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Axiomatization and Implementation of Discounted Shapley Values

Author

Listed:
  • Rene van den Brink

    (VU University Amsterdam, the Netherlands)

  • Yukihiko Funaki

    (Waseda University Tokyo, Japan)

Abstract
This discussion paper resulted in a publication in 'Social Choice and Welfare' . We generalize the null player property (satisfied by the Shapley value) and nullifying player property (satisfied by the equal division solution) to the so-called delta-reducing player property, stating that a delta-reducing player (being a player such that any coalition containing this player earns a fraction delta in [0,1] of the worth of that coalition without that player) earns a zero payoff. This property yields the null player property for delta = 1 and the nullifying player property for delta = 0. We show that efficiency, symmetry, linearity and this delta-reducing player property characterizes the corresponding delta-discounted Shapley value. Moreover, we provide a strategic implementation of these solutions where delta is a discount factor that determines the decrease in value to be distributed in the next round after the proposal is rejected and the remaining players (without the proposer) play a new round of bidding.

Suggested Citation

  • Rene van den Brink & Yukihiko Funaki, 2010. "Axiomatization and Implementation of Discounted Shapley Values," Tinbergen Institute Discussion Papers 10-065/1, Tinbergen Institute.
    • Handle: RePEc:tin:wpaper:20100065
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    File URL: https://papers.tinbergen.nl/10065.pdf
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    References listed on IDEAS

    as
    1. Yuan Ju & Peter Borm & Pieter Ruys, 2007. "The consensus value: a new solution concept for cooperative games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 28(4), pages 685-703, June.
    2. Perez-Castrillo, David & Wettstein, David, 2001. "Bidding for the Surplus : A Non-cooperative Approach to the Shapley Value," Journal of Economic Theory, Elsevier, vol. 100(2), pages 274-294, October.
    3. van den Brink, Rene, 2007. "Null or nullifying players: The difference between the Shapley value and equal division solutions," Journal of Economic Theory, Elsevier, vol. 136(1), pages 767-775, September.
    4. René Brink & Yukihiko Funaki, 2009. "Axiomatizations of a Class of Equal Surplus Sharing Solutions for TU-Games," Theory and Decision, Springer, vol. 67(3), pages 303-340, September.
    5. Sergiu Hart, 2006. "Shapley Value," Discussion Paper Series dp421, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    6. Hart, Sergiu & Mas-Colell, Andreu, 1996. "Bargaining and Value," Econometrica, Econometric Society, vol. 64(2), pages 357-380, March.
    7. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. van den Brink, René & van der Laan, Gerard & Moes, Nigel, 2013. "A strategic implementation of the Average Tree solution for cycle-free graph games," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2737-2748.
    2. Marco Rogna, 2022. "The Burning Coalition Bargaining Model," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 59(3), pages 735-768, October.
    3. Emilio Calvo & Esther Gutiérrez-López, 2018. "Discounted Solidarity Values," Discussion Papers in Economic Behaviour 0418, University of Valencia, ERI-CES.
    4. Yokote, Koji & Funaki, Yukihiko & Kamijo, Yoshio, 2016. "A new basis and the Shapley value," Mathematical Social Sciences, Elsevier, vol. 80(C), pages 21-24.
    5. Tomohiko Kawamori, 2016. "Hart–Mas-Colell implementation of the discounted Shapley value," Theory and Decision, Springer, vol. 81(3), pages 357-369, September.
    6. Emilio Calvo & Esther Gutiérrez-López, 2016. "A strategic approach for the discounted Shapley values," Theory and Decision, Springer, vol. 80(2), pages 271-293, February.
    7. Emilio Calvo & Esther Gutiérrez-López, 2016. "A strategic approach for the discounted Shapley values," Theory and Decision, Springer, vol. 80(2), pages 271-293, February.

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    More about this item

    Keywords

    Cooperative TU-game; Shapley value; equal division solution; delta-discounted Shapley value; Axiomatization; Implementation; Discounting;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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