A SIGNIFICANT APPROACH ON A SPECIAL CASE OF GAME THEORY

K. V. L. N. ACHARYULU1, MADDI. N. MURALI KRISHNA

2, SATEESH BANDIKALLA

3 & NAGU VADLANA

4

1Faculty of Mathematics, Department of Mathematics, Bapatla Engineering College, Bapatla, India

2II M.C.A, Department of M.C.A, Bapatla Engineering College, Bapatla, India

3Faculty of Computer Science, Department of MCA, Bapatla Engineering College, Bapatla, India

4Faculty 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

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.

© TJPRC Pvt. Ltd.,

© TJPRC Pvt. Ltd.,

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

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 30 31 32 33 34 35 36 37 38 39 40 41 42

17 30 3 43 44 45 46 47 48 49 50 51 50 53 54

18 31 43 4 55 56 57 58 59 60 61 62 63 64 65

19 32 44 55 5 66 67 68 69 70 71 72 73 74 75

20 33 45 56 66 6 76 77 78 79 80 81 82 83 84

21 34 46 57 67 76 7 85 86 87 88 89 90 91 92

22 35 47 58 68 77 85 8 93 94 95 96 97 98 99

23 36 48 59 69 78 86 93 9 100 101 102 103 104 105

24 37 49 60 70 79 87 94 100 10 106 107 108 109 110

25 38 50 61 71 80 88 95 101 106 11 111 112 113 114

26 39 51 62 72 81 89 96 102 107 111 12 115 116 117

27 40 52 63 73 82 90 97 103 108 112 115 13 118 119

28 41 53 64 74 83 91 98 104 109 113 116 118 14 120

29 42 54 65 75 84 92 99 105 110 114 117 119 120 15

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 Smallest sum element from step 3

Number of plays of the game thus far Number of plays of the game thus farand

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.

A Significant Approach on a Special Case of Game Theory 57

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

50 100 150 200 250

105 1352 155 2794 205 4244 255 5694 305 7144

851 2004 1651 4096 2451 6196 3251 8296 4051 10396

923 2593 1773 5285 2623 7985 3473 10685 4323 13385

993 3133 1893 6375 2793 9625 3693 12875 4593 16125

1061 3624 2011 7366 2961 11116 3911 14866 4861 18616

1127 4066 2127 8258 3127 12458 4127 16658 5127 20858

1191 4459 2241 9051 3291 13651 4341 18251 5391 22851

1253 4803 2353 9745 3453 14695 4553 19645 5653 24595

1313 5098 2463 10340 3613 15590 4763 20840 5913 26090

1371 5344 2571 10836 3771 16336 4971 21836 6171 27336

1427 5541 2677 11233 3927 16933 5177 22633 6427 28333

1481 5689 2781 11531 4081 17381 5381 23231 6681 29081

1533 3292 2883 8818 4233 14768 5583 20718 6933 26668

1478 3528 2878 9520 4278 15520 5678 21520 7078 27520

1527 5570 2977 6736 4427 7486 5877 8236 7327 8986

Table 2: By the Choice of Action A1: Player A Vs Player B from 300 - 500 Iterations

300th Iteration 350th Iteration 400th Iteration 450th Iteration 500th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

355 8594 405 10044 567 11479 617 12903 667 14353

4851 12496 5651 14596 6555 16681 7355 18755 8155 20855

5173 16085 6023 18785 7021 21470 7871 24144 8721 26844

5493 19375 6393 22625 7481 25860 8381 29084 9281 32334

5811 22366 6761 26116 7935 29851 8885 33575 9835 37325

6127 25058 7127 29258 8383 33443 9383 37617 10383 41817

6441 27451 7491 32051 8825 36636 9875 41210 10925 45810

6753 29545 7853 34495 9261 39430 10361 44354 11461 49304

7063 31340 8213 36590 9691 41825 10841 47049 11991 52299

7371 32836 8571 38336 10115 43821 11315 49295 12515 54795

7677 34033 8927 39733 10533 45418 11783 51092 13033 56792

7981 34931 9281 40781 10945 46616 12245 52440 13545 58290

8283 32618 9633 38568 11351 44503 12701 50427 14051 56377

8478 33520 9878 39520 11646 43930 13046 47174 14446 53174

8777 9736 10227 10486 11621 12811 13071 16291 14521 17041

Table 3: By the Choice of Action A2: Player A Vs Player B from 50 - 250 Iterations

50th Iteration 100th Iteration 150th Iteration 200th Iteration 250th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

135 1358 185 2800 235 4250 285 5700 335 7150

823 1981 1623 4073 2423 6173 3223 8273 4023 10373

949 2597 1799 5289 2649 7989 3499 10689 4349 13389

1019 3137 1919 6379 2819 9629 3719 12879 4619 16129

1087 3628 2037 7370 2987 11120 3937 14870 4887 18620

1153 4070 2153 8262 3153 12462 4153 16662 5153 20862

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

Table 3 - Contd.,

1217 4463 2267 9055 3317 13655 4367 18255 5417 22855

1279 4807 2379 9749 3479 14699 4579 19649 5679 24599

1339 5102 2489 10344 3639 15594 4789 20844 5939 26094

1397 5348 2597 10840 3797 16340 4997 21840 6197 27340

1453 5545 2703 11237 3953 16937 5203 22637 6453 28337

1507 5693 2807 11535 4107 17385 5407 23235 6707 29085

1559 2464 2909 7990 4259 13940 5609 19890 6959 25840

1504 4267 2904 10259 4304 16259 5704 22259 7104 28259

1553 5680 3003 6846 4453 7596 5903 8346 7353 9096

Table 4: By the Choice of Action A2: Player A Vs Player B from 300 - 500 Iterations

300th Iteration 350th Iteration 400th Iteration 450th Iteration 500th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

385 8600 435 10050 597 11493 647 12917 697 14367

4823 12473 5623 14573 6527 16666 7327 18740 8127 20840

5199 16089 6049 18789 7047 21482 7897 24156 8747 26856

5519 19379 6419 22629 7507 25872 8407 29096 9307 32346

5837 22370 6787 26120 7961 29863 8911 33587 9861 37337

6153 25062 7153 29262 8409 33455 9409 37629 10409 41829

6467 27455 7517 32055 8851 36648 9901 41222 10951 45822

6779 29549 7879 34499 9287 39442 10387 44366 11487 49316

7089 31344 8239 36594 9717 41837 10867 47061 12017 52311

7397 32840 8597 38340 10141 43833 11341 49307 12541 54807

7703 34037 8953 39737 10559 45430 11809 51104 13059 56804

8007 34935 9307 40785 10971 46628 12271 52452 13571 58302

8309 31790 9659 37740 11377 43683 12727 49607 14077 55557

8504 34259 9904 40259 11672 45517 13072 48761 14472 54761

8803 9846 10253 10596 11647 12081 13097 15561 14547 16311

Table 5: By the Choice of Action A3: Player A Vs Player B from 50 - 250 Iterations

50th Iteration 100th Iteration 150th Iteration 200th Iteration 250th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

121 1369 171 2811 221 4261 271 5711 321 7161

865 2019 1665 4111 2465 6211 3265 8311 4065 10411

909 2580 1759 5272 2609 7972 3459 10672 4309 13372

1018 3159 1918 6401 2818 9651 3718 12901 4618 16151

1086 3650 2036 7392 2986 11142 3936 14892 4886 18642

1152 4092 2152 8284 3152 12484 4152 16684 5152 20884

1216 4485 2266 9077 3316 13677 4366 18277 5416 22877

1278 4829 2378 9771 3478 14721 4578 19671 5678 24621

1338 5124 2488 10366 3638 15616 4788 20866 5938 26116

1396 5370 2596 10862 3796 16362 4996 21862 6196 27362

1452 5567 2702 11259 3952 16959 5202 22659 6452 28359

1506 5715 2806 11557 4106 17407 5406 23257 6706 29107

1558 3422 2908 8948 4258 14898 5608 20848 6958 26798

1503 3449 2903 9441 4303 15441 5703 21441 7103 27441

1552 5596 3002 6762 4452 7512 5902 8262 7352 9012

A Significant Approach on a Special Case of Game Theory 59

Table 6: By the Choice of Action A3: Player A Vs Player B from 300 - 500 Iterations

300th Iteration 350th Iteration 400th Iteration 450th Iteration 500th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

371 8611 421 10061 583 11497 633 12921 683 14371

4865 12511 5665 14611 6569 16697 7369 18771 8169 20871

5159 16072 6009 18772 7007 21458 7857 24132 8707 26832

5518 19401 6418 22651 7506 25887 8406 29111 9306 32361

5836 22392 6786 26142 7960 29878 8910 33602 9860 37352

6152 25084 7152 29284 8408 33470 9408 37644 10408 41844

6466 27477 7516 32077 8850 36663 9900 41237 10950 45837

6778 29571 7878 34521 9286 39457 10386 44381 11486 49331

7088 31366 8238 36616 9716 41852 10866 47076 12016 52326

7396 32862 8596 38362 10140 43848 11340 49322 12540 54822

7702 34059 8952 39759 10558 45445 11808 51119 13058 56819

8006 34957 9306 40807 10970 46643 12270 52467 13570 58317

8308 32748 9658 38698 11376 44634 12726 50558 14076 56508

8503 33441 9903 39441 11671 43957 13071 47201 14471 53201

8802 9762 10252 10512 11646 12732 13096 16212 14546 16962

Table 7: By the Choice of Action A4: Player A Vs Player B from 50 - 250 Iterations

50th Iteration 100th Iteration 150th Iteration 200th Iteration 250th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

122 1379 172 2821 222 4271 272 5721 322 7171

866 2029 1666 4121 2466 6221 3266 8321 4066 10421

949 2629 1799 5321 2649 8021 3499 10721 4349 13421

979 3129 1879 6371 2779 9621 3679 12871 4579 16121

1097 3670 2047 7412 2997 11162 3947 14912 4897 18662

1163 4112 2163 8304 3163 12504 4163 16704 5163 20904

1227 4505 2277 9097 3327 13697 4377 18297 5427 22897

1289 4849 2389 9791 3489 14741 4589 19691 5689 24641

1349 5144 2499 10386 3649 15636 4799 20886 5949 26136

1407 5390 2607 10882 3807 16382 5007 21882 6207 27382

1463 5587 2713 11279 3963 16979 5213 22679 6463 28379

1517 5735 2817 11577 4117 17427 5417 23277 6717 29127

1569 4274 2919 9800 4269 15750 5619 21700 6969 27650

1514 2734 2914 8726 4314 14726 5714 20726 7114 26726

1563 5510 3013 6676 4463 7426 5913 8176 7363 8926

Table 8: By the Choice of Action A4: Player A Vs Player B from 300 - 500 Iterations

300th Iteration 350th Iteration 400th Iteration 450th Iteration 500th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

372 8621 422 10071 584 11500 634 12924 684 14374

4866 12521 5666 14621 6570 16700 7370 18774 8170 20874

5199 16121 6049 18821 7047 21500 7897 24174 8747 26874

5479 19371 6379 22621 7467 25850 8367 29074 9267 32324

5847 22412 6797 26162 7971 29891 8921 33615 9871 37365

6163 25104 7163 29304 8419 33483 9419 37657 10419 41857

6477 27497 7527 32097 8861 36676 9911 41250 10961 45850

6789 29591 7889 34541 9297 39470 10397 44394 11497 49344

7099 31386 8249 36636 9727 41865 10877 47089 12027 52339

7407 32882 8607 38382 10151 43861 11351 49335 12551 54835

7713 34079 8963 39779 10569 45458 11819 51132 13069 56832

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

Table 8 - Contd.,

8017 34977 9317 40827 10981 46656 12281 52480 13581 58330

8319 33600 9669 39550 11387 45479 12737 51403 14087 57353

8514 32726 9914 38726 11682 42500 13082 45744 14482 51744

8813 9676 10263 10426 11657 13381 13107 16861 14557 17611

Table 9: By the Choice of Action A5: Player A Vs Player B from 50 - 250 Iterations

50th Iteration 100th Iteration 150th Iteration 200th Iteration 250th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

123 1389 173 2831 223 4281 273 5731 323 7181

867 2039 1667 4131 2467 6231 3267 8331 4067 10431

950 2639 1800 5331 2650 8031 3500 10731 4350 13431

1030 3189 1930 6431 2830 9681 3730 12931 4630 16181

1047 3629 1997 7371 2947 11121 3897 14871 4847 18621

1173 4131 2173 8323 3173 12523 4173 16723 5173 20923

1237 4524 2287 9116 3337 13716 4387 18316 5437 22916

1299 4868 2399 9810 3499 14760 4599 19710 5699 24660

1359 5163 2509 10405 3659 15655 4809 20905 5959 26155

1417 5409 2617 10901 3817 16401 5017 21901 6217 27401

1473 5606 2723 11298 3973 16998 5223 22698 6473 28398

1527 5754 2827 11596 4127 17446 5427 23296 6727 29146

1579 5125 2929 10651 4279 16601 5629 22551 6979 28501

1524 2018 2924 8010 4324 14010 5724 20010 7124 26010

1573 5423 3023 6589 4473 7339 5923 8089 7373 8839

Table 10: By the Choice of Action A5: Player A Vs Player B from 300 - 500 Iterations

300th Iteration 350th Iteration 400th Iteration 450th Iteration 500th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

373 8631 423 10081 585 11503 635 12927 685 14377

4867 12531 5667 14631 6571 16703 7371 18777 8171 20877

5200 16131 6050 18831 7048 21503 7898 24177 8748 26877

5530 19431 6430 22681 7518 25903 8418 29127 9318 32377

5797 22371 6747 26121 7921 29843 8871 33567 9821 37317

6173 25123 7173 29323 8429 33495 9429 37669 10429 41869

6487 27516 7537 32116 8871 36688 9921 41262 10971 45862

6799 29610 7899 34560 9307 39482 10407 44406 11507 49356

7109 31405 8259 36655 9737 41877 10887 47101 12037 52351

7417 32901 8617 38401 10161 43873 11361 49347 12561 54847

7723 34098 8973 39798 10579 45470 11829 51144 13079 56844

8027 34996 9327 40846 10991 46668 12291 52492 13591 58342

8329 34451 9679 40401 11397 46323 12747 52247 14097 58197

8524 32010 9924 38010 11692 41042 13092 44286 14492 50286

8823 9589 10273 10339 11667 14029 13117 17509 14567 18259

Table 11: By the Choice of Action A6: Player A Vs Player B from 50 - 250 Iterations

50th Iteration 100th Iteration 150th Iteration 200th Iteration 250th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

124 1391 174 2833 224 4283 274 5733 324 7183

868 2041 1668 4133 2468 6233 3268 8333 4068 10433

951 2641 1801 5333 2651 8033 3501 10733 4351 13433

1031 3191 1931 6433 2831 9683 3731 12933 4631 16183

1108 3691 2058 7433 3008 11183 3958 14933 4908 18683

A Significant Approach on a Special Case of Game Theory 61

Table 11 - Contd.,

1113 4072 2113 8264 3113 12464 4113 16664 5113 20864

1246 4534 2296 9126 3346 13726 4396 18326 5446 22926

1308 4878 2408 9820 3508 14770 4608 19720 5708 24670

1368 5173 2518 10415 3668 15665 4818 20915 5968 26165

1426 5419 2626 10911 3826 16411 5026 21911 6226 27411

1482 5616 2732 11308 3982 17008 5232 22708 6482 28408

1536 5764 2836 11606 4136 17456 5436 23306 6736 29156

1588 5239 2938 10765 4288 16715 5638 22665 6988 28615

1533 1923 2933 7915 4333 13915 5733 19915 7133 25915

1582 5433 3032 6599 4482 7349 5932 8099 7382 8849

Table 12: By the Choice of Action A6: Player A Vs Player B from 300 - 500 Iterations

300th Iteration 350th Iteration 400th Iteration 450th Iteration 500th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

374 8633 424 10083 586 11506 636 12930 686 14380

4868 12533 5668 14633 6572 16706 7372 18780 8172 20880

5201 16133 6051 18833 7049 21506 7899 24180 8749 26880

5531 19433 6431 22683 7519 25906 8419 29130 9319 32380

5858 22433 6808 26183 7982 29906 8932 33630 9882 37380

6113 25064 7113 29264 8369 33437 9369 37611 10369 41811

6496 27526 7546 32126 8880 36699 9930 41273 10980 45873

6808 29620 7908 34570 9316 39493 10416 44417 11516 49367

7118 31415 8268 36665 9746 41888 10896 47112 12046 52362

7426 32911 8626 38411 10170 43884 11370 49358 12570 54858

7732 34108 8982 39808 10588 45481 11838 51155 13088 56855

8036 35006 9336 40856 11000 46679 12300 52503 13600 58353

8338 34565 9688 40515 11406 46438 12756 52362 14106 58312

8533 31915 9933 37915 11701 41053 13101 44297 14501 50297

8832 9599 10282 10349 11676 13934 13126 17414 14576 18164

Table 13: By the Choice of Action A7: Player A Vs Player B from 50 - 250 Iterations

50th Iteration 100th Iteration 150th Iteration 200th Iteration 250th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

98 1399 175 2843 225 4293 275 5743 325 7193

844 2049 1669 4143 2469 6243 3269 8343 4069 10443

916 2649 1802 5343 2652 8043 3502 10743 4352 13443

986 3199 1932 6443 2832 9693 3732 12943 4632 16193

1054 3699 2059 7443 3009 11193 3959 14943 4909 18693

1120 4149 2183 8343 3183 12543 4183 16743 5183 20943

1107 4472 2227 9066 3277 13666 4327 18266 5377 22866

1240 4893 2416 9837 3516 14787 4616 19737 5716 24687

1295 5188 2526 10432 3676 15682 4826 20932 5976 26182

1349 5434 2634 10928 3834 16428 5034 21928 6234 27428

1402 5631 2740 11325 3990 17025 5240 22725 6490 28425

1454 5779 2844 11623 4144 17473 5444 23323 6744 29173

1505 5878 2946 11614 4296 17564 5646 23514 6996 29464

1555 1413 2941 7197 4341 13197 5741 19197 7141 25197

1499 5342 3040 6510 4490 7260 5940 8010 7390 8760

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

Table 14: By the Choice of Action A7: Player A Vs Player B from 300 - 500 Iterations

300th Iteration 350th Iteration 400th Iteration 450th Iteration 500th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

375 8643 425 10093 587 11508 637 12932 687 14382

4869 12543 5669 14643 6573 16708 7373 18782 8173 20882

5202 16143 6052 18843 7050 21508 7900 24182 8750 26882

5532 19443 6432 22693 7520 25908 8420 29132 9320 32382

5859 22443 6809 26193 7983 29908 8933 33632 9883 37382

6183 25143 7183 29343 8439 33508 9439 37682 10439 41882

6427 27466 7477 32066 8811 36631 9861 41205 10911 45805

6816 29637 7916 34587 9324 39502 10424 44426 11524 49376

7126 31432 8276 36682 9754 41897 10904 47121 12054 52371

7434 32928 8634 38428 10178 43893 11378 49367 12578 54867

7740 34125 8990 39825 10596 45490 11846 51164 13096 56864

8044 35023 9344 40873 11008 46688 12308 52512 13608 58362

8346 35414 9696 41364 11414 47279 12764 53203 14114 59153

8541 31197 9941 37197 11709 39487 13109 42731 14509 48731

8840 9510 10290 10260 11684 14685 13134 18165 14584 18915

Table 15: By the Choice of Action A8: Player A Vs Player B from 50 - 250 Iterations

50th Iteration 100th Iteration 150th Iteration 200th Iteration 250th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

99 1400 176 2844 226 4294 276 5744 326 7194

845 2050 1670 4144 2470 6244 3270 8344 4070 10444

917 2650 1803 5344 2653 8044 3503 10744 4353 13444

987 3200 1933 6444 2833 9694 3733 12944 4633 16194

1055 3700 2060 7444 3010 11194 3960 14944 4910 18694

1121 4150 2184 8344 3184 12544 4184 16744 5184 20944

1185 4550 2305 9144 3355 13744 4405 18344 5455 22944

1163 4816 2339 9760 3439 14710 4539 19660 5639 24610

1302 5195 2533 10439 3683 15689 4833 20939 5983 26189

1356 5441 2641 10935 3841 16435 5041 21935 6241 27435

1409 5638 2747 11332 3997 17032 5247 22732 6497 28432

1461 5786 2851 11630 4151 17480 5451 23330 6751 29180

1512 5885 2953 11621 4303 17571 5653 23521 7003 29471

1562 1420 2948 7204 4348 13204 5748 19204 7148 25204

1506 5349 3047 6517 4497 7267 5947 8017 7397 8767

Table 16: By the Choice of Action A8: Player A Vs Player B from 300 - 500 Iterations

300th Iteration 350th Iteration 400th Iteration 450th Iteration 500th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

376 8644 426 10094 588 11510 638 12934 688 14384

4870 12544 5670 14644 6574 16710 7374 18784 8174 20884

5203 16144 6053 18844 7051 21510 7901 24184 8751 26884

5533 19444 6433 22694 7521 25910 8421 29134 9321 32384

5860 22444 6810 26194 7984 29910 8934 33634 9884 37384

6184 25144 7184 29344 8440 33510 9440 37684 10440 41884

6505 27544 7555 32144 8889 36710 9939 41284 10989 45884

6739 29560 7839 34510 9247 39426 10347 44350 11447 49300

7133 31439 8283 36689 9761 41905 10911 47129 12061 52379

7441 32935 8641 38435 10185 43901 11385 49375 12585 54875

7747 34132 8997 39832 10603 45498 11853 51172 13103 56872

A Significant Approach on a Special Case of Game Theory 63

Table 16 - Contd.,

8051 35030 9351 40880 11015 46696 12315 52520 13615 58370

8353 35421 9703 41371 11421 47287 12771 53211 14121 59161

8548 31204 9948 37204 11716 39600 13116 42844 14516 48844

8847 9517 10297 10267 11691 14587 13141 18067 14591 18817

Table 17: By the Choice of Action A9: Player A Vs Player B from 50 - 250 Iterations

50th Iteration 100th Iteration 150th Iteration 200th Iteration 250th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

100 1401 177 2845 227 4295 277 5745 327 7195

846 2051 1671 4145 2471 6245 3271 8345 4071 10445

918 2651 1804 5345 2654 8045 3504 10745 4354 13445

988 3201 1934 6445 2834 9695 3734 12945 4634 16195

1056 3701 2061 7445 3011 11195 3961 14945 4911 18695

1122 4151 2185 8345 3185 12545 4185 16745 5185 20945

1186 4551 2306 9145 3356 13745 4406 18345 5456 22945

1248 4901 2424 9845 3524 14795 4624 19745 5724 24695

1218 5111 2449 10355 3599 15605 4749 20855 5899 26105

1362 5447 2647 10941 3847 16441 5047 21941 6247 27441

1415 5644 2753 11338 4003 17038 5253 22738 6503 28438

1467 5792 2857 11636 4157 17486 5457 23336 6757 29186

1518 5891 2959 11627 4309 17577 5659 23527 7009 29477

1568 1426 2954 7210 4354 13210 5754 19210 7154 25210

1512 5355 3053 6523 4503 7273 5953 8023 7403 8773

Table 18: By the Choice of Action A9: Player A Vs Player B from 300 - 500 Iterations

300th Iteration 350th Iteration 400th Iteration 450th Iteration 500th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

377 8645 427 10095 589 11511 639 12935 689 14385

4871 12545 5671 14645 6575 16711 7375 18785 8175 20885

5204 16145 6054 18845 7052 21511 7902 24185 8752 26885

5534 19445 6434 22695 7522 25911 8422 29135 9322 32385

5861 22445 6811 26195 7985 29911 8935 33635 9885 37385

6185 25145 7185 29345 8441 33511 9441 37685 10441 41885

6506 27545 7556 32145 8890 36711 9940 41285 10990 45885

6824 29645 7924 34595 9332 39511 10432 44435 11532 49385

7049 31355 8199 36605 9677 41821 10827 47045 11977 52295

7447 32941 8647 38441 10191 43907 11391 49381 12591 54881

7753 34138 9003 39838 10609 45504 11859 51178 13109 56878

8057 35036 9357 40886 11021 46702 12321 52526 13621 58376

8359 35427 9709 41377 11427 47293 12777 53217 14127 59167

8554 31210 9954 37210 11722 39606 13122 42850 14522 48850

8853 9523 10303 10273 11697 14593 13147 18073 14597 18823

Table 19: By the Choice of Action A10: Player A Vs Player B from 50 - 250 Iterations

50th Iteration 100th Iteration 150th Iteration 200th Iteration 250th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

101 1403 178 2843 228 4293 278 5743 328 7193

847 2053 1672 4143 2472 6243 3272 8343 4072 10443

919 2653 1805 5343 2655 8043 3505 10743 4355 13443

989 3203 1935 6443 2835 9693 3735 12943 4635 16193

1057 3703 2062 7443 3012 11193 3962 14943 4912 18693

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

Table 19 – Contd.,

1123 4153 2186 8343 3186 12543 4186 16743 5186 20943

1187 4553 2307 9143 3357 13743 4407 18343 5457 22943

1249 4903 2425 9843 3525 14793 4625 19743 5725 24693

1309 5203 2540 10443 3690 15693 4840 20943 5990 26193

1272 5358 2557 10848 3757 16348 4957 21848 6157 27348

1420 5650 2758 11340 4008 17040 5258 22740 6508 28440

1472 5798 2862 11638 4162 17488 5462 23338 6762 29188

1523 5897 2964 11837 4314 17787 5664 23737 7014 29687

1573 1537 2959 6477 4359 12477 5759 18477 7159 24477

1517 5255 3058 7055 4508 7805 5958 8555 7408 9305

Table 20: By the Choice of Action A10: Player A Vs Player B from 300 - 500 Iterations

300th Iteration 350th Iteration 400th Iteration 450th Iteration 500th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

378 8643 428 10093 478 11543 640 12971 690 14421

4872 12543 5672 14643 6472 16743 7376 18821 8176 20921

5205 16143 6055 18843 6905 21543 7903 24221 8753 26921

5535 19443 6435 22693 7335 25943 8423 29171 9323 32421

5862 22443 6812 26193 7762 29943 8936 33671 9886 37421

6186 25143 7186 29343 8186 33543 9442 37721 10442 41921

6507 27543 7557 32143 8607 36743 9941 41321 10991 45921

6825 29643 7925 34593 9025 39543 10433 44471 11533 49421

7140 31443 8290 36693 9440 41943 10918 47171 12068 52421

7357 32848 8557 38348 9757 43848 11301 49326 12501 54826

7758 34140 9008 39840 10258 45540 11864 51218 13114 56918

8062 35038 9362 40888 10662 46738 12326 52566 13626 58416

8364 35637 9714 41587 11064 47537 12782 53465 14132 59415

8559 30477 9959 36477 11359 42477 13127 46145 14527 52145

8858 10055 10308 10805 11758 11555 13152 14615 14602 15365

Table 21: By the Choice of Action A11: Player A Vs Player B from 50 - 250 Iterations

50th Iteration 100th Iteration 150th Iteration 200th Iteration 250th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

102 1404 179 2844 229 4294 279 5744 329 7194

848 2054 1673 4144 2473 6244 3273 8344 4073 10444

920 2654 1806 5344 2656 8044 3506 10744 4356 13444

990 3204 1936 6444 2836 9694 3736 12944 4636 16194

1058 3704 2063 7444 3013 11194 3963 14944 4913 18694

1124 4154 2187 8344 3187 12544 4187 16744 5187 20944

1188 4554 2308 9144 3358 13744 4408 18344 5458 22944

1250 4904 2426 9844 3526 14794 4626 19744 5726 24694

1310 5204 2541 10444 3691 15694 4841 20944 5991 26194

1368 5454 2653 10944 3853 16444 5053 21944 6253 27444

1325 5555 2663 11245 3913 16945 5163 22645 6413 28345

1476 5802 2866 11642 4166 17492 5466 23342 6766 29192

1527 5901 2968 11841 4318 17791 5668 23741 7018 29691

1577 1541 2963 6481 4363 12481 5763 18481 7163 24481

1521 5259 3062 7059 4512 7809 5962 8559 7412 9309

A Significant Approach on a Special Case of Game Theory 65

Table 22: By the Choice of Action A11: Player A Vs Player B from 300 - 500 Iterations

300th Iteration 350th Iteration 400th Iteration 450th Iteration 500th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

379 8644 429 10094 479 11544 641 12973 691 14423

4873 12544 5673 14644 6473 16744 7377 18823 8177 20923

5206 16144 6056 18844 6906 21544 7904 24223 8754 26923

5536 19444 6436 22694 7336 25944 8424 29173 9324 32423

5863 22444 6813 26194 7763 29944 8937 33673 9887 37423

6187 25144 7187 29344 8187 33544 9443 37723 10443 41923

6508 27544 7558 32144 8608 36744 9942 41323 10992 45923

6826 29644 7926 34594 9026 39544 10434 44473 11534 49423

7141 31444 8291 36694 9441 41944 10919 47173 12069 52423

7453 32944 8653 38444 9853 43944 11397 49423 12597 54923

7663 34045 8913 39745 10163 45445 11769 51124 13019 56824

8066 35042 9366 40892 10666 46742 12330 52571 13630 58421

8368 35641 9718 41591 11068 47541 12786 53470 14136 59420

8563 30481 9963 36481 11363 42481 13131 46255 14531 52255

8862 10059 10312 10809 11762 11559 13156 14514 14606 15264

Table 23: By the Choice of Action A12: Player A Vs Player B from 50 - 250 Iterations

50th Iteration 100th Iteration 150th Iteration 200th Iteration 250th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

103 1405 180 2845 230 4295 280 5745 330 7195

849 2055 1674 4145 2474 6245 3274 8345 4074 10445

921 2655 1807 5345 2657 8045 3507 10745 4357 13445

991 3205 1937 6445 2837 9695 3737 12945 4637 16195

1059 3705 2064 7445 3014 11195 3964 14945 4914 18695

1125 4155 2188 8345 3188 12545 4188 16745 5188 20945

1189 4555 2309 9145 3359 13745 4409 18345 5459 22945

1251 4905 2427 9845 3527 14795 4627 19745 5727 24695

1311 5205 2542 10445 3692 15695 4842 20945 5992 26195

1369 5455 2654 10945 3854 16445 5054 21945 6254 27445

1425 5655 2763 11345 4013 17045 5263 22745 6513 28445

1377 5703 2767 11543 4067 17393 5367 23243 6667 29093

1530 5904 2971 11844 4321 17794 5671 23744 7021 29694

1580 1544 2966 6484 4366 12484 5766 18484 7166 24484

1524 5262 3065 7062 4515 7812 5965 8562 7415 9312

Table 24: By the Choice of Action A12: Player A Vs Player B from 300 - 500 Iterations

300th Iteration 350th Iteration 400th Iteration 450th Iteration 500th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

380 8645 430 10095 480 11545 642 12974 692 14424

4874 12545 5674 14645 6474 16745 7378 18824 8178 20924

5207 16145 6057 18845 6907 21545 7905 24224 8755 26924

5537 19445 6437 22695 7337 25945 8425 29174 9325 32424

5864 22445 6814 26195 7764 29945 8938 33674 9888 37424

6188 25145 7188 29345 8188 33545 9444 37724 10444 41924

6509 27545 7559 32145 8609 36745 9943 41324 10993 45924

6827 29645 7927 34595 9027 39545 10435 44474 11535 49424

7142 31445 8292 36695 9442 41945 10920 47174 12070 52424

7454 32945 8654 38445 9854 43945 11398 49424 12598 54924

7763 34145 9013 39845 10263 45545 11869 51224 13119 56924

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

Table 24 - Contd.,

7967 34943 9267 40793 10567 46643 12231 52472 13531 58322

8371 35644 9721 41594 11071 47544 12789 53473 14139 59423

8566 30484 9966 36484 11366 42484 13134 46258 14534 52258

8865 10062 10315 10812 11765 11562 13159 14517 14609 15267

Table 25: By the Choice of Action A13: Player A Vs Player B from 50 - 250 Iterations

50th Iteration 100th Iteration 150th Iteration 200th Iteration 250th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

104 1406 181 2845 231 4295 281 5745 331 7195

850 2056 1675 4145 2475 6245 3275 8345 4075 10445

922 2656 1808 5345 2658 8045 3508 10745 4358 13445

992 3206 1938 6445 2838 9695 3738 12945 4638 16195

1060 3706 2065 7445 3015 11195 3965 14945 4915 18695

1126 4156 2189 8345 3189 12545 4189 16745 5189 20945

1190 4556 2310 9145 3360 13745 4410 18345 5460 22945

1252 4906 2428 9845 3528 14795 4628 19745 5728 24695

1312 5206 2543 10445 3693 15695 4843 20945 5993 26195

1370 5456 2655 10945 3855 16445 5055 21945 6255 27445

1426 5656 2764 11345 4014 17045 5264 22745 6514 28445

1480 5806 2870 11645 4170 17495 5470 23345 6770 29195

1428 5802 2869 11741 4219 17691 5569 23641 6919 29591

1582 1546 2968 6380 4368 12380 5768 18380 7168 24380

1526 5264 3067 7169 4517 7919 5967 8669 7417 9419

Table 26: By the Choice of Action A13: Player A Vs Player B from 300 - 500 Iterations

300th Iteration 350th Iteration 400th Iteration 450th Iteration 500th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

381 8645 431 10095 481 11545 643 12981 693 14431

4875 12545 5675 14645 6475 16745 7379 18831 8179 20931

5208 16145 6058 18845 6908 21545 7906 24231 8756 26931

5538 19445 6438 22695 7338 25945 8426 29181 9326 32431

5865 22445 6815 26195 7765 29945 8939 33681 9889 37431

6189 25145 7189 29345 8189 33545 9445 37731 10445 41931

6510 27545 7560 32145 8610 36745 9944 41331 10994 45931

6828 29645 7928 34595 9028 39545 10436 44481 11536 49431

7143 31445 8293 36695 9443 41945 10921 47181 12071 52431

7455 32945 8655 38445 9855 43945 11399 49431 12599 54931

7764 34145 9014 39845 10264 45545 11870 51231 13120 56931

8070 35045 9370 40895 10670 46745 12334 52581 13634 58431

8269 35541 9619 41491 10969 47441 12687 53377 14037 59327

8568 30380 9968 36380 11368 42380 13136 46896 14536 52896

8867 10169 10317 10919 11767 11669 13161 13889 14611 14639

Table 27: By the Choice of Action A14: Player A Vs Player B from 50 - 250 Iterations

50th Iteration 100th Iteration 150th Iteration 200th Iteration 250th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

105 1365 155 2807 205 4257 255 5707 305 7157

851 2015 1651 4107 2451 6207 3251 8307 4051 10407

923 2615 1773 5307 2623 8007 3473 10707 4323 13407

993 3165 1893 6407 2793 9657 3693 12907 4593 16157

1061 3665 2011 7407 2961 11157 3911 14907 4861 18657

A Significant Approach on a Special Case of Game Theory 67

Table 27 - Contd.,

1127 4115 2127 8307 3127 12507 4127 16707 5127 20907

1191 4515 2241 9107 3291 13707 4341 18307 5391 22907

1253 4865 2353 9807 3453 14757 4553 19707 5653 24657

1313 5165 2463 10407 3613 15657 4763 20907 5913 26157

1371 5415 2571 10907 3771 16407 4971 21907 6171 27407

1427 5615 2677 11307 3927 17007 5177 22707 6427 28407

1481 5765 2781 11607 4081 17457 5381 23307 6681 29157

1533 1497 2883 7023 4233 12973 5583 18923 6933 24873

1478 5810 2878 11802 4278 17802 5678 23802 7078 29802

1527 5223 2977 6389 4427 7139 5877 7889 7327 8639

Table 28: By the Choice of Action A14: Player A Vs Player B from 300- 500 Iterations

300th Iteration 350th Iteration 400th Iteration 450th Iteration 500th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

355 8607 405 10057 567 11466 617 12890 667 14340

4851 12507 5651 14607 6555 16666 7355 18740 8155 20840

5173 16107 6023 18807 7021 21466 7871 24140 8721 26840

5493 19407 6393 22657 7481 25866 8381 29090 9281 32340

5811 22407 6761 26157 7935 29866 8885 33590 9835 37340

6127 25107 7127 29307 8383 33466 9383 37640 10383 41840

6441 27507 7491 32107 8825 36666 9875 41240 10925 45840

6753 29607 7853 34557 9261 39466 10361 44390 11461 49340

7063 31407 8213 36657 9691 41866 10841 47090 11991 52340

7371 32907 8571 38407 10115 43866 11315 49340 12515 54840

7677 34107 8927 39807 10533 45466 11783 51140 13033 56840

7981 35007 9281 40857 10945 46666 12245 52490 13545 58340

8283 30823 9633 36773 11351 42682 12701 48606 14051 54556

8478 35802 9878 41802 11646 43456 13046 46700 14446 52700

8777 9389 10227 10139 11621 15194 13071 18674 14521 19424

Table 29: By the Choice of Action A15: Player A Vs Player B from 50 - 250 Iterations

50th Iteration 100th Iteration 150th Iteration 200th Iteration 250th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

105 1359 155 2801 205 4251 255 5701 305 7151

851 2009 1651 4101 2451 6201 3251 8301 4051 10401

923 2609 1773 5301 2623 8001 3473 10701 4323 13401

993 3159 1893 6401 2793 9651 3693 12901 4593 16151

1061 3659 2011 7401 2961 11151 3911 14901 4861 18651

1127 4109 2127 8301 3127 12501 4127 16701 5127 20901

1191 4509 2241 9101 3291 13701 4341 18301 5391 22901

1253 4859 2353 9801 3453 14751 4553 19701 5653 24651

1313 5159 2463 10401 3613 15651 4763 20901 5913 26151

1371 5409 2571 10901 3771 16401 4971 21901 6171 27401

1427 5609 2677 11301 3927 17001 5177 22701 6427 28401

1481 5759 2781 11601 4081 17451 5381 23301 6681 29151

1533 1491 2883 7017 4233 12967 5583 18917 6933 24867

1478 5174 2878 11166 4278 17166 5678 23166 7078 29166

1527 5853 2977 7019 4427 7769 5877 8519 7327 9269

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

Table 30: By the Choice of Action A15: Player A Vs Player B from 300 - 500 Iterations

300th Iteration 350th Iteration 400th Iteration 450th Iteration 500th Iteration

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

Player

A

Player

B

355 8601 405 10051 455 11501 617 12930 667 14380

4851 12501 5651 14601 6451 16701 7355 18780 8155 20880

5173 16101 6023 18801 6873 21501 7871 24180 8721 26880

5493 19401 6393 22651 7293 25901 8381 29130 9281 32380

5811 22401 6761 26151 7711 29901 8885 33630 9835 37380

6127 25101 7127 29301 8127 33501 9383 37680 10383 41880

6441 27501 7491 32101 8541 36701 9875 41280 10925 45880

6753 29601 7853 34551 8953 39501 10361 44430 11461 49380

7063 31401 8213 36651 9363 41901 10841 47130 11991 52380

7371 32901 8571 38401 9771 43901 11315 49380 12515 54880

7677 34101 8927 39801 10177 45501 11783 51180 13033 56880

7981 35001 9281 40851 10581 46701 12245 52530 13545 58380

8283 30817 9633 36767 10983 42717 12701 48646 14051 54596

8478 35166 9878 41166 11278 47166 13046 50940 14446 56940

8777 10019 10227 10769 11677 11519 13071 14474 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 69

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

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 Action A2 Action A3 Action A4 Action A5 Action A6 Action A7

A B A B A B A B A B A B A B

0.96 0 0.92 0 0.94 0 0.94 0 0.94 0 0.94 0 0.96 0

0 0 0.04 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 0 0 0 0 0.02 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 0 0 0 0 0.02 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 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 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 0 0 0 0 0 0 0 0 0 0

0 0.5 0 0.66 0 0.48 0 0.32 0 0.16 0 0.14 0 0

0.02 0.44 0.02 0.3 0.02 0.46 0.02 0.6 0.02 0.74 0.02 0.76 0 0.88

0.02 0.06 0.02 0.04 0.02 0.06 0.02 0.08 0.02 0.1 0.02 0.1 0.02 0.12

A Significant Approach on a Special Case of Game Theory 71

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

Action A8 Action A9 Action A10 Action A11 Action A12 Action A13 Action A14 Action A15

A B A B A B A B A B A B A B A B

0.96 0 0.96 0 0.96 0 0.96 0 0.96 0 0.96 0 0.96 0 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 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 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 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 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 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 0.02 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 0 0 0 0 0 0 0 0 0.02 0 0 0.86 0 0.86

0 0.88 0 0.88 0 0.86 0 0.86 0 0.86 0 0.86 0.02 0 0.02 0.14

0.02 0.12 0.02 0.12 0.02 0.14 0.02 0.14 0.02 0.14 0.02 0.14 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 Action A2 Action A3 Action A4 Action A5 Action A6 Action A7

A B A B A B A B A B A B A B

0.98 0 0.96 0 0.97 0 0.97 0 0.97 0 0.97 0 0.97 0

0 0 0.02 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

0 0 0 0 0 0 0.01 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 0 0 0 0 0 0 0.01 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 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 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 0 0 0 0 0 0 0 0 0 0 0 0

0 0.28 0 0.36 0 0.27 0 0.19 0 0.11 0 0.1 0 0.02

0.01 0.22 0.01 0.15 0.01 0.23 0.01 0.3 0.01 0.37 0.01 0.38 0.01 0.45

0.01 0.5 0.01 0.49 0.01 0.5 0.01 0.51 0.01 0.52 0.01 0.52 0.01 0.53

Table 34: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 100th Iteration

Action A8 Action A9 Action A10 Action A11 Action A12 Action A13 Action A14 Action A15

A B A B A B A B A B A B A B A B

0.97 0 0.97 0 0.97 0 0.97 0 0.97 0 0.97 0 0.98 0 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 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 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 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 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 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 0 0 0.01 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 0.02 0 0.02 0 0 0 0 0 0 0.01 0 0 0.46 0 0.46

0.01 0.45 0.01 0.45 0.01 0.52 0.01 0.52 0.01 0.52 0 0.53 0.01 0 0.01 0.07

0.01 0.53 0.01 0.53 0.01 0.48 0.01 0.48 0.01 0.48 0.01 0.47 0.01 0.54 0.01 0.47

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

Action A1 Action A2 Action A3 Action A4 Action A5 Action A6 Actio

n A7

A B A B A B A B A B A B A B

0.986 0 0.973 0 0.98 0 0.98 0 0.98 0 0.98 0 0.98 0

0 0 0.013 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

0 0 0 0 0 0 0.006 0 0 0 0 0 0 0

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

Table 35 – Contd.,

0 0 0 0 0 0 0 0 0.006 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 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0.186 0 0.24 0 0.18 0 0.126 0 0.073 0 0.066

0.013

0.006 0.146 0.006 0.1 0.006 0.153 0.006 0.2 0.006 0.246 0.006 0.253 0.006 0.3

0.006 0.666 0.006 0.66 0.006 0.666 0.006 0.673 0.006 0.68 0.006 0.68 0.006 0.686

Table 36: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 150th Iteration

Action A8 Action A9 Action A10 Action A11 Action A12 Action A13 Action A14 Action A15

A B A B A B A B A B A B A B A B

0.98 0 0.98 0 0.98 0 0.98 0 0.98 0 0.98 0 0.99 0 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 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 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 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 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 0 0 0 0

0 0 0 0 0.006 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 0

0 0 0 0 0 0 0 0 0.006 0 0 0 0 0 0 0

0 0.013 0 0.013 0 0 0 0 0 0 0.006 0 0 0.306 0 0.306

0.006 0.3 0.006 0.3 0.006 0.346 0.01 0.346 0.006 0.346 0.006 0.353 0.01 0 0.006 0.046

0.006 0.686 0.006 0.686 0.006 0.653 0.01 0.653 0.006 0.653 0.006 0.646 0.01 0.693 0.006 0.646

Table 37: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 200th Iteration

Action A1 Action A2 Action A3 Action A4 Action A5 Action A6 Action A7

A B A B A B A B A B A B A B

0.99 0 0.98 0 0.985 0 0.985 0 0.985 0 0.985 0 0.985 0

0 0 0 0 0 0 0 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 0 0 0.005 0 0 0 0 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 0 0 0.005 0 0 0

0 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 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 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 0 0 0 0 0 0 0 0

0 0.14 0 0.18 0 0.135 0 0.095 0 0.055 0 0.05 0 0.01

0.005 0.11 0.005 0.075 0.005 0.115 0.005 0.15 0.005 0.185 0.005 0.19 0.005 0.225

0.005 0.75 0.005 0.745 0.005 0.75 0.005 0.755 0.005 0.76 0.005 0.76 0.005 0.765

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

Action A8 Action A9 Action A10 Action A11 Action A12 Action A13 Action A14 Action A15

A B A B A B A B A B A B A B A B

0.985 0 0.985 0 0.985 0 0.985 0 0.985 0 0.985 0 0.99 0 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 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 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 0 0 0 0 0 0

0 0 0 0 0 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 0 0

0 0 0.005 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0.005 0 0 0 0 0 0 0 0 0 0 0

A Significant Approach on a Special Case of Game Theory 73

Table 38 – Contd., 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 0.005 0 0 0 0 0 0 0

0 0.01 0 0.01 0 0 0 0 0 0 0.005 0 0 0.23 0 0.23

0.005 0.225 0.005 0.225 0.005 0.26 0.005 0.26 0.005 0.26 0.005 0.265 0.005 0 0.005 0.035

0.005 0.765 0.005 0.765 0.005 0.74 0.005 0.74 0.005 0.74 0.005 0.735 0.005 0.77 0.005 0.735

Table 39: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 250th Iteration

Action A1 Action A2 Action A3 Action A4 Action A5 Action A6 Action A7

A B A B A B A B A B A B A B

0.992 0 0.984 0 0.988 0 0.988 0 0.988 0 0.988 0 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 0 0 0

0 0 0 0 0 0 0.004 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0.004 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 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 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 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 0 0 0 0

0 0.112 0 0.144 0 0.108 0 0.076 0 0.044 0 0.04 0 0.008

0.004 0.088 0.004 0.06 0.004 0.092 0.004 0.12 0.004 0.148 0.004 0.152 0.004 0.18

0.004 0.8 0.004 0.796 0.004 0.8 0.004 0.804 0.004 0.808 0.004 0.808 0.004 0.812

Table 40: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 250th Iteration

Action A8 Action A9 Action A10 Action A11 Action A12 Action A13 Action A14 Action A15

A B A B A B A B A B A B A B A B

0.988 0 0.988 0 0.988 0 0.988 0 0.988 0 0.988 0 0.992 0 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 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 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 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 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 0 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 0.004 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 0.008 0 0.008 0 0 0 0 0 0 0.004 0 0 0.184 0 0.184

0.004 0.18 0.004 0.18 0.004 0.208 0.004 0.208 0.004 0.208 0.004 0.212 0.004 0 0.004 0.028

0.004 0.812 0.004 0.812 0.004 0.792 0.004 0.792 0.004 0.792 0.004 0.788 0.004 0.816 0.004 0.788

Table 41: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 300th Iteration

Action A1 Action A2 Action A3 Action A4 Action A5 Action A6 Action A7

A B A B A B A B A B A B A B

0.993 0 0.986 0 0.99 0 0.99 0 0.99 0 0.99 0 0.99 0

0 0 0.006 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

0 0 0 0 0 0 0.003 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 0 0 0 0 0 0 0.003 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 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 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 0 0 0 0 0 0 0 0 0 0 0 0

0 0.093 0 0.12 0 0.09 0 0.063 0 0.036 0 0.033 0 0.006

0.003 0.073 0.003 0.05 0.003 0.076 0.003 0.1 0.003 0.123 0.003 0.126 0.003 0.15

0.003 0.833 0.003 0.83 0.003 0.833 0.003 0.836 0.003 0.84 0.003 0.84 0.003 0.843

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

Table 42: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 300th Iteration

Action A8 Action A9 Action A10 Action A11 Action A12 Action A13 Action A14 Action A15

A B A B A B A B A B A B A B A B

0.99 0 0.99 0 0.99 0 0.99 0 0.99 0 0.99 0 0.993 0 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 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 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 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 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 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 0 0 0.003 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 0.006 0 0.006 0 0 0 0 0 0 0.003 0 0 0.153 0 0.153

0.003 0.15 0.003 0.15 0.003 0.173 0.003 0.173 0.003 0.173 0.003 0.176 0.003 0 0.003 0.023

0.003 0.843 0.003 0.843 0.003 0.826 0.003 0.826 0.003 0.826 0.003 0.823 0.003 0.846 0.003 0.823

Table 43: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 350th Iteration

Action A1 Action A2 Action A3 Action A4 Action A5 Action A6 Action A7

A B A B A B A B A B A B A B

0.994 0 0.988 0 0.991 0 0.991 0 0.991 0 0.991 0 0.991 0

0 0 0.005 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

0 0 0 0 0 0 0.002 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 0 0 0 0 0 0 0.002 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 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 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 0 0 0 0 0 0 0 0 0 0 0 0

0 0.08 0 0.102 0 0.077 0 0.542 0 0.031 0 0.0285 0 0.005

0.002 0.062 0.002 0.042 0.002 0.065 0.002 0.857 0.002 0.105 0.002 0.108 0.002 0.128

0.002 0.857 0.002 0.854 0.002 0.857 0.002 0.86 0.002 0.862 0.002 0.8628 0.002 0.865

Table 44: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 350th Iteration

Action A8 Action A9 Action A10 Action A11 Action A12 Action A13 Action A14 Action A15

A B A B A B A B A B A B A B A B

0.991 0 0.991 0 0.991 0 0.991 0 0.991 0 0.991 0 0.994 0 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 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 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 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 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 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 0 0 0.002 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 0.005 0 0.005 0 0 0 0 0 0 0.002 0 0 0.131 0 0.131

0.002 0.128 0.002 0.128 0.002 0.148 0.002 0.148 0.002 0.148 0.002 0.151 0.002 0 0.002 0.02

0.002 0.865 0.002 0.865 0.002 0.851 0.002 0.851 0.002 0.851 0.002 0.848 0.002 0.868 0.002 0.848

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

Action A1 Action A2 Action A3 Action A4 Action A5 Action A6 Action A7

A B A B A B A B A B A B A B

0.985 0 0.98 0 0.982 0 0.982 0 0.982 0 0.982 0 0.982 0

0 0 0.005 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

0 0 0 0 0 0 0.002 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0.002 0 0 0 0 0

A Significant Approach on a Special Case of Game Theory 75

Table 45 – Contd.,

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 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 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 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

0 0.07 0 0.09 0 0.067 0 0.047 0 0.027 0 0.025 0 0.005

0.002 0.095 0.002 0.057 0.002 0.095 0.002 0.13 0.002 0.165 0.002 0.165 0.002 0.202

0.012 0.835 0.012 0.852 0.012 0.837 0.012 0.822 0.012 0.807 0.012 0.81 0.012 0.792

Table 46: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 400th Iteration

Action A8 Action A9 Action A10 Action A11 Action A12 Action A13 Action A14 Action A15

A B A B A B A B A B A B A B A B

0.982 0 0.982 0 0.992 0 0.992 0 0.992 0 0.992 0 0.985 0 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 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 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 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 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 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 0 0 0.002 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 0.005 0 0.005 0 0 0 0 0 0 0.002 0 0 0.115 0 0.115

0.002 0.2 0.002 0.2 0.002 0.13 0.002 0.13 0.002 0.13 0.002 0.132 0.002 0.105 0.002 0.017

0.012 0.795 0.012 0.795 0.002 0.87 0.002 0.87 0.002 0.87 0.002 0.867 0.012 0.78 0.002 0.867

Table 47: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 450th Iteration

Action A1 Action A2 Action A3 Action A4 Action A5 Action A6 Action A7

A B A B A B A B A B A B A B

0.986 0 0.937 0 0.984 0 0.984 0 0.984 0 0.984 0 0.984 0

0 0 0.004 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

0 0 0 0 0 0 0.002 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 0 0 0 0 0 0 0.002 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 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 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 0 0 0 0 0 0 0 0 0 0 0 0

0 0.062 0 0.08 0 0.06 0 0.042 0 0.024 0 0.022 0 0.004

0.002 0.14 0.002 0.106 0.002 0.14 0.002 0.171 0.002 0.202 0.002 0.202 0.002 0.235

0.011 0.797 0.011 0.813 0.011 0.8 0.011 0.786 0.011 0.773 0.011 0.775 0.011 0.72

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

Action A8 Action A9 Action A10 Action A11 Action A12 Action A13 Action A14 Action A15

A B A B A B A B A B A B A B A B

0.984 0 0.984 0 0.984 0 0.984 0 0.984 0 0.984 0 0.986 0 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 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 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 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 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 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 0 0 0.002 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 0.004 0 0.004 0 0 0 0 0 0 0.002 0 0 0.102 0 0.102

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

Table 48 – Contd.,

0.002 0.233 0.002 0.233 0.002 0.164 0.002 0.162 0.002 0.162 0.002 0.148 0.002 0.148 0.002 0.062

0.011 0.762 0.011 0.762 0.011 0.835 0.011 0.837 0.011 0.837 0.011 0.851 0.011 0.748 0.011 0.835

Table 49: Optimum Mixed Strategies of Player A & Player B from Action A1 - A7 at 500th Iteration

Action A1 Action A2 Action A3 Action A4 Action A5 Action A6 Action A7

A B A B A B A B A B A B A B

0.988 0 0.984 0 0.986 0 0.986 0 0.986 0 0.986 0 0.986 0

0 0 0.004 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

0 0 0 0 0 0 0.002 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 0 0 0 0 0 0 0.002 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 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 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 0 0 0 0 0 0 0 0 0 0 0 0

0 0.056 0 0.072 0 0.054 0 0.038 0 0.022 0 0.02 0 0.004

0.002 0.126 0.002 0.096 0.002 0.126 0.002 0.154 0.002 0.182 0.002 0.182 0.002 0.212

0.01 0.818 0.01 0.832 0.01 0.82 0.01 0.808 0.01 0.796 0.01 0.798 0.01 0.784

Table 50: Optimum Mixed Strategies of Player A & Player B from Action A8 - A15 at 500th Iteration

Action A8 Action A9 Action A10 Action A11 Action A12 Action A13 Action A14 Action A15

A B A B A B A B A B A B A B A B

0.986 0 0.986 0 0.986 0 0.986 0 0.986 0 0.986 0 0.988 0 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 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 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 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 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 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 0 0 0.002 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 0.004 0 0.004 0 0 0 0 0 0 0.002 0 0 0.092 0 0.092

0.002 0.21 0.002 0.21 0.002 0.148 0.002 0.146 0.002 0.146 0.002 0.134 0.002 0.13 0.002 0.056

0.01 0.786 0.01 0.786 0.01 0.852 0.01 0.854 0.01 0.854 0.01 0.866 0.01 0.774 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

50 100 150 200 250

LB UB LB UB LB UB LB UB LB UB

27.04 30.66 27.94 29.77 28.29 29.51 28.47 29.385 28.576 29.308

27.16 31.18 28 30.03 28.33 29.68 28.5 29.515 28.6 29.412

27.38 31.16 28.11 30.02 28.406 29.68 28.555 29.51 28.644 29.408

27.58 31.38 28.21 30.13 28.473 29.753 28.605 29.565 28.684 29.452

27.78 31.58 28.31 30.23 28.54 29.82 28.655 29.615 28.724 29.492

27.82 31.76 28.33 30.32 28.553 29.88 28.665 29.66 28.732 29.528

27.98 31.1 28.43 30.4 28.62 29.93 28.715 29.7 28.772 29.56

28 31.24 28.44 30.47 28.62 29.98 28.72 29.735 28.776 29.588

28.02 31.36 28.45 30.53 28.63 30.02 28.725 29.765 28.78 29.612

28.06 31.46 28.43 30.58 28.62 30.053 28.715 29.79 28.772 29.632

28.08 31.54 28.44 30.62 28.62 30.08 28.72 29.81 28.776 29.648

28.1 31.6 28.45 30.65 28.63 30.1 28.725 29.825 28.78 29.66

28.12 31.64 28.45 30.67 28.633 30.113 28.725 29.835 28.78 29.668

A Significant Approach on a Special Case of Game Theory 77

Table – 51:Contd.,

27.3 30.66 28.07 29.77 28.38 29.513 28.535 29.385 28.628 29.308

27.18 30.66 28.01 29.77 28.34 29.5133 28.505 29.385 28.604 29.308

Table 52: Lower Bounds and Upper Bounds at 300th Iteration – 500th Iteration

300 350 400 450 500

LB UB LB UB LB UB LB UB LB UB

28.64 29.25 28.69 29.22 28.6975 29.115 28.67 29.04 28.706 29.042

28.66 29.34 28.714 29.294 28.7325 29.18 28.704 29.104 28.734 29.094

28.703 29.34 28.74 29.29 28.7425 29.1775 28.713 29.102 28.742 29.092

28.736 29.376 28.774 29.322 28.75 29.205 28.72 29.126 28.748 29.114

28.77 29.41 28.8 29.35 28.7575 29.23 28.72 29.14 28.754 29.134

28.77 29.44 28.8 29.37 28.765 29.2525 28.73 29.16 28.76 29.152

28.81 29.46 28.83 29.4 28.77 29.2725 28.73 29.18 28.764 29.168

28.81 29.49 28.84 29.42 28.775 29.29 28.74 29.2 28.768 29.182

28.81 29.51 28.84 29.43 28.7775 29.305 28.74 29.21 28.77 29.194

28.81 29.52 28.83 29.45 28.8575 29.395 28.82 29.226 28.842 29.204

28.81 29.54 28.84 29.46 28.86 29.405 28.82 29.235 28.846 29.212

28.816 29.55 28.842 29.471 28.8625 29.4125 28.83 29.24 28.848 29.218

28.816 29.556 28.842 29.477 28.8625 29.41 28.846 29.246 28.862 29.222

28.69 29.256 28.734 29.22 28.665 29.115 28.644 29.046 28.68 29.042

28.67 29.2568 28.717 29.22 28.7525 29.1925 28.733 29.046 28.76 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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