Mathematical Models for Ranking from Paired Comparisons

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Abstract
Brief mention is made of several models in each of two categories: (I) Each possible ranking of items is assumed to have a “utility” which depends on the expected scores of the items in paired comparisons. In particular, the “worth” of an item may be defined in terms of its expected scores in comparisons with others. (II) Each item is assumed to have an intrinsic worth; these intrinsic worths determine the expected scores. A concept, “regularity” is introduced. Under (I), general linear utilities are discussed, and a necessary and sufficient condition is given in order that a linear utility may be regular. Under (II), a “minimum assumption” model is introduced. Let e(u, v) denote the expected score of an item of worth u when compared with one of worth v. The assumption is: e(u, v) is non-decreasing in u, non-increasing in v. The problem of estimating expected scores in this model is discussed.

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