In this note we consider a multicriteria decision problem where the decision maker know the the state of the world but the set of consequences is multidimensional. We suppose that a value function is specified over the attribute of the decision problem and we analyze some classes of non additive functions that can represent interaction between criteria.Multicriteria decision making, value functions, Choquet signed integral, Schur decreasing functions. functions

Marta Cardin

English

Research Papers in Economics

Preference Representation for
Multicriteria Decision Making
Marta Cardin
Dept. of Applied Mathematics
University of Venice
e-mail: mcardin@unive.it
Abstract: In this note we consider a multicriteria decision problem where the decision
maker know the the state of the world but the set of consequences is multidimensional. We
suppose that a value function is speci¯ed over the attribute of the decision problem and
we analyze some classes of non additive functions that can represent interaction between
criteria.
Keywords: Multicriteria decision making, value function, Choquet signed integral,
Schur-decreasing functions.
M.S.C. classification: 62C05, 91B08 .
J.E.L. classification: C60.
1 Introduction
In multicriteria decision making we aim at ordering multidimensional alter-
natives. We suppose that a decision can be made using a value function
which represents the preference structure of the decision maker. In this
framework the critical point of solving multi-attribute decision problem is
to determine the value function.
A traditional approach is to use a function that is a simple weight sum
where each weight represents the importance given by the decision maker to
a particular attribute. It should be noticed that the additive model implies
independence between attributes so despite its simplicity this approach suf-
fer a major drawback of not being able to take into account "inter-attribute"
relations that are present in many situations.
The problem of modelling such an interaction is a di±cult question because
there are di®erent types of dependence quite di®erent from each other such
as correlation, complementarity and preferential dependence.
In this note we study some particular classes of non additive value functions
that are usually considered in the area of decision theory under uncertainty.
So this note points out a similarity between decision under uncertainty and
1multicriteria decision making two areas which have been investigate sepa-
rately. However as shown by some recent papers under quite general hy-
pothesis decision under uncertainty and multicriteria decision making are
formally equivalent. A multicriteria decision problem is written in a deci-
sion under uncertainty framework by identifying the criteria with the states
of the world and the consequences with the acts.
In the ¯rst part of the paper we will introduce some classes of multivariate
real functions and we brie°y review Choquet nonadditive integration theory
while in the last part we prove our results concerning the preference relation
of the decision maker.
2 Preliminaries
Let us consider a multicriteria decision problem where there is a set A of
alternatives and a ¯nite set of criteria N = f1;:::;ng with respect to which
the choice between alternatives is done. Each alternative a 2 A is repre-
sented by a n-dimensional vector xa = (xa
1;:::;xa
n) 2 IRn, where for any
i 2 N, xa
n represents the partial score of a related to criterion i. We assume
that the partial scores are de¯ned according the same interval scale.
Suppose that the preferences over A of the decision maker are known and
expressed by a relation º. We also assume that the preference relation is
represented by a real value function de¯ned fon IRn so that if a;b 2 A
a º b , f(xa) ¸ f(xb)
In this note we study some particular classes of multidimensional value func-
tions de¯ned on IRn.
We also consider the concept of majorization arising as a measure of diver-
sity of the components of a n-dimensional vector. Majorization has been
comprehensively treated by[1] and [5].
We aim to formalize the idea that the components of a vector x are less"spread
out" or "more nearly equal" than the components of y.
For a vector x 2 IRn we denote its elements ranked in descending order as
x¾(1) ¸ x¾(2) ¸ :::::: ¸ x¾(n) (2.1)
where ¾is a permutation de¯ned on N.
De¯nition 1. The vector y is said to majorize the vector x, which is denoted
as x ¹ y, if
k X
i=1
x¾(i) ·
k X
i=1
y¾(i) k = 1;2:::;n¡1 and
n X
i=1
x¾(i) =
n X
i=1
y¾(i) (2.2)
2.
Majorization is a partial ordering among vectors, which applies only to vec-
tors having the same sum. It is a measure of the degree to which the vector
elements di®er. For example it can be easily shown that all vectors of sum
s majorize the uniform vector u = ( s
n;::: s
n). Intuitively, the uniform vector
is the vector with minimal di®erences between elements, so all vectors ma-
jorize it.
In order to introduce a nonadditive approach to multicriteria decision mak-
ing we introduce a non- additive integral operator. As is well known the
Choquet integral has been extensively applied in the context of decision un-
der uncertainty.
In this context the Choquet integral may be viewed as a way of aggregating
utility across di®erent states in order to arrive to a decision criterion while
in multicriteria decision making the nonadditive integral operator is a tool
for aggregating over di®erent criteria. The use of variants of the Choquet
integral allows some °exibility in the way criteria are combined. In partic-
ular in this paper we consider a signed Choquet integral as in
For the sake of our application we restrict ourselves to the ¯nite case and
for the properties of nonadditive integration we refer to [2]. We de¯ne a
nonmonotonic Choquet measure on 2N and a signed Choquet integral for a
n-dimensional vector.
De¯nition 2. A function À : 2N ! IR is called a signed capacity if
À(;) = 0 and À(N) = 1.
We will assume here À(N) = 1 as usual although this is not necessary.
We note that if S µ N, À(S) can be viewed as the importance of the set of
elements S.
De¯nition 3. Let À a signed capacity À : 2N ! IR , x 2 IRnand
¼¾(j) = À(f¾(1);::::::;¾(j)g) ¡ À(f¾(1);::::::;¾(j ¡ 1)g) (2.3)
where ¾is de¯ned by (2.2). The signed Choquet integral of x is
Z
xd¾ =
n X
j=1
¼jxj (2.4)
.
We are now able to present some aggregation operators as appropriate ex-
tensions to the weighted arithmetic mean for the aggregation of criteria.
3 Some class of multivariate value functions
Let us now recall some properties for real functions de¯ned on IRn.
3De¯nition 4. A function f, IRn ! IR is called Schur-increasing if f(x) ·
f(y) when x ¹ y. f is Schur-decreasing if ¡f is Schur-increasing.
Schur increasing functions thus preserve majorization. We note also that
a Schur increasing or decreasing function must be a symmetric function.
Moreover a symmetric convex function is Schur increasing [5] .
A Schur decreasing value function consider the attribute as symmetric and
prefer attributes that are less"spread out". In fact if x and y are interpreted
as random variables with all state equally probable the condition x ¹ y
is equivalent to the statement that x is less risky than y in the sense of
Rothschild and Stiglitz. Hence in this context Schur decreasing is a property
of risk aversion.
The following result characterize the preference relations on IRn
+ that can be
represented by a Schur-decreasing function de¯ned on IRn
+.
Theorem 1. Let ¸ a preference relation de¯ned on IRn
+. The preference
relation is represented by a function f IRn
+ ! IR that is Schur-decreasing if
and only if
(x1 :::xi + c:::xj ¡ c:::xn) · (x1 :::xi :::xj :::xn) (3.1)
for all xi;xj with xi ¸ xj and 0 < c < xj.
Proof By the proof of theorem 2 of [4] if x and y are elements of IRn
+ and
x º y then x can be derived from y via a ¯nite series of transformation of
type (3.1). Hence if we assume that(3.1)is satis¯ed x º y if and only if x ¸ y.
A nonadditive integral is a sort of weighted mean taking into account the
importance of every subset of criteria and so considering the relative impor-
tance of the criteria. It should be noticed that we consider signed capacity
that are nonmonotonic and may take negative values. We apply the signed
integral operator to multicriteria decision making modelling interactions be-
tween criteria that are so strong that monotonicity is violated.
The following result characterize the continuos preference relations on IRn
that can be represented by a signed integrals with respect to a signed ca-
pacities .
Theorem 2. Let ¸ a preference relation de¯ned on IRn
+. We suppose that
the preference relation is represented by a continuos function f IRn ! IR
that is a signed Choquet integral if and only if
i) if a;b 2 IR, a ¸ b then (a;:::;a) ¸ (b;:::;b).
4ii) if x;y;z are elements of IRn such that there exists a permutation ¾
de¯ned on N.such that x;y;z are solution of the equation :
t¾(1) ¸ t¾(2) ¸ :::::: ¸ t¾(n) (3.2)
if x ¸ y then x + z ¸ y + z.
Proof The preference relation ¸ is continuous, constant-monotonic ac-
cording to the de¯nitions in [3]. Moreover if it is veri¯ed condition ii), the
preference order satis¯es comonotonic additivity ( see [3]).Then by corollary
1 of [3] there exists a signed Choquet integral that represents the preference
order.
4 Concluding remarks
In this note we have examined multicriteria decision problem where a value
function is speci¯ed over the attributes of a deterministic decision problem.
We have studied particular classes of value functions that are usually con-
sidered in the area of decision theory under uncertainty. In particular the
Schur-increasing functions are used to compare situations according to their
level of heterogeneity while Choquet integral operator can model violations
of additivity and monotonicity. So this note points out a similarity between
decision under uncertainty and multi-attribute decision making problems,
two areas which have been developed in an almost completely independent
way. This is a potentially fruitful area for future work.
References
[1] Arnold, B.C., Majorization and the Lorenz order:A Brief Introduction.
Springer-Verlag, New York (1987).
[2] Denneberg,D., Non-additive measure and integral, Kluwer Academic
Publishers - Dordrecht/Boston/London 1994.
[3] De Waegenaere, A., Wakker, P.P,, Nonmonotonic Choquet integrals Jour-
nal of Mathematical Economics 36 (2001).
[4] Hilderman,H.J., The Lorenz Dominance order as a measure of In-
terestingness in KDD, in Chen,M.S., Yu, P.S., eds. Proceedings of the6th
Paci¯c-Asia Conference on Knowledge Discovery and Data Mining Lec-
ture Notes in Computer Science, Springer-Verlag, Taipei (2002)
[5] Marshall, A.W., Olkin I., Inequalities: Theory of Majorization and Its
Applications. Academic Press, New York (1979).
5

Inequalities: Theory of Majorization and Its Applications.

Majorization and the Lorenz order:A Brief Introduction.

Non-additive measure and integral, Kluwer Academic Publishers - Dordrecht/Boston/London

Nonmonotonic Choquet integrals

The Lorenz Dominance order as a measure of Interestingness

Preference Rapresentation for Multicriteria Decision Making

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Preference Rapresentation for Multicriteria Decision Making

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