Pairwise comparison

Pairwise comparison is any process of comparing things in pairs to judge which of two things is preferred, or has a greater amount of some something, or whether or not the two things 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.

Overview

If someone shows what they like more between two options, this preference can be expressed as a pairwise comparison. If the two alternatives are x and y, these three pairwise comparisons are possible:

The person prefers x over y: "x > y" or "xPy"

The person prefers y over x: "y > x" or "yPx"

The person has no preference between both alternatives: "x = y" or "xIy"

Transitivity

It is generally assumed that pairwise comparisons are transitive. Most agree on what transitivity is, though there is debate about the transitivity of indifference. The rules of transitivity are as follows.

1. If xPy and yPz, then xPz
2. If xPy and yIz, then xPz
3. If xIy and yPz, then xPz
4. If xIy and yIz, then xIz

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

Argument for intransitivity of indifference

Some believe that indifference (lack of preference) 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 that are the same except for the following. Suppose B is larger than A, but it is not possible to see this without an extremely sensitive measuring device. Also suppose C is larger than B, but this also can't be seen without a sensitive measuring device. However, the difference in sizes between apples A and C is large enough that you can see it. 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 a sensitive measuring device. Because of this, when shown only A and B, you have no preference between apple A and apple B; and you are indifferent between apple B and apple C when shown only B and C. However, when the pair A and C are shown, you prefer C over A.

Preference orders

If pairwise comparisons are transitive by the four rules shown above, then pairwise comparisons for a list of alternatives (A₁, A₂, A₃, ..., A_n−1, and A_n) can take the form:

A₁(>XOR=)A₂(>XOR=)A₃(>XOR=) ... (>XOR=)A_n−1(>XOR=)A_n

For example, if there are three alternatives a, b, and c, then the 13 possible preference orders are:

• [math]\displaystyle{ a\gt b\gt c }[/math]
• [math]\displaystyle{ a\gt c\gt b }[/math]
• [math]\displaystyle{ b\gt a\gt c }[/math]
• [math]\displaystyle{ b\gt c\gt a }[/math]
• [math]\displaystyle{ c\gt a\gt b }[/math]
• [math]\displaystyle{ c\gt b\gt a }[/math]
• [math]\displaystyle{ a\gt b=c }[/math]
• [math]\displaystyle{ b=c\gt a }[/math]
• [math]\displaystyle{ b\gt a=c }[/math]
• [math]\displaystyle{ a=c\gt b }[/math]
• [math]\displaystyle{ c\gt a=b }[/math]
• [math]\displaystyle{ a=b\gt c }[/math]
• [math]\displaystyle{ a=b=c }[/math]

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 number of total preorders. It can be expressed as a function of n:

[math]\displaystyle{ \sum_{k=1}^n k! S_2(n,k), }[/math]

where S₂(n, k) is the Stirling number of the second kind.

Related pages

1. Analytic Hierarchy Process
2. Law of comparative judgment
3. Potentially all pairwise rankings of all possible alternatives (PAPRIKA) method
4. PROMETHEE pairwise comparison method
5. Preference (economics)
6. Stochastic Transitivity
7. Condorcet method

References

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3. Y. Chevaleyre, P.E. Dunne, U. Endriss, J. Lang, M. Lemaître, N. Maudet, J. Padget, S. Phelps, J.A. Rodríguez-Aguilar, and P. Sousa. Issues in Multiagent Resource Allocation. Informatica, 30:3–31, 2006.

Further reading

1. Bradley, R.A. and Terry, M.E. (1952). Rank analysis of incomplete block designs, I. the method of paired comparisons. Biometrika, 39, 324–345.
2. David, H.A. (1988). The Method of Paired Comparisons. New York: Oxford University Press.
3. Luce, R.D. (1959). Individual Choice Behaviours: A Theoretical Analysis. New York: J. Wiley.
4. Thurstone, L.L. (1927). A law of comparative judgement. Psychological Review, 34, 278–286.
5. Thurstone, L.L. (1929). The Measurement of Psychological Value. In T.V. Smith and W.K. Wright (Eds.), Essays in Philosophy by Seventeen Doctors of Philosophy of the University of Chicago. Chicago: Open Court.
6. Thurstone, L.L. (1959). The Measurement of Values. Chicago: The University of Chicago Press.
7. Zermelo, E. (1928). Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift 29, 1929, S. 436–460