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, 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.
Highlights
Y ∈ Rp, x ≫ y means xi > yi for all i = 1, . . . , p
As a measure theoretic result, it implies the weaker topological result that efficient sets have empty interior.3. While this establishes the usual understanding of nullity, Lebesgue measure fails to differentiate between sets which are of measure zero, but of very different sizes
Our main result is that the Hausdorff dimension of any efficient set in Rp is no more than p − 1
Summary
As a measure theoretic result, it implies the weaker topological result that efficient sets have empty interior.3 While this establishes the usual understanding of nullity, Lebesgue measure fails to differentiate between sets which are of measure zero, but of very different sizes.. Hausdorff’s measure theoretic understanding of dimension is used to establish our second intuition It is, by design, a more sensitive concept than either Lebesgue measure or topological dimension, being defined for any extended non-negative real number. subset E of Rp with p-dimensional Lebesgue measure zero, the Hausdorff dimension can be any number between 0 and p, inclusive (Beardon, 1965). Our main result is that the Hausdorff dimension of any efficient set in Rp is no more than p − 1 This satisfies our intuition that efficient sets are (subsets of) regular curves or surfaces rather than, say, thick sets like the von Koch curve.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
