Fair Division Under Interval Uncertainty
نویسندگان
چکیده
It is often necessary to divide a certain amount of money between n participants i e to assign to each participant a certain portion wi of the whole sum so that w wn In some situations from the fairness requirements we can uniquely determine these weights wi However in some other situations general considerations do not allow us to uniquely determine these weights we only know the intervals w i w i of possible fair weights We show that natural fairness requirements enable us to choose unique weights from these intervals as a result we present an algorithm for fair division under interval uncertainty Introduction to the Problem The general problem of fair division It is often necessary to divide a certain amount of money between n participants i e to assign to each partic ipant a certain portion wi of the whole sum so that w wn In some situations we do not know the exact weights In some situa tions from the fairness requirements we can uniquely determine the weights wi However in some other situations general considerations do not allow us to uniquely determine these weights we only know the intervals wi w i w i of possible fair weights Formulation of the problem We want to select some values wi wi for which w wn and assign to each participant wi th portion of the divided sum How can we do that fairly Comment Before we choose the weights from the given intervals wi we must be sure that such a choice is possible i e that the given intervals wi are con sistent This consistency condition can be easily expressed in terms of a double inequality From w i wi w i we conclude that W w w n w wn w w n W Thus the given intervals must satisfy the inequality W W Vice versa if W W then the interval W W of possible values of w wn contains and therefore can be represented as w wn for some wi wi So this consistency condition is equivalent to W W Towards a Formalization of the Problem We want to describe a transformation T that maps every nite consistent se quence of intervals wi i n into a sequence of exactly as many real values wi wi w wn w wn in such a way that for the resulting sequence of real numbers w wn There are some natural properties that we expect from this transformation First the distribution must be fair it must not depend on the order in which we presented the participants A participant who was assigned could as well be assigned and vice versa Therefore the desired function should not change if we simply swap i th and j th participants If w wi wi wi wj wj wj wn w wi wi wi wj wj wj wn then w wi wj wi wj wi wj wn w wi wj wi wj wi wj wn The second property is related to the following fact two participants should neither gain nor lose simply by joining together If we know the exact weights w and w of each of the original participants then the weight of their combination is equal to w w If we do not know the exact weights of each participant i e if we only know the intervals of possible values w w w and w w w of these weights then the weight of their combination can take any value w w where w w and w w This set of possible values is known to be also an interval with the bounds w w w w In interval computations see e g this new interval is called the sum of the two intervals w and w and denoted by w w Ideally the division should not change if we simply combine two participants In other words If w w w wn w w w wn then w w w wn w w w wn Finally small changes in the endpoints w i or w i should not drastically a ect the resulting division In other words we want the transformation T to be continuous for any given n De nitions and the Main Result De nition We say that a sequence of intervals wi w i w i i n is consistent if w w n w w n De nition By a division under interval uncertainty we mean a transformation T that transforms every consistent nite sequence of intervals w wn into a sequence of real numbers wi wi for which w wn We say that a division is fair if it is continuous and satis es the conditions Theorem There exists exactly one fair division under interval uncertainty and this fair division has the form wi W W W w i W W W w i where W w w n and W w w n
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ورودعنوان ژورنال:
- International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems
دوره 8 شماره
صفحات -
تاریخ انتشار 2000