An Approach for Quantifying Paired Comparisons and Rank Order

Abstract

Research for the Army demobilization point system evolved a new approach to paired comparisons and rank order. Each of $N$ individuals compares or ranks $n$ things; the problem is to determine a numerical value for each of the $n$ things that will best represent the comparisons in some sense. The new criterion adopted is that the numerical values be determined so as best to distinguish between those things judged higher and those judged lower for each individual. Least-squares is employed in the analysis, and the solution appears in the form of the latent vector associated with the largest root of a matrix obtained from the comparisons or rankings. This approach applies to the conventional problem of ordinary comparisons, the numerical solution being easily obtainable by simple iterations; the conventional use of hypothetical variables and unverified hypotheses is avoided. The Army point system is an example of a new and more complicated class of problems; the same principle for the solution applies here, only more details occur in the derivations and computations.

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