Abstract
We study combinatorial auctions with n agents and m items, where the goal is to allocate the items to the agents such that the social welfare is maximized. We present a universally truthful mechanism with polynomially many queries for combinatorial auctions. Our mechanism and analysis work adaptively for all classes of valuation functions, guaranteeing \(\widetilde{O}(\min (d, \sqrt{m}))\)-approximation (where \(\widetilde{O}\) hides a polylogarithmic factor in m) of the optimal social welfare, where d is the degree of complementarity of the valuation functions. To our knowledge, this is the first mechanism that achieves an approximation guarantee better than \(\varOmega (\sqrt{m})\), when the valuations exhibit any kind of complementarity.
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Notes
- 1.
A valuation v is monotone, if for any \(S \subseteq T \subseteq [m]\), \(v(S) \le v(T)\).
- 2.
A valuation v is normalized, if \(v(\emptyset ) = 0\).
- 3.
A valuation v is submodular, if for any \(S, T \subseteq [m]\), \(v(S) + v(T) \ge v(S \cup T) + v(S \cap T)\).
- 4.
A valuation v is fractionally subadditive, if for any S, \(\{T_i\}\), and \(\{\alpha _i\}\), \(v(S) \le \sum \alpha _i v(T_i)\), whenever the following holds: for each \(j \in S\), \(\sum _{i: j \in T_i} \alpha _i \ge 1\).
- 5.
A valuation v is subadditive, if for any \(S, T \subseteq [m]\), \(v(S) + v(T) \ge v(S \cup T)\).
- 6.
\(\widetilde{O}\) hides a \(\mathrm {polylog}\,m\) factor.
- 7.
One may argue that running the state-of-the-art mechanism for each class of valuations with constant probability achieves the best approximation guarantee for all classes simultaneously. The point we try to make here is, we show how one can achieve this adaptivity with coherent design and analysis, which arguably provides more insight into the problem, and is more likely to inspire future research on the topic.
- 8.
As suggested by an anonymous reviewer, a slight modification gives all agents strict incentives to report truthfully: partition agents into two sets (\(\mathrm {STAT}\) and \(\mathrm {FIXED}\)) uniformly at random, and run a second-price auction on the grand bundle for agents in \(\mathrm {STAT}\). Then with probability 1/2, allocate the grand bundle to the highest bidder in the second-price auction, and with probability 1/2, let \(p_0\) be the highest bid, and proceed to Step 4 (the fixed-price auction) of the original mechanism.
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Acknowledgements
This work is supported by NSF award IIS-1814056. The author thanks anonymous reviewers for helpful feedback.
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Zhang, H. (2020). A Generic Truthful Mechanism for Combinatorial Auctions. In: Chen, X., Gravin, N., Hoefer, M., Mehta, R. (eds) Web and Internet Economics. WINE 2020. Lecture Notes in Computer Science(), vol 12495. Springer, Cham. https://doi.org/10.1007/978-3-030-64946-3_10
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