We initiate the study of online pricing problems in markets with “buyback,” 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, 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. Moreover, we prove that an extension of this algorithm achieves the same competitive ratio in any matroid domain, i.e., when the sets of requests which may be simultaneously satisfied constitute the independent sets of a matroid. This broad class of domains includes, for example, advertising markets in which each request is for a unit of supply coming from a specified subset of the available impressions. We also present algorithms and lower bounds for knapsack domains, i.e., when advertisers request varying quantities of a homogeneous but limited supply of impressions.