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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

Abstract

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.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:gamebe:v:64:y:2008:i:1:p:335-350
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    References listed on IDEAS

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    1. Dominique Lepelley & N. Andjiga & F. Chantreuil, 2003. "La mesure du pouvoir de vote," Post-Print halshs-00069255, HAL.
    2. Sven Berg, 1999. "On Voting Power Indices and a Class of Probability Distributions: With applications to EU data," Group Decision and Negotiation, Springer, vol. 8(1), pages 17-31, January.
    3. Dan Felsenthal & Moshé Machover, 2005. "Voting power measurement: a story of misreinvention," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 25(2), pages 485-506, December.
    4. R J Johnston, 1978. "On the Measurement of Power: Some Reactions to Laver," Environment and Planning A, , vol. 10(8), pages 907-914, August.
    5. Edward M. Bolger, 2002. "Characterizations of two power indices for voting games with r alternatives," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 19(4), pages 709-721.
    6. 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.
    7. Rae, Douglas W., 1969. "Decision-Rules and Individual Values in Constitutional Choice," American Political Science Review, Cambridge University Press, vol. 63(1), pages 40-56, March.
    8. Rubinstein, Ariel, 1980. "Stability of decision systems under majority rule," Journal of Economic Theory, Elsevier, vol. 23(2), pages 150-159, 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.
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