[论文解读] Algorithms as Mechanisms: The Price of Anarchy of Relax-and-Round

Paul Dütting, Thomas Keßelheim+1 more|arXiv (Cornell University)|Nov 30, 2015

Auction Theory and Applications被引用 1

一句话总结

该论文证明，当与支付报价定价机制结合时，具有无意识舍入的松弛-舍入算法可产生具有低纳什均衡失效率的机制。若松弛问题通过平滑性分析具有失效率界β，且舍入过程为α-近似且无意识，则所得机制的失效率为O(αβ)，将先前仅限于贪心算法的结果推广至更广泛的近似算法类别。

ABSTRACT

Many algorithms that are originally designed without explicitly considering incentive properties are later combined with simple pricing rules and used as mechanisms. The resulting mechanisms are often natural and simple to understand. But how good are these algorithms as mechanisms? Truthful reporting of valuations is typically not a dominant strategy (certainly not with a pay-your-bid, first-price rule, but it is likely not a good strategy even with a critical value, or second-price style rule either). Our goal is to show that a wide class of approximation algorithms yields this way mechanisms with low Price of Anarchy. The seminal result of Lucier and Borodin [SODA 2010] shows that combining a greedy algorithm that is an $\alpha$-approximation algorithm with a pay-your-bid payment rule yields a mechanism whose Price of Anarchy is $O(\alpha)$. In this paper we significantly extend the class of algorithms for which such a result is available by showing that this close connection between approximation ratio on the one hand and Price of Anarchy on the other also holds for the design principle of relaxation and rounding provided that the relaxation is smooth and the rounding is oblivious. We demonstrate the far-reaching consequences of our result by showing its implications for sparse packing integer programs, such as multi-unit auctions and generalized matching, for the maximum traveling salesman problem, for combinatorial auctions, and for single source unsplittable flow problems. In all these problems our approach leads to novel simple, near-optimal mechanisms whose Price of Anarchy either matches or beats the performance guarantees of known mechanisms.

研究动机与目标

• 识别能自然产生战略环境中低失效率机制的算法设计原则。
• 将近似比与失效率之间已知的关联关系——此前仅限于贪心算法——扩展至松弛-舍入范式。
• 证明当与平滑松弛结合时，无意识舍入可保持失效率保证。
• 为设计适用于组合拍卖和整数规划打包问题等多种问题的简单且近似最优机制，提供一个通用框架。
• 表明这些机制在贝叶斯均衡和基于学习的均衡下仍能保持强劲性能。

提出的方法

• 利用Roughgarden和Syrgkanis-Tardos提出的平滑性框架，通过(λ, µ)-平滑性分析机制。
• 引入无意识舍入的概念——即舍入方案不依赖于目标函数——从而实现跨问题的泛化。
• 将该框架应用于组合问题的松弛（如整数规划、旅行商问题、流问题），证明在温和条件下具有平滑性。
• 采用支付报价定价规则，并证明平滑性可保证失效率有界，即使在不完全信息环境下亦成立。
• 将结果扩展至弱平滑性情形，适用于非负且受出价限制的支付，并适用于近似松弛求解器。
• 应用Carr与Vempala提出的随机元舍入方法，从非无意识舍入方案推导出无意识方案，从而扩大适用范围。

实验结果

研究问题

• RQ1 近似比与失效率之间的关联关系能否从贪心算法推广至松弛-舍入算法？
• RQ2 在何种条件下，无意识舍入可保持机制设计中的失效率保证？
• RQ3 当松弛问题具有平滑性，且舍入过程为α-近似时，是否可得到失效率为O(αβ)的机制？
• RQ4 该框架能否应用于多单位拍卖、广义匹配和单源流问题等，实现可证明的性能？
• RQ5 由近似求解松弛问题导出的机制的失效率是多少，特别是当松弛求解器本身为诚实机制时？

主要发现

• 采用α-近似无意识舍入方案与平滑松弛的松弛-舍入机制，其失效率为O(αβ)，其中β为松弛问题的失效率。
• 该结果不仅适用于纯纳什均衡，也适用于粗相关均衡和贝叶斯均衡。
• 在多单位拍卖和广义匹配问题中，该方法可导出新颖机制，其失效率与已知界持平或更优。
• 在最大旅行商问题中，该方法首次提供了非平凡的失效率保证。
• 当使用O(log n + log L)-近似求解器处理打包线性规划时，组合机制的失效率为O(α(log n + log L))。
• 该框架可推广至非精确松弛求解器及任意非负且受出价限制的支付，弱平滑性下失效率界为(1 + µ)/λ。