Cycle Surgery: Axiomatizing Graph-Based Repairs for Arrovian Contradictions

Livson & Prokopenko (2025) formally show Arrow’s theorem generalizes Condorcet cycles even with weak preferences by explicitly constructing cycle-inducing profiles. Building on this, we design a class of “cycle surgery” rules: (1) build the majority or approval-inferred tournament/semicomplete graph; (2) detect contradictory cycles; (3) remove a minimum set of edges (feedback arc set) subject to neutrality, anonymity, and minimality axioms to yield a transitive social order. We connect to Saari’s structural explanations (2011) of paradoxes and Gladyshev’s (2019) taxonomy to prove robustness properties (e.g., minimal distortions from the base graph) and to benchmark against claims that Approval Voting “escapes Arrow” by avoiding intransitive cycles (Horn, 2024)—a claim we evaluate in this unified graph framework. Using Dobra’s empirical approach (1983), we estimate cycle frequencies in real data and quantify the manipulation-resistance of different surgery constraints. The novelty is an axiomatized, transparent post-processing layer that sits atop diverse base rules (Condorcet, approval, ranked) to systematically resolve contradictions with provable fairness guarantees. This yields implementable tie-breaking for Condorcet paradoxes and clarifies when “cycle-free” methods genuinely evade Arrow vs. merely displace contradictions.

References:

1. Arrow's Impossibility Theorem as a Generalisation of Condorcet's Paradox. Ori Livson, M. Prokopenko (2025).
2. Explaining Voting Paradoxes; Including Arrow's and Sen's Theorems - (Invited Tutorial). D. Saari (2011). International Conference on Relational and Algebraic Methods in Computer Science.
3. Vulnerability of Voting Paradoxes As a Criteria For Voting Procedure Selection. Maksim Gladyshev (2019). Social Science Research Network.
4. Three Unique Virtues of Approval Voting. Walter Horn (2024). Qeios.
5. An approach to empirical studies of voting paradoxes: An update and extension. John L. Dobra (1983).