[Paper Review] Condorcet's Principle and the Preference Reversal Paradox

Dominik Peters|arXiv (Cornell University)|Jul 27, 2017

Game Theory and Voting Systems4 citations

TL;DR

This paper proves that every Condorcet-consistent voting rule is manipulable via preference reversal—where a voter completely inverts their preference ranking—when there are at least four alternatives. Using SAT solver-aided logic encoding, the authors establish tight bounds: 15 voters for odd electorates and 24 for even ones, showing the impossibility is unavoidable in practical settings. The result strengthens the Gibbard–Satterthwaite Theorem by identifying a specific, extreme form of manipulation inherent in Condorcet extensions.

ABSTRACT

We prove that every Condorcet-consistent voting rule can be manipulated by a voter who completely reverses their preference ranking, assuming that there are at least 4 alternatives. This corrects an error and improves a result of [Sanver, M. R. and Zwicker, W. S. (2009). One-way monotonicity as a form of strategy-proofness. Int J Game Theory 38(4), 553-574.] For the case of precisely 4 alternatives, we exactly characterise the number of voters for which this impossibility result can be proven. We also show analogues of our result for irresolute voting rules. We then leverage our result to state a strong form of the Gibbard-Satterthwaite Theorem.

Motivation & Objective

• To resolve a longstanding error in Sanver and Zwicker’s (2009) proof regarding the preference reversal paradox in Condorcet extensions.
• To establish tight, practical bounds on the number of voters for which Condorcet-consistent rules remain vulnerable to preference reversal manipulation.
• To demonstrate that half-way monotonicity—a weaker form of strategyproofness—is incompatible with Condorcet-consistency, even in small electorates.
• To provide a computer-aided, logic-based proof technique using SAT solvers to derive both impossibility results and constructive examples.
• To extend the Gibbard–Satterthwaite Theorem by identifying preference reversal as a fundamental, explicit form of manipulation in Condorcet rules.

Proposed method

• The authors use a SAT solver-based approach to encode voting rules as propositional logic formulas, where models correspond to rules that are both Condorcet-consistent and half-way monotonic.
• For impossibility results, they check the unsatisfiability of the formula; when unsatisfiable, they extract a minimal unsatisfiable core (MUS) to generate human-readable impossibility proofs.
• The method treats odd and even electorates separately, using tailored encodings to minimize the number of voters required for the proof.
• The technique is applied to both resolute and irresolute (set-valued) voting rules, allowing analysis of participation and half-way monotonicity axioms.
• The authors validate their bounds by showing that for exactly 4 alternatives, Condorcet extensions satisfying half-way monotonicity exist for up to 13 (odd) and 22 (even) voters, respectively.
• They leverage these results to derive a disjunctive version of the Gibbard–Satterthwaite Theorem, distinguishing between manipulation via preference reversal and other forms.

Experimental results

Research questions

• RQ1 Can every Condorcet-consistent voting rule be manipulated by a voter who completely reverses their preference ranking when there are at least four alternatives?
• RQ2 What is the minimal number of voters for which this preference reversal paradox is unavoidable in odd and even electorates?
• RQ3 Is half-way monotonicity—defined as immunity to preference reversal—compatible with Condorcet-consistency in small electorates?
• RQ4 Can computer-aided logic methods using SAT solvers provide tight, constructive bounds on the existence of such rules?
• RQ5 Does the preference reversal paradox represent a stronger or more explicit form of manipulation than those captured by the standard Gibbard–Satterthwaite Theorem?

Key findings

• All Condorcet-consistent voting rules are manipulable via complete preference reversal when there are at least four alternatives, correcting a flawed proof in Sanver and Zwicker (2009).
• For odd electorates, the impossibility holds with as few as 15 voters; for even electorates, it requires at least 24 voters, and these bounds are tight for m=4 alternatives.
• The authors confirm via SAT solvers that Condorcet extensions satisfying half-way monotonicity exist for up to 13 voters (odd) and 22 voters (even) when m=4, but fail beyond these thresholds.
• The preference reversal paradox is incompatible with Condorcet-consistency even in small electorates, making it a practical concern rather than a theoretical curiosity.
• The result implies a strong form of the Gibbard–Satterthwaite Theorem: for anonymous, unanimous rules on the Condorcet domain, either the rule is manipulable or it is manipulable via preference reversal.
• The study shows that optimistic or pessimistic half-way monotonicity offers no advantage over resolute versions in set-valued rules, unlike participation axioms.