Optimal item pricing in online combinatorial auctions

We consider a fundamental pricing problem in combinatorial auctions. We are given a set of indivisible items and a set of buyers with randomly drawn monotone valuations over subsets of items. A decision-maker sets item prices and then the buyers make sequential purchasing decisions, taking their favorite set among the remaining items. We parametrize an instance by d , the size of the largest set a buyer may want. Our main result asserts that there exist prices such that the expected (over the random valuations) welfare of the allocation they induce is at least a factor 1/(d+1) times the expected optimal welfare in hindsight. Moreover, we prove that this bound is tight. Thus, our result not only improves upon the 1/(4d-2) bound of Dütting et al., but also settles the approximation that can be achieved by using item prices. The existence of these prices follows from the existence of a fixed point of a related mapping, and therefore, it is non-constructive. However, we show how to compute such a fixed point in polynomial time, even if we only have sample access to the valuation distributions. We provide additional results for the special case when buyers’ valuations are known (but a posted-price mechanism is still desired), and an improved impossibility result for the special case of prophet inequalities for bipartite matching.