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Preserving or removing special players: What keeps your payoff unchanged in TU-games?

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  • Béal, Sylvain
  • Rémila, Eric
  • Solal, Philippe

Abstract

If a player is removed from a game, what keeps the payoff of the remaining players unchanged? Is it the removal of a special player or its presence among the remaining players? This article answers this question in a complement study to Kamijo and Kongo (2012). We introduce axioms of invariance from player deletion in presence of a special player. In particular, if the special player is a nullifying player (resp. dummifying player), then the equal division value (resp. equal surplus division value) is characterized by the associated axiom of invariance plus efficiency and balanced cycle contributions. There is no type of special player from such a combination of axioms that characterizes the Shapley value.

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  • Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2015. "Preserving or removing special players: What keeps your payoff unchanged in TU-games?," Mathematical Social Sciences, Elsevier, vol. 73(C), pages 23-31.
    • Handle: RePEc:eee:matsoc:v:73:y:2015:i:c:p:23-31
    DOI: 10.1016/j.mathsocsci.2014.11.003
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    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.
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    3. Kamijo, Yoshio & Kongo, Takumi, 2012. "Whose deletion does not affect your payoff? The difference between the Shapley value, the egalitarian value, the solidarity value, and the Banzhaf value," European Journal of Operational Research, Elsevier, vol. 216(3), pages 638-646.
    4. Béal, Sylvain & Casajus, André & Huettner, Frank & Rémila, Eric & Solal, Philippe, 2014. "Solidarity within a fixed community," Economics Letters, Elsevier, vol. 125(3), pages 440-443.
    5. 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.
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    8. 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.
    9. Radzik, Tadeusz, 2012. "A new look at the role of players’ weights in the weighted Shapley value," European Journal of Operational Research, Elsevier, vol. 223(2), pages 407-416.
    10. Jean J. M. Derks & Hans H. Haller, 1999. "Null Players Out? Linear Values For Games With Variable Supports," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 1(03n04), pages 301-314.
    11. Yoshio Kamijo & Takumi Kongo, 2010. "Axiomatization of the Shapley value using the balanced cycle contributions property," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(4), pages 563-571, October.
    12. Casajus, André & Huettner, Frank, 2014. "Null, nullifying, or dummifying players: The difference between the Shapley value, the equal division value, and the equal surplus division value," Economics Letters, Elsevier, vol. 122(2), pages 167-169.
    13. Nowak, Andrzej S & Radzik, Tadeusz, 1994. "A Solidarity Value for n-Person Transferable Utility Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(1), pages 43-48.
    14. Casajus, André & Hüttner, Frank, 2012. "Nullifying vs. dummifying players or nullified vs. dummified players: The difference between the equal division value and the equal surplus division value," Working Papers 110, University of Leipzig, Faculty of Economics and Management Science.
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    Cited by:

    1. Sylvain Béal & André Casajus & Eric Rémila & Philippe Solal, 2021. "Cohesive efficiency in TU-games: axiomatizations of variants of the Shapley value, egalitarian values and their convex combinations," Annals of Operations Research, Springer, vol. 302(1), pages 23-47, July.
    2. Zhengxing Zou & Rene van den Brink, 2020. "Sharing the Surplus and Proportional Values," Tinbergen Institute Discussion Papers 20-014/II, Tinbergen Institute.
    3. Sylvain Béal & Eric Rémila & Philippe Solal, 2015. "A Class of Solidarity Allocation Rules for TU-games," Working Papers 2015-03, CRESE.
    4. Sylvain Béal & Eric Rémila & Philippe Solal, 2017. "Axiomatization and implementation of a class of solidarity values for TU-games," Theory and Decision, Springer, vol. 83(1), pages 61-94, June.
    5. Sylvain Béal & André Casajus & Eric Rémila & Philippe Solal, 2019. "Cohesive efficiency in TU-games: Two extensions of the Shapley value," Working Papers 2019-03, CRESE.
    6. Zhengxing Zou & René Brink & Yukihiko Funaki, 2022. "Sharing the surplus and proportional values," Theory and Decision, Springer, vol. 93(1), pages 185-217, July.
    7. Koji Yokote & Takumi Kongo & Yukihiko Funaki, 2019. "Relationally equal treatment of equals and affine combinations of values for TU games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 53(2), pages 197-212, August.
    8. Takumi Kongo, 2018. "Effects of Players’ Nullification and Equal (Surplus) Division Values," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 20(01), pages 1-14, March.
    9. Sylvain Béal & Eric Rémila & Philippe Solal, 2019. "Coalitional desirability and the equal division value," Theory and Decision, Springer, vol. 86(1), pages 95-106, February.
    10. Koji Yokote & Takumi Kongo & Yukihiko Funaki, 2021. "Redistribution to the less productive: parallel characterizations of the egalitarian Shapley and consensus values," Theory and Decision, Springer, vol. 91(1), pages 81-98, July.
    11. Sylvain Béal & André Casajus & Frank Huettner & Eric Rémila & Philippe Solal, 2016. "Characterizations of weighted and equal division values," Theory and Decision, Springer, vol. 80(4), pages 649-667, April.
    12. Li, Wenzhong & Xu, Genjiu & van den Brink, René, 2024. "Sign properties and axiomatizations of the weighted division values," Journal of Mathematical Economics, Elsevier, vol. 112(C).
    13. Sylvain Ferrières, 2017. "Nullified equal loss property and equal division values," Theory and Decision, Springer, vol. 83(3), pages 385-406, October.
    14. Sylvain Ferrières, 2016. "Smoothness, nullified equal loss property and equal division values," Working Papers 2016-01, CRESE.
    15. Emilio Calvo Ramón & Esther Gutiérrez-López, 2022. "The equal collective gains value in cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 51(1), pages 249-278, March.
    16. Xun-Feng Hu & Deng-Feng Li, 2021. "The Equal Surplus Division Value for Cooperative Games with a Level Structure," Group Decision and Negotiation, Springer, vol. 30(6), pages 1315-1341, December.
    17. Takumi Kongo, 2020. "Similarities in axiomatizations: equal surplus division value and first-price auctions," Review of Economic Design, Springer;Society for Economic Design, vol. 24(3), pages 199-213, December.

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    JEL classification:

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

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