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In the context of fair cake-cutting, a super-proportional division is a division in which each partner receives strictly more than 1/''n'' of the resource by their own subjective valuation (). A super-proportional division is better than a proportional division, in which each partner is guaranteed to receive at least 1/''n'' (). However, in contrast to proportional division, a super-proportional division does not always exist. When all partners have exactly the same value functions, the best we can do is give each partner exactly 1/''n''. A necessary condition for the existence of a super-proportional division is, therefore, that not all partners have the same value measure. A surprising fact is that, when the valuations are additive and non-atomic, this condition is also sufficient. I.e., when there are at least ''two'' partners whose value function is even slightly different, then there is a super-proportional division in which ''all'' partners receive more than 1/''n''. == Conjecture == The existence of a super-proportional division was first conjectured as early as 1948: ::It may be stated incidentally that if there are two (or more) partners with different estimations, there exists a division giving to everybody more than his due part (Knaster); this fact disproves the common opinion that differences estimations make fair division difficult. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Super-proportional division」の詳細全文を読む スポンサード リンク
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