Online Truthful Mechanisms for Combinatorial Auctions

We study combinatorial auctions in online environments with an objective to maximize the social welfare. In this problem, items arrive on each day and must be sold before the closing date of the market which is unknown to the seller and the buyers. We consider two settings, one where items must be sold immediately on arrival (called ImmediateSale), and the other where items can be stored and sold later (called DeferredSale). In the ImmediateSale setting where item must be sold immediately on arrival, we demonstrate a surprising separation between submodular and XOS valuations: on one hand, we show there is no online mechanism that can achieve Ω ((m/log m) 1/3) approximation for XOS (and subadditive) valuations in a Bayesian setting, where m is the total number of items. Our lower bound holds even if we do not require truthfulness and/or efficiency of the mechanism. On the other hand, for submodular valuations, we give a O (log m)-competitive mechanism for adversarially chosen valuations and an O (1)-competitive mechanism in a Bayesian setting. Both of our mechanisms are efficient and universally truthful. In the DeferredSale setting where the items can be stored and sold later, we provide a reduction that can convert the state-of-the-art offline mechanism into an online mechanism while preserving the approximation ratio for subadditve valuations.