[论文解读] Of the People: Voting Is More Effective with Representative Candidates

Yu Cheng, Shaddin Dughmi+1 more|arXiv (Cornell University)|May 4, 2017

Game Theory and Voting Systems参考文献 22被引用 12

一句话总结

本文通过度量空间模型研究了候选人代表性对选举结果的影响，其中选民和候选人被视为空间中的点，偏好基于距离确定。研究发现，当候选人从选民群体中独立同分布（i.i.d.）抽取时，当选者社会成本与最优候选人的社会成本之比（即失真度）从非代表性候选人的3下降至线性空间中的约1.1716，且在一般度量空间中低于2，表明代表性显著提升了选举效率。

ABSTRACT

In light of the classic impossibility results of Arrow and Gibbard and Satterthwaite regarding voting with ordinal rules, there has been recent interest in characterizing how well common voting rules approximate the social optimum. In order to quantify the quality of approximation, it is natural to consider the candidates and voters as embedded within a common metric space, and to ask how much further the chosen candidate is from the population as compared to the socially optimal one. We use this metric preference model to explore a fundamental and timely question: does the social welfare of a population improve when candidates are representative of the population? If so, then by how much, and how does the answer depend on the complexity of the metric space? We restrict attention to the most fundamental and common social choice setting: a population of voters, two independently drawn candidates, and a majority rule election. When candidates are not representative of the population, it is known that the candidate selected by the majority rule can be thrice as far from the population as the socially optimal one. We examine how this ratio improves when candidates are drawn independently from the population of voters. Our results are two-fold: When the metric is a line, the ratio improves from $3$ to $4-2\sqrt{2}$, roughly $1.1716$; this bound is tight. When the metric is arbitrary, we show a lower bound of $1.5$ and a constant upper bound strictly better than $2$ on the approximation ratio of the majority rule. The positive result depends in part on the assumption that candidates are independent and identically distributed. However, we show that independence alone is not enough to achieve the upper bound: even when candidates are drawn independently, if the population of candidates can be different from the voters, then an upper bound of $2$ on the approximation is tight.

研究动机与目标

• 研究在多数规则选举中，当候选人代表选民群体时，社会福利是否以及在多大程度上得到改善。
• 量化当候选人与选民来自同一分布时，多数规则的最坏情况失真度相较于非代表性候选人的情况。
• 在候选人具有代表性假设下，确定线性和一般度量空间中失真度的最紧上界。
• 探讨代表性与非代表性候选人之间在社会福利近似方面的差距。
• 考察当候选人与选民分布不一致时，这些边界的鲁棒性。

提出的方法

• 将选民和候选人建模为空间中的点，偏好由距离决定：距离越近的候选人越受青睐。
• 将失真度定义为当选候选人社会成本与最优候选人成本之比的最坏情况。
• 在候选人从选民分布中独立同分布抽取的两候选人选举设定下分析多数规则。
• 使用几何与概率技术，推导线性空间和一般度量空间中失真度的紧致上界。
• 通过最坏情况构造和对称性论证，证明上下界，尤其在直线情形下利用微积分最小化失真度。
• 将独立同分布候选人抽样结果与候选人与选民分布不同的情形进行比较，揭示性能上的显著差距。

实验结果

研究问题

• RQ1 当候选人从选民群体中抽取时，与任意候选人相比，多数规则的失真度如何变化？
• RQ2 在一般度量空间中，当候选人从选民分布中独立同分布抽取时，失真度的最紧上界是多少？
• RQ3 在代表性候选人假设下，线性空间与一般度量空间中的失真度边界有何不同？
• RQ4 候选人分布不匹配对失真度有何影响？是否存在代表性与非代表性候选人之间的根本性差距？
• RQ5 当候选人具有代表性时，在一般度量空间中，失真度能否被严格小于2的常数所界定？

主要发现

• 在线性空间中，当候选人独立同分布抽取时，多数规则的失真度恰好为 $4 - 2\bar{2} \approx 1.1716$，且该界是紧致的。
• 在一般度量空间中，失真度的上界为严格小于2的常数，下界为1.5。
• 当候选人未从选民分布中抽取时，即使在线性空间中，失真度的上界仍为2，表明代表性与非代表性候选人之间存在根本性差距。
• 对于非代表性候选人，失真度最高可达3，即使在线性空间中也是如此，证实了缺乏代表性时问题的严重性。
• 作者推测，在一般度量空间中，最大失真度为 $3/2$，该推测得到了计算证据以及对均匀和对称度量的局部结果的支持。
• 研究结果为抽签制（lottocracy）等制度提供了理论支持，表明代表性候选人选择可提升系统响应性与社会福利。