Simultaneous 2nd Price Item Auctions with No-Underbidding
share
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 and this is tight. 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.
READ FULL TEXT