TY - GEN

T1 - Simple mechanisms for agents with complements

AU - Feldman, Michal

AU - Friedler, Ophir

AU - Morgenstern, Jamie

AU - Reiner, Guy

N1 - Publisher Copyright:
© Copyright 2016 ACM.

PY - 2016/7/21

Y1 - 2016/7/21

N2 - We study the efficiency of simple auctions in the presence of complements. Devanur et al. [2015] introduced the single-bid auction, and showed that it has a price of anarchy (PoA) of O(log m) for complement-free (i.e., subadditive) valuations. Prior to our work, no non-trivial upper bound on the PoA of single bid auctions was known for valuations exhibiting complements. We introduce a hierarchy over valuations, where levels of the hierarchy correspond to the degree of complementarity, and the PoA of the single bid auction degrades gracefully with the level of the hierarchy. This hierarchy is a refinement of the Maximum over Positive Hypergraphs (MPH) hierarchy [Feige et al. 2015], where the degree of complementarity d is captured by the maximum number of neighbors of a node in the positive hypergraph representation. We show that the price of anarchy of the single bid auction for valuations of level d of the hierarchy is O(d2 log(m/d)), where m is the number of items. We also establish an improved upper bound of O(dlogm) for a subclass where every hyperedge in the positive hypergraph representation is of size at most 2 (but the degree is still d). Finally, we show that randomizing between the single bid auction and the grand bundle auction has a price of anarchy of at most O(√m) for general valuations. All of our results are derived via the smoothness framework, thus extend to coarse-correlated equilibria and to Bayes Nash equilibria.

AB - We study the efficiency of simple auctions in the presence of complements. Devanur et al. [2015] introduced the single-bid auction, and showed that it has a price of anarchy (PoA) of O(log m) for complement-free (i.e., subadditive) valuations. Prior to our work, no non-trivial upper bound on the PoA of single bid auctions was known for valuations exhibiting complements. We introduce a hierarchy over valuations, where levels of the hierarchy correspond to the degree of complementarity, and the PoA of the single bid auction degrades gracefully with the level of the hierarchy. This hierarchy is a refinement of the Maximum over Positive Hypergraphs (MPH) hierarchy [Feige et al. 2015], where the degree of complementarity d is captured by the maximum number of neighbors of a node in the positive hypergraph representation. We show that the price of anarchy of the single bid auction for valuations of level d of the hierarchy is O(d2 log(m/d)), where m is the number of items. We also establish an improved upper bound of O(dlogm) for a subclass where every hyperedge in the positive hypergraph representation is of size at most 2 (but the degree is still d). Finally, we show that randomizing between the single bid auction and the grand bundle auction has a price of anarchy of at most O(√m) for general valuations. All of our results are derived via the smoothness framework, thus extend to coarse-correlated equilibria and to Bayes Nash equilibria.

KW - Auctions

KW - Price of anarchy

KW - Restricted complements

UR - http://www.scopus.com/inward/record.url?scp=84983567841&partnerID=8YFLogxK

U2 - 10.1145/2940716.2940735

DO - 10.1145/2940716.2940735

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AN - SCOPUS:84983567841

T3 - EC 2016 - Proceedings of the 2016 ACM Conference on Economics and Computation

SP - 251

EP - 267

BT - EC 2016 - Proceedings of the 2016 ACM Conference on Economics and Computation

PB - Association for Computing Machinery, Inc

Y2 - 24 July 2016 through 28 July 2016

ER -