Stable Sets in multi-good pillage games are small

Alan F. Breardon and Colin Rowat

Discussion Papers from Department of Economics, University of Birmingham

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.

Keywords: pillage games; cooperative game theory; stable sets; Hausdorff dimension (search for similar items in EconPapers)
JEL-codes: C71 D51 P14 (search for similar items in EconPapers)
Pages: 23 pages
Date: 2010-02
New Economics Papers: this item is included in nep-gth
References: View complete reference list from CitEc
Citations: View citations in EconPapers (4)

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