A different perspective on a scale for pairwise comparisons

08/05/2015 ∙ by J. Fueloep, et al. ∙ 0

One of the major challenges for collective intelligence is inconsistency, which is unavoidable whenever subjective assessments are involved. Pairwise comparisons allow one to represent such subjective assessments and to process them by analyzing, quantifying and identifying the inconsistencies. We propose using smaller scales for pairwise comparisons and provide mathematical and practical justifications for this change. Our postulate's aim is to initiate a paradigm shift in the search for a better scale construction for pairwise comparisons. Beyond pairwise comparisons, the results presented may be relevant to other methods using subjective scales. Keywords: pairwise comparisons, collective intelligence, scale, subjective assessment, inaccuracy, inconsistency.



There are no comments yet.


page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

Collective intelligence (CI) practitioners face many challenges as collaboration, especially involving highly trained intellectuals, is not easy to manage. One of the important aspects of collaboration is inconsistency arising from different points of view on the same issue. According to [33], “Inconsistent knowledge management (IKM) is a subject which is the common point of knowledge management and conflict resolution. IKM deals with methods for reconciling inconsistent content of knowledge. Inconsistency in the sense of logic has been known for a long time. Inconsistency of this kind refers to a set of logical formulae which have no common model.” and “The need for knowledge inconsistency resolution arises in many practical applications of computer systems. This kind of inconsistency results from the use of various sources of knowledge in realizing practical tasks. These sources often are autonomous and they use different mechanisms for processing knowledge about the same real world. This can lead to inconsistency.”

Unfortunately, inconsistency is often taken for a synonym of inaccuracy but it is a “higher level” concept. Inconsistency indicates that inaccuracy of some sort is present in the system. Certainly, inaccuracy by itself would not take place if we were aware of it. We will illustrate it in a humorous way. When a wrong phone call is placed, the caller usually apologizes by “I am sorry, I have the wrong number” and may hear in reply: “if it is a wrong number, why have you dialed it?” Of course we would have not dialed the number if we had known that it was wrong. In fact, the respondent is the one who detects the incorrectness, not the caller.

However, a self correction may also take place in some other cases, for example, via an analysis of our own assessments for inconsistency by comparing them in pairs. Highly subjective stimuli often are present in the assessment of public safety or public satisfaction. Similarly, decision making, as an outcome of mental processes (cognitive process), is also based on mostly subjective assessments for the selection of an action among several alternatives. We can compute the inconsistency indicator of our assessments (subjective or not) rarely getting zero which stands for fully consistent assessments.

As the membership function of a fuzzy set is a generalization of the indicator function in classical sets, the inconsistency indicator is related to the degree of contradictions existing in the assessments. In fuzzy logic, the membership function represents the degree of truth. Similarly, the inconsistency indicator is related to both the degree of inaccuracy and contradiction. Degrees of truth are often confused with probabilities, although they are conceptually distinct. Fuzzy truth represents membership in vaguely defined sets but not the likelihood of some event or condition. Likewise, the inconsistency indicator is not a probability of contradictions but the degree of contradiction.

In our opinion, pairwise comparisons method is one of the most feasible representations of collective intelligence. It also allows one to measure it, for example, by comparing CI with individual intelligence. (According to the online Handbook of Collective Intelligence, hosted at the website of MIT Center of Collective Intelligence http://cci.mit.edu/research/index.html, measuring CI is one of two main projects for developing theories of CI.) Pairwise comparisons are easy to use, but may require complex computations to interpret them properly. This is why we address the fundamental issue of scales of measure, which – in particular – may have an effect on feasibility of some computational schemes.

2 Pairwise comparisons preliminaries

Comparing objects and concepts in pairs can be traced to the origin of science or even earlier – perhaps to the stone age. It is not hard to imagine that our ancestors must have compared “chicken and fish”, holding each of them in a separate hand, for trading purposes. The use of pairwise comparisons is still considered as one of the most puzzling, intriguing, and controversial scientific method although the first published use of pairwise comparisons (PC) is attributed to Condorcet in 1785 (see [12], four years before the French Revolution). Ramon Llull, or Raimundus Lullus designed an election method around 1275 in [30]. His approach promoted the use of pairwise comparisons. However, neither Llull nor Condorcet used a scale for pairwise comparisons.

Condorcet was the first who used a kind of binary version of pairwise comparisons to reflect the preference in the voting by the won-lost situation. In [37], a psychological continuum was defined by Thurstone in 1927 with the scale values as the medians of the distributions of judgments on the psychological continuum.

In [34], Saaty proposed a finite (nine point) scale in 1977. In [26]

, Koczkodaj proposed a smaller five point scale with the distance-based inconsistency indicator. This smaller scale better fits the heuristic

off by one grade or less” for the acceptable level of inconsistency proposed in [26]. We will show here that a new convexity finding for the first time supports the use of an even smaller scale.

Mathematically, an real matrix is a pairwise comparison (PC) matrix if and for all . Elements represent a result of (often subjectively) comparing the th alternative (or stimuli) with the th alternative according to a given criterion. A PC matrix is consistent if for all It is easy to see that a PC matrix is consistent if and only if there exists a positive


such that For a consistent PC matrix , the values serve as priorities or implicit weights of the importance of alternatives.

3 The pairwise comparisons scale problem

Thurstone’s approach was extensively analyzed and elaborated on in the literature, in particular by Luce and Edwards [28] in 1958. The bottom line is that subjective quantitative assessments are not easy to provide. Not only is the dependence between the stimuli and their assessments usually nonlinear, but the exact nature of the nonlinearity is in general unclear. In this context, a smaller scale is expected to generate a smaller error, for example by mitigating the deviation from nonlinearity.

On page 236 in [28], authors wrote: W.J. McGill is currently attempting to find a better way of respecting individual differences while still obtaining a “universal scale”. Authors of this study have not been able to trace any publication of the late W.J. McGill on the “universal scale” construction. However, the proposed smaller scale may be at least some kind of temporary solution as a reflection of “the small is beautiful” movement inspired by Leopold Kohr by his opposition to the “cult of bigness” in social organization. The smaller five-point scale better fits the heuristic “off by one grade or less” for the acceptable level of the distance-based inconsistency (as proposed in [26]). We will show here that the new convexity finding, for the first time, supports the use of an even smaller scale.

There are strong opponents of the pairwise comparisons method going as far as opposing the use of pairwise comparisons altogether. However, they forget that every measurement, e.g., of length, is based on pairwise comparisons since we compare the measured object with some assumed unit. For example, one meter is the basic unit of length in the International System of Units (SI). It was literally defined as a distance between two marks on a platinum-iridium bar. Evidently, we are unable to eliminate pairwise comparisons from science hence we need to improve them. As we will demonstrate, it is the issue of scale (in other words the input data) and, as such, it cannot be ignored.

4 In search of the nearest consistent pairwise comparisons matrix

Several mathematical methods have been proposed for finding the nearest consistent pairwise comparisons matrix for a given inconsistent pairwise comparisons matrix. In [34]

, the eigenvector method was proposed in which

is the principal eigenvector of . Another class of approaches is based on optimization methods and proposes different ways of minimizing (the size of) the difference between and a consistent PC matrix. If the difference to be minimized is measured in the least-squares sense, i.e. by the Frobenius norm, then we get the Least Squares Method presented by Chu et al. [10]. The problem can be written in the mathematical form (we present the normalized version, see [20]):

Since the matrices in the form of columnwise ordering can also be considered as -dimensional vectors, (say, by stacking the columns over each other), problem (4) determines a consistent PC matrix closest to in the sense of the Euclidean norm. Unfortunately, problem (4) may be a difficult nonconvex optimization problem with several possible local optima and with possible multiple isolated global optimal solutions [24, 25].

Some authors state that problem (4) has no special tractable form and is difficult to solve [10, 21, 31, 11]. In order to elude the difficulties caused by the nonconvexity of (4), several other, more easily solvable problem forms are proposed to derive priority weights from an inconsistent pairwise comparison matrix. The Weighted Least Squares Method [10, 3] in the form of

applies a convex quadratic optimization problem whose unique optimal solution is obtainable by solving a set of linear equations. The Logarithmic Least Squares Method [16, 13] in the form

is (because of constraints being linearizable) a simple optimization problem whose unique solution is the geometric mean of the rows of matrix

. For further approaches, see [5, 17, 20] and the references therein. However, we have to emphasize that the main purpose of many (if not most) optimization approaches was to exclude the difficulties caused by the possible nonconvexity of problem (4). It was usually done by sacrificing the natural approach of the Euclidean distance minimization.

As with many other real-life situations, there is no possibility to decide which solution is the best without a clear objective function. For example, a “Formula One” car is not the best vehicle for a family with five children but it may be hard to win a Grand Prix race with a family van. In fact, pairwise comparisons could be used for solving the dilemma of which approximation solution is the best for PC (and for the family transportation problem).

The distance minimization approach (4) is so natural that one may wonder why it was only recently revived in [5]. The considerable computational complexity (100 hours CPU time for ) and the possibility of having multiple solutions (and/or multiple local minima) may be the reasonable explanation for not becoming popular in the past. Problem (4) has recently been solved in [1] by reducing 100 hours of CPU (or more likely, 150 days of the CPU time) to milliseconds. It was asserted in [4, 24, 25] that the multiple solutions are far enough from the ones that appear in the real-life situations. However, it appears that these assertions are mostly based on anecdotal evidence. More (numerical and/or analytical) research to elucidate this point would be helpful.

As proved by Fülöp [20], the necessary condition for the multiple solutions to appear is that the elements of the matrix are large enough. In [20], using the classic logarithmic transformation

and the univariate function


depending on the real parameter , problem (4) can be transformed into the equivalent form


It was also proved in [20], there exists an such that for any the univariate function of (4) is strictly convex if and only if . Consequently, in the case when the condition is fulfilled for all , , then (4) can be transformed into the convex programming problem (5) with a strictly convex objective function to be minimized (see [20], Proposition ). In other words, problem (4) and the equivalent problem (5) have a unique solution which can be found using standard local search methods. The above-mentioned constant equals to , which is a reasonable bound for many real-life problems. The above is not necessarily a strict threshold since its proof is based on the convexity of univariate functions (see [20], Proposition 2, or see the Appendix of the present paper for a compact low-tech argument) and it is conceivable that the exact threshold for the sum of univariate functions is greater than . We know, however, that this threshold must be less than since, as shown by Bozóki [4], for any it is easy to construct a PC matrix with as the largest element and with multiple local minima. Finally, even if some elements of a PC matrix are relatively large, it may still happen that (4) has a single local minimum; a sample sufficient condition is given in Corollary of [20].

A nonlinear programing solver (available in Excel and described in [1]) is good enough if (4) has a single local minimum for a given a PC matrix . Our incentive for postulating a restricted ratio scale for pairwise comparisons comes both from the guaranteed uniqueness in the interval determined in [20] and from demonstrably possible (by [4]) non-uniqueness outside of a just slightly larger interval.

There have been several inconsistency indicators proposed. The distance-based inconsistency (introduced in [26]) is the maximum over all triads of elements of (with all indices distinct) of their inconsistency indicators defined as:

Convergence of this inconsistency was finally provided in [27] (an erlier attempt in [22] had a hole in the proof of Theorem 1). A modification of the distance-based inconsistency was proposed in 2002 in [18]

. Analysis of the eigenvalue-based and distance-based inconsistencies was well presented in

[6]. Paying no attention to what we really process to get the best approximation, brings us what GIGO, the informal rule of “garbage in, garbage out”, so nicely illustrates. This is why localizing the inconsistency and reducing it is so important.

5 The scale size problem

As of today, the scale size problem for the PC method has not been properly addressed. We postulate the use of a smaller rather than larger scale and more research to validate it.

As mentioned earlier, an interesting property of PC matrices has been recently found in [20]. Namely, (4) has a unique local (thus global) optimal solution and it can be easily obtained by local search techniques if holds for all , where the value is at least 3.330191 (but can not be larger than , see [4]). In our opinion, this finding has a fundamental importance for construction of any scale and we postulate the scale 1 to 3 (1/3 to 1 for inverses) should be carefully looked at before a larger scale is considered. In the light of the property from [20], finding the solution of (4) would be easier and faster. This fact should shift the research of pairwise comparisons back toward (4) for approximations of inconsistent PC matrices. This is a starting point for the distance minimization approaches. It is worth to note that PC method is for processing subjectivity expressed by quantitative data. For purely quantitative data (reflecting objectively measurable even if possibly uncertain quantities), there are usually more precise methods (e.g., equations, systems of linear equations, PDEs just to name a few of them). In general, we are better prepared for processing quantitative data (e.g., real numbers) than for qualitative data.

A comparative scale is an ordinal or rank order scale that can also be referred to as a non-metric scale. Respondents evaluate two or more objects at a time and objects are directly compared with one to another as part of the measuring process. In practice, using a moderate scale for expressing preferences makes perfect sense. When we ask someone to express his/her preference on the 0 to 100 scale, the natural tendency is to use numbers rounded to tens (e.g., 20, 40, 70,…) rather than by using finer numbers. In fact, there are situations, such as being pregnant or not, with practically nothing between. The theory of scale types was proposed by Stevens in [36]. He claimed that any measurement in science was conducted using four different types of scales that he called “nominal”, “ordinal”, “interval”, and “ratio”.

Measurement is defined as “the correlation of numbers with entities that are not numbers” by the representational theory in [32]. In the additive conjoint measurement (independently discovered by the economist Debreu in [15] and by the mathematical psychologist Luce and statistician Tukey in [29]), numbers are assigned based on correspondences or similarities between the structure of number systems and the structure of qualitative systems. A property is quantitative if such structural similarities can be established. It is a stronger form of representational theory than of Stevens, where numbers need only be assigned according to a rule. Information theory recognizes that all data are inexact and statistical in nature. Hubbard in [23], characterizes measurement as: “A set of observations that reduce uncertainty where the result is expressed as a quantity.”

In practice, we begin a measurement with an initial guess as to the value of a quantity, and then, by using various methods and instruments, try to reduce the uncertainty in the value. The information theory view, unlike the positivist representational theory, considers all measurements to be uncertain. Instead of assigning one value, a range of values is assigned to a measurement. This approach also implies that there is a continuum between estimation and measurement.

The Rasch model for measurement seems to be the relevant to PC with the decreased scale. He uses a logistic function (or logistic curve, the most common sigmoid curve): . Coincidentally, the exponential function was used in [20] for his estimations of the upper bound of .

We mentioned that the phenomenon of the scale reduction appears implicitly in the Logarithmic Least Squares Method [16, 13] as well. It is easy to see that in problem (4), it is not the original PC matrix which is approximated but which consists of the entries .

6 An example of a problem related to using two scales

Let us look at two scales: to and to :

Bigger scale 1 2 3 4 5
Smaller scale 1 1.5 2 2.5 3

The inconsistent pairwise comparisons table for the to scale generated by the triad is:

1 3 5
1 3

The inconsistency of this table is computed by as .

The triad consists of the top scale value in the middle and the middle scale value as the first and last values of the triad. Similarly, the inconsistent pairwise comparisons table for the 1 to 3 scale generated by the triad is:

1 2 3
1 2

The inconsistency of this table is computed by as .

The middle value in the triad is the upper bound of the scale to . The other two values () are equal to the middle point value of the scale to . The same goes for all values of the triad on the scale 1 to 5 hence we can see that they somehow correspond to each other yet the inconsistencies are drastically different from each other and clearly unacceptable for the heuristic assumed in [26] of for the first table and acceptable for the second table. Needless to say, there is no canonical mapping from the scale 1 to 5 to the scale 1 to 3. The table proposed above is admittedly ad hoc and we present it for demonstration purposes only.

Evidently, more research is needed for this not so recent problem. In all likelihood, it was mentioned for the first time in [28] in 1958. Most real-life projects using the pairwise comparisons method are impossible to replicate or compute for the new scale as the costs of such exercise would be substantial. It may take some time before a project with a double scale is launched and completed.

7 The power of the number three

The “use of three” for a comparison scale has a reflection in real life. Probably the greatest support for the use of three as the upper limit for a scale comes from the grammar. Our spoken and written language has evolved for thousands of years and grammar is at the core of each modern language. In his 1946 textbook [8] (which also nicely describes the degree of comparisons as they may be used in PC), Bullions defines comparisons of adjectives in as:

Adjectives denoting qualities or properties capable of increase, and so of existing in different degrees, assume different forms to express a greater or less degree of such quality or property in one object compared with another, or with several others. These forms are three, and are appropriately denominated the positive, comparative, and superlative. Some object to the positive being called a degree of comparison, because in its ordinary use it does not, like the comparative and superlative forms, necessarily involve comparison. And they think it more philosophical to say, that the degrees of comparison are only two, the comparative and superlative. This, however, with the appearance of greater exactness is little else than a change of words, and a change perhaps not for the better. If we define a degree of comparison as a form of the adjective which necessarily implies comparison, this change would be just, but this is not what grammarians mean, when they say there are three degrees of comparison. Their meaning is that there are three forms of the adjective, each of which, when comparison is intended, expresses a different degree of the quality or attribute in the things compared: Thus, if we compare wood, stone, and iron, with regard to their weight, we would say “wood is heavy, stone heavier, and iron is the heaviest.”
Each of these forms of the adjective in this comparison expresses a different degree of weight in the things compared, the positive heavy expresses one degree, the comparative heavier, another, and the superlative heaviest, a third, and of these the first is as essential an element in the comparison as the second, or the third. Indeed there never can be comparison without the statement of at least two degrees, and of these the positive form of the adjective either expressed or implied, always expresses one. When we say “wisdom is more precious than rubies,” two degrees of value are compared, the one expressed by the comparative, “more precious,” the other necessarily implied. The meaning is “rubies are precious, wisdom is more precious.” Though, therefore, it is true, that the simple form of the adjective does not always, nor even commonly denote comparison, yet as it always does indicate one of the degrees compared whenever comparison exists, it seems proper to rank it with the other forms, as a degree of comparison. This involves no impropriety, it produces no confusion, it leads to no error, it has a positive foundation in the nature of comparison, and it furnishes an appropriate and convenient appellation for this form of the adjective, by which to distinguish it in speech from the other forms.

8 Conclusions and final remarks

Expressing subjective assessments with a high accuracy is really impossible, therefore a small comparison scale is appropriate. For example, expressing our pain on the scale of 1 to 100, or even 1 to 10, seems more difficult – and arguably less meaningful – than on the scale of 1 to 3. In the past, the scale 1 to 9 was proposed in [34] and 1 to 5 in [26]. In this study, we have demonstrated that the use of the smaller 1 to 3 scale, rather than larger ones, has good mathematical foundations.

More research needs to be conducted along the measurement theory lines of [36], but with emphasis on PC. In our opinion, playing endlessly with numbers and symbols to find a precise solutions for inherently ill-defined problems should be replaced by more research towards utilization of the choice theory in pairwise comparisons. The presented strong mathematical evidence supports the use of a more restricted scale. We would like to encourage other researchers to conduct Monte Carlo simulations with the proposed scale and to compare the results with those yielded by other approaches. In particular, it would be useful to investigate more closely the relationship between the degree of inconsistency of a PC matrix, the size of the scale and the possible existence of multiple local or global optima for the Least Squares Method (cf. [4, 24, 25]).

The use of large scales (e.g., to in medicine for the pain level specification routinely asked in all Canadian hospitals upon admitting an emergency patient if he/she is still capable of talking) is a crown example of how important this problem may be for the improvement of daily life. Making inferences on the basis of meaningless numbers might have pushed other patients further in usually long emergency lineups.

Although the theoretical basis for suggesting the scale 1 to 3 hinges on the value of the constant , the importance of which was established in [20] in the context of pairwise comparisons, its applicability to the universal subjective scale is a vital possibility worth further scientific examination.


This research has been supported in part by OTKA grants K 60480, K 77420 in Hungary. Acting in the spirit of collective intelligence, we acknowledge that there are so many individuals involved in the development of our approach and in the publication process that naming them could bring us beyond the publisher’s page limit.


  • [1] Anholcer, M., Babiy, V., Bozóki, S., Koczkodaj, W.W., A simplified implementation of the least squares solution for pairwise comparisons matrices, Central European Journal of Operations, 19(4): 439-444, 2011.
  • [2] Basile L., D’Apuzzo L., Marcarelli G., Squillante M., Generalized Consistency and Representation of Preferences by Pairwise Comparisons in “Panamerican Conference of Applied Mathematics,” Huatulco, Mexico, 2006.
  • [3] Blankmeyer, E., Approaches to consistency adjustments, Journal of Optimization Theory and Applications, 54, 479–488, 1987.
  • [4] Bozóki, S., A method for solving LSM problems of small size in the AHP, Central European Journal of Operations Research, 11, 17–33, 2003
  • [5] Bozóki, S., Solution of the least squares method problem of pairwise comparisons matrices, Central European Journal of Operations Research, 16, 345–358, 2008.
  • [6] Bozóki, S, Rapcsák, T., On Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices, Journal of Global Optimization, 42(2): 157–175, 2007.
  • [7] Brunelli, M., Fedrizzi, M., Fair Consistency Evaluation in Fuzzy Preference Relations and in AHP, in “Knowledge-Based Intelligent Information and Engineering Systems,” LNCS 4693, 612–618, 2009.
  • [8] Bullions, P., The Principles of English Grammar, 16th edition, Pratt, Woodford, & Co., 1846.
  • [9] Cavallo, B., D’Apuzzo, L., A general unified framework for pairwise comparison matrices in multicriterial methods, International Journal of Intelligent Systems 24(4), pages 377–398, 2009.
  • [10] Chu, A.T.W., Kalaba, R.E., Spingarn, K., A comparison of two methods for determining the weight belonging to fuzzy sets, Journal of Optimization Theory and Applications. 4, 531–538, 1979.
  • [11] Choo, E.U., Wedley, W.C., A common framework for deriving preference values from pairwise comparison matrices. Computers and Operations Research. 31, 893–908, 2004.
  • [12] Condorcet. M., Essai sur l’Application de l’Analyse à la Probabilité des Décisions Rendues à la Pluralité des Voix, Paris, 1785.
  • [13] Crawford, G., Williams, C., A note on the analysis of subjective judgment matrices. Journal of Mathematical Psychology. 29, 387–405, 1985.
  • [14] D’Apuzzo., L., Marcarelli G., Squillante., M., Generalized consistency and intensity vectors for comparison matrices, International Journal of Intelligent Systems 22(12), 1287–1300, 2007.
  • [15] Debreu, G., Topological methods in cardinal utility theory, in “Mathematical Methods in the Social Sciences,” Arrow, K.J., Karlin, S. and Suppes, P. (eds.), Stanford University Press, 16–26, 1960.
  • [16] De Jong, P., A statistical approach to Saaty’s scaling method for priorities. Journal of Mathematical Psychology, 28, 467–478, 1984.
  • [17] Farkas, A., Lancaster, P., Rózsa, P., Consistency adjustment for pairwise comparison matrices. Numer. Linear Algebra Applications, 10, 689–700, 2003.
  • [18] Fedrizzi, Mario, Fedrizzi, Michele and Marques Pereira, R.A., On the issue of consistency in dynamical consensual aggregation, in “Technologies for Constructing Intelligent Systems,” Vol. 1, Bouchon Meunier B., Gutierrez Rios J., Magdalena L., Yager R. R. (eds), Heidelberg: Physica, Studies in Fuzziness and Soft Computing, 89, 129–137, Springer, 2002.
  • [19] Fedrizzi, M., Giove, S., Incomplete pairwise comparison and consistency optimization, European Journal of Operational Research 183(1), 303–313, 2007.
  • [20] Fülöp, J., A method for approximating pairwise comparison matrices by consistent matrices, Journal of Global Optimization, 42 (2008), 423–442.
  • [21] Golany, B., Kress, M., A multicriteria evaluation method for obtaining weights from ratio-scale matrices. European Journal of Operational Research, 69, 210–220, 1993.
  • [22] Holsztynski, W., Koczkodaj, W.W., Convergence of inconsistency algorithms for the pairwise comparisons, Information Processing Letters, 59(4), 197–202, 1996.
  • [23] Hubbard, D., How to measure anything, Wiley, 2007.
  • [24] Jensen, R.E., Comparison of eigenvector, least squares, chi squares and logarithmic least squares methods of scaling a reciprocal matrix, working paper 153, Trinity, University, 1983.
  • [25] Jensen, R.E., Alternative scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology. 28, 317–332, 1984.
  • [26] Koczkodaj, W.W., A new definition of consistency of pairwise comparisons. Mathematical and Computer Modelling, Vol. 18, 7, 79–84, 1993.
  • [27] Koczkodaj, W.W., Szarek, S.J., On distance-based inconsistency reduction algorithms for pairwise comparisons, Logic Journal of IGPL, (advance access published January 17, 2010), 2010.
  • [28] Luce, R.D., Edwards, W., The derivation of subjective scales from just noticeable differences, Psychological Review, 65(4), 222–237, 1958.
  • [29] Luce, R.D., Tukey, J.W., Simultaneous conjoint measurement: a new scale type of fundamental measurement, Journal of Mathematical Psychology, 1, 1–27, 1964.
  • [30] Llull, R., Artifitium electionis personarum (before 1283)
  • [31] Mikhailov, L., A fuzzy programming method for deriving priorities in the analytic hiarerchy process, Journal of the Operational Research Society. 51, 341–349, 2000.
  • [32] Nagel, E., Measurement, Erkenntnis, 2(1), 313–335, 1931.
  • [33] Nguyen, N.T., Advanced method in Inconsistency Knowledge Management, Springer, p.356, 2008.
  • [34] Saaty, T.L., A scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology, 15, 234–281, 1977.
  • [35] Saaty, T.L., The Analytic Hierarchy Process, McGraw-Hill, New York, 1980.
  • [36] Stevens, S.S. On the theory of scales of measurement. Science, 103, 677–680, 1946.
  • [37] Thurstone, L.L., A law of comparative judgement, Psychological Review 34, 278–286, 1927.


After a change of variables to , and a change in normalization to , the problem (4) can be rewritten as


Our goal is to provide a streamlined version of the argument from [20] for showing that if ’s are not “too large”, then this minimization problem has a unique solution.

The existence part is easy: if the norm of tends to , then – because of the constraint – we must have both and , hence for some , which forces the objective function to go to . This allows to reduce the problem to a compact subset of , where existence of a minimum follows from continuity of the objective function.

The uniqueness will follow if we show that the objective function in (6) – denote it by

– is globally convex, and strictly convex when restricted to the hyperplane given by the constraint.

For and , we set , then . Our next goal is to show that if and , then is convex. Since a composition of a linear function with a convex function (in that order) is convex, it follows that if , then each term is convex, and so is , the entire sum.

To that end, we calculate the second derivative of and obtain

Roughly, will be convex whenever the expression in the outer parentheses is negative (note that by hypothesis). Given that the expression is a quadratic function in , this will happen when is between the roots of this function, which are easily calculated to be and , where . The graphs of the functions and can be easily rendered (see Fig. 1).

Figure 1: The graphs of , and . The shaded region corresponds to “regions of convexity” of the functions .

In particular, it is apparent that there is a nontrivial range of values of , for which for all , which implies that the corresponding ’s are strictly convex on their entire domain . In view of symmetries of the problem, that range must be of the form , and it is clear from the picture that . For the extreme values and , the second derivative of will be strictly positive except at one point, which still implies strict convexity of .

It is not-too-difficult to obtain more precise results, both numerically and analytically. For the latter, we check directly (or deduce from symmetries of or ) that ; this confirms that , and so it is enough to determine . To apply the first derivative test to , we calculate

While this looks slightly intimidating, it is not hard to check that the only positive zero of is , which also shows rigorously that decreases on and increases on (both strictly). Consequently, , as asserted. All these calculations can be done by hand, or – much faster – using a computer algebra system such as Mathematica, Maple, or Maxima.

The above argument proves global convexity of , it remains to show strict convexity on the hyperplane , which is equivalent to strict convexity of the restriction to any line contained in . Given such line (with and ), consider any pair of coordinates such that and the corresponding term in the sum defining , namely . Clearly , and it can vanish for at most one value of (and only if or ). Thus is strictly convex, and since all the other terms appearing in are convex, it follows that the restriction of to the line, and hence to , are strictly convex. It is also clear that if , the above argument yields a non-trivial lower bound on the positive-definiteness of the Hessian of the restriction of to (this issue has been elaborated upon in [20]), which in particular has consequences for the speed of convergence of algorithms solving (6).