Price of anarchy for auction revenue

Abstract

This paper develops tools for welfare and revenue analyses of Bayes-Nash equilibria in asymmetric auctions with single-dimensional agents. We employ these tools to derive approximation results for social welfare and revenue. Our approach separates the smoothness framework of [e.g., Syrgkanis and Tardos, 2013] into two distinct parts, isolating the analysis common to any auction from the analysis specific to a given auction. The first part relates a bidder's contribution to welfare in equilibrium to their contribution to welfare in the optimal auction using the price the bidder faces for additional allocation. Intuitively, either an agent's utility and hence contribution to welfare is high, or the price she has to pay for additional allocation is high relative to her value. We call this condition value covering; it holds in every Bayes-Nash equilibrium of any auction. The second part, revenue covering, relates the prices bidders face for additional allocation to the revenue of the auction, using an auction's rules and feasibility constraints. Combining the two parts gives approximation results to the optimal welfare, and, under the right conditions, the optimal revenue. In mechanisms with reserve prices, our welfare results show approximation with respect to the optimal mechanism with the same reserves.As a centerpiece result, we analyze the single-item first-price auction with individual monopoly reserves (the price that a monopolist would post to sell to that agent alone; these reserves are generally distinct for agents with values drawn from distinct distributions). When each distribution satisfies the regularity condition of Myerson [1981], the auction's revenue is at least a 2e/(e − 1) ≈ 3.16 approximation to the revenue of the optimal auction. We also give bounds for matroid auctions with first-price or all-pay semantics, the generalized first-price position auction, and pay-your-bid auctions for single-minded combinatorial auctions. Finally, we give an extension theorem for simultaneous composition, i.e., when multiple auctions are run simultaneously, with single-valued, unit-demand agents. * We thank Vasilis Syrgkanis for comments on a prior version of this paper for which simultaneous composition did not hold, suggesting the study of simultaneous composition and for perspective on price-of-anarchy methodology.

Algorithms against Anarchy

“…This and similar properties have been used in previous works to upper-bound the Price of Anarchy by O(γ) (see, e.g., [Lucier and Borodin 2010;Syrgkanis and Tardos 2013;Babaioff et al 2014;Hartline et al 2014]). Our main technical achievement is to show that the above condition and a suitable generalization to randomized mechanisms yields matching upper and lower bounds.…”

“…Also relevant as a precursor and parallel literature to our work is work that analyzes the Price of Anarchy of simple mechanisms, such as [Christodoulou et al 2008;Bhawalkar and Roughgarden 2011;Feldman et al 2013], work that extends the smoothness concept so that it also yields revenue guarantees [Hartline et al 2014], and work that identifies barriers to near-optimal equilibria [Roughgarden 2014]. The first line of work differs from ours as it considers the performance of simple mechanisms for specific problems.…”

Algorithmic game theory and econometrics

“…Our approach is an empirical analogue of the smoothness approach on quantifying the worst-case inefficiency in games [Roughgarden 2009;Syrgkanis and Tardos 2013;Hartline et al 2014] and therefore inherits several robustness properties of smoothness. For instance, the lower bound on the efficiency that is derived via our method holds regardless of whether the data that we observe are the product of a Bayes-Nash equilibrium where the player valuations are stochastic, or whether they are the product of a learning process employed by an advertiser with a fixed valuation.…”

“…The smoothness approach of [Syrgkanis and Tardos 2013] and its refinement for single-parameter mechanism design environments, via the revenue and value covering formulation of [Hartline et al 2014] is based on the following argument: at any outcome of the game that satisfies a best-response or approximate best-response property either the player is getting high utility and hence high allocation probability, or the payment that he needs to make in order to achieve a high allocation probability given the competition must be high. The latter quantity is typically referred to as the threshold payment.…”

An improved welfare guarantee for first-price auctions

“…However, in the most commonly studied setting with independently distributed values over a single good, there has been a persistent gap between this welfare guarantee and the worst known example of Hartline et al (2014), in which the social welfare is a .869-fraction of the optimal social welfare. Despite the prevalence of the first-price auction format, the salience of the independent values assumption, and the ubiquity of the techniques used to prove the existing e−1 e bound, it was not clear whether this bound was tight in the independent values setting.…”