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arXiv 1509.09147cs.GTcs.DS
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基于价格外部性的组合拍卖

Combinatorial Auctions with Conflict-Based Externalities

Yun Kuen Cheung, Monika Henzinger, Martin Hoefer, Martin Starnberger

摘要:

组合拍卖是机制设计中的一个研究热点。然而,与标准模式相反,实证研究表明,投标人的估价往往不仅仅取决于分配给他的货物。例如,在adwords拍卖中,广告客户可能不希望自己的广告显示在竞争对手的广告旁边。在本文中,我们提出并分析了几个自然图论模型,将这种负外部性,投标人形成一个有向冲突图的最大出度$Δ$。我们设计的算法和真实的社会福利最大化的机制,达到近似比率取决于$Δ$。对于CA,我们的结果是双重的:(1)通过丢弃独立于投标的投标人/项目来消除冲突的彩票。它允许应用任何真实的$α$ -近似机制进行无冲突的估值,并产生一个$\mathcal{O}(αΔ)$ -近似机制。(2)对于分数次可加性赋值,我们设计了一个舍入算法,通过一个新的组合的半定规划和线性规划,从而在一个锥规划,逼近比是$\mathcal{O}((Δ\log \log Δ)/\log Δ)$。在现有硬度结果的情况下,该比率几乎是最佳的。对于adwords拍卖的突出应用,我们提出了几种算法,最相关的情况下,项目的数量是小的。特别地,我们设计了一个真实的机制与近似比$o(Δ)$时,项目的数量只是对数的投标人的数量。

英文摘要:

Combinatorial auctions (CA) are a well-studied area in algorithmic mechanism design. However, contrary to the standard model, empirical studies suggest that a bidder's valuation often does not depend solely on the goods assigned to him. For instance, in adwords auctions an advertiser might not want his ads to be displayed next to his competitors' ads. In this paper, we propose and analyze several natural graph-theoretic models that incorporate such negative externalities, in which bidders form a directed conflict graph with maximum out-degree $Δ$. We design algorithms and truthful mechanisms for social welfare maximization that attain approximation ratios depending on $Δ$. For CA, our results are twofold: (1) A lottery that eliminates conflicts by discarding bidders/items independent of the bids. It allows to apply any truthful $α$-approximation mechanism for conflict-free valuations and yields an $\mathcal{O}(αΔ)$-approximation mechanism. (2) For fractionally sub-additive valuations, we design a rounding algorithm via a novel combination of a semi-definite program and a linear program, resulting in a cone program; the approximation ratio is $\mathcal{O}((Δ\log \log Δ)/\log Δ)$. The ratios are almost optimal given existing hardness results. For the prominent application of adwords auctions, we present several algorithms for the most relevant scenario when the number of items is small. In particular, we design a truthful mechanism with approximation ratio $o(Δ)$ when the number of items is only logarithmic in the number of bidders.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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