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Sufficient Conditions for the Unique Stable Sets in Three Agent Pillage Games

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  • Manfred Kerber
  • Colin Rowat

Abstract

Pillage games [Jordan, 2006, "Pillage and Property", JET] have two features that make them richer than cooperative games in either characteristic or partition function form: they allow power externalities between coalitions; they allow resources to contribute to coalitions' power as well as to their utility. Extending von Neumann and Morgenstern's analysis of three agent games in characteristic function form to anonymous pillage games, we find: when the core is non-empty, it must take one of five forms; all such games with an empty core represent the same dominance relation. When a stable set exists, and the game also satisfies a continuity and a responsiveness axiom, it is unique and contains no more than 15 elements, a tight bound. By contrast, stable sets in three agent games in characteristic or partition function form may not be unique, and may contain continua. Finally, we provide an algorithm for computing the stable set, and can easily decide non-existence. Thus, in addition to offering attractive modelling possibilities, pillage games seem well behaved and analytically tractable, overcoming a difficulty that has long impeded use of cooperative game theory's flexibility.

Suggested Citation

  • Manfred Kerber & Colin Rowat, 2012. "Sufficient Conditions for the Unique Stable Sets in Three Agent Pillage Games," Discussion Papers 12-11, Department of Economics, University of Birmingham.
    • Handle: RePEc:bir:birmec:12-11
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    References listed on IDEAS

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    1. Manfred Kerber & Colin Rowat, 2011. "A Ramsey bound on stable sets in Jordan pillage games," International Journal of Game Theory, Springer;Game Theory Society, vol. 40(3), pages 461-466, August.
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    6. Simon MacKenzie & Manfred Kerber & Colin Rowat, 2015. "Pillage games with multiple stable sets," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(4), pages 993-1013, November.
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    Cited by:

    1. Simon MacKenzie & Manfred Kerber & Colin Rowat, 2015. "Pillage games with multiple stable sets," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(4), pages 993-1013, November.
    2. Manfred Kerber & Colin Rowat & Naoki Yoshihara, 2023. "Asymmetric majority pillage games," International Journal of Game Theory, Springer;Game Theory Society, vol. 52(4), pages 1009-1035, December.
    3. Toshiji Miyakawa, 2017. "The farsighted core in a political game with asymmetric information," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 49(1), pages 205-229, June.

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    More about this item

    Keywords

    co-operative game theory; stable sets; algorithm; core;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • P14 - Political Economy and Comparative Economic Systems - - Capitalist Economies - - - Property Rights

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