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Pairwise Comparisons, Incomparability and Partial Orders

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Pairwise Comparisons, Incomparability and Partial Orders PAIRWISE COMPARISONS, INCOMPARABILITY AND PARTIAL ORDERS Ryszard Janicki∗ Department of Computing and Software, McMaster University, Hamilton, Ontario, Canada L8S 4K1 Keywords: Decision theory, ranking, pairwise comparisons, weak and partial orders. Abstract: A new approach to Pairwise Comparisons based Ranking is presented. An abstract model based on the concept of partial order instead of numerical scale is introduced and analysed. Many faults of traditional, numerical- scale based models are discussed. The importance of the concept of equal importance or indifference is discussed. 1 INTRODUCTION rankings. The strange and contradictory examples of rank- The method of Pairwise Comparisons could be traced ings obtained by AHP cited in the literature (see to Marquis de Condorcet 1785 paper (see (Arrow, (Dyer, 1990)) follow from the following sources: 1951)), was explicitly mentioned and analysed by • When two objects are being compared, interval Fechner in 1860 (Fechner, 1860), made popular by scales are mainly used; but later, when recipro- Thurstone in 1927 (Thurstone, 1927), and was trans- cal matrices are constructed the values of interval formed into a kind of (semi) formal methodology by scales are treated as the values of ratio scales. The Saaty in 1977 (called AHP, Analytic Hierarchy Pro- concept of a scale was never formally defined or cess, see (Dyer, 1990; French, 1986; Satty, 1977)). analysed. In principle it is based on the observation that while • The domain of real numbers, R , has two distinct ranking (“weighting”) the importance of several ob- interpretations, and these two interpretations are jects is frequently problematic, it is much easier when mixed up in the calculi provided by AHP. The first restricted to two objects. The problem is then reduced interpretation is standard, numbers describe quan- to constructing a global ranking (“weighting”) from tative values that can be added, multiplied, etc. the set of partially ordered pairs. The name “Pair- The second interpretation is that this is just a rep- wise” is slightly misleading, since for every triple resentation of a total order, and for instance 1.0 some consistency rules are required (Dyer, 1990; is “better” than 3.51 and “much better” than 13.6, Koczkodaj, 1993; Satty, 1986). For the orderings of a, but that’s it! In this interpretation one must be b, and c are made independently, it frequently occurs very careful when any arithmetic operations are that the consistency rules are not satisfied. To deal performed. with this problem various consistency concepts have • Even though the input numbers are “rough” and been introduced and analysed. imprecise the results are treated as if they were At present Pairwise Comparisons are practically very precise; “incomparability” is practically dis- identified with the controversial Saaty’s AHP. On one allowed. hand AHP has respected practical applications, on the other hand it is still considered by many (see (Dyer, In the author’s opinion the problems mentioned 1990)) as a flawed procedure that produces arbitrary above stem mainly from the following sources: • The final outcome is expected to be totally ordered ∗Partially supported by NSERC of Canada grant (i.e. for all a,b, either a < b or b > a), 297 Janicki R. (2007). PAIRWISE COMPARISONS, INCOMPARABILITY AND PARTIAL ORDERS. In Proceedings of the Ninth International Conference on Enterprise Information Systems - AIDSS, pages 297-302 DOI: 10.5220/0002363102970302 Copyright c SciTePress ICEIS 2007 - International Conference on Enterprise Information Systems • Numbers are used to calculate the final outcome. ta ta .q ta J . {a} J A non-numerical solution was proposed and dis- . ? J . J cussed in (Janicki and Koczkodaj, 1996), but it as- t . b JJ^ . JJ^ sumed that on the initial level of pairwise compar- t tc .q? t tc ? Jb . {b,c} b isons, the answer could be only “a < b” or “a ≈ b”, tc . J . or “a > b”, i.e. “slightly in favour” and “very strongly J . t? J^t .q? t? in favour” were indistinguishable. In this paper we d d {d} d provide the means to make such a distinction without <1 <2 ≺2 <3 using numbers. total or weak or total neither total The model presented below is an extension of the linear stratified nor weak one from (Janicki and Koczkodaj, 1996) with some influence due the results of (Fishburn, 1985; Janicki Figure 1: Various types of partial orders (represented as and Koutny, 1993). Hase diagrams). The total order ≺2 represents the weak order <2, The partial order <3 is nether total nor weak. 2 TOTAL, WEAK AND PARTIAL classification of our rankings (defined in Section 4) ORDERS complete, as in this paper we will not use much of their rich theory. An interested reader is referred to Let X be a finite set. A relation ⊳ ⊆ X ×X isa(sharp) (Fishburn, 1985; French, 1986). partial order if it is irreflexive and transitive, i.e. if Weak orders are often defined in an alternative a ⊳ b ⇒ ¬(b ⊳ a) and a ⊳ b ⊳ c ⇒ a ⊳ c, for all way, namely (Fishburn, 1985), ⊳ a,b,c ∈ X. A pair (X, ) is called a partially ordered • a partial order (X,⊳) is a weak order iff there ex- ⊳ ⊳ set. We will often identify (X, ) with , when X is ists a total order (Y,≺) and a mapping φ : X → Y known. such that ∀x,y ∈ X. x ⊳ y ⇐⇒ φ(x) ≺ φ(y). We write a ∼⊳ b if ¬(a ⊳ b) ∧ ¬(b ⊳ a), that is if a and b are either distinct incompatible (w.r.t. ⊳) or This definition is illustrated in Figure 1, let φ φ identical elements of X. : {a,b,c,d} → {{a},{b,c},{d}} and (a) = {a}, φ φ φ We also write (b) = (c) = {b,c}, (d) = {d}. Note that for all x,y ∈ {a,b,c,d} we have a ≈⊳ b ⇐⇒ {x | x ∼⊳ a} = {x | x ∼⊳ b}. x <2 y ⇐⇒ φ(x) ≺2 φ(y). Weak orders can easily be represented by step- The relation ≈⊳ is an equivalence relation (i.e. it is reflexive, symmetric and transitive) and it is called the sequences. For instance the weak order <2 from Figure 1 is uniquely represented by the step-sequence equivalence with respect to ⊳, since if a ≈⊳ b, there is nothing in ⊳ that can distinquish between a and b {a}{b,c}{d} (c.f. (Janicki and Koutny, 1993)). (see (Fishburn, 1985) for details). We always have Following (Fishburn, 1985), in this paper a ⊳ b is a ≈⊳ b ⇒ a ∼⊳ b, and one can show that (Fishburn, 1985): interpreted as “a is less preferred than b”, and a ≈⊳ b is interpreted as “a and b are indifferent”. a ≈⊳ b ⇐⇒ {x | a ⊳ x} = {x | b ⊳ x} ∧ {x | x ⊳ a} = {x | x ⊳ b} The preferable outcome of any ranking is a total ⊳ A partial order is (Fishburn, 1985) order. For any total order , ∼⊳ is the empty rela- tion and ≈⊳ is just the equality relation. A total order • total or linear, if ∼⊳ is empty, i.e., for all a,b ∈ has two natural models, both deeply embedded in the ⊳ ⊳ X. a b ∨ b a, human perception of reality, namely: time and num- • weak or stratified, if a ∼⊳ b ∼⊳ c ⇒ a ∼⊳ c, i.e. bers. Most people consider the causality relation and if ∼⊳ is an equivalence relation, its time related version “earlier than” as total orders, • interval, if for all a,b,c,d ∈ X, a ⊳ c ∧ b ⊳ d ⇒ even though their formal models are actually only in- a ⊳ d ∨ b ⊳ c, terval orders (Janicki and Koutny, 1993). Unfortunately in many cases it is not reasonable to • semiorder, if it is interval and for all a,b,c,d ∈ X, insist that everything can or should be totally ordered. ⊳ ⊳ ⊳ ⊳ a b ∧ b c ⇒ a d ∨ d c. We may not have sufficient knowledge or such a per- Evidently, every total order is weak, every weak order fect ranking may not even exist (Arrow, 1951). Quite is a semiorder, and every semiorder is interval. We often insisting on a totally ordered ranking results in mention semiorders and interval orders to make the an artificial and misleading “global index”. 298 PAIRWISE COMPARISONS, INCOMPARABILITY AND PARTIAL ORDERS Weak (stratified) orders are a very natural gener- relation <R+ is usually not a weak order. alization of total orders. They allow the modelling of Let us assume that X is believed to be weakly or- some regular indifference, their interpretation is very dered by a relation ⊳ but the discriminatory power of simple and intuitive, and they are reluctantly accepted the data acquisition process, which seeks to uncover by decision makers. Although not as much as one this order, is limited. The acquired data establishes might expect given the huge theory of such orders (see only a partial order ⊳ which is a partial picture of (Fishburn, 1985; French, 1986)). A non-numerical the underlying order. We seek, however, an extension ranking technique proposed in (Janicki and Koczko- process which is expected to correctly identify the or- daj, 1996) produces a ranking that is weakly ordered. dered pairs that are not part of the data. If ⊳ is a weak order then a ≈⊳ b ⇐⇒ a ∼⊳ b, so Note that weak order extensions reflect the fact indifference means distinct incomparability or iden- that if x ≈⊳ y than all reasonable methods for ex- tity, and the relation ⊳ can be interpreted as a se- tending ⊳ will have x equivalent to y in the extension quence of equivalence classes of ∼⊳. For the weak since there is nothing in the data that distinguishes order <2 from Figure 1, the equivalence classes of between them (for details see (Fishburn, 1985)), ∼<2 are {a}, {b,c}, and {d}, and <2 can be inter- which leads to the definition (Janicki and Koczkodaj, preted as a sequence {a}{b,c}{d}.
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