No abstract

We study the price of anarchy (PoA) of simultaneous 2nd price auctions (S2PA) under a natural condition of no underbidding. No underbidding means that an agent's bid on every item is at least its marginal value given the outcome. In a 2nd price auction, underbidding on an item is weakly dominated by bidding the item's marginal value. Indeed, the no underbidding assumption is justified both theoretically and empirically.We establish bounds on the PoA of S2PA under no underbidding for different valuation classes, in both full-information and incomplete information settings. To derive our results, we introduce a new parameterized property of auctions, namely (γ, δ)-revenue guaranteed, and show that every auction that is (γ, δ)-revenue guaranteed has PoA at least γ/(1 + δ). An auction that is both (λ, µ)-smooth and (γ, δ)-revenue guaranteed has PoA at least (γ + λ)/(1 + δ + µ). Via extension theorems, these bounds extend to coarse correlated equilibria in full information settings, and to Bayesian PoA (BPoA) in settings with incomplete information.We show that S2PA with submodular valuations and no underbidding is (1, 1)-revenue guaranteed, implying that the PoA is at least 1 2 . Together with the known (1, 1)-smoothness (under the standard no overbididng assumption), it gives PoA of 2/3 (which extends via the extension theorems), and this is tight (even with respect to unit-demand valuations). For valuations beyond submodular valuations we employ a stronger condition of set no underbidding, which extends the no underbidding condition to sets of items. We show that S2PA with set no underbidding is (1, 1)-revenue guaranteed for arbitrary valuations, implying a PoA of at least 1/2. Together with no overbidding we get a lower bound of 2 3 on the Bayesian PoA for XOS valuations, and on the PoA for subadditive valuations.Our results also shed new light on the relative performance of S2PA and their S1PA counterparts. Specifically, under the standard no overbidding assumption, S1PA has better BPoA bounds than S2PA, but the situation flips when considering both no overbidding and no underbidding.

Walrasian equilibrium is a prominent market equilibrium notion, but rarely exists in markets with indivisible items. We introduce a new market equilibrium notion, called two-price equilibrium (2PE). A 2PE is a relaxation of Walrasian equilibrium, where instead of a single price per item, every item has two prices: one for the item's owner and a (possibly) higher one for all other buyers. Thus, a 2PE is given by a tuple (S, p, p) of an allocation S and two price vectors p, p, where every buyer i is maximally happy with her bundle S i , given prices p for items in S i and prices p for all other items. 2PE generalizes previous market equilibrium notions, such as conditional equilibrium, and is related to relaxed equilibrium notions like endowment equilibrium. We define the discrepancy of a 2PE -a measure of distance from Walrasian equilibrium -as the sum of differences pj − pj over all items (normalized by social welfare). We show that the social welfare degrades gracefully with the discrepancy; namely, the social welfare of a 2PE with discrepancy d is at least a fraction 1 d+1 of the optimal welfare. We use this to establish welfare guarantees for markets with subadditive valuations over identical items. In particular, we show that every such market admits a 2PE with at least 1/7 of the optimal welfare. This is in contrast to Walrasian equilibrium or conditional equilibrium which may not even exist. Our techniques provide new insights regarding valuation functions over identical items, which we also use to characterize instances that admit a WE.

No abstract

No abstract