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This paper provides threshold policies with tight guarantees for onlineselection with convex cost (OSCC). In OSCC, a seller wants to sell some assetto a sequence of buyers with the goal of maximizing her profit. The seller canproduce additional units of the asset, but at non-decreasing marginal costs. Ateach time, a buyer arrives and offers a price. The seller must make animmediate and irrevocable decision in terms of whether to accept the offer andproduce/sell one unit of the asset to this buyer. The goal is to develop anonline algorithm that selects a subset of buyers to maximize the seller'sprofit, namely, the total selling revenue minus the total production cost. Ourmain result is the development of a class of simple threshold policies that arelogistically simple and easy to implement, but have provable optimalityguarantees among all deterministic algorithms. We also derive a lower bound oncompetitive ratios of randomized algorithms and prove that the competitiveratio of our threshold policy asymptotically converges to this lower bound whenthe total production output is sufficiently large. Our results generalize andunify various online search, pricing, and auction problems, and provide a newperspective on the impact of non-decreasing marginal costs on real-world onlineresource allocation problems.

Threshold Policies with Tight Guarantees for Online Selection with Convex Costs