A New Method for Finding the Schulze Winner Set
For social choice theorists, this provides a new algorithmic perspective and formal foundation for the Schulze winner set, but the result is primarily theoretical and incremental.
The paper proposes a new voting algorithm that computes the Schulze winner set by successively eliminating weaker candidates based on pairwise comparisons, and proves it is equivalent to the Schulze rule. It also shows the direct sum of survival sets equals the Schwartz set, formalizing the relationship between the two sets.
We propose a new voting algorithm based on the pairwise majority-comparison matrix derived from voters' preference profiles. We show that this algorithm induces exactly the winner set of the Schulze rule (Schulze, 1997). Our algorithm successively eliminates weaker candidates in terms of all-pairs comparisons, thereby reflecting a dual spirit to Condorcet's original idea of splitting preference cycles (de Condorcet, 1785). We further show that the direct sum of the survival sets obtained at each elimination round coincides with the Schwartz set (Schwartz, 1972). These two equivalence results provide a formal mathematical foundation for the ``folklore'' relationship between the Schulze winner set and the Schwartz set, as well as a new Condorcetian interpretation of the Schulze winner set.