HAL Id: hal-02545648
https://hal.archives-ouvertes.fr/hal-02545648
Submitted on 17 Apr 2020
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
Bi-capacities
Michel Grabisch, Christophe Labreuche
To cite this version:
Michel Grabisch, Christophe Labreuche. Bi-capacities. [Research Report] lip6.2003.002, LIP6. 2003.
�hal-02545648�
Michel GRABISCH LIP6, UPMC
4, Place Jussieu, 75252 Paris, France email [email protected]
Christophe LABREUCHE Thales Research & Technology
Domaine de Corbeville, 91404 Orsay Cedex, France email Christophe.Labreuche@tha lesg rou p.co m
Abstract
We introducebi-capacities asa natural generalizationof capacities (or fuzzy mea-sures)throughtheidentityof Choquetintegral ofbinary alternativeswithfuzzy mea-sures. We examine the underlyingstructure and derive the Mobius transform of bi-capacities. Next,theChoquet andSugeno integralsw.r.tbi-capacities areintroduced. It is shown that symmetric and asymmetric integrals are recovered. Lastly, we intro-ducetheShapleyvalue andinteractionindices. Itis seenthatbesidesageneralization basedon theclassicalde nitions,a de nitioninvolvingtwo argumentsisnatural.
Keywords|bi-capacity,Choquet integral,Mobius transform, Shapley value, interaction index.
1 Introduction
Capacities [1](orfuzzymeasures [15],non-additivemeasures [3]), and integrals w.r.t capac-ities such as the Choquet integral [1] and the Sugeno integral [15], have become a central tool in decisionmaking, extendinge.g. expected utility models and linear models of multi-attribute utilitytheory. Whilethe so-calledChoquet expected utility models areamong the mostgeneralmodelsforpreference representation whenutilityfunctionsare positive,the in-troductionofnegativequantities forutilityfunctionsmakespossibleseveralextensionof the usual Choquet expected utility model. Up to now, the most general extension iscalled the CPTmodel(CumulativeProspect Theory[16]), andisadi erenceoftwoChoquetintegrals.
Morespeci cally,letusdenoteby 1
; 2
twocapacitiesona niteuniverseN ofnelements. For any real-valued function onN, the CPT modelis expressed as:
CPT 1 ; 2 (f):=C 1 (f + ) C 2 (f )
where C
i
is the Choquet integral with respect to the capacity i , and f := f _0, f = ( f) + . We denote by 1 A
the characteristic function of A, for any A N. We call these functionsbinaryfunctions,asthey takeonlyvalues0and1. ItisknownthatforanyAN, anycapacity,wehaveC
(1
A
)=(A). Hencethecapacityisuniquelydeterminedbygiving the integral (or expected utility) of all binary functions. Let us consider ternary functions, i.e. functionsvaluedonf 1;0;1g,whichweexpressundertheform(1
A ; 1 B ),forA;B N, A\B =;. Observe that CPT 1 ; 2 (1 A ; 1 B )= 1 (A) 2 (B);
which entails that, if 1
; 2
are given, we have no freedom for determining the utility of a ternary alternative (function), since this value is determined from the utility of two binary alternatives.
In order toget ridof this limitation,we introduce bi-capacities as the value assignedto a ternary function.
The report presents rst results on bi-capacities, their structure and machinery. We assume basic knowledge on capacities and Choquet integral (see e.g. [9] for background). To avoid heavy notations, we will often omit braces and commas to denote sets. Also, the cardinalof a set is denoted by the corresponding small letter,e.g. jNj=n.
2 Bi-capacities
The de nition isgiven through the following: anact withscores equalto1 onAN,to-1 on B N has an overall utility denoted v(A;B). By convention, v(;;;) = 0, v(N;;) = 1 and v(;;N)= 1. Due tobasic assumptions indecision making (dominance),we have
v(A;B)v(C;B) if AC; v(A;B)v(A;D) if B D:
Thisleadstothe followingde nition. WedenoteQ(N):=f(A;B)2P(N)P(N)jA\B = ;g.
De nition 1 A function v :Q(N) ! R is a bi-capacity if it satis es:
(i) v(;;;)=0
(ii) AB implies v(A;)v(B;) and v(;A)v(;B).
In addition, v is normalized if v(N;;)=1= v(;;N).
In the sequel, unless otherwise speci ed, we willconsider that bi-capacities are normalized. Notethat the de nition impliesthat v(;;)0 and v(;;)0.
An interesting particular case is when leftand right part can beseparated. We say that abi-capacityisoftheCPT type (referingtoCumulativeProspectTheory[16])if thereexists two (normalized) capacities
1 ; 2 such that v(A;B)= 1 (A) 2 (B);8(A;B)2Q(N):
1 2 2 1 By analogy with the classical case, a bi-capacity is said to be additive if it satis es for all(A;B)2Q(N): v(A;B)= X i2A v(i;;)+ X i2B v(;;i): (1)
They are de ned by a \positive" distributionand a \negative" distribution,and are clearly of the CPT type. More generally, decomposable bi-capacities can be de ned as well, using t-conorms. However, the problem arises of combining positiveand negative quantities. For a given t-conorm S, it is known that extensions of S on [ 1;1] so that the structure is a group can be built onlyif S is astrict t-conorm [6][...detail if necessary...].
De nition 2 Let S be a strict t-conorm. A bi-capacity is said to be S-decomposable if it satis es: v(A;B)=(S i2A v(i;;)) S (S i2B ( v(;;i))) where S
is the S-di erence (see [6]).
However the case of maximum remains interesting since despite its extension fails to build a group only due to a lack of associativity, associativity holds when negative and positive quantities are merged separately[4, 5].
De nition 3 A bi-capacity is saidto be max-decomposable or to be a bi-measure of possi-bility if v(A;B)= 6 i2A v(i;;) 6 6 i2B v(;;i) ;
where 6 is the symmetric maximum [4] de ned by
a6b := 8 > > < > > :
(jaj_jbj) if b6= a and either jaj_jbj= a or = b
0 if b= a jaj_jbj else.
To avoid heavy notations, we will often omit braces and commas to denote sets. For example, fig;fi;jg;f1;2;3gare denoted respectively i;ij;123.
3 The structure of Q(N)
We study in this section the structure of Q(N). From its de nition, Q(N) is ismorphicto the set of mappings from N to f 1;0;1g, hence jQ(N)j=3
n
. Also,any element (A;B) in Q(N) can be denoted (x 1 ;:::;x n ), with x i 2 f 1;0;1g, with x i = 1 if i 2 A, x i = 1 if i2B, and 0 otherwise.
As a preliminary remark, Q(N) is a subset of P(N) 2
, and so can be represented in a matrix form, using some total order on P(N). A natural order is the binary order, ob-tained by ordering in an increasing sequence the integers coding the elements of P(N):
generating pattern
Wegivebelowthe matrix obtained with n=3.
; 1 2 12 3 13 23 123 ; 1 2 12 3 13 23 123 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5
As for P(N), it is convenient to de ne a total order on Q(N), so as to reveal structures. A natural one is to use a ternary coding. Several are possible, but it seems that the most suitable one is to code (considering elements of Q(N) asmappings valued onf 1;0;1g)-1 by 0, 0 by 1, and 1 by 2. The increasing sequence of integers in ternary code is 0, 1, 2, 10, 11,12, 20etc., whichleads to the following order of elements of Q(N):
(2;3)(12;3) (;;12)(;;2)(1;2) (;;1) (;;;) (1;;) (2;1)(2;;)(12;;) (3;12)(3;2)
Again, weremark afractalstructure,whichisenhanced by boxes: the (k+1)thbox isbuilt form the kth box by adding to its elements (of Q(N)) element k of N, either to their left part, orto their rightpart.
Itiseasytosee thatQ(N)isalattice,whenequippedwiththe followingorder: (A;B) (C;D)if A C and B D. Supremum and in mum are respectively
(A;B)_(C;D)=(A[C;B\D) (A;B)^(C;D)=(A\C;B[D):
TheseareelementsofQ(N)since(A[C)\(B\D)=;and(A\C)\(B[D)=;. Topand bottom are respectively (N;;) and (;;N). Notice that a bi-capacity is anorder-preserving mappingfromQ(N)toR. Wecallvertices ofQ(N)anyelement(A;B)suchthatA[B =N, since they coincidewiththe verticesof [0;1]
n
. Wegivein gures1 and2 theHasse diagram of (Q(N);)for n =2and n =3.
Q(N)isinfactthe lattice3 n . Itisformedby2 n Booleansub-lattices2 n : eachsub-lattice corresponds to a given partition of N into two parts, one for positive scores, the other for negativeones, whichcontain allsubsets of non-zero scores, includingthe empty set. Hence, allthese sub-lattices have asa commonpoint (;;;).
For any ordered pair ((A;B);(A[D;BnC)) of Q(N) with C B and D (N n(A[ B))[C the interval[(A;B);(A[D;BnC)]isasub-lattice oftype2
k 3
l
@ @ @ @ @ @ @ @ b b b b @ @ @ @ @ @ @ @ b b b b @ @ @ @ @ @ @ @ b b b b @ @ @ @ @ @ @ @ b b b b 1;; 2;; ;;; ;;1 ;;2 ;;12 1;2 2;1 r r r r
Figure1: The latticeQ(N)for n =2
and l=jC\Dj. Asa particularcase, a sub-latticeof type2 k
isobtained if C\D=;, and of type 3
l
if C =D. We use the notation
"(A;B):=f(C;D)2Q(N)j(C;D)(A;B)g
todenote the up-set of (A;B).
Let us remark that the nodes of Q(N) appear in a rather unnatural way. It is possible to have a more natural structure if we replace each node (A;B) by (A;B
c ). Let us call (Q (N);
) this new lattice. A node (A;B) inQ
(N)is such that AB, and A isthe set of scores equal to 1,while B is the set of scoresbeing equalto0 or 1. Wehave
(A;B)
(C;D)if and only if AC and B D (A;B)_ (C;D)=(A[C;B[D) (A;B)^ (C;D)=(A\C;B\D): Hence,
is simply the product order on P(N) 2
. Figures3 and 4 showthe Hasse diagram of (Q
(N);) for n=2and n =3.
LetusgivesomepropertiesofQ(N)(theyarethesameforQ
(N)). Since3 n
isaproduct ofdistributivelattices,itisitselfdistributive(see e.g. [2]). Howeveritisnotcomplemented, since for example (;;;) has no complement (b is the complement of a if a ^b = ? and a_b=>). Hence, (Q(N);)isnot aBooleanlattice,andsoisnotisomorphic tothe power set of some set.
Nevertheless, it is possible to give a simpler representation of Q(N). We recall some de nitions and a fundamentalresult of lattice theory [2].
De nition 4 Let (L;) be a lattice. An element x 2 L is _-irreducible if x 6= ? and x=a_b implies x=a or x=b, 8a;b2 L.
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c c Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c c Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c c Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c c Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c c Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c c Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c c 12;3 1;3 1;23 12;; 1;; 1;2 13;; 13;2 2;3 ;;3 ;;23 2;; ;;; ;;2 23;; 3;; 3;2 2;13 ;;13 ;;123 2;1 ;;1 ;;12 23;1 3;1 3;12 s s s s s s
Figure2: The latticeQ(N)for n =3
De nition 5 Let (P;) be a partially ordered set. Q P is a downset of P if x 2Q and yx imply y 2Q.
Proposition 1 Every nite distributivelattice (L;) isisomorphictoO(J), whereJ isthe set of the _-irreducible elements of L, and O(J) is the set of all down-sets of J.
It is easyto see that the _-irreducible elements of Q(N) are (;;i c
) and (i;i c
),for alli2N. Wehave for any (A;B)2Q(N),
(A;B)= _ i2A (i;i c )_ _ j2NnB (;;j c ): In (Q (N);
), the irreducible elements are (;;i) and (i;i), 8i 2 N. On gures 1 to 4, _-irreducible elementsare indicated by black circles.
This permits to de ne layers in Q(N) as follows: (;;N) is the bottom layer (layer 0), the set of all_-irreducible elements form layer 1, and layer k, for k = 2;:::;n, contains all elementswhichcanbewrittenasthesupremum overexactlyk _-irreducibleelements. Layer k is denoted Q
[k]
(N),and contains allelements(A;B)such that jBj=n k. The toplayer (layern) is reduced to(N;;).
@ @ @ @ @ @ @ @ b b b b @ @ @ @ @ @ @ @ b b b b @ @ @ @ @ @ @ @ b b b b @ @ @ @ @ @ @ @ b b b b 1;12 2;12 ;;12 ;;2 ;;1 ;;; 1;1 2;2 r r r r
Figure 3: The lattice Q
(N) for n=2
4 Mobius transform of bi-capacities
Let us recall some basic facts about the Mobius transform (see [13]). Let us consider f;g two real-valuedfunctions ona locally nite poset (X;)such that
g(x)= X
yx
f(y): (2)
The solution of this equationin term of g isgiven through the Mobius function by
f(x)= X
yx
(y;x)g(y) (3)
where is de ned inductivelyby
(x;y)= 8 < : 1; if x=y P xt<y (x;t); if x<y 0; otherwise:
Notethat depends onlyonthe structure of(X;). When (X;)isaBooleanlattice, e.g. (P(N);), itis well known that the Mobius functionbecomes,for any A;B 2P(N)
(A;B)= ( 1) jBnAj if A B 0; otherwise: (4)
Observe that this Mobius function has the followingproperty X
ACB
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c c Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c c Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c c Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c c Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c c Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c c Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c c c c c c c c 12;12 1;12 1;1 12;123 1;123 1;13 13;123 13;13 2;12 ;;12 ;;1 2;123 ;;123 ;;13 23;123 3;123 3;13 2;2 ;;2 ;;; 2;23 ;;23 ;;3 23;23 3;23 3;3 s s s s s s
Figure 4: The lattice Q (N) for n=3 Indeed, when A(B X ACB (A;C)= X ACB ( 1) jCnAj = jBnAj X k=0 jB nAj k ( 1) k =(1 1) jBnAj =0:
Ifg isacapacity,whichwe denoteby v,thenf inEq. (2)iscalledthe Mobius transform of v, usually denoted by m orm
v
if necessary. Equations (2) and (3) write
v(A)= X BA m(B) (6) m(A)= X BA ( 1) jBnAj v(B): (7)
The Mobius transform is an important concept for capacities, as it can be viewed as the coordinates of v in the basis of unanimity games, de ned by
u B (A)= 1; if AB 0; otherwise. Then Eq. (6)can be rewritten as
v(A)= X BN m(B)u B (A):
Mobius function onQ(N).
Theorem 1 The Mobius function on Q(N) is givenby, for any (A;A 0 );(B;B 0 )2Q(N) ((A;A 0 );(B;B 0 ))= ( 1) jBnAj+jA 0 nB 0 j ; if (A;A 0 )(B;B 0 ) and A 0 \B =; 0; otherwise.
Proof: We use the fact that if P;Q are posets, then the Mobius function onP Q with the product orderis the product ofthe Mobius functions onP and Q [13]. In our case, this gives 3 n((x 1 ;y 1 );:::;(x n ;y n ))= n Y i=1 3 (x i ;y i ) where 3 n
is the Mobiusfunction onQ(N)=3 n
, 3
the Mobiusfunction on3:=f 1;0;1g, and (x 1 ;:::;x n );(y 1 ;:::;y n ) 2 f 1;0;1g n correspond to (A;A 0 );(B;B 0 ) respectively. It is easy tosee that
3 (x i ;y i )= 8 > < > : 1; if x i =y i 1; if x i =y i 1 0; otherwise. Then 3 n((x 1 ;y 1 );:::;(x n ;y n
)) = 0 i there is some i 2 N such that 3 (x i ;y i ) = 0. This conditions readsx i >y i orx i = 1;y i
=1. In termof subsets, thismeans (A;A 0 )6(B;B 0 ) orB \A 0 6=;. We have 3 n ((x 1 ;y 1 );:::;(x n ;y n
))=1 i there is noi2N such that 3 (x i ;y i )=0, and the number of i 2 N such that
3 (x
i ;y
i
)= 1 is even. We examine the second condition. Wehave: 3 (x i ;y i )= 1, 8 > < > : x i =0 and y i =1 or x i = 1 and y i =0 which in terms of subsets, reads (jB nAj = 1 and jA
0 n B 0 j = 0) or (jB n Aj = 0 and jA 0 nB 0
j= 1). Then clearly the second condition is equivalent to jB nAj+jA 0 nB 0 j = 2k. The case 3 n ((x 1 ;y 1 );:::;(x n ;y n ))= 1 works similarly.
Consequently, the Mobius transformof v isexpressed by
m(A;A 0 )= X (B;B 0 )(A;A 0 ) B 0 \A=; ( 1) jAnBj+jB 0 nA 0 j v(B;B 0 )= X BA A 0 B 0 A c ( 1) jAnBj+jB 0 nA 0 j v(B;B 0 ): (8)
By de nition of the Mobius transform,we have
v(A;A 0 )= X (B;B 0 )(A;A 0 ) m(B;B 0 ): (9)
P
(A;B)2Q(N)
m(A;B)=v(N;;)=1.
Proceeding as in[8], we may write the Mobius transform into a matrix form, using the total order we have de ned on Q(N). Denoting v;m put in vector form as v
(n) ;m (n) , Eq. (8)can berewritten as m (n) =T (n) Æv (n) where Æ is the usual matrix product, and T
(n)
is the matrix coding the Mobius transform. As withthe case of classicalcapacities, T
(n)
has aninteresting fractalstructure, asitcan be seen from the case n =2 illustrated below.
T (2) = ;;12 ;;2 1;2 ;;1 ;;; 1;; 2;1 2;; 12;; ;;12 ;;2 1;2 ;;1 ;;; 1;; 2;1 2;; 12;; 2 6 6 6 6 6 6 6 6 6 6 6 6 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 7 7 7 7 7 7 7 7 7 7 7 7 5
The generating element has the form 2 4 1 1 1 1 1 3 5
and is the concatenation of two generating elements [ 1
11
] of the Mobius transform for classical capacities [8].
Let usexamine several particular cases of bi-capacities.
Proposition 2 Letv be abi-capacityof the CPTtype, withv(A;B)= 1
(A) 2
(B). Then its Mobius transform is given by:
m(A;A c )=m 1 (A); 8AN;A6=; m(;;B)=m 2 (B c ); 8B (N m(;;N)= 1
m(A;B)=0; 8(A;B)2Q(N) such that A6=; and B 6=A c
:
Proof: Let usconsider A6=;. Wehave
m(A;A 0 )= X A 0 B 0 A c ( 1) jB 0 nA 0 j " X BA ( 1) jAnBj v(B;B 0 ) # :
BA ( 1) jAnBj v(B;B 0 )= BA ( 1) jAnBj ( 1 (B) 2 (B 0 )) = X BA ( 1) jAnBj 1 (B) 2 (B 0 ) X BA ( 1) jAnBj = X BA ( 1) jAnBj 1 (B)=m 1 (A);
where we have used (5). Putting inm(A;A 0 ) leads to m(A;A 0 )=m 1 (A) X A 0 B 0 A c ( 1) jB 0 nA 0 j :
Usingagain (5), the sum iszero unless A 0
=A c
(onlyone term in the sum). Hence we get
m(A;A 0 )= ( m 1 (A); if A 0 =A c 0; otherwise :
Let usnow take A=;. We have:
m(;;A 0 )= X A 0 B 0 N ( 1) jB 0 nA 0 j v(;;B 0 ) = X B 0 A 0 ( 1) jB 0 nA 0 j 2 (B 0 ): Let us consider A 0
6= N, since in this case we know already that m(;;N)= 1. We recall that the co-Mobius transform[....] of a capacity is de ned by
m (A)= X BA c ( 1) jNnBj (B): Weremarkthatm(;;A 0 )=( 1) jNnAj+1 m 2 (A c
). Usingthefactthatm (A)=( 1) jAj+1 m (A) for any A6=; [...], we get nallym(;;A)=m
2
(A c
).
We get as immediate corollaries the expression of the Mobius transform of symmetric and asymmetricbi-capacities. Observe inparticularthatfor asymmetricbi-capacitiesv(A;B)= (A) (B), we have for any A6=N
m(;;A)=m
(A c
):
Applying the above result leads easilyto the followingone.
Proposition 3 Let v be an additive bi-capacity on Q(N). Then its Mobius transform is non nullonly forthe _-irreducible elements and the bottom of Q(N). Speci cally,
m(;;i c )= v(;;i);8i2N m(i;i c )=v(i;;);8i2N m(;;N)= 1:
De nition 6 A bi-capacity is said to be k-additive if its Mobius transform vanishes for all elements (A;B) in Q
[l ]
(N), for l=k+1;:::;n.
Equivalently, v is k-additive i m(A;B) = 0 whenever jBj < n k. Clearly, 1-additive bi-capacities coincidewith additivebi-capacities.
5 The Choquet and Sugeno integrals with respect to bi-capacities
Theexpressionofthe Choquet integralw.r.tabi-capacityhas beenintroducedaxiomatically in [12], see alsoa presentation based onsymmetry considerationsin [7]. For any functionf onN, we denote by f
i
the value f(i), i2N.
De nition 7 Let v be a bi-capacity and f be a real-valued function on N. The Choquet integralof f w.r.t v is given by C v (f):=C N + (jfj) where N
+ is a real-valuedset function on N de ned by
N + (C):=v(C \N + ;C\N ); and N + :=fi2Njf i 0g, N =N nN + .
Observe that wehave C v
(1 A
; 1 B
)=v(A;B)for any (A;B)2Q(N). C
v
(f)can berewritten as:
C v (f)= n X i=1 jf (i) j h v(A (i) \N + ;A (i) \N ) v(A (i+1) \N + ;A (i+1) \N i (10) where A (i)
:=f(i);:::;(n)g, and is apermutationon N sothat jf (1)
j jf (n)
j. The above formulais verysimilar tothe one proposed by Greco et al. [11].
The following result shows that our de nition encompasses the CPT model, and conse-quently the symmetric and asymmetricChoquet integrals.
Proposition 4 If v isof theCPT type, with v(A;B)= 1
(A) 2
(B), the Choquet integral reduces to C v (f)= n X i=1 f + (i) h 1 (A (i) \N + ) 1 (A (i+1) \N + )] n X i=1 f (i) h 2 (A (i) \N ) 2 (A (i+1) \N )] =C 1 (f + ) C 2 (f ):
Proof: Using the de nition of v, and splittingN in N ;N ,(10) becomes: C v (f)= X (i)2N + f (i) h 1 (A (i) \N + ) 2 (A (i) \N ) 1 (A (i+1) \N + )+ 2 (A (i+1) \N ) i X (i)2N f (i) h 1 (A (i) \N + ) 2 (A (i) \N ) 1 (A (i+1) \N + )+ 2 (A (i+1) \N ) i : If (i) 2 N + , then A (i) \N = A (i+1)
\N , and if (i) 2 N , we have A (i) \N + = A (i+1) \N +
. Substitutinginthe above expression leads tothe rst relation. Letusdenote by
+
; the permutations on N such that f +
;f become non decreasing. Observe that A + (i) =A (i) \N + and A (i) =A (i)
\N , which proves the second relation.
A similar construction can be done with the Sugeno integral. Let S
(f) denotes the Sugenointegralof f :N ![0;1] w.r.t. a capacity .
De nition 8 Let v be a bi-capacityand f :N ![ 1;1]. TheSugeno integral of f w.r.t v is de ned by S v (f):=S N + (jfj) with the same notations as above.
However, since N
+ can assume negativevalues, the usual de nition of Sugenointegral has tobereplaced by the followingone [5]
S (f):=h n 6 i=1 [f (i) 7(A (i) )]i
wheref is assumed tobenon negative, isany real-valuedset functionsuch that(;)=0, and isapermutationonN such thatf becomesnon decreasing. Also,6 isthesymmetric maximum, 7 the symmetricminimum de ned by
a7b:=
(jaj^jbj) if signa6=signb jaj^jbj else,
and for any sequence a 1 ;:::;a n in [ 1;1], the expression h n 6 i=1 a i i is a shorthand for ( n 6 i=1 a + i )6( n 6 i=1 a i ).
Then, the Sugeno integral forbi-capacities becomes
S v (f)=h n 6 i=1 h jf (i) j7v(A (i) \N + ;A (i) \N ) i i: (11)
Proposition 5 Letv beabi-capacitysatisfyingv(A;B)= 1
(A)6 ( 2
(B))forall(A;B)2 Q(N), with
1 ;
2
being two normalized capacities on N (v is called a _-CPT bi-capacity). Then the Sugeno integral reduces to
S v (f):=S 1 (f + )6( S 2 (f )):
1 2 v symmetric Sugenointegral [5].
Proof: DenotebyapermutationonN suchthatjfjisnon-decreasing. Sincef +
;f ; 1
; 2 are non negative, wehave asin the proof of Prop. 4
S 1 (f + )= n _ i=1 h f + (i) ^ 1 (A (i) \N + ) i S 2 (f )= n _ i=1 h f (i) ^ 2 (A (i) \N ) i :
Usingthe de nition of v,weget
S v (f)=h n 6 i=1 h jf (i) j7[ 1 (A (i) \N + )6( 2 (A (i) \N ))] i i:
Due to the de nition of hi, we have to show that if S
1 (f
+
) is larger (resp. smaller) than S
2
(f ), then the maximum of positive terms is equal to S
1 (f
+
) and is larger in absolute valuethanthemaximumofnegativeterms(resp. themaximumofabsolutevalueofnegative termsisequaltoS
2
(f )andislargerinabsolutevaluethanthemaximumofpositiveterms). Let usconsider (i)2N
+
. Two cases can happen.
if 1 (A (i) \N + )> 2 (A (i)
\N ),thenthecorrespondingtermreducestof + (i) 7 1 (A (i) \ N +
). This term is identical to the ith term inS 1
(f +
).
if not, the ith term in S v (f)reduces to f + (i) 7 2 (A (i) \N ). Due to monotonicity of 1
,this willbealsothecase forallsubsequentindices(i+1);:::(i+k),provided they belong toN
+
. Moreover, assuming(i+k+1)2N , we have
jf (i+k+1) j7 h 1 (A (i+k+1) \N + )6( 2 (A (i+k+1) \N )) i = jf (i+k+1) j7 2 (A (i+k+1) \N ) jf (j) j7 2 (A (j) \N | {z } A (i+k +1) \N ); 8j =i;:::;i+k:
Hence, inthe negativepart of S v
(f),the term in(i+k+1)remains,while allterms in (i);:::;(i+k) are cancelled, and it coincides with the term in S
2 (f ). On the other hand, in S 1 (f + )6( S 2
(f )), the term in (i) in S
1 (f
+
) is smaller than the term in (i+k+1) of S
2
(f ), so that the term in (i) cannot be the result of the computation,and thus it can be discarded fromS
1 (f
+ ).
A similar reasoning can be done with (i)2N .
C v (f)= X BN m(;;B) ^ i2B c \N f i + X (A;B)2Q(N) A6=; m(A;B) h ^ i2(A[B) c \N f i + ^ i2A f i _0 i
with the convention ^ ;
f i
:=0.
The proof is based onthe followingresult.
Lemma 1 For any real valued function f on N, the Choquet integral of f w.r.t. a bi-unanimity game u
(A;B)
(see (14)) is given by:
C u (A;B) (f)= 8 > > > > < > > > > : 0; if (A;B)=(;;N) ( V i2B c f i )^0; if (A;B)=(;;B) ( V i2A f i )_0; if (A;B)=(A;A c );A6=; ( V i2(A[B) c \N f i + V i2A f i )_0 ; otherwise: (12)
Proof: Using (10), weget:
C u (A;B) (f)= n X i=1 jf (i) j h u (A;B) (A (i) \N + ;A (i) \N ) u (A;B) (A (i+1) \N + ;A (i+1) \N ) i
with theabove de ned notations. Foragiven i2N, the di erenceintobrackets isnon zero onlyinthefollowingcases: the lefttermis0andthesecondoneisequalto1(case1
i ),orthe converse (case 2 i ). Case 1 i
happens if and only if A (i+1) \N + A and A (i+1) \N B, andeitherA (i) \N + 6AorA (i)
\N 6B. Observingthatthe 3dcondition cannotoccur if the 1st holds,this amounts to:
Case 1 i , 8 > < > : ( i ) (i)2B c \N ( i ) A (i) \N + A ( i ) A (i+1) \N B:
Proceeding similarlywith case 2 i , we get: Case 2 i , 8 > < > : ( 0 i ) (i)2A\N + ( i ) A (i) \N + A ( i ) A (i+1) \N B:
Weremark that only the rst conditions di er.
Let us suppose that (A;B) = (;;N). Then, for any i 2 N, neither case 1 i
nor case 2 i can happen, since conditions (
i ), (
0 i
) cannot hold. Hence C u
(;;N)
(f) =0. This proves the rst linein (12).
Let us suppose (A;B) = (;;B), B 6= N. Then for any i 2 N, case 2 i
can never occur since condition (
0 i
) is not ful lled, and condition ( i
) is always ful lled. If B c
there is at least one i such that both ( i
) and ( i
) are ful lled, namely (i) 2 B \N suchthat jf
(i)
jis maximum(this forces (i+1) tobe eitherin N + orinB). Hence we get C u (;;B) (f) = ^ i2B cf i under B c
\N 6= ;, and 0 otherwise. This proves the second line of (12).
Letussuppose (A;B)=(A;A c
). Ifcondition i
holdsforsome i,thenwe haveA N +
. Foranyi2N,case1
i
cannotoccursince(i)2A\N bycondition i ,sothatbycondition i , (i) 2 N +
, which contradicts condition i
. Any (i) 2 A ful lls conditions 0 i
and i
, while ful lling also condition
i
imposes to choose (i) 2 A such that f (i)
is minimum. Hence, underthese conditionsweget C
u (A;A c ) (f)=^ i2A f i
. Thisprovesthe thirdlineof(12). Let us consider the general case, with A 6= ;;B
c
. Let us suppose that case 1 i
occurs for some i 2 N. Then necessarily, A N
+ by condition i , (i) 2 (A[B) c \N 6= ; (conditions i , i
),and moreover(i)issuchthat jf (i) j ismaximum on(A[B) c \N and jf (i) j<^ j2A f (j) (13) (conditions i , i
). Under the assumption that case 1 i
holds, let usshow that case 2 j
must occur for some j. Since ; 6=A N
+
, condition 0 j
holds for some j. For any such j, j >i by (13). Now, condition
j
imposes to choose j such that f (j)
is minimum in A\N +
. j being such de ned, let us show that
j
holds. Suppose it is not the case. This means that thereexists j 0 suchthat(j 0 )2A (j+1) \N and(j 0 )62B. SinceAN +
,this meansthat (j 0 )2 (A[B) c \N . However, i <j <j 0 , so that jf (j 0 ) j >jf (i)
j, which contradicts the de nition of (i). The reverse result canbeshown aswell,sothat either both cases holdor both fail to hold. If both hold, C
u (A;B) (f)= ^ i2(A[B) c \N f i +^ i2A f i , which is 0 by (13).
This provesthe lastlineof (12).
Using the above lemma, the proof of Prop. 6 is immediate considering Eq. (15) and convention ^
; =0.
6 The Shapley value and interaction indices
We consider now bi-capacities as bi-cooperative games, as introduced by Bilbao et al. [...]. Their de nition coincide with our de nition of bi-capacities. An example of bi-cooperative gameisthe oneof ternaryvoting games asproposedbyFelsenthaland Machover [...],where the value of v is limited to f 1;1g. In ternary games, v(S;T) for any (S;T) 2 Q(N) is interpreted as the result of voting (+1: accepted, -1: rejected) when S is the set of voters voting in favor and T the set of voters voting against. N nS [T is the set of abstainers. For(general) bi-cooperativegames,one can keep the samekindof interpretation: v(S;T)is the worth of the coalition S when T is the opposite coalition, and N nS[T is the set of indi erent (indecise) players. [....to be modi ed....]
An importantconcept in gametheory isthe Shapley value [14] andother relatedindices (e.g. Banzhaf index, probabilistic values), as well as their generalizations as interaction indices.
A directtranspositionof the notionof unanimitygame leadstothe following. Let(S;S 0
)in Q(N). The bi-unanimity game centered on(S;S
0 ) isde ned by: u (S;S 0 ) (T;T 0 )= ( 1; if T S and T 0 S 0 0; otherwise: (14)
It is easyto see by (9) that the Mobius transform of u (S;S 0 ) is m u (S;S 0 ) (T;T 0 )= ( 1; if (T;T 0 )=(S;S 0 ) 0; otherwise :
Hence, as inthe classical case, the set of allbi-unanimitygames is abasis for bi-capacities:
v(T;T 0 )= X (S;S 0 )2Q(N) m(S;S 0 )u (S;S 0 ) (T;T 0 ): (15) Remark that u (S;S 0 )
is not a normalizedbi-capacity since u (S;S
0 )
(;;N)6= 1.
6.2 Derivatives of bi-capacities
Weextend the notion of derivative ofa set function to bi-cooperative games(in factto any functiononQ(N)). Asbi-cooperativegamesarede nedonQ(N),soshouldbethevariables used in derivation. For any i 2 N, the left-derivative with respect to i of v at point (S;T) is given by:
i;;
v(S;T):=v(S[i;T) v(S;T); 8(S;T)2Q(N ni): (16) Similarly,the right-derivative isgiven by:
;;i
v(S;T):=v(S;T) v(S;T [i); 8(S;T)2Q(N ni): (17)
The monotonicityof v entailsthat the leftderivativeisnon negative, whilethe rightderiv a-tiveis non positive. One can alsointroducethe derivative w.r.t. iby
i v(S;T):= i;; v(S;T)+ ;;i v(S;T)=v(S[i;T) v(S;T [i): (18)
Left and right derivatives permit to de ne in general the derivative with respect to any element (;;;)6=(S;T) inQ(N)by the recursiverelation:
S;T v(K;L):= i;; ( Sni;T v(K;L))= ;;i ( S;Tni v(K;L));8(K;L)2Q(N n(S[T)): (19)
Wehave for example
i;j v(K;L)=v(K[i;L) v(K[i;L[j) v(K;L)+v(K;L[j) ij;; v(K;L)=v(K[ij;L) v(K[i;L) v(K[j;L)+v(K;L):
S;T v(K;L)= X S 0 S T 0 T ( 1) (s s 0 ) t 0 v(K[S 0 ;L[T 0 ); 8(K;L)2Q(N n(S[T)): (20)
As before, weintroducethe derivativew.r.t. S forany SN
S v(K;L):= S;; v(K;L)+ ;;S v(K;L):
We express the derivative in terms of the Mobius transform. The starting point is the following.
Lemma 2 For any i2N and any (S;T)2Q(N ni),
i;; v(S;T)= X (S 0 ;T 0 )2[(i;i c );(S[i;T)] m(S 0 ;T 0 ) (21) ;;i v(S;T)= X (S 0 ;T 0 )2[(;;i c );(S;T)] m(S 0 ;T 0 ) (22) i v(S;T)= X (S 0 ;T 0 )2[(;;i c );(S[i;T)] m(S 0 ;T 0 ): (23)
Proof: Let usshow (21). Forany (S;T)2Q(N ni),
i;; v(S;T)=v(S[i;T) v(S;T) = X (S 0 ;T 0 )(S[i;T) m(S 0 ;T 0 ) X (S 0 ;T 0 )(S;T) m(S 0 ;T 0 ) = X (S 0 ;T 0 )(S;T) m(S 0 [i;T 0 ):
On the otherhand,
[(i;i c );(S[i;T)]=f(S 0 ;T 0 )2Q(N)ji2S 0 S[i;T T 0 i c g =f(S 0 [i;T 0 )2Q(N)jS 0 S;T 0 Tg
hence the result. Similarly,we have
;;i v(S;T)=v(S;T) v(S;T [i) = X (S 0 ;T 0 )(S;T) m(S 0 ;T 0 ) X (S 0 ;T 0 )(S;T[i) m(S 0 ;T 0 ) = X S 0 S;T 0 T;i62T 0 m(S 0 ;T 0 )= X (S 0 ;T 0 )2[(;;i c );(S;T)] m(S 0 ;T 0 ):
Eq. (23) follows immediatelysince [(i;i c
);(S[i;T)]\[(;;i c
);(S;T)]=;.
S;T v(K;L)= X (S 0 ;T 0 )2[ W i2S (i;i c )_ W j2T (;;j c );(S[K ;L)] m(S 0 ;T 0 ); 8(K;L)2Q(N n(S[T)) (24) S v(K;L)= X (S 0 ;T 0 )2[ W i2S (i;i c );(S[K ;L)] m(S 0 ;T 0 )+ X (S 0 ;T 0 )2[ W j2S (;;j c );(K ;L)] m(S 0 ;T 0 ) = X CK DNn(S[L) h m(S[C;L[D)+m(C;L[D) i ; 8(K;L)2Q(N nS): (25)
Proof: We prove(24) by induction over(S;T). The resultholds for(i;;)and (;;i)dueto Lemma 2. We suppose that the above formulaholds up to a given cardinality of S and T. Letuscompute
S[k;T
v(K;L),forsomek2Nn(S[T),andany(K;L)2Q(Nn(S[T[k)). As a preliminaryremark, note that
_ i2S (i;i c )_ _ j2T (;;j c )=(S;N n(S[T)): Wehave S[k;T v(K;L)= (k;;) ( S;T v(K;L))= S;T v(K[k;L) S;T v(K;L) = h X (S 0 ;T 0 )2[(S;Nn(S[T));(S[K[k;L)] m(S 0 ;T 0 ) X (S 0 ;T 0 )2[(S;Nn(S[T));(S[K ;L)] m(S 0 ;T 0 ) i = h X SS 0 S[K[k LT 0 Nn(S[T) m(S 0 ;T 0 ) X SS 0 S[K LT 0 Nn(S[T) m(S 0 ;T 0 ) i = X S[kS 0 S[K[k LT 0 Nn(S[T[k) m(S 0 ;T 0 ) = X (S 0 ;T 0 )2[(S[k;Nn(S[T[k);(S[K[k;L)] m(S 0 ;T 0 ) = X (S 0 ;T 0 )2[ W i2S[k (i;i c )_ W j2T (;;j c );(S[K[k;L)] m(S 0 ;T 0 )
which is the desired result. The case of S;T[k
v(K;L) works similarly. Lastly, Eq. (25)
comes directly from(24).
Remark that for any (S;T)2Q(N),
S;T v(;;N n(S[T))=m(S;N n(S[T)) m(S;T)= S;Nn(T[S) v(;;T):
The Shapley value for bi-cooperative games can be de ned axiomatically by introducing axioms which are straightforward extensions of the originalaxioms, plus anadditional sym-metry axiom [...detail if necessary...]. For any playeri 2N, itis shown in [...] that the Shapley value of i for v is:
v (i)= X SNni (n s 1)!s! n! h v(S[i;N n(S[i)) v(S;N nS) i : (26)
Notethatthetermintobracketsissimply i
v(S;Nn(S[i)), sothat theShapleyvalue uses the value of v onlyat vertices.
It is immediate tosee that if v isof the CPT type,i.e. v(S;T)= 1 (S) 2 (T),then v (i)= 1 (i)+ 2 (i); 8i2N; where 1 ; 2
are the(classical)Shapleyvalues of 1
and 2
. Recallingthat forany game,
(i) =
(i) for any i 2 N, we get v
(i)= 2
(i) for any symmetric or asymmetric game v. Ifv is anadditive bi-capacity, we have
v
(i)=v(i;;) v(;;i).
The following expression givesthe Shapley value interms of the Mobius transform.
Proposition 8 Let v be a bi-cooperative gameon N. For any i2N,
v (i)= X (S;S 0 )(;;i c ) 1 n s 0 m(S;S 0 ):
Weneed the following lemma.
Lemma 3 k X i=0 (n i 1)!k! n!(k i)! = 1 n k : Proof: k X i=0 (n i 1)!k! n!(k i)! = 1 n + k n(n 1) ++ k! n(n 1)(n k) = (n 1)(n k)+k(n 2)(n k)+k(k 1)(n 3)(n k)++k! n(n 1)(n k) :
k(k 1)2(n k)+k!=k2(n k+1) k3(n k+1)(n k)+k2(n k+1)=k3(n k+1)(n k+2) . . . ki(n k+i 2)(n k)+ k(i 1)(n k+i 2)(n k+1)=ki(n k+i 2) (n k+1)(n k+i 1) . . . k(n 2)(n k)+k(k 1)(n 2)(n k+1)=k(n 2)(n k+1)(n 1) (n 1)(n k)+k(n 1)(n k+1)=(n 1)(n k+1)n: We can show nowProp. 8.
Proof: We have by Lemma 2
i v(S;N n(S[i))= X (S 0 ;T 0 )2[(;;i c );(S[i;Nn(S[i))] m(S 0 ;T 0 ):
Observe that for S = N ni, the interval becomes " (;;i c
), which contains each interval [(;;i
c
);(S[i;N n(S[i))] when S N ni. Hence,
v (i)= X (S 0 ;T 0 )2"(;;i c ) m(S 0 ;T 0 ) X SNni S[iS 0 Nn(S[i)T 0 (n s 1)!s! n! :
In the second summation, condition S[i S 0
is redundant. Also,due tothe symmetry of the combinatorial factor, it is equivalent to use S or N n(S[i) as variable. So the second summation can be rewritten as
X ST 0 (n s 1)!s! n! = t 0 X s=0 (n s 1)!s! n! t 0 s = t 0 X s=0 (n s 1)!t 0 ! n!(t 0 s)! :
UsingLemma 3, the result isproven.
6.4 The interaction index
As in [10], the interaction index can be obtained from the Shapley value by a recursion formula. We rst introducenecessary notions. Letv beabi-cooperativegameonN,and let
;6=K N. The restricted game v is the game v restricted to players in N nK, hence v
NnK
(S;T) = v(S;T) for any (S;T) 2 Q(N nK), and is not de ned outside. The reduced game v
[K]
is the game where allplayers in K are considered as a single player denoted [K], i.e. the set of players is then N
[K]
:=(NnK)[f[K]g. The reduced game is de ned by
v [K] (S;T):=v( [K] (S); [K] (T)) for any (S;T)2Q(N [K] ),and [K] :N [K] !N is de ned by [K] (S):= ( S; if [K]62S (Sn[K])[K; otherwise. Let us denote by I v
(S) the interaction index for coalitionS 6= ; in game v. The recursion formulais [10] I v (S)=I v [S] ([S]) X KS;K6=;;S I v NnK (SnK):
It can be shown that the interaction index writes
I v (S)= X TNnS (n s t)!t! (n s+1)! " X LS ( 1) s l v(L[T;Nn(T [S)) X LS ( 1) s l v(T;N n(L[T)) # :
Observe that this may be writtenalso
I v (S)= X TNnS (n s t)!t! (n s+1)! S v(T;N n(S[T)):
Proposition 9 Let v be a bi-cooperative gameon N. For any S N,
I v (S)= X (S 0 ;T 0 ) W i2S (i;i c ) 1 n t 0 s+1 m(S 0 ;T 0 )+ X (S 0 ;T 0 )2Q(NnS) 1 n t 0 s+1 m(S 0 ;T 0 ):
Proof: We have by Prop. 7 for any T N nS
S v(T;Nn(S[T))= X (S 0 ;T 0 )2[(S;NnS);(S[T;Nn(S[T))] m(S 0 ;T 0 )+ X (S 0 ;T 0 )2[(;;NnS);(T;Nn(S[T))] m(S 0 ;T 0 ):
Letus study the rst term. Observe that for T =N nS,the intervalbecomes "(S;N nS), whichcontains allintervals [(S;N nS);(S[T;N n(S[T)]. Hence
X TNnS (n s t)!t! (n s+1)! X (S 0 ;T 0 )2[(S;NnS);(S[T;Nn(S[T))] m(S 0 ;T 0 ) = X (S 0 ;T 0 )2"(S;NnS) m(S 0 ;T 0 ) X TNnS S[TS 0 Nn(S[T)T 0 (n s t)!t! (n s+1)! :
Observe that inthe secondsummation, condition S[T S onT is redundant. Also, due tothe symmetryof thecombinatorialfactor, wecan interchange T withNn(S[T). Sothe second summationcan be rewritten as
X TT 0 (n s t)!t! (n s+1)! = t 0 X t=0 t 0 t (n s t)!t! (n s+1)! = t 0 X t=0 (n s t)!t 0 ! (t 0 t)!(n s+1)! = 1 n s t 0 +1 by using Lemma 3.
Inthesecondterm,forT =NnS,theintervalbecomes[(;;NnS);(NnS;;)]=Q(NnS).
The rest works similarly.
Note: the aboveproof willbe subsumedby the prooffor bi-interactionindices, more clearly written.
This expression shows that if v is k-additive, then I v
(S) =0 for any S of more than k elements, and for any S of exactly k elements,I
v
(S)=m(S;N nS)+m(;;N nS).
Proposition 10 If v is of the CPT type, with v(S;T)= 1 (S) 2 (T), then I v (S)=I 1 (S)+I 2 (S); where I i
is the (classical) interaction index of i . Since I i (S) = ( 1) s+1 I i (S), we have I v
(S)=0 whens is even and v isasymmetric.
Proof: Using Prop. 2 and expression of classicalinteraction w.r.t. the Mobius transforms of 1 and 2 ,we have I v (S)= X S 0 S 1 s 0 s+1 m(S 0 ;N nS 0 )+ X T 0 NnS 1 n t 0 s+1 m(;;T 0 ) = X S 0 S 1 s 0 s+1 m 1 (S 0 )+ X T 0 NnS 1 n t 0 s+1 m 2 (N nT 0 ) =I 1 (S)+ X T 00 NnS 1 t 00 s+1 m 2 (T 00 ) =I 1 (S)+I 2 (S):
Sincebi-cooperativegamesare de ned onQ(N),the interaction index shouldbede ned for all coalitions in Q(N). The most natural de nition seems to use the derivative, as for the classicalcase. We propose the following
De nition 9 Let (S;T)2Q(N). The bi-interaction index w.r.t (S;T) is de ned by:
I v (S;T):= X KNn(S[T) (n s t k)!k! (n s t+1)! S;T v(K;N n(K[S[T)):
A rst observation isthat for any S N
I v (S)=I v (S;;)+I v (;;S): Indeed, I v (S;;)= X KNnS (n s k)!k! (n s+1)! S;; v(K;N n(K[S)) I v (;;S)= X KNnS (n s k)!k! (n s+1)! ;;S v(K;N n(K[S)) and since S v = S;; v+ ;;S
v, the result holds. This givesimmediately
X i2N I v (i;;) X i2N I v (;;i)=2 (27) since P i2N v (i)=v(N;;) v(;;N).
Proposition 11 Let v be a bi-cooperative gameon N. For any (S;T)Q(N),
I v (S;T)= X (S 0 ;T 0 )2"( W i2S (i;i c )_ W j2T (;;j c ))\Q(NnT) 1 n s t t 0 +1 m(S 0 ;T 0 ) = X (S 0 ;T 0 )2[(S;Nn(S[T));(NnT;;)] 1 n s t t 0 +1 m(S 0 ;T 0 ):
Proof: By Prop. 7,we have
S;T v(K;Nn(K[S[T))= X (S 0 T 0 )2[(S;N n(S[T));(S[K;N n(K[S[T))] | {z } SS 0 S[K Nn(K[S[T)T 0 Nn(S[T) S 0 \T 0 =; m(S 0 ;T 0 ):
"( W i2S (i;i c )_ W j2T (;;j c
))\Q(NnT). Thisintervalcontains allintervals[(S;Nn(S[T));(S[
K;N n(K[S[T))] since S[K N nT. Hence, X KNn(S[T) (n s t k)!k! (n s t+1)! S;T v(K;N n(K[S[T)) = X (S 0 ;T 0 )2[(S;Nn(S[T));(NnT;;)] m(S 0 ;T 0 ) X KNn(S[T) S[KS 0 Nn(K[S[T)T 0 (n s t k)!k! (n s t+1)! :
Observe thatinthe secondsummation,condition S[K S 0
isredundant. Indeed,wehave N n(K[S[T)T 0 ,K[S[T NnT 0 . Since N nT 0 S 0 and T \S 0 =;, wededuce S[K S 0 .
Using this fact and lettingK 0
:=N n(K[S[T),the second summation becomes: X Nn(K[S[T)T 0 (n s t k)!k! (n s t+1)! = X K 0 T 0 k 0 !(n k 0 s t)! (n s t+1)! = t 0 X k 0 =0 t 0 k 0 k 0 !(n k 0 s t)! (n s t+1)! = t 0 X k 0 =0 t 0 !(n k 0 s t)! (n s t+1)!(t 0 k 0 )! = 1 n s t t 0 +1 using Lemma 3.
The above proposition contains Prop. 9 and Prop. 8,as itcan bechecked. We examinethe case of k-additive bi-capacitiesand CPT-type bi-capacities.
Proposition 12 (i) If v is a k-additive bi-capacity, then
I v (S;T)=0; 8(S;T)2Q(N) such that jS[Tj>k (28) I v (S;T)=m(S;N n(S[T)); 8(S;T)2Q(N) such that jS[Tj=k: (29)
(ii) If v isof CPT type, then I v S;T
=0 unless S=; or T =;.
Proof: (i) v isk-additivei m(S 0 ;T 0 )=0 for allT 0 suchthat t 0 <n k. Using Prop. 11, wesee that inthesummation, T
0 Nn(S[T). Consequently, ifjS[Tj>k, m(S 0 ;T 0 )will be always 0, and soI v (S;T)=0. Now, if jS[Tj =k, only T 0
=N n(S[T) gives a non zero term. For any T 0 , we have S 0 N nT 0
. Since we have alsothe condition S S 0
N nT, the only solutionis S 0
=S, hence the result.
(ii) ByProp. 2, we know that m(S;T )6=0i S =; or S =N nT . In the expression of I
v
(S;T) of Prop. 11, the rst condition implies S = ;, while the second implies T = ;.
References
[1] G. Choquet. Theory of capacities. Annales de l'Institut Fourier, 5:131{295, 1953.
[2] B.A. Davey and H.A. Priestley. Introduction to Lattices and Orders. Cambridge Uni-versity Press, 1990.
[3] D. Denneberg. Non-Additive Measure and Integral. KluwerAcademic, 1994.
[4] M. Grabisch. Symmetricand asymmetricfuzzy integrals: the ordinal case. In 6th Int. Conf. on Soft Computing (Iizuka'2000),Iizuka,Japan, October2000.
[5] M. Grabisch. The symmetricSugeno integral. Fuzzy Sets and Systems, toappear.
[6] M. Grabisch,B. De Baets, and J. Fodor. On symmetric pseudo-additions and pseudo-multiplications. isit possible tobuild rings on[-1,1]? In 9th Int. Conf. on Information Proc.andManagementofUncertaintyinKnowledge-Basedsystems(IPMU'2002),pages 1349{1355, Annecy, France,July 2002.
[7] M. Grabisch and Ch. Labreuche. Bi-capacities for decision making on bipolar scales. In EUROFUSE Workshop on Informations Systems, pages 185{190, Varenna, Italy, September 2002.
[8] M. Grabisch,J.L. Marichal, andM. Roubens. Equivalent representations of aset func-tion with applicationsto game theory and multicriteriadecisionmaking. Mathematics of Operations Research, 25(2):157{178,2000.
[9] M. Grabisch,T. Murofushi, and M.Sugeno. Fuzzy Measures andIntegrals. Theoryand Applications (edited volume). Studiesin Fuzziness. Physica Verlag,2000.
[10] M. Grabisch and M. Roubens. An axiomatic approach to the concept of interaction among players incooperative games. Int. Journal of Game Theory,28:547{565, 1999.
[11] S. Greco, B. Matarazzo, and R. S lowinski. BipolarSugeno and Choquet integrals. In EUROFUSE Workshop on Informations Systems, Varenna, Italy, September 2002.
[12] Ch. Labreuche and M. Grabisch. Generalized Choquet-like aggregation functions for handlingratio scales. Eur. J. of Operational Research,submitted.
[13] G.C. Rota. On the foundations of combinatorialtheory I. Theoryof Mobius functions. Zeitschrift f ur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 2:340{368, 1964.
Contributions to the Theory of Games, Vol. II, number 28 in Annals of Mathematics Studies, pages307{317. Princeton University Press, 1953.
[15] M. Sugeno. Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology, 1974.
[16] A.Tversky andD.Kahneman. Advances inprospecttheory: cumulativerepresentation of uncertainty. J. of Risk and Uncertainty,1992.
