In this paper, we study the metric distortion of deterministic social choice
rules that choose a winning candidate from a set of candidates based on voter
preferences. Voters and candidates are located in an underlying metric space. A
voter has cost equal to her distance to the winning candidate. Ordinal social
choice rules only have access to the ordinal preferences of the voters that are
assumed to be consistent with the metric distances. Our goal is to design an
ordinal social choice rule with minimum distortion, which is the worst-case
ratio, over all consistent metrics, between the social cost of the rule and
that of the optimal omniscient rule with knowledge of the underlying metric
space.
The distortion of the best deterministic social choice rule was known to be
between 3 and 5. It had been conjectured that any rule that only looks at
the weighted tournament graph on the candidates cannot have distortion better
than 5. In our paper, we disprove it by presenting a weighted tournament rule
with distortion of 4.236. We design this rule by generalizing the classic
notion of uncovered sets, and further show that this class of rules cannot have
distortion better than 4.236. We then propose a new voting rule, via an
alternative generalization of uncovered sets. We show that if a candidate
satisfying the criterion of this voting rule exists, then choosing such a
candidate yields a distortion bound of 3, matching the lower bound. We
present a combinatorial conjecture that implies distortion of 3, and verify
it for small numbers of candidates and voters by computer experiments. Using
our framework, we also show that selecting any candidate guarantees distortion
of at most 3 when the weighted tournament graph is cyclically symmetric.Comment: EC 201