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Abstract
We introduce efficient sets, a class of sets in Rp in which, in each set, no element is greater in all dimensions than any other. Neither differentiability nor continuity is required of such sets, which include: level sets of utility functions, quasiindifference classes associated with a preference relation not given by a utility function, mean–variance frontiers, production possibility frontiers, and Pareto efficient sets. By Lebesgue’s density theorem, efficient sets have pdimensional measure zero. As Lebesgue measure provides an imprecise description of small sets, we then prove the stronger result that each efficient set in Rp has Hausdorff dimension at most p−1. This may exceed its topological dimension, with the two notions becoming equivalent for smooth sets. We apply these results to stable sets in multigood pillage games: for n agents and m goods, stable sets have dimension at most m(n−1)−1. This implies, and is much stronger than, the result that stable sets have m(n−1)dimensional measure zero, as conjectured by Jordan.
Original language  English 

Pages (fromto)  367374 
Journal  Journal of Mathematical Economics 
Volume  49 
Issue number  5 
Early online date  2 May 2013 
DOIs  
Publication status  Published  Oct 2013 
Keywords
 Efficient sets
 Lebesgue measure
 Hausdorff dimension
 Pillage games
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Dive into the research topics of 'Efficient sets are small'. Together they form a unique fingerprint.Projects
 1 Finished

Weak Property Rights: Financial Markets and Development
Rowat, C. & Dutta, J.
Economic & Social Research Council
1/04/05 → 31/03/09
Project: Research Councils