Stable Sets in Multi-Good Pillage Games are Small

23 Pages Posted: 23 Feb 2010

See all articles by Alan Beardon

Alan Beardon

University of Cambridge

Colin Rowat

University of Birmingham - Department of Economics

Date Written: February 16, 2010

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-indiff erence sets, which includes level sets of utility functions, quasi-indiff erence 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 R^p 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-indi fference set in R^p has Hausdor ff 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

JEL Classification: C71, D51, P14

Suggested Citation

Beardon, Alan and Rowat, Colin, Stable Sets in Multi-Good Pillage Games are Small (February 16, 2010). Available at SSRN: https://ssrn.com/abstract=1557316 or http://dx.doi.org/10.2139/ssrn.1557316

Alan Beardon

University of Cambridge ( email )

Department of Pure Mathematics and Mathematical St
Wilberforce Road
Cambridge, CB3 0WB
United Kingdom

Colin Rowat (Contact Author)

University of Birmingham - Department of Economics ( email )

Economics Department
Birmingham, B15 2TT
United Kingdom
+44 121 414 3754 (Phone)
+44 121 414 7377 (Fax)

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