[论文解读] Prophet Secretary

Hossein Esfandiari, MohammadTaghi Hajiaghayi+2 more|arXiv (Cornell University)|Jul 5, 2015

Auction Theory and Applications参考文献 26被引用 6

一句话总结

本文提出了 Prophet Secretary 问题，这是 prophet 不等式与秘书问题的混合模型，其中顾客以随机顺序到达，并从已知分布中提供价格。通过使用 n 个非自适应阈值，作者实现了接近 (1−1/e) ≈ 0.632 的竞争比，优于经典的 0.5 上限，并证明了任何在线算法都无法超过 0.75 的竞争比，显著推进了单件物品顺序报价定价机制的发展。

ABSTRACT

Optimal stopping theory is a powerful tool for analyzing scenarios such as online auctions in which we generally require optimizing an objective function over the space of stopping rules for an allocation process under uncertainty. Perhaps the most classic problems of stopping theory are the prophet inequality problem and the secretary problem. The classical prophet inequality states that by choosing the same threshold OPT/2 for every step, one can achieve the tight competitive ratio of 0.5. On the other hand, for the basic secretary problem, the optimal strategy achieves the tight competitive ratio of 1/e. In this paper, we introduce Prophet Secretary, a natural combination of the prophet inequality and the secretary problems. An example motivation for our problem is as follows. Consider a seller that has an item to sell on the market to a set of arriving customers. The seller knows the types of customers that may be interested in the item and he has a price distribution for each type: the price offered by a customer of a type is anticipated to be drawn from the corresponding distribution. However, the customers arrive in a random order. Upon the arrival of a customer, the seller makes an irrevocable decision whether to sell the item at the offered price. We address the question of finding a strategy for selling the item at a high price. We show that by using a uniform threshold one cannot break the 0.5 barrier. However, we show that i) using n distinct non-adaptive thresholds one can obtain a competitive ratio that goes to (1-1/e) as n grows; and ii) no online algorithm can achieve a competitive ratio better than 0.75. Our results improve the (asymptotic) approximation guarantee of single-item sequential posted pricing mechanisms from 0.5 to (1-1/e) when the order of agents (customers) is chosen randomly.

研究动机与目标

• 建模顾客以随机顺序到达且价格来自已知分布的在线销售场景。
• 分析此混合设置下停止规则的性能，弥合经典 prophet 不等式与秘书问题之间的差距。
• 确定在此新框架下在线算法可达到的最佳竞争比。
• 改进随机顺序设置下单件物品顺序报价定价机制的近似保证。

提出的方法

• 提出一种混合模型，其中参与者以随机顺序到达，每个参与者从已知分布中提供一个价格。
• 分析单一统一阈值的使用，表明其竞争比无法超过 0.5。
• 提出一种使用 n 个不同非自适应阈值的策略，以逐步提升竞争比。
• 推导当 n 增大时的渐近竞争比，表明其收敛于 (1−1/e)。
• 使用概率分析和随机优势论证来界定基于阈值策略的性能边界。
• 证明任何在线算法可达到的竞争比上限为 0.75。

实验结果

研究问题

• RQ1 在 Prophet Secretary 问题中，单一统一阈值策略能否实现优于 0.5 的竞争比？
• RQ2 在 Prophet Secretary 模型中，使用 n 个非自适应阈值时，可达到的渐近竞争比是多少？
• RQ3 在该设定下，(1−1/e) 是否为任何在线算法可达到的最佳竞争比？
• RQ4 与对抗性顺序相比，到达顺序的随机性如何影响报价定价机制的性能？
• RQ5 在 Prophet Secretary 问题中，任何在线算法可达到的竞争比的理论上限是多少？

主要发现

• 在 Prophet Secretary 问题中，使用单一统一阈值无法使竞争比超过 0.5。
• 采用 n 个不同的非自适应阈值可使竞争比随着 n 增大而渐近逼近 (1−1/e) ≈ 0.632。
• 在 n 趋于无穷大时，(1−1/e) 是通过非自适应阈值策略可达到的最佳竞争比。
• 任何在线算法都无法实现超过 0.75 的竞争比，从而确立了理论上限。
• 结果将单件物品顺序报价定价机制在随机顺序到达下的渐近近似保证从 0.5 提升至 (1−1/e)。