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Voters' power in voting games with abstention: Influence relation and ordinal equivalence of power theories


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  • Tchantcho, Bertrand
  • Lambo, Lawrence Diffo
  • Pongou, Roland
  • Engoulou, Bertrand Mbama


The influence relation was introduced by Isbell [Isbell, J.R., 1958. A class of simple games. Duke Math. J. 25, 423-439] to qualitatively compare the a priori influence of voters in a simple game, which by construction allows only "yes" and "no" votes. We extend this relation to voting games with abstention (VGAs), in which abstention is permitted as an intermediate option between a "yes" and a "no" vote. Unlike in simple games, this relation is not a preorder in VGAs in general. It is not complete either, but we characterize VGAs for which it is complete, and show that it is a preorder whenever it is complete. We also compare the influence relation with recent generalizations to VGAs of the Shapley-Shubik and Banzhaf-Coleman power indices [Felsenthal, D.S., Machover, M., 1997. Ternary voting games. Int. J. Game Theory 26, 335-351; Freixas, J., 2005a. The Shapley-Shubik power index for games with several levels of approval in the input and output. Dec. Support Systems 39, 185-195; Freixas, J., 2005b. The Banzhaf index for games with several levels of approval in the input and output. Ann. Operations Res. 137, 45-66]. For weakly equitable VGAs, the influence relation is a subset of the preorderings defined by these two power theories. We characterize VGAs for which the three relations are equivalent.

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Bibliographic Info

Article provided by Elsevier in its journal Games and Economic Behavior.

Volume (Year): 64 (2008)
Issue (Month): 1 (September)
Pages: 335-350

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Handle: RePEc:eee:gamebe:v:64:y:2008:i:1:p:335-350

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Keywords: Voting games with abstention Influence relation Equitableness Ordinal equivalence theorem;


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Cited by:
  1. Alaitz Artabe & Annick Laruelle & Federico Valenciano, 2012. "Preferences, actions and voting rules," SERIEs, Spanish Economic Association, vol. 3(1), pages 15-28, March.
  2. Bertrand Tchantcho & Lawrence Diffo Lambo & Roland Pongou & Joël Moulen, 2010. "On the equilibrium of voting games with abstention and several levels of approval," Social Choice and Welfare, Springer, vol. 34(3), pages 379-396, March.
  3. 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.
  4. Freixas, Josep, 2012. "Probabilistic power indices for voting rules with abstention," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 89-99.
  5. 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.
  6. L. Diffo Lambo & B. Tchantcho & J. Moulen, 2012. "Comparing influence theories in voting games under locally generated measures of dissatisfaction," International Journal of Game Theory, Springer, vol. 41(3), pages 719-731, August.
  7. René Brink & Agnieszka Rusinowska & Frank Steffen, 2013. "Measuring power and satisfaction in societies with opinion leaders: an axiomatization," Social Choice and Welfare, Springer, vol. 41(3), pages 671-683, September.
  8. Sébastien Courtin & Bertrand Tchantcho, 2013. "A note on the ordinal equivalence of power indices in games with coalition structure," THEMA Working Papers 2013-30, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
  9. Ren� van den Brink & Agnieszka Rusinowska & Frank Steffen, 2009. "Measuring Power And Satisfaction in Societies with Opinion Leaders: Dictator and Opinion Leader Properties," Tinbergen Institute Discussion Papers 09-052/1, Tinbergen Institute.
  10. Michel Grabisch & Agnieszka Rusinowska, 2010. "A model of influence with an ordered set of possible actions," Theory and Decision, Springer, vol. 69(4), pages 635-656, October.
  11. 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.
  12. Parker, Cameron, 2012. "The influence relation for ternary voting games," Games and Economic Behavior, Elsevier, vol. 75(2), pages 867-881.
  13. Freixas, Josep & Marciniak, Dorota & Pons, Montserrat, 2012. "On the ordinal equivalence of the Johnston, Banzhaf and Shapley power indices," European Journal of Operational Research, Elsevier, vol. 216(2), pages 367-375.
  14. Sebastien Courtin & Bertrand Tchantcho, 2013. "A note on the ordinal equivalence of power indices in games with coalition structure," Working Papers hal-00914910, HAL.


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