Variance Allocation and Shapley Value

V ariance Allo cation and Shapley V alue Riccardo Coli ni-Baldesc hi ∗ 1 , Marco Scarsini † 1 , and Stefa no V accari ‡ 2 1 Diparti men to di Economi a e Finanza, LUISS, Viale Romani a 32, 00197 Roma, Italy 2 Diparti men to MEMOTEF, Sapienza -U niv ersit` a di Roma, Via Del Castro Laurenzia no 9, 0016 1 Roma September 11, 20 18 Abstract Motiv ated b y the problem of utilit y allocation in a p ortfolio und er a Mark o witz mean- v ariance c hoice paradigm, w e prop ose an allo cation criterion for the v ariance of the sum of n p ossibly d e p endent random v ariables. This criterion, the Shapley v alue, requir e s to translate the p roblem int o a co op erat iv e game. The Shapley v alue has n ic e prop erties, bu t , in general, is computationally d e manding. The main r e sult of th is paper sh o ws that in our particular case the S hapley v alue has a v ery simple form that can b e easily computed. The same criterion is used also to allocate the standard deviation of the su m of n random v ariables and a co njecture ab out the r e lation of the v alues in th e t w o games is formulate d. Keyw ords : Sh a pley v alue; core; v ariance game; co v ariance m a trix; computational complexit y . AMS 2010 Sub ject Classification : 91A12, 62J10. ∗ rcolin i@luiss.i t . This author is a mem b er of is a member of GN CS-INdAM. † marco. scarsini@ lu iss.it . This author is a m ember of GNAMP A-INdAM. His work is partially supp orted by PRIN 20103S5RN3 and MOE20 13-T2-1-15 8 . ‡ stefan o.vaccari @u niroma1.it . 1 1 In tro duction In the mean-v ariance mo del of Mark o witz ( 1952 ) preferences for prospects are represen t ed b y a linear com bination of the mean and the v ariance of the prospect: U θ [ X ] = E [ X ] − θ V a r [ X ] , (1.1) for some θ > 0 (see, e.g., Stein bac h , 2001 ) . If an inv estor is dealing with a p ortfolio whose returns are X 1 , . . . , X n , it may b e imp ortant for her to allo cate t o eac h a sset of the p ortfolio its con tribution to the t otal utility score U θ [ P n i =1 X i ]. G iv en that returns are t ypically correlated, allo cating to asset i a con tr ibutio n equal to E [ X i ] − θ V a r [ X i ] w ould not solv e the problem. A similar problem is conside red and solv ed (unde r some conditions) in the Capital Ass et Pricing Mo del of Sharp e ( 1964 ) and Lintne r ( 1965 ) (see, e.g., F ama and F renc h , 2004 ). In this pap er we use to ols of co op erativ e ga me theory to solv e the problem. In particular, w e resort to the Shapley v a lue ( Shapley , 1953 ), a standard solution concept for co operative games, whic h has nice prop erties and has b een extensiv ely used in a v ariety of fields. First, we define a co opera t iv e game whe re the pla y ers are the assets in the po rtfolio, and the w o rth of eac h coa lit ion, i.e., o f ev ery subp ortfolio, is the utilit y score of this subpo rtfolio. Then, w e compute the Shapley v alue of eac h component of the p ortfolio. In general the Shapley v alue is used to allo cate costs or gains among differen t agents who con- tribute to a join t pro ject. The basic idea is that the allo cation ha s to b e fa ir for eac h pla y er. In order to a chiev e this fairness eac h pla y er is a llo cated a v alue that corresp onds to her marginal con tribution to the w orth of a coalition, the av erag e b eing suitably ta ken o v er all p ossible coalitions. Differen t solution concepts exist, based o n differen t principles. F or instance, the core, which em b odies an idea of stabilit y , is the set of all allo cations suc h that no coalition has an incen tiv e to deviate from the grand coalition. W e refer the reader to P eleg and Sudh¨ olter ( 2007 ) or Masc hler, Solan, and Zamir ( 2 013 ) for a nice treatmen t of co op e rativ e games and t heir solution concepts. One of the main drawbac ks of the Shapley v alue is its computational comple xit y as the n umber of pla y ers grows. This is due to the fact that its expression is an a v erage o v er n ! ma r g inal con tributio ns . In our problem, thoug h, the expression of the Shapley v alue is extremely easy to compute and the complexit y for the computation of the whole v ector of Shapley v alues is quadratic in n . W e will explain ho w this result fits in t o a more general result of Conitzer and Sandholm ( 20 0 4 ) ab out the Shapley v alue o f decomp osable games. 2 1.1 Related literature W e are not the first to prop ose game-theoretic to ols for the analysis of cost allo cations related to risks. In an innov ativ e pap er in actuarial scie nce, Lemaire ( 1 9 84 ) prop oses to use to ols fro m coop erativ e game theory to allo cate op erating costs a mong differen t lines of an insurance compan y . The no v elty of his pap er is to solv e a complicated accounting problem b y computing a suitable solution of a co op erativ e g ame. Whereas the a ccounting problem is typic ally quite cumbersome, once the translation in game-theoretic languag e is p erfo rme d, the solution is elegan t and easy to in terpret. Lemaire’s pap er ga v e rise to a whole literat ure on cost allo cation in insurance. In this subseq uen t literature the atten tion is fo cus ed on the allo cation of costs when dealing with a p ortfolio of risks. As Denault ( 2001 ) p o in ts out, “the problem of a llocation is in teresting and non-trivial b ec ause the sum of the risk capitals of eac h cons tituen t is usually larger than the risk capita l of the firm taken as a whole, something c alled the div ersification effect. This de crease of total costs, or ‘rebate,’ needs to b e shared fairly b et we en the constituen ts.” His goal is to prov ide an allo cation criterion that is based on fa irnes s. St a rting from the axiomatic definition of coheren t risk measures pro vided by Artzner, Delbaen, Eb er, and Heath ( 19 99 ), he prop oses a set of axioms for the coherence of risk capita l allo cation principles. He ends up w ith an allo cation that corresp onds to the Aumann-Shapley v alue of nonat omic co op era t ive games (see Aumann a nd Shapley , 1974 ). Tsanak as and Barnett ( 2003 ) and Tsanak as ( 2004 ) prop ose the Aumann-Shapley v a lue as an allo cation mechanis m when the risk measure is giv en b y a distortion premium principle. Tsanak as ( 2009 ) do es the same when con v ex risk measures ar e use d. Abbasi and Hosseinifard ( 2013 ) use the Shapley v alue to allo cate capital in the tail conditional exp ectation mo del. Our analysis is close in spirit to a stream of literature in regression analysis that deals w ith quan- tifying the relativ e imp o rtance of eac h regressor f o r the response. Con tributions on this topic that mak e use of the Shapley v alue can b e found in Lip o v etsky and Conklin ( 200 1 ), Lip o v etsky ( 2006 ), Gr¨ omping ( 2007 , 2009 ), Gr¨ omping and L a ndau ( 2010 ), Mishra ( 2016 ), among others. These ideas actually go back to Lindeman, Merenda, and Go ld ( 1980 ), Krusk al ( 1987 ), who do not explicitly men tion the connection with the Shapley v alue. A parallel more general literature studies ho w to quan tify the imp ortance of random input v ariables to a function b y using ANO V A tec hniques. One w a y to do it is through So bol ′ indices ( Sob ol ′ , 1990 , 1993 ) when the inputs are independen t and their generalizations when the inputs a re dep enden t (see, e.g., Chastaing, G am b oa, and Prieur , 2012 , 2015 , Ow en , 2 013 , Gilquin, Prieur, and Arnaud , 2015 ). A different appro a c h, based o f the Shapley v alue has b een pro p osed b y Ow en ( 2014 ) and further studied b y Song, Nels on, and Staum 3 ( 2016 ) and Ow en and Prieur ( 2016 ). It is in teresting to notice t ha t the Shapley v alue has b een emplo yed in v arious pro blems that in v olv e pro babilit y mo dels. F or instance, it has been implicitly used in reliability theory . Barlo w and Prosc han ( 19 75 ) define an imp ortance index o f system comp onen ts, whose j -th co ordinate indicates the probabilit y that the failure of comp onen t j causes the whole syste m to f ail. Maric hal and Mathonet ( 2013 ) p oin t out that this index is actually a Shapley v alue. Co op erativ e game theory to ols ha v e b een used in queueing t heory , see, e.g., Anily and Ha viv ( 2010 ) , among others, and in in v en t o ry , see, e.g., M ¨ uller, Scarsini, and Shak ed ( 2002 ) and Montrucc hio and Scarsini ( 2007 ). W e refer the reader to Moretti a nd P atrone ( 2008 ) for a nice surv ey of p ossible applications of the Shapley v alue in v arious fields. Sev eral authors hav e considered computatio na l issues related to the Shapley v alue and hav e prop osed efficien t algorithms in some sp ecial cases, whic h typically inv olve voting games or games with a g raph structure. Among them Deng and Papadimitriou ( 1994 ), Conitzer and Sandholm ( 2004 ), Ieong and Shoham ( 2005 , 200 6 ), F a tima, W o oldridge, a nd Jennings ( 2008 , 2010 ), Castro, G´ omez, and T ejada ( 2009 ), Azari Soufiani, Chic kering, Charles, and P ark es ( 2014 ). The in terested reader should consult the b o ok b y Chalkiadakis, Elkind, and W o oldridge ( 2011 ) for a nice surv ey of computational asp ect of co op e rative game theory . 1.2 Organization of the pap er In Section 2 we in tro duce some fundamen tal concepts of co op erativ e game theory and some imp ortan t solution concepts. In Section 3 v ariance games are defined and analyzed. Section 4 deals with standard- dev iation games and prop oses a conjecture ab out the comparison b e t w een the t w o classes of games. Section 5 deals with s ome computational aspects of the Shapley v alue and explains wh y the complexit y of its calculation in the v aria nce g ame is p olynomial. 2 Co op erati v e games W e start in tro ducing some basic concepts in co operat ive game theory . Giv en a set of pla y ers N = { 1 , . . . , n } , a c o op er ative ga m e is a pair h N , ν i , where ν : 2 N → R is suc h that ν ( ∅ ) = 0. Any subset J ⊂ N is called a c o alition . The set N is called t he gr and c o alition . The f unc tion ν is called the char acteristic f unc tion of the game. Giv en that the set of play ers N is fixed, for the sak e of simplicit y , we will just call game its c hara cteristic function. So, if ν represen ts utilities, then ν ( J ) 4 is the utility that coalition J can ac hieve b y itself. If ν represen ts costs, then ν ( J ) is the cost that coalition J m ust pay if it a cts by itself. W e call G ( N ) the class of all games on N . 2.1 Core and an ticore A game ξ is called additive if for all I , J ⊂ N suc h that I ∩ J = ∅ , w e ha v e ξ ( I ∪ J ) = ξ ( I ) + ξ ( J ) . W e call M ( N ) the class of additive games on N . The c or e ( antic or e ) of the game ν is defined as the set of all v ectors x = ( x 1 , . . . , x n ) suc h that ν ( J ) ≤ ( ≥ ) X i ∈ J x i for all J ⊂ N , (2.1) ν ( N ) = X i ∈ N x i . (2.2) A v ector x ∈ R n suc h that ( 2.2 ) holds represen ts a p ossible a llocation amo ng pla y ers of what ac hiev ed by the grand coalition. If ν represen ts a utility for coalitions, then the core is the set of all p ossible allo cations that are stable, that is, no p oss ible coalition has an incen tiv e to deviate. Stabilit y is g iv en by ( 2.1 ), whic h tells that no coalition by itself can ac hiev e more than what allo cated to it. If the function ν represen t s costs, the set of stable allo cations is given by the antic ore. The core (an ticore) of a game can be empt y . A game ν is called sup ermo dular ( submo dular ) if for all I , J ⊂ N ν ( I ∪ J ) + ν ( I ∩ J ) ≥ ( ≤ ) ν ( I ) + ν ( J ) . It is w ell kno wn ( Shapley , 1971/72 ) that the core of a sup ermo dular game and the an ticore of a submo dular game are non-empt y . 2.2 Shapley v alue As seen in Subsection 2.1 , the core is a set of stable allo cations. Its app eal is the stability of its allo cations. Among its shortcomings w e ha v e the fact that it may b e empty and, when it is not a singleton, it is not clear how to choose one single suitable allo cation. W e no w introduce a different solution concept that is o bta ine d axiomatically and pro duc es a single allo cation. 5 Call pla y er i a dumm y if for all J ⊂ N w e hav e ν ( J ∪ { i } ) = ν ( J ) . Call pla y ers i, j symm etric if for all J ⊂ N suc h that i 6∈ J and j 6∈ J w e hav e ν ( J ∪ { i } ) = ν ( J ∪ { j } ) . The Shapley value of ν is a function φ : G ( N ) → R n that satisfies the following prop erties: 1. Efficiency : P n i =1 φ i ( ν ) = ν ( N ). 2. Symmetry : If i and j ar e symmetric, then φ i ( ν ) = φ j ( ν ) . 3. Dummy pla y er : If play er i is a dumm y , then φ i ( ν ) = 0. 4. Line arity : for ν, µ ∈ G ( N ) and α, β ∈ R we ha v e φ ( αν + β µ ) = α φ ( ν ) + β φ ( µ ) . Shapley ( 1953 ) sho we d that the only function φ with these four prop erties has the f ollo wing form φ i ( ν ) = X J ⊂ N \{ i } | J | !( N − | J | − 1)! N ! ( ν ( J ∪ { i } ) − ν ( J )) (2.3) = 1 n ! X ψ ∈ P ( N )  ν ( P ψ ( i ) ∪ { i } ) − ν ( P ψ ( i ))  , (2.4) where P ( N ) is the set of all p erm uta t ions of N , P ψ ( i ) is the set o f play ers who precede i in the order determined by p e rm utation ψ , and | J | is the cardinalit y of J . In general the Shapley v alue of a game do es not necessarily lie in its core. If the game is super- mo dular (submodular), the Shapley v a lue lies in t he core ( an ticore) and is actually its barycen ter. 6 2.2.1 Shapley fusion prop ert y Consider a game h N , ν i . F or J ⊂ N consider the new g ame h N J , ν J i where all play ers in J are fused in to a single play er. A g ame h N , ν i suc h that, for all J ⊂ N , w e hav e φ J ( ν J ) = X i ∈ J φ i ( ν ) . is said to satisfy the S h apley fusion pr op erty . In general the Shapley fusion prop ert y do es not hold, as t he follo wing coun terexample sho ws. Example 2.1. T ak e N { 1 , 2 , 3 } and, for ev ery i, j ∈ N , with i 6 = j , ν ( { i } ) = 0 , ν ( { i, j } ) = 1 ν ( N ) = 1 . By symmetry , φ i ( ν ) = 1 / 3, but, for J = { 2 , 3 } , w e ha v e ν J ( { 1 } ) = 0 , ν J ( J ) = 1 , ν J ( { 1 , J } ) = 1 , hence φ J ( ν J ) = 1 6 = φ 2 ( ν ) + φ 3 ( ν ) . 3 V ariance games W e consider a random ve ctor X := ( X 1 , . . . , X n ) whose comp onen ts can b e see n, for instance, as the returns of n securities in a p o rtfolio. The r eturn o f the whole p ortfolio is then P n i =1 X i . If preferences are represen ted b y the utilit y score U θ defined in ( 1.1 ), w e wan t to fairly allo cate the utilit y U θ [ P n i =1 X i ] of the p ortfolio to each of its comp onen ts. T o do this w e hav e to take in to accoun t the correlation among the v arious returns. T o ac hiev e this goal w e turn to co op erativ e game theory . W e define a suitable co operative game based on U θ and we use the Shapley v alue o f this game as the allo cation criterion. Consider a random v ector X = ( X 1 , . . . , X n ) with finite second momen ts and define , for ev ery J ⊂ N S J := X i ∈ J X i . (3.1) 7 F or each J ⊂ N , define γ ( J ) := U θ [ S J ] = E [ S J ] − θ V a r [ S J ] . (3.2) The expres sion ( 3.2 ) defines a co op erativ e game h N , γ i . This game is a linear com bination of t wo other games: γ ( · ) = ε ( · ) − θ ν ( · ) , (3.3) where ε ( J ) := E [ S J ] and ν ( J ) := V ar [ S J ] . (3.4) Giv en Prop ert y 4 of the Shapley v a lue , w e hav e φ ( γ ) = φ ( ε ) − θ φ ( ν ) . Since the exp ectation is a linear op erator, the game ε is additiv e and φ i ( ε ) = E [ X i ] . Therefore, the problem of finding the Shapley v alue of γ reduce s t o finding the Shapley v alue of ν , whic h w e call a va rianc e gam e . 3.1 Main result The next result sho ws that the Shapley v alue of the g ame ν has a v ery in tuitive simple f orm in terms of the cov ariance matrix of X . Theorem 3.1. F or ν define d as in ( 3.4 ) we h a ve φ i ( ν ) = Cov [ X i , S N ] . (3.5) 8 Pr o of. F rom ( 3.4 ), for i 6∈ J , w e hav e ν ( J ∪ { i } ) − ν ( J ) = V ar   X j ∈ J ∪{ i } X j   − V ar X j ∈ J X j ! = X j,ℓ ∈ J ∪{ i } Cov [ X j , X ℓ ] − X j,ℓ ∈ J Cov [ X j , X ℓ ] = V ar [ X i ] + 2 X j ∈ J Cov [ X i , X j ] . Therefore, from ( 2.4 ) it follows t ha t φ i ( ν ) = 1 n ! X ψ ∈ P ( N )   V a r [ X i ] + 2 X j ∈ P ψ ( i ) Cov [ X i , X j ]   = V ar [ X i ] + 2 n ! X j ∈ N \{ i } X ψ : j ∈ P ψ ( i ) Cov [ X i , X j ] = V ar [ X i ] + X j ∈ N \{ i } Cov [ X i , X j ] = n X j =1 Cov [ X i , X j ] = Cov [ X i , S N ] . The allo cation in ( 3.5 ) is similar (although not equal) to the one obtained b y W ang ( 2002 , Theorem 3.2) for m ultinormally distributed risks, when the exp onen tial tilting model is used. Anal- ogously , Ow en and Prieur ( 2 0 16 ) find a similar allo cation f o r a linear regr ession mo de l with multi- normally distributed regressors. Notice that Theorem 3.1 do es not make any parametric assumption on the distribution of X . Remark 3.2. An immediate corollary of Theorem 3.1 is that the Shapley fusion prop ert y holds for v ariance games. 3.2 Additional prop erties W e examine no w some in teresting prop erties of the v ar ia nc e allo cation through the Shapley v alue. 9 Example 3.3. The Shapley v alue of the v ar ia nc e g ame can assume negative v alues. Consider the case X = ( X 1 , X 2 ) with X 2 = − 2 X 1 and V a r [ X 1 ] = 1. Then Cov [ X ] = 1 − 2 − 2 4 ! , hence φ 1 ( ν ) = − 1 and φ 2 ( ν ) = 2. The idea is that, if a random v ariable con tributes to hedge a risk , then it is “rew arded” with a negativ e Shapley v alue. The follow ing pro perty sa ys that, if p erfect hedging can b e achie v ed, then the Shapley v alue is iden tically zero, no mat t er what the individual v ar ia nc es are. Prop osition 3.4. If V a r [ S N ] = 0 , then φ ( ν ) = 0 . Pr o of. Let V a r [ S N ] = 0, that is X i ∈ N X j ∈ N Cov [ X i , X j ] = 0 . (3.6) Then S N is almo st surely a constan t, whic h, without an y loss of generalit y , w e can assume to b e zero. Hence fo r eac h i ∈ N X i = − X j ∈ N \{ i } X j , whic h implies V a r [ X i ] = X j ∈ N \{ i } X ℓ ∈ N \{ i } Cov [ X j , X ℓ ] . (3.7) Plugging ( 3.7 ) in to ( 3.6 ) we obtain X j ∈ N Cov [ X i , X j ] = 0 for a ll i ∈ N , whic h, b y Theorem 3.1 , giv es the desired result. As the follo wing example shows , it is not p o s sible to apply the result of Prop osition 3.4 to a sub v ector of the ve ctor X . Example 3.5. It is p ossible to ha v e V ar [ S J ] = 0 for some J ⊂ N , without hav ing φ j ( ν ) = 0 for all j ∈ J . F or instance, let X = ( X 1 , X 2 , X 3 , X 4 ) b e suc h that − X 1 = X 2 = X 3 = X 4 , 10 with V a r [ X 1 ] = 1. Then Cov [ X ] =       1 − 1 − 1 − 1 − 1 1 1 1 − 1 1 1 1 − 1 1 1 1       , V a r [ X 1 + X 2 ] = 0 and φ 1 ( ν ) = − 2 and φ 2 ( ν ) = 2. This is due to the f act that the Shapley v alue is computed globally , lo oking at the marginal con tributions of a random v ariable to the v ariance of all p ossible sub v ectors of the v ector X . On the other hand, what is true is that, if V a r [ S J ] = 0, t he n P j ∈ J φ j = 0. Example 3.6. Eve n if the Shapley v alue has the symmetry prop ert y , it is p oss ible to ha v e X i and X j exc ha ngeable (or eve n i.i.d.) without necessarily ha ving φ i ( ν ) = φ j ( ν ) . F or instance, consider X = ( X 1 , X 2 , X 3 , X 4 ) suc h that X 2 and X 3 are i.i.d. and X 1 = X 2 , X 4 = − X 3 . Let V a r [ X 1 ] = 1. Then Cov [ X ] =       1 1 0 0 1 1 0 0 0 0 1 − 1 0 0 − 1 1       . Therefore φ 2 ( ν ) = 2 and φ 3 ( ν ) = 0. Again, this is due to the globa l prop ert y of the Shapley v alue. Tw o exc hangeable random v aria ble s can ha v e v ery differen t relations with the other components of X , therefore their Shapley v alue can differ. Finally , w e lo ok at sup ermo dularit y (submo dularit y) of the v ariance game, whic h, as men tioned b efore, has imp ortan t implications for the nonemptiness of its cor e (an ticore). Prop osition 3.7. (a) If Cov [ X i , X j ] ≥ 0 for al l i, j ∈ N , then the gam e ν is sup ermo dular. (b) If Cov [ X i , X j ] ≤ 0 for al l i, j ∈ N , then the gam e ν is submo dular. 11 Pr o of. (a) If Cov [ X i , X j ] ≥ 0, then we ha v e ν ( I ∪ J ) + ν ( I ∩ J ) = V a r [ S I ∪ J ] + V a r [ S I ∩ J ] = X i ∈ I ∪ J X j ∈ I ∪ J Cov [ X i , X j ] + X i ∈ I ∩ J X j ∈ I ∩ J Cov [ X i , X j ] = X i ∈ I X j ∈ I Cov [ X i , X j ] + X i ∈ J X j ∈ J Cov [ X i , X j ] + 2 X i ∈ I \ J X j ∈ J \ I Cov [ X i , X j ] ≥ X i ∈ I X j ∈ I Cov [ X i , X j ] + X i ∈ J X j ∈ J Cov [ X i , X j ] = V ar [ S I ] + V a r [ S J ] = ν ( I ) + ν ( J ) . (b) If Cov [ X i , X j ] ≤ 0, then the inequality go es in the opp osite direction. 4 Standard devi a tion games Giv en a random v ector X = ( X 1 , . . . , X n ) w e can define a standard deviation game λ on N = { 1 , . . . , n } as fo llows: λ ( J ) = p V a r [ S J ] , where S J is defined as in ( 3.1 ). Computing the Shapley v alue for this ga me is muc h more difficult than for the v ariance game. W e will examine the relation b et w een these tw o types o f games. The next example sho ws that the Shapley fusion prop ert y do es not hold for standard deviation games. Example 4.1. Consider the fo llo wing cov ariance matrix Σ =    1 0 0 0 4 0 0 0 9    . 12 The corresp onding standard deviation game is λ ( { 1 } ) = 1 , λ ( { 2 } ) = 2 , λ ( { 3 } ) = 3 , λ ( { 1 , 2 } ) = √ 5 , λ ( { 1 , 3 } ) = √ 10 , λ ( { 2 , 3) } = √ 13 , λ ( { 1 , 2 , 3 } ) = √ 14 . Therefore the Shapley v alue of the ab o v e game is φ 1 ( λ ) = 1 6  2 √ 14 + √ 10 + √ 5 − 3 − 2 √ 13  , φ 2 ( λ ) = 1 6  2 √ 14 + √ 13 + √ 5 − 2 √ 10  , φ 3 ( λ ) = 1 6  2 √ 14 + √ 13 + √ 10 + 3 − 2 √ 5  . F or S = { 2 , 3 } the cov ariance matrix b ecomes Σ S = " 1 0 0 13 # and the corresp onding games is λ S ( { 1 } ) = 1 , λ S ( S ) = √ 13 , λ S ( { 1 , S } ) = √ 14 . The Shapley v alue o f the abov e game is φ 1 ( λ S ) = 1 2  1 + √ 14 − √ 13  , φ S ( λ S ) = 1 2  √ 14 + √ 13 − 1  . W e hav e φ S ( λ S ) 6 = φ 2 ( λ ) + φ 3 ( λ ) . 13 4.1 A conjecture Giv en t w o v ectors x , y ∈ R n w e sa y that x is ma j orized b y y ( x ≺ y ) if n X i = k x ( i ) ≤ n X i = k y ( i ) for all k ∈ { 1 , . . . , n − 1 } , n X i =1 x i = n X i =1 y i , where x (1) ≤ x (2) ≤ · · · ≤ x ( n ) is the increasing rearrangemen t of x . The reader is referred to Marshall, Olkin, and Arnold ( 2011 ) for prop erties of ma jorization. The follo wing prop osition sho ws that, for n = 2, the normalized Shapley v alue of the v ariance game ma jo rize s the cor r esp onding normalized Shapley v alue of the standard deviation game. Prop osition 4.2. Consider a c ovarianc e matrix Σ = " σ 2 1 σ 12 σ 12 σ 2 2 # . Cal l ν the c o rr esp onding varianc e game and λ the c orr es p onding standar d deviation g a me. Then 1 φ 1 ( λ ) + φ 2 ( λ ) φ ( λ ) ≺ 1 φ 1 ( ν ) + φ 2 ( ν ) φ ( ν ) , Pr o of. Assume, without a n y loss of generalit y , that σ 1 ≤ σ 2 . W e need to sho w that φ 1 ( λ ) φ 1 ( λ ) + φ 2 ( λ ) ≥ φ 1 ( ν ) φ 1 ( ν ) + φ 2 ( ν ) , that is σ 1 + p σ 2 1 + σ 2 2 + 2 σ 12 − σ 2 p σ 2 1 + σ 2 2 + 2 σ 12 ≥ σ 2 1 + σ 12 σ 2 1 + σ 2 2 + 2 σ 12 . After simple algebra, this corresp onds to ( σ 1 − σ 2 ) q σ 2 1 + σ 2 2 + 2 ρσ 1 σ 2 + σ 2 2 + ρσ 1 σ 2 ≥ 0 , (4.1) where σ 12 = ρ σ 1 σ 2 . 14 F or ρ = − 1, expres sion ( 4.1 ) b ecome s − σ 2 1 + σ 1 σ 2 ≥ 0 , and is therefore true. Since the right ha nd side of ( 4.1 ) is incre asing in ρ , w e ha v e t he res ult. W e conjecture the a bov e result to b e true f or all n ∈ N . Conjecture 4.3. F or any n × n c ovarianc e matrix Σ , if ν is the c orr es p onding varianc e game and λ the c orr esp onding standar d devi a t ion game, then 1 P n i =1 φ i ( λ ) φ ( λ ) ≺ 1 P n i =1 φ i ( ν ) φ ( ν ) . (4.2) W e ha v e v erified the conjecture n umerically when the matrix Σ is diagonal. The program that v erifies the conjecture w as written in C and is based on the fo llowing consideration. Let S n − 1 = ( ( σ 1 , . . . , σ n ) ∈ R n + : n X i =1 σ 2 i = 1 ) , D n =  ( σ 1 , . . . , σ n ) ∈ R n + : σ 1 ≤ σ 2 ≤ · · · ≤ σ n − 1 ≤ σ n  , M n = S n − 1 ∩ D n . Because of the normalization factors in b oth sides of ( 4.2 ), if the conjecture holds for eac h diagonal matrix Σ = diag( σ 2 1 , σ 2 2 , . . . , σ 2 n ) then it holds also f o r the diagona l matrix Σ ′ = diag( ασ 2 ψ (1) , . . . , ασ 2 ψ ( n ) ), with α > 0 and ψ an y permutation of (1 , . . . , n ). Therefore, in order to v erify the conjecture for an y diagonal co v aria nce matrix, it suffices to ve rify it f or each ( σ 1 , σ 2 , . . . , σ n ) ∈ M n . The pro cedure w orks as follows. F or a giv en n um b er of play ers n w e extract N indep enden t normally distributed random v ectors Z j = ( Z 1 ,j , Z 2 ,j , . . . , Z n,j ) ∼ N ( 0 , I n ) where I n denotes the n × n iden tit y matrix. It is w ell kno wn ( Muller , 1959 ) that X j = ( X 1 ,j , X 2 ,j , . . . , X n,j ), where X i,j = Z i,j / || Z j || , is uniformly distributed on S n − 1 . Therefore | X j | := ( | X 1 ,j | , | X 2 ,j | , . . . , | X n,j | ) is uniformly distributed on the in tersection of S n − 1 and t he nonnegative ort ha n t of R n . Call σ j = ( σ 1 ,j , σ 2 ,j , . . . , σ n,j ) the no ndecreasing rearrangemen t of X j . Then the p oin ts { σ j } N j =1 are indep en den tly uniformly distributed on the set M n . The pro cedu re chec ks for each point { σ j } N j =1 whether the n − 1 inequalities giv en b y the ma jorization conditions in ( 4.2 ) hold for the diago- nal co v ariance matrix Σ j = diag ( σ 2 1 ,j , σ 2 2 ,j , . . . , σ 2 n,j ). Conditions w ere tested when n = 3 , 4 , 5 and N = 10 9 . 15 5 Computation al asp ects Theorem 3.1 sho ws that the Shapley v alue of the v a riance game can b e easily computed in p olynomial time. F or eac h i ∈ { 1 , . . . , n } the v alue φ i ( ν ) is just the sum of n know n cov ariances. W e now w an t to fra me this r esult in a more general fra me w ork concerning computational complexit y of the Shapley v a lue in suitable classe s of games. In a g e neral coalit io n for ma t ion problem there are tw o main sources of computational complexit y: the computation of each single coalition’s v alue and how to distribute this v alue among the partic- ipan ts of eac h coa lition. The former a pp ears when eac h coalition ha s to solv e a hard optimization problem in order to compute its v alue. The la t ter depends on the c haracteristic f unction of the game and on the solution concep t. In our setting, coalitions do not face an y hard optimization problem to compute their v alues, and, among the solution concepts, w e use the Shapley v alue as an allo cation criterion. Th us, the in teresting q uestion that we address is wh y the Shapley v alue of t he v ariance game can b e computed in an efficien t w a y , whereas a similar metho d canno t b e used for the standard deviation game. The reason lies in the form o f the c haracteristic function of the tw o games. The c hara cteristic function of the v ariance game is easily decomposable in a sum of distinct functions, whereas the c haracteristic function o f the s tandard deviation g a me is not decomp osable due to the presence of the sq uare ro ot. Therefore, to the b est of our kno wledge, all algorithms to compute exactly the Shapley v alue of the standard deviation game a r e non-p olynomial. Conitzer and Sandholm ( 2004 ) pro v e that the Shapley v alue is efficien tly computable if the c haracteristic function of the ga me can b e decomposed in a sp ec ific form. In the following w e sho w that the c ha racteris tic function of the v ariance game respects their decomp osition requiremen ts. W e first in tro duce t he definition of de c o m p osition of a c haracteristic function. Definition 5.1 ( Conitzer and Sandholm ( 20 0 4 , Definition 4) ) . The v ector of characteristic functions ( ν 1 , ν 2 , . . . , ν T ), with each ν t : 2 N → R , is a decompo s ition o v er T issues of c haracteristic functions ν : 2 N → R if for an y J ⊆ N , ν ( J ) = P T t =1 ν t ( J ). The decomp osition of the original c haracteristic function is particularly conv enien t if eac h ν t restricts its fo cus on a subset of agents. Definition 5.2 ( Conitzer and Sandholm ( 2004 , D efinition 5)) . W e sa y that ν t concerns only C t ⊆ N if ν t ( J 1 ) = ν t ( J 2 ) whenev er C t ∩ J 1 = C t ∩ J 2 . In this case, we only need to define ν t o v er 2 C t . This represen tation shrinks the n umber of v alues from 2 | N | to P T t =1 2 | C t | , exponen tially few er tha n 16 the original represen tation. Notice that when | C t | is b ounded b y a small constan t, the n um b er of v alues is linear in T . No w, the c haracteristic function o f the v ariance game is represen ted by ν ( J ) = V a r " X i ∈ J X i # = X i ∈ J X j ∈ J Cov [ X i , X j ] . Th us, for eac h set J , ν can be decomp ose d into | J | ( | J | + 1) / 2 terms , considering that Cov [ X i , X j ] = Cov [ X j , X i ]. Notice that for eac h set J ′ ⊆ J all the c haracteristic functions in J ′ are also presen t in J . W e can represen t T as the set o f all pair s ( i, j ) ∈ N × N with i ≤ j . Consequen tly , | T | = | N | ( | N | + 1) / 2 < | N | 2 . F or eac h ( i, j ) ∈ T we hav e ν ( i,j ) ( J ) =          V a r [ X i ] if J = ( i, i ) , 2 Cov ( X i , X j ) if J = ( i, j ) and i 6 = j, 0 otherwise . Therefore, b y Definition 5.2 we kno w that eac h ν t ∈ T concerns at most 2 play ers, th us | C t | ≤ 2 for all t ∈ T . W e can then apply the following theorem: Theorem 5.3 ( Conitzer and Sandholm ( 2 004 , Theorem 1 )) . Supp ose we ar e given a char acteristic function with a de c o mp osition ν = P T t =1 ν t , r epr esente d a s fol lows. F or e ach t wi t h 1 ≤ t ≤ T we ar e given C t ⊆ N , so that e ach ν t c onc erns only C t . Each ν t is flatly r epr esente d over 2 C t , that is, for e ach t w it h 1 ≤ t ≤ T , we a r e given ν t ( J t ) explicitly for e ach J t ⊆ C t . Then (assuming that table lo okups for the ν t ( J t ) , as wel l c omputations of fac torial s , multiplic ations a n d subtr actions take c onstant time), we c an c ompute the Shapley value of ν for any given agent in time O ( P T t =1 2 | C t | ) , or less pr e cisely O ( T · 2 max t | C t | ) . Th is holds whether or not the cha r acteristic function is incr e asing, and whether o r not it is sup er additive. 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