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On extensions of the core and the anticore of transferable utility games

Author

Listed:
  • Derks, Jean

    () (Department of Knowledge Engineering)

  • Peters, Hans

    () (Department of Quantitative Economics)

  • Sudhölter, Peter

    () (Department of Business and Economics)

Abstract

We consider several related set extensions of the core and the anticore of games with transferable utility. An efficient allocation is undominated if it cannot be improved, in a specific way, by sidepayments changing the allocation or the game. The set of all such allocations is called the undominated set, and we show that it consists of finitely many polytopes with a core-like structure. One of these polytopes is the L1-center, consisting of all efficient allocations that minimize the sum of the absolute values of the excesses. The excess Pareto optimal set contains the allocations that are Pareto optimal in the set obtained by ordering the sums of the absolute values of the excesses of coalitions and the absolute values of the excesses of their complements. The L1-center is contained in the excess Pareto optimal set, which in turn is contained in the undominated set. For three-person games all these sets coincide. These three sets also coincide with the core for balanced games and with the anticore for antibalanced games. We study properties of these sets and provide characterizations in terms of balanced collections of coalitions. We also propose a single-valued selection from the excess Pareto optimal set, the min-prenucleolus, which is defined as the prenucleolus of the minimum of a game and its dual.

Suggested Citation

  • Derks, Jean & Peters, Hans & Sudhölter, Peter, 2012. "On extensions of the core and the anticore of transferable utility games," Discussion Papers of Business and Economics 4/2012, University of Southern Denmark, Department of Business and Economics.
  • Handle: RePEc:hhs:sdueko:2012_004
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    References listed on IDEAS

    as
    1. Bossert, Walter & Derks, Jean & Peters, Hans, 2005. "Efficiency in uncertain cooperative games," Mathematical Social Sciences, Elsevier, vol. 50(1), pages 12-23, July.
    2. Peleg, B, 1986. "On the Reduced Game Property and Its Converse," International Journal of Game Theory, Springer;Game Theory Society, vol. 15(3), pages 187-200.
    3. Guni Orshan & Peter Sudhölter, 2010. "The positive core of a cooperative game," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(1), pages 113-136, March.
    4. Derks, Jean & Peters, Hans, 1998. "Orderings, excess functions, and the nucleolus," Mathematical Social Sciences, Elsevier, vol. 36(2), pages 175-182, September.
    5. Camelia Bejan & Juan Gómez, 2009. "Core extensions for non-balanced TU-games," International Journal of Game Theory, Springer;Game Theory Society, vol. 38(1), pages 3-16, March.
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    Citations

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    Cited by:

    1. Michel Grabisch & Peter Sudhölter, 2016. "Characterizations of solutions for games with precedence constraints," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(1), pages 269-290, March.
    2. Michel Grabisch & Peter Sudhölter, 2014. "The positive core for games with precedence constraints," Documents de travail du Centre d'Economie de la Sorbonne 14036, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    3. Chen, Haoxun, 2017. "Undominated nonnegative excesses and core extensions of transferable utility games," European Journal of Operational Research, Elsevier, vol. 261(1), pages 222-233.

    More about this item

    Keywords

    Transferable utility game; core; anticore; core extension; min-prenucleolus;

    JEL classification:

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

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