Fuzzy Nash Equilibriums in Crisp and Fuzzy Games

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1 Fuzzy Nash Equlbrums n Crsp and Fuzzy Games Alreza Chaker Fard Shekholeslam Member IEEE Abstract In ths paper we ntroduce Fuzzy Nash Equlbrum to determne a graded representaton of Nash equlbrums n crsp and fuzzy games. Ths nterpretaton shows the dstrbuton of equlbrums n the matr form of a game and handles uncertantes n s. Also a new method for rankng fuzzy values wth the user's vewpont s nvestgated. By ths mean the defnton of Satsfacton Functon whch provdes the result of comparson n the form of real value s developed when users have preferences regardng the s. Inde Terms Fuzzy game fuzzy Nash equlbrum fuzzy preference relaton satsfacton functon fuzzy value. A I. INTRODUCTION game s a decson makng system that nvolves more than one decson maker each havng profts that conflct wth each other. A strategc game frst defnes each player's actons (strategy). The combnaton of all the players strateges wll determne an outcome to the game and the s to all players n whch each player tres to mamze hs own. The tradtonal game theory assumes that all data of a game are known eactly by players. However n real games the players are often not able to evaluate eactly the game due to lack of nformaton mprecson n the avalable nformaton of the envronment or the behavor of the other players. Intally fuzzy sets were used by Butnaru [] n noncooperatve game theory. He used fuzzy sets to represent the belef of each player for strateges of other players. There have been several approaches to etend fuzzy cooperatve games; Butnaru n [2] ntroduced core and stable sets n fuzzy coalton games where a degree of partcpaton of players n a coalton s assgned. Moreover Mares n [3] consdered fuzzy core n fuzzy cooperatve game where possblty of each fuzzy coalton s fuzzy nterval as an etenson of core n classc TU games. Also he dscussed Shapely value n cooperatve game wth determnstc characterstc and fuzzy coalton. Fuzzy game theory has been appled to many compettve decson-makng stuatons [4-8]. Vjay et al. [4] Manuscrpt receved Aprl 200; revsed September and October 0 20; accepted Aprl A. Chaker s wth the Electrcal and Computer Engneerng Department Isfahan Unversty of Technology Isfahan Iran (e-mal: alreza.chaker@ec.ut.ac.r). F. Shekholeslam s wth the Electrcal and Computer Engneerng Department Isfahan Unversty of Technology Isfahan Iran (e-mal: shekh@cc.ut.ac.r). Dgtal Object Identfer 0.09/TFUZZ consdered a game wth fuzzy goals and fuzzy parameters and proved that such a game s equvalent to a prmal dual par of certan fuzzy lnear programmng (FLP) problems wth fuzzy goals and parameters. Chen and Larban n [6] dscussed multple attrbute decson makng wth a two person zero-sum game and acheved smpler crtera for solvng the correspondng FLP. Lu and Kao n [7] defned the value of games n fuzzy form. Nshzak and Sakawa [8] dscussed fuzzy bmatr game and by usng a nonlnear programmng the equlbrums were searched. Maeda [9] studed zero-sum b-matr games wth fuzzy s. In [0] a fuzzy dfferental game approach was proposed to solve the N- person quadratc dfferental non-cooperatve and cooperatve game. Kma and Leeb n [3] consdered fuzzy constrants as well as fuzzy preference and proved some theorems on the estence of equlbrum. Song and Kandel n [4] used a mult-goal problem where the degree of satsfacton for each goal was a fuzzy one and the overall s a weghted sum of the satsfacton of all goals. They assumed that each player has a fuzzy knowledge about hs opponents med strateges.e. each player assgns a membershp functon to the probablty dstrbuton of hs opponents strateges. Garagc and Cruz n [5] transformed a game wth fuzzy strateges and fuzzy s to a crsp game usng Fuzzy IF-Then rules. Subsequently they dscussed the Nash equlbrums n the equvalent crsp game; they proved that ths crsp game has at least one pure strategy Nash equlbrum. L et al. n [6] employed two fuzzy approaches ncludng fuzzy multcrtera decson makng method and the theory of fuzzy moves to nvestgate the game of chcken. Ther model ncorporates the player's subjectve manner and mprecse knowledge to the game model. In [7] [8] a mathematcal programmng approach of fuzzy matr games wth ntutonstc fuzzy s and nterval-valued ntutonstc fuzzy (IVIF) s [9] was developed. It s proven that each matr game wth IVIF s has a soluton. In ths paper we develop a new approach to N-person crsp and fuzzy non-cooperatve games to obtan Nash equlbrums for these knds of problems. The most sgnfcant advantages of usng the proposed method are the range of game-theoretc problems that can be analyzed and the nformaton about equlbrums that can be obtanable to players. In the proposed approach the defnton of equlbrum n crsp and fuzzy games has been generalzed to show dstrbuton of Nash equlbrums n matr games and also present the amount of optmalty of the players strateges by a degree. In ths regard /$ IEEE

2 2 we do not need to determne whether a pure strategy s Nash equlbrum. Instead we assgn a graded membershp to any pure strategy that descrbes to what possblty t s Nash equlbrum. Hence we can consder strateges wth hgh degrees of equlbrum whch are not necessarly the equlbrum ponts. In fuzzy games fuzzy Nash equlbrum approach s more approprate for real world problems whch are modeled by game theory. The proposed approach avods loss of any nformaton that happens by defuzzfcaton method n games and handles uncertanty of s through all steps of fndng Nash equlbrum. It shall be noted that n ths approach the estence of theorem for equlbrum was not establshed snce the focus s not on estence of equlbrum but n the degree of equlbrum. The paper s structured as follows. Secton II ntroduces the concept of degree of beng Nash equlbrum n games wth crsp s usng the fuzzy preference relaton. In Secton III a new functon modelng the SF s defned. In ths defnton the weghts of the doman n the fuzzy values are consdered drectly n the formula. Also Secton III eplans a new approach n fuzzy games usng the satsfacton functon. Moreover the consequence of player's vewponts n Nash degrees of cells s dscussed. The concluson hghlghts the man fndngs of the paper. II. GAMES WITH CRISP PAYOFFS A. Non-cooperatve N-person Games Ths subsecton contans the background on game theory whch are needed to develop the fuzzy Nash equlbrum. Games have been classfed by the number of players the number of strateges the nature of the s functon and cooperatveness. A normal game conssts of a set of players ther strateges and the s avalable for all combnatons of players strateges. Non-cooperatve N-person strategc game can be formulated as follows [20]: P P... P 2 ) There are N players to be denoted by N. 2) There are a fnte number of alternatves for each player to choose from. Let s denote the number of alternatves avalable to P and further denote the nde set 2...s by X wth a typcal element of X desgnated as. 3) If P chooses a strategy X then the for P s a sngle number 2... N X X... X X.... Also X s the set of all s for P where denotes the strateges chosen by other players. 4) Players play wth a polcy of mamzng ther they take nto account the possble ratonal choces of the other players and they also make decson ndependently. N Unlke one-player decson makng where optmalty has an eplct meanng n mult person decson makng the optmalty s n the form of Nash equlbrum. A pure strategy Nash Equlbrum s a strategy set wheren f a player knows hs opponent s strategy he s totally satsfed wth hs equlbrum strategy and s reluctant to change hs strategy [2]. Defnton : Classcal game theory determnes a cell as a Nash Equlbrum f and only f t mamzes all players' s when other players refran from changng ther strateges N s the pure strategy Nash equlbrum f and only f [22] N... N X One can rephrase the classcal game theory soluton to a smple algorthm n two steps: ) Replace wth f mamzed by N () s otherwse replace t wth 0. 2) Fnd the mnmum of the elements of each cell: f ths value equals the cell s Nash equlbrum and f ths value gets zero the cell s not Nash equlbrum. For nstance consder the smple game of Table I where each cell ncludes two crsp s; the frst for player and the second for player 2. Player and 2 have three strateges namely J J 2 J 3 and K K 2 K 3 respectvely. Table II shows the Nash equlbrum cells. TABLE I A SAMPLE GAME WITH CRISP PAYOFFS ΠΠ2 K K2 K3 J ( ) ( ) ( ) J2 ( ) ( ) ( ) J3 ( ) ( ) ( ) TABLE II RESULTS OF THE GAME OF TABLE I K K2 K3 J 0 0 J J3 0 0 For each par of strateges the numbers 0 and ndcate not beng and beng a Nash equlbrum respectvely B. Fuzzy Nash Equlbrums n Games wth Crsp Payoffs Accordng to classcal game theory f a player knowng

3 3 others' strateges chooses a strategy whereby he cannot get mamum he wll completely regret hs choce. However just as the Table I depcts n real world problems there may be stuatons where the dfference between s are neglgble e.g. s and n Table I. If a player chooses he/she wll be qute satsfed wth hs/her choce. Therefore one can consder and to have appromately same value n Table I. Ths appromaton prompted the employment of fuzzy logc to make a soft measurement between s. In ths paper nstead of the logc "greater than" relaton a new measure s defned that of the amount beng greater; for nstance the potental amount of beng greater between 000 and s greater than that between 2 and.the amount of beng greater may be perceved dfferently by each partcular player. In other words t depends on the mental state and belefs of that player. The more metculous and gree a player s the greater a dfference he/she perceves between slghtly dfferent s. A new term s defned n ths paper to determne the amount of beng greater between two s n (2); ths value s nterpreted as the preference between two s. Then usng fuzzy preference relaton matr [23] prortes are calculated by the Least Devaton Method [24] n whch the prorty vector s a vector whch determnes the degree of mportance of alternatves. Here the grades of beng Nash equlbrums are modeled accordng to the prorty that players feel for ther strateges. Ths defnton for the grade of beng Nash equlbrum seems meanngful because f a player knows the opponent s strategy he s satsfed wth hs strategy to the degree that ths strategy has prorty for hm. The greater prorty the players feel for each cell the greater the possblty that the cell s the game s equlbrum. Algorthm: Frst t s necessary to normalze all s and make the range between 0 and. Let a b be the s of selectng strateges j respectvely for a player f the opponents' strateges are fed. The amount of beng greater between a b can be calculated as follows: a b mn ( a b) ( )( ).5 b p j 0.5 b a mn ( b a) ( )( ).5 a a b a b a b The algorthm shown n Fg. s desgned to determne to what degree a cell belongs to the fuzzy set of Nash Equlbrums. Ths algorthm assgns each cell the mnmum prorty of players as the degree of beng Nash equlbrum. However the mnmum operator can be replaced by any other T-norm; Eample : The algorthm descrbed n Fg. s mplemented n the game of Table I assumng The results of the frst two steps are shown n Table III. (2) Step : make a matr wth the sze of s matr and ntalze all tems to zero (ths matr s called prorty matr n ths algorthm) Step 2: for all players P - F the strategy of other players( -) - Calculate the prorty of all s of P usng fuzzy preference relaton - For all strateges of P o Put prorty of strategy n the th element of cell ( -) Step 3: determne the mnmum of the elements of each cell as the degree of beng Nash Equlbrum. (Fnd the graded membershp) Fg.. The algorthm for fndng the degree of belongng to Nash Equlbrum fuzzy set for each cell of the crsp s game In the frst row of the game (Table I) player 2 has appromately the same s and player has relatvely hgh s for all cells. Therefore one may epect they have appromately the same degrees of beng Nash equlbrum. Ths s eactly what the proposed algorthm has calculated and s obvous n the frst row of Table IV. In other words a relatvely small dfference n player 2 s has resulted n the dstrbuton of an equlbrum degree n the frst row. TABLE III PRIORITY MATRIX OF THE GAME OF TABLE I K K2 K3 J ( ) ( ) ( ) J2 ( ) ( ) ( ) J3 ( ) ( ) ( ) TABLE IV DEGREE OF BEING NASH EQUILIBRIUM FOR THE GAME OF TABLE I USING MINIMUM AS T-NORM OPERATOR K K2 K3 J Nash of deg Nash of deg Nash of deg J2 Nash of deg 0.03 Nash of deg 0.05 Nash of deg 0.03 J3 Nash of deg 0.6 Nash of deg 0.06 Nash of deg 0.59 The effect of δ on the prortes of s ( ) and ( ) s shown n Fg. 2 and Fg. 3 respectvely. The only correlaton that can be dentfed n these fgures s that as δ ncreases the prorty of the hghest decreases. As the fgures shows there s no lnear relatonshp between and the prorty.

4 ( ) ma A( h( )) (3) h( ) where h s a mult-valued mappng.e. h( V ( ) ). In the proposed method the length of -cuts of the fuzzy value s ncreased or decreased accordng user vewpont. We suggest the followng formula for comparson between fuzzy values A and B where ther modfed membershp functon s used Prorty Fg. 2. Prorty of s ( ) versus δ. SF ( A B) V y ( ) ( ) ( y) ( y) d d (4) SF ( B A) V y ( ) ( ) ( y) ( y) d d (5) Prorty Fg. 3. Prorty of s ( ) versus δ. III. GAMES WITH FUZZY PAYOFFS A. Possblty of Beng Greater Between Fuzzy Values In the fuzzy decson makng and fuzzy game theory wth fuzzy s rankng the fuzzy value s a necessary procedure. Varous methods for rankng fuzzy subsets have been planned [25-32]. Though most methods can only rank fuzzy values n [25] the credblty measure as the summaton of possblty and necessty measure s used to show the degree of greatness. However ther method can not consder the possblty dstrbuton of fuzzy values. In [26-28] the satsfacton functon (SF) as the truth value of an arthmetc comparson between fuzzy values was ntroduced. However method n [28] can only rank the fuzzy values when there are vewponts but cannot show the degree of beng greater. In addton t s unclear how a fuzzy number wth ndefnte substance can compare wth vewpont. In other words the fuzzy sets have nature of possblty but vewponts are constructed as user's preferences and nterests. Ths paper ntroduces a new method for calculatng SF when users have a vewpont. The user's vewpont s ncorporated n the doman of value. By ths means each element s etended tov where V s the user vewpont. Hence the membershp functon of fuzzy values s modfed dependent on vewpont as follows: where the operator s a T-norm e.g. t can be mn or the multplcaton operator. ( A B) and ( A B) determne the possblty of truth of the A B and A B.e. they represent the possblty that fuzzy number A s smaller than B and A s larger than B respectvely. It s obvous that SF ( A B) SF ( A B) V V. ' ' Snce A and B n defnton (3) are fuzzy values all propertes whch confrmed n [28] holds for defntons (4) and (5) n ths paper. Eample 2: The followng eample s presented to show how the user vewpont affects the value of SF. It conssts of the categores of optmsm pessmsm and neutral user. Consder two fuzzy values A and B and three vewpontsv V2 and V 3 shown n Fg. 4 and Fg. 5 respectvely. Table V shows the a comparson between fuzzy values A and B usng the approach n [28] and Table VI shows the results of a comparson between fuzzy values A and B usng (4) and (5). As shown n Table V the method for rankng fuzzy values n [28] s nsenstve to the user vewponts and t s unable to determne the effect of vewponts snce t can only rank fuzzy values when user nterests est. However t seems that the result for pessmstc players ( V ) gves A larger than B the result of proposed method s congruous as epected. If the user s a pessmstc one he s satsfed wth the lower value. Hence the low value has great mportance to the user and ( A B) s less than 0.5 because A s closer to zero. Addtonally f a user s optmstc he/she prefers to choose the hgh value and the hgh value s more mportant to hm/her than a low value. Table VI shows that ( A B)

5 5 for an optmstc user s bgger than ( A B) for a neutral user snce n optmsm one the hgh value has more weghts and fuzzy value B has value of doman n hgh value. Fg. 4. Two smple fuzzy values A and B. A V() V2() Fg. 5. Three VewpontsV V 2 and V 3 show pessmstc neutral and optmstc player respectvely. TABLE V RANKING FUZZY VALUES A AND B USING METHOD IN [28] V V2 V3 B > A B > A B > A TABLE VI RANKING FUZZY VALUES A AND B USING (4) AND (5) V V2 V3 SFV (A < B) SFV (A > B) B. Fuzzy Nash Equlbrums n Games wth Fuzzy Payoffs There have been many studes for defnng a game n fuzzy parameters. As dscussed earler a game has four man components: a set of players a set of strateges for each player a set of s and preference relatonshp. Defnng each of these components as a fuzzy component would lead to a fuzzy game. Most of the prevous works on fuzzy games s concerned wth defnng fuzzy s and as a result defnng a preference on these fuzzy s. An eample s n an electon where the canddates may select dfferent campagn ssues on whch to focus. Dfferent ssues may brng dfferent votes and the number of votes can only be estmated. For nstance canddates may thnk that f they concentrate on a specfc ssue for each of the number of votes there s possblty. Fuzzy sets theory s shown to be an approprate means to model these uncertantes. In ths paper a new method s proposed for fndng degree of beng Nash equlbrum of each cell. Ths eplanaton determnes the dstrbuton of the degree of beng Nash equlbrum n the matr game. In fact the algorthm n B V3() 8 classcal game theory mentoned n secton II s modfed n the case of havng uncertanty n s.e. t s a generalzaton of the classcal game theory algorthm. In classcal game theory a crsp s clearly greater than another one or not but n the case of a fuzzy there s uncertanty n rankng fuzzy values. These uncertantes are shown by the degree of truth of arthmetc comparson.e. the satsfacton functon. Defnton 2: f each player has vewpontv every N-tuple strategy has a possblty of beng pure strategy N Nash equlbrum wth the degree of Nash mn {... N} mn X SF V N N N Also n matr games t can be stated n two steps: ) Replace X wth mn SFV. 2) Fnd the mnmum of the elements of each cell. Ths value s the degree of beng Nash equlbrum for that cell. Eample 3: Regardng crsp games two person games are dscussed because they are easer to consder but they can be generalzed to more than two players. For nstance consder T a b denotes the fuzzy game descrbed n Table VII where a fuzzy trangular number wth a center on a and boundares a b such as Fg. 6. The results of the proposed algorthm on on the game are shown n Tables VIII and IX for neutral players. μ A() A Fg. 6. A Smple fuzzy trangular number A (T(5)). TABLE VII A SIMPLE GAME WITH FUZZY PAYOFFS ΠΠ2 K K2 K3 J T(5)T(32) T(6)T(3) T(52)T(3.5) J2 T(3)T() T(32)T(4) T(3)T(32) J3 T(4)T(42) T(52)T(32) T(7)T(62) (6)

6 6 TABLE VIII FINDING NASH EQUILIBRIUM DEGREE FOR EACH NEUTRAL PLAYERS K K2 K3 J ( ) ( ) (0.00.5) J2 (00) (00.854) (00.46) J3 ( ) ( ) ( ) TABLE IX FINDING NASH EQUILIBRIUM DEGREE OF EACH CELL FOR NEUTRAL PLAYERS K K2 K3 J Nash of deg 0.5 Nash of deg 0.5 Nash of deg 0.0 J2 Nash of deg 0 Nash of deg 0 Nash of deg 0 J3 Nash of deg 0.04 Nash of deg Nash of deg For analyzng the effect of user vewpont n the dstrbuton of Nash degrees consderv andv 3 n Fg. 5 for both players n whch V shows a pessmstc player andv 3 shows an optmstc one. Tables X and XI determne the graded Nash of each cell for pessmsm and optmsm players respectvely: TABLE X FINDING NASH EQUILIBRIUM DEGREE OF EACH CELL FOR PESSIMISM PLAYERS K K2 K3 J Nash of deg Nash of deg Nash of deg 0.3 J2 Nash of deg 0 Nash of deg 0.5 Nash of deg J3 Nash of deg Nash of deg Nash of deg 0 TABLE XI FINDING NASH EQUILIBRIUM DEGREE OF EACH CELL FOR OPTIMISM PLAYERS K K2 K3 J Nash of deg 0.54 Nash of deg Nash of deg 0.03 J2 Nash of deg 0 Nash of deg 0 Nash of deg 0 J3 Nash of deg Nash of deg Nash of deg Comparng the results of Tables X and XI one can conclude that as players become less gree.e. eperencng hgher degrees of satsfacton from lower s degrees of beng Nash equlbrum become more wdely dstrbuted n the games matr and the degrees grow closer to each other. Ths effect occurs because when players become less gree the prorty of dfferent s wll ncrease and become more smlar to others. In the frst row of the game when players are optmstc they prefer to choose T 32 and hence ths yelds more SF than when players are neutral. Moreover when players are pessmstc cells wth a hgh Nash degree n neutral and optmstc cases change to cells wth a low Nash degree and ths s because the of nverson of the user vewpont. IV. CONCLUSION In ths paper a new approach s ntroduced for analyzng games more realstcally than prevous models. In the frst part only the preference relatonshp s generalzed to a fuzzy one.e. the relatonshp of "greater than or equal" s etended to a fuzzy one whch descrbes how much a crsp number s greater than or equal to another number. In crsp games a fuzzy preference relaton was employed for comparng s and calculatng the prorty of each usng the Least Devaton method. Usng ths prorty a value of beng equlbrum s computed and t s shown that ths value yelds more realstc results. In the case of havng fuzzy s the defnton of satsfacton functon when players have vewponts s mproved. The proposed method ncorporates player vewponts n the doman of fuzzy value and transforms t to another fuzzy value. The algorthm for fndng the Nash degree of each cell s proposed. Fnally the effect of dfferent vewponts on the result of the game s studed. Comparng the results to the fuzzy Nash equlbrum the results obtaned through ths strategy were more senstve to the s. REFERENCES [] D. Butnaru "Fuzzy games: a descrpton of the concept" Fuzzy Sets Syst. vol. pp [2] D. Butnaru "Soluton concepts for n-person fuzzy games" Advances n Fuzzy Set Theory and Applcaton"' pp [3] M. Mares Fuzzy Cooperatve Games Physca-Verlag Hedelberg 200. [4] V.Vjay S. Cahandra C. R. Bector "Matr game wth fuzzy goals and fuzzy s" Omega vol 33 ssue 5 Elsever [5] C. R. Bector and S. Chandra Fuzzy Mathematcal Programmng and Fuzzy Matr Games Sprnger-Verlag [6] Y. Chen and M. Larban "Two-person zero-sum game approach for fuzzy multple attrbute decson makng problems" Fuzzy Sets Syst. vol. 57 pp [7] S.T. Lu C. Kao "Soluton of fuzzy matr games: an applcaton of etenson prncple" Internatonal Journal of Intellgent Systems vol. 22 pp [8] I. Nshzak M. Sakawa "Equlbrum solutons n multobjectve bmatr games wth fuzzy s and fuzzy goals" Fuzzy Sets Syst. vol. pp [9] T. Maeda "On characterzaton of equlbrum strategy of two-person zero-sum games wth fuzzy s" Fuzzy Sets Syst. vol. 39 pp [0] B. S. Chen C. H. Tseng H. J. Uang "Fuzzy dfferental games for nonlnear stochastc systems: suboptmal approach" IEEE Trans. Fuzzy Syst. vol. 0 pp [] R. Sharma M. Gopal "Hybrd game strategy n fuzzy markov-gamebased control" IEEE Trans. Fuzzy Syst. vol. 6 pp [2] A. Chaker A. Nour Daran C. Lucas "How Can fuzzy logc determne game equlbrums better?" n Proc. IEEE Int. Conf. Intellgent Systems Bulgara 2008 pp [3] W. Kma and K. Leeb "Generalzed fuzzy games and fuzzy equlbra" Fuzzy Sets Syst. vol. 22 pp [4] Q. Song and A. Kandel "A fuzzy approach to strategc games" IEEE Trans. Fuzzy Syst. vol. 7 pp [5] D. Garagc J.B. Cruz "An approach to fuzzy non-cooperatve Nash games" J. Optm. Theory Appl. vol. 8 pp [6] K. W. L F. Karray K. W. Hpel D. M. Klgour "Fuzzy approach to the game of chcken" IEEE Trans. Fuzzy Syst. vol. 9 pp

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