Simultaneous 2nd Price Item Auctions with No-Underbidding

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.

Two-Price Equilibrium

“…Finally, in Section 3.1 we show that every PNE of a S2PA has a corresponding 2PE with the same allocation (see Proposition 3.5). Feldman and Shabtai [11] establish bounds on the price of anarchy of S2PA under a "no underbidding" assumption for different valuation classes. It would be interesting to study whether a PNE satisfying no underbidding corresponds to a 2PE with bounded discrepancy.…”

“…The 2PE notion is related to other relaxations of WE, such as the endowment equilibrium ( [1], [9]), named after the endowment effect, discovered by Nobel laureate Richard Thaler ( [18], [19], [21]), stating that buyers tend to inflate the value of items they own. Moreover, as we show in Section 3.1, 2PE is also related to Nash equilibria of simultaneous item auctions -a simple auction format that attracted much research in the last decade ( [2], [6], [12], [11], [7], [4]).…”

Two-Price Equilibrium

“…The 2PE notion is related to other relaxations of WE, such as the endowment equilibrium ( (Babaioff, Dobzinski, and Oren 2018), (Ezra, Feldman, and Friedler 2019)), named after the endowment effect, discovered by Nobel laureate Richard Thaler ( (Kahneman, Knetsch, and Thaler 1990), (Kahneman, Knetsch, and Thaler 1991), (Knetsch, Tang, and Thaler 2001)), stating that buyers tend to inflate the value of items they own. Moreover, as we show in Section 3.1, 2PE is also related to Nash equilibria of simultaneous item auctions -a simple auction format that attracted much research in the last decade ( (Bhawalkar and Roughgarden 2011), (Christodoulou, Kovács, and Schapira 2016), , (Feldman and Shabtai 2020), (Christodoulou et al 2016), (Cai and Papadimitriou 2014)).…”