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Anonymous yes-no voting with abstention and multiple levels of approval

Author

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  • Freixas, Josep
  • Zwicker, William S.

Abstract

Symmetric (3,2) simple games serve as models for anonymous voting systems in which each voter may vote "yes," abstain, or vote "no," the outcome is "yes" or "no," and all voters play interchangeable roles. The extension to symmetric (j,2) simple games, in which each voter chooses from among j ordered levels of approval, also models some natural decision rules, such as pass-fail grading systems. Each such game is determined by the set of (anonymous) minimal winning profiles. This makes it possible to count the possible systems, and the counts suggest some interesting patterns. In the (3,2) case, the approach yields a version of May's Theorem, classifying all possible anonymous voting rules with abstention in terms of quota functions. In contrast to the situation for ordinary simple games these results reveal that the class of simple games with 3 or more levels of approval remains large and varied, even after the imposition of symmetry.

Suggested Citation

  • Freixas, Josep & Zwicker, William S., 2009. "Anonymous yes-no voting with abstention and multiple levels of approval," Games and Economic Behavior, Elsevier, vol. 67(2), pages 428-444, November.
  • Handle: RePEc:eee:gamebe:v:67:y:2009:i:2:p:428-444
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    References listed on IDEAS

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    Cited by:

    1. Pongou, Roland & Tchantcho, Bertrand & Tedjeugang, Narcisse, 2014. "Power theories for multi-choice organizations and political rules: Rank-order equivalence," Operations Research Perspectives, Elsevier, vol. 1(1), pages 42-49.
    2. Guemmegne, Juliette T. & Pongou, Roland, 2014. "A policy-based rationalization of collective rules: Dimensionality, specialized houses, and decentralized authority," Journal of Mathematical Economics, Elsevier, vol. 52(C), pages 182-193.
    3. Roland Pongou & Bertrand Tchantcho & Lawrence Diffo Lambo, 2011. "Political influence in multi-choice institutions: cyclicity, anonymity, and transitivity," Theory and Decision, Springer, vol. 70(2), pages 157-178, February.
    4. Freixas, Josep & Parker, Cameron, 2015. "Manipulation in games with multiple levels of output," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 144-151.
    5. Freixas, Josep & Tchantcho, Bertrand & Tedjeugang, Narcisse, 2014. "Achievable hierarchies in voting games with abstention," European Journal of Operational Research, Elsevier, vol. 236(1), pages 254-260.
    6. Freixas, Josep, 2012. "Probabilistic power indices for voting rules with abstention," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 89-99.
    7. Annick Lamelle & Federico Valenciano, 2011. "Majorities with a quorum," Journal of Theoretical Politics, , vol. 23(2), pages 241-259, April.
      • Laruelle, Annick & Valenciano Llovera, Federico, 2010. "Majorities with a quorum," IKERLANAK 2010-42, Universidad del País Vasco - Departamento de Fundamentos del Análisis Económico I.
    8. Matthew Gould & Matthew D. Rablen, 2016. "Equitable representation in councils: theory and an application to the United Nations Security Council," Public Choice, Springer, vol. 169(1), pages 19-51, October.
    9. Sascha Kurz & Nikolas Tautenhahn, 2013. "On Dedekind’s problem for complete simple games," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(2), pages 411-437, May.
    10. Sven Grüner & Anica Fietz & Antje Jantsch, 2015. "Float like a butterfly, $${\varvec{decide}}$$ d e c i d e like a bee," Journal of Bioeconomics, Springer, vol. 17(3), pages 243-254, October.
    11. Alaitz Artabe & Annick Laruelle & Federico Valenciano, 2012. "Preferences, actions and voting rules," SERIEs: Journal of the Spanish Economic Association, Springer;Spanish Economic Association, vol. 3(1), pages 15-28, March.
    12. Kóczy Á., László & Pintér, Miklós, 2011. "Az ellenzék ereje - általánosított súlyozott szavazási játékok
      [Minority power - generalized weighted voting games]
      ," Közgazdasági Szemle (Economic Review - monthly of the Hungarian Academy of Sciences), Közgazdasági Szemle Alapítvány (Economic Review Foundation), vol. 0(6), pages 543-551.
    13. Birkmeier Olga & Käufl Andreas & Pukelsheim Friedrich, 2011. "Abstentions in the German Bundesrat and ternary decision rules in weighted voting systems," Statistics & Risk Modeling, De Gruyter, vol. 28(1), pages 1-16, March.
    14. Sascha Kurz, 2016. "The inverse problem for power distributions in committees," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 47(1), pages 65-88, June.
    15. Annick Laruelle & Federico Valenciano, 2012. "Quaternary dichotomous voting rules," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 38(3), pages 431-454, March.
    16. Parker, Cameron, 2012. "The influence relation for ternary voting games," Games and Economic Behavior, Elsevier, vol. 75(2), pages 867-881.
    17. Courtin, Sébastien & Nganmeni, Zéphirin & Tchantcho, Bertrand, 2017. "Dichotomous multi-type games with a coalition structure," Mathematical Social Sciences, Elsevier, vol. 86(C), pages 9-17.
    18. Sébastien Courtin & Zéphirin Nganmeni & Bertrand Tchantcho, 2017. "Dichotomous multi-type games with a coalition structure," Post-Print halshs-01545772, HAL.
    19. Keith L. Dougherty, 2015. "Supermajority rules," Chapters,in: Handbook of Social Choice and Voting, chapter 7, pages 102-116 Edward Elgar Publishing.
    20. Laruelle, Annick & Valenciano Llovera, Federico, 2010. "Quaternary dichotomous voting rules," IKERLANAK 2010-41, Universidad del País Vasco - Departamento de Fundamentos del Análisis Económico I.
    21. Laruelle, Annick & Valenciano Llovera, Federico, 2010. "Majorities with a quorum," IKERLANAK 2010-42, Universidad del País Vasco - Departamento de Fundamentos del Análisis Económico I.

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