Adapting a kidney exchange algorithm to align with human values.

Adapting a kidney exchange algorithm to align with human values[☆](#aep-article-footnote-id1)
=============================================================================================

Abstract
--------

The efficient and fair allocation of limited resources is a classical problem in economics and computer science. In kidney exchanges, a central market maker allocates living kidney donors to patients in need of an organ. Patients and donors in kidney exchanges are prioritized using ad-hoc weights decided on by committee and then fed into an allocation algorithm that determines who gets what—and who does not. In this paper, we provide an end-to-end methodology for estimating weights of individual participant profiles in a kidney exchange. We first elicit from human subjects a list of patient attributes they consider acceptable for the purpose of prioritizing patients (e.g., medical characteristics, lifestyle choices, and so on). Then, we ask subjects comparison queries between patient profiles and estimate weights in a principled way from their responses. We show how to use these weights in kidney exchange market clearing algorithms. We then evaluate the impact of the weights in simulations and find that the precise numerical values of the weights we computed matter little, other than the ordering of profiles that they imply. However, compared to not prioritizing patients at all, there is a significant effect, with certain classes of patients being (de)prioritized based on the human-elicited value judgments.

Introduction
------------

As AI is deployed increasingly broadly, AI researchers are confronted with the moral implications of their work. The pursuit of simple objectives, such as minimizing error rates, maximizing resource efficiency, or decreasing response times, often results in systems that have unintended consequences when they confront the real world, such as discriminating against certain groups of people [34]. It would be helpful for AI researchers and practitioners to have a general set of principles with which to approach these problems [45], [41], [24], [16], [33].

One may ask why any moral decisions should be left to computers at all. There are multiple possible reasons. One is that the decision needs to be made so quickly that calling in a human for the decision is not feasible, as would be the case for a self-driving car having to make a split-second decision about whom to hit [13]. Another reason could be that each individual decision by itself is too insignificant to bother a human, even though all the decisions combined may be highly significant morally—for example, if we were to consider the moral impact of each advertisement shown online. A third reason is that the moral decision is hard to decouple from a computational problem that apparently exceeds human capabilities. This is the case in many machine learning applications (e.g., should this person be released on bail? [27]), but also in other optimization problems.

We are interested in one such problem: the clearing house problem in *kidney exchanges*. In a kidney exchange, patients who need a kidney transplant and have a willing but incompatible live donor may attempt to trade their donors' kidneys [40]. Once these people appear at an exchange, we face a highly complex problem of deciding who matches with whom. In some exchanges, this matching problem is solved using algorithms developed in the AI community: the United States [19], the United Kingdom [30], the Netherlands [23], and so on [9].

In this paper, we investigate the following issue. Suppose, in principle, that we prioritize certain patients over others—for example, younger patients over older patients. To do so would clearly be a morally laden decision. How should this affect the role of the AI researcher developing these systems? From a purely algorithmic perspective, it may seem that there is little more to this than to change some weights in the objective function accordingly. But we argue that our job, as AI researchers, does not end with this simple observation. Rather, we should be closely involved with the process for determining these weights, both because we can contribute technical insights that are useful for this process itself, and because it is our responsibility to understand the consequences to which these weights will lead. The methodology that we develop integrates this prioritization into our development work.

In this paper, we provide an end-to-end methodology for estimating weights of individual patient profiles in a kidney exchange, where these weights are used only for tiebreaking purposes (i.e., when multiple solutions give the maximal number of transplants).

Executing our methodology in such a way that we would advocate directly adopting the results in practice would require substantially more effort and participation from other parties. For example, we would need to consult domain experts to determine which patient characteristics should be used to determine edge weights. We would also need to involve stakeholders such as policy-makers, doctors, and kidney exchange participants in the process for determining weights. For this reason, we execute this methodology in a limited fashion as a proof-of-concept, and evaluate the results in simulations.

We first elicit from human subjects a list of patient attributes they consider acceptable for the purpose of prioritizing patients in kidney exchanges (e.g., most subjects did not find race an acceptable attribute for prioritization). Then, we ask subjects comparison queries between patient profiles that differ only on acceptable attributes, and estimate weights from their responses. We show how to use these weights in kidney exchange market clearing algorithms, to break ties among multiple maximum-sized solutions. We then evaluate the impact of the weights in simulations. We find that the precise numerical values of the weights we computed matter little, other than the ordering of profiles that they imply. However, compared to not prioritizing patients at all, there is a significant effect. Specifically, the difference is experienced by donor-patient pairs that have an “underdemanded” [6], [42] combination of blood types; for them, their chances rise or drop significantly depending on their tiebreaking weights.

Section snippets
----------------

Kidney exchange model
---------------------

We briefly review the standard mathematical model for kidney exchange and techniques from the AI community used to clear real kidney exchanges, and then give illustrative examples where tiebreaking would or would not play a role.

Determining and using prioritization weights
--------------------------------------------

In this section, we describe our procedure for computing prioritization weights and integrating them into the algorithm for clearing kidney exchanges. Because this procedure was intended as a proof-of-concept, we gathered preference data from participants recruited through the online platform Amazon Mechanical Turk (“MTurk”).3 However, if this procedure were used in a real-life kidney exchange, medical experts and other

Experiments
-----------

Having described how we obtained weights and how we integrated these weights into the IP-based algorithm, we now describe our experiments testing the effects of our prioritizing algorithm in simulations.

Discussion
----------

In this section, we discuss the potential for applying these results to real-world kidney exchanges, some of the ethical context of our work, and directions for future research.

Declaration of Competing Interest
---------------------------------

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements
----------------

This work is partially supported by the project “How to Build Ethics into Robust Artificial Intelligence” funded by the Future of Life Institute (grants 2016-158697 and 2017-174867), by the Templeton World Charity Foundation grant TWCF0321, by NSF IIS-1527434, and by Duke Bass Connections. Dickerson was supported in part by NSF CAREER Award IIS-1846237 and a Google Faculty Research Award. Conitzer was supported in part by NSF IIS-1814056. We thank Lirong Xia, Zhibing Zhao, and Kyle Burris, and

References (46)
---------------

* P. Biró *et al.*### [Inapproximability of the kidney exchange problem](/science/article/pii/S0020019006002869)

### Inf. Process. Lett.

(2007)
* R.A. Bradley### [14 paired comparisons: some basic procedures and examples](/science/article/pii/S0169716184040165)
* R. Montgomery *et al.*### [Domino paired kidney donation: a strategy to make best use of live non-directed donation](/science/article/pii/S0140673606691150)

### Lancet

(2006)
* A. Roth *et al.*### [Pairwise kidney exchange](/science/article/pii/S0022053105001055)

### J. Econ. Theory

(2005)
* P. Toulis *et al.*### [Design and analysis of multi-hospital kidney exchange mechanisms using random graphs](/science/article/pii/S0899825615000020)

### Games Econ. Behav.

(2015)
* D. Abraham *et al.*### Clearing algorithms for barter exchange markets: enabling nationwide kidney exchanges
* C. Allen *et al.*### Artificial morality: top-down, bottom-up, and hybrid approaches

### Ethics Inf. Technol.

(2005)
* M. Anderson *et al.*### Machine Ethics

(2011)
* R. Anderson *et al.*### Finding long chains in kidney exchange using the traveling salesman problem

### Proc. Natl. Acad. Sci.

(2015)
* I. Ashlagi, D. Gamarnik, M. Rees, A.E. Roth, The need for (long) chains in kidney exchange, 2017, initial version...

- I. Ashlagi *et al.*### Free riding and participation in large scale, multi-hospital kidney exchange

### Theor. Econ.

(2014)
- C. Barnhart *et al.*### Branch-and-price: column generation for solving huge integer programs

### Oper. Res.

(1998)
- K. Baum *et al.*### From machine ethics to machine explainability and back
- P. Biró *et al.*### Kidney exchange practices in Europe
- P. Biró *et al.*### Maximum weight cycle packing in directed graphs, with application to kidney exchange programs

### Discrete Math. Algorithms Appl.

(2009)
- A. Blum *et al.*### Opting into optimal matchings
- J.-F. Bonnefon *et al.*### The social dilemma of autonomous vehicles

### Science

(2016)
- S. Bringsjord *et al.*### Contextual deontic cognitive event calculi for ethically correct robots
- V. Conitzer *et al.*### Moral decision making frameworks for artificial intelligence
- J.P. Dickerson *et al.*### Position-indexed formulations for kidney exchange
- J.P. Dickerson *et al.*### Failure-aware kidney exchange

### Manag. Sci.

(2019)
- J.P. Dickerson *et al.*### FutureMatch: combining human value judgments and machine learning to match in dynamic environments
- E. Elkind *et al.*### Rationalizations of voting rules
View more referencesCited by (43)
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[View full text](/science/article/pii/S0004370220300229)

Rachel Freedman
Jana Schaich Borg
Walter Sinnott-Armstrong
John P
Dickerson
Vincent Conitzer


Inconsistency evaluation in pairwise comparison using norm-based distances.

Decisions in Economics and Finance (2020) 43:657–672
https://doi.org/10.1007/s10203-020-00304-9
Inconsistency evaluation in pairwise comparison using
norm-based distances
Michele Fedrizzi1·Nino Civolani2·Andrew Critch3
Received: 14 December 2019 / Accepted: 11 August 2020 / Published online: 30 August 2020
© The Author(s) 2020
Abstract
This paper studies the properties of an inconsistency index of a pairwise comparison
matrix under the assumption that the index is deﬁned as a norm-induced distance fromthe nearest consistent matrix. Under additive representation of preferences, it is proved
that an inconsistency index deﬁned in this way is a seminorm in the linear space of
skew-symmetric matrices and several relevant properties hold. In particular, this linearspace can be partitioned into equivalence classes, where each class is an afﬁne subspace
and all the matrices in the same class share a common value of the inconsistency
index. The paper extends in a more general framework some results due, respectively,to Crawford and to Barzilai. It is also proved that norm-based inconsistency indices
satisfy a set of six characterizing properties previously introduced, as well as an upper
bound property for group preference aggregation.
Keywords Inconsistency index ·Pairwise comparison matrix ·Norm ·Distance
JEL Classiﬁcation C44·D7
1 Introduction
Pairwise comparison over a set of alternatives X={x1,..., xn}is a well known
and powerful method for preference elicitation in a decision problem. An important
characteristic of this method is the capability of dealing with the imprecision of the
collected data due to the unavoidable inconsistency of human judgements. Each entry
aijof a pairwise comparison matrix, PCM in the following, A=(aij)n×nquantiﬁes
B Michele Fedrizzi
michele.fedrizzi@unitn.it
1Department of Industrial Engineering, University of Trento, Via Sommarive 77, 38123 Trento,
Italy
2University of Trento, Trento, Italy
3Center for Human-Compatible AI, University of California, Berkeley, California, USA
123
658 M. Fedrizzi et al.
the degree of preference of alternative xiover alternative xj. Two widely used rep-
resentations of preferences are the so-called multiplicative and additive ones. In the
multiplicative approach (Saaty 1977 ),aijis the relative preference of alternative xi
over alternative xj, and therefore, it estimates the ratio between the weight wiofxiand
the weight wjofxj,aij≈wi
wjConversely, in the additive approach, aijestimates the
difference between the weights of xiandxj, respectively, aij≈wi−wj. Therefore,
different assumptions in preference quantiﬁcation correspond to different meaning
of the entries aij. It has been proved that the multiplicative and the additive repre-
sentations are isomorphic and, therefore, equivalent. In fact, a multiplicative PCM
(aij)n×ncan be easily transformed into a corresponding additive PCM by compo-
nentwise applying the logarithmic function, thus obtaining (ln(aij))n×n. Details on
this isomorphism can be found in (Barzilai 1998 ; Cavallo and D’Apuzzo 2009 ). In the
multiplicative approach, a pairwise comparison matrix PCM is a positive real-valued
matrix A=(aij)n×nwith aii=1∀iandaijaji=1∀i,j. Multiplicative reciprocity
aijaji=1 derives fromwi
wjwj
wi=1, and this property is always required. A PCM is
said consistent if and only if
aijajk=aik,i,j,k=1,..., n. (1)
Consistency condition ( 1) corresponds to the ideal situation where the decision maker
is perfectly coherent in her/his judgements and it is in general not required, since it is
well known that in making paired comparisons people do not have the intrinsic logical
ability to always be consistent (Saaty 1994 ). Despite a violation of ( 1) to some extent
is necessarily accepted, consistency has always been regarded as a desirable property,
since coherent judgements are clearly considered more preferable than contradictory
ones. Therefore, a correct inconsistency evaluation is regarded as a crucial task andseveral indices have been proposed in order to quantify the deviation from the condition
of full consistency ( 1).
The problem of inconsistency evaluation has been addressed by means of many
different proposals. We brieﬂy cite some of them. Saaty ( 1977 ) proposed, in his sem-
inal paper, the ﬁrst and still most popular inconsistency index. Koczkodaj ( 1993 )
and Duszak and Koczkodaj ( 1994 ) introduced an inconsistency index which became
very popular too. Bozóki and Rapcsák ( 2008 ) summarized the relationship between
some weighting methods and the corresponding inconsistency indices. Then, the
authors compared Saaty’s and Koczkodaj’s inconsistency indices. Recently, Brunelliand Fedrizzi ( 2019 ) proposed a general formulation for inconsistency indices that
includes many among the known indices. Cavallo ( 2020 ) studied the functional rela-
tions between some important consistency indices, mainly focusing on the case n=3.
Nevertheless, the detailed description of all the relevant contributions on the consis-
tency evaluation is beyond the scope of this paper. For a comprehensive survey, see(Brunelli 2018 ). Most of these studies assume the multiplicative representation of the
preferences, i.e., they deal with multiplicative PCM s. Some papers take into account
both the multiplicative and the additive representations (Fichtner 1984 ;C h u 1998 ).
Brunelli et al. ( 2013 ) compared ten known indices numerically, while Brunelli ( 2017 ),
Brunelli and Fedrizzi ( 2015a ) and Csató ( 2018 ,2019 ) proposed a more theoretical
approach by studying some axioms for inconsistency indices. In Cavallo and D’Apuzzo
123
Inconsistency evaluation using norm-based distances 659
(2009 ) and in some following papers by the same authors, a general framework for
inconsistency evaluation is proposed, based on the algebraic structure of group. Ramík
and Korviny ( 2010 ) proposed a distance-based evaluation of the inconsistency in the
framework of fuzzy sets. Other interesting distance-based approaches to the study of
inconsistency have been proposed more recently by Brunelli and Cavallo ( 2020 ) and
Mazurek and Ramík ( 2019 ).
This paper proposes a geometric-oriented unifying point of view for inconsistency
evaluation based on norm-induced metrics in matrix spaces. Our proposal can be con-
sidered as a generalization of the approach by Crawford and Williams ( 1985 ), where
the logarithmic least square method (LLSM) is applied. In fact, their method corre-
sponds to the Euclidean norm minimization after passing to the additive representation
of preferences. This justiﬁes, in our view, the numerous good properties of the LLSMand the related geometric mean solution.
The rest of the document is organized as follows. After some preliminaries in Sect. 2,
we assume the additive representation of preferences, and in Sect. 3, we introduce an
inconsistency index for a PCM deﬁned as a norm-induced distance from the nearest
consistent PCM . More precisely, in Sect. 3.1we partition the subspace of PCM s (skew-
symmetric matrices) into equivalence classes, so that each class will correspond to a
single inconsistency value. In Sect. 3.2, we deﬁne our index and we prove that it is
a seminorm in the vector space of PCM s. We prove, respectively, in Sect. 3.3 and
in Sect. 3.4, that our index satisﬁes a set of six axioms previously introduced and an
upper bound property for group preference aggregation. Finally, in Sect. 4, we discuss
some conclusions and future work.
2 Preliminaries
For a ﬁxed n>2, let Rn×nbe the vector space of n-order real matrices. Let A⊂Rn×n
be the set of multiplicative PCM s of order n,
A={ A=(aij)n×n|aij>0,aijaji=1∀i,j}.
Similarly, the set of consistent multiplicative PCM sA∗⊂Ais deﬁned as
A∗={ A=(aij)n×n|A∈A,aik=aijajk∀i,j,k}.
An inconsistency index is a function which associates a real number to each PCM
A∈A,
I:A→ R
The number I(A)quantiﬁes the inconsistency of A.
APCM A=(aij)n×ncan be viewed as a point in the vector space Rn×n. Closeness
to consistency condition ( 1) can therefore be interpreted as closeness of Ato a consis-
tent matrix ACin the same space. It is interesting to observe that choosing the ‘closest’
consistent matrix ACexactly corresponds to compute the weight vector wfrom A.I n
123
660 M. Fedrizzi et al.
fact, ACis consistent if and only if there exists a weight vector w=(w1,...,w n)such
that AC=(wi
wj). Every method for deriving a weight vector w=(w1,...,w n)from
an inconsistent PCM Aidentiﬁes a consistent matrix AC=(wi
wj)which is considered
the ‘closest’ one to Aaccording to a certain metric in Rn×n. Then, deriving a weight
vector and measuring inconsistency can be integrated into one process by minimizing
an appropriate objective function such that its minimum point is the weight vector,
while the optimal value of the objective function gives the inconsistency of the pair-wise comparison matrix. These observations suggest to characterize an inconsistency
index by means of the notion of distance. Nevertheless, it is convenient to ﬁrst shift to
the additive representation of preferences. This can be done by a simple component-
wise logarithmic transformation and allows working with the powerful tools of linear
algebra. Note that the multiplicative and the additive representation of preferencesare isomorphic, as described in detail by Barzilai ( 1998 ) and Cavallo and D’Apuzzo
(2009 ).
By componentwise applying a logarithmic function to a PCM A=(a
ij),
ln(aij)=rij, (2)
a skew-symmetric matrix R=(rij)is obtained, since multiplicative reciprocity
aijaji=1 is transformed into additive reciprocity
rij+rji=0,i,j=1,..., n. (3)
Being ( 3) a homogenous linear system, the image set of Athrough the logarithmic
function is the linear subspace of Rn×nof the skew-symmetric matrices, say L=
{ln(A);A∈A}= ln(A). The dimension of Lisn(n−1)/2, i.e., the number of the
upper-diagonal entries.
Aconsistent matrix A=(aij)∈A∗is transformed into a skew-symmetric matrix
R=ln(A)=(ln(aij))=(rij)satisfying
rij+rjk=rik,i,j,k=1,..., n. (4)
Property ( 4) clearly follows from ( 1) and proves that the image set of A∗through
the logarithmic function is a linear subspace of L, and therefore a linear subspace of
Rn×n,s a y L∗={ln(A);A∈A∗}= ln(A∗)(Kocakodaj 1997 ). The dimension of L∗
isn−1 since it is known that n−1 entries are necessary and sufﬁcient to completely
identify a consistent n-order matrix, provided that they are adequately chosen (Chu
1998 ; Fedrizzi et al. 2019 ).
Since Ais isomorphic to Land linear spaces are simple structures where it is possi-
ble working with tools of linear algebra, then, from now on, we will study preference
inconsistency in the subspace L. Therefore, we will use the usual notation Aalso to
denote a skew-symmetric matrix in L. An inconsistency index will be a function
I:L→ R, (5)
and we will call ‘additive PCM’ a matrix in L.
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Inconsistency evaluation using norm-based distances 661
3 Distances and norms
As remarked above, an inconsistency index measures to which extent a preference
matrix is far from consistency. Since distance is the most suitable mathematical tool
to evaluate how much two objects are different, it is quite natural to deﬁne an incon-
sistency index I(A)of a matrix A∈Las the distance of Afrom the linear subspace
L∗of consistent matrices,
Id(A)=d(A,L∗). (6)
Nevertheless, the notion of distance is too general to conveniently characterize incon-
sistency, and deﬁnition ( 6) can lead to unsatisfactory inconsistency indices (Fichtner
1986 ).
In the following, we will prove that the problem can be overcome by restricting
metrics to those deﬁned by norms. The resulting inconsistency indices satisfy a relevantset of good properties. Before proceeding, let us recall the basic notions of distance,
norm and seminorm.
Deﬁnition 1 (Distance ) A distance, or metric, on a given set Zis a function d:
Z×Z→ Rthat satisﬁes the following three conditions,
1.d(x,y)≥0∀x,y∈Zand d(x,y)=0⇔ x=y.
2.d(x,y)=d(y,x)∀x,y∈Z (symmetry).
3.d(x,z)≤d(x,y)+d(y,z)∀x,y,
z∈Z (triangle inequality).
Deﬁnition 2 (Norm ) A norm on a vector space Vis a function ||·||:V→ Rthat
satisﬁes the following conditions,
1.||x||≥0∀x∈V (nonnegativity).
2.||λx||=|λ|||x||∀x∈V,∀λ∈R (homogeneity).
3.||x+y||≤||x||+||y||∀x,y∈V (triangle inequality).
4.||x||=0⇔x=0 (strict positivity)
Aseminorm onVis a function s:V→ Rthat satisﬁes the ﬁrst three conditions
above. In other words, a seminorm has the same properties as a norm, except that it
may have a zero value for some nonzero vectors.
3.1 A partition of Linto equivalence classes
The vector space structure of Land L∗naturally induces a partition of Linto equiv-
alence classes. Since L∗is a subspace of L, it is possible to consider the quotient
space L/L∗. More formally, the quotient space L/L∗is deﬁned as follows, using the
corresponding equivalence relation.
Deﬁnition 3 (Quotient space ) Consider the equivalence relation in L
A∼B⇔B−A∈L∗A,B∈L. (7)
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662 M. Fedrizzi et al.
Fig. 1 Example of an equivalence class [A]forn=3
The quotient set L/L∗is deﬁned as
L/L∗=L/∼ (8)
and it is a vector space with the operations naturally induced by the operations in L.
The elements of L/L∗are the equivalence classes [A]obtained by adding to A∈L
an arbitrary matrix in L∗,
[A]={ A+C,C∈L∗}. (9)
Every equivalence class [A]is an afﬁne subspace of Land can also be represented as
[A]= A+L∗. Figure 1illustrates an example of [A]in the case n=3.
We assume that every inconsistency index ( 5) assigns the same value, say 0 for
simplicity, to every consistent matrix, see axiom 1 in (Brunelli and Fedrizzi 2015a )
and Sect. 3.3.1 . From what precedes, it is natural to assume that an inconsistency index
assigns the same value to every matrix in a ﬁxed equivalence class,
A,B∈[A]⇒ I(B)=I(A). (10)
Now, in order to assign an inconsistency value I(·)to each equivalence class coherently
with the assumption ( 6), the following must hold. For all [A]∈ L/L∗, every matrix
A∈[A]must have the same distance from L∗. Corollary 1will prove that this result
can be achieved assuming, as in the following, that distance ( 6) is induced by a norm.
123
Inconsistency evaluation using norm-based distances 663
3.2 Inconsistency index as a seminorm
Let||A||be a norm in Rn×nand let
d(A,B)=||A−B|| (11)
be the corresponding distance between matrices Aand B.
It is now possible to give our main deﬁnition.
Deﬁnition 4 Given a skew-symmetric matrix A∈Land a norm ||·||inRn×n,t h e
inconsistency index of Abased on distance ( 11)i s
Id(A)=d(A,L∗)=min
B∈L∗d(A,B)=min
B∈L∗||A−B||=/vextendsingle/vextendsingle/vextendsingle/vextendsingleA−A∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (12)
where A∗∈L∗is a solution of the minimization problem in ( 12).
Note that, by choosing the 1-norm ||·||1, the optimization problem ( 12) is equivalent
to the optimization problem (24) in (Brunelli and Cavallo 2020 ), and the same holds
for the inconsistency index proposed there, except for a numerical constant. Most of
the results in this paper are based on the following theorem.
Theorem 1 An inconsistency index I d(A)deﬁned as a norm-based distance (12)is a
seminorm in L.
Proof It must be proved that
1.Id(kA)=|k|Id(A)∀A∈L,∀k∈R
2.Id(A+A/prime)≤Id(A)+Id(A/prime)∀A,A/prime∈L.
Let A∗∈L∗be a d-nearest consistent matrix to A∈L,a si n( 12). Then, it is possible
to prove that kA∗is ad-nearest consistent matrix to kA. Equality 1. is obviously true
fork=0. Then, let us assume k/negationslash=0. It is
/vextendsingle/vextendsingle/vextendsingle/vextendsinglekA−kA∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle=|k|/vextendsingle/vextendsingle/vextendsingle/vextendsingleA−A∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤|k|||A−B||∀B∈L∗.
Since
|k|||A−B||=||kA−kB||,
then it is
/vextendsingle/vextendsingle/vextendsingle/vextendsinglekA−kA∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤||kA−kB||∀B∈L∗. (13)
It follows, from ( 13), that
/vextendsingle/vextendsingle/vextendsingle/vextendsinglekA−kA∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle=min
kB∈L∗||kA−kB||=Id(kA). (14)
123
664 M. Fedrizzi et al.
The last equality follows from the fact that, for a ﬁxed k/negationslash=0, every matrix of L∗can
be expressed as kBwith B∈L∗. Then,
Id(kA)=d(kA,L∗)=|k|d(A,L∗)=|k|Id(A)∀k∈R, (15)
so that the ﬁrst property is proved. The second property can be proved as follows,
Id(A+A/prime)=min
B∈L∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle(A+A/prime)−B/vextendsingle/vextendsingle/vextendsingle/vextendsingle=
= min
C,C/prime∈L∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle(A+A/prime)−(C+C/prime)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
= min
C,C/prime∈L∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle(A−C)+(A/prime−C/prime)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
≤min
C∈L∗||(A−C)||+min
C/prime∈L∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle(A/prime−C/prime)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
=Id(A)+Id(A/prime) (16)
where B=C+C/primeand the arbitrariness of Cand C/primeinL∗follows from the one of B.
/intersectionsq/unionsq
Clearly, an inconsistency index Id(A)is not a norm in L, since Id(A)=0∀A∈L∗.
Theorem 2 IfA∈L,
1.Every inconsistency index I d(A)given by (12)is invariant with respect to addition
of a consistent skew-symmetric matrix,
Id(A)=Id(A+B)∀B∈L∗. (17)
2.IfA∗∈L∗is a d-nearest consistent matrix to Aand B∈L∗, then A∗+Bis a
d-nearest consistent matrix to A+B
Proof For all A∈Land B∈L∗,i ti s Id(B)=0 and therefore
Id(A+B)≤Id(A)+Id(B)=Id(A), (18)
but also
Id(A)=Id((A+B)−B)≤Id(A+B)+Id(−B)=Id(A+B), (19)
since B∈L∗⇒− B∈L∗. Equality ( 17) follows from ( 18) and ( 19).
To prove the second statement, let us ﬁx B∈ L∗and evaluate
||(A+B)−(A∗+B)||,
/vextendsingle/vextendsingle/vextendsingle/vextendsingle(A+B)−(A∗+B)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingleA−A∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle (20)
≤||A−C||∀C∈L∗. (21)
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Inconsistency evaluation using norm-based distances 665
Fig. 2 Representation of Theorem 2in the case n=3
Since
||A−C||=||(A+B)−(C+B)||,
then it is
/vextendsingle/vextendsingle/vextendsingle/vextendsingle(A+B)−(A∗+B)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤||(A+B)−(C+B)||∀C∈L∗
/intersectionsq/unionsq
Figure 2illustrates Theorem 2in the case n=3.
Crawford ( 1987 ) introduced the so-called Geometric Consistency Index, GCI , and
proved a theorem (Theorem 2 in the cited paper) which states, for his index, the same
result as in Theorem 2of this paper. He used the multiplicative representation of
preferences, so that he proved the invariance with respect to the Hadamard productA·B=(a
ijbij)n×ninstead of invariance with respect to the sum as in ( 17). Theorem 2
can therefore be viewed as an extension of Crawford’s theorem to the general case
of norm-based consistency indices in the additive representation of preferences. Thesemantic of Theorem 2is as follows. The inconsistency degree I
d(A)of a PCM A
remains unchanged by adding consistent preferences.
From Theorem 2, the following corollary can be stated.
Corollary 1 If matrices A,B∈Lare in the same equivalence class, then they have
the same inconsistency index (12),
A∼B⇒Id(A)=Id(B), (22)
where ∼is deﬁned in (7).
The proof of ( 22) directly follows from ( 17) and ( 9).
Since L∗is the subspace of Lwhere the seminorm Id(·)is null, then this seminorm
induces a norm in L/L∗. The result is described in the following corollary.
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666 M. Fedrizzi et al.
Corollary 2 The function I d:L/L∗→ R, deﬁned as follows
Id([A])=Id(A), A∈[A], (23)
is a norm on L/L∗.
Note that, for sake of simplicity, we use the same notation Id(·)for the inconsistency
index deﬁned in L, where Idis a seminorm, and the one deﬁned in L/L∗,where Idis
a norm.
If the norm in ( 11) arises from an inner product, ||A||=√/angbracketleftA,A/angbracketright, then it is possible
to deﬁne the orthogonal complement (L∗)⊥ofL∗, where (L∗)⊥={ A∈L|A⊥L∗}.
Therefore, quotient space L/L∗is isomorphic to (L∗)⊥and Lcan be obtained as
direct sum of L∗and(L∗)⊥,
L=L∗⊕/parenleftbig
L∗/parenrightbig⊥. (24)
Property ( 24) extends the result of Barzilai ( 1998 ), where he proposed the orthogonal
decomposition of a matrix A∈Linto its ‘consistent’ and ‘totally inconsistent’ com-
ponents, A=C+E, and orthogonality C⊥Erefers to the standard dot product on
Rn×ncorresponding to the Euclidean norm. Note that the decomposition proposed by
Barzilai has been further investigated and generalized by Cavallo ( 2019 ).
3.3 Axioms satisfaction
Brunelli and Fedrizzi ( 2015a ) proposed ﬁve characterizing properties, or axioms, for
an inconsistency index. Then, Brunelli ( 2017 ) completed the set of axioms with a
sixth one. The axioms can be formulated both in the additive and in the multiplicativerepresentation of preferences. We clearly refer to the former. In this section, we prove
that an inconsistency index I
d(A), given by ( 12), satisﬁes all the six axioms, provided
that ( 11)i sa permutation-invariant norm on Rn×n, that is a norm invariant with respect
to permutations on the coordinates.
3.3.1 Axiom (A1): Existence of a unique element representing consistency
The ﬁrst characterizing property (A1) requires that an inconsistency index must be
associated to a unique real number which represents perfectly consistent preferences.
More formally, it is demanded that
∃!ν∈Rsuch that I(A)=ν⇔A∈L∗.
An inconsistency index ( 12) satisﬁes axiom (A1) with ν=0, since d(A,L∗)=0⇔
A∈L∗.
3.3.2 Axiom (A2): Invariance under permutation of alternatives
Axiom (A2) requires that an inconsistency index must be independent from the order
of the alternatives, i.e., from the order to which the alternatives are associated with the
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Inconsistency evaluation using norm-based distances 667
rows and columns of A,
I(PAPT)=I(A)∀A∈L, (25)
for any permutation matrix P.
If the norm in ( 12)i spermutation invariant , then axiom (A2) is satisﬁed.
3.3.3 Axiom (A3): Monotonicity under reciprocity-preserving mapping
We ﬁrst brieﬂy summarize axiom (A3). For a more comprehensive description, please
refer to Brunelli and Fedrizzi ( 2015a ). The idea underlying axiom (A3) is the following:
if inconsistent preferences are intensiﬁed, then a better value of an inconsistencyindex cannot be obtained. By ‘preference intensiﬁcation,’ we mean going farther from
complete indifference a
ij=0∀i,j, which is clearly fully consistent. Going farther
from this, uniformity means having stronger judgments and this should not make
their possible inconsistency less evident. It can be proved that the only transformation
aij→f(aij)which can intensify preferences by preserving the necessary additive
reciprocity structure ( 3)i s
f(aij)=kaij,k>1, (26)
or, equivalently,
A→kA,k>1. (27)
The formalization of Axiom (A3) is as follows.
Axiom (A3) An inconsistency index I(·)satisﬁes axiom (A3) if and only if
I(kA)≥I(A)∀k>1∀A∈L. (28)
Proposition 1 An inconsistency index deﬁned by (12)satisﬁes axiom (A3) .
Proof Theorem 1implies that Id(kA)=| k|Id(A)∀k∈ R. Then, Id(kA)>
Id(A)∀k>1 and axiom (A3) is satisﬁed. /intersectionsq/unionsq
3.3.4 Axiom (A4): Monotonicity on single entries
We brieﬂy summarize axiom (A4). The reader can refer to Brunelli and Fedrizzi
(2015a ) for a more comprehensive description. The idea underlying axiom (A4) is that
an inconsistency index is non-decreasing with respect to an elementary modiﬁcation of
a consistent matrix. More formally, given a consistent skew-symmetric matrix A∈L∗
and b∈R,b/negationslash=0, let Apq(b)∈Lbe the inconsistent skew-symmetric matrix
obtained from Aby replacing the single entry apqwith apq+b,p,q∈{1,..., n},
p/negationslash=q. Clearly, aqpmust be replaced with aqp−bin order to preserve skew-symmetry.
Axiom (A4) requires that the larger the change of apqfrom its consistent value, the
123
668 M. Fedrizzi et al.
more inconsistent the obtained matrix is. The formalization of Axiom (A4) is as
follows.
Axiom (A4) An inconsistency index I(·)satisﬁes axiom (A4) if and only if
I(Apq(b))is a non-decreasing function of bforb>0 and a non-increasing function
ofbforb<0,∀A∈L∗,∀p,q∈{1,..., n},p/negationslash=q.
Proposition 2 An inconsistency index deﬁned by ( 12) satisﬁes axiom (A4) .
Proof The proof follows from the homogeneity of seminorms. Given a consistent
skew-symmetric matrix A∈L∗andb∈R,b/negationslash=0, let Apq(b)∈Lbe the inconsistent
skew-symmetric matrix obtained as described above. Matrix Apq(b)can be obtained
as follows,
Apq(b)=A+bMpq,
where Mpqis the n×nmatrix with all null entries except Mpq(p,q)=1,
Mpq(q,p)=− 1. Theorem 2implies that all the inconsistency of matrix Apq(b)
is due to matrix bMpq, when inconsistency is evaluated by means of Id(·),
Id(Apq(b))=Id(A+bMpq)=Id(bMpq).
Theorem 1implies that Id(bMpq)=|b|Id(Mpq)∀b∈R. Since Id(Mpq)> 0, then
axiom (A4) is satisﬁed. /intersectionsq/unionsq
3.3.5 Axiom (A5): Continuity
Axiom (A5) requires continuity of an inconsistency index. Continuity of ( 12) directly
follows from the continuity of norms.
3.3.6 Axiom (A6): Invariance under inversion of preferences
Axiom (A6) (Brunelli 2017 ) requires that an inconsistency index does not change
when preferences expressed in the form of a pairwise comparison matrix Aare inverted
by taking its transpose AT. The idea underlying axiom (A6) is that by inverting all
the preferences we change their polarity, but leave their structure unchanged. Theformalization of Axiom (A6) is as follows. Axiom (A6) An inconsistency index I(·)
satisﬁes axiom (A6) if and only if I(A)=I(A
T)∀A∈L.
Proposition 3 An inconsistency index deﬁned by (12)satisﬁes axiom (A6) .
Proof Since Ais skew-symmetric, it is AT=− A. Then, an inconsistency index I(·)
satisﬁes axiom (A6) if and only if I(A)=I(AT)=I(−A)∀A∈L. By deﬁning
B/prime:= − B,i ti s
Id(−A)=min
B∈L∗d(−A,B)=min
B∈L∗||−A−B|| (29)
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Inconsistency evaluation using norm-based distances 669
=min
B/prime∈L∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle−A+B/prime/vextendsingle/vextendsingle/vextendsingle/vextendsingle=min
B/prime∈L∗|−1|/vextendsingle/vextendsingle/vextendsingle/vextendsingleA−B/prime/vextendsingle/vextendsingle/vextendsingle/vextendsingle (30)
=min
B/prime∈L∗/vextendsingle/vextendsingle/vextendsingle/vextendsingleA−B/prime/vextendsingle/vextendsingle/vextendsingle/vextendsingle=Id(A), (31)
Similarly to ( 14), the proof follows from the fact that every matrix BofL∗can be
expressed as B=− B/primewith B/prime∈L∗. Then, axiom (A6) is satisﬁed by ( 12). /intersectionsq/unionsq
3.4 Boundary property for group decision making
In this section, we prove that inconsistency index Id(A)given by ( 12) satisﬁes the
upper bound property described by Brunelli and Fedrizzi ( 2015b ).
If the preferences of k=1,..., mdecision makers are expressed by means of m
PCM sAk=(ak
ij), it is a relevant problem to study the group PCM AG=(aG
ij)
obtained by aggregating the mindividual PCM s. Dijkstra ( 2012 ) proved that, if the m
PCM s are expressed in the multiplicative representation of preferences, the weighted
geometric mean is the unique aggregation method that guarantees some importantproperties of the group PCM to hold. Let us consider
a
G
ij=m/productdisplay
k=1(ak
ij)λk, (32)
where (λ1,...,λ m)is the weight vector of the decision makers such that λk≥0,/summationtextm
k=1λk=1 and Ak=(ak
ij)∈A. In particular, ( 32) is the unique consistency-
preserving non-trivial aggregation method. Brunelli and Fedrizzi ( 2015b ) studied how
the inconsistency of AGdepends on the inconsistency of Ak,k=1,...., m.T h e y
proved, in particular, that some known inconsistency indices satisfy the followingupper-bound property.
Deﬁnition 5 (Brunelli and Fedrizzi 2015b ) A function I:A→ Ris upper bounded
w.r.t the geometric mean if:
I(A
G)≤max{I(A1) ,..., I(Am)}. (33)
The following property is more restrictive than ( 33), it was also introduced by Brunelli
and Fedrizzi ( 2015b ) and studied for some known inconsistency indices.
Deﬁnition 6 (Brunelli and Fedrizzi 2015b ) A function I:A→ Ris strongly upper
bounded w.r.t the geometric mean if:
I(AG)≤m/summationdisplay
k=1λkI(Ak), (34)
where (λ1,...,λ m)is the same weight vector used in ( 32) to obtain AG.
Let us study the upper-bound properties ( 33) and ( 34) from the additive approach
described above, that is, in the vector space Lof skew-symmetric preference matrices.
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670 M. Fedrizzi et al.
By componentwise applying the logarithmic function ( 2) as described in Section 2,t h e
aggregation ( 32)i n Aclearly corresponds to a linear combination in L. By using, for
simplicity, the same notation as in ( 32), the group skew-symmetric preference matrix
is given by
aG
ij=m/summationdisplay
k=1λk(ak
ij)i,j=1,..., n (35)
AG=m/summationdisplay
k=1λkAk, (36)
where Ak=(ak
ij)∈L. The following result holds.
Theorem 3 An inconsistency index I d(A)deﬁned as a norm-based distance (12)is
strongly upper bounded, i.e., it satisﬁes the upper boundary property (34).
Proof Theorem 1implies that
Id(AG)=Id(m/summationdisplay
k=1λkAk)≤m/summationdisplay
k=1Id(λkAk)
=m/summationdisplay
k=1|λk|Id(Ak)=m/summationdisplay
k=1λkId(Ak) (37)
/intersectionsq/unionsq
4 Conclusions and future work
In this paper, we showed that by deﬁning an inconsistency index Id(A)by means
of a distance induced by a norm, as in ( 12), it is possible to prove many relevant
properties of this index. Note that the property of homogeneity of a (semi)norm isa crucial assumption in proving several Theorems and Propositions. Nevertheless, a
norm is, in our view, a sufﬁciently general notion to satisfy different requirements of
a decision maker. Let us clarify our point with the example of the p-norm of a vector
x=(x
1,..., xn),
||x||p=/parenleftBiggn/summationdisplay
i=1|xi|p/parenrightBigg1
p
,p≥1. (38)
Ap-norm is clearly a permutation-invariant norm, so that all the results in Sect. 3hold
ifId(A)is deﬁned by means of a p-norm ( 38). Note that the well-known Geometric
Consistency Index GCI introduced by Crawford ( 1987 ) corresponds, in the additive
representation of preferences, to the usual Euclidean norm ||·||2. More generally, if a
decision maker uses p-norm ( 38), he/she is provided with a ﬂexible tool in emphasizing
123
Inconsistency evaluation using norm-based distances 671
inconsistent preferences. More precisely, the larger p, the more emphasis is paid
in order to avoid large differences between the elicited preferences and the closest
consistent ones.
In future research, we will study the relationship between the ‘natural’ inconsistency
ranking deﬁned by Csató ( 2019 ) and the ranking on the set of triads induced by the
norm-based inconsistency indices deﬁned in this paper. More precisely, we will studyhow the choice of a norm affects this relationship. A second research topic will be
the possible relationship between our inconsistency index and the index introduced
by Koczkodaj ( 1993 ).
Funding Open access funding provided by Universitá degli Studi di Trento within the CRUI-CARE Agree-
ment.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you giveappropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence,
and indicate if changes were made. The images or other third party material in this article are included
in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If
material is not included in the article’s Creative Commons licence and your intended use is not permitted
by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from thecopyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ .
References
Barzilai, J.: Consistency measures for pairwise comparison matrices. J. Multi-Criteria Decis. Anal. 7(3),
123–132 (1998)
Bozóki, S., Rapcsák, T.: On Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices. J.
Glob. Optim. 42(2), 157–175 (2008)
Brunelli, M.: Studying a set of properties of inconsistency indices for pairwise comparisons. Ann. Oper.
Res. 248(1–2), 143–161 (2017)
Brunelli, M.: A survey of inconsistency indices for pairwise comparisons. Int. J. Gen. Syst. 47(8), 751–771
(2018)
Brunelli, M., Cavallo, B.: Distance-based measures of incoherence for pairwise comparisons. Knowl. Based
System. 187, 104808 (2020)
Brunelli, M., Fedrizzi, M.: Axiomatic properties of inconsistency indices for pairwise comparisons. J. Oper.
Res. Soc. 61(1), 1–15 (2015a)
Brunelli, M., Fedrizzi, M.: Boundary properties of the inconsistency of pairwise comparisons in group
decisions. Eur. J. Oper. Res. 240(3), 765–773 (2015b)
Brunelli, M., Fedrizzi, M.: A general formulation for some inconsistency indices of pairwise comparisons.
Ann. Oper. Res. 274(1–2), 155–169 (2019)
Brunelli, M., Canal, L., Fedrizzi, M.: Inconsistency indices for pairwise comparison matrices: a numerical
study. Ann. Oper. Res. 211(1), 493–509 (2013)
Cavallo, B.: G-distance and G-decomposition for improving G-consistency of a Pairwise Comparison
Matrix. Fuzzy Optim. Decis. Mak. 18(1), 57–83 (2019)
Cavallo, B.: Functional relations and Spearman correlation between consistency indices. J. Oper. Res. Soc.
71(2), 301–311 (2020)
Cavallo, B., D’Apuzzo, L.: A general uniﬁed framework for pairwise comparison matrices in multicriterial
methods. Int. J. Intell. Syst. 24(4), 377–398 (2009)
Chu, M.T.: On the optimal consistent approximation to pairwise comparison matrices. Linear Algebra Appl.
272, 155–168 (1998)
Crawford, G.: The geometric mean procedure for estimating the scale of a judgement matrix. Math. Model.
9(3–5), 327–334 (1987)
123
672 M. Fedrizzi et al.
Crawford, G., Williams, C.: A note on the analysis of subjective judgement matrices. J. Math. Psychol. 29,
25–40 (1985)
Csató, L.: Characterization of an inconsistency ranking for pairwise comparison matrices. Ann. Oper. Res.
261(1–2), 155–165 (2018)
Csató, L.: Axiomatizations of inconsistency indices for triads. Ann. Oper. Res. 280(1–2), 99–110 (2019)
Dijkstra, T.K.: On the extraction of weights from pairwise comparison matrices. Cent. Eur. J. Oper. Res.
21(1), 103–123 (2013)
Duszak, Z., Koczkodaj, W.W.: Generalization of a new deﬁnition of consistency for pairwise comparisons.
Inf. Process. Lett. 52(5), 273–276 (1994)
Fedrizzi, M., Brunelli, M., Caprila, A.: The linear algebra of pairwise comparisons. Int. J. Approx. Reason.
118, 190–207 (2019)
Fichtner, J.: Some thoughts about the mathematics of the analytic hierarchy process. Report 8403, Institut
für Angewandte Systemforschung und Operations Research, Hochschule der Bundeswehr München
(1984)
Fichtner, J.: On deriving priority vectors from matrices of pairwise comparisons. Socio-Econ. Plan. Sci. 20,
341–345 (1986)
Koczkodaj, W.W.: A new deﬁnition of consistency of pairwise comparisons. Math. Comput. Model. 18(7),
79–84 (1993)
Koczkodaj, W.W., Orlowski, M.: An orthogonal basis for computing a consistent approximation to a pairwise
comparison matrix. Comput. Math. Appl. 34(10), 41–47 (1997)
Mazurek, J., Ramík, J.: Some new properties of inconsistent pairwise comparisons matrices. Int. J. Approx.
Reason. 113, 119–132 (2019)
Ramík, J., Korviny, P .: Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric
mean. Fuzzy Sets Syst. 161(11), 1604–1613 (2010)
Saaty, T.L.: A scaling method for priorities in hierarchical structures. J. Math. Psychol. 15, 234–281 (1977)
Saaty, T.L.: Highlights and critical points in the theory and application of the Analytic Hierarchy Process.
Eur. J. Oper. Res. 74, 426–447 (1994)
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Michele Fedrizzi
Nino Civolani
Andrew Critch