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Three-Valued Simple Games

Author

Listed:
  • Musegaas, M.

    (Tilburg University, Center For Economic Research)

  • Borm, P.E.M.

    (Tilburg University, Center For Economic Research)

  • Quant, M.

    (Tilburg University, Center For Economic Research)

Abstract

In this paper we study three-valued simple games as a natural extension of simple games. We analyze to which extent well-known results on the core and the Shapley value for simple games can be extended to this new setting. To describe the core of a three-valued simple game we introduce (primary and secondary) vital players, in analogy to veto players for simple games. Moreover, it is seen that the transfer property of Dubey (1975) can still be used to characterize the Shapley value for three-valued simple games. We illustrate three-valued simple games and the corresponding Shapley value in a parliamentary bicameral system.
(This abstract was borrowed from another version of this item.)
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Musegaas, M. & Borm, P.E.M. & Quant, M., 2015. "Three-Valued Simple Games," Discussion Paper 2015-026, Tilburg University, Center for Economic Research.
    • Handle: RePEc:tiu:tiucen:473afd5c-99b1-4073-888f-2006e20fbf5f
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    References listed on IDEAS

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    1. Kjell Hausken & Matthias Mohr, 2001. "The value of a player in n-person games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(3), pages 465-483.
    2. Josep Freixas, 2005. "Banzhaf Measures for Games with Several Levels of Approval in the Input and Output," Annals of Operations Research, Springer, vol. 137(1), pages 45-66, July.
    3. Freixas, Josep, 2012. "Probabilistic power indices for voting rules with abstention," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 89-99.
    4. Shapley, L. S. & Shubik, Martin, 1954. "A Method for Evaluating the Distribution of Power in a Committee System," American Political Science Review, Cambridge University Press, vol. 48(3), pages 787-792, September.
    5. Xiaotie Deng & Toshihide Ibaraki & Hiroshi Nagamochi, 1999. "Algorithmic Aspects of the Core of Combinatorial Optimization Games," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 751-766, August.
    6. Bilbao, J. M. & Fernandez, J. R. & Jimenez, N. & Lopez, J. J., 2002. "Voting power in the European Union enlargement," European Journal of Operational Research, Elsevier, vol. 143(1), pages 181-196, November.
    7. M. Musegaas & P. E. M. Borm & M. Quant, 2016. "Simple and three-valued simple minimum coloring games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(2), pages 239-258, October.
    8. J. Bilbao & J. Fernández & A. Losada & J. López, 2000. "Generating functions for computing power indices efficiently," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 8(2), pages 191-213, December.
    9. Hsiao Chih-Ru & Raghavan T. E. S., 1993. "Shapley Value for Multichoice Cooperative Games, I," Games and Economic Behavior, Elsevier, vol. 5(2), pages 240-256, April.
    10. 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.
    11. Ichiishi, Tatsuro, 1981. "Super-modularity: Applications to convex games and to the greedy algorithm for LP," Journal of Economic Theory, Elsevier, vol. 25(2), pages 283-286, October.
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    Cited by:

    1. M. Musegaas & P. E. M. Borm & M. Quant, 2016. "Simple and three-valued simple minimum coloring games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(2), pages 239-258, October.
    2. Wilms, Ingo, 2020. "Dynamic programming algorithms for computing power indices in weighted multi-tier games," Mathematical Social Sciences, Elsevier, vol. 108(C), pages 175-192.

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    More about this item

    Keywords

    simple games; three-valued simple games; core; Shapley value;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory

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