $2/e$ of this benchmark. In the course of doing so, we design mechanisms for a single bidder problem of independent interest: how should you repeatedly sell an item to a (per-round IR) buyer with adversarial valuations if you know their total value over all rounds is $V$ but not how their value changes over time? We demonstrate a mechanism that achieves revenue $V/e$ and show that this is tight.