A sequence of recent studies show that even in the simple setting of a single seller and a single buyer with additive, independent valuations over m items, the revenuemaximizing mechanism is prohibitively complex. This problem has been addressed using two main approaches:• Approximation: the best of two simple mechanisms (sell each item separately, or sell all the items as one bundle) gives 1/6 of the optimal revenue [BILW14].• Enhanced competition: running the simple VCG mechanism with additional m buyers extracts at least the optimal revenue in the original market [EFF + 17a].Both approaches, however, suffer from severe drawbacks: On the one hand, losing 83% of the revenue is hardly acceptable in any application. On the other hand, attracting a linear number of new buyers may be prohibitive. Our main result is that by combining the two approaches one can achieve the best of both worlds. Specifically, for any constant ǫ one can obtain a (1 − ǫ) fraction of the optimal revenue by running simple mechanisms -either selling each item separately or selling all items as a single bundle -with substantially fewer additional buyers: logarithmic, constant, or even none in some cases.

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We study the efficiency of simple auctions in the presence of complements. Devanur et al. [11] 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 [16], 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(d 2 log(m/d)), where m is the number of items. We also establish an improved upper bound of O(d log m) 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.

Losses loom larger than gains" -Daniel Kahneman; Amos TverskyThe endowment effect, coined by Nobel Laureate Richard Thaler, posits that people tend to inflate the value of items they own. This bias has been traditionally studied mainly using experimental methodology. Recently, Babaioff et al. proposed a specific formulation of the endowment effect in combinatorial markets, and showed that the existence of Walrasian equilibrium with respect to the endowed valuations extends from gross substitutes to submodular valuations, but provably fails to extend to XOS valuations.We propose to harness the endowment effect further. To this end, we introduce a principlebased framework that captures a wide range of different formulations of the endowment effect (including that of Babaioff et al.). We equip our framework with a partial order over the different formulations, which (partially) ranks them from weak to strong, and provide algorithms for computing endowment equilibria with high welfare for sufficiently strong endowment effects, as well as non-existence results for weaker ones.Our main results are the following: