Title: Of the People: Voting Is More Effective with Representative Candidates,
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.