Tight Bounds for the Price of Anarchy of Simultaneous First-Price Auctions

Abstract

We study the price of anarchy (PoA) of simultaneous first-price auctions (FPAs) for buyers with submodular and subadditive valuations. The current best upper bounds for the Bayesian price of anarchy (BPoA) of these auctions are e /( e − 1) [Syrgkanis and Tardos 2013] and 2 [Feldman et al. 2013], respectively. We provide matching lower bounds for both cases even for the case of full information and for mixed Nash equilibria via an explicit construction. We present an alternative proof of the upper bound of e /( e − 1) for FPAs with fractionally subadditive valuations that reveals the worst-case price distribution, which is used as a building block for the matching lower bound construction. We generalize our results to a general class of item bidding auctions that we call bid-dependent auctions (including FPAs and all-pay auctions) where the winner is always the highest bidder and each bidder’s payment depends only on his own bid. Finally, we apply our techniques to discriminatory price multiunit auctions. We complement the results of de Keijzer et al. [2013] for the case of subadditive valuations by providing a matching lower bound of 2. For the case of submodular valuations, we provide a lower bound of 1.109. For the same class of valuations, we were able to reproduce the upper bound of e /( e − 1) using our nonsmooth approach.