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International Journal of Computer Science Engineering and Information Technology Research (IJCSEITR) ISSN 2249-6831 Vol. 3, Issue 2, Jun 2013, 55-78 © TJPRC Pvt. Ltd.

A SIGNIFICANT APPROACH ON A SPECIAL CASE OF GAME THEORY

© TJPRC Pvt. Ltd.,

K. V. L. N. ACHARYULU1, MADDI. N. MURALI KRISHNA2, SATEESH BANDIKALLA3 & NAGU VADLANA4 © TJPRC Pvt. Ltd., 1

Faculty of Mathematics, Department of Mathematics, Bapatla Engineering College, Bapatla, India 2

II M.C.A, Department of M.C.A, Bapatla Engineering College, Bapatla, India

3

Faculty of Computer Science, Department of MCA, Bapatla Engineering College, Bapatla, India

4

Faculty of Computer Science, Department of MCA, Bapatla Engineering College, Bapatla, India

ABSTRACT A special case of game theory problem is identified and investigated for getting optimal pure mixed strategies in a non both row and column dominant game with the assistance of Brown’s Algorithm in this paper. The problem presents a non dominance nature for both rows and columns. It is instituted with the premise of having same quantity in (i,j) and (j,i) actions.Few noteworthy determinations are found by computing

maximum number of possible iterations with the

classical Java program .The results are also shown in the graphs where ever necessary and feasible. The Lower bounds and Upper bounds are also traced in each scientific computation.The consequences are observed at each computational level.

KEYWORDS: Game Theory, Players, Strategy, Pay-Off Matrix, Optimal Solution, Lower Bound, Upper Bound AMS Classification: 91A05, 91A18, 91A43, 91A90

INTRODUCTION Real Life problems need firm decision making in a competition situation even though they have many opposing parties with mutual conflicting interests. In a competition, the course of actions for each competitor may be finite or infinite. In pure strategies each player cognizes incisively what other player is going to do. But in mixed strategies, the players have a set of strategies and each player is always prevented to imagine the other players selected course of action. The main objective in any game problem is to maximize expected gains or to minimize expected losses. Sometimes it is also noticed that one of the pure strategies of a player is always inferior to at least one of the remaining ones. Then the superior strategies dominate the inferior ones. In this investigation, the authors have considered a special case in which any of the strategies does not dominate on the other. Brown’s algorithm yields an approximate solution for the value of the game and exact value will be obtained at some high degree of accuracy. It is also acknowledged as Iterative Method of approximate solution. K.V.L.N.Acharyulu and Maddi.N.Murali Krishna[1,2] investigated few game theory special problems and established some fruitful results. McKinsey [7] formulated theory of Games in 1952. Raiffa, R. D [6] hashed out the nature of games and possible decisions in 1958.Later Dresher, M [5] focused on strategies and applications of game theory in 1961.Afterwards Rapoport [4], Levin and Desjardins [3] explicated the conceptions of game theory to make a good path in operations research. Billy E.Gillett [2] discussed how to solve the large size of problems in the games by employing Brown’s algorithm. In the continuation of their work, the authors constructed a 15x15 game problem which is a special case of game theory and evaluated with the aid of Brown’s Algorithm. It has no dominance nature on both the rows and columns. The


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precept of this model is followed by taking the same action on ((i,j) and (j,i) components. Noteworthy results are found by computing maximum number of possible iterations. The incurred results are given in the conclusions and also the necessary graphs are illustrated. The iterations are calculated from 50 th iteration to 500th iteration. The authors utilized Brown's algorithm with the help of programming language of Java for this investigation. The influences among the actions of Player A and the actions of Player B are established. The errors are also calculated in each iteration and shown in a table. Upper bounds and Lower bounds are estimated for classifying the nature of the game. The uttermost possible iterations have been reckoned to reach the best optimum mixed strategies for the players.

BASIC FORMATION OF 15x15 GAME The game with 15 rows and 15 columns is construted with the 15 possible actions of player A & Player B. One player selects only one single action from his/her set possible actions. It consists of fifteen possible actions of A i.eA1,A2,A3,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A15 which will effect on the other fifteen possible actions of player B i.e B1,B2,B3,B4,B5,B6,B7,B8,B9,B10,B11, B12,B13, B14,B15.The pay off matrix can be represented as 1 16  17  18 19   20  21   22  23   24   25  26   27  28   29

16 2

17 30

18 31

19 32

20 33

21 34

22 35

23 36

24 37

30 31

3 43

43 4

44 55

45 56

46 57

47 58

48 59

49 60

32 33

44 45

55 56

5 66

66 6

67 76

68 77

69 78

70 79

34

46

57

67

76

7

85

86

87

35 36

47 48

58 59

68 69

77 78

85 86

8 93

93 9

94 100

37 38

49 50

60 61

70 71

79 80

87 88

94 95

100 10 101 106

39 40

51 52

62 63

72 73

81 82

89 90

96 97

102 107 103 108

41

53

64

74

83

91

98

104

109

42

54

65

75

84

92

99

105

110

29  42  50 51 50 53 54   61 62 63 64 65  71 72 73 74 75   80 81 82 83 84  88 89 90 91 92   95 96 97 98 99  101 102 103 104 105   106 107 108 109 110   11 111 112 113 114  111 12 115 116 117   112 115 13 118 119  113 116 118 14 120   114 117 119 120 15  25 38

26 39

27 40

28 41

MATERIAL AND METHODS The authors adopted Brown’s algorithm to solve this special case of 15x15 game in which row and columns both dominated. Brown’s Algorithm: Step 1: Player A chooses one of the possible actions(Ai1) from A1-A15 to play, and Player B then plays with the possible action Bj1 corresponding to the smallest element in the selected action Ai1. Step 2: Player A then picks out the possible action (Ai2) from A1 - A15 to play corresponding to the largest element in the possible action (Bj1) selected by Player B in step 1. Step 3: Player B sums the actions of Player A has played thus far, and plays with the possible action of Bj2 corresponding to a smallest sum element. Step 4: Player A sums the actions of Player B has played thus far, and plays the possible action (Ai3) corresponding to a largest sum element. After the required iterations are computed,then go to step 5; otherwise, come back to step 3. Step 5: Compute an upper and lower bound  and  respectively.  

Largest sum element from step 4 Number of plays of the game thus far

and  

Smallest sum element from step 3 Number of plays of the game thus far

Step 6: let Xi be the portion of the time Player A played row i with i=1,2,...,m and let Yi be the proportion of the time Player B played column j with j=1,2,...,n. These strategies approximate the optimal mini max strategies. Upper and Lower bounds on the value of the game where     are calculated in step 5. The Process completes.


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RESULTS Brown's algorithm is applied on constituted game to derive the best optimum mixed strategies for both the players from 50th iteration to 500 th iteration with the help of Java Program. The effects on all possible actions of player A from player B are given in the following tables from Table(1) to Table(0). Table 1: By the Choice of Action A1: Player A Vs Player B from 50 - 250 Iterations 50 105 851 923 993 1061 1127 1191 1253 1313 1371 1427 1481 1533 1478 1527

1352 2004 2593 3133 3624 4066 4459 4803 5098 5344 5541 5689 3292 3528 5570

155 1651 1773 1893 2011 2127 2241 2353 2463 2571 2677 2781 2883 2878 2977

100 2794 4096 5285 6375 7366 8258 9051 9745 10340 10836 11233 11531 8818 9520 6736

205 2451 2623 2793 2961 3127 3291 3453 3613 3771 3927 4081 4233 4278 4427

150 4244 6196 7985 9625 11116 12458 13651 14695 15590 16336 16933 17381 14768 15520 7486

255 3251 3473 3693 3911 4127 4341 4553 4763 4971 5177 5381 5583 5678 5877

200 5694 8296 10685 12875 14866 16658 18251 19645 20840 21836 22633 23231 20718 21520 8236

305 4051 4323 4593 4861 5127 5391 5653 5913 6171 6427 6681 6933 7078 7327

250 7144 10396 13385 16125 18616 20858 22851 24595 26090 27336 28333 29081 26668 27520 8986

Table 2: By the Choice of Action A1: Player A Vs Player B from 300 - 500 Iterations 300th Iteration Player Player A B 355 8594 4851 12496 5173 16085 5493 19375 5811 22366 6127 25058 6441 27451 6753 29545 7063 31340 7371 32836 7677 34033 7981 34931 8283 32618 8478 33520 8777 9736

350th Iteration Player Player A B 405 10044 5651 14596 6023 18785 6393 22625 6761 26116 7127 29258 7491 32051 7853 34495 8213 36590 8571 38336 8927 39733 9281 40781 9633 38568 9878 39520 10227 10486

400th Iteration Player Player A B 567 11479 6555 16681 7021 21470 7481 25860 7935 29851 8383 33443 8825 36636 9261 39430 9691 41825 10115 43821 10533 45418 10945 46616 11351 44503 11646 43930 11621 12811

450th Iteration Player Player A B 617 12903 7355 18755 7871 24144 8381 29084 8885 33575 9383 37617 9875 41210 10361 44354 10841 47049 11315 49295 11783 51092 12245 52440 12701 50427 13046 47174 13071 16291

500th Iteration Player Player A B 667 14353 8155 20855 8721 26844 9281 32334 9835 37325 10383 41817 10925 45810 11461 49304 11991 52299 12515 54795 13033 56792 13545 58290 14051 56377 14446 53174 14521 17041

Table 3: By the Choice of Action A2: Player A Vs Player B from 50 - 250 Iterations 50th Iteration Player Player A B 135 1358 823 1981 949 2597 1019 3137 1087 3628 1153 4070

100th Iteration Player Player A B 185 2800 1623 4073 1799 5289 1919 6379 2037 7370 2153 8262

150th Iteration Player Player A B 235 4250 2423 6173 2649 7989 2819 9629 2987 11120 3153 12462

200th Iteration Player Player A B 285 5700 3223 8273 3499 10689 3719 12879 3937 14870 4153 16662

250th Iteration Player Player A B 335 7150 4023 10373 4349 13389 4619 16129 4887 18620 5153 20862


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1217 1279 1339 1397 1453 1507 1559 1504 1553

4463 4807 5102 5348 5545 5693 2464 4267 5680

2267 2379 2489 2597 2703 2807 2909 2904 3003

9055 9749 10344 10840 11237 11535 7990 10259 6846

Table 3 - Contd., 3317 13655 3479 14699 3639 15594 3797 16340 3953 16937 4107 17385 4259 13940 4304 16259 4453 7596

4367 4579 4789 4997 5203 5407 5609 5704 5903

18255 19649 20844 21840 22637 23235 19890 22259 8346

5417 5679 5939 6197 6453 6707 6959 7104 7353

22855 24599 26094 27340 28337 29085 25840 28259 9096

Table 4: By the Choice of Action A2: Player A Vs Player B from 300 - 500 Iterations 300th Iteration Player Player A B 385 8600 4823 12473 5199 16089 5519 19379 5837 22370 6153 25062 6467 27455 6779 29549 7089 31344 7397 32840 7703 34037 8007 34935 8309 31790 8504 34259 8803 9846

350th Iteration Player Player A B 435 10050 5623 14573 6049 18789 6419 22629 6787 26120 7153 29262 7517 32055 7879 34499 8239 36594 8597 38340 8953 39737 9307 40785 9659 37740 9904 40259 10253 10596

400th Iteration Player Player A B 597 11493 6527 16666 7047 21482 7507 25872 7961 29863 8409 33455 8851 36648 9287 39442 9717 41837 10141 43833 10559 45430 10971 46628 11377 43683 11672 45517 11647 12081

450th Iteration Player Player A B 647 12917 7327 18740 7897 24156 8407 29096 8911 33587 9409 37629 9901 41222 10387 44366 10867 47061 11341 49307 11809 51104 12271 52452 12727 49607 13072 48761 13097 15561

500th Iteration Player Player A B 697 14367 8127 20840 8747 26856 9307 32346 9861 37337 10409 41829 10951 45822 11487 49316 12017 52311 12541 54807 13059 56804 13571 58302 14077 55557 14472 54761 14547 16311

Table 5: By the Choice of Action A3: Player A Vs Player B from 50 - 250 Iterations 50th Iteration Player Player A B 121 1369 865 2019 909 2580 1018 3159 1086 3650 1152 4092 1216 4485 1278 4829 1338 5124 1396 5370 1452 5567 1506 5715 1558 3422 1503 3449 1552 5596

100th Iteration Player Player A B 171 2811 1665 4111 1759 5272 1918 6401 2036 7392 2152 8284 2266 9077 2378 9771 2488 10366 2596 10862 2702 11259 2806 11557 2908 8948 2903 9441 3002 6762

150th Iteration Player Player A B 221 4261 2465 6211 2609 7972 2818 9651 2986 11142 3152 12484 3316 13677 3478 14721 3638 15616 3796 16362 3952 16959 4106 17407 4258 14898 4303 15441 4452 7512

200th Iteration Player Player A B 271 5711 3265 8311 3459 10672 3718 12901 3936 14892 4152 16684 4366 18277 4578 19671 4788 20866 4996 21862 5202 22659 5406 23257 5608 20848 5703 21441 5902 8262

250th Iteration Player Player A B 321 7161 4065 10411 4309 13372 4618 16151 4886 18642 5152 20884 5416 22877 5678 24621 5938 26116 6196 27362 6452 28359 6706 29107 6958 26798 7103 27441 7352 9012


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A Significant Approach on a Special Case of Game Theory

Table 6: By the Choice of Action A3: Player A Vs Player B from 300 - 500 Iterations 300th Iteration Player Player A B 371 8611 4865 12511 5159 16072 5518 19401 5836 22392 6152 25084 6466 27477 6778 29571 7088 31366 7396 32862 7702 34059 8006 34957 8308 32748 8503 33441 8802 9762

350th Iteration Player Player A B 421 10061 5665 14611 6009 18772 6418 22651 6786 26142 7152 29284 7516 32077 7878 34521 8238 36616 8596 38362 8952 39759 9306 40807 9658 38698 9903 39441 10252 10512

400th Iteration Player Player A B 583 11497 6569 16697 7007 21458 7506 25887 7960 29878 8408 33470 8850 36663 9286 39457 9716 41852 10140 43848 10558 45445 10970 46643 11376 44634 11671 43957 11646 12732

450th Iteration Player Player A B 633 12921 7369 18771 7857 24132 8406 29111 8910 33602 9408 37644 9900 41237 10386 44381 10866 47076 11340 49322 11808 51119 12270 52467 12726 50558 13071 47201 13096 16212

500th Iteration Player Player A B 683 14371 8169 20871 8707 26832 9306 32361 9860 37352 10408 41844 10950 45837 11486 49331 12016 52326 12540 54822 13058 56819 13570 58317 14076 56508 14471 53201 14546 16962

Table 7: By the Choice of Action A4: Player A Vs Player B from 50 - 250 Iterations 50th Iteration Player Player A B 122 1379 866 2029 949 2629 979 3129 1097 3670 1163 4112 1227 4505 1289 4849 1349 5144 1407 5390 1463 5587 1517 5735 1569 4274 1514 2734 1563 5510

100th Iteration Player Player A B 172 2821 1666 4121 1799 5321 1879 6371 2047 7412 2163 8304 2277 9097 2389 9791 2499 10386 2607 10882 2713 11279 2817 11577 2919 9800 2914 8726 3013 6676

150th Iteration Player Player A B 222 4271 2466 6221 2649 8021 2779 9621 2997 11162 3163 12504 3327 13697 3489 14741 3649 15636 3807 16382 3963 16979 4117 17427 4269 15750 4314 14726 4463 7426

200th Iteration Player Player A B 272 5721 3266 8321 3499 10721 3679 12871 3947 14912 4163 16704 4377 18297 4589 19691 4799 20886 5007 21882 5213 22679 5417 23277 5619 21700 5714 20726 5913 8176

250th Iteration Player Player A B 322 7171 4066 10421 4349 13421 4579 16121 4897 18662 5163 20904 5427 22897 5689 24641 5949 26136 6207 27382 6463 28379 6717 29127 6969 27650 7114 26726 7363 8926

Table 8: By the Choice of Action A4: Player A Vs Player B from 300 - 500 Iterations 300th Iteration Player Player A B 372 8621 4866 12521 5199 16121 5479 19371 5847 22412 6163 25104 6477 27497 6789 29591 7099 31386 7407 32882 7713 34079

350th Iteration Player Player A B 422 10071 5666 14621 6049 18821 6379 22621 6797 26162 7163 29304 7527 32097 7889 34541 8249 36636 8607 38382 8963 39779

400th Iteration Player Player A B 584 11500 6570 16700 7047 21500 7467 25850 7971 29891 8419 33483 8861 36676 9297 39470 9727 41865 10151 43861 10569 45458

450th Iteration Player Player A B 634 12924 7370 18774 7897 24174 8367 29074 8921 33615 9419 37657 9911 41250 10397 44394 10877 47089 11351 49335 11819 51132

500th Iteration Player Player A B 684 14374 8170 20874 8747 26874 9267 32324 9871 37365 10419 41857 10961 45850 11497 49344 12027 52339 12551 54835 13069 56832


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K. V. L. N. Acharyulu, Maddi. N. Murali Krishna, Sateesh Bandikalla & Nagu Vadlana

8017 8319 8514 8813

34977 33600 32726 9676

9317 9669 9914 10263

40827 39550 38726 10426

Table 8 - Contd., 10981 46656 11387 45479 11682 42500 11657 13381

12281 12737 13082 13107

52480 51403 45744 16861

13581 14087 14482 14557

58330 57353 51744 17611

Table 9: By the Choice of Action A5: Player A Vs Player B from 50 - 250 Iterations 50th Iteration Player Player A B 123 1389 867 2039 950 2639 1030 3189 1047 3629 1173 4131 1237 4524 1299 4868 1359 5163 1417 5409 1473 5606 1527 5754 1579 5125 1524 2018 1573 5423

100th Iteration Player Player A B 173 2831 1667 4131 1800 5331 1930 6431 1997 7371 2173 8323 2287 9116 2399 9810 2509 10405 2617 10901 2723 11298 2827 11596 2929 10651 2924 8010 3023 6589

150th Iteration Player Player A B 223 4281 2467 6231 2650 8031 2830 9681 2947 11121 3173 12523 3337 13716 3499 14760 3659 15655 3817 16401 3973 16998 4127 17446 4279 16601 4324 14010 4473 7339

200th Iteration Player Player A B 273 5731 3267 8331 3500 10731 3730 12931 3897 14871 4173 16723 4387 18316 4599 19710 4809 20905 5017 21901 5223 22698 5427 23296 5629 22551 5724 20010 5923 8089

250th Iteration Player Player A B 323 7181 4067 10431 4350 13431 4630 16181 4847 18621 5173 20923 5437 22916 5699 24660 5959 26155 6217 27401 6473 28398 6727 29146 6979 28501 7124 26010 7373 8839

Table 10: By the Choice of Action A5: Player A Vs Player B from 300 - 500 Iterations 300th Iteration Player Player A B 373 8631 4867 12531 5200 16131 5530 19431 5797 22371 6173 25123 6487 27516 6799 29610 7109 31405 7417 32901 7723 34098 8027 34996 8329 34451 8524 32010 8823 9589

350th Iteration Player Player A B 423 10081 5667 14631 6050 18831 6430 22681 6747 26121 7173 29323 7537 32116 7899 34560 8259 36655 8617 38401 8973 39798 9327 40846 9679 40401 9924 38010 10273 10339

400th Iteration Player Player A B 585 11503 6571 16703 7048 21503 7518 25903 7921 29843 8429 33495 8871 36688 9307 39482 9737 41877 10161 43873 10579 45470 10991 46668 11397 46323 11692 41042 11667 14029

450th Iteration Player Player A B 635 12927 7371 18777 7898 24177 8418 29127 8871 33567 9429 37669 9921 41262 10407 44406 10887 47101 11361 49347 11829 51144 12291 52492 12747 52247 13092 44286 13117 17509

500th Iteration Player Player A B 685 14377 8171 20877 8748 26877 9318 32377 9821 37317 10429 41869 10971 45862 11507 49356 12037 52351 12561 54847 13079 56844 13591 58342 14097 58197 14492 50286 14567 18259

Table 11: By the Choice of Action A6: Player A Vs Player B from 50 - 250 Iterations 50th Iteration Player Player A B 124 1391 868 2041 951 2641 1031 3191 1108 3691

100th Iteration Player Player A B 174 2833 1668 4133 1801 5333 1931 6433 2058 7433

150th Iteration Player Player A B 224 4283 2468 6233 2651 8033 2831 9683 3008 11183

200th Iteration Player Player A B 274 5733 3268 8333 3501 10733 3731 12933 3958 14933

250th Iteration Player Player A B 324 7183 4068 10433 4351 13433 4631 16183 4908 18683


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A Significant Approach on a Special Case of Game Theory

1113 1246 1308 1368 1426 1482 1536 1588 1533 1582

4072 4534 4878 5173 5419 5616 5764 5239 1923 5433

2113 2296 2408 2518 2626 2732 2836 2938 2933 3032

8264 9126 9820 10415 10911 11308 11606 10765 7915 6599

Table 11 - Contd., 3113 12464 3346 13726 3508 14770 3668 15665 3826 16411 3982 17008 4136 17456 4288 16715 4333 13915 4482 7349

4113 4396 4608 4818 5026 5232 5436 5638 5733 5932

16664 18326 19720 20915 21911 22708 23306 22665 19915 8099

5113 5446 5708 5968 6226 6482 6736 6988 7133 7382

20864 22926 24670 26165 27411 28408 29156 28615 25915 8849

Table 12: By the Choice of Action A6: Player A Vs Player B from 300 - 500 Iterations 300th Iteration Player Player A B 374 8633 4868 12533 5201 16133 5531 19433 5858 22433 6113 25064 6496 27526 6808 29620 7118 31415 7426 32911 7732 34108 8036 35006 8338 34565 8533 31915 8832 9599

350th Iteration Player Player A B 424 10083 5668 14633 6051 18833 6431 22683 6808 26183 7113 29264 7546 32126 7908 34570 8268 36665 8626 38411 8982 39808 9336 40856 9688 40515 9933 37915 10282 10349

400th Iteration Player Player A B 586 11506 6572 16706 7049 21506 7519 25906 7982 29906 8369 33437 8880 36699 9316 39493 9746 41888 10170 43884 10588 45481 11000 46679 11406 46438 11701 41053 11676 13934

450th Iteration Player Player A B 636 12930 7372 18780 7899 24180 8419 29130 8932 33630 9369 37611 9930 41273 10416 44417 10896 47112 11370 49358 11838 51155 12300 52503 12756 52362 13101 44297 13126 17414

500th Iteration Player Player A B 686 14380 8172 20880 8749 26880 9319 32380 9882 37380 10369 41811 10980 45873 11516 49367 12046 52362 12570 54858 13088 56855 13600 58353 14106 58312 14501 50297 14576 18164

Table 13: By the Choice of Action A7: Player A Vs Player B from 50 - 250 Iterations 50th Iteration Player Player A B 98 1399 844 2049 916 2649 986 3199 1054 3699 1120 4149 1107 4472 1240 4893 1295 5188 1349 5434 1402 5631 1454 5779 1505 5878 1555 1413 1499 5342

100th Iteration Player Player A B 175 2843 1669 4143 1802 5343 1932 6443 2059 7443 2183 8343 2227 9066 2416 9837 2526 10432 2634 10928 2740 11325 2844 11623 2946 11614 2941 7197 3040 6510

150th Iteration Player Player A B 225 4293 2469 6243 2652 8043 2832 9693 3009 11193 3183 12543 3277 13666 3516 14787 3676 15682 3834 16428 3990 17025 4144 17473 4296 17564 4341 13197 4490 7260

200th Iteration Player Player A B 275 5743 3269 8343 3502 10743 3732 12943 3959 14943 4183 16743 4327 18266 4616 19737 4826 20932 5034 21928 5240 22725 5444 23323 5646 23514 5741 19197 5940 8010

250th Iteration Player Player A B 325 7193 4069 10443 4352 13443 4632 16193 4909 18693 5183 20943 5377 22866 5716 24687 5976 26182 6234 27428 6490 28425 6744 29173 6996 29464 7141 25197 7390 8760


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Table 14: By the Choice of Action A7: Player A Vs Player B from 300 - 500 Iterations 300th Iteration Player Player A B 375 8643 4869 12543 5202 16143 5532 19443 5859 22443 6183 25143 6427 27466 6816 29637 7126 31432 7434 32928 7740 34125 8044 35023 8346 35414 8541 31197 8840 9510

350th Iteration Player Player A B 425 10093 5669 14643 6052 18843 6432 22693 6809 26193 7183 29343 7477 32066 7916 34587 8276 36682 8634 38428 8990 39825 9344 40873 9696 41364 9941 37197 10290 10260

400th Iteration Player Player A B 587 11508 6573 16708 7050 21508 7520 25908 7983 29908 8439 33508 8811 36631 9324 39502 9754 41897 10178 43893 10596 45490 11008 46688 11414 47279 11709 39487 11684 14685

450th Iteration Player Player A B 637 12932 7373 18782 7900 24182 8420 29132 8933 33632 9439 37682 9861 41205 10424 44426 10904 47121 11378 49367 11846 51164 12308 52512 12764 53203 13109 42731 13134 18165

500th Iteration Player Player A B 687 14382 8173 20882 8750 26882 9320 32382 9883 37382 10439 41882 10911 45805 11524 49376 12054 52371 12578 54867 13096 56864 13608 58362 14114 59153 14509 48731 14584 18915

Table 15: By the Choice of Action A8: Player A Vs Player B from 50 - 250 Iterations 50th Iteration Player Player A B 99 1400 845 2050 917 2650 987 3200 1055 3700 1121 4150 1185 4550 1163 4816 1302 5195 1356 5441 1409 5638 1461 5786 1512 5885 1562 1420 1506 5349

100th Iteration Player Player A B 176 2844 1670 4144 1803 5344 1933 6444 2060 7444 2184 8344 2305 9144 2339 9760 2533 10439 2641 10935 2747 11332 2851 11630 2953 11621 2948 7204 3047 6517

150th Iteration Player Player A B 226 4294 2470 6244 2653 8044 2833 9694 3010 11194 3184 12544 3355 13744 3439 14710 3683 15689 3841 16435 3997 17032 4151 17480 4303 17571 4348 13204 4497 7267

200th Iteration Player Player A B 276 5744 3270 8344 3503 10744 3733 12944 3960 14944 4184 16744 4405 18344 4539 19660 4833 20939 5041 21935 5247 22732 5451 23330 5653 23521 5748 19204 5947 8017

250th Iteration Player Player A B 326 7194 4070 10444 4353 13444 4633 16194 4910 18694 5184 20944 5455 22944 5639 24610 5983 26189 6241 27435 6497 28432 6751 29180 7003 29471 7148 25204 7397 8767

Table 16: By the Choice of Action A8: Player A Vs Player B from 300 - 500 Iterations 300th Iteration Player Player A B 376 8644 4870 12544 5203 16144 5533 19444 5860 22444 6184 25144 6505 27544 6739 29560 7133 31439 7441 32935 7747 34132

350th Iteration Player Player A B 426 10094 5670 14644 6053 18844 6433 22694 6810 26194 7184 29344 7555 32144 7839 34510 8283 36689 8641 38435 8997 39832

400th Iteration Player Player A B 588 11510 6574 16710 7051 21510 7521 25910 7984 29910 8440 33510 8889 36710 9247 39426 9761 41905 10185 43901 10603 45498

450th Iteration Player Player A B 638 12934 7374 18784 7901 24184 8421 29134 8934 33634 9440 37684 9939 41284 10347 44350 10911 47129 11385 49375 11853 51172

500th Iteration Player Player A B 688 14384 8174 20884 8751 26884 9321 32384 9884 37384 10440 41884 10989 45884 11447 49300 12061 52379 12585 54875 13103 56872


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A Significant Approach on a Special Case of Game Theory

8051 8353 8548 8847

35030 35421 31204 9517

9351 9703 9948 10297

40880 41371 37204 10267

Table 16 - Contd., 11015 46696 11421 47287 11716 39600 11691 14587

12315 12771 13116 13141

52520 53211 42844 18067

13615 14121 14516 14591

58370 59161 48844 18817

Table 17: By the Choice of Action A9: Player A Vs Player B from 50 - 250 Iterations 50th Iteration Player Player A B 100 1401 846 2051 918 2651 988 3201 1056 3701 1122 4151 1186 4551 1248 4901 1218 5111 1362 5447 1415 5644 1467 5792 1518 5891 1568 1426 1512 5355

100th Iteration Player Player A B 177 2845 1671 4145 1804 5345 1934 6445 2061 7445 2185 8345 2306 9145 2424 9845 2449 10355 2647 10941 2753 11338 2857 11636 2959 11627 2954 7210 3053 6523

150th Iteration Player Player A B 227 4295 2471 6245 2654 8045 2834 9695 3011 11195 3185 12545 3356 13745 3524 14795 3599 15605 3847 16441 4003 17038 4157 17486 4309 17577 4354 13210 4503 7273

200th Iteration Player Player A B 277 5745 3271 8345 3504 10745 3734 12945 3961 14945 4185 16745 4406 18345 4624 19745 4749 20855 5047 21941 5253 22738 5457 23336 5659 23527 5754 19210 5953 8023

250th Iteration Player Player A B 327 7195 4071 10445 4354 13445 4634 16195 4911 18695 5185 20945 5456 22945 5724 24695 5899 26105 6247 27441 6503 28438 6757 29186 7009 29477 7154 25210 7403 8773

Table 18: By the Choice of Action A9: Player A Vs Player B from 300 - 500 Iterations 300th Iteration Player Player A B 377 8645 4871 12545 5204 16145 5534 19445 5861 22445 6185 25145 6506 27545 6824 29645 7049 31355 7447 32941 7753 34138 8057 35036 8359 35427 8554 31210 8853 9523

350th Iteration Player Player A B 427 10095 5671 14645 6054 18845 6434 22695 6811 26195 7185 29345 7556 32145 7924 34595 8199 36605 8647 38441 9003 39838 9357 40886 9709 41377 9954 37210 10303 10273

400th Iteration Player Player A B 589 11511 6575 16711 7052 21511 7522 25911 7985 29911 8441 33511 8890 36711 9332 39511 9677 41821 10191 43907 10609 45504 11021 46702 11427 47293 11722 39606 11697 14593

450th Iteration Player Player A B 639 12935 7375 18785 7902 24185 8422 29135 8935 33635 9441 37685 9940 41285 10432 44435 10827 47045 11391 49381 11859 51178 12321 52526 12777 53217 13122 42850 13147 18073

500th Iteration Player Player A B 689 14385 8175 20885 8752 26885 9322 32385 9885 37385 10441 41885 10990 45885 11532 49385 11977 52295 12591 54881 13109 56878 13621 58376 14127 59167 14522 48850 14597 18823

Table 19: By the Choice of Action A10: Player A Vs Player B from 50 - 250 Iterations 50th Iteration Player Player A B 101 1403 847 2053 919 2653 989 3203 1057 3703

100th Iteration Player Player A B 178 2843 1672 4143 1805 5343 1935 6443 2062 7443

150th Iteration Player Player A B 228 4293 2472 6243 2655 8043 2835 9693 3012 11193

200th Iteration Player Player A B 278 5743 3272 8343 3505 10743 3735 12943 3962 14943

250th Iteration Player Player A B 328 7193 4072 10443 4355 13443 4635 16193 4912 18693


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K. V. L. N. Acharyulu, Maddi. N. Murali Krishna, Sateesh Bandikalla & Nagu Vadlana

1123 1187 1249 1309 1272 1420 1472 1523 1573 1517

4153 4553 4903 5203 5358 5650 5798 5897 1537 5255

2186 2307 2425 2540 2557 2758 2862 2964 2959 3058

8343 9143 9843 10443 10848 11340 11638 11837 6477 7055

Table 19 – Contd., 3186 12543 3357 13743 3525 14793 3690 15693 3757 16348 4008 17040 4162 17488 4314 17787 4359 12477 4508 7805

4186 4407 4625 4840 4957 5258 5462 5664 5759 5958

16743 18343 19743 20943 21848 22740 23338 23737 18477 8555

5186 5457 5725 5990 6157 6508 6762 7014 7159 7408

20943 22943 24693 26193 27348 28440 29188 29687 24477 9305

Table 20: By the Choice of Action A10: Player A Vs Player B from 300 - 500 Iterations 300th Iteration Player Player A B 378 8643 4872 12543 5205 16143 5535 19443 5862 22443 6186 25143 6507 27543 6825 29643 7140 31443 7357 32848 7758 34140 8062 35038 8364 35637 8559 30477 8858 10055

350th Iteration Player Player A B 428 10093 5672 14643 6055 18843 6435 22693 6812 26193 7186 29343 7557 32143 7925 34593 8290 36693 8557 38348 9008 39840 9362 40888 9714 41587 9959 36477 10308 10805

400th Iteration Player Player A B 478 11543 6472 16743 6905 21543 7335 25943 7762 29943 8186 33543 8607 36743 9025 39543 9440 41943 9757 43848 10258 45540 10662 46738 11064 47537 11359 42477 11758 11555

450th Iteration Player Player A B 640 12971 7376 18821 7903 24221 8423 29171 8936 33671 9442 37721 9941 41321 10433 44471 10918 47171 11301 49326 11864 51218 12326 52566 12782 53465 13127 46145 13152 14615

500th Iteration Player Player A B 690 14421 8176 20921 8753 26921 9323 32421 9886 37421 10442 41921 10991 45921 11533 49421 12068 52421 12501 54826 13114 56918 13626 58416 14132 59415 14527 52145 14602 15365

Table 21: By the Choice of Action A11: Player A Vs Player B from 50 - 250 Iterations 50th Iteration Player Player A B 102 1404 848 2054 920 2654 990 3204 1058 3704 1124 4154 1188 4554 1250 4904 1310 5204 1368 5454 1325 5555 1476 5802 1527 5901 1577 1541 1521 5259

100th Iteration Player Player A B 179 2844 1673 4144 1806 5344 1936 6444 2063 7444 2187 8344 2308 9144 2426 9844 2541 10444 2653 10944 2663 11245 2866 11642 2968 11841 2963 6481 3062 7059

150th Iteration Player Player A B 229 4294 2473 6244 2656 8044 2836 9694 3013 11194 3187 12544 3358 13744 3526 14794 3691 15694 3853 16444 3913 16945 4166 17492 4318 17791 4363 12481 4512 7809

200th Iteration Player Player A B 279 5744 3273 8344 3506 10744 3736 12944 3963 14944 4187 16744 4408 18344 4626 19744 4841 20944 5053 21944 5163 22645 5466 23342 5668 23741 5763 18481 5962 8559

250th Iteration Player Player A B 329 7194 4073 10444 4356 13444 4636 16194 4913 18694 5187 20944 5458 22944 5726 24694 5991 26194 6253 27444 6413 28345 6766 29192 7018 29691 7163 24481 7412 9309


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Table 22: By the Choice of Action A11: Player A Vs Player B from 300 - 500 Iterations 300th Iteration Player Player A B 379 8644 4873 12544 5206 16144 5536 19444 5863 22444 6187 25144 6508 27544 6826 29644 7141 31444 7453 32944 7663 34045 8066 35042 8368 35641 8563 30481 8862 10059

350th Iteration Player Player A B 429 10094 5673 14644 6056 18844 6436 22694 6813 26194 7187 29344 7558 32144 7926 34594 8291 36694 8653 38444 8913 39745 9366 40892 9718 41591 9963 36481 10312 10809

400th Iteration Player Player A B 479 11544 6473 16744 6906 21544 7336 25944 7763 29944 8187 33544 8608 36744 9026 39544 9441 41944 9853 43944 10163 45445 10666 46742 11068 47541 11363 42481 11762 11559

450th Iteration Player Player A B 641 12973 7377 18823 7904 24223 8424 29173 8937 33673 9443 37723 9942 41323 10434 44473 10919 47173 11397 49423 11769 51124 12330 52571 12786 53470 13131 46255 13156 14514

500th Iteration Player Player A B 691 14423 8177 20923 8754 26923 9324 32423 9887 37423 10443 41923 10992 45923 11534 49423 12069 52423 12597 54923 13019 56824 13630 58421 14136 59420 14531 52255 14606 15264

Table 23: By the Choice of Action A12: Player A Vs Player B from 50 - 250 Iterations 50th Iteration Player Player A B 103 1405 849 2055 921 2655 991 3205 1059 3705 1125 4155 1189 4555 1251 4905 1311 5205 1369 5455 1425 5655 1377 5703 1530 5904 1580 1544 1524 5262

100th Iteration Player Player A B 180 2845 1674 4145 1807 5345 1937 6445 2064 7445 2188 8345 2309 9145 2427 9845 2542 10445 2654 10945 2763 11345 2767 11543 2971 11844 2966 6484 3065 7062

150th Iteration Player Player A B 230 4295 2474 6245 2657 8045 2837 9695 3014 11195 3188 12545 3359 13745 3527 14795 3692 15695 3854 16445 4013 17045 4067 17393 4321 17794 4366 12484 4515 7812

200th Iteration Player Player A B 280 5745 3274 8345 3507 10745 3737 12945 3964 14945 4188 16745 4409 18345 4627 19745 4842 20945 5054 21945 5263 22745 5367 23243 5671 23744 5766 18484 5965 8562

250th Iteration Player Player A B 330 7195 4074 10445 4357 13445 4637 16195 4914 18695 5188 20945 5459 22945 5727 24695 5992 26195 6254 27445 6513 28445 6667 29093 7021 29694 7166 24484 7415 9312

Table 24: By the Choice of Action A12: Player A Vs Player B from 300 - 500 Iterations 300th Iteration Player Player A B 380 8645 4874 12545 5207 16145 5537 19445 5864 22445 6188 25145 6509 27545 6827 29645 7142 31445 7454 32945 7763 34145

350th Iteration Player Player A B 430 10095 5674 14645 6057 18845 6437 22695 6814 26195 7188 29345 7559 32145 7927 34595 8292 36695 8654 38445 9013 39845

400th Iteration Player Player A B 480 11545 6474 16745 6907 21545 7337 25945 7764 29945 8188 33545 8609 36745 9027 39545 9442 41945 9854 43945 10263 45545

450th Iteration Player Player A B 642 12974 7378 18824 7905 24224 8425 29174 8938 33674 9444 37724 9943 41324 10435 44474 10920 47174 11398 49424 11869 51224

500th Iteration Player Player A B 692 14424 8178 20924 8755 26924 9325 32424 9888 37424 10444 41924 10993 45924 11535 49424 12070 52424 12598 54924 13119 56924


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7967 8371 8566 8865

34943 35644 30484 10062

9267 9721 9966 10315

40793 41594 36484 10812

Table 24 - Contd., 10567 46643 11071 47544 11366 42484 11765 11562

12231 12789 13134 13159

52472 53473 46258 14517

13531 14139 14534 14609

58322 59423 52258 15267

Table 25: By the Choice of Action A13: Player A Vs Player B from 50 - 250 Iterations 50th Iteration Player Player A B 104 1406 850 2056 922 2656 992 3206 1060 3706 1126 4156 1190 4556 1252 4906 1312 5206 1370 5456 1426 5656 1480 5806 1428 5802 1582 1546 1526 5264

100th Iteration Player Player A B 181 2845 1675 4145 1808 5345 1938 6445 2065 7445 2189 8345 2310 9145 2428 9845 2543 10445 2655 10945 2764 11345 2870 11645 2869 11741 2968 6380 3067 7169

150th Iteration Player Player A B 231 4295 2475 6245 2658 8045 2838 9695 3015 11195 3189 12545 3360 13745 3528 14795 3693 15695 3855 16445 4014 17045 4170 17495 4219 17691 4368 12380 4517 7919

200th Iteration Player Player A B 281 5745 3275 8345 3508 10745 3738 12945 3965 14945 4189 16745 4410 18345 4628 19745 4843 20945 5055 21945 5264 22745 5470 23345 5569 23641 5768 18380 5967 8669

250th Iteration Player Player A B 331 7195 4075 10445 4358 13445 4638 16195 4915 18695 5189 20945 5460 22945 5728 24695 5993 26195 6255 27445 6514 28445 6770 29195 6919 29591 7168 24380 7417 9419

Table 26: By the Choice of Action A13: Player A Vs Player B from 300 - 500 Iterations 300th Iteration Player Player A B 381 8645 4875 12545 5208 16145 5538 19445 5865 22445 6189 25145 6510 27545 6828 29645 7143 31445 7455 32945 7764 34145 8070 35045 8269 35541 8568 30380 8867 10169

350th Iteration Player Player A B 431 10095 5675 14645 6058 18845 6438 22695 6815 26195 7189 29345 7560 32145 7928 34595 8293 36695 8655 38445 9014 39845 9370 40895 9619 41491 9968 36380 10317 10919

400th Iteration Player Player A B 481 11545 6475 16745 6908 21545 7338 25945 7765 29945 8189 33545 8610 36745 9028 39545 9443 41945 9855 43945 10264 45545 10670 46745 10969 47441 11368 42380 11767 11669

450th Iteration Player Player A B 643 12981 7379 18831 7906 24231 8426 29181 8939 33681 9445 37731 9944 41331 10436 44481 10921 47181 11399 49431 11870 51231 12334 52581 12687 53377 13136 46896 13161 13889

500th Iteration Player Player A B 693 14431 8179 20931 8756 26931 9326 32431 9889 37431 10445 41931 10994 45931 11536 49431 12071 52431 12599 54931 13120 56931 13634 58431 14037 59327 14536 52896 14611 14639

Table 27: By the Choice of Action A14: Player A Vs Player B from 50 - 250 Iterations 50th Iteration Player Player A B 105 1365 851 2015 923 2615 993 3165 1061 3665

100th Iteration Player Player A B 155 2807 1651 4107 1773 5307 1893 6407 2011 7407

150th Iteration Player Player A B 205 4257 2451 6207 2623 8007 2793 9657 2961 11157

200th Iteration Player Player A B 255 5707 3251 8307 3473 10707 3693 12907 3911 14907

250th Iteration Player Player A B 305 7157 4051 10407 4323 13407 4593 16157 4861 18657


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A Significant Approach on a Special Case of Game Theory

1127 1191 1253 1313 1371 1427 1481 1533 1478 1527

4115 4515 4865 5165 5415 5615 5765 1497 5810 5223

2127 2241 2353 2463 2571 2677 2781 2883 2878 2977

8307 9107 9807 10407 10907 11307 11607 7023 11802 6389

Table 27 - Contd., 3127 12507 3291 13707 3453 14757 3613 15657 3771 16407 3927 17007 4081 17457 4233 12973 4278 17802 4427 7139

4127 4341 4553 4763 4971 5177 5381 5583 5678 5877

16707 18307 19707 20907 21907 22707 23307 18923 23802 7889

5127 5391 5653 5913 6171 6427 6681 6933 7078 7327

20907 22907 24657 26157 27407 28407 29157 24873 29802 8639

Table 28: By the Choice of Action A14: Player A Vs Player B from 300- 500 Iterations 300th Iteration Player Player A B 355 8607 4851 12507 5173 16107 5493 19407 5811 22407 6127 25107 6441 27507 6753 29607 7063 31407 7371 32907 7677 34107 7981 35007 8283 30823 8478 35802 8777 9389

350th Iteration Player Player A B 405 10057 5651 14607 6023 18807 6393 22657 6761 26157 7127 29307 7491 32107 7853 34557 8213 36657 8571 38407 8927 39807 9281 40857 9633 36773 9878 41802 10227 10139

400th Iteration Player Player A B 567 11466 6555 16666 7021 21466 7481 25866 7935 29866 8383 33466 8825 36666 9261 39466 9691 41866 10115 43866 10533 45466 10945 46666 11351 42682 11646 43456 11621 15194

450th Iteration Player Player A B 617 12890 7355 18740 7871 24140 8381 29090 8885 33590 9383 37640 9875 41240 10361 44390 10841 47090 11315 49340 11783 51140 12245 52490 12701 48606 13046 46700 13071 18674

500th Iteration Player Player A B 667 14340 8155 20840 8721 26840 9281 32340 9835 37340 10383 41840 10925 45840 11461 49340 11991 52340 12515 54840 13033 56840 13545 58340 14051 54556 14446 52700 14521 19424

Table 29: By the Choice of Action A15: Player A Vs Player B from 50 - 250 Iterations 50th Iteration Player Player A B 105 1359 851 2009 923 2609 993 3159 1061 3659 1127 4109 1191 4509 1253 4859 1313 5159 1371 5409 1427 5609 1481 5759 1533 1491 1478 5174 1527 5853

100th Iteration Player Player A B 155 2801 1651 4101 1773 5301 1893 6401 2011 7401 2127 8301 2241 9101 2353 9801 2463 10401 2571 10901 2677 11301 2781 11601 2883 7017 2878 11166 2977 7019

150th Iteration Player Player A B 205 4251 2451 6201 2623 8001 2793 9651 2961 11151 3127 12501 3291 13701 3453 14751 3613 15651 3771 16401 3927 17001 4081 17451 4233 12967 4278 17166 4427 7769

200th Iteration Player Player A B 255 5701 3251 8301 3473 10701 3693 12901 3911 14901 4127 16701 4341 18301 4553 19701 4763 20901 4971 21901 5177 22701 5381 23301 5583 18917 5678 23166 5877 8519

250th Iteration Player Player A B 305 7151 4051 10401 4323 13401 4593 16151 4861 18651 5127 20901 5391 22901 5653 24651 5913 26151 6171 27401 6427 28401 6681 29151 6933 24867 7078 29166 7327 9269


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Table 30: By the Choice of Action A15: Player A Vs Player B from 300 - 500 Iterations 300th Iteration Player Player A B 355 8601 4851 12501 5173 16101 5493 19401 5811 22401 6127 25101 6441 27501 6753 29601 7063 31401 7371 32901 7677 34101 7981 35001 8283 30817 8478 35166 8777 10019

350th Iteration Player Player A B 405 10051 5651 14601 6023 18801 6393 22651 6761 26151 7127 29301 7491 32101 7853 34551 8213 36651 8571 38401 8927 39801 9281 40851 9633 36767 9878 41166 10227 10769

400th Iteration Player Player A B 455 11501 6451 16701 6873 21501 7293 25901 7711 29901 8127 33501 8541 36701 8953 39501 9363 41901 9771 43901 10177 45501 10581 46701 10983 42717 11278 47166 11677 11519

450th Iteration Player Player A B 617 12930 7355 18780 7871 24180 8381 29130 8885 33630 9383 37680 9875 41280 10361 44430 10841 47130 11315 49380 11783 51180 12245 52530 12701 48646 13046 50940 13071 14474

500th Iteration Player Player A B 667 14380 8155 20880 8721 26880 9281 32380 9835 37380 10383 41880 10925 45880 11461 49380 11991 52380 12515 54880 13033 56880 13545 58380 14051 54596 14446 56940 14521 15224

CONCLUSIONS 

The player B influences partially on all available actions of player A.

It has obtained good correlation among the iterations.

Sufficient accuracy is enhanced by successive iterations.

INFLUENCES OF PLAYER B ON THE AVAILABLE ACTIONS OF PLAYER A The effect of Player B on the possible actions of Player A are illustrated from Fig.1 to Fig.15 at each computation.


A Significant Approach on a Special Case of Game Theory

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CONCLUSIONS 

In any scientific computation the influence on one from other has gradual gain only up to some period of time

There is sudden fluctuation observed in the considerable duration

Steep declines have been identified in the course of action at the end of the game

Systematic and coherent influences are formed

The strong compatibility is established between the competences

From the middle of the game normal distribution growth has been traced

OPTIMUM MIXIED STRATEGIES OF PLAYER A AND PLAYER B The optimum mixed strategies of the playerA from the iteration 50-500 are obtained as shown in the following tables from Table-31 to Table-50. Table 31: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 50th Iteration Action A1 A B 0.96 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.5 0.02 0.44 0.02 0.06

Action A2 A B 0.92 0 0.04 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.66 0.02 0.3 0.02 0.04

Action A3 A B 0.94 0 0 0 0.02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.48 0.02 0.46 0.02 0.06

Action A4 A B 0.94 0 0 0 0 0 0.02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.32 0.02 0.6 0.02 0.08

Action A5 A B 0.94 0 0 0 0 0 0 0 0.02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.16 0.02 0.74 0.02 0.1

Action A6 A B 0.94 0 0 0 0 0 0 0 0 0 0.02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.14 0.02 0.76 0.02 0.1

Action A7 A B 0.96 0 0 0 0 0 0 0 0 0 0 0 0.02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.88 0.02 0.12


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Table 32: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 50th Iteration Action A8 A B 0.96 0 0 0 0 0 0 0 0 0 0 0 0 0 0.02 0 0 0 0 0 0 0 0 0 0 0 0 0.88 0.02 0.12

Action A9 A B 0.96 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.02 0 0 0 0 0 0 0 0 0 0 0.88 0.02 0.12

Action A10 A B 0.96 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.02 0 0 0 0 0 0 0 0 0.86 0.02 0.14

Action A11 A B 0.96 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.02 0 0 0 0 0 0 0.86 0.02 0.14

Action A12 A B 0.96 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.02 0 0 0 0 0.86 0.02 0.14

Action A13 A B 0.96 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.02 0 0 0.86 0.02 0.14

Action A14 A B 0.96 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.86 0.02 0 0.02 0.14

Action A15 A B 0.96 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.86 0.02 0.14 0.02 0

Table 33: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 100th Iteration Action A1 A B 0.98 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.28 0.01 0.22 0.01 0.5

Action A2 A B 0.96 0 0.02 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.36 0.01 0.15 0.01 0.49

Action A3 A B 0.97 0 0 0 0.01 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.27 0.01 0.23 0.01 0.5

Action A4 A B 0.97 0 0 0 0 0 0.01 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.19 0.01 0.3 0.01 0.51

Action A5 A B 0.97 0 0 0 0 0 0 0 0.01 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.11 0.01 0.37 0.01 0.52

Action A6 A B 0.97 0 0 0 0 0 0 0 0 0 0.01 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1 0.01 0.38 0.01 0.52

Action A7 A B 0.97 0 0 0 0 0 0 0 0 0 0 0 0.01 0 0 0 0 0 0 0 0 0 0 0 0 0.02 0.01 0.45 0.01 0.53

Table 34: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 100th Iteration Action A8 A B 0.97 0 0 0 0 0 0 0 0 0 0 0 0 0 0.01 0 0 0 0 0 0 0 0 0 0 0.02 0.01 0.45 0.01 0.53

Action A9 A B 0.97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.01 0 0 0 0 0 0 0 0 0.02 0.01 0.45 0.01 0.53

Action A10 A B 0.97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.01 0 0 0 0 0 0 0 0.01 0.52 0.01 0.48

Action A11 A B 0.97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.01 0 0 0 0 0 0.01 0.52 0.01 0.48

Action A12 A B 0.97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.01 0 0 0 0.01 0.52 0.01 0.48

Action A13 A B 0.97 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.01 0 0 0.53 0.01 0.47

Action A14 A B 0.98 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.46 0.01 0 0.01 0.54

Action A15 A B 0.98 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.46 0.01 0.07 0.01 0.47

Table 35: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 150th Iteration Action A1 A 0.986 0 0 0

B 0 0 0 0

Action A2 A 0.973 0.013 0 0

B 0 0 0 0

Action A3 A 0.98 0 0.006 0

B 0 0 0 0

Action A4 A 0.98 0 0 0.006

B 0 0 0 0

Action A5 A 0.98 0 0 0

B 0 0 0 0

Action A6 A 0.98 0 0 0

B 0 0 0 0

A 0.98 0 0 0

Actio n A7 B 0 0 0 0


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0 0 0 0 0 0 0 0 0 0.006 0.006

0 0 0 0 0 0 0 0 0.186 0.146 0.666

0 0 0 0 0 0 0 0 0 0.006 0.006

0 0 0 0 0 0 0 0 0.24 0.1 0.66

0 0 0 0 0 0 0 0 0 0.006 0.006

0 0 0 0 0 0 0 0 0.18 0.153 0.666

Table 35 – Contd., 0 0 0.006 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.126 0 0.006 0.2 0.006 0.006 0.673 0.006

0 0 0 0 0 0 0 0 0.073 0.246 0.68

0 0.006 0 0 0 0 0 0 0 0.006 0.006

0 0 0 0 0 0 0 0 0.066 0.253 0.68

0 0 0.006 0 0 0 0 0 0.006 0.006

0 0 0 0 0 0 0 0 0.013 0.3 0.686

Table 36: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 150th Iteration Action A8 A B 0.98 0 0 0 0 0 0 0 0 0 0 0 0 0 0.006 0 0 0 0 0 0 0 0 0 0 0.013 0.006 0.3 0.006 0.686

Action A9 A B 0.98 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.006 0 0 0 0 0 0 0 0 0.013 0.006 0.3 0.006 0.686

Action A10 A B 0.98 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.006 0 0 0 0 0 0 0 0.006 0.346 0.006 0.653

Action A11 A B 0.98 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.01 0 0 0 0 0 0.01 0.346 0.01 0.653

Action A12 A B 0.98 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.006 0 0 0 0.006 0.346 0.006 0.653

Action A13 A B 0.98 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.006 0 0.006 0.353 0.006 0.646

Action A14 A B 0.99 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.306 0.01 0 0.01 0.693

Action A15 A B 0.986 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.306 0.006 0.046 0.006 0.646

Table 37: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 200th Iteration Action A1 A B 0.99 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.14 0.005 0.11 0.005 0.75

Action A2 A B 0.98 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.18 0.005 0.075 0.005 0.745

Action A3 A B 0.985 0 0 0 0.005 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.135 0.005 0.115 0.005 0.75

Action A4 A B 0.985 0 0 0 0 0 0.005 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.095 0.005 0.15 0.005 0.755

Action A5 A B 0.985 0 0 0 0 0 0 0 0.005 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.055 0.005 0.185 0.005 0.76

Action A6 A B 0.985 0 0 0 0 0 0 0 0 0 0.005 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.05 0.005 0.19 0.005 0.76

Action A7 A B 0.985 0 0 0 0 0 0 0 0 0 0 0 0.005 0 0 0 0 0 0 0 0 0 0 0 0 0.01 0.005 0.225 0.005 0.765

Table 38: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 200th Iteration Action A8 A B 0.985 0 0 0 0 0 0 0 0 0 0 0 0 0 0.005 0 0 0 0 0

Action A9 A B 0.985 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.005 0 0 0

Action A10 A B 0.985 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.005 0

Action A11 A B 0.985 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Action A12 A B 0.985 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Action A13 A B 0.985 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Action A14 A B 0.99 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Action A15 A B 0.99 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


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A Significant Approach on a Special Case of Game Theory

Table 38 – Contd., 0 0 0 0.005 0.005

0 0 0.01 0.225 0.765

0 0 0 0.005 0.005

0 0 0.01 0.225 0.765

0 0 0 0.005 0.005

0 0 0 0.26 0.74

0.005 0 0 0.005 0.005

0 0 0 0.26 0.74

0 0.005 0 0.005 0.005

0 0 0 0.26 0.74

0 0 0.005 0.005 0.005

0 0 0 0.265 0.735

0 0 0 0.005 0.005

0 0 0.23 0 0.77

0 0 0 0.005 0.005

0 0 0.23 0.035 0.735

Table 39: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 250th Iteration Action A1 A B 0.992 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.112 0.004 0.088 0.004 0.8

Action A2 A B 0.984 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.144 0.004 0.06 0.004 0.796

Action A3 A B 0.988 0 0 0 0.004 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.108 0.004 0.092 0.004 0.8

Action A4 A B 0.988 0 0 0 0 0 0.004 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.076 0.004 0.12 0.004 0.804

Action A5 A B 0.988 0 0 0 0 0 0 0 0.004 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.044 0.004 0.148 0.004 0.808

Action A6 A B 0.988 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.04 0.004 0.152 0.004 0.808

Action A7 A B 0.988 0 0 0 0 0 0 0 0 0 0 0 0.004 0 0 0 0 0 0 0 0 0 0 0 0 0.008 0.004 0.18 0.004 0.812

Table 40: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 250th Iteration Action A8 A B 0.988 0 0 0 0 0 0 0 0 0 0 0 0 0 0.004 0 0 0 0 0 0 0 0 0 0 0.008 0.004 0.18 0.004 0.812

Action A9 A B 0.988 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.004 0 0 0 0 0 0 0 0 0.008 0.004 0.18 0.004 0.812

Action A10 A B 0.988 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.004 0 0 0 0 0 0 0 0.004 0.208 0.004 0.792

Action A11 A B 0.988 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.004 0 0 0 0 0 0.004 0.208 0.004 0.792

Action A12 A B 0.988 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.004 0 0 0 0.004 0.208 0.004 0.792

Action A13 A B 0.988 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.004 0 0.004 0.212 0.004 0.788

Action A14 A B 0.992 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.184 0.004 0 0.004 0.816

Action A15 A B 0.992 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.184 0.004 0.028 0.004 0.788

Table 41: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 300th Iteration Action A1 A B 0.993 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.093 0.003 0.073 0.003 0.833

Action A2 A B 0.986 0 0.006 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.12 0.003 0.05 0.003 0.83

Action A3 A B 0.99 0 0 0 0.003 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.09 0.003 0.076 0.003 0.833

Action A4 A B 0.99 0 0 0 0 0 0.003 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.063 0.003 0.1 0.003 0.836

Action A5 A B 0.99 0 0 0 0 0 0 0 0.003 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.036 0.003 0.123 0.003 0.84

Action A6 A B 0.99 0 0 0 0 0 0 0 0 0 0.003 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.033 0.003 0.126 0.003 0.84

Action A7 A B 0.99 0 0 0 0 0 0 0 0 0 0 0 0.003 0 0 0 0 0 0 0 0 0 0 0 0 0.006 0.003 0.15 0.003 0.843


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Table 42: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 300th Iteration Action A8 A B 0.99 0 0 0 0 0 0 0 0 0 0 0 0 0 0.003 0 0 0 0 0 0 0 0 0 0 0.006 0.003 0.15 0.003 0.843

Action A9 A B 0.99 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.003 0 0 0 0 0 0 0 0 0.006 0.003 0.15 0.003 0.843

Action A10 A B 0.99 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.003 0 0 0 0 0 0 0 0.003 0.173 0.003 0.826

Action A11 A B 0.99 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.003 0 0 0 0 0 0.003 0.173 0.003 0.826

Action A12 A B 0.99 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.003 0 0 0 0.003 0.173 0.003 0.826

Action A13 A B 0.99 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.003 0 0.003 0.176 0.003 0.823

Action A14 A B 0.993 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.153 0.003 0 0.003 0.846

Action A15 A B 0.993 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.153 0.003 0.023 0.003 0.823

Table 43: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 350th Iteration Action A1 A B 0.994 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.08 0.002 0.062 0.002 0.857

Action A2 A B 0.988 0 0.005 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.102 0.002 0.042 0.002 0.854

Action A3 A B 0.991 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.077 0.002 0.065 0.002 0.857

Action A4 A B 0.991 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.542 0.002 0.857 0.002 0.86

Action A5 A B 0.991 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.031 0.002 0.105 0.002 0.862

Action A6 A B 0.991 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0285 0.002 0.108 0.002 0.8628

Action A7 A B 0.991 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0.005 0.002 0.128 0.002 0.865

Table 44: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 350th Iteration Action A8 A B 0.991 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0.005 0.002 0.128 0.002 0.865

Action A9 A B 0.991 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0.005 0.002 0.128 0.002 0.865

Action A10 A B 0.991 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0.002 0.148 0.002 0.851

Action A11 A B 0.991 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0.002 0.148 0.002 0.851

Action A12 A B 0.991 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0.002 0.148 0.002 0.851

Action A13 A B 0.991 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0.002 0.151 0.002 0.848

Action A14 A B 0.994 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.131 0.002 0 0.002 0.868

Action A15 A B 0.994 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.131 0.002 0.02 0.002 0.848

Table 45: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 400th Iteration Action A1 A B 0.985 0 0 0 0 0 0 0 0 0

Action A2 A B 0.98 0 0.005 0 0 0 0 0 0 0

Action A3 A B 0.982 0 0 0 0.002 0 0 0 0 0

Action A4 A B 0.982 0 0 0 0 0 0.002 0 0 0

Action A5 A B 0.982 0 0 0 0 0 0 0 0.002 0

Action A6 A B 0.982 0 0 0 0 0 0 0 0 0

Action A7 A B 0.982 0 0 0 0 0 0 0 0 0


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A Significant Approach on a Special Case of Game Theory

0 0 0 0 0 0 0 0 0.002 0.012

0 0 0 0 0 0 0 0.07 0.095 0.835

0 0 0 0 0 0 0 0 0.002 0.012

0 0 0 0 0 0 0 0.09 0.057 0.852

0 0 0 0 0 0 0 0 0.002 0.012

0 0 0 0 0 0 0 0.067 0.095 0.837

Table 45 – Contd., 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.047 0 0.002 0.13 0.002 0.012 0.822 0.012

0 0 0 0 0 0 0 0.027 0.165 0.807

0.002 0 0 0 0 0 0 0 0.002 0.012

0 0 0 0 0 0 0 0.025 0.165 0.81

0 0.002 0 0 0 0 0 0 0.002 0.012

0 0 0 0 0 0 0 0.005 0.202 0.792

Table 46: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 400th Iteration Action A8 A B 0.982 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0.005 0.002 0.2 0.012 0.795

Action A9 A B 0.982 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0.005 0.002 0.2 0.012 0.795

Action A10 A B 0.992 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0.002 0.13 0.002 0.87

Action A11 A B 0.992 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0.002 0.13 0.002 0.87

Action A12 A B 0.992 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0.002 0.13 0.002 0.87

Action A13 A B 0.992 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0.002 0.132 0.002 0.867

Action A14 A B 0.985 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.115 0.002 0.105 0.012 0.78

Action A15 A B 0.995 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.115 0.002 0.017 0.002 0.867

Table 47: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 450th Iteration Action A1 A B 0.986 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.062 0.002 0.14 0.011 0.797

Action A2 A B 0.937 0 0.004 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.08 0.002 0.106 0.011 0.813

Action A3 A B 0.984 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.06 0.002 0.14 0.011 0.8

Action A4 A B 0.984 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.042 0.002 0.171 0.011 0.786

Action A5 A B 0.984 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.024 0.002 0.202 0.011 0.773

Action A6 A B 0.984 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.022 0.002 0.202 0.011 0.775

Action A7 A B 0.984 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0.004 0.002 0.235 0.011 0.72

Table 48: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 450th Iteration Action A8 A B 0.984 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0.004

Action A9 A B 0.984 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0.004

Action A10 A B 0.984 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0

Action A11 A B 0.984 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0

Action A12 A B 0.984 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0

Action A13 A B 0.984 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0

Action A14 A B 0.986 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.102

Action A15 A B 0.986 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.102


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K. V. L. N. Acharyulu, Maddi. N. Murali Krishna, Sateesh Bandikalla & Nagu Vadlana

0.002 0.011

0.233 0.762

0.002 0.011

0.233 0.762

0.002 0.011

0.164 0.835

0.002 0.011

Table 48 – Contd., 0.162 0.002 0.162 0.837 0.011 0.837

0.002 0.011

0.148 0.851

0.002 0.011

0.148 0.748

0.002 0.011

0.062 0.835

Table 49: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 500th Iteration Action A1 A B 0.988 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.056 0.002 0.126 0.01 0.818

Action A2 A B 0.984 0 0.004 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.072 0.002 0.096 0.01 0.832

Action A3 A B 0.986 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.054 0.002 0.126 0.01 0.82

Action A4 A B 0.986 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.038 0.002 0.154 0.01 0.808

Action A5 A B 0.986 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.022 0.002 0.182 0.01 0.796

Action A6 A B 0.986 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.02 0.002 0.182 0.01 0.798

Action A7 A B 0.986 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0.004 0.002 0.212 0.01 0.784

Table 50: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 500th Iteration Action A8 A B 0.986 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0 0 0.004 0.002 0.21 0.01 0.786

Action A9 A B 0.986 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0 0.004 0.002 0.21 0.01 0.786

Action A10 A B 0.986 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0 0 0.002 0.148 0.01 0.852

Action A11 A B 0.986 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0 0 0.002 0.146 0.01 0.854

Action A12 A B 0.986 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0 0 0.002 0.146 0.01 0.854

Action A13 A B 0.986 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002 0 0.002 0.134 0.01 0.866

Action A14 A B 0.988 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.092 0.002 0.13 0.01 0.774

Action A15 A B 0.988 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.092 0.002 0.056 0.01 0.852

UPPER BOUNDS AND LOWER BOUNDS AT ALL COMPUTATIONS At each play of the game the smallest sum element selected by player B divided by the number of place of the game is known as lower bound.Similarly At each play of the game the largest sum element selected by player A divided by the number of place of the game is called as upper bound.The Values of U.Bs and L.Bs in 15x15 game are shown in the tables from Table (51) to Table (52). Table 51: Lower Bounds and Upper Bounds at 50th Iteration – 250th Iteration 50 LB 27.04 27.16 27.38 27.58 27.78 27.82 27.98 28 28.02 28.06 28.08 28.1 28.12

UB 30.66 31.18 31.16 31.38 31.58 31.76 31.1 31.24 31.36 31.46 31.54 31.6 31.64

100 LB UB 27.94 29.77 28 30.03 28.11 30.02 28.21 30.13 28.31 30.23 28.33 30.32 28.43 30.4 28.44 30.47 28.45 30.53 28.43 30.58 28.44 30.62 28.45 30.65 28.45 30.67

150 LB 28.29 28.33 28.406 28.473 28.54 28.553 28.62 28.62 28.63 28.62 28.62 28.63 28.633

200 UB 29.51 29.68 29.68 29.753 29.82 29.88 29.93 29.98 30.02 30.053 30.08 30.1 30.113

LB 28.47 28.5 28.555 28.605 28.655 28.665 28.715 28.72 28.725 28.715 28.72 28.725 28.725

250 UB 29.385 29.515 29.51 29.565 29.615 29.66 29.7 29.735 29.765 29.79 29.81 29.825 29.835

LB 28.576 28.6 28.644 28.684 28.724 28.732 28.772 28.776 28.78 28.772 28.776 28.78 28.78

UB 29.308 29.412 29.408 29.452 29.492 29.528 29.56 29.588 29.612 29.632 29.648 29.66 29.668


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A Significant Approach on a Special Case of Game Theory

27.3 27.18

30.66 30.66

28.07 28.01

29.77 29.77

Table – 51:Contd., 28.38 29.513 28.535 28.34 29.5133 28.505

29.385 29.385

28.628 28.604

29.308 29.308

Table 52: Lower Bounds and Upper Bounds at 300th Iteration – 500th Iteration 300 LB 28.64 28.66 28.703 28.736 28.77 28.77 28.81 28.81 28.81 28.81 28.81 28.816 28.816 28.69 28.67

UB 29.25 29.34 29.34 29.376 29.41 29.44 29.46 29.49 29.51 29.52 29.54 29.55 29.556 29.256 29.2568

350 LB UB 28.69 29.22 28.714 29.294 28.74 29.29 28.774 29.322 28.8 29.35 28.8 29.37 28.83 29.4 28.84 29.42 28.84 29.43 28.83 29.45 28.84 29.46 28.842 29.471 28.842 29.477 28.734 29.22 28.717 29.22

400 LB UB 28.6975 29.115 28.7325 29.18 28.7425 29.1775 28.75 29.205 28.7575 29.23 28.765 29.2525 28.77 29.2725 28.775 29.29 28.7775 29.305 28.8575 29.395 28.86 29.405 28.8625 29.4125 28.8625 29.41 28.665 29.115 28.7525 29.1925

450 LB UB 28.67 29.04 28.704 29.104 28.713 29.102 28.72 29.126 28.72 29.14 28.73 29.16 28.73 29.18 28.74 29.2 28.74 29.21 28.82 29.226 28.82 29.235 28.83 29.24 28.846 29.246 28.644 29.046 28.733 29.046

500 LB 28.706 28.734 28.742 28.748 28.754 28.76 28.764 28.768 28.77 28.842 28.846 28.848 28.862 28.68 28.76

UB 29.042 29.094 29.092 29.114 29.134 29.152 29.168 29.182 29.194 29.204 29.212 29.218 29.222 29.042 29.042

CONCLUSIONS 

In general the pure optimum mixed strategy may not be obtained directly in a non both row-column. dominant game.But this peculiar game proves that there is a chance of evidence to gain the required approximate pure mixed strategy in a game problem.

Component wise influences on a set of actions of Player A are identified in any computation.

Player B has initially not influenced by player A, but finally effected.

Player A is a strong competent than Player B at the beginning of the game but the dominance is reversed at the end of game.

In the considered maximum iteration the lower bound is obtained as   28.26 and upper bound is   29.042 .

The error is gradually decreasing as 3.48,1.76,1.17,0.88,0.704,0.5868,0.503,0.44,0.313,0.282 from the first iteration to last iteration.

The value of the game lies between 28.26 and 29.042 ie 28.26       29.042.

The difference between the bounds is 0.282 only at the last computation.By the continuation of the problem the game tends to become strictly deterministic game.

ACKNOWLEGDEMENTS The authors are grateful to the principal, HOD & the Faculty members of Dept. of M.C.A, Bapatla Engineering College for their encouragement.

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K.V.L.N.Acharyulu and Maddi.N.Murali Krishna,(2013). Some Remarkable Results In Row And Column Both Dominance Game With Rown’s Algorithm, International Applications Research , Vol. 3, No.1, pp.139-150

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Dresher, M., Games of Strategy (1961). Theory and Applications,Prentice-Hall, Inc., Englewood Cliffs,N.J.

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Raiffa, R. D.(1958). Games and Decisions,John Wiley & Sons, Inc., NewYork.

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McKinsey,J.C.C.(1952).Introduction of the Theory of Games,McGraw-Hill Book Company, NewYork.

7. - A significant - full  
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