We examine strategy-proof elections to select a winner amongst a set of
agents, each of whom cares only about winning. This impartial selection problem
was introduced independently by Holzman and Moulin and Alon et al. Fisher and
Klimm showed that the permutation mechanism is impartial and 1/2-optimal,
that is, it selects an agent who gains, in expectation, at least half the
number of votes of most popular agent. Furthermore, they showed the mechanism
is 7/12-optimal if agents cannot abstain in the election. We show that a
better guarantee is possible, provided the most popular agent receives at least
a large enough, but constant, number of votes. Specifically, we prove that, for
any ϵ>0, there is a constant Nϵ​ (independent of the number
n of voters) such that, if the maximum number of votes of the most popular
agent is at least Nϵ​ then the permutation mechanism is
(43​−ϵ)-optimal. This result is tight.
Furthermore, in our main result, we prove that near-optimal impartial
mechanisms exist. In particular, there is an impartial mechanism that is
(1−ϵ)-optimal, for any ϵ>0, provided that the maximum number
of votes of the most popular agent is at least a constant Mϵ​