Respecting Linear Orders for Supermajority Rules

We consider linear orders of finite alternatives constructed by aggregating individual preferences. Specifically, we focus on linear orders that respect modified collective preference relations derived from supermajority rules, where modifications are introduced through two procedures if cycles occur. One procedure utilizes the transitive closure, while the other employs the Suzumura-consistent closure, ensuring the elimination of cycles through consistency adjustments. We derive two sets of linear orders that respect these modified collective preference relations derived from all supermajority rules, and show that these sets are generally nonempty. We show that these sets of linear orders closely relate to those obtained by the ranked pairs method and the Schulze method, thereby providing new insights into these influential methods. Finally, we show that any linear order belonging to either set satisfies two important properties: the extended Condorcet criterion and the Pareto principle.