International Journal of Game Theory

, Volume 26, Issue 3, pp 335–351 | Cite as

Ternary voting games

  • Dan S. Felsenthal
  • Moshé Machover


We defineternary voting games (TVGs), a generalization ofsimple voting games (SVGs). In a play of an SVG each voter has just two options: voting ‘yes’ or ‘no’. In a TVG a third option is added: abstention. Every SVG can be regarded as a (somewhat degenerate) TVG; but the converse is false. We define appropriate generalizations of the Shapley-Shubik and Banzhaf indices for TVGs. We define also theresponsiveness (ordegree of democratic participation) of a TVG and determine, for eachn, the most responsive TVGs withn voters. We show that these maximally responsive TVGs are more responsive than the corresponding SVGs.


Economic Theory Game Theory Vote Game Democratic Participation Banzhaf Index 
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  1. Banzhaf JF (1965) Weighted voting doesn't work: a mathematical analysis. Rutgers Law Review 19: 317–343Google Scholar
  2. Brams SJ (1975) Game theory and politics. Free Press, New YorkGoogle Scholar
  3. Brams SJ, Affuso PJ, Kilgour DM (1989) Presidential power: a game-theoretic analysis. In Brace P, Harrington CB, King G (eds.) The presidency in American politics. New York University Press, New York, pp. 55–72Google Scholar
  4. Coleman JS (1986) Individual interests and collective action: Selected essays. Cambridge University Press, CambridgeGoogle Scholar
  5. Dubey P, Shapley LS (1979) Mathematical properties of the Banzhaf power index. Mathematics of Operations Research 4: 99–131Google Scholar
  6. Felsenthal DS (1991) Averting the quorum paradox. Behavioral Science 36: 57–63Google Scholar
  7. Felsenthal DS, Machover M (1995) Postulates and paradoxes of relative voting power: a critical re-appraisal. Theory and Decision 38: 195–229Google Scholar
  8. Felsenthal DS, Machover M (1996) Alternative forms of the Shapley value and the Shapley-Shubik index. Public Choice 87: 315–318Google Scholar
  9. Felsenthal DS, Machover M, Zwicker W (1997) The bicameral postulates and indices of a priori relative voting power. Theory and Decision (forthcoming)Google Scholar
  10. Fishburn PC (1973) The theory of social choice. Princeton University Press, PrincetonGoogle Scholar
  11. Lambert JP (1988) Voting games, power indices and presidential elections. UMAP Journal 9: 216–277Google Scholar
  12. Lucas WF (1982) Measuring power in weighted voting systems. In: Brams SJ, Lucas WF, Straffin PD (eds.) Political and related models, models in applied mathematics. Springer-Verlag, New York: 183–255Google Scholar
  13. Rae DW (1969) Decision rules and individual values in constitutional choice. American Political Science Review 63: 40–56Google Scholar
  14. Rapoport A (1970) N-person game theory: Concepts and applications. University of Michigan Press, Ann ArborGoogle Scholar
  15. Riker WH (1982) Liberalism against populism: A confrontation between the theory of democracy and the theory of social choice. WH Freeman and Company, San FranciscoGoogle Scholar
  16. Roth AE (ed.) (1988) The Shapley value: Eassys in honor of Lloyd S. Shapley. Cambridge University Press, CambridgeGoogle Scholar
  17. Shapley LS (1962) Simple games: An outline of the descriptive theory. Behavioral Science 7: 59–66Google Scholar
  18. Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. American Political Science Review 48: 787–792Google Scholar
  19. Simma B (ed.) (1994) The charter of the United Nations-A commentary. Oxford University Press, New YorkGoogle Scholar
  20. Straffin PD (1982) Power indices in politics. In Brams SJ, Lucas WF, Straffin PD (eds.) Political and related models, models in applied mathematics. Springer-Verlag, New York: 256–321Google Scholar
  21. Taylor A (1995) Mathematics and politics: Strategy, voting, power and proof. Springer-Verlag, New YorkGoogle Scholar

Copyright information

© Physica-Verlag 1997

Authors and Affiliations

  • Dan S. Felsenthal
    • 1
  • Moshé Machover
    • 2
  1. 1.Department of Political ScienceUniversity of HaifaHaifaIsrael
  2. 2.Department of PhilosophyKing's College LondonStrandUK

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