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Pillage Games with Multiple Stable Sets

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  • Simon MacKenzie
  • Manfred Kerber
  • Colin Rowat

Abstract

We prove that pillage games (Jordan in J Econ Theory 131.1:26–44, 2006 , “Pillage and property”, JET) can have multiple stable sets, constructing pillage games with up to $$2^{\tfrac{n-1}{3}}$$ 2 n - 1 3 stable sets, when the number of agents, $$n$$ n , exceeds four. We do so by violating the anonymity axiom common to the existing literature to establish a power dichotomy: for all but a small exceptional set of endowments, powerful agents can overcome all the others; within the exceptional set, the lesser agents can defend their resources. Once the allocations giving powerful agents all resources are included in a candidate stable set, deriving the rest proceeds by considering dominance relations over the finite exceptional sets—reminiscent of stable sets’ derivation in classical cooperative game theory. We also construct a multi-good pillage game with only three agents that also has two stable sets. Copyright The Author(s) 2015
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Suggested Citation

  • Simon MacKenzie & Manfred Kerber & Colin Rowat, 2013. "Pillage Games with Multiple Stable Sets," Discussion Papers 13-07, Department of Economics, University of Birmingham.
  • Handle: RePEc:bir:birmec:13-07
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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.
    2. Beardon, Alan F. & Rowat, Colin, 2013. "Efficient sets are small," Journal of Mathematical Economics, Elsevier, vol. 49(5), pages 367-374.
    3. 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.
    4. R.J. Aumann & S. Hart (ed.), 2002. "Handbook of Game Theory with Economic Applications," Handbook of Game Theory with Economic Applications, Elsevier, edition 1, volume 3, number 3.
    5. J. Jordan, 2009. "Power and efficiency in production pillage games," Review of Economic Design, Springer;Society for Economic Design, vol. 13(3), pages 171-193, September.
    6. R. M. Thrall & W. F. Lucas, 1963. "N‐person games in partition function form," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 10(1), pages 281-298, March.
    7. Rowat, Colin & Kerber, Manfred, 2014. "Sufficient conditions for unique stable sets in three agent pillage games," Mathematical Social Sciences, Elsevier, vol. 69(C), pages 69-80.
    8. Leonard,Robert, 2010. "Von Neumann, Morgenstern, and the Creation of Game Theory," Cambridge Books, Cambridge University Press, number 9780521562669.
    9. Alan F. Breardon & Colin Rowat, 2010. "Stable Sets in multi-good pillage games are small," Discussion Papers 10-05, Department of Economics, University of Birmingham.
    10. Lucas, William F., 1992. "Von Neumann-Morgenstern stable sets," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 17, pages 543-590, Elsevier.
    11. Jordan, J.S., 2006. "Pillage and property," Journal of Economic Theory, Elsevier, vol. 131(1), pages 26-44, November.
    12. Manfred Kerber & Colin Rowat, 2009. "Stable Sets in Three Agent Pillage Games," Discussion Papers 09-07, Department of Economics, University of Birmingham.
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    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. Rowat, Colin & Kerber, Manfred, 2014. "Sufficient conditions for unique stable sets in three agent pillage games," Mathematical Social Sciences, Elsevier, vol. 69(C), pages 69-80.
    3. Núñez, Marina & Vidal-Puga, Juan, 2022. "Stable cores in information graph games," Games and Economic Behavior, Elsevier, vol. 132(C), pages 353-367.

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    More about this item

    Keywords

    pillage games; cooperative game theory; core; stable sets;
    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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