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Axiomatizations of the proportional division value

Author

Listed:
  • Zhengxing Zou

    (Vrije Universiteit Amsterdam
    Beijing Jiaotong University)

  • René Brink

    (Vrije Universiteit Amsterdam)

  • Youngsub Chun

    (Seoul National University)

  • Yukihiko Funaki

    (Waseda University)

Abstract

We present axiomatic characterizations of the proportional division value for TU-games, which distributes the worth of the grand coalition in proportion to the stand-alone worths of the players. First, a new proportionality principle, called proportional-balanced treatment, is introduced by strengthening Shapley’s symmetry axiom, which states that if two players make the same contribution to any nonempty coalition, then they receive the amounts in proportion to their stand-alone worths. We characterize the family of values satisfying efficiency, weak linearity, and proportional-balanced treatment. We also show that this family is incompatible with the dummy player property. However, we show that the proportional division value is the unique value in this family that satisfies the dummifying player property. Second, we propose appropriate monotonicity axioms, and obtain axiomatizations of the proportional division value without both weak linearity and the dummifying player property. Third, from the perspective of a variable player set, we show that the proportional division value is the only one that satisfies proportional standardness and projection consistency. Finally, we provide a characterization of proportional standardness.

Suggested Citation

  • Zhengxing Zou & René Brink & Youngsub Chun & Yukihiko Funaki, 2021. "Axiomatizations of the proportional division value," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 57(1), pages 35-62, July.
    • Handle: RePEc:spr:sochwe:v:57:y:2021:i:1:d:10.1007_s00355-020-01299-3
    DOI: 10.1007/s00355-020-01299-3
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    Cited by:

    1. David Lowing & Kevin Techer, 2022. "Marginalism, egalitarianism and efficiency in multi-choice games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 59(4), pages 815-861, November.
    2. Zhengxing Zou & Rene van den Brink & Yukihiko Funaki, 2020. "Compromising between the proportional and equal division values: axiomatization, consistency and implementation," Tinbergen Institute Discussion Papers 20-054/II, Tinbergen Institute.
    3. Zhengxing Zou & Rene van den Brink, 2020. "Sharing the Surplus and Proportional Values," Tinbergen Institute Discussion Papers 20-014/II, Tinbergen Institute.
    4. Zou, Zhengxing & van den Brink, René & Funaki, Yukihiko, 2021. "Compromising between the proportional and equal division values," Journal of Mathematical Economics, Elsevier, vol. 97(C).
    5. Rene van den Brink & Youngsub Chun & Yukihiko Funaki & Zhengxing Zou, 2021. "Balanced Externalities and the Proportional Allocation of Nonseparable Contributions," Tinbergen Institute Discussion Papers 21-024/II, Tinbergen Institute.
    6. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2022. "Lexicographic solutions for coalitional rankings based on individual and collective performances," Journal of Mathematical Economics, Elsevier, vol. 102(C).
    7. Erfang Shan & Zhiqiang Yu & Wenrong Lyu, 2023. "Union-wise egalitarian solutions in cooperative games with a coalition structure," 4OR, Springer, vol. 21(3), pages 533-545, September.
    8. van den Brink, René & Chun, Youngsub & Funaki, Yukihiko & Zou, Zhengxing, 2023. "Balanced externalities and the proportional allocation of nonseparable contributions," European Journal of Operational Research, Elsevier, vol. 307(2), pages 975-983.
    9. David Lowing & Kevin Techer, 2021. "Marginalism, Egalitarianism and E ciency in Multi-Choice Games," Working Papers halshs-03334056, HAL.
    10. Besner, Manfred, 2022. "The grand surplus value and repeated cooperative cross-games with coalitional collaboration," Journal of Mathematical Economics, Elsevier, vol. 102(C).
    11. Wenzhong Li & Genjiu Xu & Hao Sun, 2020. "Maximizing the Minimal Satisfaction—Characterizations of Two Proportional Values," Mathematics, MDPI, vol. 8(7), pages 1-17, July.
    12. Besner, Manfred, 2021. "The grand dividends value," MPRA Paper 107615, University Library of Munich, Germany.
    13. Besner, Manfred, 2021. "The grand dividends value," MPRA Paper 106638, University Library of Munich, Germany.
    14. Zhengxing Zou & René Brink & Yukihiko Funaki, 2022. "Sharing the surplus and proportional values," Theory and Decision, Springer, vol. 93(1), pages 185-217, July.

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

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