Generalized Veto Core and a Practical Voting Rule with Optimal Metric Distortion
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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 Plurality Matching. The rule picks an arbitrary candidate for whom a certain candidate-specific bipartite graph contains a perfect matching, and thus, it is not neutral (i.e, symmetric with respect to candidates). Subsequently, a much simpler rule called Plurality Veto was shown to achieve distortion 3 as well. This rule only constructs such a matching implicitly but the winner depends on the order that voters are processed, and thus, it is not anonymous (i.e., symmetric with respect to voters). We provide an intuitive interpretation of this matching by generalizing the classical notion of the (proportional) veto core in social choice theory. This interpretation 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 leading to different sets of candidates as winners. For a broad generalization of the veto core, we show that the generalized veto core is equal to the set of candidates who can emerge as winners under a natural class of matching algorithms reminiscent of Serial Dictatorship. Extending these matching algorithms into continuous time, we 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, this rule satisfies many other axioms desirable in practice.
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