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A proportional division is a kind of fair division in which a resource is divided among ''n'' partners with subjective valuations, giving each partner at least 1/''n'' of the resource by his/her own subjective valuation. For example, consider a land asset that has to be divided among 3 heirs: Alice and Bob who think that it's worth 3 million dollars, and George who thinks that it's worth $4.5M. In a proportional division, Alice receives a land-plot that she believes to be worth at least $1M, Bob receives a land-plot that ''he'' believes to be worth at least $1M (even though Alice may think it is worth less), and George receives a land-plot that he believes to be worth at least $1.5M. Proportionality was the first fairness criterion studied in the literature; hence it is sometimes called "simple fair division". == Existence == A proportional division does not always exist. For example, if the resource contains several indivisible items and the number of people is larger than the number of items, then some people will get no item at all and their value will be zero. A proportional division is guaranteed to exist if the following conditions hold: * The valuations of the players are ''non-atomic'', i.e., there are no indivisible elements with positive value. * The valuations of the players are ''additive'', i.e., when a piece is divided, the sum of a piece is equal to the sum of its parts. Hence, proportional division is usually studied in the context of fair cake-cutting. See proportional cake-cutting for detailed information about procedures for achieving a proportional division in the context of cake-cutting. A more lenient fairness criterion is ''partial proportionality'', in which each partner receives a certain fraction ''f''(''n'') of the total value, where ''f''(''n'') ≤ 1/''n''. Partially proportional divisions exist (under certain conditions) even for indivisible items. 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「proportional division」の詳細全文を読む スポンサード リンク
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