In light of the classic impossibility results of Arrow and Gibbard and Sa erthwaite 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 a ention to the most fundamental and common social choice se ing: a population of voters, two 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; this holds even when the underlying metric is a line. 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 √ 2) ≈ 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 be er than 2 on the distortion of majority rule. e aforementioned positive results depend in part on the assumption that the two candidates are independently and identically distributed. However, we show that i.i.d. candidates do not su ce for our upper bounds: if the population of candidates can be di erent from that of voters, an upper bound of 2 on the distortion is tight for both general metric spaces and the line. us, we show a constant gap between representative and non-representative candidates in both cases.
Cheng et al. REF restricted their attention to the distribution of voters and candidates and justified that the expected distortion is small when candidates are drawn from the same distribution of voters.
Of the People: Voting Is More Effective with Representative Candidates
EC '17
Mathematics Computer Science