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| Schulze method | |
|---|---|
| Name | Schulze method |
| Type | Voting system |
| Inventor | Markus Schulze |
| Introduced | 1997 |
| Criterion | Condorcet, Smith |
| Usage | Organizations, political groups, private elections |
Schulze method
The Schulze method is a Condorcet-compliant single-winner and multiwinner voting procedure designed to select a most-preferred candidate in elections by comparing pairwise preferences among alternatives. It was developed to satisfy several voting criteria and to resist strategic anomalies observed in methods such as First-past-the-post, Instant-runoff voting, Borda count and plurality systems used in jurisdictions like United States, United Kingdom, Germany, Australia and organizations like Wikipedia and the Free Software Foundation. The method has been adopted by associations, political parties, and professional bodies seeking a Condorcet-consistent mechanism distinct from runoff or scoring systems.
The Schulze method arose in the late 1990s as part of a broader movement among electoral reformers, theorists, and practitioners associated with figures and institutions such as Condorcet, Kenneth Arrow, Donald G. Saari, Tideman, Arrow's impossibility theorem, Paul N. Cohen and reform efforts linked to entities like FairVote and Electoral Reform Society. It builds on pairwise majority comparisons explored in classical texts like The Federalist Papers debates and later formal treatments by scholars at universities including University of Oxford, Harvard University, Massachusetts Institute of Technology, University of Cambridge and Stanford University.
At its core the method constructs a directed graph of pairwise victories among candidates, where nodes correspond to candidates and edges represent majority margins calculated from ballots such as those used in Australian Electoral Commission contests or organizational elections of the Internet Engineering Task Force. The algorithm computes the strongest paths between every pair of nodes using a variant of the widest path problem related to algorithms by Edmonds, Dijkstra, Floyd–Warshall and implementations inspired by work at Bell Labs and in textbooks from MIT Press. Winners are determined by comparing strengths of strongest paths: a candidate X defeats candidate Y if the strongest path from X to Y is stronger than the strongest path from Y to X. The method accommodates weighted ballots, ties, and partial rankings frequently encountered in elections organized by European Union institutions, non-profits like Amnesty International, or tech communities such as the Apache Software Foundation.
The method satisfies the Condorcet criterion, the Smith criterion, reversal symmetry, monotonicity in many practical settings, and independence of clones under specific definitions explored in literature by Kenneth O. May, Nicholas Rescher, John H. Smith and voting theorists at Princeton University and Yale University. It fails certain criteria implied by impossibility results such as variants of Arrow's theorem and can exhibit non-monotonic behavior in contrived profiles related to paradoxes studied by Bertrand Russell and Jean-Charles de Borda. The method is compatible with participating organizations that require compliance with principles similar to those in bylaws of United Nations agencies or codes of conduct enforced by International Committee of the Red Cross.
Computationally, the central step requires computing strongest paths for all ordered pairs of candidates: this can be executed with O(n^3) time using a modified Floyd–Warshall algorithm, or improved with heuristics and sparse matrix techniques inspired by implementations at Google, Microsoft, Facebook and academic projects at Carnegie Mellon University and ETH Zurich. Practical implementations exist in software libraries for languages promoted by organizations like Python Software Foundation, GNU Project, Apache Foundation and in election platforms used by groups such as Linux Foundation and Mozilla Foundation. Memory usage scales with O(n^2) for the pairwise matrix; multiwinner adaptations increase complexity and require additional combinatorial handling similar to algorithms in Combinatorica and research at INRIA.
The Schulze method has been used in internal elections of Wikimedia Foundation, committees within European Parliament study groups, open-source project governance such as Debian Project votes, professional societies including Association for Computing Machinery, and in voting for awards like those overseen by Hugo Awards administrators and organizational contests run by IEEE. Empirical studies comparing outcomes against IRV and plurality in case studies from New Zealand, Ireland and municipal experiments reveal different winners and trade-offs documented by researchers at University of California, Berkeley and London School of Economics.
Extensions include multiwinner adaptations incorporating concepts from the Smith set and Condorcet k-winner frameworks, proportional implementations inspired by Single transferable vote research, and weighted or score-ballot hybrids that intersect with methods developed in contexts such as California Proposition campaigns and corporate governance at firms listed on New York Stock Exchange. Theoretical extensions draw on graph-theoretic refinements from work by Edsger W. Dijkstra, flow algorithms by Jack Edmonds, and combinatorial optimization studied at Institute for Operations Research and the Management Sciences.
Criticisms center on complexity for voters and administrators, potential non-monotonic instances, and opacity relative to simple plurality or two-round system mechanisms; these critiques have been voiced by commentators in venues like The Economist, panels at International Institute for Democracy and Electoral Assistance, and analysts at think tanks such as Brookings Institution and Cato Institute. Debates continue over trade-offs between normative axioms championed by scholars at Princeton University and pragmatic considerations applied by election officials in jurisdictions including France, Germany, and municipal bodies in the United States. Some controversy has also arisen in adoption debates within organizations like Wikimedia Foundation and open-source governance bodies where different stakeholder groups reference historical reforms from entities such as Association of Computing Machinery and Free Software Foundation.