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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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    File URL: ftp://ftp.bham.ac.uk/pub/RePEc/pdf/13-07.pdf
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    References listed on IDEAS

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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. 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.
    3. 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.
    4. Leonard,Robert, 2010. "Von Neumann, Morgenstern, and the Creation of Game Theory," Cambridge Books, Cambridge University Press, number 9780521562669, May.
    5. 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.
    6. 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.
    7. Manfred Kerber & Colin Rowat, 2009. "Stable Sets in Three Agent Pillage Games," Discussion Papers 09-07, Department of Economics, University of Birmingham.
    8. Beardon, Alan F. & Rowat, Colin, 2013. "Efficient sets are small," Journal of Mathematical Economics, Elsevier, vol. 49(5), pages 367-374.
    9. 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.
    10. Jordan, J.S., 2006. "Pillage and property," Journal of Economic Theory, Elsevier, vol. 131(1), pages 26-44, 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, 2012. "Sufficient Conditions for the Unique Stable Sets in Three Agent Pillage Games," Discussion Papers 12-11, Department of Economics, University of Birmingham.
    3. 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.

    More about this item

    Keywords

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

    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 - Economic Systems - - Capitalist Systems - - - Property Rights

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