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The influence relation for ternary voting games


  • Parker, Cameron


Although simple games are very useful in modeling decision-making bodies, they allow each voter only two choices: to support or oppose a measure. This restriction ignores that voters often can abstain from voting, which is effectively different from the other two options. Following the approach of Felsenthal and Machover (1997), for modeling voting with abstentions, we will look at the extension of the influence relation for simple games to the Ternary Voting Game given in Tchantcho et al. (2008). That paper showed that the influence relation is ordinally equivalent to the classical Banzhaf and Shapley–Shubik indices in a class of games called weakly equitable. In this paper, we will show that this result does hold true for all Ternary Voting Games. Also we will show that adding a third voting option allows for asymmetric distribution of power that cannot be achieved by any simple game.

Suggested Citation

  • Parker, Cameron, 2012. "The influence relation for ternary voting games," Games and Economic Behavior, Elsevier, vol. 75(2), pages 867-881.
  • Handle: RePEc:eee:gamebe:v:75:y:2012:i:2:p:867-881
    DOI: 10.1016/j.geb.2012.02.007

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    References listed on IDEAS

    1. Josep Freixas, 2010. "On ordinal equivalence of the Shapley and Banzhaf values for cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(4), pages 513-527, October.
    2. Jane Friedman & Lynn Mcgrath & Cameron Parker, 2006. "Achievable Hierarchies In Voting Games," Theory and Decision, Springer, vol. 61(4), pages 305-318, December.
    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. Dwight Bean & Jane Friedman & Cameron Parker, 2008. "Simple Majority Achievable Hierarchies," Theory and Decision, Springer, vol. 65(4), pages 285-302, December.
    5. Josep Freixas & William S. Zwicker, 2003. "Weighted voting, abstention, and multiple levels of approval," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 21(3), pages 399-431, December.
    6. Carreras, Francesc & Freixas, Josep, 2008. "On ordinal equivalence of power measures given by regular semivalues," Mathematical Social Sciences, Elsevier, vol. 55(2), pages 221-234, March.
    7. Rubinstein, Ariel, 1980. "Stability of decision systems under majority rule," Journal of Economic Theory, Elsevier, vol. 23(2), pages 150-159, October.
    8. Carreras, Francesc & Freixas, Josep, 1996. "Complete simple games," Mathematical Social Sciences, Elsevier, vol. 32(2), pages 139-155, October.
    9. Lawrence Diffo Lambo & Joël Moulen, 2002. "Ordinal equivalence of power notions in voting games," Theory and Decision, Springer, vol. 53(4), pages 313-325, December.
    10. repec:cup:apsrev:v:48:y:1954:i:03:p:787-792_00 is not listed on IDEAS
    11. 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.
    12. Tchantcho, Bertrand & Lambo, Lawrence Diffo & Pongou, Roland & Engoulou, Bertrand Mbama, 2008. "Voters' power in voting games with abstention: Influence relation and ordinal equivalence of power theories," Games and Economic Behavior, Elsevier, vol. 64(1), pages 335-350, September.
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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. Sébastien Courtin & Bertrand Tchantcho, 2015. "A note on the ordinal equivalence of power indices in games with coalition structure," Theory and Decision, Springer, vol. 78(4), pages 617-628, April.
    3. Sébastien Courtin & Bertrand Tchantcho, 2015. "A note on the ordinal equivalence of power indices in games with coalition structure," Post-Print hal-00914910, HAL.
    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. repec:spr:annopr:v:244:y:2016:i:2:d:10.1007_s10479-016-2124-5 is not listed on IDEAS

    More about this item


    Cooperative games; Ternary voting games; Ordinal equivalence; Hierarchies;

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D72 - Microeconomics - - Analysis of Collective Decision-Making - - - Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior


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