A New Method for Finding the Schulze Winner Set

Satoru Fujishige, Leo Goto and Satoshi Nakada

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Abstract: 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.

Date: 2026-06, Revised 2026-06
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