Pairwise comparison

Pairwise comparison generally refers to any process of comparing entities in pairs to judge which of each pair is preferred, or has a greater amount of some quantitative property. The method of pairwise comparison is used in the scientific study of preferences, attitudes, voting systems, social choice, public choice, 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.

Overview
If a decision agent, such as an individual or organization or even an agent within a multiagent AI system, must express a preference between two mutually distinct alternatives to meet some objective, 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"

Probabilistic models for pairwise comparison
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:

$$ \Pr \{X_{ji}=1\} =\frac{e^{{\delta_j} - {\delta_i}}}{1 + e^{{\delta_j} - {\delta_i}}}, $$

where $$\delta_i$$ is the scale location of object i. For example, the scale location might represent the perceived quality of a product, or the perceived weight of an object.

The BTL is very closely related to the Rasch model for measurement.

Thurstone used the method of pairwise comparisons as an approach to measuring perceived intensity of physical stimuli, attitudes, preferences, choices, and values. He also studied implications of the theory he developed for opinion polls and political voting (Thurstone, 1959).

Transitivity of Pairwise Comparisons
For a given decision agent, if the information, objective, and alternatives used by the agent remain constant, then it is generally assumed that pairwise comparisons over those alternatives by the decision agent are transitive. Most agree upon what transitivity is, though there is debate about the transitivity of indifference. The rules of transitivity are as follows for a given decision agent.

If xPy and yPz, then xPz

If xPy and yIz, then xPz

If xIy and yPz, then xPz

If xIy and yIz, then xIz

This corresponds to (xPy or xIy) being a total preorder, P being the corresponding strict weak order, and I being the corresponding equivalence relation.

Probabilistic models require transitivity only within the bounds of errors of estimates of scale locations of entities. Thus, decisions need not be deterministically transitive in order to apply probabilistic models. However, transitivity will generally hold for a large number of comparisons if models such as the BTL can be effectively applied.

Argument for Intransitivity of Indifference
Some contend that indifference is not transitive. Consider the following example. Suppose you like apples and you prefer apples that are larger. Now suppose there exists an apple A, an apple B, and an apple C which have identical intrinsic characterisics except for the following. Suppose B is larger than A, but it is not discernible without an extremely sensitive scale. Further suppose C is larger than B, but this also is not discernible without an extremely sensitive scale. However, the difference in sizes between apples A and C is large enough that you can discern that C is larger than A without a sensitive scale. In psychophysical terms, the size difference between A and C is above the just noticeable difference ('jnd') while the size differences between A and B and B and C are below the jnd.

You are confronted with the three apples in pairs without the benefit of a sensitive scale. Therefore, when presented A and B alone, you are indifferent between apple A and apple B; and you are indifferent between apple B and apple C when presented B and C alone. However, when the pair A and C are shown, you prefer C over A.

Preference orders
If pairwise comparisons are in fact transitive in respect to the four mentioned rules, then pairwise comparisons for a list of alternatives (A1, A2, A3,...,An-1, and An) can take the form:

A1(>XOR=)A2(>XOR=)A3(>XOR=)...(>XOR=)An-1(>XOR=)An

For example, if there are three alternatives a, b, and c, then the possible preference orders are:

a>b>c

a>c>b

b>a>c

b>c>a

c>a>b

c>b>a

a>b=c

b=c>a

b>a=c

a=c>b

c>a=b

a=b>c

a=b=c

If the number of alternatives is n, and indifference is not allowed, then the number of possible preference orders for any given n-value is n!. If indifference is allowed, then the number of possible preference orders is the. It can be expressed as a function of n:

$$\sum_{k=1}^n k!*S_2(n,k)$$, where S2(n, k) is the Stirling number of the second kind.