Econ Papers Daily / 期刊论文
尊重超多数规则下的线性顺序
Respecting linear orders for supermajority rules
摘要
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.
中文摘要
本文研究如何将个体偏好聚合为有限备选方案集合上的无并列集体排序（线性序）。我们提出一个基于超多数规则构建此类排序的框架，该规则由阈值参数 $$\alpha \in \left[ 1/2,1\right) $$ 定义。对于每个 $$\alpha $$，我们考虑两两比较中产生的击败二元关系，并应用两种闭包运算——传递闭包和铃村一致闭包——以消除循环。随后，我们研究尊重这些关系的线性序集合；即，在保留所有严格击败关系的同时将其扩展为完整排序的线性序。为避免固定单一 $$\alpha $$ 的任意性，我们取所有 $$\alpha $$ 的交集。由此得到两组线性序：T-序集（来自传递闭包）和S-序集（来自铃村一致闭包），两者均非空，前者为后者的子集，且共同涵盖超多数规则的完整谱系。我们证明，这一视角统一了多种主流方法：Schulze方法等同于T-序集，Split Cycle方法等同于S-序集，而Ranked Pairs方法的每个结果均属于S-序集。此外，S-序集中的每个线性序均满足扩展的孔多塞准则与强帕累托原则。因此，我们的研究将这些方法置于统一的超多数规则框架下，并厘清了它们之间的关联。