[论文解读] Sequential Posted Price Mechanisms with Correlated Valuations

Marek Adamczyk, Allan Borodin+3 more|arXiv (Cornell University)|Mar 7, 2015

Auction Theory and Applications参考文献 30被引用 9

一句话总结

本文研究在买方估值存在相关性的情境下的顺序出价机制（SPPMs），表明当估值相关时，标准SPPMs仅能提取最优收益的可忽略部分。本文提出增强型SPPMs，通过向部分买方探查估值并基于其他买方的估值设定价格，实现了最优收益的Ω(1/d)比例，其中d衡量估值依赖程度。

ABSTRACT

We study the revenue performance of sequential posted price mechanisms and some natural extensions, for a general setting where the valuations of the buyers are drawn from a correlated distribution. Sequential posted price mechanisms are conceptually simple mechanisms that work by proposing a take-it-or-leave-it offer to each buyer. We apply sequential posted price mechanisms to single-parameter multi-unit settings in which each buyer demands only one item and the mechanism can assign the service to at most k of the buyers. For standard sequential posted price mechanisms, we prove that with the valuation distribution having finite support, no sequential posted price mechanism can extract a constant fraction of the optimal expected revenue, even with unlimited supply. We extend this result to the the case of a continuous valuation distribution when various standard assumptions hold simultaneously. In fact, it turns out that the best fraction of the optimal revenue that is extractable by a sequential posted price mechanism is proportional to ratio of the highest and lowest possible valuation. We prove that for two simple generalizations of these mechanisms, a better revenue performance can be achieved: if the sequential posted price mechanism has for each buyer the option of either proposing an offer or asking the buyer for its valuation, then a Omega(1/max{1,d}) fraction of the optimal revenue can be extracted, where d denotes the degree of dependence of the valuations, ranging from complete independence (d=0) to arbitrary dependence (d=n-1). Moreover, when we generalize the sequential posted price mechanisms further, such that the mechanism has the ability to make a take-it-or-leave-it offer to the i-th buyer that depends on the valuations of all buyers except i's, we prove that a constant fraction (2-sqrt{e})/4~0.088 of the optimal revenue can be always be extracted.

研究动机与目标

• 评估当买方估值任意相关时，顺序出价机制（SPPMs）的收益表现。
• 确定在估值相关的情况下，SPPMs是否能实现最优收益的常数比例，特别是当估值存在依赖时。
• 设计增强型SPPM变体，通过整合部分买方的估值信息来提升收益。
• 量化在估值相关环境中，探查成本与收益增益之间的权衡。
• 探讨随机化在SPPM设计中的必要性，以及无序机制的潜在可能性。

提出的方法

• 提出一个广义的SPPM框架，机制可选择向买方提供“要么接受要么放弃”的价格，或要求买方提供其估值。
• 引入买方的随机划分，分为探查组与出价组，其中出价组中每位买方获得的价格取决于探查组的估值。
• 使用概率分析，以估值依赖程度d为参数，界定增强机制的期望收益。
• 应用集中与期望论证，证明期望收益至少为最优机制收益的(1−q)q^d倍。
• 利用(1−1/d)^d随d增加而增大，并在d→∞时趋近于1/e的性质，优化探查概率q。
• 证明当d≥2时，设定q=1−1/d可获得近似比(1−1/d)^d，该值在d=2时至少为1/4，并随d增大趋近于1/e。

实验结果

研究问题

• RQ1 当估值相关时，标准顺序出价机制能否提取最优收益的常数比例？
• RQ2 在估值相关条件下，任何顺序出价机制所能实现的最佳收益近似比是多少？
• RQ3 估值之间依赖程度d的高低如何影响增强型SPPMs的收益表现？
• RQ4 为实现O(1/d)的收益近似保证，探查过程是否必须依赖随机化？
• RQ5 无序SPPMs是否能在不控制买方选择顺序的情况下，实现类似的收益保证？

主要发现

• 当估值相关时，标准顺序出价机制即使在无限供应条件下，也无法提取最优收益的常数比例。
• 对于连续、正则、MHR及依存分布，可提取的最优收益比例与最高估值和最低估值之比成正比。
• 一种探查d名买方并基于其估值设定价格的增强型SPPM，可实现最优收益的Ω(1/d)比例。
• 当d=0（估值独立）时，机制可实现α-近似，其中α为原始机制的近似比。
• 当d≥2时，设定探查概率q=1−1/d可获得(1−1/d)^d的收益保证，该值在d=2时至少为1/4，并随d增大趋近于1/e≈0.368。
• 当价格基于所有其他买方估值设定时，即使在最坏的相关分布下，机制仍能实现最优收益的常数比例——具体为(2−√e)/4≈0.088。