Simultaneous auctions are (almost) efficient

Simultaneous item auctions are simple and practical proce- dures for allocating items to bidders with potentially com- plex preferences. In a simultaneous auction, every bidder submits independent bids on all items simultaneously. The allocation and prices are then resolved for each item sepa- rately, based solely on the bids submitted on that item. We study the efficiency of Bayes-Nash equilibrium (BNE) out- comes of simultaneous first- and second-price auctions when bidders have complement-free (a.k.a. subadditive) valua- tions. While it is known that the social welfare of every pure Nash equilibrium (NE) constitutes a constant fraction of the optimal social welfare, a pure NE rarely exists, and moreover, the full information assumption is often unreal- istic. Therefore, quantifying the welfare loss in Bayes-Nash equilibria is of particular interest. Previous work established a logarithmic bound on the ratio between the social welfare of a BNE and the expected optimal social welfare in both first-price auctions (Hassidim et al. [11]) and second-price auctions (Bhawalkar and Roughgarden [2]), leaving a large gap between a constant and a logarithmic ratio. We intro- duce a new proof technique and use it to resolve both of these gaps in a unified way. Specifically, we show that the expected social welfare of any BNE is at least 1/2 of the op- timal social welfare in the case of first-price auctions, and at least 1/4 in the case of second-price auctions.