Combinatorial Auctions Do Need Modest Interaction

Abstract

We study the necessity of interaction for obtaining efficient allocations in combinatorial auctions with subadditive bidders . This problem was originally introduced by Dobzinski, Nisan, and Oren (STOC’14) as the following simple market scenario: m items are to be allocated among n bidders in a distributed setting where bidders valuations are private and hence communication is needed to obtain an efficient allocation. The communication happens in rounds: In each round, each bidder, simultaneously with others, broadcasts a message to all parties involved. At the end, the central planner computes an allocation solely based on the communicated messages. Dobzinski et al. showed that (at least some) interaction is necessary for obtaining even an approximately efficient allocation: No non-interactive (1-round) protocol with polynomial communication (in the number of items and bidders) can achieve approximation ratio better than Ω( m 1/4 ), while for any r ≥ 1, there exists r -round protocols that achieve Õ ( r · m 1/( r +1) ) approximation with polynomial communication. This in particular implies that O (log m ) rounds of interaction suffice to obtain an approximately efficient allocation—namely, a polylog(m)-approximation. A natural question at this point is to identify the "right" level of interaction (i.e., number of rounds) necessary to obtain approximately efficient allocations. In this article, we resolve this question by providing an almost tight round-approximation tradeoff for this problem: We show that for any r ≥ 1, any r -round protocol that uses poly( m , n ) bits of communication can only approximate the social welfare up to a factor of Ω(1/ r · m 1/(2 r +1) ). This in particular implies that Ω(log m /log log m ) rounds of interaction are necessary for obtaining any allocation with a reasonable welfare approximation—namely, a constant or even a polylog( m )-approximation. Our work builds on the multi-party round-elimination technique of Alon, Nisan, Raz, and Weinstein (FOCS’15)—used to prove similar-in-spirit lower bounds for round-approximation tradeoff in unit-demand markets—and settles an open question posed initially by Dobzinski et al. (STOC’14) and subsequently by Alon et al. (FOCS’15).