Mathl. Compui. Modelling Vol. 17, No. 413, pp. l-12, 1993 0695-7177193 $6.00 + 0.00

Printed in Great Britain. All rights reserved Copyright@ 1993 Pergamon Press Ltd

WHAT IS RELATIVE MEASUREMENT? THE RATIO SCALE PHANTOM

THOMAS L. SAATY

Joseph M. Katz Graduate School of Business, 322 Mervis Hall

University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.

Abstract-Traditionally, measurement scales with a unit are assumed to be available to measure and rank alternatives. However, such scales are scarce and it is often desired to rank alternatives on many tangible and intangible criteria. How ? Relative measurement is a method for deriving ratio scales from paired comparisons represented by absolute numbers. This approach to measurement is useful in multicriteria decision-malting where the concept of dominance is used to represent the comparisons numerically and derive and synthesize ratio scales to obtain an overall ratio scale ranking of the alternatives. Mathematical and structural issues relating to relative measurement and to the overall ranks of the alternatives are discussed in this paper.

1. INTRODUCTION

When one speaks of relative measurement, those of us trained in the physical sciences and in mathematics are likely to think of measuring things, for example, on a scale such as the yard or the meter, each with its units and dividing corresponding lengths to get the relative lengths. But that is not what I mean by relative measurement. First, I ask what would I do if I did not have a scale to measure length to define the relative length of two objects? Mary Blocksma [l] has given us, in her book, a list of things we know how to measure. It is large, but far from giving us ways to deal with intangibles in psychology, politics and all those aspects of human nature that we need to understand and track. Henri Lebesgue [2] wrote:

It would seem that the principle of economy would always require that we evaluate ratios directly and not as ratios of measurements. However, in practice, all lengths are measured in meters, all angles in degrees, etc.; that is we employ auxiliary units and, as it seems, with only the disadvantage of having two measurements to make instead of one. Sometimes, this is because of experimental difficulties or impossibilities that prevent the direct comparison of lengths or angles. But there is also another reason.

In geometrical problems, one needs to compare two lengths, for example, and only those two. It is quite different in practice when one encounters a hundred lengths and may expect to have to compare these lengths two at a time in all possible manners. thus it is desirable and economical procedure to measure each new length. One single measurement for each length, made as precisely as possible, gives the ratio of the length in question to each other length. This explains the fact that, in practice, comparisons are never, or almost never, made directly but through comparisons with a standard scale.

But when we have no standard scales to measure things absolutely, we must make comparisons and derive relative measurements from them. The question is how, and what have we learned in this process? Regrettably, there are those who do not share Lebesgue’s depth of understanding and insist that there is no legitimacy for measurement through paired comparisons. R.D. Holder, a British physicist, writes: “It is in fact difficult to see when.. . there is ever any justification for using ‘relative measurement.’ ”

We should note that we are not talking about a proposed theory that we can accept or reject, but about a talent of our brains that has been neglected in science because we have not learned

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2 T.L. SAATY

how to formalize it in harmony with this other way of creating standard scales and comparing or measuring things on them one at a time.

The cognitive psychologist Blumenthal [3] writes:

Absolute judgment is the identification of the magnitude of some simple stimulus,. . . , whereas comparative judgment is the identification of some relation between two stimuli both present to the observer. Absolute judgment involves the relation between a single stimulus and some information held in short-term memory-information about some former comparison stimuli or about some previously experienced measurement scale.. To make the judgment, a person must compare an immediate impression with a memory impression of similar stimuli.. . .

Thus, relative measurement through comparative judgment is intrinsic to our thinking and should not be carried by us as an appendage whose real function is not understood well or at all and should be kept outside. It is not difficult to see that relative measurement predates and is necessary for creating and understanding absolute measurement. Some of the work reported on here is now well-known, but we need it for the subsequent discussion that lays the foundation for relative measurement.

2. THE PARADIGM CASE; CONSISTENCY

Assume that n activities are being considered by a group of interested people. We assume that the group’s goals are:

(a) to provide judgments on the relative importance of these activities; (b) to ensure that the judgments are quantified to an extent which also permits a quantita-

tive interpretation of the judgments among all activities. Clearly, this goal will require appropriate technical assistance.

Our goal is to describe a method for deriving, from the quantified judgments (i.e., from the relative values associated with pairs of activities), a set of weights to be associated with individual activities. What this approach achieves is to put the information resulting from (a) and (b) into usable form.

Let Al,Az,..., A,, be the activities. The quantified judgments on pairs of activities (Ai, Aj) are represented by an n x n matrix

A = (aij), (i,j = 1,2,. . . ,n).

The problem now is to assign to the n activities Al, AZ,. , . , A,, a set of numerical weights

W,Wzr..., w, that would “reflect the recorded quantified judgments.”

It is convenient to first get a simple question out of the way. The matrix A of quantified judgments eij may have several, or only few, non-zero entries. The question arises: how many entries (i.e., how many quantifiable judgments) are necessary in order to ensure the existence of a set of weights that is meaningful in the context of the problem? The obvious answer is: it is sufficient that there be a set of entries that interconnects all activities in the sense that for every two indices, i,j, there should be some chain of (positive) entries connecting i with j:

%i, 9 aili, aiai, 9 . . . , Ui,j .

Note that eij itself is such a chain of length 1. (Such a matrix A = (aij) corresponds to a strongly connected graph.) This gives precise content to the formulation of goal b.

If two people, a tall adult and a little child are compared according to height, the adult may be estimated to be two and a half times taller, demonstrated by marking off several heights of the child end to end or by using a surrogate device. If we actually have a scale of measurement for height with the child measuring wi units and the adult w2 units, then the comparison would assign the adult the relative value ws/wi and the child wr/wz, the reciprocal value. These ratios also give us the paired comparison values (w2/w1)/1 and l/(ws/wi), respectively, in which the

Relative measurement 3

height of the child serves as the unit of comparison. Such a representation is valid only if wi

and w2 belong to a ratio scale so that the ratio wr/wz is independent of the unit used, be it in inches or in centimeters, for example. In this way, we can interpret all ratios as absolute numbers or dominance units.

Let us form the matrix whose rows give the ratios of the weights of each to all others and multiply it by the vector w = (wr, . . . , w,). We have:

DEFINITION. A = (aij) is consistent if:

aijajk = aik, i,j,k= l,..., n;

otherwise A is said to be inconsistent.

activity with respect

(1)

We see that the entire matrix can be constructed from a set of n elements which form a chain (or more generally, a spanning tree, in graph-theoretic terminology) across the rows and columns.

It is easy to prove the following.

THEOREM 1. A positive n x n matrix has the ratio form A = (wi/wj), i, j = 1, . . . , n, if and only

if it is consistent.

COROLLARY. If (1) is true then A is reciprocal: aijaji = 1.

The foregoing formulation has the advantage of exhibiting the solution. But it also gives rise to an interesting theoretical interpretation. We have multiplied A on the right by the vector of weights w = (WI, ~2, . . . , w,,)~. The result of this multiplication is nw. We have the following.

THEOREM 2. The matrix of ratios A = (wi/wj) is consistent if and only if n is its principal eigenvafue and Aw = nw. Further, w > 0 is unique to within a multiplicative constant.

PROOF. The “if’ part of the proof is clear. Now for the other half: if A is consistent, then n and w are one of its eigenvalues and its corresponding eigenvector, respectively. Now A has rank one because every row is a constant multiple of the first row. Thus, all its eigenvalues except one are equal to zero. The sum of the eigenvalues of a matrix is equal to its trace, the sum of the diagonal elements, and in this case, the trace of A is equal to n. Therefore, n is a simple eigenvalue of A. It is also the largest, or principal, eigenvalue of A. Alternatively, A = DeeTD-’ where D

is a diagonal matrix with dii = wi, and e = (1, . . . . 1) T. Therefore, A and eeT are similar and have the same eigenvalues [4]. The characteristic equation of eeT is obviously A” - nX”-l = 0, and the result follows.

The solution w of Aw = nw, the principal right eigenvector of A, consists of positive entries and is obviously unique to within a positive multiplicative constant (a similarity transformation) thus defining a ratio scale. To ensure uniqueness, we normalize w by dividing by the sum of its entries.

Given the comparison matrix A, we can directly recover w as the normalized version of any column of A; A = WV, v = (~/WI,. . . , l/w,). It is interesting to note that for A = (wi/wj), ~11 the conclusions of the well-known theorem of Perron are valid without recourse to that theorem.

Perron’s theorem says that a matrix of positive entries has a simple positive real eigenvalue which dominates all other eigenvalues in modulus and a corresponding eigenvector whose entries are positive that is unique to within multiplication by a constant.

Here, we concern ourselves only with right eigenvectors because of the nature of dominance. In paired comparisons, the smaller element of a pair serves as the unit of comparison. There is no way of starting with the larger of a pair and decomposing it to determine what fraction of it the smaller is, without first using the smaller one as a standard for the decomposition.

If A is consistent, then aij may be represented as a ratio from an existing ratio scale, such as the kilogram scale for weight, or by using a judgment estimate as to how many times more

4 T.L. SAATY

the dominant member of the pair has a property for which no scale exists, such as smell or customer satisfaction. Of course, if the measurements from an actual scale are used in the pairwise comparisons, the derived scale of relative magnitudes is not a new scale-it is the same one used to do the measuring. We note that any finite set of n readings WI,. . . , wn from a ratio scale defines the principal eigenvector of a consistent n x n matrix W = (wi/wj).

With regard to the order induced in w by W, in general, we would expect for an arbitrary positive matrix A = (aij), that if for some i and j, (lik 2 ajk for all Jz, then wi 2 wj should hold. But when A is inconsistent, i.e., it does not satisfy (l), what is an appropriate order condition to be satisfied by the aij, and how general can such a condition be? We now develop conditions for order preservation that are essentially observations on the behavior of a consistent matrix later generalized to the inconsistent case. The ratio (wi/wj)/l may be interpreted as assigning

the ith activity the unit value of a scale and the jth activity the absolute value wj/wi. In the consistent case, order relations on UJ~, i = 1,. . . , n, can be inferred from the aij as follows: we factor out 201 from the first row, w2 from the second and so on, leaving us with a matrix of identical rows and w; 2 wj is both necessary and sufficient for A * w.

C. Berge [5] reports on a proposal by T. H. Wei [6] on the measurement of dominance or power of a player in a tournament through a pairwise comparison matrix B = (bij). Each row of B defines the standing of one player relative to the other players in the tournament. We have:

{

0 if i loses to j

bij = 1 if i ties j (in particular bii = 1) ,

2 if i wins over j

and thus bij + bji = 2. The overall power of each player i is defined as the ith component of limk,, Bke/eTBke, where Bk is the bth power of B. This coincides with a constant multiple of the ith component of the solution of Bw = Amax w where Amaw is the principal eigenvalue of the matrix B. From a set of arbitrary non-negative numbers one obtains a ratio scale w. But under what conditions is the solution relevant to the bij? This is what concerns us below.

There is a canon about order relations in A and correspondingly in w, when A is consistent, that we need to observe when A is inconsistent. We begin with a consistent matrix A. By successive application of the consistency condition (1) to each factor on the left of the condition itself, we obtain:

0 1 A2 1

0

k-l

A= =...= - Ak = . . . . n n

and in normalized form A A2 A”

- = eT = . ..= eT Ae

p-j& = “‘, (2) which shows that every power of A must be considered in the preservation of consistency. When A is consistent, the consistency condition (1) can be stated in equivalent terms for an arbitrary power of A. This is a useful observation for developing an order condition to be satisfied when A is inconsistent.

Five Conditions on A For Preserving Order

(i) A weaker condition for order preservation than

(Ali 1 (A)j implies Wi > Wj is

(ii)

(Aoh L (Ao)j

implies wi > wj, where (A)i and (Ae)i denote the ith row and ith row sum of A, and its generalization to powers of A given in the normalized form:

(iii)

(Ameli , (Ame)j - - eTAme - eTAme

implies Wi 2 Wj.

Relative measurement 5

The condition for order preservation must include all powers of A, and is given here in terms of their sum. For sufficiently large integer N > 0, and for p 2 N,

implies wi 1 wj, and

implies Wi 2 Wj.

THEOREM 3. Zf A is consistent, then

-+ cwj > 0

and (i)-(v) are true.

PROOF. The proof follows from A” = nmalA and A = (wi/wi).

It appears that the problem of constructing ratio scales from aij has a natural principal eigen- value structure. Our task is to extend this formulation to the case where A is no longer consistent.

3. SMALL PERTURBATIONS AND RATIO SCALE APPROXIMATION

Because we are interested in the construction of an appropriate matrix W of ratios that serves

as a “good” approximation to a given reciprocal matrix A, we begin by assuming that A itself is a perturbation of W. We need the following kind of background information.

For an unrepeated eigenvalue of a positive matrix A it is known [7-91 that a small perturbation A(C) of A gives rise to a small perturbation X(E) that is analytic in the neighborhood of c = 0. The following known theorems give us a part of what we need.

THEOREM 4. (Existence): If X is a simple eigenvalue of A, then for small E > 0, there is an eigenvalue A(c) of A(r) such that

X(E) = x + EX(l) + c2x(2) + . . .

and corresponding right and left eigenvectors w(c) and ~(6) such that

w(c) = w + (W(l) + &l(2) + . .

v(c) = 2, + uJ(‘) + c2J2) + ’ . .

Let 8ij be a perturbation of a reciprocal matrix A such that B = (aij +tlij) is also positive [IO].

THEOREM 5. If a positive reciprocal matrix A has the eigenvalues Al, As,. . . , An where the mul- tiplicity of Xj is mj with C,“,, mj = n, then given E > 0 there is a 6(c) > 0 such that if

jaij +6ij -aij 1 5 b for all i and j the matrix B has exactly mj eigenvalues in the circle lpj -Xi 1 < E for each j = 1,. . . , s where ~1,. . . , p, are the eigenvalues of B.

If A is a consistent matrix, then it has one positive eigenvalue Xi = R and all other eigenvalues are zero. For a suitable c > 0 there is a 6(c) > 0 such that for l0ij) < 6 the perturbed matrix B has one eigenvalue in the circle Ipi - nl < c and the remaining eigenvalues fall in a circle (pj -01 < E, j- ,...,n. ‘-2

THEOREM 6. If n is a simple eigenvalue of A which dominates the remaining eigenvalues in modulus, for sufficiently small c, n(c) s X,, dominates the remaining eigenvalues of A(c) in modulus.

6 T.L. SAATY

When A is inconsistent, several conditions on aij and on wi, along with uniqueness, must be met to enable us to approximate A by ratios. Our conditions are divided into two categories: those dealing with the order induced by aij as absolute numbers (wi/wj)/l or l/(wi/wi) from a standard scale, on the components of the scale w, and those dealing with the equality or near equality of the aij to the ratios wi/wj formed from the derived scale w.

When A is inconsistent, how do we construct W so that the order preservation condition (v) still holds? Later we address the other question; what conditions must A satisfy to ensure that wi/wj is a “good” approximation to aij?

Let us consider estimates of ratios given by an expert who may make small perturbations cij in W = (wi/wj). Comparison by ratios allows US to write aij = (wi/wj) Eij, cij > 0, i,j = 1,. . . , n.

In that case, A takes the form A = WOE = DED-’ where W = (wi/wj), E = (cij), D a diagonal matrix with w as diagonal vector, and o refers to the Hadamard or elementwise product of the two matrices. The principal eigenvalue of A coincides with that of E. The principal eigenvector of A is the elementwise product of the principal eigenvectors w = (WI,. . . , w,,)~, and e = (1,. . . , l)T of W and of E, respectively [4].

The distinction we make between an arbitrary positive matrix and a reciprocal matrix is that we can control a step by step modification of a reciprocal matrix so that in the representation A = W o E = DED-‘, the cij, i, j = 1,. . . , n are small. The purpose is to ensure that perturbing the principal eigenvalue and eigenvector of W yields the principal eigenvalue and eigenvector of A.

Why do we need such a perturbation ? Because we assume that there is an underlying ratio scale that we attempt to approximate. By improving the consistency of the matrix, we obtain an approximation of the underlying scale by the principal eigenvector of the resulting matrix.

THEOREM 7. w is the principal eigenvector of a positive matrix A, if and only if, Ee = X,, e.

Note that e is the principal eigenvector of E and E is a perturbation of the matrix eTe. When Ee # Amax e, the principal eigenvector of A is another vector w’ # w and A = W’ o E’ where E’e = X,, e.

COROLLARY. w is the principal eigenvector of a positive reciprocal matrix A = W o E, if and only if, Ee = Amax e and cji = (yj)-l s

If A is an arbitrary positive matrix that is a small perturbation E of W = (Wi/Wj), then:

THEOREM 8. (order preservation): A positive matrix A satisfies condition (v), if and only if, the

derived scale w is the principal right eigenvector of A, i.e., Aw = ,imaxw.

PROOF. We give two proofs of this theorem; the first is based on Perron’s theorem and the second, which is more appropriate for our purpose, is based on perturbation.

Let Ake

Sk = - eTAke (3)

and

The convergence of the components of t, to the same limit as the components of s,,, is the standard Cesaro summability. Since

Ake Sk = &&yW ask+co

where w is the normalized principal right eigenvector of A, we have

tm2e Ake

m---+w asm+oo. m

k=l

(6)

Relative measurement 7

For the second proof, first assume that A has only simple eigenvalues. Using Sylvester’s formula:

njzi(A - AI) f(A) = 2 f(xi) njgicAj _ x.) ) &ax = x1

i=l I

we have on writing f(A) = Ak , dividing through by Ah,, multiplying on the left by (A - A,,I)

to obtain the characteristic polynomial of A then multiplying on the right by e we obtain:

Ake iirnW r = cw,

- max

c > 0.

Sylvester’s formula for multiple eigenvalues of multiplicity mi shows that one must consider derivatives of f(A) of order no more than mi. However, it is easy to verify by interchanging derivative and limit, that when each term is divided by A;, its value tends to zero as Ic ---* 00, and the result again follows.

Therefore, it is necessary to obtain the principal eigenvector w to capture order properties from A, but not sufficient to ensure that W = (wi/wj) is a good approximation to A. The method we use to derive the scale w from a positive inconsistent matrix must also satisfy the following four conditions on what constitutes a good numerical approximation to the oij by ratios. The first two are local conditions on each aij , the second two are global conditions on all eij through the principal eigenvalue and eigenvector as functions of the eij.

3.1. Reciprocal

The reciprocal condition is a local relation between pairs of elements: aji = l/eij, needed to ensure that, as perturbations of ratios, oij and eji can be approximated by ratios from a ratio scale that are themselves reciprocal. It is a necessary condition for consistency.

3.2. Homogeneous-Uniformly Bounded Above and Below

Homogeneity is also a local condition on each aij. To ensure consistency in the paired compar- isons, the elements must be of the same order of magnitude which means that our perceptions in comparing them should be of nearly the same order of magnitude. Thus we require that the eij be uniformly bounded above by a positive constant Ii’ and, because of the reciprocal condition, they are automatically uniformly bounded below away from zero:

f 5 aij 5 Ii, K>O, i,j=l,..., n.

It is a fact that people are unable to directly compare widely disparate objects such as an apple and a watermelon according to weight. If they are not comparable, it should be possible to aggregate them in such homogeneous clusters to make the comparisons. For example, we put the apple with a grapefruit and a cantaloupe in one cluster, then the cantaloupe again, a honeydew melon and varying sizes of watermelons in another cluster. The relative measurements in the clusters can be combined because we included the largest element (the cantaloupe) in the small cluster as the smallest element of the adjacent larger cluster. Then the relative weights of the elements in the second cluster are all divided by the relative weight of the common element and multiplied by its relative weight in the smaller cluster. In this manner, relative measurement of the elements in the two clusters can be related and the two clusters combined after obtaining relative measurement by paired comparisons in each cluster. The process is continued from cluster to adjacent cluster.

3.3. Near Consistent

The near consistency condition, which is global, is formed in terms of the (structural param- eters) A,,, and n of A and W. It is a less familiar and more intricate condition that we need to discuss at some length. The requirement that comparisons be carried out on homogeneous

8 T.L. SAATY

elements ensures that the coefficients in the comparison matrix are not too large and generally of the same order of magnitude, i.e., from 1 to 9. Knowing this constrains the size of the per- turbations cij, whose sum as we shall see below, is measured in terms of the near consistency condition Amax - n.

The object then is to apply this condition to develop algorithms to explore changing the judgments and their approximation by successively decreasing the inconsistency of the judgments and then approximating them with ratios from the derived scale. The simplest such algorithm is one which identifies that eij for which aij wj/wi is maximum and indicates decreasing it in the direction of wi/wj. Another algorithm, due to Harker [ll], utilizes the gradient of the oij. In the end, we obtain either a consistent matrix or a closer approximation to a consistent one depending on whether the information available allows for making the proposed revisions in eij.

Because consistency is necessary and sufficient for A to have the form A = (wi/wj), we use w to explore possible changes in aij to modify A “closer” to that form. We form a consistent matrix W’ = (w:/wi), h w ose elements are approximations to the corresponding elements of A.

We have oij = (w:/w~) ‘ii, Cij > 0. We have the converse of: given a problem, find a good approximation to its solution; given a problem with its exact solution, use the properties of this solution to revise the problem, i.e., the judgments which give rise to oij. Repeat the process to a level of admissible consistency (see below).

3.4. Uniformly Continuous

Uniform continuity implies that wi, i = 1,. . . ,n as a function of oij should be relatively insensitive to small changes in the aij in order that the ratios wi/wj remain good approximations

to the eij. For example, it holds in wi as the ith component of the principal eigenvector because it is an algebraic function of X,,, (whose value is shown to lie near n because of 3), and of the oij and l/eij, which are bounded.

Let us now further elaborate on the near consistency condition in (3). We first show the interesting result, that inconsistency or violation of (1) by different oij can be captured by a single number X,,, - n, which measures the deviation of all eij from wi/wj.

Assume that the reciprocal condition eji = l/aij and boundedness l/K 5 oji 5 I<, where I< > 0 is a constant, hold. Let oij = (1+6ij) wi/wj, 6ij > -1, be a perturbation of W = (wi/wj), where w is the principal eigenvector of A.

THEOREM 9.

A max 2 n.

PROOF. Using aji = l/aij, and AW = Xmaxw, we have

x *ax -n=- : ,<ig<, is L O. i) ---

(7)

THEOREM 10. A is consistent ifand only if&,-,, = n.

PROOF. If A is consistent, then because of (l), each row of A is a constant multiple of a given row. This implies that the rank of A is one, and all but one of its eigenvalues &, i = 1,. . . , n, are zero. However, it follows from our earlier argument that, Cy=i & = Trace (A) = n. Therefore, x max = n. Conversely, Amax = n, implies cij = 0, and aij = wi/wj.

From (2) we can determine the magnitude of the “greatest” perturbation by setting one of the terms equal to Amax - n and solving for 6ij in the resulting quadratic. An average perturbation value is obtained by replacing Amax - n in the previous result by (X,,, - n)/(n - 1).

A measure of inconsistency is obtained by taking the ratio of X,,, - n to its average value over a large number of reciprocal matrices of the same order n, whose entries are randomly chosen in the interval [l/1(, K]. If this ratio is small (e.g., 10% or less-for example 5% for 3 x 3 matrices) [12], we accept the estimate of w. Otherwise, we attempt to improve consistency and derive a new w. After each iteration, we assume that the new matrix is a perturbation of W and its eigenvalue and eigenvector are perturbations of n and w, respectively.

Relative measurement 9

In his experimental work in the 1950’s, the psychologist George Miller [13] found that, in general, people (such as chess experts looking ahead a few moves to decide on a good next move) could deal with information involving simultaneously only a few facts: seven plus or minus two. With more, they become confused and cannot handle the information. This is in harmony with the established fact that for a reciprocal matrix (though not in general) the principal eigenvalue is stable for small perturbations when n is small.

We have seen that only order-preserving derived scales w are of interest. There are many ways to obtain w from A. Most of them are error minimizing procedures such as the method of least squares:

n

CC

2 Wi

aij --

wj > (8)

i,j=l

which also produces non-unique answers. Only the principal eigenvector satisfies order preserving requirements when there is inconsistency. We summarize with the following.

THEOREM 11. If a positive n x n matrix A is reciprocal, homogeneous, and near consistent, then the scale w derived from Aw = X,,, w is order preserving, unique to within a similarity transformation and uniformly continuous in the aij, i, j, = 1, . . . , n.

Similar results can be obtained when A is non-negative. Also, we have extended this discrete approximation of A by W to the continuous case of A reciprocal kernel and its eigenfunction [14].

4. STRUCTURAL PROPERTIES OF POSITIVE RECIPROCAL MATRICES

We make the following observations on the structure of reciprocal matrices. The elementwise product of two n x n reciprocal matrices is a reciprocal matrix. It follows that the set of reciprocal matrices is closed under the operation Hadamard product. The matrix eTe is the identity: eTe = eTeoeTe = eeT and AT is the inverse of A, AoAT = AToA = eTe. Thus, the set G of n x n reciprocal matrices is an abelian group. Because every subgroup of an abelian group is normal, in particular, the set of n x n consistent matrices is a normal subgroup (E o W o ET = W) of the group of positive reciprocal matrices.

Two matrices A and B are R-equivalent (A R B) if and only if, there is a vector w and positive constants a and b such that (l/a) Aw = (l/b) Bw. The set of all consistent matrices can be partitioned into disjoint equivalence classes. Given a consistent matrix W and a perturbation

matrix E such that Ee = ae, a > 0 a constant, we use the Hadamard product to define A’ = W o E such that (l/a)A’w = (l/n) Ww. A’ and W are R-equivalent. There is a l-l correspondence between the set of all consistent matrices and the set of all matrices A’ defined by such Hadamard products. An R-equivalence class Q(W) is the set of all A’ such that A’R W. The set of equivalence classes Q(W) forms a partition of the set of reciprocal matrices. It is known that all the elements in Q(W) are connected by perturbations E, E’, E”, . . . , corresponding to a fixed value of a > 0 such that (E o E’Oe” . . . )e = ae. Thus, given an arbitrary reciprocal matrix A, there exists an equivalence class to which A belongs.

DeTurck [15] has proved that: The structure group G of the set of positive reciprocal n x n matrices has 2n! connected components. It consists of non-negative matrices which have ex- actly one nonzero entry in each row and column, i.e., the matrices can be expressed as D. S, where D is a diagonal matrix with positive diagonal entries and S is a permutation matrix, and the negatives of such matrices. The connected component Go of the identity consists of diagonal matrices with positive entries on the diagonal. If A is a positive reciprocal matrix with principal right eigenvector w = (~1, ~2,. . . , w,,)~ and D E Go is a diagonal matrix with positive diag- onal entries dl , d2, . . . , d, , then ID(A) = DAD-’ is a positive reciprocal matrix with principal eigenvector w’ = (dl WI, . . . , d,w,)T. The principal eigenvalue is the same for both matrices. If 21 = (~1,. . . ,v,)~ and w = (WI,. . . , w,,)~ are two positive column vectors, then conjugation by the diagonal matrix D,, with entries v1/wl,. . . ,v,/w,, on the diagonal maps A, onto A,. The corresponding diagonal matrix D ,,,” provides the inverse map. Moreover, D,, maps the consistent matrix of A, to the consistent matrix of A,.

10 T.L. SAATY

5. THE HILBERT METRIC

There are many ways to judge when two ratio scales are close. But there is one that is particularly attractive for a number of reasons. The theorem of Perron says that if A is a positive linear transformation on Rm, then there is an x0 > 0 such that for all x 2 0, A”+ converges in direction to z, so that

A”* X,?

According to Birkhoff [16,17], this theorem is a special case of the contraction mapping theorem which says that if A is a contraction on a complete metric space (X, 0) mapping X into X, i.e., for some k < 1, D(Az,Ay) 5 kD(z,y) f or all x, y E X, then there exists x, E X such that A”x + x0, for all 2 E X.

Birkhoff observed that there is a metric D on x in which all positive linear transformations act- ing on the set of rays X in R$’ satisfy the contraction condition because convergence in rays is also convergence in direction. The unique metric D, invented by Hilbert for non-Euclidean geometries specialized to RT that makes a positive (or even a non-negative) matrix into a contraction that satisfies the theorem, is given by:

where xi and yi are the ith coordinates of the vectors x and y. It is a metric on rays because

D(oz, by) = D(t, Y) f or a, b > 0 and thus it is order preserving. A method that needs to preserve order must converge along rays and thus cannot use an arbitrary metric.

Hilbert’s metric is useful in analyzing closeness when ratio scales are involved. For example, in medical diagnosis, priorities can be established for the symptoms of a disease. This is done for several diseases. An arriving patient is questioned and priorities established for his/her symptoms. This vector is then compared with each of the priority vectors of the different diseases to determine the most likely illness for the patient. The question is, what is a reliable way for doing this? The Hilbert metric is one indicator of closeness. So is the Euclidean metric. My colleague Luis Vargas and I have performed experiments on matrices of judgments whose entries

are perturbed in a prescribed range. We find that Hilbert’s metric is useful for discrimination

among vectors in a neighborhood of a known answer, in this case the principal right eigenvector of the original matrix. The ratios in the Hilbert metric are more sensitive to small variations than are differences in the Euclidean metric. We found experimentally that both metrics taken with respect to a given vector follow a normal and log normal distribution respectively. These distributions can then be used to construct standards to determine if a given point falls in a range close to the given vector. Multivariate analysis could be used to compare the results of

both metrics as to whether the points falling in admissible ranges also fall in the n-dimensional confidence ellipsoid.

6. THE RANK PRESERVATION PARADOX

There is a long held belief by the practitioners of Utility Theory and more generally by those of Multiatribute Utility Theory (MAUT) that in ranking a set of alternatives, if one should later add or delete a dominated (often called irrelevant) alternative to the set, the rank of the original set must remain the same. This is a reasonable assumption when the alternatives are ranked on one property or criterion. It holds when alternatives are rated one at a time by using a scale. It also holds in relative measurement when the resulting matrix is consistent. However, it has been observed in practice that with multiple criteria (where the overall ranking of the alternatives is obtained by combining their ranks with respect to the individual criteria by using the weight of the criteria themselves) this assumption is no longer valid. For example, a copy or copies of an alternative can depreciate the desirability of that alternative. It has also been found that the presence of a highly rated alternative can affect preference among other alternatives. For example, a phantom alternative is a higher priced product advertised to be available only to cause people to shift from a lower priced to an intermediate priced product. Thus, a nonexistent

Relative measurement 11

irrelevant alternative causes change in people’s choices. Such information cannot be displayed as one of the criteria because it is not an intrinsic attribute of any single alternative but depends on all of them and because the importance of such a criterion would then depend on the alternatives, contradicting the assumption that the criteria are independent from the alternatives. This kind of precise quantitative information is captured in relative measurement through normalization which reflects the influence of the number of alternatives and the relative dominance of an alternative in the overall rank of the others for all the criteria. There are also cases where such an effect is considered unnecessary. For example, in choosing a best car, it is often unimportant how many copies of that car there may be. Instead of normalization, relative measurement uses the notion of an ideal alternative by dividing the weights of the alternatives with respect to each criterion by the weight of the highest rated one which need not be the same alternative for all the criteria. In that case, a copy or an everywhere dominated alternative cannot affect the choice of the more preferred one. Only an alternative that is more preferred than some ideal alternative can.

Here is an example from the literature showing that rank reversals occur in practice but are unaccounted for by rating alternatives one at a time. Regularity is a condition of choice theory that has to do with rank preservation. R. Corbin and A. Marley [20] provide an example that,

concerns a lady in a small town, who wishes to buy a hat. She enters the only hat store in town, and finds two hats, A and B, that she likes equally well, and so might be considered equally likely to buy. However, now suppose that the sales clerk discovers a third hat, C, identical to A. Then the lady may well choose hat B for sure (rather than risk the possibility of seeing someone else wearing a hat just like hers), a result that contradicts regularity.

Let us show how relative measurement can be used to explain this Utility Theory paradox. Here is a plausible set of judgements in the case of the two hats A and B, with A slightly preferred to B that explain why this phenomenon can happen. Adding C that is a copy of A changes the preference to B over A with respect to uniqueness and thus makes it the more desired choice overall. The numerical values used may, for this purpose, be regarded as perturbations of the identity comparison (of an element with itself).

Style (.4) Uniqueness (.6)

A B A B

A 1 3

I I

.75 A 1 1 .5

B + 1 .25 I I B 1 1 .5

A = .75 x .4 + .5 x .6 = .6

B = .25 x .4 + .5 x .6 = .4

and A is preferred to B.

Style (.4)

A B C

A 1 1 1

B 3 f 3

c 1 i 1

.42 A

.16 B

.42 C

Uniqueness (.6)

A B C

1 6 1 .125

1. 6 1 6 .750

1 6 1 ,125

A = .42 x .4 + .125 x .6 = .238

B= .16 x .4 + .750 x .6 = .514

C = .42 x .4 + .125 x .6 = .238

and now B is preferred to A. Another example of rank reversal with an irrelevant alternative is when a state has 60% con-

servatives and 40% liberals. If there is one liberal and one conservative candidate, the result would favor the conservative 60-40. If a third less popular conservative candidate is brought in, the conservative vote is split and the liberal candidate becomes the winner. Vote-splitting is a time honored political tactic.

12 T.L. SAATY

There are also situations where we would want to enforce rank preservation by rating alter- natives in conformity with standards learned through experience. For example, in admitting students to a university, the rank of students already judged accepted and notified for the fall term should remain unchanged if a new application is added to the collection. This is a clear instance where we wish to minimize regret by not changing a former decision. To preserve rank, one can rate the students on intensities such as high, medium, low on each criterion. Weights for these semantic intensities can be established through paired comparisons. Here relative measure- ment is applied to the criteria and also to the intensities which are then standardized according to the ideal (described above) and the alternatives rated on the standards of each criterion and their overall ratio scale ranks are then derived by weighing and addition.

7. CONCLUSIONS

When we compare two alternatives, we often refer to their measurements on a scale with a unit if there is one. But measurements on a scale are simply indicators. They must be interpreted in some human value system to make sense. At times, small readings are interpreted in a positive way, larger readings negatively and still larger positively or conversely. For example, very cold and very hot temperatures are good for preserving food but bad for our health. The opposite is true for moderate temperatures. Thus, in the final analysis, relative interpretation is essential. The question then is, particularly when dealing with intangibles, whether it is possible from the start to make these interpretations without need for a scale.

Relative measurement is a tool of our mind. We need to learn more about it and use it efficiently and more often when intangibles are involved and also when we must interpret the measurements of tangibles. That is where relative measurement and hierarchies must come together as in the Analytic Hierarchy Process [21].

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