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.
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