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

TL;DR

This paper establishes tight bounds for the Price of Anarchy in simultaneous first-price auctions with submodular and subadditive valuations, providing both upper bounds and matching lower bounds through explicit constructions.

Contribution

It offers the first matching lower bounds for these auctions, introduces an alternative proof for the upper bound, and extends results to bid-dependent and multi-unit auctions.

Findings

Matching lower bounds for Bayesian Price of Anarchy: e/(e-1) and 2.

Alternative proof revealing worst-case price distribution.

Lower bounds for subadditive and submodular valuations in multi-unit auctions.

Abstract

We study the Price of Anarchy of simultaneous first-price auctions for buyers with submodular and subadditive valuations. The current best upper bounds for the Bayesian Price of Anarchy 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 first-price auctions with fractionally subadditive valuations which reveals the worst-case price distribution, that 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 first-price auctions 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 multi-unit 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 non-smooth approach.