On Simultaneous Two-player Combinatorial Auctions.

We consider the following communication problem: Alice and Bob each have some valuation functions υ1(·) and υ2(·) over subsets of m items, and their goal is to partition the items into [EQUATION] in a way that maximizes the welfare, [EQUATION]. We study both the allocation problem, which asks for a welfare-maximizing partition and the decision problem, which asks whether or not there exists a partition guaranteeing certain welfare, for binary XOS valuations. For interactive protocols with poly(m) communication, a tight 3/4-approximation is known for both [29, 23]. For interactive protocols, the allocation problem is provably harder than the decision problem: any solution to the allocation problem implies a solution to the decision problem with one additional round and log m additional bits of communication via a trivial reduction. Surprisingly, the allocation problem is provably easier for simultaneous protocols. Specifically, we show: • There exists a simultaneous, randomized protocol with polynomial communication that selects a partition whose expected welfare is at least 3/4 of the optimum. This matches the guarantee of the best interactive, randomized protocol with polynomial communication. • For all ε > 0, any simultaneous, randomized protocol that decides whether the welfare of the optimal partition is ≥ 1 or ≤ 3/4 − 1/108 + ε correctly with probability > 1/2 + 1/poly(m) requires exponential communication. This provides a separation between the attainable approximation guarantees via interactive (3/4) versus simultaneous (≤ 3/4 − 1/108) protocols with polynomial communication. In other words, this trivial reduction from decision to allocation problems provably requires the extra round of communication. We further discuss the implications of our results for the design of truthful combinatorial auctions in general, and extensions to general XOS valuations. In particular, our protocol for the allocation problem implies a new style of truthful mechanisms.