Pairwise comparison

Pairwise comparison generally is any process of comparing entities in pairs to judge which of each entity is preferred, or has a greater amount of some quantitative property, or whether or not the two entities are identical. The method of pairwise comparison is used in the scientific study of preferences, attitudes, voting systems, social choice, public choice, requirements engineering and multiagent AI systems. In psychology literature, it is often referred to as paired comparison. Pairwise comparison generally is any process of comparing entities in pairs to judge which of each entity is preferred, or has a greater amount of some quantitative property, or whether or not the two entities are identical. The method of pairwise comparison is used in the scientific study of preferences, attitudes, voting systems, social choice, public choice, requirements engineering and multiagent AI systems. In psychology literature, it is often referred to as paired comparison. Prominent psychometrician L. L. Thurstone first introduced a scientific approach to using pairwise comparisons for measurement in 1927, which he referred to as the law of comparative judgment. Thurstone linked this approach to psychophysical theory developed by Ernst Heinrich Weber and Gustav Fechner. Thurstone demonstrated that the method can be used to order items along a dimension such as preference or importance using an interval-type scale. If an individual or organization expresses a preference between two mutually distinct alternatives, this preference can be expressed as a pairwise comparison. If the two alternatives are x and y, the following are the possible pairwise comparisons: The agent prefers x over y: 'x > y' or 'xPy' The agent prefers y over x: 'y > x' or 'yPx' The agent is indifferent between both alternatives: 'x = y' or 'xIy' In terms of modern psychometric theory, Thurstone's approach, called the law of comparative judgment, is more aptly regarded as a measurement model. The Bradley–Terry–Luce (BTL) model (Bradley & Terry, 1952; Luce, 1959) is often applied to pairwise comparison data to scale preferences. The BTL model is identical to Thurstone's model if the simple logistic function is used. Thurstone used the normal distribution in applications of the model. The simple logistic function varies by less than 0.01 from the cumulative normal ogive across the range, given an arbitrary scale factor. In the BTL model, the probability that object j is judged to have more of an attribute than object i is: where δ i {displaystyle delta _{i}} is the scale location of object i {displaystyle i} ; σ {displaystyle sigma } is the logistic function (the inverse of the logit). For example, the scale location might represent the perceived quality of a product, or the perceived weight of an object.