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Harsanyi Power Solutions for Games on Union Stable Systems

Author

Listed:
  • Encarnacion Algaba

    (University of Seville)

  • Jesus Mario Bilbao

    (University of Seville)

  • Rene van den Brink

    (VU University Amsterdam)

Abstract

This discussion paper resulted in a publication in Annals of Operations Research , 2015, 225, 27-44. This paper analyzes Harsanyi power solutions for cooperative games in which partial cooperation is based on union stable systems. These structures contain as particular cases the widely studied communication graph games and permission structures, among others. In this context, we provide axiomatic characterizations of the Harsanyi power solutions which distribute the Harsanyi dividends proportional to weights determined by a power measure for union stable systems. Moreover, the well-known Myerson value is exactly the Harsanyi power solution for the equal power measure, and on a special subclass of union stable systems the position value coincides with the Harsanyi power solution obtained for the influence power measure.

Suggested Citation

  • Encarnacion Algaba & Jesus Mario Bilbao & Rene van den Brink, 2011. "Harsanyi Power Solutions for Games on Union Stable Systems," Tinbergen Institute Discussion Papers 11-182/1, Tinbergen Institute.
    • Handle: RePEc:tin:wpaper:20110182
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    References listed on IDEAS

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    1. E. Algaba & J. M. Bilbao & P. Borm & J. J. López, 2000. "The position value for union stable systems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 52(2), pages 221-236, November.
    2. van den Nouweland, Anne & Borm, Peter & Tijs, Stef, 1992. "Allocation Rules for Hypergraph Communication Situations," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(3), pages 255-268.
    3. Jean Derks & Gerard Laan & Valery Vasil’ev, 2010. "On the Harsanyi payoff vectors and Harsanyi imputations," Theory and Decision, Springer, vol. 68(3), pages 301-310, March.
    4. Borm, P.E.M. & Owen, G. & Tijs, S.H., 1992. "On the position value for communication situations," Other publications TiSEM 5a8473e4-1df7-42df-ad53-f, Tilburg University, School of Economics and Management.
    5. Gilles, Robert P & Owen, Guillermo & van den Brink, Rene, 1992. "Games with Permission Structures: The Conjunctive Approach," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(3), pages 277-293.
    6. Faigle, U. & Grabisch, M. & Heyne, M., 2010. "Monge extensions of cooperation and communication structures," European Journal of Operational Research, Elsevier, vol. 206(1), pages 104-110, October.
    7. René Brink & Ilya Katsev & Gerard Laan, 2011. "Axiomatizations of two types of Shapley values for games on union closed systems," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 47(1), pages 175-188, May.
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    9. E. Algaba & J. M. Bilbao & J. J. López, 2004. "The position value in communication structures," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 59(3), pages 465-477, July.
    10. Encarnacion Algaba & Jesus Mario Bilbao & Rene van den Brink & Jorge J. Lopez, 2011. "The Myerson Value and Superfluous Supports in Union Stable Systems," Tinbergen Institute Discussion Papers 11-127/1, Tinbergen Institute.
    11. René Brink & Gerard Laan & Vitaly Pruzhansky, 2011. "Harsanyi power solutions for graph-restricted games," International Journal of Game Theory, Springer;Game Theory Society, vol. 40(1), pages 87-110, February.
    12. E. Algaba & J. M. Bilbao & P. Borm & J. J. López, 2001. "The Myerson value for union stable structures," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 54(3), pages 359-371, December.
    13. Rene van den Brink & Ilya Katsev & Gerard van der Laan, 2011. "Games on Union Closed Systems," Tinbergen Institute Discussion Papers 11-036/1, Tinbergen Institute.
    14. Martin J. Conyon & Mark R. Muldoon, 2006. "The Small World of Corporate Boards," Journal of Business Finance & Accounting, Wiley Blackwell, vol. 33(9‐10), pages 1321-1343, November.
    15. Faigle, U & Kern, W, 1992. "The Shapley Value for Cooperative Games under Precedence Constraints," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(3), pages 249-266.
    16. Martin J. Conyon & Mark R. Muldoon, 2006. "The Small World of Corporate Boards," Journal of Business Finance & Accounting, Wiley Blackwell, vol. 33(9-10), pages 1321-1343.
    17. E. Algaba & J. M. Bilbao & R. Brink & J. J. López, 2012. "The Myerson Value and Superfluous Supports in Union Stable Systems," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 650-668, November.
    18. 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.
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    Citations

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    Cited by:

    1. Encarnación Algaba & Stefano Moretti & Eric Rémila & Philippe Solal, 2021. "Lexicographic solutions for coalitional rankings," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 57(4), pages 817-849, November.
    2. Encarnacion Algaba & Rene van den Brink, 2021. "Networks, Communication and Hierarchy: Applications to Cooperative Games," Tinbergen Institute Discussion Papers 21-019/IV, Tinbergen Institute.
    3. Liying Kang & Anna Khmelnitskaya & Erfang Shan & Dolf Talman & Guang Zhang, 2021. "The average tree value for hypergraph games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(3), pages 437-460, December.
    4. Encarnaciön Algaba & Sylvain Béal & Eric Rémila & Phillippe Solal, 2018. "Harsanyi power solutions for cooperative games on voting structures," Working Papers 2018-05, CRESE.
    5. C. Manuel & E. Ortega & M. del Pozo, 2023. "Marginality and the position value," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(2), pages 459-474, July.
    6. Clinton Gubong Gassi, 2024. "A Characterization of the Myerson value for cooperative games on voting structures," Working Papers 2024-10, CRESE.
    7. Rene van den Brink & Ilya Katsev & Gerard van der Laan, 2023. "Properties of Solutions for Games on Union-Closed Systems," Mathematics, MDPI, vol. 11(4), pages 1-16, February.
    8. Encarnacion Algaba & Rene van den Brink, 2019. "The Shapley Value and Games with Hierarchies," Tinbergen Institute Discussion Papers 19-064/II, Tinbergen Institute.
    9. Encarnacion Algaba & Jesus Mario Bilbao & Rene van den Brink & Jorge J. Lopez, 2011. "The Myerson Value and Superfluous Supports in Union Stable Systems," Tinbergen Institute Discussion Papers 11-127/1, Tinbergen Institute.
    10. Encarnación Algaba & René Brink & Chris Dietz, 2018. "Network Structures with Hierarchy and Communication," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 265-282, October.
    11. E. Algaba & J. M. Bilbao & R. Brink & J. J. López, 2012. "The Myerson Value and Superfluous Supports in Union Stable Systems," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 650-668, November.
    12. Zhengxing Zou & Qiang Zhang, 2018. "Harsanyi power solution for games with restricted cooperation," Journal of Combinatorial Optimization, Springer, vol. 35(1), pages 26-47, January.

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    More about this item

    Keywords

    Cooperative TU-game; Union stable system; Harsanyi dividend; Power measure; Harsanyi power solution; Myerson value; Position value;
    All these keywords.

    JEL classification:

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

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