Evaluation of rankings with regard to the possible number of agreements and conflicts

An interesting problem in group decision analysis is how many different agreements can occur, or conversely disagreements may exist, between two or more different rankings of a set of alternatives. In this paper it is assumed that a reference ranking has been established for the set of alternatives. This reference ranking may represent the ranking of a high authority decision maker or be just a virtual ranking to be used in determining the discrepancy between pairs of rankings. Then, the problem examined here is to evaluate the number of possible rankings when the ranking method is the number of agreements with some reference ranking. The analysis presented here illustrates that this problem is not trivial and moreover, its simple context conceals complexity in its depth. The purpose of this paper is to provide an evaluation of the number of possible agreements in rankings given to a set of concepts, alternatives or ideas, by two or more decision makers. The number of possible agreements takes on the values 0, 1, 2, ..., n–2, or n when n concepts are compared. This paper develops a recursive closed form formula for calculating the frequencies for the various numbers of agreements.