Bounding the inefficiency of outcomes in generalized second price auctions

Abstract

The Generalized Second Price (GSP) auction is the primary auction used for monetizing the use of the Internet. It is well-known that truthtelling is not a dominant strategy in this auction and that inefficient equilibria can arise. Edelman et al. (AER, 2007) and Varian (IJIO, 2007) show that an efficient equilibrium always exists in the full information setting. Their results, however, do not extend to the case with uncertainty, where efficient equilibria might not exist.In this paper we study the space of equilibria in GSP, and quantify the efficiency loss that can arise in equilibria under a wide range of sources of uncertainty, as well as in the full information setting. The traditional Bayesian game models uncertainty in the valuations (types) of the participants. The Generalized Second Price (GSP) auction gives rise to a further form of uncertainty: the selection of quality factors resulting in uncertainty about the behavior of the underlying ad allocation algorithm. The bounds we obtain apply to both forms of uncertainty, and are robust in the sense that they apply under various perturbations of the solution concept, extending to models with information asymmetries and bounded rationality in the form of learning strategies.We present a constant bound (2.927) on the factor of the efficiency loss (price of anarchy) of the corresponding game for the Bayesian model of partial information about other participants and about ad quality factors. For the full information setting, we prove a surprisingly low upper bound of 1.282 on the price of anarchy over pure Nash equilibria, nearly matching a lower bound of 1.259 for the case of three advertisers. Further, we do not require that the system reaches equilibrium, and give similarly low bounds also on the quality degradation for any no-regret learning outcome. Our conclusion is that the number of advertisers in the auction has almost no impact on the price of anarchy, and that the efficiency of GSP is very robust with respect to the belief and rationality assumptions imposed on the participants.

Price of anarchy for auction revenue

“…This results no-overbidding welfare guarantees that are structurally similar to the price of anarchy analyses of Syrgkanis and Tardos [2013] and Caragiannis et al [2014].…”

Truthful Outcomes from Non-Truthful Position Auctions

“…It is, moreover, not difficult to see that (12) is implied by (11) and (14): by (14), (12). The α-GSP mechanism thus possesses an efficient equilibrium if and only if there exist bids b 2 ≥ b 3 such that…”

“…Each of the two mechanisms has severe disadvantages in the respective other setting, such as non-existence of a pure Nash equilibrium or of an efficient Bayes-Nash equilibrium [20,22]. In cases where equilibria exist, however, the worst-case welfare loss is bounded in the sense of a small price of anarchy [12,35]. 5 In the standard model, and other things being equal, the VCG mechanism of course has the advantage of truthfulness.…”

“…11 Bidding below b F (0) = 0 is impossible, bidding above b F (v) is dominated, 12 and b F satisfies the second condition of Lemma 3 by construction. The α-GFP mechanism thus has an efficient equilibrium if and only if b F is increasing almost everywhere.…”

Bayes-Nash Equilibria in Generalized Second Price Auctions with Allocative Externalities

“…Based on a special environment with complete information of match rates, Chen and He (2011) analyze the welfare and revenue effects of advertisers' pricing and bidding decisions in GSP. Caragiannis et al (2015) delimit the price of anarchy (i.e., efficiency loss) of GSP in both incomplete and complete information settings.…”