The incompatibility of the Condorcet winner and loser criteria with positive involvement and resolvability

2026-01-15

AI Summary

This study investigates whether four fundamental axioms—Condorcet winner criterion, Condorcet loser criterion, positive participation, and resolvability—can be simultaneously satisfied in a five-candidate preferential voting system. Through axiomatic analysis and formal proof within the framework of social choice theory, the paper establishes, for the first time, an impossibility result that does not rely on the assumption of ordinal neutrality. The findings demonstrate that these four desirable properties are mutually incompatible, thereby yielding a new impossibility theorem. This result reveals an inherent tension among fairness, incentive compatibility, and decisiveness in preference aggregation mechanisms and delineates theoretical limits for the design of voting rules.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Knowledge Representation and Reasoning: Preferences
• Humans and AI: Voting

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We prove that there is no preferential voting method satisfying the Condorcet winner and loser criteria, positive involvement (if a candidate $x$ wins in an initial preference profile, then adding a voter who ranks $x$ uniquely first cannot cause $x$ to lose), and $n$-voter resolvability (if $x$ initially ties for winning, then $x$ can be made the unique winner by adding some set of up to $n$ voters). This impossibility theorem holds for any positive integer $n$. It also holds if either the Condorcet loser criterion is replaced by independence of clones or positive involvement is replaced by negative involvement.

Problem

• Condorcet winner
• Condorcet loser
• positive involvement
• resolvability
• voting method

Innovation

• Condorcet winner
• positive involvement
• resolvability
• impossibility theorem
• preferential voting