Approximation algorithms and online mechanisms for item pricing (original) (raw)
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Optimal bidding in online auctions
Journal of Revenue and Pricing Management, 2009
On-line auctions are arguably one of the most important and distinctly new applications of the Internet. The predominant p l a yer in on-line auctions, eBay, has over 42 million users, and it was the host of over $9.3 billion worth of goods sold in the year 2001. Using methods from approximate dynamic programming and integer programming, we design algorithms for optimally bidding for a single item in an on-line auction, and in simultaneous or overlapping multiple on-line auctions. We report computational evidence using data from eBay's web site from 1772 completed auctions for personal digital assistants and from 4208 completed auctions for stamp collections that shows that (a) the optimal dynamic policy outperforms simple but widely used static heuristic rules for a single auction, and (b) a new approach f o r t h e m ultiple auctions problem that uses the value functions of single auctions found by dynamic programming in an integer programming framework produces high quality solutions fast and reliably.
Competitive Algorithms for Online Pricing
Lecture Notes in Computer Science, 2011
Given a seller with m amount of items, a sequence of users {u1, u2, ...} come one by one, the seller must set the unit price and assign some amount of items to each user on his/her arrival. Items can be sold fractionally. Each ui has his/her value function vi(•) such that vi(x) is the highest unit price ui is willing to pay for x items. The objective is to maximize the revenue by setting the price and amount of items for each user. In this paper, we have the following contributions: if the highest value h among all vi(x) is known in advance, we first show the lower bound of the competitive ratio is O(log h), then give an online algorithm with competitive ratio O(log h); if h is not known in advance, we give an online algorithm with competitive ratio O(h 3 log −1/2 h).
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.
An efficient approximation algorithm for combinatorial auctions
2002
Abstract We present a mathematical programming approximation approach to the winner determination problem for multi-round combinatorial auctions. The winner determination problem is a set packing problem, and hence NP-complete. Most methods developed recently rely on exhaustive search based methods which are exponential in worst case time complexity.
Seller-Focused Algorithms for Online Auctioning
Lecture Notes in Computer Science, 2001
In this paper we provide an algorithmic approach to the study of online auctioning. From the perspective of the seller we formalize the auctioning problem as that of designing an algorithmic strategy that fairly maximizes the revenue earned by selling¨identical items to bidders who submit bids online. We give a randomized online algorithm that is © !-competitive against an oblivious adversary, where the bid values vary between " and per item. We show that this algorithm is optimal in the worst-case and that it performs significantly better than any worst-case bounds achievable via deterministic strategies. Additionally we present experimental evidence to show that our algorithm outperforms conventional heuristic methods in practice. And finally we explore ways of modifying the conventional model of online algorithms to improve competitiveness of other types of auctioning scenarios while still maintaining fairness.
Incentives in Online Auctions via Linear Programming
Lecture Notes in Computer Science, 2010
Online auctions in which items are sold in an online fashion with little knowledge about future bids are common in the internet environment. We study here a problem in which an auctioneer would like to sell a single item, say a car. A bidder may make a bid for the item at any time but expects an immediate irrevocable decision. The goal of the auctioneer is to maximize her revenue in this uncertain environment. Under some reasonable assumptions, it has been observed that the online auction problem has strong connections to the classical secretary problem in which an employer would like to choose the best candidate among n competing candidates [HKP04]. However, a direct application of the algorithms for the secretary problem to online auctions leads to undesirable consequences since these algorithms do not give a fair chance to every candidate and candidates arriving early in the process have an incentive to delay their arrival. In this work we study the issue of incentives in the online auction problem where bidders are allowed to change their arrival time if it benefits them. We derive incentive compatible mechanisms where the best strategy for each bidder is to first truthfully arrive at their assigned time and then truthfully reveal their valuation. Using the linear programming technique introduced in Buchbinder et al [BJS10], we first develop new mechanisms for a variant of the secretary problem. We then show that the new mechanisms for the secretary problem can be used as a building block for a family of incentive compatible mechanisms for the online auction problem which perform well under different performance criteria. In particular, we design a mechanism for the online auction problem which is incentive compatible and is 3/16 ≈ 0.187-competitive for revenue, and a (different) mechanism that is 1 2 √ e ≈ 0.303-competitive for efficiency. ⋆ This work was done while at Microsoft Research, New England. ⋆⋆ Part of the work done while at Microsoft Research, New England. 2 √ e ≈ 0.303-competitive for efficiency.
Pricing combinatorial auctions
European Journal of Operational Research, 2004
Single-item auctions have many desirable properties. Mechanisms exist to ensure optimality, incentive compatibility and market-clearing prices. When multiple items are offered through individual auctions, a bidder wanting a bundle of items faces an exposure problem if the bidder places a high value on a combination of goods but a low value on strict subsets of the desired collection. To remedy this, combinatorial auctions permit bids on bundles of goods. However, combinatorial auctions are hard to optimize and may not have incentive compatible mechanisms or market-clearing individual item prices. Several papers give approaches to provide incentive compatibility and imputed, individual prices. We find the relationships between these approaches and analyze their advantages and disadvantages.
An Approximate Truthful Mechanism for Combinatorial Auctions with Single Parameter Agents
Internet Mathematics, 2004
Mechanism design seeks algorithms whose inputs are provided by selfish agents who would lie if it were to their advantage. Incentive-compatible mechanisms c o m p e lt h ea g e n t st ot e l lt h et r u t hb ym a k i n gi ti nt h e i rs e l f -i n t e r e s tt od os o . O ften, as in combinatorial auctions, such mechanisms involve the solution of NP-hard problems. Unfortunately, approximation algorithms typically destroy incentive compatibility. Randomized rounding is a commonly used technique for designing approximation algorithms. We devise a version of randomized rounding that is incentivecompatible, giving a truthful mechanism for combinatorial auctions with single parameter agents (e.g., "single minded bidders") that approximately maximizes the social value of the auction. We discuss two orthogonal notions of truthfulness for a randomized mechanism-truthfulness with high probability and in expectation-and give a mechanism that achieves both simultaneously.