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

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  • Wesley H. Holliday

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.

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  • Wesley H. Holliday, 2026. "The incompatibility of the Condorcet winner and loser criteria with positive involvement and resolvability," Papers 2601.10506, arXiv.org, revised Feb 2026.
    • Handle: RePEc:arx:papers:2601.10506
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    References listed on IDEAS

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    1. Wesley H. Holliday & Eric Pacuit, 2020. "Split Cycle: A New Condorcet Consistent Voting Method Independent of Clones and Immune to Spoilers," Papers 2004.02350, arXiv.org, revised Nov 2023.
    2. Moulin, Herve, 1988. "Condorcet's principle implies the no show paradox," Journal of Economic Theory, Elsevier, vol. 45(1), pages 53-64, June.
    3. Wesley H. Holliday & Eric Pacuit, 2023. "Split Cycle: a new Condorcet-consistent voting method independent of clones and immune to spoilers," Public Choice, Springer, vol. 197(1), pages 1-62, October.
    4. Wesley H. Holliday & Eric Pacuit, 2021. "Measuring Violations of Positive Involvement in Voting," Papers 2106.11502, arXiv.org.
    5. Young, H. P., 1977. "Extending Condorcet's rule," Journal of Economic Theory, Elsevier, vol. 16(2), pages 335-353, December.
    6. Kasper, Laura & Peters, Hans & Vermeulen, Dries, 2019. "Condorcet Consistency and the strong no show paradoxes," Mathematical Social Sciences, Elsevier, vol. 99(C), pages 36-42.
    7. Joaquin Perez, 1995. "Incidence of no-show paradoxes in Condorcet choice functions," Investigaciones Economicas, Fundación SEPI, vol. 19(1), pages 139-154, January.
    8. Holliday, Wesley H., 2024. "An impossibility theorem concerning positive involvement in voting," Economics Letters, Elsevier, vol. 236(C).
    9. Joaqui´n Pérez, 2001. "The Strong No Show Paradoxes are a common flaw in Condorcet voting correspondences," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(3), pages 601-616.
    10. Kramer, Gerald H., 1977. "A dynamical model of political equilibrium," Journal of Economic Theory, Elsevier, vol. 16(2), pages 310-334, December.
    11. Wesley H. Holliday & Eric Pacuit, 2023. "An extension of May's Theorem to three alternatives: axiomatizing Minimax voting," Papers 2312.14256, arXiv.org, revised Apr 2025.
    12. Paul B. Simpson, 1969. "On Defining Areas of Voter Choice: Professor Tullock on Stable Voting," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 83(3), pages 478-490.
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