Generalized Veto Core and a Practical Voting Rule with Optimal Metric Distortion

We revisit the recent breakthrough result of Gkatzelis et al. on (single-winner) metric voting, which showed that the optimal distortion of 3 can be achieved by a mechanism called PluralityMatching. The rule picks an arbitrary candidate for whom a certain candidate-specific bipartite graph contains a perfect matching. Subsequently, a much simpler rule called PluralityVeto was shown to achieve distortion 3 as well; this rule only constructs such a matching implicitly, but it, too, makes some arbitrary decisions affecting the outcome. Our point of departure is the question whether there is an intuitive interpretation of this matching, with the goal of identifying the underlying source of arbitrariness in these rules. We first observe directly from Hall's condition that a matching for candidate c certifies that there is no coalition of voters that can jointly counterbalance the number of first-place votes c received, along with the first-place votes of all candidates ranked lower than c by any voter in this coalition. This condition closely mirrors the classical definition of the (proportional) veto core in social choice theory, except that coalitions can veto a candidate c whenever their size exceeds the plurality score of c, rather than the average number of voters per candidate. Based on this connection, we define a general notion of the veto core with arbitrary weights for voters and candidates which respectively represent the veto power and the public support they have. This connection opens up a number of immediate consequences. Previous methods for electing a candidate from the veto core can be interpreted simply as matching algorithms. Different election methods realize different matchings, in turn witnessing different sets of candidates as winners. Viewed through this lens, we first resolve nontrivial tie breaking issues contributing to the inherent arbitrariness of the above rules. Our approach to ties reveals a novel characterization of the (general) veto core, showing it to be identical to the set of candidates who can emerge as winners under a natural class of matching algorithms reminiscent of SerialDictatorship. Then, we extend this class of voting rules into continuous time, and obtain a highly practical voting rule with optimal distortion 3, which is also intuitive and easy to explain: Each candidate starts off with public support equal to his plurality score. From time 0 to 1, every voter continuously brings down, at rate 1, the support of her bottom choice among not-yet-eliminated candidates. A candidate is eliminated if he is opposed by a voter after his support reaches 0. On top of being anonymous and neutral, the absence of arbitrary non-deterministic choices in this rule allows for the study of other axiomatic properties that are desirable in practice. We show that the canonical voting rule we propose for electing from the (plurality) veto core also satisfies resolvability, monotonicity, majority, majority loser, mutual majority and reversal symmetry, in addition to still guaranteeing metric distortion 3.