arXiv 1705.017362017-05cs.GTcs.AI

人民：有代表性的候选人投票更有效

Of the People: Voting Is More Effective with Representative Candidates

University of Southern California（南加州大学）

Yu Cheng, Shaddin Dughmi, David Kempe

摘要：

鉴于阿罗和Gibbard和Satterthwaite关于顺序规则投票的经典不可能性结果，最近人们对描述普通投票规则如何接近社会最优状态感兴趣。为了量化近似的质量，很自然地将候选人和投票人视为嵌入在一个公共度量空间中，并询问与社会最优候选人相比，所选候选人离总体有多远。我们使用这个度量偏好模型来探索一个基本而及时的问题：当候选人代表人口时，人口的社会福利是否会改善？如果是这样，那么这个问题的答案取决于度量空间的复杂性，取决于多少？我们将注意力限制在最基本和最常见的社会选择设置上：一群选民，两个独立的候选人，以及多数人统治的选举。当候选人不代表人口时，已知由多数规则选择的候选人可以是社会最优候选人的三倍。我们研究了当候选人独立于选民群体时，这一比例如何提高。我们的结果是双重的：当度量是一条线时，比率从$3$提高到$4-2\sqrt{2}$，大致为$1.1716$ ;这个界限是紧的。当度量为任意时，我们给出了$1.5$的一个下界和一个常数上界，在多数规则的逼近比上严格优于$2$.肯定的结果部分取决于候选人是独立同分布的假设。然而，我们表明，独立性本身是不够的，以实现上限：即使当候选人是独立绘制的，如果候选人的人口可以从选民不同，然后上界的$2$的近似是紧的。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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