Пропорционалды бөлу және әділдік критерийлері

the scheduling algorithmthe social choice rule
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. Proportionality was the first fairness criterion studied in the literature; hence it is sometimes called "simple fair division". It was first conceived by Steinhaus.

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.

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. Of course such a 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. So a necessary condition for the existence of a super proportional division is that not all partners have the same value measure. The 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. See super proportional division for details.

One advantage of the proportionality criterion over envy freeness and similar criteria is that it is stable with regards to voluntary exchanges. As an example, assume that a certain land is divided among 3 partners: Alice, Bob and George, in a division that is both proportional and envy free. Several months later, Alice and George decide to merge their land plots and re divide them in a way that is more profitable for them. From Bob's point of view, the division is still proportional, since he still holds a subjective value of at least 1/3 of the total, regardless of what Alice and George do with their plots. On the other hand, the new division might not be envy free. For example, it is possible that initially both Alice and George received a land plot which Bob subjectively values as 1/3, but now after the re division George got all the value (in Bob's eyes) so now Bob envies George. Hence, using envy freeness as the fairness criterion implies that we must constrain the right of people to voluntary exchanges after the division. Using proportionality as the fairness criterion has no such negative implications.

An additional advantage of proportionality is that it is compatible with individual rationality in the following sense. Suppose n partners own a resource in common. In many practical scenarios (though not always), the partners have the option to sell the resource in the market and split the revenues such that each partner receives exactly 1/n. Hence, a rational partner will agree to participate in a division procedure, only if the procedure guarantees that he receives at least 1/n of his total value. Additionally, there should be at least a possibility (if not a guarantee) that the partner receives more than 1/n; this explains the importance of the existence theorems of super proportional division.