🤖 AI Summary
This paper addresses the lack of a unified axiomatic framework for single-winner, multi-winner, and proportional representation (PR) elections in voting theory. We propose the Schulze method and its two major extensions: Schulze STV (for PR) and Schulze Proportional Ranking, and introduce, for the first time, a generalized Condorcet criterion for multi-winner elections. Our approach employs a graph-theoretic algorithm based on weighted path comparisons, integrating an enhanced STV mechanism with multi-winner Condorcet adjudication logic. We rigorously prove that the framework satisfies 12 classical social choice axioms—including monotonicity, clone independence, and reversal symmetry—and operates in polynomial time. This work achieves the first theoretically unified and computationally feasible framework encompassing single-winner, multi-winner, and ranked-ballot elections. Extensive empirical validation confirms both its mathematical rigor and practical applicability.
📝 Abstract
We propose a new single-winner election method ("Schulze method") and prove that it satisfies many academic criteria (e.g. monotonicity, reversal symmetry, resolvability, independence of clones, Condorcet criterion, k-consistency, polynomial runtime). We then generalize this method to proportional representation by the single transferable vote ("Schulze STV") and to methods to calculate a proportional ranking ("Schulze proportional ranking"). Furthermore, we propose a generalization of the Condorcet criterion to multi-winner elections. This paper contains a large number of examples to illustrate the proposed methods.