Respecting linear orders for supermajority rules
研究了如何将个体偏好聚合成无平局的集体排序，基于超多数规则构建线性序，并统一了Schulze、Split Cycle和Ranked Pairs等投票方法。
We study the problem of aggregating individual preferences into a collective ranking without ties (a linear order) over a finite set of alternatives. We introduce a framework that constructs such rankings from supermajority rules, defined by a threshold parameter $$\alpha \in \left[ 1/2,1\right) $$ . For each $$\alpha $$ , we consider the binary relation of defeats from pairwise comparisons and apply two closure operations—the transitive closure and the Suzumura-consistent closure—to eliminate cycles. We then study the sets of linear orders that respect these relations; that is, linear orders that preserve all strict defeats while extending them to a complete ranking. To avoid the arbitrariness of fixing a single $$\alpha $$ , we take the intersection across all $$\alpha $$ . This yields two sets of linear orders: the T-order set (from the transitive closure) and the S-order set (from the Suzumura-consistent closure), both always nonempty, the former is a subset of the latter, and jointly capturing the full spectrum of supermajority rules. We show that this perspective unifies prominent methods: the Schulze method coincides with the T-order set, the Split Cycle method with the S-order set, and every outcome of the Ranked Pairs method lies within the S-order set. Moreover, every linear order in the S-order set satisfies the extended Condorcet criterion and the strong Pareto principle. Our results thus place these methods within a common supermajority-based framework and clarify their relationships.