A New Method for Finding the Schulze Winner Set
2026-06-01 • Discrete Mathematics
Discrete Mathematics
AI summaryⓘ
The authors introduce a new voting method that looks at how candidates compare to each other one-on-one, based on voter preferences. They prove that their method finds the same winners as a well-known rule called the Schulze rule. Their approach also breaks down why these winners relate to another concept called the Schwartz set, giving a clearer math explanation of how these ideas connect. Overall, they provide new insights into how to fairly choose winners when preferences cycle in complicated ways.
pairwise majorityvoting algorithmSchulze rulepreference profilesCondorcet methodpreference cyclesSchwartz setwinner setelimination method
Authors
Satoru Fujishige, Leo Goto, Satoshi Nakada
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.