Efficient sets are small

Alan F. Beardon, Colin Rowat

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3 Citations (Scopus)
191 Downloads (Pure)


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, quasi-indifference 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 p-dimensional 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 multi-good 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 languageEnglish
Pages (from-to)367-374
JournalJournal of Mathematical Economics
Issue number5
Early online date2 May 2013
Publication statusPublished - Oct 2013


  • Efficient sets
  • Lebesgue measure
  • Hausdorff dimension
  • Pillage games


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