We study the so-called signed discrete Choquet integral (also called
non-monotonic discrete Choquet integral) regarded as the Lov\'asz extension of
a pseudo-Boolean function which vanishes at the origin. We present
axiomatizations of this generalized Choquet integral, given in terms of certain
functional equations, as well as by necessary and sufficient conditions which
reveal desirable properties in aggregation theory

Cardin, Marta

Couceiro, Miguel

Giove, Silvio

Marichal, Jean-Luc

English

arXiv

We study the so-called signed discrete Choquet integral
(also called non-monotonic discrete Choquet integral) regarded as the Lovasz extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations,  as well as by necessary and sufficient conditions which reveal desirable properties in aggregation theory

Cardin M.

Couceiro M.

Giove S.

Marichal J.L.

Archivio Istituzionale della Ricerca

Axiomatizations of signed discrete Choquet integrals

We study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lovasz extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations, as well as by necessary and sufficient conditions which reveal desirable properties in aggregation theor

Marichal J.-L.

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AXIOMATIZATIONS OF SIGNED DISCRETE CHOQUET
INTEGRALS
MARTA CARDIN, MIGUEL COUCEIRO, SILVIO GIOVE,
AND JEAN-LUC MARICHAL
Abstract. We study the so-called signed discrete Choquet integral
(also called non-monotonic discrete Choquet integral) regarded as the
Lova´sz extension of a pseudo-Boolean function which vanishes at the
origin. We present axiomatizations of this generalized Choquet integral,
given in terms of certain functional equations, as well as by necessary
and sufficient conditions which reveal desirable properties in aggregation
theory.
1. Introduction
This paper deals with the so-called “signed (discrete) Choquet integral”
(also called non-monotonic Choquet integral) which naturally generalizes the
Choquet integral [1]. Traditionally, the Choquet integral is defined in terms
of a capacity (also called fuzzy measure [10, 11]), i.e., a set function µ : 2[n] →
R such that µ(∅) = 0 and µ(S) 6 µ(T ) whenever S ⊆ T . Dropping the
monotonicity requirement in the definition of µ, we obtain what is referred to
as a signed capacity (also called non-monotonic fuzzy measure). The signed
Choquet integral is then defined exactly the same way but replacing the
underlying capacity by a signed capacity. This extension has been considered
by several authors, e.g., [3, 7, 8].
A convenient way to introduce the signed Choquet integral is via the no-
tion of Lova´sz extension. Indeed, the signed Choquet integral can be thought
of as the Lova´sz extension of a pseudo-Boolean function f : {0, 1}n → R
which vanishes at the origin. Moreover, we retrieve the classical Choquet
integral by further assuming that f : {0, 1}n → R is nondecreasing.
In this paper we consider the latter approach to the signed Choquet in-
tegral. In Section 2 we recall the basic notions and terminology concerning
Choquet integrals and Lova´sz esxtensions needed throughout the paper. In
Section 3 we present various characterizations of the signed Choquet inte-
gral. First, we recall the piecewise linear nature of Lova´sz extensions which
particularizes to the signed Choquet integral (Theorem 3.1). Then we gen-
eralize Schmeidler’s axiomatization of the signed discrete Choquet integral
Date: October 21, 2010.
2010 Mathematics Subject Classification. Primary 39B22, 39B72; Secondary 26B35.
Key words and phrases. Signed discrete Choquet integral, signed capacity, Lova´sz ex-
tension, functional equation, comonotonic additivity, homogeneity, axiomatization.
1
2 MARTA CARDIN, MIGUEL COUCEIRO, SILVIO GIOVE, AND JEAN-LUC MARICHAL
given in terms of continuity and comonotonic additivity, showing that pos-
itive homogeneity can be replaced for continuity (Theorem 3.2). The main
result of this paper, Theorem 3.3, presents a characterization of families
of signed Choquet integrals in terms of necessary and sufficient conditions
which:
(1) reveal the linear nature of these generalized Choquet integrals with
respect to the underlying signed capacities,
(2) express properties of the family members defined on the standard
basis of signed capacites, and
(3) make apparent the meaningfulness with respect to interval scales of
signed Choquet integrals.
We also discuss the independence of axioms given in Theorem 3.3.
Throughout this paper, the symbols ∧ and ∨ denote the minimum and
maximum functions, respectively.
2. Choquet integrals and Lova´sz extensions
A capacity on [n] is a set function µ : 2[n] → R such that µ(∅) = 0 and
µ(S) 6 µ(T ) whenever S ⊆ T . A capacity µ on [n] is said to be normalized
if µ([n]) = 1.
Definition 2.1. Let µ be a capacity on [n] and let x ∈ [0,∞[n. The Choquet
integral of x with respect to µ is defined by
Cµ(x) =
n∑
i=1
(µpii − µ
pi
i+1)xpi(i),
where pi is a permutation on [n] such that xpi(1) 6 · · · 6 xpi(n) and µ
pi
i =
µ({pi(i), . . . , pi(n)}) for i ∈ [n+ 1], with the convention that µpin+1 = µ(∅).
The concept of Choquet integral can be formally extended to more general
set functions and n-tuples of Rn as follows. A signed capacity (or game) on
[n] is a set function v : 2[n] → R such that v(∅) = 0.
Definition 2.2. Let v be a signed capacity on [n] and let x ∈ Rn. The
signed Choquet integral of x with respect to v is defined by
Cv(x) =
n∑
i=1
(vpii − v
pi
i+1)xpi(i),
where pi is a permutation on [n] such that xpi(1) 6 · · · 6 xpi(n) and v
pi
i =
v({pi(i), . . . , pi(n)}) for i ∈ [n+ 1], with the convention that vpin+1 = v(∅).
The more general concept of a set function v : 2[n] → R (without any
constraint) leads to the notion of the Lova´sz extension of a pseudo-Boolean
function, which we now briefly describe. For general background, see [4, 9].
Let Sn denote the symmetric group on [n] and, for each pi ∈ Sn, define
Ppi = {x ∈ R
n : xpi(1) 6 · · · 6 xpi(n)}.
AXIOMATIZATIONS OF SIGNED DISCRETE CHOQUET INTEGRALS 3
Let v : 2[n] → R be a set function and let f : {0, 1}n → R be the correspond-
ing pseudo-Boolean function, that is, such that f(1S) = v(S). The Lova´sz
extension of f is the continuous function fˆ : Rn → R which is defined on
each Ppi as the unique affine function that coincides with f at the n + 1
vertices of the standard simplex [0, 1]n ∩ Ppi of [0, 1]
n. In fact, fˆ can be
expressed as
(1) fˆ(x) = f(0) +
n∑
i=1
(fpii − f
pi
i+1)xpi(i) (x ∈ Ppi).
where fpii = f(1{pi(i),...,pi(n)}) = v({pi(i), . . . , pi(n)}) for i ∈ [n] and f
pi
n+1 =
f(0). Thus fˆ is a continuous function whose restriction to each Ppi is an
affine function.
It follows from (1) that the Lova´sz extension of a pseudo-Boolean function
f : {0, 1}n → R is a signed Choquet integral if and only if f(0) = 0. Its
restriction to [0,∞[n is a Choquet integral if, in addition, f is nondecreasing.
It was also shown [6] that the Lova´sz extension fˆ can also be written as
(2) fˆ(x) =
∑
S⊆[n]
m(S)
∧
i∈S
xi (x ∈ R
n),
where the set function m : 2[n] → R is the Mo¨bius transform of v, given by
m(S) =
∑
T⊆S(−1)
|S|−|T | v(T ). Thus, a signed Choquet integral has the
form (2) with m(∅) = 0.
3. Axiomatizations of Lova´sz extensions
We have a first characterization that immediately follows from the defi-
nition of Lova´sz extensions.
Theorem 3.1. A function g : Rn → R is a Lova´sz extension if and only if
(3) g(λx+ (1− λ)x′) = λ g(x) + (1− λ) g(x′) (0 6 λ 6 1)
for all comonotonic vectors x,x′ ∈ Rn. The function g is a signed Choquet
integral if additionally g(0) = 0.
Proof. The condition stated in the theorem means that g is affine (since it
is both convex and concave) on each Ppi. Hence, it is continuous on R
n and
thus it is a Lova´sz extension. 
The following theorem is inspired from a characterization of the Choquet
integral by de Campos and Bolan˜os [2].
Theorem 3.2. A function g : Rn → R is a Lova´sz extension if and only if
the function h : Rn → R, defined by h = g − g(0),
(i) is comonotonic additive.
(ii) is continuous or satisfies h(rx) = rh(x) for all r > 0.
The function g is a signed Choquet integral if additionally g(0) = 0.
4 MARTA CARDIN, MIGUEL COUCEIRO, SILVIO GIOVE, AND JEAN-LUC MARICHAL
Proof. It is not difficult to see that the conditions are necessary. So let us
prove the sufficiency. Fix pi ∈ Sn and x ∈ Ppi. Then we have
x = xpi(1)1[n] +
n∑
i=2
(xpi(i) − xpi(i−1))1{pi(i),...,pi(n)}.
By comonotonic additivity, we get
h(x) = h
(
xpi(1)1[n]
)
+
n∑
i=2
h
(
(xpi(i) − xpi(i−1))1{pi(i),...,pi(n)}
)
.
Also by comonotonic additivity, we have
0 = h(0) = h
(
1[n] − 1[n]
)
= h
(
1[n]
)
+ h
(
−1[n]
)
and hence h
(
−1[n]
)
= −h
(
1[n]
)
. Moreover, if h(rx) = rh(x) for all r > 0
(and even for r = 0 since h(0) = 0), then h
(
r1[n]
)
= rh
(
1[n]
)
for all r ∈ R
and hence
h(x) = xpi(1) h
(
1[n]
)
+
n∑
i=2
(xpi(i) − xpi(i−1))h
(
1{pi(i),...,pi(n)}
)
=
n∑
i=1
(hpii − h
pi
i+1)xpi(i)
where hpii = h(1{pi(i),...,pi(n)}) for i ∈ [n] and h
pi
n+1 = h(1∅).
Let us now show that h satisfies the positive homogeneity property as soon
as it is continuous. Comonotonic additivity implies that g(nx) = ng(x) for
every x ∈ Rn and every positive integer n. For any positive integers n,m,
we then have
m
n
h(x) =
m
n
h
(
n
x
n
)
= mh
(x
n
)
= h
(m
n
x
)
which means that h(rx) = rh(x) for every positive rational r and even for
every positive real r by continuity. 
In the following characterization of the signed Choquet integral, we will
assume that the function to axiomatize is constructed from a signed capacity.
More precisely, denoting the set of signed capacities on [n] by Σn, we now
regard our function as a map f : Rn ×Σn → R, or equivalently, as the class
{fv : R
n → R : v ∈ Σn}. We will adopt the latter terminology to state our
result, which is inspired from a characterization given in [5].
For every T ⊆ [n], let vT ∈ Σn be the unanimity game defined by vT (S) =
1, if S ⊇ T , and 0, otherwise. Note that the vT (T ⊆ [n]) form a basis
(actually, the standard basis) for Σn. Indeed, for every v ∈ Σn, we have
v =
∑
T⊆[n]
mv(T ) vT ,
where mv is the Mo¨bius transform of v.
AXIOMATIZATIONS OF SIGNED DISCRETE CHOQUET INTEGRALS 5
Theorem 3.3. If the class {fv : R
n → R : v ∈ Σn} satisfies the following
properties
(i) There exist 2n functions gT : R
n → R (T ⊆ [n]) such that
fv =
∑
T⊆[n]
v(T ) gT ;
(ii) For every S ⊆ [n], we have fvS (x) = 0 whenever xi = 0 for some
i ∈ S;
(iii) For every S ⊆ [n], r > 0, s ∈ R, and x ∈ Rn, we have
fvS (rx+ s1[n]) = rfvS(x) + s ;
then and only then fv = Cv for all v ∈ Σn.
Proof. The sufficiency is straightforward, so let us prove the necessity. Given
the relation between v and mv, condition (i) is equivalent to assuming the
existence of 2n functions hT : R
n → R (T ⊆ [n]) such that
fv =
∑
T⊆[n]
mv(T )hT .
Thus fvT = hT . Therefore, it suffices to prove the following claim.
Claim. For any fixed T ⊆ [n], if the function fvT : R
n → R satisfies condi-
tions (ii) and (iii), then fvT (x) = ∧i∈Txi for all x ∈ R
n.
Let x ∈ Rn. If x1 = · · · = xn, then
fvT (x) = fvT
(( ∧
i∈[n]
xi
)
1[n]
)
=
∧
i∈[n]
xi ,
since fvT (0) = 0 by (iii).
Otherwise, if
∨
i∈[n] xi −
∧
i∈[n] xi 6= 0, then by (iii) we have
(4) fvT (x) =
(∨
i∈[n] xi −
∧
i∈[n] xi
)
fvT (x
′) +
∧
i∈[n] xi,
where
x′ =
x−
(∧
i∈[n] xi
)
1[n]∨
i∈[n] xi −
∧
i∈[n] xi
∈ [0, 1]n.
By (iii) and (ii),
fvT (x
′) = fvT
(
x′ −
(∧
i∈T x
′
i
)
1[n]
)
+
∧
i∈T x
′
i =
∧
i∈T x
′
i.
By (4), fvT (x) =
∧
i∈T xi. 
Note that the conditions of Theorem 3.3 are independent. Indeed,
(i), (iii) 6⇒ (ii): Consider the class {fv : R
n → R : v ∈ Σn} given by
the weighted arithmetic mean functions
fv(x) =
∑
T⊆[n]
mv(T )
( 1
|T |
∑
i∈T
xi
)
,
where mv is the Mo¨bius transform of v.
6 MARTA CARDIN, MIGUEL COUCEIRO, SILVIO GIOVE, AND JEAN-LUC MARICHAL
(i), (ii) 6⇒ (iii): Consider the class {fv : R
n → R : v ∈ Σn} given by
the multilinear polynomial functions
fv(x) =
∑
T⊆[n]
mv(T )
∏
i∈T
xi ,
where mv is the Mo¨bius transform of v.
(ii), (iii) 6⇒ (i): Define the normalized capacity v∗ ∈ Σ3 by v
∗({1, 2}) =
v∗({3}) = 0 and v∗({1, 3}) = v∗({2, 3}) = 1/2 and consider the class
{fv : R
3 → R : v ∈ Σ3} given by fv = Cv for every v ∈ Σ3 \ {v
∗},
and
fv∗(x1, x2, x3) =
(x1 + x2
2
)
∧ x3.
Remark 1. (a) The conditions in Theorem 3.3 can be justified as fol-
lows. Condition (i) expresses the fact that the aggregation model is
linear with respect to the underlying signed capacities. Condition
(ii) expresses minimal requirements on the functions defined on the
standard basis {vS : S ⊆ [n]} of Σn. Condition (iii) expresses the
fact that fvS is meaningful with respect to interval scales.
(b) The characterization given in Theorem 3.3 does not use the fact that
v(∅) = 0. Therefore they can be immediately adapted to Lova´sz
extensions by redefining Σn as the set of set functions on [n].
References
[1] G. Choquet. Theory of capacities. Ann. Inst. Fourier, Grenoble, 5:131–295 (1955),
1953–1954.
[2] L. M. de Campos and M. J. Bolan˜os. Characterization and comparison of Sugeno and
Choquet integrals. Fuzzy Sets and Systems, 52(1):61–67, 1992.
[3] A. De Waegenaere and P. Wakker. Nonmonotonic Choquet integrals. J. Mathematical
Economics, 36:45–60, 2001.
[4] L. Lova´sz. Submodular functions and convexity. In Mathematical programming, 11th
int. Symp., Bonn 1982, 235–257. 1983.
[5] J.-L. Marichal. An axiomatic approach of the discrete Choquet integral as a tool to
aggregate interacting criteria. IEEE Trans. Fuzzy Syst., 8(6):800–807, 2000.
[6] J.-L. Marichal. Aggregation of interacting criteria by means of the discrete Choquet in-
tegral. In Aggregation operators: new trends and applications, pages 224–244. Physica,
Heidelberg, 2002.
[7] T. Murofushi, M. Sugeno, and M. Machida. Non-monotonic fuzzy meansures and the
Choquet integral. Fuzzy Sets and Systems, 64:73–86, 1994.
[8] D. Schmeidler. Integral representation without additivity. Proc. Amer. Math. Soc.,
97(2):255–261, 1986.
[9] I. Singer. Extensions of functions of 0-1 variables and applications to combinatorial
optimization. Numer. Funct. Anal. Optimization, 7:23–62, 1984.
[10] M. Sugeno. Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute
of Technology, Tokyo, 1974.
[11] M. Sugeno. Fuzzy measures and fuzzy integrals—a survey. In M. M. Gupta, G. N.
Saridis, and B. R. Gaines, editors, Fuzzy automata and decision processes, pages 89–
102. North-Holland, New York, 1977.
AXIOMATIZATIONS OF SIGNED DISCRETE CHOQUET INTEGRALS 7
Department of Applied Mathematics, University Ca’ Foscari of Venice,
Dorsoduro 3825/E–30123, Venice, Italy
E-mail address: mcardin[at]unive.it
Mathematics Research Unit, FSTC, University of Luxembourg, 6, rue Coudenhove-
Kalergi, L-1359 Luxembourg, Luxembourg
E-mail address: miguel.couceiro[at]uni.lu
Department of Applied Mathematics, University Ca’ Foscari of Venice,
Dorsoduro 3825/E–30123, Venice, Italy
E-mail address: sgiove[at]unive.it
Mathematics Research Unit, FSTC, University of Luxembourg, 6, rue Coudenhove-
Kalergi, L-1359 Luxembourg, Luxembourg
E-mail address: jean-luc.marichal[at]uni.lu


peer reviewedWe study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lovász extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations, as well as by necessary and sufficient conditions which reveal desirable properties in aggregation theory.F1R-MTH-PUL-09MRDO > MRDO > 01/01/2009 - 31/12/2011 > MARICHAL Jean-Lu

Open Repository and Bibliography - Luxembourg

AXIOMATIZATIONS OF SIGNED DISCRETE CHOQUETINTEGRALSMARTA CARDIN, MIGUEL COUCEIRO, SILVIO GIOVE,AND JEAN-LUC MARICHALAbstract. We study the so-called signed discrete Choquet integral(also called non-monotonic discrete Choquet integral) regarded as theLovász extension of a pseudo-Boolean function which vanishes at theorigin. We present axiomatizations of this generalized Choquet integral,given in terms of certain functional equations, as well as by necessaryand sufficient conditions which reveal desirable properties in aggregationtheory.1. IntroductionThis paper deals with the so-called “signed (discrete) Choquet integral”(also called non-monotonic Choquet integral) which naturally generalizes theChoquet integral [1]. Traditionally, the Choquet integral is defined in termsof a capacity (also called fuzzy measure [10, 11]), i.e., a set function µ : 2[n] →R such that µ(∅) = 0 and µ(S) 6 µ(T ) whenever S ⊆ T . Dropping themonotonicity requirement in the definition of µ, we obtain what is referred toas a signed capacity (also called non-monotonic fuzzy measure). The signedChoquet integral is then defined exactly the same way but replacing theunderlying capacity by a signed capacity. This extension has been consideredby several authors, e.g., [3, 7, 8].A convenient way to introduce the signed Choquet integral is via the no-tion of Lovász extension. Indeed, the signed Choquet integral can be thoughtof as the Lovász extension of a pseudo-Boolean function f : {0, 1}n → Rwhich vanishes at the origin. Moreover, we retrieve the classical Choquetintegral by further assuming that f : {0, 1}n → R is nondecreasing.In this paper we consider the latter approach to the signed Choquet in-tegral. In Section 2 we recall the basic notions and terminology concerningChoquet integrals and Lovász esxtensions needed throughout the paper. InSection 3 we present various characterizations of the signed Choquet inte-gral. First, we recall the piecewise linear nature of Lovász extensions whichparticularizes to the signed Choquet integral (Theorem 3.1). Then we gen-eralize Schmeidler’s axiomatization of the signed discrete Choquet integralDate: October 21, 2010.2010 Mathematics Subject Classification. Primary 39B22, 39B72; Secondary 26B35.Key words and phrases. Signed discrete Choquet integral, signed capacity, Lovász ex-tension, functional equation, comonotonic additivity, homogeneity, axiomatization.12 MARTA CARDIN, MIGUEL COUCEIRO, SILVIO GIOVE, AND JEAN-LUC MARICHALgiven in terms of continuity and comonotonic additivity, showing that pos-itive homogeneity can be replaced for continuity (Theorem 3.2). The mainresult of this paper, Theorem 3.3, presents a characterization of familiesof signed Choquet integrals in terms of necessary and sufficient conditionswhich:(1) reveal the linear nature of these generalized Choquet integrals withrespect to the underlying signed capacities,(2) express properties of the family members defined on the standardbasis of signed capacites, and(3) make apparent the meaningfulness with respect to interval scales ofsigned Choquet integrals.We also discuss the independence of axioms given in Theorem 3.3.Throughout this paper, the symbols ∧ and ∨ denote the minimum andmaximum functions, respectively.2. Choquet integrals and Lovász extensionsA capacity on [n] is a set function µ : 2[n] → R such that µ(∅) = 0 andµ(S) 6 µ(T ) whenever S ⊆ T . A capacity µ on [n] is said to be normalizedif µ([n]) = 1.Definition 2.1. Let µ be a capacity on [n] and let x ∈ [0,∞[n. The Choquetintegral of x with respect to µ is defined byCµ(x) =n∑i=1(µπi − µπi+1)xπ(i),where π is a permutation on [n] such that xπ(1) 6 · · · 6 xπ(n) and µπi =µ({π(i), . . . , π(n)}) for i ∈ [n+ 1], with the convention that µπn+1 = µ(∅).The concept of Choquet integral can be formally extended to more generalset functions and n-tuples of Rn as follows. A signed capacity (or game) on[n] is a set function v : 2[n] → R such that v(∅) = 0.Definition 2.2. Let v be a signed capacity on [n] and let x ∈ Rn. Thesigned Choquet integral of x with respect to v is defined byCv(x) =n∑i=1(vπi − vπi+1)xπ(i),where π is a permutation on [n] such that xπ(1) 6 · · · 6 xπ(n) and vπi =v({π(i), . . . , π(n)}) for i ∈ [n+ 1], with the convention that vπn+1 = v(∅).The more general concept of a set function v : 2[n] → R (without anyconstraint) leads to the notion of the Lovász extension of a pseudo-Booleanfunction, which we now briefly describe. For general background, see [4, 9].Let Sn denote the symmetric group on [n] and, for each π ∈ Sn, definePπ = {x ∈ Rn : xπ(1) 6 · · · 6 xπ(n)}.AXIOMATIZATIONS OF SIGNED DISCRETE CHOQUET INTEGRALS 3Let v : 2[n] → R be a set function and let f : {0, 1}n → R be the correspond-ing pseudo-Boolean function, that is, such that f(1S) = v(S). The Lovászextension of f is the continuous function f̂ : Rn → R which is defined oneach Pπ as the unique affine function that coincides with f at the n + 1vertices of the standard simplex [0, 1]n ∩ Pπ of [0, 1]n. In fact, f̂ can beexpressed as(1) f̂(x) = f(0) +n∑i=1(fπi − fπi+1)xπ(i) (x ∈ Pπ).where fπi = f(1{π(i),...,π(n)}) = v({π(i), . . . , π(n)}) for i ∈ [n] and fπn+1 =f(0). Thus f̂ is a continuous function whose restriction to each Pπ is anaffine function.It follows from (1) that the Lovász extension of a pseudo-Boolean functionf : {0, 1}n → R is a signed Choquet integral if and only if f(0) = 0. Itsrestriction to [0,∞[n is a Choquet integral if, in addition, f is nondecreasing.It was also shown [6] that the Lovász extension f̂ can also be written as(2) f̂(x) =∑S⊆[n]m(S)∧i∈Sxi (x ∈ Rn),where the set function m : 2[n] → R is the Möbius transform of v, given bym(S) =∑T⊆S(−1)|S|−|T | v(T ). Thus, a signed Choquet integral has theform (2) with m(∅) = 0.3. Axiomatizations of Lovász extensionsWe have a first characterization that immediately follows from the defi-nition of Lovász extensions.Theorem 3.1. A function g : Rn → R is a Lovász extension if and only if(3) g(λx+ (1− λ)x′) = λ g(x) + (1− λ) g(x′) (0 6 λ 6 1)for all comonotonic vectors x,x′ ∈ Rn. The function g is a signed Choquetintegral if additionally g(0) = 0.Proof. The condition stated in the theorem means that g is affine (since itis both convex and concave) on each Pπ. Hence, it is continuous on Rn andthus it is a Lovász extension. ¤The following theorem is inspired from a characterization of the Choquetintegral by de Campos and Bolaños [2].Theorem 3.2. A function g : Rn → R is a Lovász extension if and only ifthe function h : Rn → R, defined by h = g − g(0),(i) is comonotonic additive.(ii) is continuous or satisfies h(rx) = rh(x) for all r > 0.The function g is a signed Choquet integral if additionally g(0) = 0.4 MARTA CARDIN, MIGUEL COUCEIRO, SILVIO GIOVE, AND JEAN-LUC MARICHALProof. It is not difficult to see that the conditions are necessary. So let usprove the sufficiency. Fix π ∈ Sn and x ∈ Pπ. Then we havex = xπ(1)1[n] +n∑i=2(xπ(i) − xπ(i−1))1{π(i),...,π(n)}.By comonotonic additivity, we geth(x) = h(xπ(1)1[n])+n∑i=2h((xπ(i) − xπ(i−1))1{π(i),...,π(n)}).Also by comonotonic additivity, we have0 = h(0) = h(1[n] − 1[n])= h(1[n])+ h(−1[n])and hence h(−1[n])= −h(1[n]). Moreover, if h(rx) = rh(x) for all r > 0(and even for r = 0 since h(0) = 0), then h(r1[n])= rh(1[n])for all r ∈ Rand henceh(x) = xπ(1) h(1[n])+n∑i=2(xπ(i) − xπ(i−1))h(1{π(i),...,π(n)})=n∑i=1(hπi − hπi+1)xπ(i)where hπi = h(1{π(i),...,π(n)}) for i ∈ [n] and hπn+1 = h(1∅).Let us now show that h satisfies the positive homogeneity property as soonas it is continuous. Comonotonic additivity implies that g(nx) = ng(x) forevery x ∈ Rn and every positive integer n. For any positive integers n,m,we then havemnh(x) =mnh(nxn)= mh(xn)= h(mnx)which means that h(rx) = rh(x) for every positive rational r and even forevery positive real r by continuity. ¤In the following characterization of the signed Choquet integral, we willassume that the function to axiomatize is constructed from a signed capacity.More precisely, denoting the set of signed capacities on [n] by Σn, we nowregard our function as a map f : Rn ×Σn → R, or equivalently, as the class{fv : Rn → R : v ∈ Σn}. We will adopt the latter terminology to state ourresult, which is inspired from a characterization given in [5].For every T ⊆ [n], let vT ∈ Σn be the unanimity game defined by vT (S) =1, if S ⊇ T , and 0, otherwise. Note that the vT (T ⊆ [n]) form a basis(actually, the standard basis) for Σn. Indeed, for every v ∈ Σn, we havev =∑T⊆[n]mv(T ) vT ,where mv is the Möbius transform of v.AXIOMATIZATIONS OF SIGNED DISCRETE CHOQUET INTEGRALS 5Theorem 3.3. If the class {fv : Rn → R : v ∈ Σn} satisfies the followingproperties(i) There exist 2n functions gT : Rn → R (T ⊆ [n]) such thatfv =∑T⊆[n]v(T ) gT ;(ii) For every S ⊆ [n], we have fvS (x) = 0 whenever xi = 0 for somei ∈ S;(iii) For every S ⊆ [n], r > 0, s ∈ R, and x ∈ Rn, we havefvS (rx+ s1[n]) = rfvS (x) + s ;then and only then fv = Cv for all v ∈ Σn.Proof. The sufficiency is straightforward, so let us prove the necessity. Giventhe relation between v and mv, condition (i) is equivalent to assuming theexistence of 2n functions hT : Rn → R (T ⊆ [n]) such thatfv =∑T⊆[n]mv(T )hT .Thus fvT = hT . Therefore, it suffices to prove the following claim.Claim. For any fixed T ⊆ [n], if the function fvT : Rn → R satisfies condi-tions (ii) and (iii), then fvT (x) = ∧i∈Txi for all x ∈ Rn.Let x ∈ Rn. If x1 = · · · = xn, thenfvT (x) = fvT(( ∧i∈[n]xi)1[n])=∧i∈[n]xi ,since fvT (0) = 0 by (iii).Otherwise, if∨i∈[n] xi −∧i∈[n] xi 6= 0, then by (iii) we have(4) fvT (x) =(∨i∈[n] xi −∧i∈[n] xi)fvT (x′) +∧i∈[n] xi,wherex′ =x− (∧i∈[n] xi)1[n]∨i∈[n] xi −∧i∈[n] xi∈ [0, 1]n.By (iii) and (ii),fvT (x′) = fvT(x′ − (∧i∈T x′i)1[n])+∧i∈T x′i =∧i∈T x′i.By (4), fvT (x) =∧i∈T xi. ¤Note that the conditions of Theorem 3.3 are independent. Indeed,(i), (iii) 6⇒ (ii): Consider the class {fv : Rn → R : v ∈ Σn} given bythe weighted arithmetic mean functionsfv(x) =∑T⊆[n]mv(T )( 1|T |∑i∈Txi),where mv is the Möbius transform of v.6 MARTA CARDIN, MIGUEL COUCEIRO, SILVIO GIOVE, AND JEAN-LUC MARICHAL(i), (ii) 6⇒ (iii): Consider the class {fv : Rn → R : v ∈ Σn} given bythe multilinear polynomial functionsfv(x) =∑T⊆[n]mv(T )∏i∈Txi ,where mv is the Möbius transform of v.(ii), (iii) 6⇒ (i): Define the normalized capacity v∗ ∈ Σ3 by v∗({1, 2}) =v∗({3}) = 0 and v∗({1, 3}) = v∗({2, 3}) = 1/2 and consider the class{fv : R3 → R : v ∈ Σ3} given by fv = Cv for every v ∈ Σ3 \ {v∗},andfv∗(x1, x2, x3) =(x1 + x22)∧ x3.Remark 1. (a) The conditions in Theorem 3.3 can be justified as fol-lows. Condition (i) expresses the fact that the aggregation model islinear with respect to the underlying signed capacities. Condition(ii) expresses minimal requirements on the functions defined on thestandard basis {vS : S ⊆ [n]} of Σn. Condition (iii) expresses thefact that fvS is meaningful with respect to interval scales.(b) The characterization given in Theorem 3.3 does not use the factthat v(∅) = 0. Therefore it can be immediately adapted to Lovászextensions by redefining Σn as the set of set functions on [n].References[1] G. Choquet. Theory of capacities. Ann. Inst. Fourier, Grenoble, 5:131–295 (1955),1953–1954.[2] L. M. de Campos and M. J. Bolaños. Characterization and comparison of Sugeno andChoquet integrals. Fuzzy Sets and Systems, 52(1):61–67, 1992.[3] A. De Waegenaere and P. Wakker. Nonmonotonic Choquet integrals. J. MathematicalEconomics, 36:45–60, 2001.[4] L. Lovász. Submodular functions and convexity. In Mathematical programming, 11thint. Symp., Bonn 1982, 235–257. 1983.[5] J.-L. Marichal. An axiomatic approach of the discrete Choquet integral as a tool toaggregate interacting criteria. IEEE Trans. Fuzzy Syst., 8(6):800–807, 2000.[6] J.-L. Marichal. Aggregation of interacting criteria by means of the discrete Choquet in-tegral. In Aggregation operators: new trends and applications, pages 224–244. Physica,Heidelberg, 2002.[7] T. Murofushi, M. Sugeno, and M. Machida. Non-monotonic fuzzy meansures and theChoquet integral. Fuzzy Sets and Systems, 64:73–86, 1994.[8] D. Schmeidler. Integral representation without additivity. Proc. Amer. Math. Soc.,97(2):255–261, 1986.[9] I. Singer. Extensions of functions of 0-1 variables and applications to combinatorialoptimization. Numer. Funct. Anal. Optimization, 7:23–62, 1984.[10] M. Sugeno. Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Instituteof Technology, Tokyo, 1974.[11] M. Sugeno. Fuzzy measures and fuzzy integrals—a survey. In M. M. Gupta, G. N.Saridis, and B. R. Gaines, editors, Fuzzy automata and decision processes, pages 89–102. North-Holland, New York, 1977.AXIOMATIZATIONS OF SIGNED DISCRETE CHOQUET INTEGRALS 7Department of Applied Mathematics, University Ca’ Foscari of Venice,Dorsoduro 3825/E–30123, Venice, ItalyE-mail address: mcardin[at]unive.itMathematics Research Unit, FSTC, University of Luxembourg, 6, rue Coudenhove-Kalergi, L-1359 Luxembourg, LuxembourgE-mail address: miguel.couceiro[at]uni.luDepartment of Applied Mathematics, University Ca’ Foscari of Venice,Dorsoduro 3825/E–30123, Venice, ItalyE-mail address: sgiove[at]unive.itMathematics Research Unit, FSTC, University of Luxembourg, 6, rue Coudenhove-Kalergi, L-1359 Luxembourg, LuxembourgE-mail address: jean-luc.marichal[at]uni.lu

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CARDIN, Marta

GIOVE, Silvio

Marichal J. L.

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MARTA CARDIN

MIGUEL COUCEIRO

SILVIO GIOVE

JEAN-LUC MARICHAL

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AXIOMATIZATIONS OF SIGNED DISCRETE CHOQUET INTEGRALS

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Axiomatizations of signed discrete Choquet integrals

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