On the axiomatization of some classes of discrete universal integrals
Erich Peter Klement
a,
⇑
, Radko Mesiar
b,c
a
Department of Knowledge-Based Mathematical Systems, Johannes Kepler University, Linz, Austria
b
Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovakia
c
Institute of Theory of Information and Automation, Czech Academy of Sciences, Prague, Czech Republic
article info
Article history:
Received 20 June 2011
Received in revised form 25 October 2011
Accepted 26 October 2011
Available online 7 November 2011
Keywords:
Comonotone modularity
Copula
Universal integral
Choquet integral
Sugeno integral
abstract
Following the ideas of the axiomatic characterization of the Choquet integral due to [D. Schmeidler, Inte-
gral representation without additivity, Proc. Amer. Math. Soc. 97 (1986) 255–261] and of the Sugeno inte-
gral given in [J.-L. Marichal, An axiomatic approach of the discrete Sugeno integral as a tool to aggregate
interacting criteria in a qualitative framework, IEEE Trans. Fuzzy Syst. 9 (2001) 164–172], we provide a
general axiomatization of some classes of discrete universal integrals, including the case of discrete cop-
ula-based universal integrals (as usual, the product copula corresponds just to the Choquet integral, and
the minimum to the Sugeno integral).
Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction
In this contribution, we restrict ourselves to a fixed finite space
X = {1, ...,n}, and we will deal with functions from X to [0,1] which
we identify with n-dimensional vectors x =(x
1
, ...,x
n
). From the
application point of view, we can look at x as a score vector of some
alternative characterized by n criteria. To be able to decide which
of the alternatives described by the score vectors x and y, respec-
tively, is to be preferred, a typical approach is to evaluate both x
and y by means of some utility function U.
The utility function U is often constructed from a boolean utility
function B acting on x 2 {0,1}
n
. However, each such boolean utility
function B can be seen as a capacity m :2
X
? [0, 1], m(E)=B(1
E
).
Typical extension approaches are related to integration, i.e.,
U(x)=I(m,x), where I(m, Á) is some integral on X with respect to
the capacity m.
Another approach is based on some axiomatization (and bool-
ean utility function B). It is well-known that the additivity of the
utility function U : [0, 1]
n
? [0, 1] is related to the application of
Lebesgue integral, UðxÞ¼
R
x dm, and then also the capacity m
should be additive. Putting m({i}) = w
i
, we obtain the well-known
weighted arithmetic mean, UðxÞ¼
P
n
i¼1
w
i
Á x
i
.
Our contribution recalls some classes of universal integrals
(including, among others, the Choquet, the Sugeno and the Lebes-
gue integral) and provides corresponding axiomatizations.
Because of the link to utility functions, we restrict ourselves to
(normed) capacities and to input values from [0, 1], although many
integrals mentioned here (including the Choquet and Sugeno inte-
gral) can be considered in a more general (unbounded) framework
[14].
However, we do not consider any further restriction concerning
the underlying capacity, such as additivity or pseudo-additivity,
and thus we will not deal with integrals based on such special
capacities (compare, e.g., [20,22,27]).
The paper is organized as follows. In the following section, the
Choquet and the Sugeno integral as well as their axiomatizations
are summarized. In Section 3, we recall (discrete) copula-based
integrals and some other classes of discrete universal integrals,
including some examples. In Section 4, the axiomatization of these
discrete universal integrals is given. As a special case, symmetric
discrete copula-based universal integrals (generalizing OWA oper-
ators) are discussed.
2. Choquet and Sugeno integrals, and their axiomatization
Though all integrals discussed in this paper can be defined on an
arbitrary measurable space, in this paper we consider (as already
mentioned) the finite space X = {1, ..., n} only, equipped with the
r
-algebra 2
X
={EjE # X}.
Definition 2.1. A capacity on X is a set function m :2
X
? [0, 1]
which is non-decreasing, i.e., we have m(E) 6 m(F) whenever
E # F # X, and satisfies the boundary conditions m(;) = 0 and
m(X)=1.
0950-7051/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.knosys.2011.10.015
⇑
Corresponding author.
E-mail addresses: ep.klement@jku.at (E.P. Klement), mesiar@math.sk (R. Mesiar).
Knowledge-Based Systems 28 (2012) 13–18
Contents lists available at SciVerse ScienceDirect
Knowledge-Based Systems
journal homepage: www.elsevier.com/locate/knosys