On Revenue Monotonicity in Combinatorial Auctions

Abstract

Along with substantial progress made recently in designing near-optimal mechanisms for multi-item auctions, interesting structural questions have also been raised and studied. In particular, is it true that the seller can always extract more revenue from a market where the buyers value the items higher than another market? In this paper we obtain such a revenue monotonicity result in a general setting. Precisely, consider the revenue-maximizing combinatorial auction for m items and n buyers in the Bayesian setting, specified by a valuation function v and a set F of nm independent item-type distributions. Let REV(v, F) denote the maximum revenue achievable under F by any incentive compatible mechanism. Intuitively, one would expect that $REV(v, G)\ge REV(v, F)$ if distribution G stochastically dominates F. Surprisingly, Hart and Reny (2012) showed that this is not always true even for the simple case when v is additive. A natural question arises: Are these deviations contained within bounds? To what extent may the monotonicity intuition still be valid? We present an approximate monotonicity theorem for the class of fractionally subadditive (XOS) valuation functions v, showing that $REV(v, G)\ge c\,REV(v, F)$ if G stochastically dominates F under v where $c>0$ is a universal constant. Previously, approximate monotonicity was known only for the case $n=1$: Babaioff et al. (2014) for the class of additive valuations, and Rubinstein and Weinberg (2015) for all subaddtive valuation functions.