The Maximin Share Dominance Relation
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Given a finite set X and an ordering ≽ over its subsets, the l-out-of-d maximin-share of X is the maximal (by ≽) subset of X that can be constructed by partitioning X into d parts and picking the worst union of l parts. A pair of integers (l,d) dominates a pair (l',d') if, for any set X and ordering ≽, the l-out-of-d maximin-share of X is at least as good (by ≽) as the l'-out-of-d' maximin-share of X. This note presents a necessary and sufficient condition for deciding whether a given pair of integers dominates another pair, and an algorithm for finding all non-dominated pairs. It compares the l-out-of-d maximin-share to some other criteria for fair allocation of indivisible objects among people with different entitlements.
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