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:''For rank reversals in voting, see Voting paradox.'' In decision-making, a rank reversal is a change in the rank ordering of the preferability of alternative possible decisions when, for example, the method of choosing changes or the set of other available alternatives changes. The issue of rank reversals lies at the heart of many debates in decision-making and multi-criteria decision-making, in particular. Unlike most other computational procedures, it is hard to tell if a particular decision-making method has derived the correct answer or not. Such methods analyze a set of alternatives described in terms of some criteria. They determine which alternative is the best one, or they provide relative weights of how the alternatives perform, or just how the alternatives should be ranked when all the criteria are considered simultaneously. This is exactly where the challenge with decision making exists. Often it is hard, if not practically impossible, to determine whether a ''correct'' answer has been reached or not. With other computational methods, for instance with a job scheduling method, one can examine a set of different answers and then categorize the answers according to some metric of performance (for instance, a project's completion time). But this may not be possible to do with the answers derived by most decision making methods. After all, determining the ''best'' decision making method leads to a decision making paradox. Thus the following question emerges: How can one evaluate decision-making methods? This is a very difficult issue and may not be answered in a globally accepted manner. A critical part in answering this fundamental question is played by what is known as rank reversals. == Rank reversal == One way to test the validity of decision-making methods is to construct special test problems and then study the solutions they derive. If the solutions exhibit some logic contradictions (in the form of undesirable rank reversals of the alternatives), then one may argue that something is wrong with the method that derived them. To see the above point more clearly, suppose that three candidates are evaluated for some job opening. Let us designate these candidates as A, B, and C. Suppose that some decision making method has determined that the best candidate for that job is person A, followed by B, who is followed by C. This is the first ranking and it is indicated as follows: A > B > C (where ''>'' means ''better than''). Next, suppose that candidate B (who is not the best one) is replaced by an even worse candidate, say person D. That is, now we have B > D, and candidate B is replaced by D while candidates A and C remain in the pool of candidates with exactly the same characteristics as before. When the new set of alternatives (i.e., candidates A, D and C) are ranked together and by assuming that the criteria have exactly the same weights as before, then should not candidate A still be the best one? It turns out that under some decision making methods the best alternative may be different now. This is known as a rank reversal and it is one of the types of rank reversals. The first type of rank reversal in the above context was observed by Belton and Gear in 1983 as part of a study of the analytic hierarchy process (AHP). They first considered a simple decision problem comprised by 3 alternatives and 2 criteria. Next a copy of a non-optimal alternative was introduced. When the 4 alternatives (i.e., the previous 3 plus the copy) were evaluated, and under the assumption that the criteria weights are exactly the same as before, it was observed that now the indication of the best alternative can change. That is, a rank reversal may occur with the AHP. A few years later it was observed that the AHP, as well as a new variant to it that was introduced by Professor Thomas Saaty (the inventor of the AHP) in response to the previous observation by Belton and Gear, may exhibit rank reversals when a non-optimal alternative is replaced by a worse one (and not a copy of an alternative as in Belton and Gear’s experiment). The issue of rank reversals has captured the interest of many researchers and practitioners in the field of decision-making. It is something that continues to be considered controversial by many and is debated a lot. 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「Rank reversals in decision-making」の詳細全文を読む スポンサード リンク
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