An Analysis Framework for Metric Voting based on LP Duality

Distortion-based analysis has established itself as a fruitful framework for comparing voting mechanisms. The assumption is that the m voters and n candidates are jointly embedded in an (unknown) metric space, and the voters submit rankings of candidates by non-decreasing distance from themselves. Based on the submitted rankings, the social choice rule chooses a winning candidate; the quality of the winner is the sum of the (unknown) distances to the voters. Since it is missing the information about the actual distances, the rule's choice will in general be suboptimal, and the worst-case ratio between the cost of its chosen candidate and the optimal candidate is called the rule's distortion. It was shown in prior work that every deterministic rule has distortion at least 3, while the Copeland rule and related rules guarantee distortion at most 5, and a very recent result gave a generalization of Copeland with distortion 2 + √ 5 ≈ 4.236. We provide a framework based on LP-duality and flow interpretations of the dual which provides a simpler and more unified way for proving upper bounds on the distortion of social choice rules. Rather than having to reason about all possible metric spaces, to establish an upper bound, it is sufficient to exhibit a certain type of flow with small cost. We illustrate the utility of this approach with three examples. First, we give a fairly simple proof of a strong generalization of the upper bound of 5 on the distortion of Copeland, to social choice rules with short paths from the winning candidate to the optimal candidate in generalized weak preference graphs. A special case of this result recovers the recent 2 + √ 5 guarantee. Next, we use this generalization to show that the Ranked Pairs and Schulze rules have distortion Θ( √ n). Finally, our framework naturally suggests a combinatorial rule that is a strong candidate for achieving distortion 3, which had also been proposed in recent work. We prove that the distortion bound of 3 would follow from any of three combinatorial conjectures we formulate (and have verified by computer for n ≤ 7 candidates).