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Prosperity properties of TU-games

Author

Listed:
  • J. R. G. van Gellekom

    (Department of Mathematics, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands)

  • J. A. M. Potters

    (Department of Mathematics, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands)

  • J. H. Reijnierse

    (Department of Mathematics, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands)

Abstract

An important open problem in the theory of TU-games is to determine whether a game has a stable core (Von Neumann-Morgenstern solution (1944)). This seems to be a rather difficult combinatorial problem. There are many sufficient conditions for core-stability. Convexity is probably the best known of these properties. Other properties implying stability of the core are subconvexity and largeness of the core (two properties introduced by Sharkey (1982)) and a property that we have baptized extendability and is introduced by Kikuta and Shapley (1986). These last three properties have a feature in common: if we start with an arbitrary TU-game and increase only the value of the grand coalition, these properties arise at some moment and are kept if we go on with increasing the value of the grand coalition. We call such properties prosperity properties. In this paper we investigate the relations between several prosperity properties and their relation with core-stability. By counter examples we show that all the prosperity properties we consider are different.

Suggested Citation

  • J. R. G. van Gellekom & J. A. M. Potters & J. H. Reijnierse, 1999. "Prosperity properties of TU-games," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(2), pages 211-227.
    • Handle: RePEc:spr:jogath:v:28:y:1999:i:2:p:211-227
    Note: Received: June 1998/Revised version: December 1998
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    Citations

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    Cited by:

    1. Schouten, Jop & Dietzenbacher, Bas & Borm, Peter, 2019. "The Nucleolus and Inheritance of Properties in Communication Situations," Discussion Paper 2019-008, Tilburg University, Center for Economic Research.
    2. Núñez, Marina & Vidal-Puga, Juan, 2022. "Stable cores in information graph games," Games and Economic Behavior, Elsevier, vol. 132(C), pages 353-367.
    3. Dietzenbacher, Bas, 2019. "The Procedural Egalitarian Solution and Egalitarian Stable Games," Discussion Paper 2019-007, Tilburg University, Center for Economic Research.
    4. Estévez-Fernández, Arantza, 2012. "New characterizations for largeness of the core," Games and Economic Behavior, Elsevier, vol. 76(1), pages 160-180.
    5. Marina Núñez & Tamás Solymosi, 2017. "Lexicographic allocations and extreme core payoffs: the case of assignment games," Annals of Operations Research, Springer, vol. 254(1), pages 211-234, July.
    6. Shellshear, Evan & Sudhölter, Peter, 2009. "On core stability, vital coalitions, and extendability," Games and Economic Behavior, Elsevier, vol. 67(2), pages 633-644, November.
    7. Jesús Getán & Josep Izquierdo & Jesús Montes & Carles Rafels, 2015. "The bargaining set for almost-convex games," Annals of Operations Research, Springer, vol. 225(1), pages 83-89, February.
    8. Hirai, Toshiyuki & Watanabe, Naoki, 2018. "von Neumann–Morgenstern stable sets of a patent licensing game: The existence proof," Mathematical Social Sciences, Elsevier, vol. 94(C), pages 1-12.
    9. Jesús Getán & Jesús Montes & Carles Rafels, 2014. "A note: characterizations of convex games by means of population monotonic allocation schemes," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(4), pages 871-879, November.
    10. Dietzenbacher, Bas, 2020. "Monotonicity and Egalitarianism (revision of CentER DP 2019-007)," Discussion Paper 2020-003, Tilburg University, Center for Economic Research.
    11. Dinko Dimitrov & Emiliya A. Lazarova & Shao-Chin Sung, 2016. "Inducing stability in hedonic games," University of East Anglia School of Economics Working Paper Series 2016-09, School of Economics, University of East Anglia, Norwich, UK..
    12. Calleja, Pedro & Llerena, Francesc & Sudhölter, Peter, 2021. "Constrained welfare egalitarianism in surplus-sharing problems," Mathematical Social Sciences, Elsevier, vol. 109(C), pages 45-51.
    13. Calleja, Pedro & Rafels, Carles & Tijs, Stef, 2009. "The aggregate-monotonic core," Games and Economic Behavior, Elsevier, vol. 66(2), pages 742-748, July.
    14. van Velzen, Bas & Hamers, Herbert & Solymosi, Tamas, 2008. "Core stability in chain-component additive games," Games and Economic Behavior, Elsevier, vol. 62(1), pages 116-139, January.
    15. Pedro Calleja & Carles Rafels & Stef Tijs, 2006. "The Aggregate-Monotonic Core," Working Papers 280, Barcelona School of Economics.
    16. J. Schouten & B. Dietzenbacher & P. Borm, 2022. "The nucleolus and inheritance of properties in communication situations," Annals of Operations Research, Springer, vol. 318(2), pages 1117-1135, November.
    17. Dylan Laplace Mermoud, 2023. "Geometry of Set Functions in Game Theory: Combinatorial and Computational Aspects," Papers 2301.02950, arXiv.org, revised Oct 2023.
    18. Thomas Bietenhader & Yoshio Okamoto, 2006. "Core Stability of Minimum Coloring Games," Mathematics of Operations Research, INFORMS, vol. 31(2), pages 418-431, May.
    19. Dietzenbacher, Bas, 2021. "Monotonicity and egalitarianism," Games and Economic Behavior, Elsevier, vol. 127(C), pages 194-205.

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