  Simultaneous item auctions are simple procedures for allocating items to
bidders with potentially complex preferences over different item sets. In a
simultaneous auction, every bidder submits bids on all items simultaneously.
The allocation and prices are then resolved for each item separately, based
solely on the bids submitted on that item. Such procedures occur in practice
(e.g. eBay) but are not truthful. We study the efficiency of Bayesian Nash
equilibrium (BNE) outcomes of simultaneous first- and second-price auctions
when bidders have complement-free (a.k.a. subadditive) valuations. We show that
the expected social welfare of any BNE is at least 1/2 of the optimal social
welfare in the case of first-price auctions, and at least 1/4 in the case of
second-price auctions. These results improve upon the previously-known
logarithmic bounds, which were established by [Hassidim, Kaplan, Mansour and
Nisan '11] for first-price auctions and by [Bhawalkar and Roughgarden '11] for
second-price auctions.
