Obtaining linear scores from preference pairs

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Abstract
A score is to be constructed as a linear combination of variables that will reflect an ordering based on expert judgement. Available data are a set of preference-ordered pairs together with values of the variables. The numerical problem is treated as a maximum likelihood solution of a probabilistic formulation; the maximization details are expressed in terms of non-linear least squares. Necessary and sufficient conditions are given for a set of data to determine uniquely an order-preserving linear score.