The incomplete Analytic Hierarchy Process and Bradley-Terry model inconsistency and information retrieval

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The incomplete Analytic Hierarchy Process and

Bradley-Terry model: (in)consistency and information

retrieval

László Gyarmati1,∗, Éva Orbán-Mihálykó1, Csaba Mihálykó1,

Sándor Bozóki2,3, Zsombor Szádoczki2,3

1Department of Mathematics, University of Pannonia, 8200 Veszprém, Hungary

2Research Group of Operations Research and Decision Systems,

Research Laboratory on Engineering & Management Intelligence

Institute for Computer Science and Control (SZTAKI), Eötvös Loránd Research Network

(ELKH), Budapest, Hungary

3Department of Operations Research and Actuarial Sciences

Corvinus University of Budapest, Hungary

Abstract

Several methods of preference modeling, ranking, voting and multi-criteria decision

making include pairwise comparisons. It is usually simpler to compare two objects at

a time, furthermore, some relations (e.g., the outcome of sports matches) are naturally

known for pairs. This paper investigates and compares pairwise comparison models

and the stochastic Bradley-Terry model. It is proved that they provide the same

priority vectors for consistent (complete or incomplete) comparisons. For incomplete

comparisons, all ﬁlling in levels are considered. Recent results identiﬁed the optimal

subsets and sequences of multiplicative/additive/reciprocal pairwise comparisons for

small sizes of items (up to n= 6). Simulations of this paper show that the same

subsets and sequences are optimal in case of the Bradley-Terry and the Thurstone

models as well. This, somehow surprising, coincidence suggests the existence of a

more general result. Further models of information and preference theory are subject

to future investigation in order to identify optimal subsets of input data.

Keywords: paired comparison, pairwise comparison, consistency, Bradley-Terry model,

information retrieval, graph of graphs;

arXiv:2210.03700v1 [math.OC] 7 Oct 2022

1 Introduction

Comparison in pairs is a frequently used method in ranking and rating objects when scaling

is diﬃcult due to its subjective nature. From a methodological point of view, two main

types of models can be distinguished: the ones based on pairwise comparison matrices

(PCMs) and the stochastic models motivated by Thurstone. The aim of this paper is to

present some linkages between these approaches. We establish a direct relation between

them via the concept of consistency. Evaluating consistent data, we mainly focus on the

similarities of the results in the case of incomplete comparisons. In (Bozóki and Szádoczki,

2022) and (Szádoczki et al., 2022), the authors investigate incomplete pairwise compar-

isons evaluated via two diﬀerent weight calculation techniques from an information retrieval

point of view. Incompleteness means that the comparisons between some pairs of objects

are missing. In case of consistent data, this absent information is preserved in the results

of the compared pairs. However, in real problems, the results of comparisons are usually

not consistent, therefore this missing information can cause signiﬁcant modiﬁcations in the

evaluations. The main question of the (Bozóki and Szádoczki, 2022) and (Szádoczki et al.,

2022) is which structure of the comparisons is optimal for ﬁxed numbers of comparisons

investigating pairwise comparison matrices via the logarithmic least squares and the eigen-

vector methods (LLSM and EM). We ask the same question in case of the Bradley-Terry

(BT) model, when the evaluation is performed via maximum likelihood estimation (MLE).

What are the similarities and the diﬀerences of the optimal comparisons’ arrangements in

case of LLSM and BT? Do the ﬁndings of the papers remain valid in the case of a sub-

stantially diﬀerent paired comparison method? Are the conclusions method-speciﬁc, i.e.,

do the results of the paper (Bozóki and Szádoczki, 2022) apply only for the method LLSM

and EM or in general as well, for other models based on paired comparisons?

The rest of the paper is organized as follows. Section 2 presents the closely related

literature and the research gap that we would like to consider in the current study. Sec-

tion 3 describes the preliminary methods, namely the AHP with LLSM, EM and the

Bradley-Terry model with MLE. In Section 4 we describe the connection between these

models. We pay special attention to consistency, which is deeply investigated in PCM-

based methods, but not in stochastic models. Subsection 4.1 contains theoretical results

about the connection of the models in consistent cases, while Subsection 4.2 presents ex-

amples demonstrating the diﬀerences if the data are inconsistent. Section 5 details the

simulation methodology that is used to ﬁnd optimal solutions concerning information re-

trieval, while Section 6 contains the main results of the numerical experiments. Finally,

Section 7 concludes and discusses further research questions.

2 Literature review

Applying the method of paired comparisons is essential in psychology (Thurstone, 1927),

sports (Csató, 2021; Orbán-Mihálykó et al., 2022), preference modelling (Choo and Wed-

ley, 2004; Mantik et al., 2022), ranking (Fürnkranz and Hüllermeier, 2011; Shah and Wain-

wright, 2018), and decision making methods (Stewart, 1992).

Through the concept of consistency/inconsistency, we compare some recent results

(Bozóki and Szádoczki, 2022) gained on the domain of PCMs used by the popular multi-

attribute decision making method Analytic Hierarchy Process (AHP) (Saaty, 1977) to the

outcomes provided by the also widely used Bradley-Terry model (Bradley and Terry, 1952).

The latter one is a special case of the more general Thurstone motivated stochastic

models. Both these stochastic methods and the generalization of the AHP can be used on

incomplete data, when some of the paired comparisons are missing (Harker, 1987; Ishizaka

and Labib, 2011), which is often demonstrated on sport examples (Orbán-Mihálykó et al.,

2019; Bozóki et al., 2016). However, in this regard, several theoretical questions have been

investigated in the most recent literature (Chen et al., 2022).

AHP-based and stochastic Thurstone-motivated models are signiﬁcantly diﬀerent in

their fundamental concepts. In case of two diﬀerent principles of a problem’s solution,

the linkage between them is always motivational: what are the common features and the

diﬀerences of the methods. As far as the authors know, only few publications are devoted

to this question. Researchers usually deal with one of the methods. However, in (MacKay

et al., 1996), the authors recognize the following: ‘the two branches resemble each other

in that both may be used to estimate unidimensional scale values for decision alternatives

or stimuli from pairwise preference judgments about pairs of stimuli. The models diﬀer in

other respects.’ Nevertheless, in (Genest and M’lan, 1999) the authors compare the AHP

based methods and the Bradley-Terry model in case of complete comparisons and they

prove that in special cases some diﬀerent types of techniques provide equal solutions. In

(Orbán-Mihálykó et al., 2015), the authors compared numerically the AHP and Thurstone

methods: evaluating a real data set on a 5-value scale, the numerical results provided by

the diﬀerent methods were very close to each other. Further numerical comparisons for

incomplete data can be found in (Orbán-Mihálykó et al., 2019).

Consistency/inconsistency of PCMs is a focal issue in the case of pairwise comparisons

(Brunelli, 2018; Duleba and Moslem, 2019), but it is not investigated in stochastic models.

The results’ compatibility with real experiences are related to inconsistency of the PCM:

discrepancy may appear even in the case of complete comparisons. The question necessarily

raises: what does the consistency mean in BT model?

Nowadays, more and more attention is paid to incomplete comparisons, as it is a part

of information recovery. In case of missing comparisons there are two further aspects that

have crucial eﬀect on the results in every model, namely, the number of known entries,

and their arrangement. Our approach is strongly relying on the graph representation

of incomplete paired comparisons (Gass, 1998). We are determining the best representing

graphs (the best pattern of known comparisons) in the Bradley-Terry model for all possible

number of comparisons (edges) for given number of alternatives (vertices). The importance

of the pattern of known comparisons in PCMs has been investigated for some special

cases by (Szádoczki et al., 2022), who emphasized the eﬀect of (quasi-)regularity and the

minimal diameter (longest shortest path) property of the representing graphs. (Szádoczki

et al., 2022) also examined some additional ordinal information in the examples studied

by them. Finally, (Bozóki and Szádoczki, 2022) have investigated all the possible ﬁlling

in patterns of incomplete PCMs, and determined the best ones for all possible (n, e)pairs

up until 6 alternatives with the help of simulations, where nis the number of items to

be compared, while eis the number of compared pairs. Their main ﬁndings (besides

the concrete graphs) are (i) the star-graph is always optimal among spanning trees; (ii)

regularity and bipartiteness are important properties of optimal ﬁlling patterns.

To the best of the authors’ knowledge, there has been no similar study in case of the

family of stochastic models, thus in this paper we would like to ﬁll in this research gap,

too.

3 Preliminaries of the applied methods

3.1 Analytic Hierarchy Process

The AHP methodology is based on PCMs, which can be used to evaluate alternatives

according to a criterion or to compare the importance of the diﬀerent criteria.

Deﬁnition 1 (Pairwise comparison matrix (PCM)) Let us denote the number of items

to be compared (usually criteria or alternatives) in a decision problem by n. The n×n

matrix A= [aij ]is called a pairwise comparison matrix, if it is positive (aij >0for ∀i

and jand reciprocal (1/aij =aji for ∀iand j).

aij , the general element of a PCM, shows how many times item iis better/more im-

portant than item j. In an ideal case these elements are not contradicting to each other,

thus we are dealing with a consistent PCM.

Deﬁnition 2 (Consistent PCM) A PCM is called consistent if aik =aij ajk for ∀i, j, k.

If a PCM is not consistent, then it is said to be inconsistent.

In practical problems, the PCMs ﬁlled in by decision makers are usually not consistent,

and because of that, there is a large literature of how to measure the inconsistency of

these matrices (Brunelli, 2018). Recently even a general framework has been proposed for

deﬁning inconsistency indices of reciprocal pairwise comparisons (Bortot et al., 2022).

In case of consistent PCMs all the diﬀerent weight calculation techniques result in the

same weight (prioritization/preference) vector that determines the ranking of the compared

items. However, for inconsistent data, the results of diﬀerent weight calculation methods

can vary. Two of the most commonly used techniques are the logarithmic least squares

method (Crawford and Williams, 1985) and the eigenvector method (Saaty, 1977).

Deﬁnition 3 (Logarithmic Least Squares Method (LLSM)) Let Abe an n×nPCM.

The weight vector wof Adetermined by the LLSM is given as follows:

min

i=1

j=1 ln(aij )−ln wi

wj2

,(1)

where wiis the i-th element of w,0< wiand Pn

i=1 wi= 1.

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TheincompleteAnalyticHierarchyProcessandBradley-Terrymodel:(in)consistencyandinformationretrievalLászlóGyarmati1;,ÉvaOrbán-Mihálykó1,CsabaMihálykó1,SándorBozóki2;3,ZsomborSzádoczki2;31DepartmentofMathematics,UniversityofPannonia,8200Veszprém,Hungary2ResearchGroupofOperationsResearchandDecisionSyste...

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