Optimal Bi-Valued Auctions

📝 Abstract

We investigate \emph{bi-valued} auctions in the digital good setting and construct an explicit polynomial time deterministic auction. We prove an unconditional tight lower bound which holds even for random superpolynomial auctions. The analysis of the construction uses the adoption of the finer lens of \emph{general competitiveness} which considers additive losses on top of multiplicative ones. The result implies that general competitiveness is the right notion to use in this setting, as this optimal auction is uncompetitive with respect to competitive measures which do not consider additive losses.

💡 Analysis

We investigate \emph{bi-valued} auctions in the digital good setting and construct an explicit polynomial time deterministic auction. We prove an unconditional tight lower bound which holds even for random superpolynomial auctions. The analysis of the construction uses the adoption of the finer lens of \emph{general competitiveness} which considers additive losses on top of multiplicative ones. The result implies that general competitiveness is the right notion to use in this setting, as this optimal auction is uncompetitive with respect to competitive measures which do not consider additive losses.

📄 Content

arXiv:1106.4677v1 [cs.DS] 23 Jun 2011 Optimal Bi-Valued Auctions Oren Ben-Zwi∗ Ilan Newman† Abstract We investigate bi-valued auctions in the digital good setting and construct an explicit poly- nomial time deterministic auction. We prove an unconditional tight lower bound which holds even for random superpolynomial auctions. The analysis of the construction uses the adoption of the finer lens of general competitiveness which considers additive losses on top of multiplicative ones. The result implies that general competitiveness is the right notion to use in this setting, as this optimal auction is uncompetitive with respect to competitive measures which do not consider additive losses. 1 introduction Marketing a digital good may suffer from a low revenue due to incomplete knowledge of the mar- keter. Consider, for example, a major sport event with some 108 potential TV viewers. Assume further that every potential viewer is willing to pay $10 or more to watch the event, and that no more than 106 are willing to pay $100 for that. If the concessionaire will charge $1 or $100 as a fixed pay per view price for the event, the overall collected revenue will be $108 at the most. This is worse than the revenue that can be collected, having known the valuations beforehand. This lack of knowledge motivates the study of unlimited supply, unit demand, single item auc- tions. Goldberg et al. [11] studied these auctions; in order to obtain a prior free, worst case analysis framework, they suggested to compare the revenue of these auctions to the revenue of optimal fixed price auctions. They adopted the online algorithms terminology [20] and named the revenue of the fixed price auction the offline revenue and the revenue of a multi price truthful auction, i.e., an auction for which every bidder has an incentive to bid its own value, online revenue. The com- petitive ratio of an auction for a bid vector b is defined to be the ratio between the best offline revenue for b to the revenue of that auction on b. The competitive ratio of an auction is just the worst competitive ratio of that auction on all possible bid vectors. For random auctions, a similar notion is defined by taking the expected revenue. If an auction has a constant competitive ratio it is said to be competitive. If an auction has a constant competitive ratio, possibly with some small additive loss, it is said to be general competitive (see Section 2 for definitions). We remark that later, Koutsoupias and Pierrakos [13] used online auctions in the usual context of online algorithms, but here we will stick to Goldberg et al.’s [11] notion. ∗Faculty of Industrial Engineering and Management, Technion, Haifa, Israel. Email: orenb@technion.ac.il. This research was supported in part by the Google Inter-university center for Electronic Markets and Auctions. †Department of Computer Science, Haifa University, Haifa, Israel. Email: ilan@cs.haifa.ac.il. 1 This attempt to carefully select the right benchmark in order to obtain a prior free, worst case analysis was a posteriori justified when Hartline and Roughgarden defined a general benchmark for the analysis of single parameter mechanism design problems [12]. This general benchmark, which bridges Bayesian analysis [16] from economics and worst case analysis from the theory of computer science, collides for our setting with the optimal fixed price benchmark [12]. A further justification for taking the offline fixed price auction as a benchmark is the following. Although an online auction seems less restricted than the offline fixed price auction, as it can assign different players different prices, it was shown [9] that the online truthful revenue is no more than the offline revenue. In fact, there even exists a lower bound of 2.42 on the competitive ratio of any truthful online auction [10]. Note, however, that the optimal offline revenue (or more precisely the optimal fixed price) is unknown to the concessionaire in advance. It is well known (see for example [15]) that in order to achieve truthfulness one can use only the set of bid independent auctions, i.e., auctions in which the price offered to a bidder is independent of the bidder’s own bid value. Hence, an intuitive auction that often comes to one’s mind is the Deterministic Optimal Price (DOP) auction [2, 9, 19]. In this auction the mechanism computes and offers each bidder the price of an optimal offline auction for all other bids. This auction preforms well on most bid vectors. In fact, it was even proved by Segal [19] that if the input is chosen uniformly at random, then this auction is asymptotically optimal. For a worst case analysis, however, it preforms very poor. Consider, for example, an auction in which there are n bidders and only two possible bid values: 1 and h, where h ≫1. We denote this setting as bi-valued auctions. Let nh be the number of bidders who bid h. Applying DOP on a bid vector for which nh = n/h will result in a revenue of nh instead of the n revenue of an offline auct

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