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Stable Sets in multi-good pillage games are small

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  • Alan F. Breardon
  • Colin Rowat

Abstract

It is known that, in one-good pillage games, stable sets are finite. For m goods, it has been conjectured that the stable sets have measure zero. We introduce a class of sets, termed pseudo-indifference sets, which includes level sets of utility functions, quasi-indifference classes associated with a preference relation not given by a utility function, production possibility frontiers, and Pareto efficient sets. We establish the truth of the conjecture by proving that pseudo-indifference sets in Rp have p-dimensional measure zero. This implies that stable sets in n-agent, m-good pillage games have m(n - 1)-dimensional measure zero. We then prove that each pseudo-indifference set in Rp has Hausdorff dimension at most p - 1, a much stronger result than measure zero. Finally, we establish a stronger version of the conjecture: stable sets in n-agent, m-good pillage games have dimension at most m(n-1)-1.

Suggested Citation

  • 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.
    • Handle: RePEc:bir:birmec:10-05
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    Cited by:

    1. Schwarz Mordechai E., 2019. "From Jungle to Civilized Economy: The Power Foundation of Exchange Economy Equilibrium," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 19(2), pages 1-14, June.
    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. Manfred Kerber & Colin Rowat, 2009. "Stable Sets in Three Agent Pillage Games," Discussion Papers 09-07, Department of Economics, University of Birmingham.
    4. 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.

    More about this item

    Keywords

    pillage games; cooperative game theory; stable sets; Hausdorff dimension;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies
    • P14 - Political Economy and Comparative Economic Systems - - Capitalist Economies - - - Property Rights

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