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We consider the following communication problem: Alice and Bob each have some valuation functions $$v_1(\cdot)$$ and $$v_2(\cdot)$$ over subsets of $$m$$ items, and their goal is to partition the items into $$S, \bar{S}$$ in a way that maximizes the welfare, $$v_1(S) + v_2(\bar{S})$$. 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 [Fei06,DS06]. 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: 1) 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. 2) For all $$\varepsilon > 0$$, any simultaneous, randomized protocol that decides whether the welfare of the optimal partition is $$\geq 1$$ or $$\leq 3/4 - 1/108+\varepsilon$$ 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 ($$\leq 3/4-1/108$$) 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.
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On the hardness of dominant strategy mechanism design
We study the communication complexity of dominant strategy implementations of combinatorial auctions. We start with two domains that are generally considered “easy”: multi-unit auctions with decreasing marginal values and combinatorial auctions with gross substitutes valuations. For both domains we have fast algorithms that find the welfare-maximizing allocation with communication complexity that is poly-logarithmic in the input size. This immediately implies that welfare maximization can be achieved in ex-post equilibrium with no significant communication cost, by using VCG payments. In contrast, we show that in both domains the communication complexity of any dominant strategy implementation that achieves the optimal welfare is polynomial in the input size. We then move on to studying the approximation ratios achievable by dominant strategy mechanisms. For multi-unit auctions with decreasing marginal values, we provide a dominant-strategy communication FPTAS. For combinatorial auctions with general valuations, we show that there is no dominant strategy mechanism that achieves an approximation ratio better than m1−є that uses poly(m,n) bits of communication, where m is the number of items and n is the number of bidders. In contrast, a randomized dominant strategy mechanism that achieves an O(√m) approximation with poly(m,n) communication is known. This proves the first gap between computationally efficient deterministic dominant strategy mechanisms and randomized ones. En route, we answer an open question on the communication cost of implementing dominant strategy mechanisms for more than two players, and also solve some open problems in the area of simultaneous combinatorial auctions.
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- Award ID(s):
- 2127781
- PAR ID:
- 10352728
- Editor(s):
- Stefano Leonardi
- Date Published:
- Journal Name:
- STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing
- Page Range / eLocation ID:
- 690 to 703
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3519935.3520013
