Influence and interaction indexes for pseudo-Boolean functions: a unified least squares approach

INFLUENCE AND INTERA CTION INDEXES F OR PSEUDO-BOOLEAN FUNCTIONS: A UNIFIED LEAST SQ UARES APPR OA CH JEAN-LUC MARICHAL AND PIERRE MA THONET Abstract. The Banzhaf pow er and interaction indexes f or a pseudo-Boolean function (or a cooperative game) appear naturally as leading coefficien ts in the standard least squ ares approximation o f the function b y a pseudo-Boolean function of a specified degree. W e first observ e that this proper ty still holds if we consider approximations by pseudo-Boolean functions depending only on sp ecified v ariables. W e then sho w that the Banzhaf influenc e i ndex can also be obtained from the latter appro ximation problem. Considering cer- tain weigh ted ve r sions of this approximation problem, we introduce a class of we i gh ted Banzha f influence indexes, analyze their most im portant pr operties, and point out similar i ties betw een the w eighted Banzhaf influence index and the corresponding weigh ted Banzhaf inte r action index. W e also discuss the issue of reconstructing a pseudo-Bo olean function fr om prescrib ed influence s and point out very different behaviors in the weigh ted and non-we i gh ted cases. 1. I n troduction Let f ∶ { 0 , 1 } n → R b e an n -v aria ble pseudo-Bo o lean function and let S b e a subset of its v ar iables. Define the influenc e of S over f a s the exp ected v alue, denoted I f ( S ) , of the highest v aria tion of f when assigning v alues indep endent ly and uniformly at ra ndom to the v ariables not in S (see [12] for a norma lized version of this definition). That is, I f ( S ) = 1 2 n −∣ S ∣  T ⊆ N ∖ S  max R ⊆ S f ( T ∪ R ) − min R ⊆ S f ( T ∪ R ) , where N = { 1 , . . . , n } . 1 This notion was fir s t introduced for Bo o lean functions f ∶ { 0 , 1 } n → { 0 , 1 } by Ben-Or and Linial [2] (s e e a ls o [10]). There the influence I f ( S ) w a s (equiv alently) defined a s the probability that, assig ning v a lues indepen- dent ly and uniformly at random to the v ariables not in S , the v alue of f remains undetermined. Since its in tro duction, this concept ha s found man y applications in discrete mathematics, coop erative g ame t heo ry , theoretical computer science, and so cial c ho ice theory (see, e.g ., the survey a rticle [11]). Date : March 29, 2014. 2010 M athematics Subje ct Classific ation. Primary 91A12, 93E24; Secondary 39A70, 41A10. Key wor ds a nd phr ases. Coop erative game; pseudo-Boolean function; p ow er index; influence index; interaction index; least squares approximation. 1 Throughout we i dentify Bo ol ean v ectors x ∈ { 0 , 1 } n and subsets T ⊆ N by set ting x i = 1 if and only if i ∈ T . W e th us use the same symbol to denot e b oth a pseudo-Boolean function f ∶ { 0 , 1 } n → R and the corresponding set function f ∶ 2 N → R interc hangeably . 1 2 JEAN-LUC MARICHAL AND PIERRE M A THONET When the function f is nondecrea s ing in each v a riable, the form ula above r educes to (1) I f ( S ) = 1 2 n −∣ S ∣  T ⊆ N ∖ S  f ( T ∪ S ) − f ( T )  . The latter expression has an in tere s ting in terpreta tio n even if f is not nondecreas - ing. In coo per ative game theor y for ins tance, wher e f ( T ) r e pr esents the w or th of coalition T in the game f , this expressio n is precisely the average v alue of the marginal contributions f ( T ∪ S ) − f ( T ) of coa lition S to outer coalitions T ⊆ N  S . Thu s , it measures an ov era ll influence (which can be positive or negative) of co ali- tion S in the game f . In par ticular, when S = { i } is a singleton it reduces to the Banzhaf p ow er index I f ({ i }) = 1 2 n − 1  T ⊆ N ∖{ i }  f ( T ∪ { i } ) − f ( T )  . Thu s , the expre ssion in (1) ca n b e seen as a v ariant of the orig ina l concept of influence that simply extends the Banzhaf p ow er index to coalitions. W e ca ll it the Banzhaf influenc e index and denote it by Φ B ( f , S ) . Actually , this index was int r o duced, axiomatized, and even generalized to weigh ted versions in [13]. The Banzhaf inter action index [17], another index which extends the Banzhaf power index to coalitio ns, is defined for a pseudo-Bo olean function f ∶ { 0 , 1 } n → R and a subset S ⊆ N by (2) I B ( f , S ) = 1 2 n −∣ S ∣  T ⊆ N ∖ S ( ∆ S f )( T ) , where ∆ S f deno tes the S -difference (or discrete S -der iv a tive) o f f . 2 When  S   2, this index measures an o verall degree o f interaction amo ng the v ariables of f that are in S . When f is a game, it mea s ures an ov era ll degree o f interaction among the play ers of coalition S in the game f (see, e.g., [5, 6, 7]). It is known that the Banzhaf p ow er and in tera ction indexes can b e obtained fro m the solution of a s tandard leas t squar es approximation pro blem for pseudo- Bo olean functions (see [6, 8]). W eighted versions of this approximation pro blem recently enabled us to define a class o f weighted Banzhaf in tera ction indexes having several nice pr op erties (see [14]). How ever, w e observe that there is no such least squares construction for the Banzha f influence index in the literature. In this paper we fill this gap in the following w ay . In Section 2 we firs t show that the Banzhaf in teractio n index can b e obtained from a differ e nt, mor e natural (but still elementary) least squares approximation problem. Sp ecifically , I B ( f , S ) app ears as the leading coefficient in the m ultilinear repres ent a tion of the best ap- proximation f S of f by a pseudo-Bo olean function that de p ends only on the v ari- ables in S . W e then prov e that the Banzhaf influence index Φ B ( f , S ) can b e ob- tained from the same appr oximation problem simply b y considering the difference f S ( S ) − f S () . In Section 3 w e intro duce a class o f weigh ted Banzha f influence indexes from the solution o f a weigh ted version of this approximation problem. W e show that these indexes define a subclas s of the family of gener alize d values , give their most impo rtant prope r ties, and p oint out similar ities b etw een the weigh ted Banzhaf influence index and the cor resp onding w eig h ted Banzhaf int er action index. 2 The differences of f ar e defined as ∆ ∅ f = f , ∆ { i } f ( x ) = f ( x ∣ x i = 1 ) − f ( x ∣ x i = 0 ) , and ∆ S f = ∆ { i } ∆ S ∖{ i } f for i ∈ S . INFLUENCE AND INTERACT ION INDEXES 3 In Section 4 we discuss the issue of r epresenting pseudo-B o olean functions in terms of Banzha f influence indexes. More pr ecisely , we show that in the generic w eig ht ed case any pseudo-Bo olean function ca n b e reco nstructed, up to an additive constant, from prescribed influences. By con tra st, in the non-w eig h ted case only half of the information con tained in t he pseudo-Bo o lean function can be reconstructed. This impo rtant obser v atio n fully motiv ates the in vestigation o f the w eig ht ed case, whic h therefore is not a straightforw a rd extension of the non-weighted c a se. Finally , in Section 5 we present an application of the weigh ted Banz ha f influence index in system reliability theory and give a couple o f concluding remar ks. 2. I n teractions, influences, and least squares appro xima tions In this section w e reca ll how the Banzhaf int er action index ca n b e obtained from the so lution o f a standard least squar es appr oximation problem and w e sho w ho w a v ariant of this a pproximation problem ca n b e used to define b oth the Banzhaf int er action and influence indexes . It is w ell kno wn (see, e.g., [9]) that a ny pseudo-Bo olean function f ∶ { 0 , 1 } n → R can b e uniquely r epresented by a m ultilinear polynomial function f =  T ⊆ N a ( T ) u T , where u T ( x ) = ∏ i ∈ T x i is the unanimity game (or unanimity function ) for T ⊆ N (with the conven tion u ∅ = 1) and the set function a ∶ 2 N → R , called the M¨ obius tr ansform of f , is defined through the conv er sion formulas (M¨ obius in version for- m ula s) (3) a ( S ) =  T ⊆ S (− 1 ) ∣ S ∣−∣ T ∣ f ( T ) and f ( S ) =  T ⊆ S a ( T ) . By extending formally any pseudo-Bo olea n function f ∶ { 0 , 1 } n → R to the unit hypercub e [ 0 , 1 ] n by linear interpola tion, Owen [15, 16] introduced the multiline ar extension of f , i.e., the multilinear p o lynomial ¯ f ∶ [ 0 , 1 ] n → R defined b y ¯ f ( x ) =  S ⊆ N a ( S )  i ∈ S x i , where a is the M¨ o bius transform of f . Denote by F N the set o f pseudo- Bo olean functions on N (i.e., with v a r iables in N ). Recall that the Banzhaf inter action index [7, 17] is the mapping I B ∶ F N × 2 N → R defined in Eq. (2). Extending the S - difference op era tor ∆ S to multilinear po lynomials on [ 0 , 1 ] n , w e can sho w the fo llowing identities (see [6, 16]) I B ( f , S ) = ( ∆ S ¯ f ) 1 2  =  [ 0 , 1 ] n ∆ S ¯ f ( x ) d x , where 1 2 stands for  1 2 , . . . , 1 2  . Since the S -difference operato r has the same effect as the S - de r iv a tive op erato r D S (i.e., the par tia l deriv a tive op era to r with respect to the v ariables in S ) when applied to multilinear polyno mials on [ 0 , 1 ] n , w e a lso hav e (4) I B ( f , S ) = ( D S ¯ f ) 1 2  =  [ 0 , 1 ] n D S ¯ f ( x ) d x . W e now reca ll how the index I B can be obtained from a n a pproximation problem. F or k ∈ { 0 , . . . , n } define V k = span { u T ∶ T ⊆ N ,  T   k } , 4 JEAN-LUC MARICHAL AND PIERRE M A THONET that is, V k is the linear subspace of all multilinear polyno mials g ∶ { 0 , 1 } n → R of degree at most k , i.e., of the form g =  T ⊆ N ∣ T ∣⩽ k c ( T ) u T , c ( T ) ∈ R . The b est k th a pproximation of a function f ∶ { 0 , 1 } n → R is the function f k ∈ V k that minimizes the square d dista nc e (5)  x ∈{ 0 , 1 } n  f ( x ) − g ( x )  2 =  T ⊆ N  f ( T ) − g ( T )  2 among all functions g ∈ V k . The following pr o po sition, whic h w as pr ov ed in [6] (see [8] for an ear lier work), expresses the num b er I B ( f , S ) in terms of the best  S  th a pproximation f ∣ S ∣ of f . Prop ositio n 2.1 ([6]) . F or every f ∶ { 0 , 1 } n → R and every S ⊆ N , the nu mb er I B ( f , S ) is the c o efficient of u S in the multiline ar r epr esen t ation of t he b est  S  th appr oximation f ∣ S ∣ of f . An alternative (and perhaps more natural) a pproach to measur e the influence on f of its i th v ariable consists in considering the coefficient of u { i } in the best approximation of f by a function of the form g = c (  ) u ∅ + c ({ i } ) u { i } (instead of a function in V 1 ), as class ically done for linear mo dels in statistics . More generally , for every S ⊆ N define V S = { u T ∶ T ⊆ S } , that is, V S is the linear s ubspace of all mu ltilinea r p olynomia ls g ∶ { 0 , 1 } n → R that depend only on the v aria ble s in S , i.e., of the form g =  T ⊆ S c ( T ) u T , c ( T ) ∈ R . The b est S -appr oximation of a function f ∶ { 0 , 1 } n → R is then the function f S ∈ V S that minimizes the squa r ed distance (5) among all fu nctio ns g ∈ V S . W e no w sho w that I B ( f , S ) is a lso the co efficient of u S in the m ultilinear rep- resentation of f S . On the o ne hand, f S is the orthogonal pr o jection of f onto V S with resp ect to the inner pr o duct (6)  f , g  = 1 2 n  T ⊆ N f ( T ) g ( T ) . 3 On the other hand, it is w ell known and easy to prove that the 2 n functions v T ( x ) =  i ∈ T ( 2 x i − 1 ) , T ⊆ N , form an orthonormal set with resp ect to this inner pr o duct. Thu s , the bes t k th- and S -approximations o f f are respe c tiv ely g iven by (7) f k =  T ⊆ N ∣ T ∣⩽ k  f , v T  v T and f S =  T ⊆ S  f , v T  v T . These fo rmulas enable us to prove the following simple but impor tant result, which expresse s the num b er I B ( f , S ) in terms of the best S -a pproximation f S of f . 3 Note that the multiplicativ e normalization of the i nner pro duct does not cha nge the pro jection problem. INFLUENCE AND INTERACT ION INDEXES 5 Prop ositio n 2. 2. F or every f ∶ { 0 , 1 } n → R and every S ⊆ N , the numb er I B ( f , S ) is the c o efficient of u S (i.e., the le ading c o efficient) in the m ultiline ar r epr esentation of t he b est S -appr oximation f S of f . Pr o of. Since I B ( f , S ) is the co efficient of u S in the multilinear repre sentation of f ∣ S ∣ , from the firs t equality in (7 ) we obta in (8) I B ( f , S ) = 2 ∣ S ∣  f , v S  . W e th en conclude b y the sec o nd equality in (7).  Thu s , co m bining Pro po sition 2 .2 with Eq. (3), we immediately see that the nu mber I B ( f , S ) can b e expressed in terms of the approximation f S as I B ( f , S ) =  T ⊆ S ( − 1 ) ∣ S ∣−∣ T ∣ f S ( T ) . Recall that the Banzhaf influenc e index [13] is the mapping Φ B ∶ F N × 2 N → R defined by (9) Φ B ( f , S ) = 1 2 n −∣ S ∣  T ⊆ N ∖ S  f ( T ∪ S ) − f ( T )  . Since the map f  Φ B ( f , S ) is linear for every S ⊆ N , it can b e express ed by means of the inner pro duct (6 ). T o this aim, consider th e function g S ∶ { 0 , 1 } n → R defined by (10) g S ( x ) = 2 ∣ S ∣   i ∈ S x i −  i ∈ S ( 1 − x i ) . Prop ositio n 2.3. F or every f ∶ { 0 , 1 } n → R and every S ⊆ N , we have Φ B ( f , S ) =  f , g S  . Pr o of. Using (6), w e obtain  f , g S  = 1 2 n −∣ S ∣   T ⊇ S f ( T ) −  T ⊆ N ∖ S f ( T ) , which is precisely the righ t-hand side of (9).  F ro m Prop osition 2.3 w e can easily derive an explicit expressio n for Φ B ( f , S ) in terms of the Banzhaf int er action index I B . T his expression was already found in [12]. W e firs t consider a lemma. Lemma 2.4. F or ev ery S ⊆ N , we have g S = 2 ∑ T ⊆ S, ∣ T ∣ odd v T . Pr o of. Since the functions v T ( T ⊆ N ) for m a n orthonormal basis for F N , w e hav e g S = ∑ T ⊆ N  g S , v T  v T . Using (8), (10), a nd then (4 ), we o btain  g S , v T  = 2 −∣ T ∣ I B ( g S , T ) = 2 −∣ T ∣ ( D T ¯ g S ) 1 2  . The result then follows directly from the co mputation of the deriv ative D T ¯ g S .  Prop ositio n 2.5 ([12, P rop osition 4.1 ]) . F or every f ∶ { 0 , 1 } n → R and every S ⊆ N , we ha ve Φ B ( f , S ) =  T ⊆ S ∣ T ∣ o dd  1 2  ∣ T ∣− 1 I B ( f , T ) . 6 JEAN-LUC MARICHAL AND PIERRE M A THONET Pr o of. By Pr op osition 2.3 and Lemma 2 .4, w e o btain (11) Φ B ( f , S ) =  f , g S  = 2  T ⊆ S ∣ T ∣ o dd  f , v T  . W e th en conclude b y (8).  The follo wing propo s ition gives a n expr ession for Φ B ( f , S ) in terms of the be s t S -approximation f S of f . This pro po sition together with Pro p os ition 2.2 show that the indexes I B ( f , S ) and Φ B ( f , S ) are actually tw o face ts of the same cons tr uction, namely the b est S -a pproximation of f . Prop ositio n 2.6. F or every f ∶ { 0 , 1 } n → R and eve ry S ⊆ N , we have Φ B ( f , S ) = f S ( S ) − f S (  ) . Pr o of. By (7), w e hav e f S ( S ) − f S (  ) =  T ⊆ S  f , v T   v T ( S ) − v T (  )  =  T ⊆ S  f , v T   1 − ( − 1 ) ∣ T ∣  . Using (11), we see that the latter expr ession is precisely Φ B ( f , S ) .  Prop ositio n 2.6 is actually one of the key results of this pap er. Indeed, as w e will now see, it will enable us to define w eig h ted Banzhaf influence indexes from a weigh ted version of the a pproximation problem in complete analogy with the w ay the weigh ted Banzhaf interaction index was defined in [14]. 3. Weighted influences defined by least squares In [14] we inv estig ated weighted versions of the best k th approximation problem for pseudo-Bo ole an functions (e.g., to a llow nonuniform assignments of the v ari- ables). This study enabled us to define a class of weigh ted Banzhaf in tera ction indexes. In the present section we show that the corresp onding weighted v er s ion of the b est S -appr oximation pr oblem des crib ed in Sec tion 2 not only yields the same w eighted Banzhaf interaction index but also pro vides a na tural definition of a weigh ted Banzhaf influence index. Given a weigh t function w ∶ { 0 , 1 } n → ] 0 , ∞ [ and a pse udo -Bo olean function f ∶ { 0 , 1 } n → R , we define the b est S -appr oximation of f as the unique multilin- ear po lynomial in V S that minimizes the squared dista nce (12)  x ∈{ 0 , 1 } n w ( x )  f ( x ) − g ( x )  2 =  T ⊆ N w ( T )  f ( T ) − g ( T )  2 among all functions g ∈ V S . Assuming without loss of generality that ∑ T ⊆ N w ( T ) = 1, we see that w defines a probability distribution ov er 2 N . C o nsidering the game theory context, w e can int er pret w ( T ) as the probability that coalitio n T forms, that is, w ( T ) = P r ( C = T ) , where C represents a r andom coalition. W e also as sume that the v a riables are set indep endently of eac h other . In ga me theory , this means that the play er s b ehav e indep endently of ea ch other to for m coalitions, i.e., the even ts ( C ∋ i ) ( i ∈ N ) a re indep endent. 4 Setting p i = Pr ( C ∋ i ) = ∑ S ∋ i w ( S ) , we then hav e (13) w ( S ) =  i ∈ S p i  i ∈ N ∖ S ( 1 − p i ) , 4 In Section 5 we give a justification for this independence assumption. INFLUENCE AND INTERACT ION INDEXES 7 which implies 0 < p i < 1. Thus, the probabilit y distribution w is completely deter - mined by the n -tuple p = ( p 1 , . . . , p n ) ∈ ] 0 , 1 [ n . W e now provide an explicit expressio n for the b e st S -approximation of a pseudo- Bo olean function. O n the one hand, the squared distance (1 2) is induced by the weigh ted Euclidean inner pro duct  f , g  =  x ∈{ 0 , 1 } n w ( x ) f ( x ) g ( x ) . On the o ther hand, as observed in [3] the functions v T , p ∶ { 0 , 1 } n → R ( T ⊆ N ) defined by (14) v T , p ( x ) =  i ∈ T x i − p i  p i ( 1 − p i ) are pairwise orthogonal and nor malized. This provides the following immediate solution to the weigh ted appr oximation pro ble m. Prop ositio n 3.1. The b est S -appr oximation of f ∶ { 0 , 1 } n → R is giv en by (15) f S, p =  T ⊆ S  f , v T , p  v T , p . F ro m P rop osition 3.1 w e immediately deduce that the co efficient of u S (i.e., the leading co efficient) in the m ultilinear repr esentation of f S, p is given by (16) I B , p ( f , S ) =  f , v S, p  ∏ i ∈ S  p i ( 1 − p i ) , which is precisely the weigh ted Banzhaf interaction index intro duced in [14] by means of the corr esp onding k th approximation problem. In the non-w eig h ted case (i.e., when p = 1 2 ), Eq. (16) reduces to (8). By a nalogy with Prop ositio n 2.6 we now prop ose the following definition of weigh ted Banzhaf influence index. Definition 3.2. Le t Φ B , p ∶ F N × 2 N → R be defined as Φ B , p ( f , S ) = f S, p ( S ) − f S, p (  ) . W e no w provide v ar io us explic it expressions for Φ B , p ( f , S ) in ter ms o f the weigh ted Banzhaf interaction index, the M¨ obius transform of f , and the f v a lues. W e start with the follo wing r esult, which is the weighted co un ter pa rt of Pr o po - sition 2.5. Prop ositio n 3.3. F or every f ∶ { 0 , 1 } n → R and eve ry S ⊆ N , we have (17) Φ B , p ( f , S ) =  T ⊆ S I B , p ( f , T )   i ∈ T ( 1 − p i ) − ( − 1 ) ∣ T ∣  i ∈ T p i  . Pr o of. Using Definition 3 .2 and Eq s . (1 5) and (14), we obtain (18) Φ B , p ( f , S ) =  T ⊆ S  f , v T , p    i ∈ T 1 − p i  p i ( 1 − p i ) − ( − 1 ) ∣ T ∣  i ∈ T p i  p i ( 1 − p i )  . W e th en conclude b y (16).  Using the ex pr ession o f the weigh ted Banzhaf interaction index in terms of the M¨ obius transform of f , that is, (19) I B , p ( f , S ) =  T ⊇ S a ( T )  i ∈ T ∖ S p i 8 JEAN-LUC MARICHAL AND PIERRE M A THONET (see [14]), we can obtain the corres po nding expression for the w eig ht ed Banzhaf influence index. T o this extent, r ecall the binomial pro duct formula (20)  T ⊆ N  i ∈ T a i  i ∈ N ∖ T b i =  i ∈ N ( a i + b i ) . Prop ositio n 3.4. F or every f ∶ { 0 , 1 } n → R and eve ry S ⊆ N , we have (21) Φ B , p ( f , S ) =  T ⊆ N T ∩ S ≠∅ a ( T )  i ∈ T ∖ S p i . Pr o of. Combining (17) with (19), we obtain Φ B , p ( f , S ) =  R ⊆ S  T ⊇ R a ( T )  i ∈ T ∖ R p i   i ∈ R ( 1 − p i ) −  i ∈ R ( − p i ) =  T ⊆ N T ∩ S ≠∅ a ( T )  i ∈ T ∖ S p i  R ⊆ T ∩ S  i ∈( T ∩ S )∖ R p i   i ∈ R ( 1 − p i ) −  i ∈ R ( − p i ) . (22) Using the binomial pro duct formula (20), we see that the inner sum in (22) beco mes 1 − ∏ i ∈ T ∩ S ( p i − p i ) = 1 . This completes the pro of of the prop osition.  Int er estingly , Eqs . (19) and (21) show that b oth I B , p ( f , S ) a nd Φ B , p ( f , S ) are independent of those p i such that i ∈ S . A gener alize d value [13] is a ma pping G ∶ F N × 2 N → R defined b y (23) G ( f , S ) =  T ⊆ N ∖ S p S T ( f ( T ∪ S ) − f ( T )) , where the co efficients p S T are real num b ers for every S ⊆ N and ev er y T ⊆ N  S . The following lemma g ives an expres sion fo r G ( f , S ) in terms of the M¨ obius transform of f . The pro o f is given in App endix A . Lemma 3.5. A mapping G ∶ F N × 2 N → R of the f orm (24) G ( f , S ) =  R ⊆ N R ∩ S ≠∅ q S R a ( R ) , wher e a is the M¨ obius tr ansform of f , defines a gener alize d value if and only if the c o efficients q S R dep end only on S and R  S . In this c ase, the c onversion b etwe en (23) a n d (24) is given by q S R =  T ∶ R ∖ S ⊆ T ⊆ N ∖ S p S T and p S T =  R ∶ T ⊆ R ⊆ N ∖ S ( − 1 ) ∣ R ∣−∣ T ∣ q S R ∪ S . The following prop osition shows that the weigh ted Banzhaf influence index Φ B , p is a particular g eneralized v a lue. Prop ositio n 3.6. F or every f ∶ { 0 , 1 } n → R and eve ry S ⊆ N , we have Φ B , p ( f , S ) =  T ⊆ N ∖ S p S T  f ( T ∪ S ) − f ( T )  , wher e the c o efficients (25) p S T =  i ∈ T p i  i ∈ N ∖( S ∪ T ) ( 1 − p i ) satisfy t he c onditions p S T  0 and ∑ T ⊆ N ∖ S p S T = 1 . INFLUENCE AND INTERACT ION INDEXES 9 Pr o of. Pr op osition 3.4 and Lemma 3 .5 show that Φ B , p is a generalized v alue with q S R = ∏ i ∈ R ∖ S p i . By Lemma 3.5 we then hav e p S T =  R ∶ T ⊆ R ⊆ N ∖ S ( − 1 ) ∣ R ∣−∣ T ∣  i ∈ R p i =  i ∈ T p i  R ∶ T ⊆ R ⊆ N ∖ S  i ∈ R ∖ T ( − p i ) . The result then follows from the binomial pr o duct form ula (20).  The co efficients p S T given in (2 5) coincide with those o f the corres po nding expres- sion for the weighted Banzha f interaction index (see [1 4, Theorem 10]). Therefore, we immediately der ive the following in terpre ta tions of these co efficients (see [14, Prop ositio n 11]). F or e very S ⊆ N and every T ⊆ N  S , we hav e p S T = Pr ( T ⊆ C ⊆ S ∪ T ) = Pr ( C = S ∪ T  C ⊇ S ) = P r ( C = T  C ⊆ N  S ) , where C denotes a random coa lition. F or every S ⊆ N , define the linear ope r ator σ S for functions o n { 0 , 1 } n or [ 0 , 1 ] n by σ S f ( x ) = f ( x  x i = 1 ∀ i ∈ S ) − f ( x  x i = 0 ∀ i ∈ S ) . F or instance, when applied to the unanimity ga me u T ( T ⊆ N ), we obta in (26) σ S u T =        u T ∖ S , if S ∩ T ≠  , 0 , otherwise . The next result gives v arious expr essions for Φ B , p ( f , S ) in ter ms of the function σ S f . Recall first that, for every function f ∶ { 0 , 1 } n → R , w e have (27) ¯ f ( p ) =  x ∈{ 0 , 1 } n w ( x ) f ( x ) = E [ f ( C )] , where C denotes a random coa lition (see [16] or [14, Pr op osition 4]). Prop ositio n 3.7. F or every f ∶ { 0 , 1 } n → R and eve ry S ⊆ N , we have (28) Φ B , p ( f , S ) = ( σ S ¯ f )( p ) =  x ∈{ 0 , 1 } n w ( x ) σ S f ( x ) = E  ( σ S f )( C )  , wher e C denotes a r andom c o alition. Pr o of. The first equality immedia tely follows fro m Eqs. (21) and (26). The other equalities immediately follow from (27).  Int er estingly , (28) sho ws a strong a na logy with the ident ities (see [14, Prop osi- tions 4 and 9]) (29) I B , p ( f , S ) = ( D S ¯ f )( p ) =  x ∈{ 0 , 1 } n w ( x ) ∆ S f ( x ) = E  ( ∆ S f )( C )  . W e also hav e the following expre ssion for Φ B , p ( f , S ) as an integral. W e o mit the pro of since it follows exactly the same steps as in the pro of of the corr esp onding expression for I B , p ( f , S ) (see [1 4, Prop ositio n 1 2]). Prop ositio n 3.8. L et F 1 , . . . , F n b e cu mulative distribution functions on [ 0 , 1 ] . Then Φ B , p ( f , S ) =  [ 0 , 1 ] n ( σ S ¯ f )( x ) dF 1 ( x 1 )  dF n ( x n ) for every f ∶ { 0 , 1 } n → R and every S ⊆ N if and only if p i = ∫ 1 0 x dF i ( x ) for every i ∈ N . 10 JEAN-LUC MARICHAL AND PIERRE M A THONET W e now generalize Prop osition 2.3 to the w eighted cas e. T o this aim, co ns ider the function g S, p ∶ { 0 , 1 } n → R defined b y g S, p ( x ) =  i ∈ S x i p i −  i ∈ S 1 − x i 1 − p i . Prop ositio n 3.9. F or every f ∶ { 0 , 1 } n → R and eve ry S ⊆ N , we have (30) Φ B , p ( f , S ) =  f , g S, p  =  x ∈{ 0 , 1 } n w ( x ) f ( x )   i ∈ S x i p i −  i ∈ S 1 − x i 1 − p i  and (31) Φ B , p ( f , S ) =  x ∈{ 0 , 1 } n f ( x ) g S ( x ) 2 ∣ S ∣  i ∈ N ∖ S p x i i ( 1 − p i ) 1 − x i . Pr o of. On the one hand, by substituting (14) int o (18), we obtain Φ B , p ( f , S ) =  f , g ′ S, p  , where g ′ S, p ( x ) =  T ⊆ S   i ∈ T x i − p i p i − ( − 1 ) ∣ T ∣  i ∈ T x i − p i 1 − p i  . Using the binomial pr o duct formula (20), we immediately see that g ′ S, p = g S, p , which proves (30). On the o ther hand, for every x ∈ { 0 , 1 } n we hav e g S, p ( x ) w ( x ) = g S, p ( x )  i ∈ N p x i i ( 1 − p i ) 1 − x i = g S ( x ) 2 ∣ S ∣  i ∈ N ∖ S p x i i ( 1 − p i ) 1 − x i , which, when co mb ined with ( 3 0), immediately leads to (31).  W e end this section b y giving an in ter pretation of the Banzhaf influence index Φ B as a center of mass of weigh ted Banzhaf influence indexes Φ B , p . As already mentioned, th e index Φ B can b e expressed in terms of Φ B , p simply by s etting p = 1 2 . How ever, by Prop osition 3.6 we als o hav e the following expres s ion (32) Φ B ( f , S ) =  [ 0 , 1 ] n Φ B , p ( f , S ) d p . This formula ca n be in ter pr eted in the game theo ry context in the same w ay a s the corres p onding fo rmula for the in teraction index (see [14, § 5.1 ]). W e ha ve assumed that the play ers b ehav e independently of each other to form coalitions, e ach player i with pro bability p i ∈ ] 0 , 1 [ . Assuming further tha t this pr obability is not known a priori, to define an influence index it is then natural to consider the average (cen ter of mass) of the weigh ted indexes ov er all p ossible choices of the probabilities p i . Eq. (32) then shows that w e obtain the non-weigh ted influence index Φ B . The Shapley gener alize d value [12, 1 3] for a function f ∶ { 0 , 1 } n → R and a co alition S ⊆ N is defined b y Φ Sh ( f , S ) =  T ⊆ N T ∩ S ≠∅ a ( T )  T  S  + 1 , where a is the M¨ obius transform of f . Using (21) we obta in the following expression for Φ Sh in terms of Φ B , p , namely (33) Φ Sh ( f , S ) =  1 0 Φ B , ( p,...,p ) ( f , S ) dp . INFLUENCE AND INTERACT ION INDEXES 11 Here the play ers still b ehav e indep endently of ea ch o ther to form coalitions but with the s ame pro ba bilit y p . The in tegral in (3 3) s imply repres e nts the average o f the weigh ted indexes ov e r all the p os sible probabilities. 4. Weighted influences as al terna tive represent a tions of pseudo-Boolean functions It is w ell known that the v a lues I B ( f , S ) ( S ⊆ N ) of the no n-weigh ted Banzhaf int er action index for a function f ∶ { 0 , 1 } n → R provide an alternative representation of f (see [6]). This observ ation still holds in the w eighted case. Indeed, combining the T aylor expansio n formula with (29 ) yields (see [14, Eq. (16)]) (34) f ( x ) =  S ⊆ N I B , p ( f , S )  i ∈ S ( x i − p i ) . Thu s , for every p t he map f  { I B , p ( f , S ) ∶ S ⊆ N } is a linear bijection. In this section w e disc uss the issue o f re pr esenting pseudo-Bo olean functions in terms of Banzhaf influence indexes. In fact, we compare the non- w e ig ht ed and weigh ted v ers ions of the Banz ha f influence indexes and show that they hav e dif- ferent b ehaviors in terms of re construction of the o riginal pseudo -Bo olean function from pres c rib ed influences. In the non-weigh ted v ersio n w e sho w tha t the index is degenerate: roughly spea king, the v alues Φ B ( f , S ) (  ≠ S ⊆ N ) enco de o nly half o f the information contained in the function f . In con trast, in the weight e d v er sion, for a generic w eig ht p the v a lues Φ B , p ( f , S ) (  ≠ S ⊆ N ) allow to reconstruct f up to an additive constant. The degener a cy of the non-weigh ted influence index Φ B follows from linear re- lations among the linear functiona ls Φ B ( ⋅ , S ) ( S ⊆ N ) o n the space F N . F or instance, for every i , j ∈ N we have g { i,j } = g { i } + g { j } , which, by Prop osition 2.3, translates into Φ B ( ⋅ , { i, j }) = Φ B ( ⋅ , { i }) + Φ B ( ⋅ , { j }) , i , j ∈ N . The following r esult generalizes this linea r dep e ndence relatio n. Prop ositio n 4.1. F or every f ∶ { 0 , 1 } n → R and eve ry S ⊆ N , we have (35) I B , p ( f , S )  v S, p ( S ) − v S, p ( ∅ )   i ∈ S  p i ( 1 − p i ) =  T ⊆ S ( − 1 ) ∣ S ∣−∣ T ∣ Φ B , p ( f , T ) . Pr o of. Just apply the M¨ obius in version formula to (1 7).  F or mu la (35) shows that if p is s uch that ( v S, p ( S ) − v S, p ( ∅ )) = 0 for s ome S ∈ 2 N ∖ { ∅ } , the linea r functional Φ B , p ( ⋅ , S ) on the space F N is a linear combination of the functionals Φ B , p ( ⋅ , T ) for T  S . Mo r eov er, by definition we alw ays ha ve Φ B , p ( ⋅ , ∅ ) = 0. Therefore repla c ing a pseudo-Bo o lean function f with the v alues Φ B , p ( f , S ) ( S ⊆ N ) results in a loss of informatio n which dep ends on p . Assuming a total order on 2 N , we may regar d Φ B , p as the linear ma p Φ B , p ∶ F N → R 2 n defined by f  ( Φ B , p ( f , S ) ∶ S ⊆ N ) . W e can mea sure the degree of dependence among the functionals Φ B , p ( ⋅ , S ) ( S ⊆ N ) by computing the rank rk ( Φ B , p ) o f Φ B , p . Similarly , the resulting loss of infor mation corres p onds to the kernel k er ( Φ B , p ) of Φ B , p . 12 JEAN-LUC MARICHAL AND PIERRE M A THONET Prop ositio n 4.2. We have ker ( Φ B , p ) = span { v S, p ∶ S ⊆ N and v S, p ( S ) = v S, p ( ∅ )} and rk ( Φ B , p ) = 2 n − { S ⊆ N ∶ v S, p ( S ) = v S, p ( ∅ )} . Pr o of. Combining (14) with (18), we obtain Φ B , p ( v T , p , S ) =        0 , if T  S , v T , p ( T ) − v T , p ( ∅ ) , otherwise . Thu s , if v T , p ( T ) − v T , p ( ∅ ) = 0, then v T , p ∈ ker ( Φ B , p ) . F or the conv erse inclusion, take f ∈ F N . By (34), we ha ve f =  S ⊆ N I B , p ( f , S )  i ∈ S  p i ( 1 − p i ) v S, p . If f ∈ k er ( Φ B , p ) , then I B , p ( f , S ) ( v S, p ( S ) − v S, p ( ∅ )) = 0 for every S ⊆ N by (35). This provides the conv erse inclusion. The v alue o f rk ( Φ B , p ) immediately follows.  W e observ e that the c o ndition v S, p ( S ) = v S, p ( ∅ ) also rea ds (36)  i ∈ S ( 1 − p i ) = ( − 1 ) ∣ S ∣  i ∈ S p i . 5 Since w e hav e p ∈ ] 0 , 1 [ n , this co ndition cannot b e fulfilled when  S  is o dd. There- fore b y Pr op osition 4.2 the rank of Φ B , p ranges within the interv a l [ 2 n − 1 , 2 n − 1 ] . This motiv a tes the following definition. Definition 4.3. A tuple p ∈ ] 0 , 1 [ n is n onde gener ate if for every S ∈ 2 N ∖ { ∅ } we hav e v S, p ( S ) ≠ v S, p ( ∅ ) , i.e., if r k ( Φ B , p ) = 2 n − 1 . Otherwise, it is said to be de gener ate . A tuple p is m ax imal ly de gener ate if rk ( Φ B , p ) = 2 n − 1 . Prop ositio n 4.4. The set of n onde gener ate t uples is an op en dense su bset in ] 0 , 1 [ n . F or n  3 ther e is a unique maximal ly de gener ate tuple, namely p = 1 2 . Pr o of. F or S ≠ ∅ , Eq. (36) is a no n tr ivial p olynomial e quation on the comp onents of the tuple p . This prov es the first statement. T o see that the second statemen t holds w e note that p is maximally degenera te if Eq. (36) holds for every S such that  S  is even. In particular it m ust hold for S = { i, j } , so that p i + p j = 1 for a ll i, j ∈ N . This implies p = 1 2 whenever n  3. Finally , we can ea sily c heck that for this tuple we ha ve r k ( Φ B , p ) = 2 n − 1 .  In the following tw o subsections w e fur ther a nalyze both the maximally degen- erate and nondegener ate cases. 4.1. Behav i or of the non - w eighted Banzhaf i nfluence indexes. By Prop o- sition 4.4 the non-weigh ted Banzha f influence index Φ B is maximally degenerate. Let us now in terpret its k ernel. Definition 4.5. L e t ∗∶ F N → F N be the opera tor that carries f int o f ∗ defined by f ∗ ( S ) = − f ( N ∖ S ) . Set a lso S = { f ∈ F N ∶ f ∗ = f } and A = { f ∈ F N ∶ f ∗ = − f } . The spaces S and A c an be described in terms of the functions v S as follows. 5 Or equiv alently , ∏ i ∈ S ( 1 − 1 / p i ) = 1. INFLUENCE AND INTERACT ION INDEXES 13 Prop ositio n 4. 6. We have k er ( Φ B ) = A = span { v S ∶  S  even } and S = spa n { v S ∶  S  o dd } . The sp ac e F N is t he dir e ct sum of the ortho gonal subsp ac es S and A . F or every S ⊆ N , we have g S ∈ S . Final ly, { g S ∶  S  o dd } is a b asis of S . Pr o of. On the one hand, by Prop osition 4.2, we have ker ( Φ B ) = span { v S ∶  S  even } . On the other hand, we clearly ha ve v ∗ S = ( − 1 ) ∣ S ∣+ 1 v S for every S ⊆ N . Therefore we hav e (37) span { v S ∶  S  ev en } ⊆ A and span { v S ∶  S  o dd } ⊆ S . It follows that dim ( A )  2 n − 1 and dim ( S )  2 n − 1 . But since we hav e A ∩ S = { 0 } , we must ha ve dim ( A ) = dim ( S ) = 2 n − 1 and this proves the co n verse inclusions in (37). This descriptio n of A and S pr ov es the second ass e rtion. The last ass e rtions follow easily fro m Lemma 2.4.  Combining P rop osition 2.3 and Eq. (8) with Prop o s ition 4.6 s hows that the linea r functionals Φ B ( ⋅ , S ) with S ⊆ N a nd I B ( ⋅ , S ) with  S  o dd a re combinations of the functionals Φ B ( ⋅ , T ) with T ⊆ N and  T  o dd. These relatio ns are g iven explicitly in the next prop os ition. Let E n ( x ) denote the n th Euler p olynomial and E n = 2 n E n ( 1 2 ) the n th Euler nu mber . Prop ositio n 4.7. F or every f ∶ { 0 , 1 } n → R and eve ry S ⊆ N , we have (38) Φ B ( f , S ) = −  T ⊆ S ∣ T ∣ o dd E ∣ S ∣−∣ T ∣ ( 0 ) 2 ∣ S ∣−∣ T ∣ Φ B ( f , T ) , if  S  is even , and (39) I B ( f , S ) = 2 ∣ S ∣− 1  T ⊆ S ∣ T ∣ o dd E ∣ S ∣−∣ T ∣ Φ B ( f , T ) , if  S  is o dd . Pr o of. By Pr op osition 2.3 we ca n prov e (38) by sho wing that (40) g S = −  T ⊆ S ∣ T ∣ o dd E ∣ S ∣−∣ T ∣ ( 0 ) 2 ∣ S ∣−∣ T ∣ g T , if  S  is even , or equiv a lent ly (using the basic prop erties of Euler p oly no mials), (41)  T ⊆ S E ∣ S ∣−∣ T ∣ ( 0 ) 2 −∣ T ∣ g T = 0 . T o s e e that (41) holds , w e show that  T ⊆ S E ∣ S ∣−∣ T ∣ ( 0 ) 2 −∣ T ∣  g T , v K  = 0 , K ⊆ N . If  K  is even, then  g T , v K  = 0 since g T ∈ S a nd v K ∈ A by Prop osition 4.6. If  K  is o dd, then b y Lemma 2.4 w e hav e  g T , v K  = 2 if K ⊆ T , a nd 0, other wise. Thu s , it remains to show that  T ∶ K ⊆ T ⊆ S E ∣ S ∣−∣ T ∣ ( 0 ) 2 1 −∣ T ∣ = 0 , for o dd  K  . Using the classica l translation form ula for Euler polyno mials, we c an rewr ite this sum as 2 1 −∣ K ∣ ∣ S ∣−∣ K ∣  t = 0   S  −  K  t   1 2  t E ∣ S ∣−∣ K ∣− t ( 0 ) = 2 1 −∣ K ∣ E ∣ S ∣−∣ K ∣  1 2  14 JEAN-LUC MARICHAL AND PIERRE M A THONET and the latter expres sion is zero since  S  −  K  is o dd. This completes the pro of of (38). E q. (39) can b e pro ved similar ly .  According to the results a bove, the influences Φ B ( f , S ) ( S ⊆ N ) o f a function f ∈ F N determine only the orthogonal pro jection of f onto S . On the other hand, due to Eq. (40), not all v ector s in R 2 n − 1 are influences o f a function in F N : the bes t we can do is to build a unique function in S with prescrib ed “ o dd” influences. This is done in the follo wing result. Prop ositio n 4 . 8. F or every set { i T ∈ R ∶  T  o dd } , the unique function f S ∈ S such that Φ B ( f S , T ) = i T for every T ⊆ N ,  T  o dd, is gi ven by f S = 1 2  S ⊆ N ∣ S ∣ o dd    T ⊆ S ∣ T ∣ o dd E ∣ S ∣−∣ T ∣ i T   v S . Pr o of. By Prop osition 2.3, the conditions r equired o n f S ∈ S re duce to the equalities  f S , g T  = i T for o dd  T  . Pr op osition 4.6 then ensures ex istence and uniqueness of f S . Since the set { v S ∶  S  o dd } is an o rthonormal basis for S we can write f S =  S ⊆ N ∣ S ∣ o dd  f S , v S  v S . F or o dd  S  , b y (8) we hav e  f S , v S  = 2 −∣ S ∣ I B ( f S , S ) and then we compute I B ( f S , S ) by using (39).  4.2. Behav i or of the w ei gh ted Banzhaf influence indexes . The prop erties of the weigh ted influence index Φ B , p for a nondeg enerate p a re completely different from those o f the non- w e ig ht ed influence index Φ B . By Pro po sition 4 .2, for a nondegenera te p the kernel of Φ B , p is one- dimensional and reduced to the constant functions. Moreover, the functionals Φ B , p ( ⋅ , S ) for S ≠ ∅ are linearly indep endent. Therefore, we can build a function f from its influences Φ B , p ( f , S ) for S ≠ ∅ , up to a n additiv e constant. Requiring a pre scrib ed v a lue of f on the empt y set, or a prescrib ed interaction I B , p ( f , ∅ ) , a llows us to build a unique function. This is the aim of the next res ult. Prop ositio n 4.9. A ssume that p ∈ ] 0 , 1 [ n is nonde gener ate and c onsider a set function i ∶ 2 N → R . Ther e exists a u nique function f ∈ F N such that Φ B , p ( f , S ) = i ( S ) for every nonempty S ⊆ N and I B , p ( f , ∅ ) = i ( ∅ ) . I t is given by f = i ( ∅ ) +  S ≠ ∅ v S, p v S, p ( S ) − v S, p ( ∅ )  T ⊆ S ( − 1 ) ∣ S ∣−∣ T ∣ i ( T ) . Ther e exists a unique set fun ction g ∈ F N such that Φ B , p ( g , S ) = i ( S ) for every nonempty S ⊆ N and g ( ∅ ) = i ( ∅ ) . It is given by g = i ( ∅ ) +  S ≠ ∅ v S, p − v S, p ( ∅ ) v S, p ( S ) − v S, p ( ∅ )  T ⊆ S ( − 1 ) ∣ S ∣−∣ T ∣ i ( T ) . Pr o of. W e compute f b y substituting (3 5) in (34). Then we hav e immediately g = f − f ( ∅ ) + i ( ∅ ) .  INFLUENCE AND INTERACT ION INDEXES 15 5. A pplica tion and final remarks W e now e nd our inv es tigation with a n application of the concept of weighted Banzhaf influence index in r eliability enginee ring. W e also give a justification for our independence a s sumption, in tr o duce a normalized influence index, and deriv e tight upp er b ounds on influence s. 5.1. An appli cation in system rel iability theory. Consider a system made up of n interconnected comp onents. Let C = { 1 , . . . , n } b e the set of comp onents a nd let φ ∶ { 0 , 1 } n → { 0 , 1 } be the stru ctur e function whic h expr esses the sta te o f the s ystem in terms of the states of its comp onents. W e ass ume that the system is semic oher ent , i.e., the structure function φ is nondecrea sing in each v ariable and sa tisfies the conditions φ ( 0 , . . . , 0 ) = 0 and φ ( 1 , . . . , 1 ) = 1. W e also a ssume that, at any time, the co mpone nt states X 1 , . . . , X n are statistically indep endent. The r eliability of every comp onent i ∈ C is then defined as the pro bability p i = Pr ( X i = 1 ) . F or general background on sy stem reliability theor y , s e e, e.g., Barlow and Pro schan [1]. According to the definition g iven b y Ben- Or and Linial [2] (as rec a lled in the int r o duction), for every subset S of comp onents, the index I φ ( S ) = Φ B ( φ, S ) = 1 2 n −∣ S ∣  T ⊆ C ∖ S  φ ( T ∪ S ) − φ ( T )  measures, at a given time, the probability that the state of the system is unde- termined once the sta te of each co mpone nt i not in S is s e t to one or zero with probability p i = 1  2. In practice, how ever, the probabilities Pr ( X i = 1 ) and Pr ( X i = 0 ) need not b e equal. The w eig ht ed version Φ B , p of the Banzhaf influence index then provides a straightforward genera liz ation of Be n- Or a nd L inia l’s definition to the general case of arbitrar y reliabilities p 1 , . . . , p n . More sp ecifically , the weigh ted index Φ B , p ( φ, S ) =  T ⊆ C ∖ S p S T  φ ( T ∪ S ) − φ ( T )  , where p S T =  i ∈ T p i  i ∈ C ∖( S ∪ T ) ( 1 − p i ) , (as described in Prop osition 3.6) pr ecisely measures, at a given time, the probability that the state of the system r emains undetermined once the state of e a ch co mp onent i not in S is set to one with pr obability p i and to zero with probabilit y 1 − p i . In a sense this pr obability measures , at a given time, the influence of the subset of comp onents in S over the sy stem. When S r educes to a singleton { i } a nd p i = 1  2, we r etrieve the classica l Banzhaf power index, also k nown in r eliability theory as the Birnbaum str uctur al measure of comp onent impo rtance. 5.2. On the indep endence assumpti o n. W e ha ve made the impo rtant assump- tion tha t the v a riables are set independently o f each other. F ro m this assumption we derived condition (13). Let us now show that this assumption is rather natura l. F or ev er y proba bilit y distribution w such that p i = ∑ S ∋ i w ( S ) ∈ ] 0 , 1 [ , the b est { i } -appr oximation o f f ∶ { 0 , 1 } n → R with resp ect to the squared distance (12) asso - ciated with w is given by f { i } =  f , v { i } , p  v { i } , p +  f , 1  , 16 JEAN-LUC MARICHAL AND PIERRE M A THONET where v { i } , p ( x ) = ( x i − p i )  p i ( 1 − p i ) . 6 Therefore, w e can define the power/influence index asso ciated with w by I w ( f , { i }) =  f , v { i } , p   p i ( 1 − p i ) =  T ⊆ N ∖{ i }  w ( T ∪ { i }) p i f ( T ∪ { i } ) − w ( T ) 1 − p i f ( T ) . How ever, w e know from the liter ature on co op era tive ga me theor y (see, e.g., [4 , 18]) that “go o d” power indexes should b e of the form (42) I ( f , { i }) =  T ⊆ N ∖{ i } c i T ∆ { i } f ( T ) , c i T ∈ R . It follows that the index I w ( ⋅ , { i }) is of the form (4 2) if and only if w ( T ∪{ i }) p i = w ( T ) 1 − p i for every T ⊆ N ∖ { i } . Th us, we hav e prov ed the following result. Prop ositio n 5.1. The index I w ( ⋅ , { i }) is of the form (42) for every i ∈ N if and only i f (13) hold s. 5.3. Normali zed index and upp er b ounds on in flue nces. Since the index Φ B , p is a linea r map, it ca nnot b e consider ed a s an absolute influence index but rather as a relative index constructed to asse s s and compare influences for a given function. If w e wan t to compa r e influences for differ ent functions, we need to consider an absolute, no rmalized influence index. Such an index can be defined as follows. Considering a gain 2 N as a proba bilit y s pace with res p ect to the measur e w , we see that, for every S ⊆ N the n umber Φ B , p ( f , S ) is the cov aria nc e cov ( f , g S, p ) of the random v aria bles f and g S, p . In fact, denoting the exp ecta tio n of f b y E [ f ] = ¯ f ( p ) (see (27)), we ha ve Φ B , p ( f , S ) =  f , g S, p  =  f − E [ f ] , g S, p − E [ g S, p ] = cov ( f , g S, p ) since E [ g S, p ] = ¯ g S, p ( p ) = 0 and  E [ f ] , g S, p  = Φ B , p ( E [ f ] , S ) = 0. T o define a normalized influence index, we naturally cons ider the P ea rson co rre- lation co efficient instead o f the cov aria nce. 7 First observe that, for every no nempt y subset S ⊆ N , the standard devia tion of g S, p is given by (43) σ ( g S, p ) =      i ∈ S 1 p i +  i ∈ S 1 1 − p i . In fact, since g S, p ∈ V S , we have σ 2 ( g S, p ) = cov ( g S, p , g S, p ) = Φ B , p ( g S, p , S ) = g S, p ( S ) − g S, p ( ∅ ) , which immediately leads to (43). Definition 5.2. The normali ze d influenc e index is the mapping r ∶ { f ∶ { 0 , 1 } n → R ∶ σ ( f ) ≠ 0 } × ( 2 N ∖ { ∅ }) → R defined by r ( f , S ) = cov ( f , g S, p ) σ ( f ) σ ( g S, p ) = Φ B , p ( f , S ) σ ( f ) σ ( g S, p ) . 6 Indeed, the functions 1 and v { i } , p form an orthonormal basis for V { i } . 7 This approach was also considered f or the interact ion index (see [14, § 5]). INFLUENCE AND INTERACT ION INDEXES 17 By definition the nor malized influence index rema ins unchanged under interv al scale transformations, that is, r ( af + b, S ) = r ( f , S ) for a ll a > 0 and b ∈ R . Thus, it do es not depe nd on the “size ” of f and therefore can b e use d to co mpare differen t functions in terms o f influence. Moreov er , as a correlatio n co e fficie nt, the normalized influence index satisfies the inequality  r ( f , S )  1, that is,  Φ B , p ( f , S ) σ ( f )  σ ( g S, p ) . The equality ho lds if and only if there exis t a, b ∈ R such that f = a g S, p + b . Int er estingly , this prop erty shows that (43) is a tigh t upp er bound on the influ- ence of a normalized function f  σ ( f ) . Thus, for every nonempty subset S ⊆ N , those normalized functions for which S has the greatest influence are of the form f = ( ± g S, p + c ) σ ( g S, p ) , where c ∈ R . Ackno wledgments The authors gr atefully ackno wledge partial supp ort by the resea rch pro ject F1R- MTH-PUL-12RDO2 of the University of Luxembour g. Appendix A. P roof of Lemma 3.5 Pr o of of L emma 3.5. Using the definition o f the M¨ o bius transfor m in (23), w e o b- tain G ( f , S ) =  T ⊆ N ∖ S p S T   R ⊆ T ∪ S a ( R ) −  R ⊆ T a ( R ) =  R ⊆ N a ( R )   T ∶ R ∖ S ⊆ T ⊆ N ∖ S p S T −  T ∶ R ⊆ T ⊆ N ∖ S p S T  , which shows that G has the form (24) with the prescrib ed q S R . Conv ersely , substituting (3) into (24) and ass uming S ≠ ∅ , w e obtain  R ⊆ N R ∩ S ≠ ∅ q S R a ( R ) =  R ⊆ N R ∩ S ≠ ∅ q S R  T ⊆ R ( − 1 ) ∣ R ∣−∣ T ∣ f ( T ) =  T ⊆ N f ( T )  R ⊇ T R ∩ S ≠ ∅ ( − 1 ) ∣ R ∣−∣ T ∣ q S R . Partitioning every R in to R ′ = R ∖ S a nd R ′′ = R ∩ S , the latter expre s sion b ecomes  T ⊆ N f ( T )  T ∖ S ⊆ R ′ ⊆ N ∖ S  T ∩ S ⊆ R ′′ ⊆ S R ′′ ≠ ∅ ( − 1 ) ∣ R ′ ∣+∣ R ′′ ∣−∣ T ∣ q S R ′ ∪ R ′′ . Since our as sumption on the co efficients q S R implies q S R ′ ∪ R ′′ = q S R ′ ∪ S , the latter ex- pression b ecomes  T ⊆ N f ( T )  T ∖ S ⊆ R ′ ⊆ N ∖ S ( − 1 ) ∣ R ′ ∣−∣ T ∖ S ∣ q S R ′ ∪ S  T ∩ S ⊆ R ′′ ⊆ S R ′′ ≠ ∅ ( − 1 ) ∣ R ′′ ∣−∣ T ∩ S ∣ , where the inner sum equals ( 1 − 1 ) ∣ S ∖ T ∣ , if T ∩ S ≠ ∅ , a nd − 1, o therwise. Setting T ′ = T ∖ S for every T co n ta ining S , the latter expression finally b ecomes  T ′ ⊆ N ∖ S   T ′ ⊆ R ′ ⊆ N ∖ S ( − 1 ) ∣ R ′ ∣−∣ T ′ ∣ q S R ′ ∪ S   f ( T ′ ∪ S ) − f ( T ′ )  , which completes the proof of the lemma.  18 JEAN-LUC MARICHAL AND PIERRE M A THONET References [1] R. E. Barlow and F. Pr osc han. St ati st ic al the ory of r eliability and life t esting . T o Begin With, Silver Spring, MD, 1981. [2] M. Ben-Or and N. Linial. Coll ectiv e coin flipping. In: S. Micali (Ed.), R andomness and Compu- tation . Acade m ic Press, N ew Y ork, pp. 91–11 5, 1990. (Earlier version: Coll ectiv e coin flipping, robust voting games, and mini m a of Banzhaf v alues. Pr o c . 26th IEE E Symp. on F oundations of Comp. Sci. , Portland, pp. 408–416, 1985.) [3] G. Ding, R.F. Lax, J. Chen, P . P . Chen, and B.D. Marx. T ransforms of pseudo-Boolean random v ariables. Discr ete Appl. Math. , 158(1):13–24 , 2010. [4] P . Dub ey , A. Neyman, and R. J. W eb er. V alue theory without efficiency. Math. Op er. R es. , 6:122–128, 1981. [5] K. F ujimoto, I. Ko jadinovic, and J.-L. Maric hal. Axiomatic cha r acterizations of probabilistic and cardinal-probabilistic i n teraction indices. Games Ec onom. Be hav. , 55( 1):72–99, 2006. [6] M. Grabisch, J.-L. Marichal, and M. Roub ens. Equiv alent represen tations of set functions. Math. O p e r. Res. , 25(2):157–1 78, 2000. [7] M. Grabisc h a nd M. Roub ens. An axiomatic approac h to the conc ept of in teraction among play ers in co op erativ e games. Internat. J. Game The ory , 28(4):547–56 5, 1999. [8] P . L. Hammer and R. Holzman. Approximations of pseudo-Bo olean functions; applications to game theory . Z. Op er. R es. , 36(1):3–21, 1992 . [9] P . Hamm er and S. Rudea nu. Bo ole an metho ds in op er ations r ese ar ch and r elate d ar e as . Berlin- Heidelber g- New Y ork: Springer-V er lag, 1968. [10] J. Kahn, G. Kalai, and N. Linial. The influenc e of v ariables on Boolean functions. Pr o c. 29th Ann. Symp. on F oundations of Comp. Sci., Computer So c i ety Pr ess , page s 68–80, 1988. [11] G. Kalai and S. Safra. Threshold phenomena and influence: P ers pectives from mathematics, computer science, and economics. In: A.G. Percu s , G. Istrate, C. Mo ore (Eds.), Computational Complexity and Statistic al Physics . San ta F e Institute Studies on the Sciences of Complexity , Oxford Univ. Press, New Y ork, pp. 25–60, 2006. [12] J.-L. Maricha l . The influence of v ariables on pseudo-Bo olean functions with applications to game theory and multicriteria dec i sion making. Discr e te Appl. Math. , 107(1-3):139 –164, 2000 . [13] J.-L. M aricha l , I. Ko j adinovic, and K. F ujimoto. Axiomatic characterizations of generalized v alues. Discr ete Appl. Math. , 155(1):26–43, 2007. [14] J.-L. Marichal and P . M athonet. W eigh ted Banz haf pow er and interaction i ndexes through we i gh ted approximations of games. Eur. J. of Op er. R es. , 211(2):352–3 58, 2011. [15] G. Owe n. M ultilinear extensions of games. Management Sci. , 18:P64–P79, 1972. [16] G. Owen. Multilinear extensions of games. In: A.E. Roth , editor. The Shapley V alue. Essays in Honor of L loyd S. Shapl ey , pages 139–151. Cambridge Universit y Press, 1988. [17] M. Roub ens. I nteraction b et ween cri teria and definition of w eights in MCDA problems. In Pr o c. 44th Me eting of the Eur. Working Gr oup “Multiple Crit e ria De cisi on Aiding” , pp. 693– 696, O ct. 1996. [18] R. J. W eber. Probabilis tic v alues f or games. In The Shapl e y value. Essays in honor of L loyd S. Shap ley , pages 101–119. Cambridge Universit y Press, 1988. Ma thema tics Research Unit, FSTC, University of Luxembourg, 6, rue Coudenhove- Kalergi, L-13 5 9 Luxembourg, Luxemb ou rg E-mail addr ess : Jean-l uc.marichal[at] u ni.lu University of Li ` ege, Dep art m ent of Ma thema tics, Grande Tra verse, 12 - B37, B-4000 Li ` ege, Belgium E-mail addr ess : p.math onet[at]ulg.ac. b e