We study online pricing problems in markets with cancellations, i.e., markets in which prior allocation decisions can be revoked, but at a cost. In our model, a seller receives requests online and chooses which requests to accept, subject to constraints on the subsets of requests which may be accepted simultaneously. A request, once accepted, can be canceled at a cost which is a fixed fraction of the request value. This scenario models a market for web advertising campaigns, in which the buyback cost represents the cost of canceling an existing contract.
We analyze a simple constant-competitive algorithm for a single-item auction in this model, and we prove that its competitive ratio is optimal among deterministic algorithms, but that the competitive ratio can be improved using a randomized algorithm. We then model ad campaigns using knapsack domains, in which the requests differ in size as well as in value. We give a deterministic online algorithm that achieves a bi-criterion approximation in which both approximation factors approach 1 as the buyback factor and the size of the maximum request approach 0. We show that the bi-criterion approximation is unavoidable for deterministic algorithms, but that a randomized algorithm is capable of achieving a constant competitive ratio. Finally, we discuss an extension of our randomized algorithm to matroid domains (in which the sets of simultaneously satisfiable requests constitute the independent sets of a matroid) as well as present results for domains in which the buyback factor of different requests varies.