AUTOMATIC GENERATION OF FUZZY PAYOFF MATRIX IN GAME THEORY Dr. Farha I. D. Al Ai * ad Dr. Muhaed Alfarras ** * College of Egieerig ** College of Coputer Egieerig ad scieces Gulf Uiversity * Dr.farha@gulfuiversity.et; ** Dr.uhaed@gulfuiversity.et ABSTRACT I fact, all the updated available research o gae theory cosider oly the explicitly playoff atrix; where both oppoets are assued to be very clever such that the strategies of each oppoet are kow to the other. The solutios obtaied are restricted to these give data. I reality, eve if the player is pioeer, we do ot expect hi to kow all the strategies of his oppoet, hece the obtaied solutio is ot accurate. I this paper, a succeeded attept to predict the i-betwee data, i.e. the data which are give iplicitly could be geerated. Hece a ore accurate solutios are obtaied. Key words: Gae theory, fuzzy, strategies, payoff atrix, players. KEYWORDS Gae Theory, Tuzzy, Strategies, Payoff Matrix, Players. 1 INTRODUCTION The well kow: gae theory is defied as the theory which deals with situatios i which two oppoets have equal or uequal uber of strategies. These strategies iclude a set of coflictig objectives. Strategies at voices are selected by the oppoets or players such that each of the is tryig to icrease his profit ad wi at the ed of takig the actio. I fact, all the updated available researches o gae theory cosider oly the give playoff atrix; see for exaples [1],[2],[3] ad [4]. The etries of these atrices are give explicitly i.e. both oppoets are very clever such that the strategies of each oppoet are kow to the other; hece the solutios obtaied are restricted to these give data. I this paper, a succeeded attept to predict the i-betwee data, i.e. the data which are give iplicitly could be geerated. These explicit data are expected to be a extra iforatio help the oppoets to take the right decisios ad the better strategies tha the exist. As a results the solutios are expected to be ore accurate tha the available. The proposed research geerates the payoff atrix to a reasoable size. The size depeds o the required accuracy. A progra writte i ath lap is developed to geerate the payoff atrices autoatically. The geerated atrix ca be solved by the available techiques; such as siplex ad Big M ethods. 2 ASSUMPTIONS I gae theory ad i order to fid the solutio of the gae, such that the strategies (pure or ixed) ust be deteried for both oppoets ad the value of the gae is calculated. To do so the etries of the payoff atrix are give ad the followig assuptios should be iposed. 364
Both oppoets have set of strategies or uber of choices which are equal or ot. Both oppoets are very clever such that ai strategies of each oppoet are kow to the other but there are a set of ukow strategies i betwee. Both oppoets take their decisios which strategies they will be adopted at the sae tie. See [4],[5] ad [6] ad [7]. 3 GAME THEORY MODELS The odel orally adopted accordig to the above assuptio ca be represeted by the followig payoff atrix (we will use players A & B as the two oppoets). Table 1.The Origial Payoff Matrix Player B The proble ca be rewritte as a liear prograig proble as Where Subject to: is the value of the gae. Player B is optial strategies ca be deteried by solvig Play er B There are & strategies for players A & B respectively. Play A will use his strategy with probabilities, ad player B will use his strategies with probabilities. The atheatical odel ca be writte accordig to player A's probabilities etioed above. Optial probabilities ca be deteried by solvig: This proble ca be rewritte: Dividig all the above equatios by V ad a certai arrageets yield: Player A: Mi(1lv) s.t i 1 pi 365
i 1 c ij pi 1 j 1,2... A(2K,1) A(2K,2) A(2K,) Player B: Mi(1lv) i Q 1 i s.t i 1 c ijqi 1 j 1,2... Where Q i =q i /v ad P i =p i /v. I fact both probles ca be cosidered such that ay of the is the prial ad the secod is dual. As proble ca be solved by usig Big M ethod. B's proble ca be solved by usig siple ethod which is sipler tha the above. I both cases the value of the gae is to be optial. It is ot eed to solve both probles istead; oe solutio is et the requireets. 4 THE PROPOSED MATHEMATICAL MODELS I fact i ost real probles ot all the assuptio iposed i the previous sectio ca be et. For exaple it is ot always valid, eve whe the players are very clever, that they kow the strategies of their oppoets, hece a fuzzy gae is raised. I this paper, a proposed oppoet is subjected o estiatig the etries of the oppoet strategies. The expected payoff atrix of the fuzzy gae as show i table 2. Table 2. the Expected Geerated Payoff Matrix Player A Player B A(1,1) A(1,2) A(1,) A(2,1) A(1,2) A(1,) A(3,1) A(1,2) A(1,) A(2K,1) A(2K,2) A(2K,) A(k+1,1) A(k+1,2) A(k+1,) A(k+2,1) A(k+2,2) A(k+2,) A(k+3,1) A(k+3,2) A(k+3,) A(2K+1,1) A(2K+1,2) A(2K+1,) A(2K+2,1) A(2K+2,2) A(2K+2,) A(2K+3,1) A(2K+3,2) A(2K+3,) A(3K,1) A(3K,2) A(3K,) 1)K,1) 2)K+1,1) 2)K+2,1) 2)K+1,2) 2)K+2,2) 2)K,2) Note that the etries 2)K+1,) 2)K+1,) 2)K,) are kow explicitly. While Aij are kow iplicitly ad have to be calculated. I fact there are ifiite ubers of strategies of both players ca be estiated ad calculated with the rage of the explicitly give etries values. It is expected that there are differeces attitude towards egative risk. These egative risks ca be represeted by assigig those etries betwee the explicitly give strategies. These ca be doe by usig what is called utilized criteria. Accordig to this criteria a weights to each two strategies of player A ad the of player B. i fact weights ca be assiged to two or ore strategies to produce the fuzzy strategies. I this paper, the weights will be assiged to the two strategies. By doig so uliited uber of strategies for each player ca be geerated. I reality, the uber ust liited. By the days developed coputers, a huge payoff atrix ca be geerated ad solved. As a result if uliited payoff atrix will be geerated the value of the gae ca ot be calculated. Istead, a appropriate uber of strategies for 366
each player will be geerated. Matheatically we will assue that K cobiatio of weights will be assiged such that for each 2 etries, the precedig ad the followig oe will assig firstly. The first will assig w1 ad will assig w2 ad vice-versa i.e. i the ext stage, Cij will assig w1. That is for player A, for B, C ij will assig w1, ad will assig w2 ad vice versa. The geeratig strategies for player. A ( K 1)( 1) ad B ( K 1)( 1) 5 THE MATHEMATICAL MODELS Four odels will be cosidered. 5.1 Model 1: * This is the odel cosists of the give atrix with o chage ad o utilizatio o their etries. 5.2 Model 2: [( 1) k ][ ] This is the odel whe the utilizatio o A's strategies have bee doe but othig doe o B's strategies. 5.3 Model III: [( 1) k ][ ] This odel whe utilizatio o B's strategies have bee but othig doe o A's strategies w assiged that B ca predict the fuzzy strategies but ot A. 5.4 Model 4: [( 1) k ][( 1) k ] This is the odel whe both of the players ca predict the sae uber of ew strategies. I reality we ca iterpret a huge uber of differet ubers of strategies ad they ca be predicted. 6. SOLUTION METHODS: Solutios ca be obtaied by solvig a huge uber of geeratig payoff atrix etries together with origial give atrix as liear prograig probles. I fact, A solutios ca be obtaied for the origial payoff atrix, the ew geerated etries of the atrix or the copoud oe which iclude the origial together with ew geerated etries. The origial atrix ad the copoud will be solved i order to ake a copariso betwee the two solutios. Oly odel 4 will be described atheatically, because a certai restrictio o it ca geerate ay of the above odels: I or II or III. i ( k 1)( 1) Cij Pi Ckl P k, j 1,2..., k 1 k 1,2,...,( k 1)( 1) For all i=1,2,3..., ad k=+1,+2, +3...,(K-1)(-1) 7. PSEUDO CODE A Pseudo code is writte i Matlap ad it is available whe it is required by researchers. 8. EXAMPLE Cosider the payoff Matrix show below : 367
Player B I II III Q 1 Q 2 Q 3 player I P 1 4 1-3 A II P 2 3 1 6 III P 3-3 4-2 Solve it by usig Liear Prograig, geerate the fuzzy atrix. a. Solutio. It ca be show that the solutio, see [8]: The strategies of player A are { p 1 = 0.25, p 2 =0.5,p 3 =0.25} ad the value of the gae is = (7/4). The strategies of player B are { q 1 =(27/92),q 2 =(62/92),q 3 =(3/92) ad the value of the gae =(7/4). b. The geerated Fuzzy atrix. It is obtaied by usig the Pseudo code give i sectio 7. 4.0000 3.0000-3.0000 3.1000 3.2000-2.3000-1.6000-2.4000-1.8000 1.0000 1.0000 4.0000 1.0000 1.0000 3.7000 3.4000 3.7000 3.4000-3.0000 6.0000-2.0000 5.1000 4.2000-2.1000-2.2000-1.2000-0.4000 1.3000 1.2000 3.3000 1.2100 1.2200 3.1000 2.9000 3.0900 2.8800 1.6000 1.4000 2.6000 1.4200 1.4400 2.5000 2.4000 2.4800 2.3600-2.3000 5.7000-2.1000 4.9000 4.1000-2.1200-2.1400-1.3200-0.5400-1.6000 5.4000-2.2000 4.7000 4.0000-2.1400-2.0800-1.4400-0.6800-2.6000 5.5000-1.4000 4.6900 3.8800-1.5200-1.6400-0.7100-0.0200-2.2000 5.0000-0.8000 4.2800 3.5600-0.9400-1.0800-0.2200 0.3600 It ca be see that the geerated atrix is 9*9 =81 eleets while the origial oe is 3*3= 9 eleets.it eas that 6 extra strategies are to be cosidered for each player. As have bee etioed that extra eleets ca be icreased to uliited uber.by solvig this ew atrix, it ca be see that the uber of strategies of the players are greater. The ew strategies of player A are { p 1, p 2, p 3,..., ad p 9 }. The strategies of player B are { q 1,q 2, q 2,,...,ad q 9 } ad the value of the gae is certaily differet. For ore accurate values a huge fuzzy atrix ca be geerated ad a huge uber of strategies will be geerated as a result with a ew ad accurate solutios. 9. CONCLUSION Ulike to the up dated researches, ot oly the explicitly give atrix is cosidered, but a extra part is cosidered. This part iclude the iplicitly strategies have to be cosidered by both players. To do so, what is called a fuzzy payoff atrix is autoatically geerated.as a result, the obtaied solutio is expected to be ore accurate. REFERENCES 1. 1Schotter, A.(2000) The Ecooic Theory of Social Istitutios, Cabridge. 2. 2Shubik,M.(2006) Gae Theory i the Social Scieces : Cocepts ad solutios. 3. 3Stor,j. ad Goerec, JK ad Holt,C.A(2001) Evolutioary Gae Theory,books We,bull. 4. Bbos,R.G.(1992) A pieer i Gae Theory. 5. Frieda, D.(1998) O Ecooic Applicatios of Evolutioary Gae theory, J. of Evolutioary Ecooics,Vol.22. 6. Morto D. Davis(1997) Gae Theory: A Notechical Itroductio to the Aalysis of strategy; Dover Publicatios: (Revised Editio). 7. Joh C. Harsayi ad Reihard Selte (1988)A Geeral Theory of Equilibriu Selectio i Gaes. MIT Press Books with uber 0262582384. Volue: 1,Editio: 1,ISBN: 0-262-58238-4,Hadle: RePEc:tp:titles:0262582384. 8. Vera,A.P.(2010)Operatios Research, S.K. Kataria &Sos,Delhi- 110092. 368