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Player splitting, players merging, the Shapley set value and the Harsanyi set value

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  • Besner, Manfred

Abstract

Shapley (1953a) introduced the weighted Shapley values as a family of values, also known as Shapley set. For each exogenously given weight system exists a seperate TU-value. Shapley (1981) and Dehez (2011), in the context of cost allocation, and Radzik (2012), in general, presented a value for weighted TU-games that covers the hole family of weighted Shapley values all at once. To distinguish this value from a weighted Shapley value in TU-games we call it Shapley set value. This value coincides with a weighted Shapley value only on a subdomain and allows weights which can depend on coalition functions. Hammer (1977) and Vasil’ev (1978) introduced independently the Harsanyi set, also known as selectope (Derks, Haller and Peters, 2000), containing TU-values which are referred to as Harsanyi-payoffs. These values are obtained by distributing the dividends from all coalitions by a sharing system that is independent from the coalition function. In this paper we introduce the Harsanyi set value that, similar to the Shapley set value, covers the hole family of Harsanyi payoffs at once, allows not exogenously given share systems and coincides thus also with non linear values on some subdomains. We present some new axiomatizations of the Shapley set value and the Harsanyi set value containing a player splitting or a players merging property respectively as a main characterizing element that recommend these values for profit distribution and cost allocation.

Suggested Citation

  • Besner, Manfred, 2018. "Player splitting, players merging, the Shapley set value and the Harsanyi set value," MPRA Paper 87125, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:87125
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    File URL: https://mpra.ub.uni-muenchen.de/87125/1/MPRA_paper_87125.pdf
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    References listed on IDEAS

    as
    1. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer;Game Theory Society, vol. 17(2), pages 89-99.
    2. repec:wsi:igtrxx:v:01:y:1999:i:03n04:n:s0219198999000220 is not listed on IDEAS
    3. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
    4. Besner, Manfred, 2017. "Axiomatizations of the proportional Shapley value," MPRA Paper 82990, University Library of Munich, Germany.
    5. Pierre Dehez, 2011. "Allocation of fixed costs: characterization of the (dual) weighted Shapley value," Working Papers of BETA 2011-03, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
    6. 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.
    7. Besner, Manfred, 2018. "Proportional Shapley levels values," MPRA Paper 87120, University Library of Munich, Germany.
    8. repec:bla:joares:v:19:y:1981:i:2:p:299-312 is not listed on IDEAS
    9. repec:wsi:igtrxx:v:13:y:2011:i:02:n:s0219198911002915 is not listed on IDEAS
    10. Jean Derks & Hans Haller & Hans Peters, 2000. "The selectope for cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 29(1), pages 23-38.
    11. Nowak, A.S. & Radzik, T., 1995. "On axiomatizations of the weighted Shapley values," Games and Economic Behavior, Elsevier, vol. 8(2), pages 389-405.
    12. DEHEZ, Pierre, 2011. "Allocation of fixed costs: characterization of the (dual) weighted Shapley value," CORE Discussion Papers RP 2405, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    More about this item

    Keywords

    Cost allocation · Profit distribution · Player splitting · Players merging · Shapley set value · Harsanyi set value;

    JEL classification:

    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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