Interdependent Relationships in Game Theory: A Generalized Model
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1 Interdependent elatonshps n Game Theory: A Generalzed Model Jawe L Independent researcher, Nottngham, UK Emal: Jawemchaell@gmalcom Abstract: A generalzed model of games s proposed, wth whch both cooperatve games and non-cooperatve games can be epressed and analyzed, as well as those games that are nether cooperatve nor non-cooperatve The model s based on relatonshps between players and supposed relatonshps A relatonshp s a numercal value that denotes how one player cares for the payoffs of another player A supposed relatonshp s a player s belef about the relatonshp between two players The players choose ther strateges by tang nto consderaton not only the materal payoffs but also relatonshps and ther change Two games, a prsoner s dlemma and a repeated ultmatum game, are analyzed as eamples of applcaton of ths model Keywords: game theory, cooperatve games, non-cooperatve games, relatonshp Introducton There have been two nds of researches n game theory: cooperatve games theory poneered by Von Neumann and Morgenstern (95) and non-cooperatve games theory developed by Nash (95) When analyzng a game, one frst needs to confrm what type the game s because absolutely dfferent methods wll be used for two types of games Methods for non-cooperatve games are based on Nash equlbrum, varous perfects of Nash equlbrum (eg, strong Nash equlbrum by Aumann (959), Subgame Perfect Nash Equlbrum and Tremblng Hand Perfect Equlbrum by Selten (965, 975), Bayesan Nash Equlbrum and Strct Nash Equlbrum by Harsany (967, 97)) and the fol theorems, whle cooperatve games are analysed by means of coaltons, core, and Shapley value (Shapley (95)) The most mportant wor comes from non-cooperatve analyss, although some scholars regan nterest n cooperatve games recently A queston s whether or not any games can be categorzed nto these two groups? onsder contract brdge, the card game that s played by four players n two competng partnershps Ths game s nether cooperatve nor non-cooperatve because the relatons between ndvdual players are not dentcal There are both cooperaton between partners and competton between two partnershps The players n a non-cooperatve game only care for ther own payoffs whlst the players n a cooperatve game care for the payoffs of other players n the coalton as equal mportant as ther own payoffs If we use a numercal value to denote how much one player cares for another player s payoff, ths value wll be zero for noncooperatve games and t wll be one for cooperatve games One may naturally as what f ths value s set to be nether zero nor one In ths paper, we ntroduce the concepts of relatonshp and supposed relatonshp A relatonshp s a numercal value denotng how much one player cares for another player s payoff A supposed relatonshp s a numercal value denotng a player s belef about how much one player cares for another player s payoff elatonshps and supposed relatonshps are determned by the players and they are changeable n dfferent stages of games We propose a relatonshp model of games, n whch
2 strategc nteracton among players s determned by the materal payoffs, relatonshps, and the players belef about relatonshps ooperatve games and noncooperatve games, as well as those games that are nether cooperatve nor noncooperatve can be epressed and analysed by usng ths model Interpersonal relatonshp has not attracted nterests of research n game theory although t has long been an mportant topc of research n many socal scence dscplnes such as psychology and poltcs (Kelley (0), Heder (0)) Game theorsts tae t for granted that the relatonshps between players are predetermned and they wll never change durng the strategc nteractons of players Interdependent preference, whch denotes that a player s preference depends on hs opponent s payoff as well as hs own payoff, has been used to eplan cooperaton phenomena n epermental economcs (eg Bolton 99, 000, Bnmore et al 00, Ochs and oth 989, Samuelson 00) It s qute smlar to the relatonshp concept defned n ths paper eputaton effect poneered by Kreps and Wlson (98) and Mlgrom and oberts (98) s ntroduced nto game theory to eplan cooperaton n repeated non-cooperatve games The relatonshp s obvously dfferent from reputaton n that reputaton s ndependent of the game model and then has less effect n one-shot games ooper and et al (996) and han (000) have suggested that reputaton s unnecessarly the unque factor leadng to cooperaton n ether nfntely or fntely repeated games In the followng paper, we ntroduce a novel game model tang nto consderaton the relatonshp and relatonshp change among players We also show that there ests a large set of games that are nether cooperatve nor non-cooperatve A elatonshp Model Defnton : For players and j n a game, a relatonshp s a numercal value denotng how much player cares for player j s payoff Specally, there s = Player s atttude toward j s non-cooperatve when =0, and cooperatve when = We call t a sub-cooperatve atttude when 0< <0, or a dedcated atttude when > <, a hostle atttude when Defnton : For players, j and n a game, a supposed relatonshp s a numercal value denotng how much player thns player cares for player j s payoff Specally, there s = Player thns that player s atttude toward j s non-cooperatve when =0, cooperatve when =, sub-cooperatve when 0< <, hostle when <0, or dedcated when > In a game wth complete nformaton, there are =, whch means that the relatonshps between players are nown to all players A supposed relatonshp s not necessarly equvalent to the correspondng relatonshp n a game of ncomplete nformaton Obvously cooperatve and non-cooperatve games are specal cases n the set of games defned by dfferent values of } and } For eample, a game s non- { {
3 cooperatve when there are when = = = for any, j and =0 for any j whle a game s cooperatve Note that { } can be consdered as a subset of { } because every player nows ther own relatonshp and thus there must be We eep both varables n ths paper n order to dstngush a relatonshp from other player s belef about t Defnton : elatonshp model of a n-player game G s a -tuple, G { I, S, U, }, where I = {,, n} s a player set, S s a strategy set, U s a payoff set, and s a supposed relatonshp set Defnton : A supposed payoff of player, j u, denotes how much player thns player j s payoff s by tang player s supposed relatonshps nto consderaton Gven a strategy profle s,, s ), player j s payoff u s,, s ), and player s ( n j ( n supposed relatonshps { }, the supposed payoff, u j, s computed by n u f j u j( s,, sn) () n ju f j Under the relatonshp model, the players n a game choose ther strateges accordng to ther supposed payoffs For eample, player s choce s determned by s supposed payoffs, { u} We prove n the followng theorem that there must est a Nash equlbrum for every game when players mae choces accordng to {u } Theorem : Under relatonshp model, there must est a Nash equlbrum for every game Proof: onsder an arbtrary player n a n-player game ( N ) Let { u} denote s supposed payoffs Snce { u} s a complete payoff matr for n players, there must be a Nash equlbrum strategy profle for { u}, as Nash had proved n [] Let s be player s strategy n ths profle The strateges of all players form a new strategy profle s }, {,, n} It s obvous that s } s a Nash equlbrum because { { every player has no ncentve to devate from s Let s analyze the prsoner s dlemma as an eample to show how to use the relatonshp game model The payoff matr of a prsoner s dlemma s shown n Fg Player D, 0, 5 D 5, 0, Fgure Two players choose between ooperate () and Defect (D) n the prsoner s dlemma The numbers n each cell denote the payoffs of players and y respectvely
4 Let and denote the relatonshps between the players and y, and and y the correspondng supposed relatonshps of y and The supposed payoffs of player can be computed accordng to () They are epressed as a matr as shown n Fg, Player D +, + 5, 5 D 5, 5 +, + Fgure The supposed payoffs of player What player chooses between and D depends on the values of and s the domnant strategy for player when, whle D s domnant when In the case of, s domnant when whle D s domnant when No strategy s domnant for player when there are, Smlarly, the supposed payoffs of player y are computed and epressed as a matr as shown n Fg, Player D +, + y 5, 5 y D 5, 5 +, + y Fgure The supposed payoffs of player y omparng Fg wth Fg, the supposed payoffs of two players wll be dentcal f and y are replaced by and s the domnant strategy for player y when, whle D s domnant when In the case of, s domnant when y whle D s domnant when y No strategy s domnant when there are, y If we consder the prsoner s dlemma as a game of complete nformaton, there should be = and y =, whch means that the relatonshps are nown to both players The supposed payoffs of dfferent players are dentcal n a game of complete nformaton The players choces depend on the eact values of and For eample, the strategy profle (, ) s domnant when and ; (D, D) s domnant when and ; (, D) s domnant when and ; and (D, ) s domnant when and The game has multple equlbra and no strategy profle s domnant when there are,
5 The strateges of players n a game of ncomplete nformaton can be much dverse and comple because the relatonshps of one player are unnown to others The supposed relatonshps are not necessarly equvalent to the correspondng relatonshp n a game of ncomplete nformaton Snce the supposed relatonshps are prvate nformaton, the players would tae advantage of them n strategc nteractons For eample, player attempts to eplot the opponent by settng < On the other hand, = j > reflects an altrustc atttude of player toward j There should be j f player adopts a tt-for-tat strategy, or n other words, player wants to treat the opponent eactly same as what the opponent treats hm/her If we consder the prsoner s dlemma as a game of ncomplete nformaton, equlbrum analyss can be made gven the values of,,, and y Suppose that there are = and = y, the supposed payoffs can 5 be computed as shown n Fg From Fg (a), D s domnant strategy for player y Player wll choose gven that he/she beleves that the other player wll choose D Smlarly, player y wll also choose gven that he/she thns the opponent would choose D Thus, strategy profle (, ) wll be the outcome j Player D, 5, 5 D 5,, 5 Player D 5,, 5 D 5, 5, (a) (b) Fgure (a) Supposed payoffs of player (b) Supposed payoffs of player y epeated games T T A repeated game G s a -tuple, G { I, S, U, }, where T s the number of teraton The players n a repeated game wll have to tae relatonshp change nto consderaton when choosng ther strateges elatonshp change reflects the complety of ntellgent decson mang It could be a comple acton dependng on how the players retreve nformaton from prevous nteractons wth other players and how they update ther supposed relatonshps One reason for relatonshp change n repeated games les n the fact that prevous strategc nteractons provde new nformaton about relatonshps so that the players should update ther supposed relatonshps Tae the terated prsoner s dlemma as an eample Suppose that two players play the prsoner s dlemma wth payoff matr as shown n Fg repeatedly and there are = and = y at the 5 begnnng of game Accordng to two players supposed payoffs shown n Fg, they choose (, ) n the frst round After playng the frst round, two players realze that they have underestmated and y If the orgnal values of and y matched the correspondng relatonshps eactly, each player should have chosen D 5
6 nstead of Two players should then ncrease ther supposed relatonshps to some values greater than elatonshp change may tae place n games of complete nformaton as well Assume that there are n the above eample A tt-for-tat player wll update hs/her relatonshp to mae sure In the followng eample we analyze a repeated ultmatum game n order to show how players tae nto consderaton relatonshp change n game playng Let s consder an nfntely repeated ultmatum game In each round, ow player proposes an offer of dvdng one dollar between two players If olumn player accepts the offer, they receve the correspondng share Otherwse, both players receve nothng The mnmum dvson of one dollar s one cent Fg 5 shows the payoff matr of ths game The supposed payoffs of olumn player are shown n Fg 6 ow player olumn player Accept eject 0, 0 0, , 00 0, 0 0, 0 α -α, α 0, 0 0, , 099 0, 0 Fgure 5 A ultmatum game n whch ow player maes an offer of dvdng a dollar to olumn player If olumn player accepts the offer, two players share the dollar as the offer suggests Otherwse, both receve zero ow player olumn player Accept eject 0, 0 0, 0 0, 0 α, ( ) 0, 0 0, 0 Fgure 6 The supposed payoffs of olumn player The supposed payoffs of two players can be computed accordng to the relatonshps and supposed relatonshps of,,, and For an offer, ow player s supposed payoff profle s (, ( ) ) whle olumn player s supposed payoff profle s (, ( ) ) In order for olumn player to accept, there must be ( ) >0, or On the other hand, for ow player there must be >0, or Tang nto consderaton the supposed relatonshps of two players, ow player wll offer satsfyng satsfyng and olumn player wll accept When 0, ow player thns olumn player would accept any offer >0 so ow player would offer =00 to mamze the payoff Ths s what non- 6
7 cooperatve game theory has predcted However, olumn player could choose <0 and would probably reject the offer of =00 In an nfntely repeated game, two players wll reach an agreement wth the offer n range of even f two players cannot communcate wth each other onsder the case of = =-05 and each player does not now the eactly relatonshp of the other player, for eample olumn player wll reject any offer wth, whch transfers the nformaton about olumn player s relatonshp to ow player who wll have to ncrease the offer Ths process s smlar to a two-player barganng game n whch ratonal players wll reach an agreement wth n the barganng elatonshps have a sgnfcant nfluence on strategc nteracton n games Players could choose relatonshps n order to mamze payoffs For eample, olumn player n the above game chooses a mnus relatonshp How players tae advantage of relatonshps n game playng depends on specfc materal payoffs of the game and t could be a comple problem 6 onclusons A generalzed model of games that taes nto consderaton the relatonshps between players s proposed ooperatve games and non-cooperatve games are specal cases n ths model There ests a sgnfcant set of games that are nether cooperatve nor non-cooperatve, whch have not been nvestgated n game theory We prove that there must est Nash equlbrum for every game under the relatonshp model so that equlbrum analyss developed n non-cooperatve game theory can be appled A prsoner s dlemma and a repeated Ultmatum game are analyzed One advantage of the relatonshp model les n that t provdes an accurate descrpton of the players atttudes toward others n game playng A player s atttude toward another player could be sub-cooperatve, hostle, dedcated, as well as cooperatve and non-cooperatve By tang supposed relatonshps nto consderaton, the atttudes of altrusm and eplotaton can be consdered elatonshps and relatonshp changes mae the strateges of players nterdependent How the players tae advantage of relatonshps and relatonshp changes wll be my future research eference [] Aumann J (959) Acceptable ponts n general cooperatve n-person games n ontrbutons to the Theory of Games, Vol IV, Prnceton Unv Press, Prnceton [] Harsany J (967) Games wth ncomplete nformaton played by Bayesan players part I II and III Management Scence : 59-8, 0-, [] Harsany J (97) Games wth randomly dstrebed payoffs: a new ratonale for med strategy equlbrum ponts Internatonal Journal of Game Theory, : - [] Nash J (95) Non-cooperatve games Ann Math 5, [5] Selten (965) Speltheoretsche Behandlung enes Olgopolmodells mt Nachfragetraghet Zetschrft fur Gesamte Staatswssenschaft,, 0- [6] Selten (97) A smple model of mperfect competton where are few and 6 are many InternatonalJournal of Game Theory (): 0 [7] Shapley LS (95) A value for n-person games In: Tucer AW, Kuhn HW (eds) ontrbutons to the theory of games II, Prnceton Unversty Press, Prnceton [8] Von Neumann and Morgenstern J (95) Theory of Games and Economc Behavor rd ed Prnceton Unversty Press, Prnceton 7
8 [9] Aelrod (98) The emergence of cooperaton among egosts Amercan Poltcal Scence evew, 75: 06-8 [0] Kreps D and Wlson (98) Sequental equlbrum Econometrca, 50: 86-9 [] Mlgrom P and oberts J (98) Lmt prcng and entry under ncomplete nformaton: an equlbrum analyss Econometrca, 0: -59 [] ooper and et al (996) ooperaton wthout reputaton: epermental evdence from prsoner s dlemma games Games and Economc Behavor, : 87-8 [] han J (000) On the Non-Estence of eputaton Effects n Two-Person epeated Games mmeo, Johns Hopns Unversty [] ogers, (96) The nterpersonal relatonshp: The core of gudance Harvard Educatonal evew [] Bnmore K, Mcarthy J, Pont G, Samuelson L and Shaed A (00) A bacward nducton eperment, Journal of Economc Theory, 0: -88 [5] Bolton G (99) A comparatve model of barganng: Theory and evdence, Amercan Economc evew, 8: [6] Bolton G and Ocenfels A (000) E: A theory of equty, recprocty and competton, Amercan Economc evew, 90: 66-9 [7] Samuelson L (00) Analoges, adaptaton, and anomales, Journal of Economc Theory, 97: 0-67 [8] Ochs J and oth E (989) An epermental study of sequental barganng, Amercan Economc evew, 79: 55-8 [9] Heder, F (0) The psychology of nterpersonal relatons Psychology Press [0] Kelley, H H (0) Personal relatonshps: Ther structures and processes Psychology Press 8
Interdependent Relationships in Game Theory: A Generalized Model
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