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Cross invariance, the Shapley value, and the Shapley–Shubik power index

Author

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  • Chun-Ting Chen

    (National Taipei University)

  • Wei-Torng Juang

    (Institute of Economics, Academia Sinica)

  • Ching-Jen Sun

    (Deakin University)

Abstract

In this paper we propose a simple axiom which, along with the axioms of additivity (transfer) and dummy player, characterizes the Shapley value (the Shapley–Shubik power index) on the domain of TU (simple) games. The new axiom, cross invariance, demands payoff invariance on symmetric players across “quasi-symmetric games,” that is, games where excluding null players, all players are symmetric. Additionally, we demonstrate that the axiom of additivity can be replaced by a new axiom called strong monotonicity, or it can be completely dropped if a stronger version of cross invariance is employed. We also show that the weighted Shapley values can be characterized using a weighted variant of cross invariance. Efficiency is derived rather than assumed in our characterizations. This fresh perspective contributes to a deeper understanding of the Shapley value and its applicability.

Suggested Citation

  • Chun-Ting Chen & Wei-Torng Juang & Ching-Jen Sun, 2024. "Cross invariance, the Shapley value, and the Shapley–Shubik power index," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 62(2), pages 397-418, March.
    • Handle: RePEc:spr:sochwe:v:62:y:2024:i:2:d:10.1007_s00355-023-01490-2
    DOI: 10.1007/s00355-023-01490-2
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    References listed on IDEAS

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    1. Sylvain Béal & Eric Rémila & Philippe Solal, 2015. "Axioms of invariance for TU-games," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(4), pages 891-902, November.
    2. Chun, Youngsub, 1991. "On the Symmetric and Weighted Shapley Values," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(2), pages 183-190.
    3. Roth, Alvin E, 1977. "The Shapley Value as a von Neumann-Morgenstern Utility," Econometrica, Econometric Society, vol. 45(3), pages 657-664, April.
    4. Pradeep Dubey & Abraham Neyman & Robert James Weber, 1981. "Value Theory Without Efficiency," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 122-128, February.
    5. Gérard Hamiache, 2001. "Associated consistency and Shapley value," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(2), pages 279-289.
    6. Blair, Douglas H. & McLean, Richard P., 1990. "Subjective evaluations of n-person games," Journal of Economic Theory, Elsevier, vol. 50(2), pages 346-361, April.
    7. Nowak, A.S. & Radzik, T., 1995. "On axiomatizations of the weighted Shapley values," Games and Economic Behavior, Elsevier, vol. 8(2), pages 389-405.
    8. Annick Laruelle & Federico Valenciano, 2001. "Shapley-Shubik and Banzhaf Indices Revisited," Mathematics of Operations Research, INFORMS, vol. 26(1), pages 89-104, February.
    9. 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.
    10. Ezra Einy, 1987. "Semivalues of Simple Games," Mathematics of Operations Research, INFORMS, vol. 12(2), pages 185-192, May.
    11. Casajus, André, 2014. "The Shapley value without efficiency and additivity," Mathematical Social Sciences, Elsevier, vol. 68(C), pages 1-4.
    12. Chun, Youngsub, 1989. "A new axiomatization of the shapley value," Games and Economic Behavior, Elsevier, vol. 1(2), pages 119-130, June.
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