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Stable Sets in multi-good pillage games are small

Listed author(s):
  • Alan F. Breardon
  • Colin Rowat

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.

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Paper provided by Department of Economics, University of Birmingham in its series Discussion Papers with number 10-05.

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Length: 23 pages
Date of creation: Feb 2010
Handle: RePEc:bir:birmec:10-05
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