ELIMINATION OF DOMINATED STRATEGIES AND INESSENTIAL PLAYERS

Transcription

1 OPERATIONS RESEARCH AND DECISIONS No DOI: /ord1513 Mamoru KANEKO 1 Shuge LIU 1 ELIMINATION OF DOMINATED STRATEGIES AND INESSENTIAL PLAYERS We udy he proce, called he IEDI proce, of eraed elmnaon of (rcly) domnaed raege and neenal player for fne raegc game. Such elmnaon may reduce he ze of a game conderably, for example, from a game wh a large number of player o one wh a few player. We exend wo exng reul o our conex; he preervaon of Nah equlbra and orderndependence. Thee gve a way of compung he e of Nah equlbra for an nal uaon from he endgame. Then, we revere our perpecve o ak he queon of wha nal uaon end up a a gven fnal game. We ae wha uaon underle an endgame. We gve condon for he paern of player e requred for a reulng equence of he IEDI proce o an endgame. We llurae our developmen wh a few exenon of he bale of he exe. Keyword: domnaed raege, neenal player, eraed elmnaon, order-ndependence, emaon of nal game 1. Inroducon Elmnaon of domnaed raege a bac noon n game heory, and relaonhp o oher oluon concep, uch a raonalzably, have been exenvely dcued [5, 11]. I naure, however, dffer from oher oluon concep; ugge negavely wha would/hould no be played, whle oher concep ugge/predc wha would/hould be choen n game. In h paper, we alo conder he elmnaon of neenal player whoe unlaeral change of raege do no affec any player payoff ncludng h own. Th concep a bac a ha of domnaed raege. We 1 Waeda Unvery, Shnuku-ku, Tokyo, Japan, e-mal addree: mkanekoep@waeda.p, huge_lu@aag.waeda.p

2 34 M. KANEKO, S. LIU conder he proce of eraed elmnaon of domnaed raege and of neenal player, whch we call he IEDI proce. Thee wo ype of elmnaon nerac wh each oher, and he uaon dffer from ha of only elmnaon of domnaed raege. To ee uch neracon, a well a her negave naure, we conder hree example here. The fr decrbed n a prece manner bu he oher wo n an ndcave manner. Example 1.1. Bale of he exe wh a econd boy. Conder a bale of he exe uaon conng of boy 1, grl 2, and anoher boy 3. Each boy 1, 3 ha wo raege, 1, 2and grl 2 ha four raege, 21,..., 24. Boy 1 and grl 2 can dae a he boxng arena ( ) or he cnema ( ) bu make decon ndependenly. Now, boy 3 ener h cene. Grl 2 can dae boy 3 n a dfferen arena ( 23 31) or cnema ( 24 32). When 1 and 2 conder her dae, hey would be happy even f hey fal o mee; 3 choce doe no affec her payoff a all. Alo, we aume ha when 3 hnk abou he cae ha 2 chooe o dae boy 1, boy 3 adly ndfferen beween h arena and cnema. The ame ndfference aumed for 1 when 2 chooe o dae 3. Aumng h, her payoff are decrbed a Table 1 and 2. The number n he parenhee n Table 1 are 3 payoff. The dang uaon for 3 and 2 parallel and decrbed n Table 2; bu grl 2 much le happy. Table 1. Beween 1 and 2 1\2 (3) , 1 ( 1) 5, 5 ( 5) 5, 5 ( 5) 1, 15 ( 1) 12 Table 2. Beween 3 and 2 3\2 (1) , 1 ( 1) 5, ( 5) 5, ( 5) 1, 2 ( 1) 32 In h game, 2 raege 23 and 24 are domnaed by 21 and, 22 nce he wan o dae boy 1. Elmnang hee domnaed raege, we oban a maller game. Now, 3 neenal n he ene ha 3 choce doe no affec any of he player. Thu, we elmnae 3 a an neenal player, and oban he 2-peron bale of he exe. In he game heory leraure, andard o ar wh a gven game, and analyze wh ome oluon concep. Some abracon ake place before reachng h gv-

3 Elmnaon of domnaed raege and neenal player 35 en game. In he above cae, elmnaon of he domnaed raege for grl 2 and of boy 3 a an neenal player conue h proce o oban he 2-peron bale of he exe. In Example 1.1, elmnaon of domnaed raege generae neenal player. However, he poble neracon beween elmnaon of domnaed raege and of neenal player are more complcaed and can be ummarzed a follow: (a) elmnaon of domnaed raege may generae boh new domnaed raege and new neenal player; (b) elmnaon of neenal player can only generae new neenal player. Hence, we oban a proce of eraed elmnaon of domnaed raege and of neenal player, whch our IEDI proce. Th an exenon of he proce known a eraed elmnaon of domnaed raege n he leraure [5, 11]. Elmnaon of boh may reduce a large game no a mall game n he ene of he ze of he player e and raegy e. Alo, he followng example how very dfferen ocal uaon underlyng he ame bale of he exe. Example 1.2. A game wh many player quckly reduced o a mall game. We add 99 boy o Example 1.1, who are he ame a boy 3 from he dang perpecve. Now, he uaon con of 12 player bu all could be eenal unle 2 gnore hee 1 boy. Her raege o dae any one of hem are domnaed by her dang raege nvolvng boy 1. Once hee domnaed raege are elmnaed, he boy from 3 o 12 all become neenal and can be elmnaed. Agan, we have he 2-peron bale of he exe. In Example 1.2, we need only wo ep f we allow mulaneou elmnaon of mulple domnaed raege and mulple neenal player. However, here are dfferen uaon where many ep are requred o reach an endgame. In he nex example, he reulng oucome he ame 2-peron bale of he exe bu he proce nrncally longer. Example 1.3. Reducon ake many ep. Agan, we add 99 boy o Example 1.1, where hey are onlooker. We aume ha player k + 1 a frend of k and k + 1 opnon affec only k payoff (k = 3,..., 11); k + 1 ha wo acon: eher o encourage k o ell h opnon o k 1 or no ( k 4) and 4 can encourage 3 o cheer up. We aume ha f 2 chooe o dae 1, hen 3 would be ndfferen beween h choce wh or whou 4 encouragemen. The argumen n Example 1.1 appled o h;.e., elmnang 2 domnaed raege 23 and 24, boy 3 become neenal and elmnaed. Then, boy 4 loe a frend o cheer up and become neenal. Smlarly, f k dappear, hen k + 1 neenal. Afer 1 erave elmnaon, we have he 2-peron bale of he exe. In h example, elmnaon of neenal player only generae new neenal player. The hree example above have dfferen nal uaon and how dfferen elmnaon procee, whle he endgame he ame. Such procee can have dfferen

4 36 M. KANEKO, S. LIU poble combnaon for he elmnaon of domnaed raege and neenal player. In an IEDI proce, we ake he order n whch domnaed raege and hen neenal player are elmnaed no conderaon. The equence reulng from h proce called an IEDI equence. Among uch poble equence, one ype repreenave, whch we call he rc IEDI equence; n each ucceve round, fr all domnaed raege are elmnaed and hen all neenal player are elmnaed. Two exng reul n he leraure are convered o our conex. One he preervaon heorem ([5], Theorem 4.35), ang ha he Nah equlbra are preerved n he proce of elmnang domnaed raege. Th exended o he IEDI proce (Theorem 2.1). The oher he order-ndependence heorem [1, 3]: he proce reul n he ame endgame regardle of he order n whch domnaed raege are elmnaed. Th alo exended o our conex (Theorem 3.1), and addonally hown ha he rc IEDI equence he hore and malle among poble IEDI equence. Thee wo reul gve a mple way of compung he e of Nah equlbra from he endgame o ha of he nal game; he mehod gven explcly a (8) n Secon 3.1. The IEDI proce can be regarded a an abracon proce from a ocal uaon no a mple decrpon by elmnang ome rrelevan facor. The above example how ha here are very dfferen underlyng uaon ha end up a he ame endgame. In Secon 4, we ak he revere queon of wha are poble underlyng uaon ha end up a a gven game. We focu on a equence of par of e of player, whch pecfe he player e and he ube of player wh domnaed raege o be elmnaed. Once uch a equence and an endgame are gven, we reconruc an IEDI equence. The characerzaon heorem (Theorem 4.1) gve condon o reconruc a rc IEDI equence. Ung h, we can nfer he poble underlyng uaon behnd a gven endgame. The paper organzed a follow: Secon 2 gve bac defnon of domnance, an neenal player, and preen our preervaon heorem. Secon 3 defne he IEDI proce, and prove our order-ndependence heorem. Secon 4 gve and prove he characerzaon heorem. In Secon 5, we gve a ummary and dcu ome remanng problem. 2. Elmnaon of domnaed raege and neenal player We defne hree way of reducng a game by elmnaon of domnaed raege and of neenal player bu we how ha one way more effecve han he oher. We alo how ha he Nah equlbra are preerved n hee reducon.

5 Elmnaon of domnaed raege and neenal player Bac defnon Le G ( N,{ S },{ h} ) be a fne raegc game, where N a e of player, N N S a nonempy e of raege, and h : Π S N he payoff funcon for player N. We allow N o be empy, n whch cae he game he empy game, denoed a SN : Π NS a ( I ; N I), where I { } I and N I { } NI. When I {}, we wre S for SN {} and ( ; ) for ( ; ). { } N { } Le G be gven, and, S. We ay ha domnae n G ff h( ; ) h( ; ) for all S. When domnaed by ome, we mply ay ha domnaed n G. We ay ha an neenal player n G ff for all N, h ( ; ) h ( ; ) for all, S and S (1) A choce by player doe no affec any player payoff ncludng own, provded ha he oher raege are fxed. Noe ha when S 1, player already neenal. We fnd a weaker veron of h concep n Mouln [7], who defned he concep of d-olvably by only requrng (1) for. Once player become neenal n h ene, he may op hnkng abou h choce bu may ll affec he oher payoff; n h cae, he ll relevan o hem. Alhough (1) an arbue of a ngle player, we can rea a group of uch player a neenal, whch aed n he followng lemma 2. Lemma 2.1. Le I be a e of neenal player n G. Then, for all N, h ( ; ) h ( ; ) forall, S and S (2) I N I I NI I I I NI NI Proof. Le I { 1,..., k }, and I {,... 1, } for 1,..., k. Alo, le, SN be fxed. We prove h ( I ; ) ( ; ) NI h I NI by nducon on 1,..., k. The bae cae,.e., h( ; ) ( ; ), 1 h 1 1 obaned from (1). Suppoe ha h ( ; ) 1 I N I h( I ; ). N I Snce ( ; ) ( ; ), I NI I 1 NI we have h 1 ( I ; ) 1 N I1 h( I ; ). N I By (1), h ( ; ) I N I h ( I ; ). 1 N I By he uppoon, we oban 1 2 The concep of an neenal player concepually dffer from ha of a dummy player n cooperave game heory (cf., Oborne Rubnen [11], p. 28). Ung he maxmn defnon of a characerc funcon game, we have example o how he logcal ndependence of hoe wo concep.

6 38 M. KANEKO, S. LIU h( I ; ) 1 N I h ( ; ) 1 I N I h ( ; ) I N I h ( ; ). I 1 N I Thu, he aeron 1 hold for +1. Le I be a e of neenal player n G, N N I, and any player n N. The rercon h of h on Π N S wh S S for N defned by h( ) h( ; ) forall S and S (3) N I N N N I I The well-defnedne of h guaraneed by Lemma 2.1. Thu, ( N,{ S } N,{ h } N) he raegc game obaned from G ( N,{ S} N,{ h} N) by elmnang a e of neenal player I and ome raege from S for N. We fr gve a general defnon: We ay ha G ( N,{ S } N,{ h } N) a D- -reducon of G ( N, { S },{ h} ) ff N N DR1. N N and any N N an neenal player n G ; DR2. For all N, S S and any S S a domnaed raegy n G ; DR3. h he rercon of h o Π N S. Some domnaed raege and neenal player n G may no be elmnaed durng he reducon o G. Such domnaed raege and neenal player reman domnaed and neenal n G, whch aed n Lemma and Clam mple ha elmnaon of neenal player generae no new domnaed raege. Lemma 2.2. Le G ( N,{ S } N,{ h } N) be a D-reducon of G If S ( N) domnaed n G, hen domnaed n G If N an neenal player n G, hen neenal n G Suppoe ha S S for all N. Le N and S. Then, domnaed n G f and only f domnaed n G. Proof of Suppoe ha domnaed by n G. Then, h( ; N ) h( ; N ) for all N SN. We can aume whou lo of generaly ha no a domnaed raegy n G, o S. We have, by (3), for all NN SNN, h( ; ) h( ; ; ) h( ; ; ) h( ; ) for all S. N N NN N NN N domnaed by n. Thu, G The proof of (2.2.2) mlar. N N

7 Elmnaon of domnaed raege and neenal player The only-f par mmedae. Conder he f par. Suppoe ha domnaed by n G. Then, h ( ; N) h ( ; N) for all N S N. Le N be any elemen n S N SN. By (3), for all NN SNN, h( ; N; NN) h ( ; N ) h ( ; N ) h( ; N ; N N). Thu, domnaed by n G. A D-reducon allow mulaneou elmnaon of boh domnaed raege and neenal player. However, would be eaer o eparae hee ype of elmnaon. Fr, le N N hold n DR1,.e., G reul from G by elmnang ome domnaed raege; n h cae, G called a d-reducon of G, denoed a G d G. Second, le S S for all N n DR2,.e., G reul from G by elmnang ome neenal player; n h cae, G called an p-reducon of G, denoed by G p G. When all domnaed raege are elmnaed n Gd G, called he rc d-reducon, and mlarly, when all neenal player are elmnaed n G p G called he rc p-reducon. We focu on he order n whch d-reducon appled and hen p-reducon done. We ay ha G a DI-(compound) reducon of G ff here an nerpolang game G uch ha G d G and G p G. We ay ha G he rc DI-reducon of G ff boh G d G and Gp G are rc. Even f G G, poble ha G G or G G. For comparon, we conder anoher compound reducon; G an ID-reducon of G ff Gp Gd G for ome G. Lemma ae ha ID -reducon are equvalen o D-reducon bu ha a DI-reducon allow more poble. The convere of doe no hold; n Example 1.1, 3 become neenal afer elmnaon of 2 domnaed raege. Lemma G a D-reducon of G f and only f G an ID-reducon of G If G a D-reducon of G, hen G a DI -reducon of G. Proof of Only-If. Le G be a D-reducon of G. I follow from Lemma ha we can popone elmnaon of domnaed raege unl he elmnaon of neenal player ha been carred ou. Hence, G can be an ID-reducon. (If): Le G be an ID-reducon of G,.e., G p G d G for ome G. Lemma ae ha G ha he ame e of domnaed raege a G. Hence, we can combne hee wo reducon no one, whch yeld he D-reducon of G.

8 4 M. KANEKO, S. LIU Th can be proved by a mlar argumen o he only-f par of poponng p-reducon, nead of d-reducon Preervaon of Nah equlbra The concep of a D-reducon reduce a game by elmnang rrelevan player a well a rrelevan acon for ome player. I derable ha uch a reducon loe no eenal feaure of he ocal uaon beng modeled. Th wha Merern [6] mall world axom requre for a oluon concep. Here, we how ha h hold for he concep of Nah equlbrum wh repec o a D-reducon. In addon, he convere hold n our cae. We ay ha S a Nah equlbrum n a nonempy game G ff for all N, h() h( ; ) for all S. Le be he null ymbol,.e., for any S we e ( ; ), and pulae ha he rercon of o he empy game G he null ymbol. Alo, we pulae ha he Nah equlbrum n G. We have he followng heorem. The fr clam correpond o Meren [6], p. 733, mall world axom, for he cae of Nah equlbrum. Boh clam are preened n [5], Theorem 4.35, p. 19, for he cae of elmnaon of domnaed raege only. Theorem 2.1. Preervaon of Nah equlbra. Le G be a D-reducon of G. A. If N an NE n G, hen rercon N on G an NE n G. B. If N an NE n G, hen ( N ; N N) an NE n G for any N N n Π N NS. Proof of A. Le be an NE n G. If N, h( ; ) h( ; ) for any S. Le N. Then S, nce no domnaed n G. Le S. Snce G a D- -reducon, we have h ( ; N) h( ; N) h( ; N) h ( ; N); o N an NE n G. o o B. Le N be an NE n G. We chooe any N N S N N. We le G ( N,{ S } N, o o { h} N) where S S f N; and S S f N N. The rercon of h on o o Π NS denoed by h elf. Fr, we how ha ( N ; N N) an NE n G. Le N. Then, h ( N ) h( N ; N N) for any N S N by Lemma 2.1, nce he player n N N are neenal n G. Snce N an NE n G, we have h( ; ; ) h( ; ) h( ; ) h( ; ; ) for all S. Le N NN N N N NN

9 Elmnaon of domnaed raege and neenal player 41 o N N. Then, h( ; ; ) h( ; ; ) for all S. Hence, N NN N NN o ( N ; N N) an NE n G. Now we how ha ( N ; N N) an NE n G. Le N. Suppoe ha N ha a raegy n G uch ha h( ; N) h( ; N). We can chooe uch an gvng he maxmum h( ; N ). Th no domnaed n G. Hence, reman n G, whch conradc he aemen ha N an NE n G. Le NE( G ) and NE( G ) be he e of Nah equlbra for a game G and D-reducon G. I follow from Theorem 2.1 ha NE( G ) and NE( G ) are conneced by: NE( G) Π NNS NEG ( ) (4) Here, we pulae ha when N N, Π N N S he un e wh repec o he e mulplcaon,.e., NE( G) NE( G). When G an empy game G, he Nah equlbrum of G he null ymbol, and Theorem 2.1.B ae ha any raegy profle ( ; ) a Nah equlbrum n G. I follow from Lemma ha (4) hold when G an ID-reducon of G. For a DI-reducon G of G, (4) hould be modfed lghly. Le G G and G G, where G ( N,{ S},{ h} ) he nerpolang game. Then, p N N NE( G) S NE( G ) (5) N N Snce N N and S S for all N, we fr have NE( G) Π N N S NE( G) by (4), and hen we oban NE( G) NEG ( ) Π NNS NEG ( ). Noe NE( G ) NE( G), nce he domnaed raege n G are no n NE( G ). The formula (5) wll be ued o gve a way of compung he e of NE of an nal game from he endgame n he IEDI proce. Theorem 2.1 hold wh repec o mxed raegy Nah equlbra, a well a raonalzably, correlaed equlbra and Nah [8] non-cooperave oluon. So far, we only have pove reul a long a he concep of purely non-cooperave oluon are concerned 3. d 3 The oluon concep called he nraperonal coordnaon equlbrum n Kaneko Klne [4] regarded a a concep of a non-cooperave oluon bu ncompable wh he elmnaon of domnaed raege. I capure ome cooperave apec hrough an ndvdual nraperonal hnkng abou

10 42 M. KANEKO, S. LIU 3. The IEDI proce and generaed equence Here, we conder he proce of eraed elmnaon of domnaed raege and neenal player (he IEDI proce). In Secon 3.1, we preen an exenon of he order-ndependence heorem, and n Secon 3.2, we gve a heorem dvdng elmnaon of neenal player from ha of domnaed raege IEDI equence and order-ndependence We ay ha 1 Γ( ),,..., G G G G an IEDI equence from a game G ff 1 1 G DI G G G a -reducon of and for each,..., 1; (6) G ha no domnaed raege and no neenal player (7) We ay ha (G ) = G 1,..., G he rc IEDI equence ff G he rc DIreducon of G for,..., 1. The rc IEDI equence unquely deermned byg. Example 3.1. Conder Example 1.1. The rc IEDI equence gven n Fg. 1. Player 2 raege 23 and 24 are domnaed by 21 and. 22 Then, by elmnang 23 and, 24 we ge he econd nerpolang 3-peron game. Now, 1 and 2 focu on her dang, gnorng player 3 a neenal. Elmnang hm, we oban a 2-peron bale of he exe. Th a DI-reducon of G G. Th IEDI ha lengh 1. There are wo oher IEDI ; 23 and 24 are elmnaed equenally, hen player 3 elmnaed a neenal. Each ha lengh 2. G d 1\2\ , 1, 1 5, 5, 5 p , 5, 5 1, 1, or 32 Fg. 1. The rc IEDI from Example 1.1 1\ , 1 5, 5 5, 5 1, 15 oher hnkng. An example of he non-preervaon of uch equlbrum occur n he Proner Dlemma.

11 Elmnaon of domnaed raege and neenal player 43 I known a he order-ndependence heorem [3, 1] ha wh eraed elmnaon of only domnaed raege, he order n whch raege are elmnaed doe no affec he endgame. Here, we exend h reul o he above defnon ncludng elmnaon of neenal player. We focu no only on he endgame of IEDI equence bu alo on comparon beween hee equence. We ay ha G ( N,{ S } N,{ h } N) a ubgame of G ( N,{ S} N,{ h} N) ff N N and S S for all N. If G a D-reducon of G, hen G a ubgame of k G. For an IEDI equence Γ( G ) G,..., G, f k, G a ubgame of G. We have he followng heorem, whch wll be proved a he end of h econ. Theorem 3.1. Order-ndependence, hore, and malle. Le G be a game, and Γ ( G ) G,, G he rc IEDI equence from G G. Then for any IEDI equence Γ( G ) G,, G from G, (A) G G ; (B) ; (C) for each, G a ubgame of G. Clam (A) order-ndependence 4. Clam (B) and (C) mean ha he rc IEDI equence he hore and malle wh repec o he lengh of IEDI equence and he ze of her componen game, repecvely. In Example 1.2, he rc IEDI equence ha lengh 1. However, here are many non-rc IEDI equence wh much longer lengh. In h example, grl 2 ha many dang choce, e.g., 2 (choce) 11 (boy) 22 choce. Hence, he longe IEDI equence con of he equenal elmnaon of 2 domnaed raege and 1 neenal player; he lengh hu 3. There are alo many poble order of hee elmnaon. Example 1.3 doe no requre player 2 o have more raege. Here, he rc IEDI ha lengh 1, and he longe IEDI equence ha lengh 12, nce ake wo ep o elmnae 23 and 24, and hen player from 3 o 12 are elmnaed equenally. If we focu nally only on elmnaon of domnaed raege, he 1 player reman n hee game. Elmnang hem, he game are reduced o he 2-peron bale of he exe. We have oher elmnaon procee adopng dfferen reducon uch a D- and ID-reducon. From Lemma 2.3, he rc IEDI Γ ( G ) horer and maller han he equence baed on D- or ID-reducon. I would alo be poble o apply only d-reducon unl all domnaed raege are elmnaed and hen o apply p- -reducon. The rc IEDI equence horer han or equal o h equence, a long 4 The order-ndependence heorem doe no hold for weak domnance (cf. [1], p. 6). See [1] for comprehenve dcuon on order-ndependence heorem for varou ype of domnance relaon.

12 44 M. KANEKO, S. LIU a we coun each of he DI-reducon n he rc IEDI a one ep. However, ome IEDI mgh be horer han he rc IEDI f we coun each DI-reducon conng of nonrval ub-reducon a 2 ep. We can ee Theorem 3.1 from he vewpon of he preervaon of Nah equlbra. By applyng (4) o Γ ( G ) G,..., G repeaedly, we oban he reul ha f G G ha a Nah equlbrum, hen o doe G G. Th hold even f he empy game. If G ha no Nah equlbra, he nal game G G ha no Nah equlbra eher. Th gve a mehod for compung he NE e, NE( G ), for any gven game G. Le Γ( ),, G G G be an IEDI, and G ( N,{ S },{ h } ) he nerpolang game beween G and 1 G for,..., 1. The e N N ) NE( G wren a: NE( G) Π S... Π S NEG ( ) N N N N (8) I follow from (5) ha ( NE G ) NE( G ) and NE 1 ( G ) Π S NE 1 ( G ) for N N,..., 1. Repeang h proce from 1, we oban (8). Thu, we have an algorhm for compung NE( G ) along he IEDI proce. Formula (8) gve he e NE G ( ) regardle of an IEDI equence ued bu he rc IEDI gve he hore compuaon. In Example 1.1, (8) gve NE( G ) { ( 11, 21), ( 12, 22)} { 31, 32}. Smlarly, we oban NE( G ) {(, ), (, )} S S for Example 1.2 and Table 3. d-olvable bu nonempy 1\ , 1, 1 1,, 12 Fnally, we look a Mouln [7] concep of d-olvably; a game G d-olvable ff a equence 1 G,..., G wh G d G for 1,...,, uch ha n G, each N ha conan payoff when he oher raege are fxed. If G ha an IEDI G,..., G wh G G, hen G d-olvable. The convere doe no hold; Table 3, gven n [7], ha no domnaed raege and no neenal player bu d-olvable.

13 Elmnaon of domnaed raege and neenal player 45 Now le u prove Theorem 3.1. Fr, we refer o Newman lemma (ee alo [1]). An abrac reducon yem a par ( X, ), where X an arbrary nonempy e and a bnary relaon on X. We ay ha { x ;,...} a equence n ( X, ) ff for all, x X and x x 1 (a long a x 1 defned). We ue o denoe he ranve reflexve cloure of. We ay ha ( X, ) weakly confluen ff for each x, y, z X wh x y and x z, here ome x X uch ha y x and z x. Lemma 3.1 ([9]). Le ( X, ) be an abrac reducon yem afyng N1: each equence n X fne; and N2: ( X, ) weakly confluen. Then, for any x X, here a unque endpon y wh x y. Proof of Theorem 3.1 (A). Le be he e of all fne raegc game. Then (, DI ) an abrac reducon yem, where we wre G DI G for G d G and Gp G for ome nerpolang G and G G. The relaon ID reflexve. Each equence fne,.e., N1 hold. Le u how N2. Le GG,, G wh DI DI G G and G G. Le -reducon of boh G and G. Hence, from Lemma 3.1 ha for any, IEDI equence Γ ( G ) G,, G DI G be he rc DI-reducon of G. Then, G G DI and G here a unque endpon DI G. G a DI- G Thu, follow G.. Hence, he rc ha he ame endgame,.e., G G G. Now, we prove (C) n a weaker form. Then, we prove (B), from whch (C) follow. C.We prove by nducon on ha G a ubgame of G for each mn(, ). Th hold by defnon for. Suppoe ha h hold for 1 mn(, ). Le G G G and G G G 1. From Lemma 2.2.1, f a raegy n 2.2.2, f a player n d p G domnaed n, G neenal n d p G alo domnaed n G. From Lemma G he alo neenal n G. We oban 1 G by elmnang all domnaed raege n G and all neenal player n G ; 1 o G a ubgame of G 1. B. Le Γ( G ) G,, G be any IEDI equence. From (A), G G. If, hen, from (6), G G a rc ubgame of G. From ( C ), G a ubgame of G, whch a conradcon. Thu,,.e., (B) hold. Th mple (C).

14 46 M. KANEKO, S. LIU 3.2. The elmnaon dvde 1 An IEDI equence paroned no wo egmen, 1,, m G G m and G,, G, o ha n he fr egmen boh domnaed raege and neenal player can be elmnaed, and n he econd only neenal player are elmnaed. Theorem 3.2. Paron of an IEDI equence. Le Γ( G ) G,, G be an IE- DI equence from G. There exacly one m ( m ) uch ha (): ome domnaed raegy elmnaed gong from G m o G ; (): for each ( m 1), no m 1 domnaed raege are elmnaed bu ome neenal player elmnaed gong from G m 1 m o G. 1 Proof. Suppoe ha G ha no domnaed raege. Then, G obaned from 1 G by elmnang neenal player. I follow from Lemma ha G ha no domnaed raege. Thu, for any, G ha no domnaed raege. Hence, we chooe m o be he malle value among uch. We call he m gven by Theorem 3.2 he elmnaon dvde. In Example 2.1, m, and he egmen afer m may have he lengh greaer han 1. The elmnaon dvde m play an mporan role n Secon Characerzaon of nal uaon We have uded IEDI equence generaed from a gven nal game G, and have een ha here are many dfferen nal uaon, a well a many IEDI equence ha lead o he ame endgame G. G G 1 G 2 Reverng he Focu l G = G G G G G l 1 G l 1 G l 1 G Fg. 2. Sarng from he fnal game

15 Elmnaon of domnaed raege and neenal player 47 Here, we udy he cla of hoe nal uaon ha lead o a gven endgame G;.e., we revere our pon of vew from he op of Fg. 2 o he boom. We characerze wha underlyng ocal uaon can le behnd he ame G. We gve condon for a gven paern of player e correpondng o a equence of he IEDI proce ha lead o a gven game Evolvng player confguraon and he correpondng rc IEDI equence We ar wh a equence = [(N, T ),..., ( N, T )] of par of e of player, whch we call a equence of evolvng player confguraon (EPC). Here, N,..., N are he player e and T,...,T are he ube of player wh domnaed raege correpondng o ome IEDI equence Γ( G ) G,..., G. We wh o deermne wha condon on guaranee he exence of ome rc IEDI equence (G ) correpondng o. We gve four condon on, and he fr hree are a follow: PC. T N for,, ; and N... N wh N 1; PC1. For any, f T, hen N N 1 ; m PC2. For ome m ( m ), T and T f m. PC bac. I nend o mean ha he player e are decreang wh he elmnaon of 1 neenal player. N N he e of neenal player o be elmnaed and T a e of player n N wh domnaed raege o be elmnaed. I alo requre ha he 1 change do no op wh a ngle player. PC1 correpond o he requremen G G n (6). The number m n PC2 he elmnaon dvde dcued n Secon 3.2. The fourh condon for a rc IEDI equence. We ay ha an EPC equence [( N, T ),, ( N, T )] rc ff 1 PC3. For 1,..., m, f T 1 1, hen T T. Th ae ha f a ngle player domnaed raege are o be elmnaed, h elmnaon hould no generae any new domnaed raege for hm. Acually, PC PC3 are uffcen o guaranee he exence of a rc IEDI equence. To connec he EPC and IEDI equence, we defne he concep of a D-group. Le G be a DI-reducon of G wh G d G p G. We ay ha T { N : S S} he D-group from G o G. When G he rc DI-reducon of G, T he e of all player wh domnaed raege n G. We have he followng lemma. Lemma 4.1. Neceary condon for an EPC equence. Le be an IEDI equence wh elmnaon dvde m, Γ( G ) N he player e of G,, G G for,,,

16 48 M. KANEKO, S. LIU T he D-group from G o 1 G for,, 1, and T. Then [( N, T ),, ( N, T )] afe PC PC2. If ( G ) a rc IEDI equence, hen PC3 hold, oo. Proof. Le G ( N,{ S}, { h} ) for,,. PC follow from (6) and N N 1 (7), and PC1 correpond o G G n (6). PC2 follow from he defnon of he elmnaon dvde m. Conder PC3: Le 1 Γ( G ) be he rc IEDI from G. Le T {}. If 1 N, hen T, o T T. Suppoe N. 1 1 Le G G G. Then, 1 G all of he domnaed raege for player n are elmnaed n formng From 1 Lemma 2.2.3, ha no domnaed raege n G. Hence, T T. We ay ha [( N, T ),, ( N, T )] gven n h lemma he EPC equence aocaed wh Γ( G ) G,, G. The convere of Lemma 4.1 our preen concern. Here, we confne ourelve o recoverably by rc IEDI equence. We have he followng heorem, whch proved n Secon 4.2. d p G 1. Theorem 4.1. Characerzaon. Le G ( N,{ S} N,{ h} N) be a game wh S 2 for all N, whch ha no domnaed raege and no neenal player. Le [( N, T ),,( N, T )] be a rc EPC equence wh N N. Then, here ex a game G and a rc IEDI equence Γ( G ) G,, G uch ha (A) G G; (B) S 2for all N,,, 1; and (C) he EPC equence aocaed wh ( G ). Th heorem mple ha here are a grea mulude of poble underlyng uaon behnd a gven game G. Le u look a he EPC equence aocaed wh Example Example 1.1 ha he rc IEDI equence G, G wh aocaed EPC equence: [( N, T ), ( N, T )] [({1, 2, 3}, {2}), ({1, 2}, )]. In Example 1.2, we 1 1 have [( N, T ), ( N, T )] = [({1, 2,...,12}, {2}), ({1, 2}, )]. In Example 1.3, he rc IEDI equence ha lengh 1. The aocaed EPC equence gven a 1 1 [( N, T ),..., ( N, T )] o ha N 1, 23,..., 12 for,, 1, and T {2}, T for 1,, 1. We have many oher EPC equence. For example, for,, 1, le N {1, 2} 9 ( k {1k 3,,1k 12}), T {2}, and T for 1,, 1. Player from 3 o 12 are dvded no 1 group {3, 4,, 12}, {13, 14,, 22},,

17 Elmnaon of domnaed raege and neenal player 49 {93, 94,, 12}. Each ha he ame rucure a he onlooker n Example 1.3 bu each of 3, 13, 23,, 93 wan o dae grl 2, and 4 a frend of 3, 14 a frend of 13,, and 94 a frend of 93, and o on. In he rc IEDI aocaed wh h EPC equence, player 3, 13, 23,, 93 become neenal and are elmnaed n he fr round, and hen player 4, 14,, 94 become neenal and are elmnaed, and o on. The reulng game afer 1 round he ame a he 2-peron bale of he exe. The nal game of h IEDI equence very dfferen from hoe n Example 1.2 and 1.3. We can hnk abou more complcaed nework. A long a PC PC3 are afed by a gven EPC equence, Theorem 4.1 ugge a game uaon wh uch a nework. In h ene, we regard ypcal example n game heory a beng abraced from many dfferen uaon. Condon PC3 no ued n hee example. We can exend Example 4.2 wh 1 1 [( N, T ), ( N, T )] o a uaon ncludng more ep. Suppoe ha afer elmnang all he boy from 3 o 12, 1 and 2 fnd more raege relevan o hemelve. Then, here a longer EPC equence [( N, T ),..., ( N, T )] wh N {1, 2} and T for all 1,,. When G,..., G a rc IEDI equence, PC3 mple ha for ome k (2 k ), T {1, 2} for ( 2 k ), and 1 T for ( k ). Up o ep k hey agree o elmnae her domnaed raege ogeher bu afer k,, 1 T T,.e., hey alernangly elmnae domnaed raege. In Theorem 4.1, we have no condered he raegy e n Γ ( G ). However, poble o ar wh a gven equence of game form (whou pecfyng payoff) raher han an EPC equence. A dealed analy reman open Proof of Theorem 4.1 Conder an EPC equence [( N, T ),, ( N, T )] and G ( N,{ S} N, { h } ) afyng he condon of he heorem wh N N. By nducon from N ( N, T ) o ( N, T ), we conruc a equence how ha for each 1,,, G,, G a rc IEDI generaed from G, G,, G 1 from G G, and 1 G a rc DI-reducon of G ; G. hu, Lemma 4.2 decrbe he conrucon of he nerpolang G from G 1,.e., G 1 p G. Snce G G ha no neenal player, we can aume ha S 2 for all N. In he followng lemma, we ue he ame noaon G ( N,{ S},{ h} ) N N

18 5 M. KANEKO, S. LIU for a generc game, whch hould no be confued wh he gven game G n Theorem Alo, we conder he revere drecon from G G o G G. 1 G G G (9) Lemma 4.3 Lemma 4.2 Lemma 4.2. Le G ( N,{ S} N,{ h} N) be a game wh S 2 for all N, and le I be a nonempy e of new player. Then, here a game G ( N,{ S } N,{ h } N ) uch ha (): N N I ; (): S 2 for all N; and (): G he rc p-reducon of G. Proof. We chooe raegy e S, N o ha S S for all N and S {, } for all I, where, are new raege no n G. Then we defne he payoff funcon { h } N o ha he player n I are neenal n G, bu no player n N are neenal n G. Le I be he e of neenal player n G. For each I, we chooe an arbrary raegy, ay 1 from S. Then we defne { h } N a follow: (a): f I, h( N) { I : 1} for N SN; (b): f N, h ( N) h( N) for N SN, where he rercon of N N o N. For any I, raegy doe no appear ubanvely n h for any N I. Thu, he player n I are all neenal n G. On he oher hand, each player I, a far a uch a player ex n G, affec payoff for I becaue of (a) and S 2. Th mean ha no I neenal n G. Alo, no N I neenal n G by (b). Thu, only he player n I are neenal. In um, G he rc p-reducon of G. Now, conder he conrucon from G o G n (9). For h, fr we how he followng fac: Le G ( N, { S} N, { h} N) be an n-peron game, and N a fxed player. Then, here are real number { ( )} S uch ha f domnae, hen ( ) ( ) (1) 1 Such { ( )} S are defned by nducon a follow: Fr, we le H G. Le k be k a naural number wh 1 k S 1. Suppoe ha a game H gven. Take an arbrary raegy k k k for player n H o ha no domnaed a all n H. Then, k ( ) k. Then, k 1 H obaned from k H by elmnang k from he raegy e

19 Elmnaon of domnaed raege and neenal player 51 k S for player n H. Addonally, we le ( ) S. Thu, we have { ( )}. S I reman o how ha (1) hold. Suppoe ha domnae n G H 1. Then, occur before n he equence 1 S,..., above conruced. Hence, ( ) ( ). Now, conder he ep from G o uppoed o be G and G, repecvely. G n (9). In he nex lemma, G and G are Lemma 4.3. Le G ( N,{ S} N,{ h} N) be a game, and T a nonempy ube of N wh S 2 for all N T There a game G ( N,{ S } N,{ h} N) uch ha G a d-reducon of G, T he D-group from G o G, and S 2 for all N If he followng condon hold for T, f T { }, no par of raege, S ex uch ha domnae (11) hen G he rc d-reducon of he game Ggven by Proof of Le be a new raegy for each T. We defne { S } N a follow: S { } f T S S f N T (12) Π For each N, we exend h o h : Π S N o ha he rercon of h o N S h elf and G he rc d-reducon of G, a follow: Le N. Fr, h he ame a h over Π NS,.e., h( ) h( ) f Π NS. Le S S. If N T, and f T, h( ) ( ), where ( ) above defned for G (13)

20 52 M. KANEKO, S. LIU ( ) f h () mn{ ( ): S} 1 f (14) Now le N T, and le, S S. Suppoe ha domnae n G. Conder, S S uch ha he -h componen of and are and. From (13), we ge h( ) ( ) ( ) h( ). Hence, doe no domnae n G, whch mple ha ha no domnaed raege n G. Second, le T. We chooe an S wh. From (14), we have, for any S, h ( ; ) ( ) S ( ) h( ; ). mn{ : } 1 Th doe no depend upon. Thu, domnae n G. From he analy of hee wo cae, we conclude ha T he D-group from G o G Fnally, we how ha under condon (11), for any, S S { } and T. If doe no domnae n G doe no domnae n G, hen h doe no hold n G eher. Now le domnae n G. From (11), we have T > 1. Th guaranee he exence of, S S uch ha her -h componen are and. From (14), h() ( ) ( ) h( ). Hence, doe no domnae. I follow ha G he rc d-reducon of G. Proof of Theorem 4.1. We conruc a rc IEDI equence Γ( G ) G,..., G along [( N, T ),..., ( N, T )] from he endgame G G by backward nducon. Le G G. By aumpon, condon (7) hold. Alo, S 2 for all N. 1 Suppoe ha G 1 defned wh S 2 for all N 1. From Lemma 4.2, we 1 fnd an nerpolang game G uch ha G he rc p-reducon of G wh player e N and S 2 for all N. From Lemma 4.3.1, we fnd anoher game G uch ha G a d-reducon of G wh D-group T afyng S 2 for all N. Now we obaned an IEDI Γ( G ) G,..., G wh aocaed EPC equence [( N, T ),..., ( N, T )]. When PC3 hold, condon (11) n Lemma 4.3 afed. Thu, from Lemma G he rc d-reducon of G.

21 Elmnaon of domnaed raege and neenal player Concluon We have condered he proce of eraed elmnaon of domnaed raege and of neenal player. The laer newly nroduced, and neracve wh he former. Th nroducon change he analy conderably. We have gven modfcaon of exng reul; Theorem 2.1 (preervaon) and Theorem 3.1 (order- -ndependence). Then, we preened Theorem 4.1 (characerzaon). Reul (4) on he preervaon of Nah equlbra follow Theorem 2.1. Theorem 3.1 ae ha any equence generaed from a gven game by he IEDI proce end up a he ame game and ha he rc IEDI equence he malle and hore among he IEDI equence. Combnng hee reul, we oban a mple way o compue he e of Nah equlbra for he nal game from he equlbra for he endgame, whch expreed a (8). Then we argued n Secon 4 ha ypcal example condered n game heory are repreenave of game abraced from many dfferen uaon. Theorem 4.1 gve condon for he form of IEDI equence from poble uaon ha end up a a gven game. Thee condon mply ha here are many underlyng uaon behnd a gven game. Example , ogeher wh h heorem, how ha he nroducon of neenal player gve new perpecve abou poble underlyng ocal uaon behnd a game. Alo, Theorem 2.1 and 3.1 gve a way of compung he e of Nah equlbra. (8) aocae he approprae equence of e of Nah equlbra wh he rc IEDI decrbed by Theorem 4.1. We have no ouched upon ome mporan problem. One problem o relax he concep of neenal player. The defnon of an neenal player here oo rngen n ha unlaeral change n h raege have no effec on any player payoff. One pobly o nroduce -neenal player or -nfluence for an. Player defned o be -neenal wh repec o player ff unlaeral change n raege only affec payoff whn an -magnude. Ung h defnon, we may allow boy 3 n Example 1.1 o be -ndfferen beween he arena and cnema when grl 2 chooe o dae boy 1. Our hree example ugge dfferen problem. The payoff for player depend only upon a e of neghbor. Th compable wh -neenal player or -nfluence. Th alo along he reearch lne of he preen paper. Anoher problem he complexy of aeng preference for an IEDI equence. The reul n h paper faclae uch conderaon, nce, n general, he rc IEDI equence requre le analy han any oher IEDI equence. However, baed on he raghforward defnon of complexy for preference comparon, we have an example of a game where ome IEDI equence need a maller number of preference comparon han he rc IEDI equence. A dealed udy an open problem.

22 54 M. KANEKO, S. LIU Acknowledgemen The auhor hank he referee of h ournal for many helpful commen. They alo hank he Gran-n-Ad for Scenfc Reearch No , and No , he Mnry of Educaon, Scence and Culure, and he Japan Cener of Economc Reearch for fnancal uppor. Reference [1] APT K.R., Drec proof of order ndependence, Economc Bullen, 211, 31, [2] BÖRGERS T., Pure raegy domnance, Economerca, 1993, 61, [3] GILBOA I., KALAI E., ZEMEL E., On he order of elmnang domnaed raege, Operaon Reearch Leer, 199, 9, [4] KANEKO M., KLINE J.J., Underandng he oher hrough ocal role, o be pblhed n Inernaonal Game Theory Revew, 215. [5] MASCHLER M., SOLAN E., ZAMIR S., Game Theory, Cambrdge Unvery Pre, Cambrdge 213. [6] MERTENS J.F., Sable equlbra a reformulaon II. The geomery, and furher reul, Mahemac of Operaon Reearch, 1991, 16, [7] MOULIN H., Game Theory for he Socal Scence, 2nd reved Ed., New York Unvery Pre, New York [8] NASH J.F., Non-cooperave game, Annal of Mahemac, 1951, 54, [9] NEWMAN M.H.A., On heore wh a combnaoral defnon of equvalence, Annal of Mahemac, 1942, 43, [1] MYERSON R.B., Game Theory, Harvard Unvery Pre, Cambrdge [11] OSBORNE M., RUBINSTEIN A., A Coure n Game Theory, The MIT Pre, Cambrdge Receved 3 Sepember 214 Acceped 19 January 215