José Correa, Andrés Cristi, Andrés Fielbaum, Tristan Pollner & S. Matthew Weinberg
Part of the Lecture Notes in Computer Science book series (LNCS,volume 13265)
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. We further show how to compute our prices in polynomial time. We provide additional results for the special case when buyers’ valuations are known (but a posted-price mechanism is still desired).