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

The method of Pairwise Comparisons could be traced

to Marquis de Condorcet 1785 paper (see (Arrow,

1951)), was explicitly mentioned and analysed by

Fechner in 1860 (Fechner, 1860), made popular by

Thurstone in 1927 (Thurstone, 1927), and was trans-

formed into a kind of (semi) formal methodology by

Saaty in 1977 (called AHP, Analytic Hierarchy Pro-

cess, see (Dyer, 1990; French, 1986; Satty, 1977)).

In principle it is based on the observation that while

ranking (“weighting”) the importance of several ob-

jects is frequently problematic, it is much easier when

restricted to two objects. The problem is then reduced

to constructing a global ranking (“weighting”) from

the set of partially ordered pairs. The name “Pair-

wise” is slightly misleading, since for every triple

some consistency rules are required (Dyer, 1990;

Koczkodaj, 1993; Satty, 1986). For the orderings of a,

b, and c are made independently, it frequently occurs

that the consistency rules are not satisﬁed. To deal

with this problem various consistency concepts have

been introduced and analysed.

At present Pairwise Comparisons are practically

identiﬁed with the controversial Saaty’s AHP. On one

hand AHP has respected practical applications, on the

other hand it is still considered by many (see (Dyer,

1990)) as a ﬂawed procedure that produces arbitrary

∗

Partially supported by NSERC of Canada grant

rankings.

The strange and contradictory examples of rank-

ings obtained by AHP cited in the literature (see

(Dyer, 1990)) follow from the following sources:

• When two objects are being compared, interval

scales are mainly used; but later, when recipro-

cal matrices are constructed the values of interval

scales are treated as the values of ratio scales. The

concept of a scale was never formally deﬁned or

analysed.

• The domain of real numbers,

R , has t

wo distinct

interpretations, and these two interpretations are

mixed up in the calculi provided by AHP. The ﬁrst

interpretation is standard, numbers describe quan-

tative values that can be added, multiplied, etc.

The second interpretation is that this is just a rep-

resentation of a total order, and for instance 1.0

is “better” than 3.51 and “much better” than 13.6,

but that’s it! In this interpretation one must be

very careful when any arithmetic operations are

performed.

• Even though the input numbers are “rough” and

imprecise the results are treated as if they were

very precise; “incomparability” is practically dis-

allowed.

In the author’s opinion the problems mentioned

above stem mainly from the following sources:

• The ﬁnal outcome is expected to be totally ordered

(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

• Numbers are used to calculate the ﬁnal outcome.

A non-numerical solution was proposed and dis-

cussed in (Janicki and Koczkodaj, 1996), but it as-

sumed that on the initial level of pairwise compar-

isons, the answer could be only “a < b” or “a ≈ b”,

or “a > b”, i.e. “slightly in favour” and “very strongly

in favour” were indistinguishable. In this paper we

provide the means to make such a distinction without

using numbers.

The model presented below is an extension of the

one from (Janicki and Koczkodaj, 1996) with some

inﬂuence due the results of (Fishburn, 1985; Janicki

and Koutny, 1993).

2 TOTAL, WEAK AND PARTIAL

ORDERS

Let X be a ﬁnite set. A relation ⊳ ⊆ X ×X is a (sharp)

partial order if it is irreﬂexive and transitive, i.e. if

a ⊳ b ⇒ ¬(b ⊳ a) and a ⊳ b ⊳ c ⇒ a ⊳ c, for all

a, b, c ∈ X. A pair (X, ⊳) is called a partially ordered

set. We will often identify (X, ⊳) with ⊳, when X is

known.

We write a ∼

⊳

b if ¬(a ⊳ b) ∧ ¬(b⊳ a), that is if

a and b are either distinct incompatible (w.r.t. ⊳) or

identical elements of X.

We also write

a ≈

⊳

b ⇐⇒ {x | x ∼

⊳

a} = {x | x ∼

⊳

b}.

The relation ≈

⊳

is an equivalence relation (i.e. it is

reﬂexive, symmetric and transitive) and it is called the

equivalence with respect to ⊳, since if a ≈

⊳

b, there

is nothing in ⊳ that can distinquish between a and b

(see (Fishburn, 1985) for details). We always have

a ≈

⊳

b ⇒ a ∼

⊳

b, and one can show that (Fishburn,

1985):

a ≈

⊳

b ⇐⇒ {x | a⊳ x} = {x | b⊳ x}

∧ {x | x⊳ a} = {x | x⊳ b}

A partial order is (Fishburn, 1985)

• total or linear, if ∼

⊳

is empty, i.e., for all a, b ∈

X. a⊳ b∨ b⊳ a,

• weak or stratiﬁed, if a ∼

⊳

b ∼

⊳

c ⇒ a ∼

⊳

c, i.e.

if ∼

⊳

is an equivalence relation,

• interval, if for all a, b, c, d ∈ X, a ⊳ c ∧ b ⊳ d ⇒

a⊳ d ∨ b⊳ c,

• semiorder, if it is interval and for all a, b, c, d ∈ X,

a⊳ b∧ b⊳ c ⇒ a⊳ d ∨ d ⊳ c.

Evidently, every total order is weak, every weak order

is a semiorder, and every semiorder is interval. We

mention semiorders and interval orders to make the

t

t

t

t

?

?

?

a

b

c

d

<

1

total or

linear

t

t t

t

J

J

J

J^

J

J

J

J^

a

b

c

d

<

2

weak or

stratiﬁed

q

q

q

?

?

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

{a}

{b, c}

{d}

≺

2

total

t

t t

t

J

J

J

J^

?

a

b

c

d

<

3

neither total

nor weak

Figure 1: Various types of partial orders (represented as

Hase diagrams). The total order ≺

2

represents the weak

order <

2

, The partial order <

3

is nether total nor weak.

classiﬁcation of our rankings (deﬁned in Section 4)

complete, as in this paper we will not use much of

their rich theory. An interested reader is referred to

(Fishburn, 1985; French, 1986).

Weak orders are often deﬁned in an alternative

way, namely (Fishburn, 1985),

• a partial order (X, ⊳) is a weak order iff there ex-

ists a total order (Y, ≺) and a mapping φ : X → Y

such that ∀x, y ∈ X. x⊳ y ⇐⇒ φ(x) ≺ φ(y).

This deﬁnition is illustrated in Figure 1, let

φ : {a, b, c, d} → {{a}, {b, c}, {d}} and φ(a) = {a},

φ(b) = φ(c) = {b, c}, φ(d) = {d}. Note that for all

x, y ∈ {a, b, c, d} we have

x <

2

y ⇐⇒ φ(x) ≺

2

φ(y).

Weak orders can easily be represented by step-

sequences. For instance the weak order <

2

from

Figure 1 is uniquely represented by the step-sequence

{a}{b, c}{d} (c.f. (Janicki and Koutny, 1993)).

Following (Fishburn, 1985), in this paper a⊳ b is

interpreted as “a is less preferred than b”, and a ≈

⊳

b

is interpreted as “a and b are indifferent”.

The preferable outcome of any ranking is a total

order. For any total order ⊳, ∼

⊳

is the empty rela-

tion and ≈

⊳

is just the equality relation. A total order

has two natural models, both deeply embedded in the

human perception of reality, namely: time and num-

bers. Most people consider the causality relation and

its time related version “earlier than” as total orders,

even though their formal models are actually only in-

terval orders (Janicki and Koutny, 1993).

Unfortunately in many cases it is not reasonable to

insist that everything can or should be totally ordered.

We may not have sufﬁcient knowledge or such a per-

fect ranking may not even exist (Arrow, 1951). Quite

often insisting on a totally ordered ranking results in

an artiﬁcial and misleading “global index”.

ICEIS 2007 - International Conference on Enterprise Information Systems

298

Weak (stratiﬁed) orders are a very natural gener-

alization of total orders. They allow the modelling of

some regular indifference, their interpretation is very

simple and intuitive, and they are reluctantly accepted

by decision makers. Although not as much as one

might expect given the huge theory of such orders (see

(Fishburn, 1985; French, 1986)). A non-numerical

ranking technique proposed in (Janicki and Koczko-

daj, 1996) produces a ranking that is weakly ordered.

If ⊳ is a weak order then a ≈

⊳

b ⇐⇒ a ∼

⊳

b, so

indifference means distinct incomparability or iden-

tity, and the relation ⊳ can be interpreted as a se-

quence of equivalence classes of ∼

⊳

. For the weak

order <

2

from Figure 1, the equivalence classes of

∼

<

2

are {a}, {b, c}, and {d}, and <

2

can be inter-

preted as a sequence {a}{b, c}{d}.

There are, however, cases where insisting on weak

orders may not be reasonable. Physiophysical mea-

surements of perceptions of length, pitch, loudness,

and so forth, provides other examples of qualitative

comparisons that might be analysed from the perspec-

tive of semiorders and interval orders rather than the

more precise but less realistic weak and total orders.

The reader is referred to (Fishburn, 1985; Janicki and

Koutny, 1993) for more details. In this paper we will

only use total, weak, and general partial orders.

3 PARTIAL AND WEAK ORDER

APPROXIMATIONS

Let X be a set of objects to be ranked. The problem

is that X is believed to be partially or weakly ordered

but the data acquisition process is so inﬂuenced by

informational noise, imprecision, randomness, or ex-

pert ignorance that the collected data R is only some

relation on X. We may say that R gives a fuzzy pic-

ture, and to focus it, we must do some pruning and/or

extending. Without loss of generality we may as-

sume that R is irreﬂexive, i.e. (x,x) 6∈ R. Suppose

that R is not transitive. The “best” transitive approx-

imation of R is its transitive closure R

+

=

∞

i=1

R

i

,

where R

i+1

= R

i

◦R (c.f. (Fishburn, 1985)). Evidently

R ⊆ R

+

and R

+

is transitive. The relation R

+

may not

be irreﬂexive, but in such a case we can use the fol-

lowing classical result (which is due to E. Schr

¨

oder,

1895, see (Janicki and Koczkodaj, 1996)) .

Lemma 1 Let Q⊆ X ×X be a transitive relation. De-

ﬁne: x <

Q

y ⇐⇒ xQy∧ ¬yQx. The relation <

Q

is a

partial order.

Following (Janicki and Koczkodaj, 1996) we will call

<

R

+

, a partial order approximation of (ranking rela-

tion) R. If R is a partial order then <

R

+

equals R. The

relation <

R

+

is usually not a weak order.

Let us assume that X is believed to be weakly or-

dered by a relation ⊳ but the discriminatory power of

the data acquisition process, which seeks to uncover

this order, is limited. The acquired data establishes

only a partial order ⊳ which is a partial picture of

the underlying order. We seek, however, an extension

process which is expected to correctly identify the or-

dered pairs that are not part of the data.

Note that weak order extensions reﬂect the fact

that if x ≈

⊳

y than all reasonable methods for ex-

tending ⊳ will have x equivalent to y in the extension

since there is nothing in the data that distinguishes

between them (for details see (Fishburn, 1985)),

which leads to the deﬁnition (Janicki and Koczkodaj,

1996) below (for both weak an total orders).

A weak (or total) order ⊳

w

⊆ X × X is a proper

weak (or total) order extension of ⊳ if and only if :

(x⊳ y ⇒ x⊳

w

y) and (x ≈

⊳

y ⇒ x ∼

⊳

w

y).

If X is ﬁnite then for every partial order ⊳ its

proper weak extension always exists. If ⊳ is weak,

than its only proper weak extension is ⊳

w

= ⊳. If ⊳ if

not weak, there are usually more than one such exten-

sions. Various methods were proposed and discussed

in (Fishburn, 1985). For our purposes, the best seem

to be the method based on the concept of a global

score function, which is deﬁned as:

g

⊳

(x) = |{z | z⊳ x}| − |{z | x⊳ z}|.

Given the global score function g

⊳

(x), we deﬁne

the relation ⊳

g

w

⊆ X ×X as

a⊳

g

w

b ⇐⇒ g

⊳

(a) < g

⊳

(b).

Proposition 1 ((Fishburn, 1985)) The relation ⊳

g

w

is a proper weak extension of a partial order ⊳.

Some other variations of g

⊳

and their interpreta-

tions were analyzed in (Janicki and Koczkodaj, 1996).

From Proposition 2 it follows that every ﬁnite partial

order has a proper weak extension. The well known

procedure “topological sorting”, popular in schedul-

ing problems, guarantees that every partial order has

a total extension, but even ﬁnite partial orders may not

have proper total extensions. Note that the total order

⊳

t

is a proper total extension of ⊳ if and only if the re-

lation ≈

⊳

equals the identity, i.e a ≈

⊳

b ⇐⇒ a = b.

For example no weak order has a proper total ex-

tension unless it is also already total. This indicates

that while expecting a ﬁnal ordering to be weak may

be reasonable, expecting a ﬁnal total ordering is of-

ten unreasonable. It may however happen, and often

does, that a proper weak extension is a total order,

which suggests that we should stop seeking a priori

total orderings since weak orders appear to be more

natural models of preferences than total orders.

PAIRWISE COMPARISONS, INCOMPARABILITY AND PARTIAL ORDERS

299

4 THE MODEL

Let X be a ﬁnite set of objects to be “ranked”, and let

≈, ⊏, ⊂, <, and ≺ be a family of disjoint relations

on X. The interpretation of these relations is the fol-

lowing (compare (Satty, 1986)), a ≈ b : a and b are

of equal importance, a ⊏ b : slightly in favour of b,

a ⊂ b : in favour of b, a < b: b is strongly better, a ≺ b

: b is extremely better. The lists ⊏, ⊂, <, ≺ may be

shorter or longer, but not empty and not much longer

(due to limitations of the human mind(Satty, 1986)).

We deﬁne the relations

b

⊏,

b

⊂,

b

<, and

b

≺ as follows:

b

≺ = ≺

b

< = ≺ ∪ <

b

⊂ = ≺ ∪ < ∪ ⊂

b

⊏ = ≺ ∪ < ∪ ⊂ ∪ ⊏

The relations

b

⊏,

b

⊂,

b

<, and

b

≺ are interpreted as

combined preferences, i.e. a

b

⊏b : at least slightly in

favour of b, a

b

⊂b : at least in favour of b, a

b

<b: at

least strongly in favour of b, and a

b

≺b : at least b is

far superior than a.

A relational structure Rank = (X, ≈, ⊏, ⊂, <, ≺) is

called a ranking if the following axioms (consistency

rules) are satisﬁed:

1.

b

⊏,

b

⊂,

b

<,

b

≺ are partial orders,

2. ≈ = ∼

b

⊏

, i.e. ≈ ∪

b

⊏ ∪

b

⊏

−1

= X ×X,

3. (a ≈ b∧ b ≈ c) ⇒ (a ≈ c∨ a ⊏ c∨ c ⊏ a),

4.1. (a ≈ b∧b⊏ c)∨(a⊏ b∧b≈ c) ⇒ (a⊏ c∨a ⊂ c),

4.2. (a ≈ b∧b⊂ c)∨(a⊂ b∧b≈ c) ⇒ (a⊂ c∨a < c),

4.3. (a ≈ b∧b< c)∨(a< b∧b≈ c) ⇒ (a< c∨a ≺ c),

5. (a ≈ b∧ b ≺ c) ∨ (a ≺ b∧ b ≈ c) ⇒ a ≺ c,

6.1. (a ⊏ b∧ b ⊏ c) ⇒ (a ⊏ c∨ a ⊂ c),

6.2. (a ⊂ b∧b⊏ c)∨(a⊏ b∧b⊂ c) ⇒ (a⊂ c∨a < c),

6.3. (a < b∧b⊏ c)∨(a⊏ b∧b< c) ⇒ (a< c∨a ≺ c),

7.1. (a ⊏ b∧ b ≺ c) ∨ (a ≺ b∧ b ⊏ c) ⇒ a ≺ c,

7.2. (a ⊂ b∧ b ≺ c) ∨ (a ≺ b∧ b ⊂ c) ⇒ a ≺ c,

7.3. (a < b∧ b ≺ c) ∨ (a ≺ b∧ b < c) ⇒ a ≺ c,

7.4. (a ≺ b∧ b ≺ c) ∨ (a ≺ b∧ b ≺ c) ⇒ a ≺ c.

The axioms 1 − 7.4 follow from the interpretation of

≈ as equal importance and ⊏, ⊂, <, ≺ as increasing

preferences.

We will say that a ranking (X, ≈, ⊏, ⊂, <, ≺) is to-

tally (weakly, semi-, intervally) ordered if the relation

b

⊏ is a total (weak, semi-, interval) order.

Due to the nature of stronger preferences it is

unreasonable to expect any speciﬁc ordering of

b

<, or

b

≺, however if such a speciﬁc (for example semiorder)

ordering occurs, it may give some important informa-

tion about the nature of hierarchy that is modelled by

a given ranking.

Let

b

⊏

w

be a proper weak extension of

b

⊏ and let

⊏

w

=

b

⊏

w

\

b

⊂,

Rank

w

= (X, ≈, ⊏

w

, ⊂, <, ≺).

Proposition 2 The relational structure Rank

w

=

(X, ≈, ⊏

w

, ⊂, <, ≺) is a weakly ordered ranking.

We will call Rank

w

= (X, ≈, ⊏

w

, ⊂, <, ≺) a weak

order extension of Rank = (X, ≈, ⊏, ⊂, <, ≺).

When transforming Rank into Rank

w

, we change only

the weakest preference, so if Rank = (X, ≈, ⊂, <, ≺)

then Rank

w

= (X, ≈, ⊂

w

, <, ≺), and if

Rank = (X, ≈, <, ≺) then Rank

w

= (X, ≈, <

w

, ≺).

We proceed similarly when the list of preferences is

longer.

As mentioned above (see (French, 1986; Satty,

1986)) deﬁning a proper ranking is problematic if

|X| > 2, but it usually can be done if |X| = 2. Note

that if |X| = 2 only one of the relations ≈, ⊏, ⊂, <,

≺ is not empty.

A relational structure (X, ≈, ⊏, ⊂, <, ≺) is called

a pairwise comparisons pre-ranking, if the following

properties are satisﬁed:

1. the relations ≈, ⊏, ⊂, <, ≺ are deﬁned by pair-

wise comparisons,

2. ≈, ⊏, ⊂, <, ≺ are disjoint and their union equals

X ×X,

3. ≈ is interpreted as equal importance and a ≈ a for

each a ∈ X,

4. ⊏, ⊂, <, ≺ are interpreted as increasing prefer-

ences.

The pre-ranking is not usually a ranking. Our goal is

to ﬁnd such a, preferably weakly ordered, ranking that

is “the best” approximation of a given pre-ranking. In

the classical, “numerical” approach this is handled by

the concept of consistency (Koczkodaj, 1993; Satty,

1986). The case (X, ≈, <) was solved in (Janicki and

Koczkodaj, 1996). The technique used in (Janicki

and Koczkodaj, 1996) does not require any assump-

tions about the pre-ranking relations ≈ and < (except

a ≈ b∨ a < b∨ a <

−1

b, for all a, b). The technique

can be extended to the general case, however, at the

present stage, the algorithms are complex and lacking

the elegance of the simpler case of (X,≈, <). Instead,

we propose a more interactive approach.

First notice that it is very unlikely that a given pre-

ranking is “random”, i.e. it does not even resemble a

ranking. When the process of classiﬁcation of pairs

is well designed, the outcome is a pre-ranking that is

ICEIS 2007 - International Conference on Enterprise Information Systems

300

“almost” a ranking. Only some pairs violate the rank-

ing axioms, the majority of pairs satisfy the ranking

axioms. Checking those axioms is in principle a triad

analysis, and the major violators are usually easy to

detect.

After ﬁnding the pairs that violate more axioms

than the other pairs (violations of axioms propagate,

but “innocent” pairs violate much less frequently), we

can either:

• Repeat the pairwise comparison process for those

pairs and change the judgment. Or

• Introduce a “moderator” process which will

change the relationship between those pairs, to

satisfy the axioms.

In both cases, the changes are usually minor, like from

⊂ to ⊏, etc.

The resulting ranking Rank is usually not weak, in

most cases it is only semi- or intervally ordered, so, if

we expect the outcome to be weak or total ranking,

we need to compute Rank

w

.

Computing Rank

w

is an extension process which

is expected to identify correctly the ordered pairs that

are not part of the data. The order identiﬁcation power

of weak extension procedures is substantial and vastly

underestimated. If the ranked set of objects is, by its

nature, expected to be totally ordered, the weak ex-

tension can detect it, even if the pairwise compari-

son process is not very precise, and often results in

“indifference” (see example from Table 3 in the next

section). It is a serious error to attempt to ﬁnd a

total extension without going through a weak exten-

sion process (see comments at the end of the previous

chapter). In general, admitting incomparability on the

level of pairwise comparisons is better than insisting

on an order at any cost. The latter approach leads to

an arbitrary and often incorrect total ordering.

5 AN EXAMPLE

The following experiment has been conducted. A

blindfolded person compared the weights of the eight

different stones, named A, B, C, D, E, F, G, H. The

person put one stone in their left hand and another in

their right, and then decided which of the relations

≈, ⊏, ⊂, <, or ≺ held. The experiment was re-

peated for the same set of stones by various people;

and then again for different stones and different num-

ber of stones. The results were very similar, and Table

1 presents the results of one such an experiment.

The pre-ranking (X, ≈, ⊏, ⊂, <, ≺), where X =

{A,B,C,D,E,F,G,H}, described by Table 1 is not a

ranking. However, a simple veriﬁcation of axioms

Table 1: The ﬁrst pre-ranking. It is not a ranking, the gray

cells indicate problematic relations.

A B C D E F G H

A ≈ ⊂ ≈ ⊏ ⊃ ≈ < ⊐

B ⊃ ≈ > ≈ ≻ ⊐ ≈ <

C ≈ < ≈ ⊂ ⊐ ⊏ < ⊏

D ⊐ ≈ ⊃ ≈ ≻ ≈ ⊏ >

E ⊂ ≺ ⊏ ≺ ≈ < ≺ ≈

F ≈ ⊏ ⊐ ≈ > ≈ ⊂ ⊃

G > ≈ > ⊐ ≻ ⊃ ≈ ≻

H > ≺ ⊐ < ≈ ⊂ ≺ ≈

Table 2: The second (revised) pre-ranking which is a rank-

ing. The grey cells were revised.

A B C D E F G H

A ≈ ⊂ ≈ ⊏ ⊃ ≈ < ⊐

B ⊃ ≈ > ≈ ≻ ⊐ ≈ <

C ≈ < ≈ ⊂ ⊐ ⊏ < ≈

D ⊐ ≈ ⊃ ≈ > ≈ ⊏ >

E ⊂ ≺ ⊏ < ≈ < ≺ ≈

F ≈ ⊏ ⊐ ≈ > ≈ ⊂ ⊃

G > ≈ > ⊐ ≻ ⊃ ≈ ≻

H > ≺ ≈ < ≈ ⊂ ≺ ≈

shows that the only violations are caused by relations

between C and H, and D and E (grey cells in Table

1). The person was asked to repeat the comparison

of pairs (C,H), and (D,E). This time he produced a

slightly different answers. The new pre-ranking pre-

sented by Table 2, is now a ranking. However, it is not

a weakly ordered ranking. The partial orders

b

⊏,

b

⊂,

b

<, and

b

≺ are presented in Figure 2. The relations

b

⊏,

b

⊂,

b

< are semiorders while

b

≺ is an interval order, i.e.

the ranking is semiordered. This was expected due to

the nature of this type of experiments (c.f. (Fishburn,

1985)).

The stones were weighed and their weights cre-

ated an increasing total order E, H,C, A, F, D, B, G.

Note that this is the same order as weak extensions

b

⊏

w

,

b

⊂

w

and

b

<

w

. The fact that

b

<

w

correctly describes

the real ordering is very interesting since it means that

the very rough ranking described in Table 3, where

only high preferences were recorded, all weak ones

were discarded and coded as “equal importance”, was

sufﬁcient to produce a correct total ordering. It also

shows the order identiﬁcation power of the weak or-

der extension procedure. Note also that if in the pre-

ranking from Table 1, which is not a ranking, we will

use only high preferences and replace weak by indif-

ference (i.e. we replace ⊏ and ⊂ with ≈, and leave

only < and ≺), the new pre-ranking will be a ranking.

It will differ slightly from the one of Table 3 (E ≺ D in

PAIRWISE COMPARISONS, INCOMPARABILITY AND PARTIAL ORDERS

301

Table 3: The third pre-ranking, where only ≈, < and ≺

were recorded. It is a ranking. It can be obtained from

Table 2 by replacing ⊏ and ⊂ with ≈.

A B C D E F G H

A ≈ ≈ ≈ ≈ ≈ ≈ < ≈

B ≈ ≈ > ≈ ≻ ≈ ≈ <

C ≈ < ≈ ≈ ≈ ≈ < ≈

D ≈ ≈ ≈ ≈ > ≈ ≈ >

E ≈ ≺ ≈ < ≈ < ≺ ≈

F ≈ ≈ ≈ ≈ > ≈ ≈ ≈

G > ≈ > ≈ ≻ ≈ ≈ ≻

H > ≺ ≈ < ≈ ≈ ≺ ≈

t t

t t

t t t t

?

A

A

A

A

AU

G

B

E H

C

A F D

b

≺

t t t t

t t t t

? ? ? ?

=

=

B

G

F D

E H

C A

b

<

t

t

t t t

t t t

? ? ?

A

A

A

A

A

A

AU

A

A

A

A

A

A

AU

@

@

@

@

@

@

@R

=

E H

C

D B

G

F

A

b

⊂

t t

t t

t t

t t

? ?

? ?

? ?

A

A

A

A

A

A

AU

A

A

A

A

A

A

AU

B

F

C

E

G

D

A

H

b

⊏

s

s

s

s

s

s

s

s

?

?

?

?

?

?

?

G

B

D

F

A

C

H

E

b

⊏

w

=

b

⊂

w

=

b

<

w

s s s s

s

s

s

s

?

?

+

Q

Q

Qs

A

AU

Q

Q

Qs

A

AU

+

G

B

E

H

A F

D

C

b

≺

w

Figure 2: Partial orders

b

⊏,

b

⊂,

b

<, and

b

≺ deﬁned by the rank-

ing from Table 2, and their proper weak order extensions

b

⊏

w

,

b

⊂

w

,

b

<

w

, and

b

≺

w

(created using global score function).

The orders

b

⊏,

b

⊂,

b

< are semiorders,

b

≺ is an interval order,

b

⊏

w

,

b

⊂

w

,

b

<

w

are the same total. order, and

b

≺

w

is a weak

order.

Table 1 and E < D in Table 3), but the order

b

< will be

the same in both cases! This suggests that if an origi-

nal pairwise comparison pre-ranking is not a ranking,

a “moderator” might replace lower preferences by in-

difference, and check if the outcome is a ranking. If

it is, a weak extension may be constructed and on its

bases the initial pre-ranking might be modiﬁed. Al-

ternately, a problematic decision could be re-judged.

6 FINAL COMMENTS

Apparently, all of the popular techniques based on

pairwise comparison principles suffer from many se-

rious problems and may lead to strange and contradic-

tive results. These problems follow mainly from their

treatment of imprecision, knowledge incompleteness

and indifference (incomparability) (Dyer, 1990; Jan-

icki and Koczkodaj, 1996).

The method proposed in this paper does not use

numbers at all, it is entirely based on the concept of

partial orders. It emphasizes and advocates using in-

comparability and weak orderings, as opposed to in-

sistence on the comparability of all objects and a ﬁnal

total ordering. The order identiﬁcation power of the

weak order extension procedure is discussed.

REFERENCES

Arrow K. J., 1951, Social Choice and Individual Values, J.

Wiley, New York.

Dyer J. S., 1990, Remarks on the Analytic Hierarchy Pro-

cess, Management Science, Vol. 36, No. 3, pp. 244-

258.

Fechner G. T., 1860, Elemente der Psychophysik, Breitkopf

und H

¨

artel, Leipzig.

Fishburn, P. C., 1985, Interval Orders and Interval Graphs,

J. Wiley, New York.

French S., 1986, Decision Theory, Ellis Horwood, New

York.

Janicki R., Koczkodaj W. W., 1996, Weak Order Approach

to Group Ranking, Computers Math. Applic., Vol. 32,

No. 2, pp. 51-59.

Janicki R., Koutny M., 1993, Structure of Concurrency,

Theor. Computer Science 112, pp. 5-52.

Koczkodaj W. W., 1993, A new deﬁnition of consistency of

pairwise comparisons, Mathematical and Computer

Modelling, Vol. 18, pp. 79-84.

Saaty T. L., 1977, A Scaling Methods for Priorities in Hier-

archical Structure, Journal of Mathematical Psychol-

ogy, 15, pp. 234-281.

Saaty T. L., 1986, Axiomatic Foundations of the Analytic

Hierarchy Process, Management Science, Vol. 32, No.

7, pp. 841-855.

Thurstone L. L., 1927, A Law of Comparative Judgments,

Psychological Reviews, 34, pp. 273-286.

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