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The proportional value for positive cooperative games

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  • K. Michael Ortmann

Abstract

In this article a new value for positive cooperative games is introduced. It generalizes the “proportional division of surplus” idea for two person games. A characterization by means of preservation of ratios and efficiency provides existence and uniqueness. Moreover, this value is the only one that is both proportional for two person games and consistent with respect to the reduced game of the Shapley value. Copyright Springer-Verlag Berlin Heidelberg 2000

Suggested Citation

  • K. Michael Ortmann, 2000. "The proportional value for positive cooperative games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 51(2), pages 235-248, April.
    • Handle: RePEc:spr:mathme:v:51:y:2000:i:2:p:235-248
    DOI: 10.1007/s001860050086
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    Citations

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    Cited by:

    1. Zhengxing Zou & Rene van den Brink, 2020. "Sharing the Surplus and Proportional Values," Tinbergen Institute Discussion Papers 20-014/II, Tinbergen Institute.
    2. Rene (J.R.) van den Brink & Rene Levinsky & Miroslav Zeleny, 2018. "The Shapley Value, Proper Shapley Value, and Sharing Rules for Cooperative Ventures," Tinbergen Institute Discussion Papers 18-089/II, Tinbergen Institute.
    3. Behzad Hezarkhani & Marco Slikker & Tom Woensel, 2016. "A competitive solution for cooperative truckload delivery," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 38(1), pages 51-80, January.
    4. Manfred Besner, 2019. "Axiomatizations of the proportional Shapley value," Theory and Decision, Springer, vol. 86(2), pages 161-183, March.
    5. Rene van den Brink & Rene Levinsky & Miroslav Zeleny, 2007. "The balanced solution for cooperative transferable utility games," Jena Economics Research Papers 2007-073, Friedrich-Schiller-University Jena.
    6. Florian Kellner & Andreas Otto, 2012. "Allocating CO 2 emissions to shipments in road freight transportation," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 22(4), pages 451-479, January.
    7. Zhengxing Zou & René Brink & Yukihiko Funaki, 2022. "Sharing the surplus and proportional values," Theory and Decision, Springer, vol. 93(1), pages 185-217, July.
    8. Manfred Besner, 2020. "Value dividends, the Harsanyi set and extensions, and the proportional Harsanyi solution," International Journal of Game Theory, Springer;Game Theory Society, vol. 49(3), pages 851-873, September.
    9. Béal, Sylvain & Ferrières, Sylvain & Rémila, Eric & Solal, Philippe, 2018. "The proportional Shapley value and applications," Games and Economic Behavior, Elsevier, vol. 108(C), pages 93-112.
    10. 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.
    11. Besner, Manfred, 2018. "Weighted Shapley hierarchy levels values," MPRA Paper 88160, University Library of Munich, Germany.
    12. Kellner, Florian & Schneiderbauer, Miriam, 2019. "Further insights into the allocation of greenhouse gas emissions to shipments in road freight transportation: The pollution routing game," European Journal of Operational Research, Elsevier, vol. 278(1), pages 296-313.
    13. Karl Ortmann, 2013. "A cooperative value in a multiplicative model," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 21(3), pages 561-583, September.
    14. 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.
    15. René Brink & René Levínský & Miroslav Zelený, 2015. "On proper Shapley values for monotone TU-games," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(2), pages 449-471, May.
    16. Tadeusz Radzik, 2017. "On an extension of the concept of TU-games and their values," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(1), pages 149-170, August.
    17. Li, Xun & Rey, David & Dixit, Vinayak V., 2018. "An axiomatic characterization of fairness in transport networks: Application to road pricing and spatial equity," Transport Policy, Elsevier, vol. 68(C), pages 142-157.
    18. Yoshio Kamijo & Takumi Kongo, 2015. "Properties based on relative contributions for cooperative games with transferable utilities," Theory and Decision, Springer, vol. 78(1), pages 77-87, January.
    19. Frank Huettner, 2015. "A proportional value for cooperative games with a coalition structure," Theory and Decision, Springer, vol. 78(2), pages 273-287, February.
    20. Besner, Manfred, 2019. "Value dividends, the Harsanyi set and extensions, and the proportional Harsanyi payoff," MPRA Paper 92247, University Library of Munich, Germany.
    21. Sergei Pechersky, 2001. "On Proportional Excess for NTU Games," EUSP Department of Economics Working Paper Series 2001/02, European University at St. Petersburg, Department of Economics, revised 30 Oct 2001.
    22. Besner, Manfred, 2017. "Axiomatizations of the proportional Shapley value," MPRA Paper 82990, University Library of Munich, Germany.
    23. Barry Feldman, 2002. "A Dual Model of Cooperative Value," Game Theory and Information 0207001, University Library of Munich, Germany.
    24. Cubukcu, K. Mert, 2020. "The problem of fair division of surplus development rights in redevelopment of urban areas: Can the Shapley value help?," Land Use Policy, Elsevier, vol. 91(C).

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