[论文解读] Metric Distortion of Social Choice Rules: Lower Bounds and Fairness Properties

Ashish Goel, A. Krishnaswamy+1 more|arXiv (Cornell University)|Dec 9, 2016

Game Theory and Voting Systems参考文献 26被引用 25

一句话总结

本文通过证明Ranked Pairs的最坏情况扭曲度至少为5，解决了社会选择规则度量扭曲度的开放性问题，推翻了此前猜想的3的上界。此外，本文表明任何随机化锦标赛规则都无法实现低于3的期望最坏情况扭曲度，并确立了在近似主要化下，Copeland和Randomized Dictatorship的常数因子公平比率分别为5和3。

ABSTRACT

We study social choice rules under the utilitarian distortion framework, with an additional metric assumption on the agents' costs over the alternatives. In this approach, these costs are given by an underlying metric on the set of all agents plus alternatives. Social choice rules have access to only the ordinal preferences of agents but not the latent cardinal costs that induce them. Distortion is then defined as the ratio between the social cost (typically the sum of agent costs) of the alternative chosen by the mechanism at hand, and that of the optimal alternative chosen by an omniscient algorithm. The worst-case distortion of a social choice rule is, therefore, a measure of how close it always gets to the optimal alternative without any knowledge of the underlying costs. Under this model, it has been conjectured that Ranked Pairs, the well-known weighted-tournament rule, achieves a distortion of at most 3 [Anshelevich et al. 2015]. We disprove this conjecture by constructing a sequence of instances which shows that the worst-case distortion of Ranked Pairs is at least 5. Our lower bound on the worst case distortion of Ranked Pairs matches a previously known upper bound for the Copeland rule, proving that in the worst case, the simpler Copeland rule is at least as good as Ranked Pairs. And as long as we are limited to (weighted or unweighted) tournament rules, we demonstrate that randomization cannot help achieve an expected worst-case distortion of less than 3. Using the concept of approximate majorization within the distortion framework, we prove that Copeland and Randomized Dictatorship achieve low constant factor fairness-ratios (5 and 3 respectively), which is a considerable generalization of similar results for the sum of costs and single largest cost objectives. In addition to all of the above, we outline several interesting directions for further research in this space.

研究动机与目标

• 解决关于Ranked Pairs在度量偏好下最坏情况扭曲度不超过3的开放猜想。
• 确定随机化锦标赛规则是否能在期望最坏情况扭曲度方面优于Randomized Dictatorship。
• 为在扭曲模型下分析社会选择中的公平性（特别是非求和型成本目标）建立一个框架。
• 利用近似主要化方法，为Copeland和Randomized Dictatorship等知名机制建立常数因子公平比率。

提出的方法

• 通过构造一系列度量实例，使用复杂的几何与组合论证，证明Ranked Pairs的最坏情况扭曲度至少为5。
• 应用近似主要化概念，定义并分析各类成本函数下社会选择规则的公平比率。
• 通过利用三角不等式和度量空间中首选人选集合的性质，推导出公平比率的上界。
• 使用概率分析和期望值边界，比较不同成本目标下Randomized Dictatorship等随机化规则的性能。
• 通过将随机化独裁者规则的公平比率上限控制在3以内，证明其性能优越，即相对于最优备选项，其期望前k项成本之和的比值有界。
• 通过展示随机化独裁者规则在凸成本函数（如成本之和的平方）下扭曲度无界，凸显Copeland规则性能的稳健性。

实验结果

研究问题

• RQ1 Ranked Pairs在度量扭曲框架下是否实现最坏情况扭曲度不超过3？
• RQ2 随机化锦标赛规则能否实现低于3的期望最坏情况扭曲度？
• RQ3 社会选择中的公平性是否可以超越成本之和进行有意义的量化？如果是，如何量化？
• RQ4 像Copeland和Randomized Dictatorship这样的机制是否在广泛的成本函数类上保持良好性能？
• RQ5 是否存在随机化锦标赛规则期望最坏情况扭曲度的根本限制？

主要发现

• Ranked Pairs的最坏情况扭曲度至少为5，推翻了此前猜想的3的上界。
• 任何随机化锦标赛规则都无法实现低于3的期望最坏情况扭曲度，为该类机制确立了下界。
• 在近似主要化下，Copeland的公平比率至多为5，将公平性推广至非求和型成本目标。
• Randomized Dictatorship的公平比率至多为3，表明其在多种成本函数下均表现优异。
• Randomized Dictatorship在凸成本函数（如成本之和的平方）下扭曲度无界，凸显该规则的局限性。
• Copeland的性能具有鲁棒性，既达到了目前已知的确定性扭曲度最优边界（5），又在广义成本函数类上保持了强公平比率。