On The Computation Of Weighted Shapley Values For Cooperative TU Games

Transcription

1 O he Computatio Of Weighted hapley Value For Cooperative U Game Iriel Draga echical Report

2 Computatio of Weighted hapley Value O HE COMPUAIO OF WEIGHED HAPLEY VALUE FOR COOPERAIVE U GAME IRIEL DRAGA Departmet of Mathematic,Uiverity of exa, 4.edderma Drive. Arligto, exa , UA draga@uta.edu hi paper i coiderig the problem of dividig fairly the worth of the grad coalitio i a traferable utilitie game, i cae that the coalitio i formed. he computatioal experiece for the hapley Value, the mot famou olutio, i exteive, but the cae of the Weighted hapley Value ad that of the Kalai-amet Value have bee barely coidered. Baed upo ome reult coected to the ull pace of the firt of thee lat two operator, a algorithm for computig the Weighted hapley Value i developed. he cae of the Kalai-amet Value, a more geeral value, that i reducible to a vector of weighted value, i alo coidered. A ice ew algorithm to be ued for the particular cae of the hapley Value, i derived from the Weighted hapley Value algorithm. Example are illutratig the tated algorithm applied to all cae. Keyword: hapley Value; Weighted hapley Value; Kalai-amet Value; ull pace of a liear operator. ubect Claificatio: 90 A.. Itroductio. For a fiite et of player, =, i G, the vector pace of cooperative traferable utilitity game, (U game), coider the bae: a) tadard bai of Liear Algebra, deote it by E = { E G :, }, where E ( ) =, if =, ad E ( ) = 0, otherwie; b) the uaimity bai, ued by hapley, (95b), to defie what i called owaday the hapley Value, the bai uually deoted by U = { U G :, }, where U ( ) =, if, ad U ( ) = 0, otherwie. Ay game v G i expreed i each of the two bae a v= v( ) E, or v= ( ) v U, () where v ( ) i the worth of, give by the characteritic fuctio of the game, ad ( ) i the Harayi divided of, (Harayi, (959)). If the characteritic fuctio i give, the we ay that the game i i coalitioal form, if the divided are give, the the game i i divided form. From () ad the defiitio of the bae, the relatiohip betwee the coordiate of the game i the two bae are t v ( ) = v( ), ad ( ) ( ) ( ), v = v,. () Ay fuctioal Φ: G R, i called a Value of the cooperative U game v G, or ( v, ), ad Φ i ( v, ) are the payoff offered by the value to the player i. ote that for a game ( v, ), ad,, we deote by ( v, ), the ub game i G obtaied by retrictig v to. he hapley Value i the mot famou value, defied axiomatically by L..hapley (95b), who wa provig that there i a uique value defied by hi group of axiom. hi value i give by the formula v

3 ( )!( )! Hi ( v, ) = [ v ( ) v ( {})], i i, () i :! which ca be ued for computatio, for mall value of. It hould be oted that a iteretig algorithm for computig the hapley Value wa give by Machler (9), baed upo the idea of buildig recurively a equece of game, tartig with the give game, by allocatig i each tep the worth of a coalitio to the member of that coalitio, util all coalitio have a zero worth. he, the um of allocatio i proved to be equal to the hapley Value. I a paper by Draga et al. (99), a bai of the ull pace of the hapley Value allowed the author to derive aother algorithm for computig the hapley Value, by buildig recurively a equece of game which have all the ame hapley Value a the give game, but i the lat game all value of the characteritic fuctio for coalitio of ize at mot equal zero, o that the um () ha oly term, ad i coequece the hapley Value ca be eaily computed. I the preet paper, we ited to give uch a algorithm for the Weighted hapley Value. hi i a liear value itroduced alo by L..hapley (95a), aociated with a poitive weight vector λ R++. Let u follow Kalai ad amet (97, 9), i defiig the Weighted hapley Value, H (, v, λ ), by mea of it value o the baic vector of the uaimity bai, ad the liearity, ut a hapley did for the hapley Value. For each coalitio,, defie λi Hi(, U, λ) =, i, Hi(, U, λ ) = 0, i, (4) λ( ) ad aume that the value i liear. hi mea that for ay game v G, by liearity, () ad (4), we have v ( ) Hi(, v, λ) = λi, i, (5) i : λ( ) givig the Weighted hapley Value i term of the divided form. ow, from () ad (5), we get a formula imilar to (), (hapley 95a, h.4, p.5), preciely H ( v i,, λ) = λi γ [ v ( ) v ( { i })], i ; i, (6) where t ( ) γ =,,, (7) : = λ( ) givig the Weighted hapley Value i term of the coalitioal form. Obviouly, the hapley Value i obtaied for λ i =, i, a how by (4), where λ ( ) =. Formula (6) ad (7) offer a firt alterative for computig the Weighted hapley Value for game i coalitioal form, a ecod alterative i to ue (5), if the game i i divided form. Let u illutrate the firt alterative, the the ecod. Example. Coider a game i coalitioal form v () = 00, v () = 00, v () = 00, v (, ) = 400, v (, ) = 500, v (,) = 600, v (,,) = 900. By uig () we get the hapley Value H (, v ) = (00,00,400). If we take the weight vector λ = (,, ), ad we ited to compute the Weighted hapley Value of the ame game, by uig (7), we compute firt 7 γ =, γ =, γ = 9, γ =, γ =, γ =, γ =

4 he, from (6) we obtai H (, v, λ ) = (45,, ). 7 7 Clearly, the computatio i more difficult tha for the hapley Value, where the coefficiet are eaily obtaied, a how i (). Of coure, the difficulty i icreaig with the umber of player. o how the ecod alterative we eed the divided, to be obtaied by mea of (); we have v() = 00, v() = 00, v() = 00, v(, ) = v(,) = v(,) = 00, v(,,) = 0. ow, by (5) with all weight equal oe we get the hapley Value ad if we ue (5) for the give weight we get the Weighted hapley Value, the ame a above.. A Family of Bae ad the ull pace of the Weighted hapley Value. I the paper by Draga et al., (99), a bai for the ull pace of the hapley Value wa ued to derive a algorithm for computig the hapley Value. hi algorithm wa geeralized i the ubequet paper by Draga (99), baed upo a more geeral bai for the ull pace of the Weighted hapley Value, ito a algorithm for computig thi value. However, i the preet paper the ame bai will be ued to derive a accelerated algorithm, which complete the computatio i two tage, where i the firt tage there are oly tep, ad i the ecod tage the formula (6) ad (7) are ued, but oly + coefficiet are eeded, followed by the applicatio of the formula. Moreover, the accelerated algorithm will be tated i computatioal detail of elemetary operatio. o make the paper elf cotaied, the above metioed bai i dicued i thi ectio, while the algorithm will be give i the ext ectio. I the proof, a third alterative method to compute the value ad the weighted value will be decribed, ad ued for computig the Weighted hapley Value of the baic vector. Let R be a weight vector, ad coider the et of game λ ++ defied a follow: for, W = { W G :, }, () W ( ) = λ( ), if =, the W ( ) λ, = {}, ; further W ( ) = 0 otherwie. For = if = the middle cae ca ot occur. We illutrate the et of vector W for a three pero game with traferable utilitie, i the cae of the weight vector coidered i Example. Example. he eve vector aociated with the coalitio ca be put i the table {} W W W W W W W {} {} 0 0 {,} {,} {,} {,,}

5 where i W the idex i writte a a equece of elemet of. he table ha bee writte i order to ee clearly that the eve vector are liearly idepedet, o that they form a bai i liearly R 7, ad a imilar thig happe for a higher umber of player, whe there are idepedet vector. ow, we ited to compute the Weighted hapley Value of the baic vector i W. ote that i the earlier paper by Draga (99) we computed thee value by uig the relatiohip betwee the game i W ad the game i U. Further, we derived algebraically the recurive relatiohip betwee the potetial of the Weighted hapley Value, give by Hart ad Ma- Colell (9, 99), itead of givig ome axiomatic proof of thee relatiohip, ad derived alo the Weighted hapley Value i term of the potetial. o make the preet paper horter, here we compute the Weighted hapley Value of the game W by computig firt the potetial, ad uig further the reult of Hart ad Ma-Colell about the relatiohip of the Weighted hapley Value with the potetial. For a game v G ad it ub game ( v, ),, a potetial fuctio P for the Weighted hapley Value, aociated with the weight vector λ, i defied by the equatio λi [ P (, v, λ) P ( {}, i v, λ)] = v ( ),, (9) i with P(, v, λ) = 0. (0) By ummatio, from (9), ad uig the tadard otatio λ( ) = λ, we obtai the recurio formula i P (, v, λ ) = [ v ( ) + ip ( {}, i v, }], i λ( ) λ λ () which together with (0) uiquely defie a fuctio to be coidered the potetial of the Weighted hapley Value. Ideed, Hart ad Ma-Colell (9, 99) proved that the Weighted hapley Value i give by Hi(, v, λ) = λi[ P(, v, λ) P( {}, i v, λ)], i, () (99, h.5.). Clearly, beide (6) ad (7), which allow the computatio of the Weighted hapley Value for game i coalitioal form, ad (5) which allow the computatio for game i divided form, formula (0) ad () provide a third method of computatio for game i potetial form. Example. Retur to the game ad the weight vector ued i the previou example, ad ue the third method. From () ad the game give i thoe example, by computatio we obtai P({}, v, λ ) = 00, 600 P({}, v, λ ) =, P({}, v, λ ) = 600, P({, } v, λ ) =, P({,}, v, λ ) = 560, P({,}, v, λ ) =, P({,, }, v, λ ) =. By uig (), we obtai the ame reult a i Example. heorem. For ay baic game W G,,, we have for : PW (,, λ ) =, if =, PW (,, λ ) = 0, if. Proof. For ay, i () with, by (0) ad (), ad the defiitio of W, PW (,, λ ) = 0, whe. If =, we get PW (,, λ ) =. ow, if, we have = the 4

6 we have the reult proved; if, the for = +, we have from () that PW (,, λ ) = 0, whe, becaue W ( ) = 0; if we have = {},, the formula () how that agai PW (,, λ ) = 0. Further, for all with +, o term i () i differet of zero, o that the theorem hold. λ ++ heorem. For ay weight vector R, ad ay game W W, we have Hi(, W, λ ) = 0, if, the Hi(, W {} i, λ) = λi, i, ad H (, W, λ) = λ, i. i i Proof. Formula () applied to the game W for ay coalitio, that i Hi(, W, λ) = λi[ P(, W, λ) P( {}, i W, λ)], i, (4) together with heorem, how that both term i the bracket are zero whe we have, the firt i zero ad the ecod i whe = {}, i ad the firt term i ad the ecod i zero whe =. Earlier (Draga,99), we derived from heorem a reult which helped u olve the o called ivere problem: give L R, fid out the et of game uch that H (, v, λ ) = L.. Here, heorem will help for utifyig the algorithm to be preeted i the ext ectio, via the followig reult: heorem. I G, the vector pace of cooperative U game with the et of player, for ay poitive weight vector λ R++, the et of game W = { W W :,, } { W + W {} i }, i (5) i a bai of the ull pace of the liear operator the Weighted hapley Value. Proof. I term of liear algebra, by heorem, all the vector W,, belog to the ull pace of the Weighted hapley Value. Moreover, a how by the ame heorem, the vector W + W {} i belog alo to the ull pace. A the game i the et W are i liearly idepedet vector, i the ull pace of H (, v, λ) R, while the rage of the operator i, by the fudametal theorem of liear algebra (ee, for example, K.Hoffma ad R.Kuze, 97), the et W i a bai of the ull pace of the Weighted hapley Value. (the ullity of the liear operator hould equal the differece betwee dim G = ad, the dimeio of the rage, that i exactly, the umber of liearly idepedet vector i W ).. Computig the Weighted hapley Value of U game. A how i the previou ectio, for ay game i coalitioal form, the Weighted hapley Value may be computed, either by formula (6) ad (7), or by computig the divided ad uig formula (5), a i Example, or by computig the potetial ad uig the relatiohip () betwee the Weighted hapley Value ad the potetial, a i Example. A fourth method, the aim of the preet paper, will be baed upo the ull pace of thi liear operator, determied i 5

7 the ecod ectio. he baic idea of the algorithm ca be tated a follow: to ay game v G we ca add ay liear combiatio of vector from W ad the ew game will have the ame Weighted hapley Value a the origial game. he liear combiatio i choe i each ub tep uch that all coalitio of ize at mot have the worth zero; all coalitio of ize at leat + have the ame worth a i v. he coalitio of ize + have their worth provided by the algorithm. Further, the algorithm i topped after tep ad the Weighted hapley Value of the lat game obtaied equal the Weighted hapley Value of v, ad i computed either by formula (6) ad (7), or by () ad (). ote that i the firt cae oly + coefficiet γ, =,, hould be computed by (7), the (6) will be ued. Obviouly, i geeral coefficiet hould be computed, which i a very large umber, whe i large; for example, if = 0, the = 0, while + =. he algorithm may be orgaized i two phae: the firt oe with tep, ad the ecod oe uig (6) ad (7), or () ad (). he detail of the two phae hould be explaied; we tart with the ecod phae, i which i fact, i the proof, we ued () ad (), baed upo the followig heorem 4. Let w G be a U game atifyig w ( ) = 0,,,. (6) Coider w ( {}) i xi =, λ( ) λ i i, (7) ad x = [ w ( ) + λ i iw ( {})]; i () λ( ) the, the Weighted hapley Value of w i give by i (9) H (, w, λ) = λ ( x x ),. i i i Proof. akig ito accout (6), the weighted potetial of the game ( {}, i w) are x i, for all player i, a how by (); the, the weighted potetial of the game ( w, ) i x, by uig agai formula (). Further, (9) follow from (). ow, it remai to be how how ca we build i the firt phae the liear traformatio able to traform ay game v G ito a game w G atifyig (6), uch that the Weighted hapley Value i uchaged. he procedure, to be ued i the firt phae of the algorithm i a equecial oe, baed upo the ull pace of the Weighted hapley Value, a explaied above. Let be a iteger,, uch that either =, or, if, uppoe that a game v G derived from v 0 v = v i available, atifyig ( ) = 0,,,, (0) ad H (, v, λ) = H (, v, λ), () where v G i a give U game. uppoe that v ( ) 0 for ome coalitio, with =, ad. he, the derivatio of the game v G atifyig coditio imilar to (0) ad () i explaied by the followig 6

8 heorem 5. Let where v G be a game atifyig (0) ad (), ad. v ( ) v v :. W =, he, the game = () λ( ) W are the game i W explaied i (), ad λ( ) = λ, for all, with =, atifie the coditio obtaied from (0) ad () by chagig ito +. Proof. A the Weighted hapley Value i a liear operator i G, ad accordig to heorem, we have H (, W, λ ) = 0, for all with, by () ad () we get that () hold whe i replaced by +. It remai to how that coditio imilar to (0) will alo hold. By compoet, () mea v ( ) v ( U) = v ( U). W ( ),. : U U () = λ( ) If U, the W ( U ) = 0 for all U, whe =, hece from (0) ad (), we get v ( U ) = 0. If U =, ad =, the W ( U) 0 oly whe U =, ad i thi cae we get W ( U) = W ( ) = λ( ), o that from (), takig ito accout (0), we have v ( U ) = 0. Hece, for all coalitio U with U we got v ( U ) = 0. Example 4. Retur to the game coidered i Example ; a =, we ca ue heorem 5 oly oce, for =, to get a game v G with the Weighted hapley Value uchaged ad with all worth of coalitio of ize oe equal to zero. Formula () i where obtai the game deoted above by, v() v() v()..., v = v W W W λ λ λ W, W, W are the firt three colum i the table of Example. A =, we w amely w = (0,0,0;,975, ). ow, we ue heorem 4 to compute the Weighted hapley Value for the game (7) we get 600 x =, x = 560, x = from () ad the value already computed we obtai x =, 4600, ad (9) give the value of the Weighted hapley Value foud i Example. i i w= v. From ote that for larger we hould work with huge vector W, preciely vector. herefore, it i more reaoable to traform the ketched algorithm ito a more detailed form of each tep, by howig how the ew value of v ( U ) for coalitio U with U = +, obtaied from () could be eaily computed. Recall that if U, the W ( U ) = 0, ad if 7

9 U = {}, the W( U) = λ. Hece, if U i a fixed coalitio of ize +, the i the um of formula () appear oly the term for coalitio with =, obtaied whe we have = U {}, ad thi happe for all U. If U +, the v ( U) = v ( U). herefore, formula () for a coalitio U of ize + become v ( U { }) v ( U) = v ( U) +. λ. (4) U λ( U ) λ o ummarize: i tep, the worth of all coalitio of ize to, (if ), ad + to, (if ), are uchaged; the worth of coalitio of ize become zero; the worth of coalitio of ize + are computed by formula (4). the computatio of the traformed game i doe a follow Example 5. Retur to the computatio doe i Example 4. We have v() = v() = v() = 0 ad v (,,) = 900, while the coalitio of ize two get the ew worth provided by (4): v() v() 00 v (, ) = v(, ) + λ+ λ =, λ λ v() v() v (,) = v(,) + λ+ λ = 975, λ λ v() v() 75 v (,) = v(,) + λ + λ =. λ λ ow, the computatio of the Weighted hapley Value cotiue with tage two, like i Example 4, becaue =. he tep of the algorithm may be decribed a follow 0 Iitializatio: =, v = v = v, k = 0. tage. A log a, for each coalitio of ize +, ay U = { i,..., i, i + }, compute λ ( U ), the equetially the ratio v ( U { i}) λi, for =,...,, +, (5) ( U ) λ λ i ad for each ratio computed add it to the umber k, the um of v ( U) with the um of the ratio previouly computed, util all =,..., + have bee coidered. he, either take aother coalitio U with U = +, ad, chage ito + ad repeat, or if all coalitio have bee exhauted, go to tage two. tage. Compute xi, i, the x, the H (, v, λ ), by mea of formula (7), (), ad (9), repectively, ad top. ote that thi algorithm may be ued to compute the hapley Value, which i the particular cae λ i =, i. I thi cae, the ratio (5) become v ( U { }), U. (6) he algorithm decribed above will be akig u to compute

10 v ( U) = v ( U) + v ( U { }), U, U = +. (7) U Example 6. Retur to the game for which the hapley Value wa computed i Example. A =, oly oe tep i eeded i tage ; we have v () = v () = v () = 0, v (,,) = 900, a i Example 5, ad we ue (7) for the coalitio of ize two: v (,) = v (,) + v () + v () = 700, v (, ) = v(, ) + v() + v() = 900, v (,) = v(,) + v() + v() = 00. ow, by (7) ad (), where w= v, we obtai: v (,) v (, ) v (, ) x = = 550, x = = 450, x = = 50, ad [ (,,) (, ) (,) x= v + v + v + v (,)] = 750, the, from (9), we get the ame hapley Value a i Example.. Applicatio: computig the Kalai-amet Value. Let = (,..., m ) be a ordered partitio of ad λ R++ be a poitive weight vector. he pair ( λ, ) i called a weight ytem. If = ( ), the the weight ytem i called imple. We follow Kalai ad amet (97,9) i defiig their weighted value: For a weight ytem ϖ = ( λ, ), the Kalai-amet weighted value of a game, i a fuctioal κ : G R, liear o G, ad defied by it value o the uaimity bai: for each player i, we have λi κi( U, ϖ) =, () λ( ') where ' = k, with k determied by k = max{ : =,..., m, } ad λ ( ') the um of all λ h for h ', whe i ', ad κi( U, ϖ ) = 0 whe i '. If the weight ytem i imple, the, a () ad (4) how, the Kalai-amet Value i the Weighted hapley Value. I word, the above defiitio i cotructive, o that we ca compute the Kalai- amet Value a follow: coider ome coalitio ; fid out,...,, m to determie the idex k, the larget for which we get a oempty iterectio (we got alo ' = k ); for each i ' we compute κ i by (), take all other compoet equal to zero, ad ue the liearity. I the followig, he Kalai-amet Value will be deoted by κ(, v ϖ ), where v G ad ϖ i a weight ytem. From () ad (), baed upo the liearity, we get κ( v, ϖ) = v( ) κ( U, ϖ). (9) Apparetly, thi formula together with () i the oly way to compute a Kalai-amet Value for a give game i divided form. If we chooe thi path, we have to compute the divided of the game, the the Kalai-amet Value of the baic vector of the uaimity bai, ad ue (9). We ited to how that a ecod path i poible by uimg a algorithm for computig the Weighted hapley Value.. he algorithm will be ued to compute the Weighted hapley Value of m 9

11 auxiliary game, where m i the umber of block i the ordered partitio aociated with the value. he game may be give either i coalitioal form, or i divided form. Let ( λ,..., λ ) m be the partitio of the give weight vector correpodig to the give ordered partitio = (,..., m ) of, ad ϖ = ( λ, ), be the imple weight ytem defied o all, =,..., m. We ited to how that the Kalai-amet Weighted Value i a vector compriig m ub vector where each ub vector i a Weighted hapley Value of a auxiliary game v G, =,..., m, aociated with v. Firt, otice that the expaio of ay game v G i the uaimity bai ca be writte a a um of game = m v= v, = (0) where v = v( Q) UQ, Q v = ( R Q) U,,..., m. Q, Q R... v R = Q () I the um foud i the iterior bracket we have alo a term for R =. For example, if we have a four pero game ad = (, ), with = {,}, = {,4}, the the expaio () ca be writte v = v() U+ v() U + v(, ) U, v = [ v() U+ v(,) U+ v (,) U + + v (,, ) U] + [ v(4) U4 + v(, 4) U4 + v(, 4) U4 + v(,, 4) U4] + + [ v(,4) U4 + v(,,4) U4 + v(,,4) U4 + v(,,,4) U4]. I the ecod formula, we eparated the term correpodig to the coalitio Q= {}, Q= {4}, ad Q = {, 4}, which will make clear the dicuio here below. Obviouly, a imilar ituatio occur if there are more tha two block i the ordered partitio. ow, i (0) ad (), we ca ue the liearity to compute the Kalai-amet Value a uggeted by formula (9), a how i the ext example Example 7. Coider the four pero game v () = 0, v () = 0, v () = 0, v (4) = 0, v (, ) = 5, v (, ) = 0, v (, 4) = 0, v (,) = 0, v (,4) = 0, v (, 4) = 0, v (,,) = 50, v (,,4) = 0, v (,, 4) = 50, v (,,4) = 40, v (,,, 4) = 00. uppoe that we are give the above partitio ad the weight vector λ = (,,, ). 4 6 he divided, computed by formula () are: 0, 0,0,0 for igleto, 5,0,0, 0, 0, 0 for coalitio of ize two, 5, 5, 0, 0 for coalitio of ize three ad 45 for the grad coalitio. he, we get κ( U, ϖ ) = (,0,0,0), κ( U, ϖ ) = (0,,0,0), κ( U, ϖ ) = (,,0,0), 4 4 ad, by uig the divided ad (9), we obtai for v : 05 5 κ( v, ϖ ) = (,,0,0). 4 4 Whe we compute the value for the uaimity vector appearig i the ame bracket we oticed that we get the ame reult for all of them, a follow 0

12 κ( U, ϖ) = κ( U, ϖ) = κ( U, ϖ) = κ( U, ϖ) = (0,0,,0), κ( U, ϖ) = κ( U, ϖ) = κ( U, ϖ) = κ( U, ϖ) = (0,0,0,), κ( U4, ϖ) = κ( U4, ϖ) = κ( U4, ϖ) = κ( U4, ϖ) = (0,0,, ), 4 4 where the weight how above have bee ued. I thi way, from each bracket we ca factor out the Kalai-amet Value of the correpodig uaimity vector ad the um of the divided are 5, 5, ad 45. I thi way, we get for v : κ( v, ϖ ) = 5(0,0,,0) + 5(0,0,0,) + 45(0,0,, ) = (0,0,, ) he Kalai-amet Value of the give game, obtaied from (0) by liearity ad the above computatio i κ(, v ϖ ) = (.,, ) he above example 7 ugget a ecod algorithm for computig the Kalai-amet Value. O G, =,,..., m, defie auxiliary game, v, by uig the divided form, relative to the uaimity bae of the pace G, deoted, a follow: ( Q) = ( Q), Q, ( Q) = ( ),, R... v R Q Q =,..., m, () where i the um R = i icluded. Of coure, takig ito accout formula (), we ca write alo the coalitioal form of the auxiliary game: v( ) = v( ),, v( ) = v (... ) v (... ),, =,..., m. () ow, if the uaimity bai i G i deoted by U = { x Q G : Q }, =,..., m, each game ca be writte a v = ( Q) xq. (4) Q Deote by ϖ = ( λ, ), =,..., m, the imple weight ytem o each block, ad we have κ( v, ϖ ) = ( Q ) κ( x Q, ϖ ), =,..., m. (5) Q A ϖ, =,..., m, are imple weight ytem, the Kalai-amet Value for the baic uaimity vector are Weighted hapley Value, to be computed by (4) ad (5). I thi way we get κ( v, ϖ ) = H(, v, λ), =,..., m. (6) We proved the mai reult of thi ectio: heorem 6. For a U game v G, i coalitioal form, ad the weight ytem ϖ = ( λ, ), where λ R++ i a poitive weight vector ad = (,..., m ) i a ordered partitio of, let v G, =,..., m, be the auxiliary game derived from v by (), or (), ad let λ = ( λ,..., λ ) m be the correpodig partitio of the weight vector λ. he, the Kalai- amet Weighted Value κ(, v ϖ ) i give by κ( v, ϖ) = [ H(, v, λ ),..., H(, v, λ )]. (7). m m m

13 Example. Retur to the game ued i the Example 7, ad ue the ew procedure for computig the Kalai-amet Value. he Weighted hapley Value of the baic vector correpodig to the weight vector λ = (, ) ad λ = (, ), are 4 6 H (, x, λ ) = (,0), H (, x, λ ) = (0,), H (, x, λ ) = (, ), 4 4 H (, x, λ ) = (,0), H (, x4, λ ) = (0,), H (, x4, λ ) = (, ). 4 4 ow, by (5), where we ue the divided already computed i Example 7, preciely () = 0, () =0, (,) = 5, for v, ad thoe computed by formula () from the oe of Example 7, preciely () = 5, (4) = 5, (, 4) = 45, for v, we obtai H (, v, λ ) = (, ), H (, v, λ ) = (, ), o that by puttig them together we get the ame reult a i example 7. Obviouly, we were able to compute the coalitioal form of the auxiliary game by uig formula (), the ue the algorithm developed i ectio to compute the Weighted hapley Value, i cae that at leat oe of the block had three or more player. Remark. Beide the referece coected to our paper, we poit out that there are a few importat referece for the reader who would like to fid out more about the relatiohip betwee the Weighted hapley Value ad other cocept of olutio for cooperative U game, amely: D.Moderer et all., (99), Weighted hapley Value ad the Core, Iteratioal Joural of Game heory,, 7-9. V.Vailiev, (007), Weber Polyhedro ad Weighted hapley Value, Iteratioal Game heory Review, 9,, R e f e r e c e I.Draga, J.Potter, ad.i, (99), uperadditivity of olutio of coalitioal game, Liberta matematica, vol.ix, 0-0. I.Draga, (99), he potetial bai ad the Weighted hapley Value, Liberta Matematica, Vol.XI, I.Draga, (990), O the weighted hapley Value ad the Kalai/amet value, echical Report #6, Uiverity of exa at Arligto, April 990. J.Harayi, (959), A bargaiig model for the cooperative -pero game, i Cotributio to the heory of Game, vol.4, A.W.ucker ad R.D.Luce (ed.), Hart ad A.Ma-Colell, (9), he potetial of the hapley value, i he hapleyvlalue, Eay i Hoor of L..hapley, A.E.Roth (ed.), Cambridge Uiverity Pre, 7-7..Hart ad A.Ma-Colell, (99), Potetial, Value, ad Coitecy, Ecoometrica, vol. 57,, K.Hoffma ad R.Kuhze, (97), Liear Algebra, d editio, Pretice Hall, ew Jerey E.Kalai ad D.amet, (97), O Weighted hapley Value, IJG, 6, 05-. E.Kalai ad D.amet, (9), Weighted hapley Value, i he hapley Value, Eay i Hoor of L..hapley, A.E.Roth (ed.), Cambridge Uiverity Pre, -00. M.Machler, (9), he worth of a cooperative eterprie to each member i Game, Ecoomic Dyamic ad ime erie Aalyi, Phyica Verlag, L..hapley, (95a), Additive ad o-additive et Fuctio, Ph.D.hei, Priceto.. L..hapley, (95b), A value for -pero game, Aal of Mathematic tudie, vol., 07-7.

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