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Individual weighted excess and least square values

Author

Listed:
  • Xia Zhang

    (NingboTech University
    Northwestern Polytechnical University
    VU University)

  • René van den Brink

    (VU University)

  • Arantza Estévez-Fernández

    (VU University)

  • Hao Sun

    (Northwestern Polytechnical University)

Abstract

This work deals with the weighted excesses of players in cooperative games which are obtained by summing up all the weighted excesses of all coalitions to which they belong. We first show that the resulting payoff vector is the corresponding least square value by lexicographically minimizing the individual weighted excesses of players over the preimputation set, and thus give an alternative characterization of the least square values. Second, we show that these results give rise to lower and upper bounds for the core payoff vectors and, using these bounds, we show that the least square values can be seen as the center of a polyhedron defined by these bounds. This provides a second new characterization of the least square values. Third, we show that the individually rational least square value is the solution that lexicographically minimizes the individual weighted excesses of players over the imputation set.

Suggested Citation

  • Xia Zhang & René van den Brink & Arantza Estévez-Fernández & Hao Sun, 2022. "Individual weighted excess and least square values," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 95(2), pages 281-296, April.
    • Handle: RePEc:spr:mathme:v:95:y:2022:i:2:d:10.1007_s00186-022-00781-1
    DOI: 10.1007/s00186-022-00781-1
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    References listed on IDEAS

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    1. Elisenda Molina & Juan Tejada, 2000. "note: The least square nucleolus is a general nucleolus," International Journal of Game Theory, Springer;Game Theory Society, vol. 29(1), pages 139-142.
    2. Tijs, S., 1981. "Bounds for the core of a game and the t-value," Other publications TiSEM ebc650eb-f25e-4802-ba0b-2, Tilburg University, School of Economics and Management.
    3. Luis Ruiz & Federico Valenciano & José Zarzuelo, 1998. "Some new results on least square values for TU games," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 6(1), pages 139-158, June.
    4. SCHMEIDLER, David, 1969. "The nucleolus of a characteristic function game," LIDAM Reprints CORE 44, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Ruiz, Luis M & Valenciano, Federico & Zarzuelo, Jose M, 1996. "The Least Square Prenucleolus and the Least Square Nucleolus. Two Values for TU Games Based on the Excess Vector," International Journal of Game Theory, Springer;Game Theory Society, vol. 25(1), pages 113-134.
    6. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
    7. Ruiz, Luis M. & Valenciano, Federico & Zarzuelo, Jose M., 1998. "The Family of Least Square Values for Transferable Utility Games," Games and Economic Behavior, Elsevier, vol. 24(1-2), pages 109-130, July.
    8. Krishna Chaitanya Vanam & N. Hemachandra, 2013. "Some Excess-Based Solutions For Cooperative Games With Transferable Utility," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 15(04), pages 1-14.
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