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The Owen–Shapley Spatial Power Index in Three-Dimensional Space

Author

Listed:
  • M. J. Albizuri

    (University of the Basque Country)

  • A. Goikoetxea

    (University of the Basque Country)

Abstract

Inspired by Owen’s (Nav Res Logist Quart 18:345–354, 1971) previous work on the subject, Shapley (A comparison of power indices and a non-symmetric generalization. Rand Corporation, Santa Monica, 1977) introduced the Owen–Shapley spatial power index, which takes the ideological location of individuals into account, represented by vectors in the Euclidean space $${\mathbb {R}}^{m}$$ R m , to measure their power. In this work we study the Owen–Shapley spatial power index in three-dimensional space. Peters and Zarzuelo (Int J Game Theory 46:525–545, 2017) carried out a study of this index for individuals located in two-dimensional space, but pointed out the limitation of the two-dimensional feature. In this work focusing on three-dimensional space, we provide an explicit formula for spatial unanimity games, which makes it possible to calculate the Owen–Shapley spatial power index of any spatial game. We also give a characterization of the Owen–Shapley spatial power index employing two invariant positional axioms among others. Finally, we calculate this power index for the Basque Parliament, both in the two-dimensional and three-dimensional cases. We compare these positional indices against each other and against those that result when classical non-positional indices are considered, such as the Shapley–Shubik power index (Am Polit Sci Rev 48(3):787–792, 1954) and the Banzhaf-normalized index (Rutgers Law Rev 19:317–343, 1965).

Suggested Citation

  • M. J. Albizuri & A. Goikoetxea, 2021. "The Owen–Shapley Spatial Power Index in Three-Dimensional Space," Group Decision and Negotiation, Springer, vol. 30(5), pages 1027-1055, October.
    • Handle: RePEc:spr:grdene:v:30:y:2021:i:5:d:10.1007_s10726-021-09746-x
    DOI: 10.1007/s10726-021-09746-x
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    References listed on IDEAS

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    1. Owen, G & Shapley, L S, 1989. "Optimal Location of Candidates in Ideological Space," International Journal of Game Theory, Springer;Game Theory Society, vol. 18(3), pages 339-356.
    2. Einy, Ezra & Haimanko, Ori, 2011. "Characterization of the Shapley–Shubik power index without the efficiency axiom," Games and Economic Behavior, Elsevier, vol. 73(2), pages 615-621.
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    6. 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.
    7. Francesc Carreras & María Albina Puente, 2015. "Multinomial Probabilistic Values," Group Decision and Negotiation, Springer, vol. 24(6), pages 981-991, November.
    8. Giulia Bernardi, 2018. "A New Axiomatization of the Banzhaf Index for Games with Abstention," Group Decision and Negotiation, Springer, vol. 27(1), pages 165-177, February.
    9. Guillermo Owen, 1972. "Multilinear Extensions of Games," Management Science, INFORMS, vol. 18(5-Part-2), pages 64-79, January.
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    More about this item

    Keywords

    Power index; Owen–Shapley spatial index;

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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