Lower Bounds on Revenue of Approximately Optimal Auctions (original) (raw)
Related papers
Proceedings of the forty-sixth annual ACM symposium on Theory of computing, 2014
We study the design of truthful auctions for selling identical items in unlimited supply (e.g., digital goods) to n unit demand buyers. This classic problem stands out from profit-maximizing auction design literature as it requires no probabilistic assumptions on buyers' valuations and employs the framework of competitive analysis. Our objective is to optimize the worst-case performance of an auction, measured by the ratio between a given benchmark and revenue generated by the auction. We establish a sufficient and necessary condition that characterizes competitive ratios for all monotone benchmarks. The characterization identifies the worst-case distribution of instances and reveals intrinsic relations between competitive ratios and benchmarks in the competitive analysis. With the characterization at hand, we show optimal competitive auctions for two natural benchmarks. The most well-studied benchmark F (2) (•) measures the envy-free optimal revenue where at least two buyers win. Goldberg et al. [13] showed a sequence of lower bounds on the competitive ratio for each number of buyers n. They conjectured that all these bounds are tight. We show that optimal competitive auctions match these bounds. Thus, we confirm the conjecture and settle a central open problem in the design of digital goods auctions. As one more application we examine another economically meaningful benchmark, which measures the optimal revenue across all limited-supply Vickrey auctions. We identify the optimal competitive ratios to be (n n−1) n−1 − 1 for each number of buyers n, that is e − 1 as n approaches infinity.
Revenue-Capped Efficient Auctions
Journal of the European Economic Association
We study an auction that maximizes the expected social surplus under an upper-bound constraint on the seller’s expected revenue, which we call a revenue cap. Such a constrained-efficient auction may arise, for example, when (i) the auction designer is “pro-buyer”, that is, he maximizes the weighted sum of the buyers’ and seller’s auction payoffs, where the weight for the buyers is greater than that for the seller; (ii) the auction designer maximizes the (unweighted) total surplus in a multiunit auction in which the number of units the seller owns is private information; or (iii) multiple sellers compete to attract buyers before the auction. We characterize the mechanisms for constrained-efficient auctions and identify their important properties. First, the seller sets no reserve price and sells the good for sure. Second, with a nontrivial revenue cap, “bunching” is necessary. Finally, with a sufficiently severe revenue cap, the constrained-efficient auction has a bid cap, so that bu...
Revenue monotonicity in combinatorial auctions
ACM SIGecom Exchanges, 2007
Intuitively, one might expect that a seller's revenue from an auction weakly increases as the number of bidders grows, as this increases competition. However, it is known that for combinatorial auctions that use the VCG mechanism, a seller can sometimes increase revenue by dropping bidders. In this paper we investigate the extent to which this problem can occur under other dominant-strategy combinatorial auction mechanisms. Our main result is that such failures of "revenue monotonicity" are not limited to mechanisms that achieve efficient allocations. Instead, they can occur under any dominant-strategy direct mechanism that sets prices using critical values, and that always chooses an allocation that cannot be augmented to make some bidder better off, while making none worse off.
Asymptotically Efficient Multi-Unit Auctions via Posted Prices
ArXiv, 2018
We study the asymptotic average-case efficiency of static and anonymous posted prices for nnn agents and m(n)m(n)m(n) multiple identical items with m(n)=oleft(fracnlognright)m(n)=o\left(\frac{n}{\log n}\right)m(n)=oleft(fracnlognright). When valuations are drawn i.i.d from some fixed continuous distribution (each valuation is a vector in Re+m\Re_+^mRe+m and independence is assumed only across agents) we show: (a) for any "upper mass" distribution there exist posted prices such that the expected revenue and welfare of the auction approaches the optimal expected welfare as nnn goes to infinity; specifically, the ratio between the expected revenue of our posted prices auction and the expected optimal social welfare is 1−Oleft(fracm(n)lognnright)1-O\left(\frac{m(n)\log n}{n}\right)1−Oleft(fracm(n)lognnright), and (b) there do not exist posted prices that asymptotically obtain full efficiency for most of the distributions that do not satisfy the upper mass condition. When valuations are complete-information and only the arrival order is adversarial, we provide a "tiefree" conditi...
Optimal auctions with correlated bidders are easy
Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11, 2011
We consider the problem of designing a revenue-maximizing auction for a single item, when the values of the bidders are drawn from a correlated distribution. We observe that there exists an algorithm that finds the optimal randomized mechanism that runs in time polynomial in the size of the support. We leverage this result to show that in the oracle model introduced by Ronen and Saberi [FOCS'02], there exists a polynomial time truthful in expectation mechanism that provides a ( 3 2 + ǫ)-approximation to the revenue achievable by an optimal truthful-in-expectation mechanism, and a polynomial time deterministic truthful mechanism that guarantees 5 3 approximation to the revenue achievable by an optimal deterministic truthful mechanism.
Optimal and Efficient Parametric Auctions
Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, 2013
Consider a seller who seeks to provide service to a collection of interested parties, subject to feasibility constraints on which parties may be simultaneously served. Assuming that a distribution is known on the value of each party for service-arguably a strong assumption-Myerson's seminal work provides revenue optimizing auctions [12]. We show instead that, for very general feasibility constraints, only knowledge of the median of each party's value distribution, or any other quantile of these distributions, or approximations thereof, suffice for designing simple auctions that simultaneously approximate both the optimal revenue and the optimal welfare. Our results apply to all downward-closed feasibility constraints under the assumption that the underlying, unknown value distributions are monotone hazard rate, and to all matroid feasibility constraints under the weaker assumption of regularity of the underlying distributions. Our results jointly generalize the single-item results obtained by Azar and Micali [2] on parametric auctions, and Daskalakis and Pierrakos [6] for simultaneously approximately optimal and efficient auctions.
Stepwise Randomized Combinatorial Auctions Achieve Revenue Monotonicity
Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, 2009
In combinatorial auctions that use VCG, a seller can sometimes increase revenue by dropping bidders (see e.g. ). In our previous work , we showed that such failures of "revenue monotonicity" occur under an extremely broad range of deterministic strategyproof combinatorial auction mechanisms, even when bidders have "known single-minded" valuations. In this work we consider the question of whether revenue monotonic, strategyproof mechanisms for such bidders can be found in the broader class of randomized mechanisms. We demonstrate that-surprisingly-such mechanisms do exist, show how they can be constructed, and consider algorithmic techniques for implementing them in polynomial time.
Near Optimal Non-truthful Auctions
2011
In several e-commerce applications, non-truthful auctions have been preferred over truthful weakly dominant strategy ones partly because of their simplicity and scalability. Although non-truthful auctions can have weaker incentive constraints than truthful ones, the question of how much more revenue they can generate than truthful auctions is not well understood. We study this question for natural and broad classes of non-truthful mechanisms, including quasi-proportional sharing and weakly monotonic auctions. Quasi-proportional sharing mechanisms allocate to each bidder i an amount of resource proportional to a monotonic and concave function f (b i) where b i is the bid of bidder i and ask for a payment of b i. Weakly monotonic auctions refer to a more general class of auctions which satisfy some natural continuity and monotonicity conditions. We prove that although weakly monotonic auctions are much broader and require weaker incentive constraints than dominant strategy auctions, they are not more powerful with respect to the revenue in the setting of selling a single item. Furthermore, we show that quasi-proportional sharing with multiple bidders cannot guarantee a revenue that is larger than the second highest valuation, asymptotically as the number of bidders grows large. However, in a more general single-parameter setting modeled by a downwardclosed set system, a version of the proportional sharing mechanism can obtain a constant factor of the optimal social welfare of the game where the highest valuation is replaced by the second highest valuation, which is essentially the best revenue benchmark in the prior-free framework. This is in sharp contrast to weakly dominant strategy mechanisms that cannot achieve better than log n approximation for this benchmark.
Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions
Proceedings of the 9th ACM conference on Electronic commerce - EC '08, 2008
We provide tight information-theoretic lower bounds for the welfare maximization problem in combinatorial auctions. In this problem the goal is to partition m items between k bidders in a way that maximizes the sum of bidders' values for their allocated items. Bidders have complex preferences over items expressed by valuation functions that assign values to all subsets of items.