4,988 research outputs found

    Optimal Competitive Auctions

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    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)()\mathcal{F}^{(2)}(\cdot) 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 (nn1)n11(\frac{n}{n-1})^{n-1}-1 for each number of buyers n, that is e1e-1 as nn approaches infinity

    Online Independent Set Beyond the Worst-Case: Secretaries, Prophets, and Periods

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    We investigate online algorithms for maximum (weight) independent set on graph classes with bounded inductive independence number like, e.g., interval and disk graphs with applications to, e.g., task scheduling and spectrum allocation. In the online setting, it is assumed that nodes of an unknown graph arrive one by one over time. An online algorithm has to decide whether an arriving node should be included into the independent set. Unfortunately, this natural and practically relevant online problem cannot be studied in a meaningful way within a classical competitive analysis as the competitive ratio on worst-case input sequences is lower bounded by Ω(n)\Omega(n). As a worst-case analysis is pointless, we study online independent set in a stochastic analysis. Instead of focussing on a particular stochastic input model, we present a generic sampling approach that enables us to devise online algorithms achieving performance guarantees for a variety of input models. In particular, our analysis covers stochastic input models like the secretary model, in which an adversarial graph is presented in random order, and the prophet-inequality model, in which a randomly generated graph is presented in adversarial order. Our sampling approach bridges thus between stochastic input models of quite different nature. In addition, we show that our approach can be applied to a practically motivated admission control setting. Our sampling approach yields an online algorithm for maximum independent set with competitive ratio O(ρ2)O(\rho^2) with respect to all of the mentioned stochastic input models. for graph classes with inductive independence number ρ\rho. The approach can be extended towards maximum-weight independent set by losing only a factor of O(logn)O(\log n) in the competitive ratio with nn denoting the (expected) number of nodes

    Improved Online Algorithms for Knapsack and GAP in the Random Order Model

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    The knapsack problem is one of the classical problems in combinatorial optimization: Given a set of items, each specified by its size and profit, the goal is to find a maximum profit packing into a knapsack of bounded capacity. In the online setting, items are revealed one by one and the decision, if the current item is packed or discarded forever, must be done immediately and irrevocably upon arrival. We study the online variant in the random order model where the input sequence is a uniform random permutation of the item set. We develop a randomized (1/6.65)-competitive algorithm for this problem, outperforming the current best algorithm of competitive ratio 1/8.06 [Kesselheim et al. SIAM J. Comp. 47(5)]. Our algorithm is based on two new insights: We introduce a novel algorithmic approach that employs two given algorithms, optimized for restricted item classes, sequentially on the input sequence. In addition, we study and exploit the relationship of the knapsack problem to the 2-secretary problem. The generalized assignment problem (GAP) includes, besides the knapsack problem, several important problems related to scheduling and matching. We show that in the same online setting, applying the proposed sequential approach yields a (1/6.99)-competitive randomized algorithm for GAP. Again, our proposed algorithm outperforms the current best result of competitive ratio 1/8.06 [Kesselheim et al. SIAM J. Comp. 47(5)]

    The Timing of Bid Placement and Extent of Multiple Bidding: An Empirical Investigation Using eBay Online Auctions

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    Online auctions are fast gaining popularity in today's electronic commerce. Relative to offline auctions, there is a greater degree of multiple bidding and late bidding in online auctions, an empirical finding by some recent research. These two behaviors (multiple bidding and late bidding) are of ``strategic'' importance to online auctions and hence important to investigate. In this article we empirically measure the distribution of bid timings and the extent of multiple bidding in a large set of online auctions, using bidder experience as a mediating variable. We use data from the popular auction site \url{www.eBay.com} to investigate more than 10,000 auctions from 15 consumer product categories. We estimate the distribution of late bidding and multiple bidding, which allows us to place these product categories along a continuum of these metrics (the extent of late bidding and the extent of multiple bidding). Interestingly, the results of the analysis distinguish most of the product categories from one another with respect to these metrics, implying that product categories, after controlling for bidder experience, differ in the extent of multiple bidding and late bidding observed in them. We also find a nonmonotonic impact of bidder experience on the timing of bid placements. Experienced bidders are ``more'' active either toward the close of auction or toward the start of auction. The impact of experience on the extent of multiple bidding, though, is monotonic across the auction interval; more experienced bidders tend to indulge ``less'' in multiple bidding.Comment: Published at http://dx.doi.org/10.1214/088342306000000123 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Packing Returning Secretaries

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    We study online secretary problems with returns in combinatorial packing domains with nn candidates that arrive sequentially over time in random order. The goal is to accept a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All 2n2n arrivals occur in random order. We propose a simple 0.5-competitive algorithm that can be combined with arbitrary approximation algorithms for the packing domain, even when the total value of candidates is a subadditive function. For bipartite matching, we obtain an algorithm with competitive ratio at least 0.5721o(1)0.5721 - o(1) for growing nn, and an algorithm with ratio at least 0.54590.5459 for all n1n \ge 1. We extend all algorithms and ratios to k2k \ge 2 arrivals per candidate. In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed (returned into the pool). We mainly focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of Θ(nlogn)\Theta(n \log n) is always sufficient. For matroids, we show that the expected number can be reduced to O(rlog(n/r))O(r \log (n/r)), where rn/2r \le n/2 is the minimum of the ranks of matroid and dual matroid. For bipartite matching, we show a bound of O(rlogn)O(r \log n), where rr is the size of the optimum matching. For general packing, we show a lower bound of Ω(nloglogn)\Omega(n \log \log n), even when the size of the optimum is r=Θ(logn)r = \Theta(\log n).Comment: 23 pages, 5 figure
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