The Shapley value for fuzzy games on vague sets

Transcription

1 WEA TRANACTIN on INFRMATIN CIENCE APPLICATIN The hapley value or uzzy games on vague ses Fan-Yong Meng* (Correspondng Auhor chool o Managemen Qngdao Technologcal nversy Qngdao hong Provnce P R Chna menganyongje@163com Yan Wang Busness chool Cenral ouh nversy Changsha Hunan Provnce P R Chna kmus_wy@snacom Absrac: - In hs paper a general expresson o he hapley value or uzzy games on vague ses s proposed where he player parcpaon levels are vague ses The exsence unqueness o he gven hapley value are showed by esablshng axomac sysem When he uzzy games on vague ses are convex he gven hapley value s a vague populaon monoonc allocaon uncon (VPMAF an elemen n he core Furhermore we sudy a specal knd o hs class o uzzy games whch can be seen as an exenson o uzzy games wh mullnear exenson orm An applcaon o he proposed model n jon producon problem s provded ey-words: - uzzy game; vague se; hapley value; core; mullnear exenson 1 Inroducon Cooperave uzzy games [1] descrbe he suaons ha some players do no ully parcpae n a coalon bu o a ceran degree In hs suaon a coalon s called a uzzy coalon whch s ormed by some players wh paral parcpaon A specal knd o uzzy games whch s called uzzy games wh mullnear exenson orm [9] was dscussed In [2] he auhor dened a class o uzzy games wh proporonal values gave he expresson o he hapley uncon on hs lmed class o games Laer reerence [12] poned he uzzy games gven n [2] are neher monoone nondecreasng nor connuous wh regard o raes o players' parcpaon dened a knd o uzzy games wh Choque negral orms The hapley uncon dened on hs class o games s gven Recenly reerence [3] exped he uzzy games wh proporonal values o uzzy games wh weghed uncon And he correspondng hapley uncon s also gven The uzzy games wh mullnear exenson orm Choque negral orm are boh monoone nondecreasng connuous wh respec o raes o players parcpaon As a well-known soluon concep n cooperave game heory he hapley value or uzzy games [ ] has been suded by many researchers Besdes he hapley value he uzzy core o uzzy games [13 14] he lexcographcal soluon or uzzy games [10] are also dscussed When every player s parcpaon level s 1 a uzzy game reduces o be a radonal game Namely he radonal game s a specal case o he uzzy game [1] Furhermore he hapley value or uzzy games wh uzzy payos [4 15] s also consdered All above menoned researches only consder he suaon where he player parcpaon s deermned There are many unceran acors durng he process o negoaon coalon ormng In order o reduce rsk ge more payos when he players ake par n cooperaon someme hey only know he deermnng parcpaon levels he parcpaon levels ha hey do no parcpae A hs suaon uzzy games can no be appled Bu he vague ses [5] can well descrbe he parcpaon levels o he players Based on above analyss we shall sudy uzzy games on vague ses where he player parcpaon levels are vague ses E-IN: Issue 2 Volume 9 February 2012

2 WEA TRANACTIN on INFRMATIN CIENCE APPLICATIN Ths paper s organzed as ollows In he nex secon we recall some noaons basc denons whch wll be used n he ollowng In secon 3 he expresson o he hapley value s gven an axomac denon s oered ome properes are researched In secon 4 we pay a specal aenon o dscuss a specal knd o uzzy games on vague ses gve nvesgae he explc orms o he hapley value or hs knd o uzzy games 2 Prelmnares 21 The concep o vague ses Le X be an nal unverse se X= {x 1 x 2 x n } A vague se over X s characerzed by a ruhmembershp uncon v a alse-membershp uncon v v : X [0 1] v : X [0 1]sasyng v + v 1 where v (x s a lower bound on he grade o membershp o x derved rom he evdence or x v (x s a lower bound on he negaon o x derved rom he evdence agans x The grade o membershp o x n he vague se s bounded o a subnerval [ v (x 1 v (x ] o [0 1] The vague value [ v (x 1 v (x ] ndcaes ha he exac grade o membershp v (x o x may be unknown bu s bounded by v (x v (x 1 v (x where v + v 1 When he unverse X s dscree a vague se A can be wren as [ A( 1 A( ]/ 1 n A x x x x X Le he vague ses x=[(x 1 (x] y=[(y 1 (y] where 0 (x+(x 1 0 (y+(y 1 hen (1 x y=[(x (y 1 ( (x (y]; (2 x y=[(x (y 1 ( (x (y]; (3 x y (x (y (x (y; (4 x=y (x=(y (x=(y 22 ome basc conceps or uzzy games on vague ses Le N {1 2 n} be he player se P(N be he se o all crsp subses n N The coalons n P(N are denoed by 0 T 0 For 0 P(N he cardnaly o 0 s denoed by he correspondng lower case s A uncon v 0 : PN ( sasyng v ( =0 s called a crsp game Le G denoe 0 0 he se o all crsp games n N A vague se n N s denoed by e [ ( 1 ( ] / [ ] where ={N (>0}denoes he suppor o e {(1 ( } 0 (+ ( 1 or any ( ndcaes he rue membershp grade o he player n vague se ( denoes he alse membershp grade o he player n vague se e denoes a n-dmenson vecor where e ( = 1 or any oherwse e ( = 0 The se o all vague ses n N s denoed by TF(N Le TF(N he cardnaly o s denoed by ( e ndcaes he cardnaly o ( e { 1 ( 0} For any TF(N we use he noaon only ( = ( ( = ( or ( = ( =0 or any N A uncon v: TF( N sasyng v ( 0 s called a uzzy game on vague ses Le ( N denoe he se o all uzzy games on TF(N For any TF(N v ( s sad o be rue value or ve ( s sad o be upper value or Le TF(N we have or any N ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Denon 1 Le vg V TF(N v s sad o be convex n vague se v ( v ( v ( v ( ( ( ve ve ve ve ( ( or any E-IN: Issue 2 Volume 9 February 2012

3 WEA TRANACTIN on INFRMATIN CIENCE APPLICATIN Denon 2 Le vg V TF(N he vecor n n x x x x x x x s sad o be an mpuaon or v n (1 x v ( ( x v(1 ( ; (2 x v x ve ( ( Denon 3 Le vg V TF(N he core ( v or v n s dened by CV CV( v x x x ( x v x ve x v ( ( x ve ( Denon 4 Le vg V TF(N he vague se s sad o be a carrer or v n we have or any v ( v ( ( ve ve From Denon 4 we know he vague se s a carrer or v n only e s a carrer or v n e respecvely mlar o he denon o populaon monoonc allocaon uncon (PMAF [11] we gve he denon o VPMAF as ollows: Denon 5 Le vg V TF(N he vecor y y y s sad o be a VPMAF or v n ysases 1 y v( ; y ( v e 2 y ( y ( y ( e y ( e s 3 The hapley value or uzzy games on vague ses mlar o he denon o he hapley value or radonal games uzzy games we gve he denon o he hapley value or uzzy games on vague ses as ollows: Denon 6 Le vg V TF(N A uncon : s sad o be he hapley value or v n sases he ollowng axoms: Axom 1: I s a carrer or v n hen we have v ( ( v ve e v ; ( ( Axom 2: For j we have v ( ( v ( ( j ve ve j ( ( ( ( j ( ( or any wh j hen we ge ( v ( v j e v e v ; ( j( Axom 3: Le v w G we have V ( v w ( v ( w ( e v w e v e w ( ( Theorem 1 Le vg V ( N TF(N he uncon ( v ( e v : dened by E-IN: Issue 2 Volume 9 February 2012

4 WEA TRANACTIN on INFRMATIN CIENCE APPLICATIN ( v ( v( ( v( (1 e ( e v e ( e ve ve (2 ( ( ( ( ( or any where!( 1!! e ( e! e ( ( e ( e 1! ( e! only ( ( or ( 0 or any only ( ( or ( 0or any Then ( v ( e v s he unque hapley value or v n Remark 1 ( v s sad o be he player rue hapley values ( e v s sad o be he player upper hapley values When he gven uzzy games are convex he nerval number ( v ( e v s called he player possble payos wh respec o he hapley uncon Proo Axom 1: For any \ rom Denon 4 we have v ( ( v ( ( ( v(( ( ( v ( v ( or any wh From (1 we ge v ( v ( v ( ( v ( v ; mlarly we have (( v ( v ( (( v ( v ( ve e v ( ( Axom 2: From (1 we ge ( v (( v ( v ( j (( v ( v ( ( j j j ( v ( ( ( j v ( ( j (( v ( j v ( ( j j ( v ( ( ( j v ( ( (( v ( j v ( j ( v ; mlarly we oban e v e v ( j( From (1 (2 we can easly ge Axom 3 nqueness: For any vg V TF(N snce v resrced n can be unquely expressed by where u v u e e e E-IN: Issue 2 Volume 9 February 2012

5 WEA TRANACTIN on INFRMATIN CIENCE APPLICATIN e u e v ( 1 ( e e ( 1 ( e 1 u ( 0 oherwse v 1 ( e 0 oherwse From Axom 3 we only need o show he unqueness o on unanmy game u u e where nce s a carrer or u rom Axom 1 Axom 2 we ge 1 ( u 0 oherwse mlarly we have ( e u e 1 ( e e ( The proo s nshed 0 oherwse Theorem 2 Le vg V TF(N v s convex n hen s a VPMAF or v n ( v v ( e Proo From (1 (2 we can easly ge v v( ( e v v e ( ( In he ollowng we shall show he second condon n Denon 5 holds From Denon 1 we have v ( ( v ( v ( ( v ( ( ( ve ve v e v e ( ( ( or any where When 1 For anyw we have W WH where H\ W W!( W 1!! W (!( ( 1! H W H W H! From (1 we ge ( v W (( v ( v ( W W W W W W W (( v ( v( W H W H (( ( ( W H v W H v W H W H\ W W (( v ( v ( W W ( v W W or any By nducon we have ( v ( v or any any mlarly we have ( e v ( e v or any any Thus we oban ( v ( e v s a VPMAF or v n p u p Theorem 3 Le vg V TF(N I v s convex n hen E-IN: Issue 2 Volume 9 February 2012

6 WEA TRANACTIN on INFRMATIN CIENCE APPLICATIN ( ( e v v C ( v Proo From Theorem 1 we oban v v( ( e v v e From Theorem 2 we have Namely ( ( v ( ( v ( v ve e v ( ( ( e v ( ( e v v C ( v Corollary 1 Le vg V TF(N ose v s convex n hen ( v v ( e p u p s an mpuaon or v n Proposon 1 Le vg V TF(N ose we have v ( ( v ( v ( ( ve ve v ( ( ( ( (1 ( or any wh Then we have ( v v( ( e v v ( (1 ( V V Corollary 2 Le vg V TF(N ose we have v ( ( v ( ve ve ( ( ( ( or any wh Then we have ( v ( e v 0 Proposon 2 Le v wg V TF(N ose we have v ( ( v ( w ( ( w ( ve ve ( ( ( ( we we ( ( ( ( or any any wh Then we have or any ( v ( w ( e v ( e w 4 The hapley value or a specal knd o uzzy games on vague ses In hs secon we wll dscuss a specal knd o uzzy games whch s named as uzzy games wh mullnear exenson The uzzy coalon value or hs class o uzzy games s wren as n [9]: v ( R { R ( (1 R ( } v( T T0 R\ T0 0 0 T0 R where R s a uzzy coalon as usual Le denoe he se o all uzzy games on vague ses wh mullnear exenson orm For any TF(N we have \ 0 0 (3 v ( { ( (1 ( } v ( H (4 E-IN: Issue 2 Volume 9 February 2012

7 WEA TRANACTIN on INFRMATIN CIENCE APPLICATIN H 0 ( e v ( e (1 ( v H (5 e H ( ( \ 0( 0 0 When we resrc he doman o n he seng o ( N rom denons o VPMAF mpuaon gven n secon 3 we can ge he denons o VPMAF mpuaon or v n Here we om hem Denon 6 v ( 0 G0 N s sad o be convex v ( T v ( v ( T v ( T or all 0 T P( N 0 Denon 7 For v TF(N he core CV ( v or v n s dened by C ( ( V v x x x x v x v e x v ( ( x v( e Theorem 4 Le v ( N TF(N he uncon ( v ( e v : G V ( N dened by j ( v ( ( j (1 ( j ( v ( H v ( H (6 j \ ( e v e e ( e ( e jh 0 j( e \ H 0 (1 ( (1 ( j ( j ( v ( H v ( H ( or any where e e are lke n Theorem 1 Then ( v ( e v s he unque hapley value or v n Proo From (4 (5 we have v ( ( v ( \ ( ( (1 ( ( v ( H v ( H v e v e ( ( ( ( j ( e (1 ( (1 ( j j v H v H j e H ( ( ( \ 0( 0 0( 0 0 From Theorem 1 (6 (7 we know he exsence holds In he ollowng we shall show he unqueness holds nce v ( N he resrced n can be unquely expressed by v where e e u e Thus we ge u e u e ( 1 v( e e ( 1 ( e u as gven n Theorem 1 1 ( u 0 oherwse v E-IN: Issue 2 Volume 9 February 2012

8 WEA TRANACTIN on INFRMATIN CIENCE APPLICATIN ( e u e 1 ( e ( e 0 oherwse Theorem 5 Le v TF(N I he assocaed crsp game v 0 G 0 o v s convex hen ( v ( e v s a VPMAF or v n Proo From (6 (7 we can easly ge v v( ( ( e v v( e Nex we shall show he second condon holds From he convexy o v 0 G 0 ( N (4 (5 we ge v resrced n s convex From Denon 1 we ge v ( ( v ( v ( ( v ( ( ( v e v e v e v e ( ( ( or any where From (6 (7 Theorem 2 we oban ( v ( v ( e v ( e v or any any Namely ( v ( e v VPMAF or v n s a Theorem 6 Le v TF(N I he assocaed crsp game v 0 G 0 o v s convex hen ( v ( e v C ( V v Proo The proo o Theorem 6 s smlar o ha o Theorem 3 Corollary 3 Le v TF(N I he assocaed crsp game v 0 G 0 o v s convex hen ( v ( e v s an mpuaon or v n Theorem 7 Le v TF(N I he assocaed crsp game v 0 G 0 o v s convex hen he core CV ( v can be expressed by CV ( v x x x x H \ H y ( (1 ( 0 0 R 0 R0 ( e x (1 ( 0 ( ( \ y R H 0 e R0 y C( H v R ( e T R0 y C( R0 v 0 where CH ( 0 v0 denoes he core n H 0 or v0 CR ( 0 v0 denoes he core n R 0 or v 0 Proo The proo o Theorem 7 s smlar o ha o Proposon 41 gven n [15] nce he uzzy games n esablsh he specc relaonshp beween he uzzy coalon values ha o her assocaed crsp coalons The properes or hs class o uzzy games can be obaned by researchng her assocaed crsp games 5 Numercal example There are hree companes named ha decde o cooperae wh her resources They can combne reely For example 0 = {1 2} denoes he E-IN: Issue 2 Volume 9 February 2012

9 WEA TRANACTIN on INFRMATIN CIENCE APPLICATIN cooperaon beween company 1 2 nce here are many uncerany acors durng he cooperaon each player s no wllng o oer all s resources o hs specc cooperaon In anoher word hey only supply par o her resources In order o reduce rsk ge more payos when he players ake par n hs cooperaon hey only know he deermnng parcpaon levels he parcpaon levels ha hey do no parcpae For example he company 1 has uns o resources he deermnng parcpaon level s 3000 uns 2000 uns are no devoed o cooperaon Namely he rue membershp grade o he player 1 s 03 = 3000/ he alse membershp grade o he player 1 s 02 = 2000/10000 In such a way a vague se s nerpreed Consder a vague coalon dened by = {123} [ ( 1 ( ] / = [03 06]/1+ [02 03]/2 + [06 08]/3 I he crsp coalon values are gven by able 1 Table 1 The uzzy payos o he crsp coalons (mllons o dollars 0 v 0 ( 0 0 v 0 ( 0 {1} {2} {3} {12} {13} {23} {123} From able 1 we know ha when he company 1 2 cooperae wh all her resources hen her payo s 6 mllons o dollars When he relaonshp beween he values o he uzzy coalons ha o her assocaed crsp coalons as gven n (4 (5 Namely hs uzzy game belongs o ( N From (6 we ge he player rue hapley values are ( v ( v ( v From (7 we ge he player upper hapley values are ( e v ( e v e v ( nce he assocaed crsp game s convex we know ha he player hapley values s a VMPAF an elemen n s core Furhermore he player possble payos wh respec o he hapley uncon are [ ] [ ] [ ] 6 Concluson In some cooperave games he players only know he deermnaon parcpaon levels he levels ha hey do no parcpae The uzzy games on vague ses can well solve hs suaon For hs purpose we research he uzzy games on vague ses dscuss he hapley value or uzzy games on vague ses When he gven uzzy games on vague ses are convex some properes are nvesgaed Furhermore we sudy a specal knd o uzzy games on vague ses The hapley value he core or hs knd o uzzy games are suded However we only sudy he hapley value or uzzy games on vague ses wll be neresng o sudy oher payo ndces 7 Acknowledgmen The auhors graeully hank he Che Edor Pro Panos osaraks anonymous reerees or her valuable commens whch have much mproved he paper Ths work was suppored by he Naonal Naural cence Foundaon o Chna (Nos Reerences: [1] Aubn JP Mahemacal Mehods o Game Economc Theory Norh-Holl Amserdam 1982 [2] Bunaru D ably hapley value or an n-persons uzzy game Fuzzy es ysems Vol4 No pp [3] Bunaru D roupa T hapley mappngs he cumulave value or n- person games wh uzzy coalons European Journal o peraonal Research Vol186 No pp [4] Borkookey Cooperave games wh uzzy coalons uzzy characersc uncons Fuzzy es ysems Vol159 No pp [5] Gau WL Buehrer DJ Vague ses IEEE Transacons on ysems Man Cybernecs Vol23 No pp [6] L J Zhang Q A smpled expresson o he hapley uncon or uzzy game European Journal o peraonal Research Vol196 No pp E-IN: Issue 2 Volume 9 February 2012

10 WEA TRANACTIN on INFRMATIN CIENCE APPLICATIN [7] Meng FY Zhang Q The hapley uncon or uzzy cooperave games wh mullnear exenson orm Appled Mahemacs Leers Vol23 No pp [8] Meng FY Zhang Q The hapley value on a knd o cooperave uzzy games Journal o Compuaonal Inormaon ysems Vol7 No pp [9] wen G Mullnear exensons o games Managemen cences Vol18 No pp [10] akawa M Nshzalz I A lexcographcal soluon concep n an n-person cooperave uzzy game Fuzzy es ysems Vol61 No pp [11] prumon Y Populaon monoonc allocaon schemes or cooperave games wh ranserable uly Games Economc Behavor Vol2 No pp [12] Tsurum M Tanno T Inuguch M A hapley uncon on a class o cooperave uzzy games European Journal o peraonal Research Vol129 No pp [13] Tjs Branze R Ishhara Muo n cores sable ses or uzzy games Fuzzy es ysems Vol146 No pp [14] Yu XH Zhang Q The uzzy core n games wh uzzy coalons Journal o Compuaonal Appled Mahemacs Vol 230 No pp [15] Yu XH Zhang Q An exenson o cooperave uzzy games Fuzzy es ysems Vol161 No pp E-IN: Issue 2 Volume 9 February 2012