A computer program to evaluate paired-comparison transitivity

Abstract

The method of paired comparisons is a useful assessment technique when a simple task is required or when the underlying transitivity of elements being compared is suspect. The transitivity of elements can be assessed by evaluating all possible triples or, where n signifies the number of elements being compared, n(n -1)(n 2)16 triples. The number of transitive triples (e.g., A> B, B > C, A> C) can be determined from the row marginals of the pairedcomparison scoring matrix (see Harary, Norman, & Cartwright, 1965), but determining exactly which triples are transitive or cyclic (A > B, B > C, C > A) can be a formidable task. When n is 5, the number of triples is a manageable 10, but when n is 10, the number of triples is 120, and when n is 20, the number of triples is 1,140. The present program, written in FORTRAN IV, evaluates all possible triples, prints all triples that are cyclic, and provides several statistics relating to transitivity. Input. The first card indicates the size (i.e., order) of the paired-comparison scoring matrix (n can range from 3 to 25), followed by cards representing the paired-comparison scoring matrix; each card signifies one row of the matrix. In a paired-comparison scoring matrix, an entry of 1 in the ith row and jth column indicates that the ith element was preferred to the jth element; an entry of 0 indicates the opposite. Output. The output consists of the printing of input data, followed by a listing of all cyclic triples and indexes relating to transitivity, The following is an example of a cyclic triple: