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A SCIENTIFIC COMPUTATION ON A PECULIAR CASE OF GAME THEORY IN

OPERATIONS RESEARCH

K. V. L. N. ACHARYULU1 & MADDI. N. MURALI KRISHNA2

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

ABSTRACT

A peculiar case of game theory problem is investigated with the help of Browns Algorithm in this paper. The

problem acquaints the dominance nature for both rows and columns. It is constituted with the scenario of increasing in an

action B1 to B15 of player B with the addition of successive natural numbers according to the influences of action A1 to

A15 of Player A. Some remarkable conclusions are established by computing maximum number of possible iterations in

the classical Java program .The results are given in the conclusions and also shown in the graphs. The errors are estimated

for each computation and the Lower bounds and Upper bounds are also traced.The corelations between the iterations are

obtained.

KEYWORDS: Game Theory, Players, Strategy, Pay-Off Matrix, Optimal Solution, Lower Bound, Upper Bound

AMS Classification: 91A05, 91A18, 91A43, 91A90

INRTRODUCTION

A positive decision is needed for classifying any problem. The best tools of recognizing the required decision are

available in operations research. In general, operations research intends to investigate the extremities (i.e. maximum or

minimum) of the objective function value in any linear or non linear system. It also helps us to provide optimum feasible

solution with many classical scientific computational procedures. Game theory is a part and parcel of operation research

which supplies conservative strategies to try to maximize the players gain. If it has conflict of interests of the players,

suitable conservative strategies are utilized by both the players. The resolution of this conflict is the main substance of the

game theory.

K.V.L.N.Acharyulu and Maddi.N.Murali Krishna [1] obtained some remarkable results in a special case of game

theory in their earlier work. In continuation of this work, It is beneficial to discuss higher size for investing some more

fruitful results. McKinsey [7] explicated theory of Games in 1952. Raiffa, R. D [6] discussed the nature of games and

possible decisions in 1958.Later Dresher, M [5] concentrated on strategies and applications of game theory in

1961.Afterwards Rapoport [4], Levin and Desjardins [3] developed the concepts of game theory to open new eras in

operations research. Billy E.Gillett [2] explained how to solve the large size of problems in the games by using Browns

algorithm.

In this present work, the authors probed a 15x15 game problem which is a peculiar case of game theory and

discussed with the help of Browns Algorithm. The problem has the dominance nature for both the rows and columns. The

principle is adopted for constructing this model by increasing in an action B1 to B15 of player B with the addition of

successive natural numbers according to the influences of action A1 to A15 of Player A. Remarkable results are obtained

by computing maximum number of possible iterations. The obtained results are specified in the conclusions and also the

International Journal of Computer Science Engineering

and Information Technology Research (IJCSEITR)

ISSN 2249-6831Vol. 3, Issue 1 Mar 2013, 175-190

TJPRC Pvt. Ltd.

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176 K. V. L. N. Acharyulu& Maddi. N. Murali Krishna

graphs are depicted where ever necessary and feasible. The errors are identified in each iteration and tabulated in a table.

Lower bounds and Upper bounds are calculated in each iteration for classifying the nature of the game. The iterations and

the optimum mixed strategies have the highest value of correlation in both (rows and columns) of dominant game. The

maximum possible iterations have been computed to obtain the best optimum mixed strategies for the players.The

iterations are computed from 50 th iteration to 500th iteration. The authors used 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 observed.

BASIC FORMATION OF 15x15 GAME

A special gameis constituted with 15 rows and 15 columns of player A & Player B with all 15 possible opposing

actions on one and another. 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 of

considered game having the size of 15x15 is given below in matrix representation form.

1 2 4 7 11 16 22 29 37 46 56 67 79 92 106

121 137 154 172 191 211 232 254 277 301 326 352 379 407 436

466 497 529 562 596 631 667 704 742 781 821 862 904 947 991

1036 1082 1129 1177 1226 1276 1327 1379 1432 1486 1541 1597 1654 1712 1771

1831 1892 1954 2017

Player A

2081 2146 2212 2279 2347 2416 2486 2557 2629 2702 2776

2851 2927 3004 3082 3161 3241 3322 3404 3487 3571 3656 3742 3829 3917 4006

4096 4187 4279 4372 4466 4561 4657 4754 4852 4951 5051 5152 5254 5357 5461

5566 5672 5779 5887 5996 6106 6217 6329 6442 6556 6671 6787 6904 7022 7141

7261 7387 7504 7627 7751 7876 8002 8129 8257 8386 8516 8647 8779 8912 9046

9181 9317 9454 9592 9731 9871 10012 10154 10297 10441 10586 10732 10879 11027 11176

11326 11477 11629 11782 11936 12091 12247 12404 12562 12721 12881 13042 13204 13367 1353113696 13862 14029 14197 14366 14536 14707 14879 15052 15226 15401 15577 15754 15932 16111

16291 16472 16654 16837 17021 17206 17392 17579 17767 17956 18146 18337 18529 18722 18916

19111 19307 19504 19702 19

Player B

901 20101 20302 20504 20707 20911 21116 21322 21529 21737 21946

22156 22367 22579 22792 23006 23221 23437 23654 23872 24091 24311 24532 24754 24977 25201

MATERIAL AND METHODS

The authors adopted Browns algorithm to solve this special case of 15x15 game in which row and columns both

dominated. Browns 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.

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A Scientific Computation on a Peculiar Case of Game Theory in Operations Research 177

Step 5: Compute an upper and lower bound and respectively.

Largest sum elem ent from step 4 Sm allest sum elem ent from step 3

N um ber o f p la ys of th e g am e th us far N um ber o f p lays o f th e g am e th us faran d = =

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 thetime 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.

RESULTS

The game is solved by Brown's algorithm to gain the best optimum mixed strategies for both the players from

50th iteration to 500 iteration with the aid of Java Program.The effects on all possible actions of player A from the player

B are given in the following tables from Table (1) to Table (20).

Table 1: Player A vs Player B at 50th Iteration from Action A1 to A8

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8

50 1085645 1085765 1086110 1086680 1087475 1088495 1089740 1091210

6050 1095985 1096120 1096480 1097065 1097875 1098910 1100170 1101655

23300 1106375 1106525 1106900 1107500 1108325 1109375 1110650 1112150

51800 1116815 1116980 1117370 1117985 1118825 1119890 1121180 1122695

91550 1127305 1127485 1127890 1128520 1129375 1130455 1131760 1133290

142550 1137845 1138040 1138460 1139105 1139975 1141070 1142390 1143935

204800 1148435 1148645 1149080 1149740 1150625 1151735 1153070 1154630

278300 1159075 1159300 1159750 1160425 1161325 1162450 1163800 1165375

363050 1169765 1170005 1170470 1171160 1172075 1173215 1174580 1176170

459050 1180505 1180760 1181240 1181945 1182875 1184030 1185410 1187015566300 1191295 1191565 1192060 1192780 1193725 1194895 1196290 1197910

684800 1202135 1202420 1202930 1203665 1204625 1205810 1207220 1208855

814550 1213025 1213325 1213850 1214600 1215575 1216775 1218200 1219850

955550 1223965 1224280 1224820 1225585 1226575 1227790 1229230 1230895

1107800 1234955 1235285 1235840 1236620 1237625 1238855 1240310 1241990

Table 2: Player A vs Player B at 50th Iteration from Action A9 to A15

Player A Player B

A9 A10 A11 A12 A13 A14 A15

50 1092905 1094825 1096970 1099340 1101935 1104755 1107800

6050 1103365 1105300 1107460 1109845 1112455 1115290 1118350

23300 1113875 1115825 1118000 1120400 1123025 1125875 112895051800 1124435 1126400 1128590 1131005 1133645 1136510 1139600

91550 1135045 1137025 1139230 1141660 1144315 1147195 1150300

142550 1145705 1147700 1149920 1152365 1155035 1157930 1161050

204800 1156415 1158425 1160660 1163120 1165805 1168715 1171850

278300 1167175 1169200 1171450 1173925 1176625 1179550 1182700

363050 1177985 1180025 1182290 1184780 1187495 1190435 1193600

459050 1188845 1190900 1193180 1195685 1198415 1201370 1204550

566300 1199755 1201825 1204120 1206640 1209385 1212355 1215550

684800 1210715 1212800 1215110 1217645 1220405 1223390 1226600

814550 1221725 1223825 1226150 1228700 1231475 1234475 1237700

955550 1232785 1234900 1237240 1239805 1242595 1245610 1248850

1107800 1243895 1246025 1248380 1250960 1253765 1256795 1260050

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178 K. V. L. N. Acharyulu& Maddi. N. Murali Krishna

Table 3: Player A vs Player B at 100th Iteration from Action A1to A8

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8

100 2193445 2193565 2193910 2194480 2195275 2196295 2197540 2199010

12100 2214335 2214470 2214830 2215415 2216225 2217260 2218520 2220005

46600 2235325 2235475 2235850 2236450 2237275 2238325 2239600 2241100103600 2256415 2256580 2256970 2257585 2258425 2259490 2260780 2262295

183100 2277605 2277785 2278190 2278820 2279675 2280755 2282060 2283590

285100 2298895 2299090 2299510 2300155 2301025 2302120 2303440 2304985

409600 2320285 2320495 2320930 2321590 2322475 2323585 2324920 2326480

556600 2341775 2342000 2342450 2343125 2344025 2345150 2346500 2348075

726100 2363365 2363605 2364070 2364760 2365675 2366815 2368180 2369770

918100 2385055 2385310 2385790 2386495 2387425 2388580 2389960 2391565

1132600 2406845 2407115 2407610 2408330 2409275 2410445 2411840 2413460

1369600 2428735 2429020 2429530 2430265 2431225 2432410 2433820 2435455

1629100 2450725 2451025 2451550 2452300 2453275 2454475 2455900 2457550

1911100 2472815 2473130 2473670 2474435 2475425 2476640 2478080 2479745

2215600 2495005 2495335 2495890 2496670 2497675 2498905 2500360 2502040

Table 4: Player A vs Player B at 100th Iteration from Action A9 to A15

Player A Player B

A9 A10 A11 A12 A13 A14 A15

100 2200705 2202625 2204770 2207140 2209735 2212555 2215600

12100 2221715 2223650 2225810 2228195 2230805 2233640 2236700

46600 2242825 2244775 2246950 2249350 2251975 2254825 2257900

103600 2264035 2266000 2268190 2270605 2273245 2276110 2279200

183100 2285345 2287325 2289530 2291960 2294615 2297495 2300600

285100 2306755 2308750 2310970 2313415 2316085 2318980 2322100

409600 2328265 2330275 2332510 2334970 2337655 2340565 2343700

556600 2349875 2351900 2354150 2356625 2359325 2362250 2365400

726100 2371585 2373625 2375890 2378380 2381095 2384035 2387200

918100 2393395 2395450 2397730 2400235 2402965 2405920 2409100

1132600 2415305 2417375 2419670 2422190 2424935 2427905 2431100

1369600 2437315 2439400 2441710 2444245 2447005 2449990 2453200

1629100 2459425 2461525 2463850 2466400 2469175 2472175 2475400

1911100 2481635 2483750 2486090 2488655 2491445 2494460 2497700

2215600 2503945 2506075 2508430 2511010 2513815 2516845 2520100

Table 5: Player A vs Player B at 150th Iteration from Action A1 to A8

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8

150 3301245 3301365 3301710 3302280 3303075 3304095 3305340 3306810

18150 3332685 3332820 3333180 3333765 3334575 3335610 3336870 3338355

69900 3364275 3364425 3364800 3365400 3366225 3367275 3368550 3370050

155400 3396015 3396180 3396570 3397185 3398025 3399090 3400380 3401895

274650 3427905 3428085 3428490 3429120 3429975 3431055 3432360 3433890

427650 3459945 3460140 3460560 3461205 3462075 3463170 3464490 3466035

614400 3492135 3492345 3492780 3493440 3494325 3495435 3496770 3498330

834900 3524475 3524700 3525150 3525825 3526725 3527850 3529200 3530775

1089150 3556965 3557205 3557670 3558360 3559275 3560415 3561780 3563370

1377150 3589605 3589860 3590340 3591045 3591975 3593130 3594510 3596115

1698900 3622395 3622665 3623160 3623880 3624825 3625995 3627390 3629010

2054400 3655335 3655620 3656130 3656865 3657825 3659010 3660420 3662055

2443650 3688425 3688725 3689250 3690000 3690975 3692175 3693600 3695250

2866650 3721665 3721980 3722520 3723285 3724275 3725490 3726930 3728595

3323400 3755055 3755385 3755940 3756720 3757725 3758955 3760410 3762090

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Table 6: Player A vs Player B at 150th Iteration from Action A9 to A15

Player A Player B

A9 A10 A11 A12 A13 A14 A15

150 3308505 3310425 3312570 3314940 3317535 3320355 3323400

18150 3340065 3342000 3344160 3346545 3349155 3351990 3355050

69900 3371775 3373725 3375900 3378300 3380925 3383775 3386850155400 3403635 3405600 3407790 3410205 3412845 3415710 3418800

274650 3435645 3437625 3439830 3442260 3444915 3447795 3450900

427650 3467805 3469800 3472020 3474465 3477135 3480030 3483150

614400 3500115 3502125 3504360 3506820 3509505 3512415 3515550

834900 3532575 3534600 3536850 3539325 3542025 3544950 3548100

1089150 3565185 3567225 3569490 3571980 3574695 3577635 3580800

1377150 3597945 3600000 3602280 3604785 3607515 3610470 3613650

1698900 3630855 3632925 3635220 3637740 3640485 3643455 3646650

2054400 3663915 3666000 3668310 3670845 3673605 3676590 3679800

2443650 3697125 3699225 3701550 3704100 3706875 3709875 3713100

2866650 3730485 3732600 3734940 3737505 3740295 3743310 3746550

3323400 3763995 3766125 3768480 3771060 3773865 3776895 3780150

Table 7: Player A vs Player B at 200th Iteration from Action A1 to A8

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8

200 4409045 4409165 4409510 4410080 4410875 4411895 4413140 4414610

24200 4451035 4451170 4451530 4452115 4452925 4453960 4455220 4456705

93200 4493225 4493375 4493750 4494350 4495175 4496225 4497500 4499000

207200 4535615 4535780 4536170 4536785 4537625 4538690 4539980 4541495

366200 4578205 4578385 4578790 4579420 4580275 4581355 4582660 4584190

570200 4620995 4621190 4621610 4622255 4623125 4624220 4625540 4627085

819200 4663985 4664195 4664630 4665290 4666175 4667285 4668620 4670180

1113200 4707175 4707400 4707850 4708525 4709425 4710550 4711900 4713475

1452200 4750565 4750805 4751270 4751960 4752875 4754015 4755380 4756970

1836200 4794155 4794410 4794890 4795595 4796525 4797680 4799060 4800665

2265200 4837945 4838215 4838710 4839430 4840375 4841545 4842940 48445602739200 4881935 4882220 4882730 4883465 4884425 4885610 4887020 4888655

3258200 4926125 4926425 4926950 4927700 4928675 4929875 4931300 4932950

3822200 4970515 4970830 4971370 4972135 4973125 4974340 4975780 4977445

4431200 5015105 5015435 5015990 5016770 5017775 5019005 5020460 5022140

Table 8: Player A vs Player B at 200th Iteration from Action A9 to A15

Player A Player B

A9 A10 A11 A12 A13 14 A15

200 4416305 4418225 4420370 4422740 4425335 4428155 4431200

24200 4458415 4460350 4462510 4464895 4467505 4470340 4473400

93200 4500725 4502675 4504850 4507250 4509875 4512725 4515800

207200 4543235 4545200 4547390 4549805 4552445 4555310 4558400

366200 4585945 4587925 4590130 4592560 4595215 4598095 4601200570200 4628855 4630850 4633070 4635515 4638185 4641080 4644200

819200 4671965 4673975 4676210 4678670 4681355 4684265 4687400

1113200 4715275 4717300 4719550 4722025 4724725 4727650 4730800

1452200 4758785 4760825 4763090 4765580 4768295 4771235 4774400

1836200 4802495 4804550 4806830 4809335 4812065 4815020 4818200

2265200 4846405 4848475 4850770 4853290 4856035 4859005 4862200

2739200 4890515 4892600 4894910 4897445 4900205 4903190 4906400

3258200 4934825 4936925 4939250 4941800 4944575 4947575 4950800

3822200 4979335 4981450 4983790 4986355 4989145 4992160 4995400

4431200 5024045 5026175 5028530 5031110 5033915 5036945 5040200

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Table 9: Player A vs Player B at 250th Iteration from Action A1 to A8

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8

250 5516845 5516965 5517310 5517880 5518675 5519695 5520940 5522410

30250 5569385 5569520 5569880 5570465 5571275 5572310 5573570 5575055

116500 5622175 5622325 5622700 5623300 5624125 5625175 5626450 5627950259000 5675215 5675380 5675770 5676385 5677225 5678290 5679580 5681095

457750 5728505 5728685 5729090 5729720 5730575 5731655 5732960 5734490

712750 5782045 5782240 5782660 5783305 5784175 5785270 5786590 5788135

1024000 5835835 5836045 5836480 5837140 5838025 5839135 5840470 5842030

1391500 5889875 5890100 5890550 5891225 5892125 5893250 5894600 5896175

1815250 5944165 5944405 5944870 5945560 5946475 5947615 5948980 5950570

2295250 5998705 5998960 5999440 6000145 6001075 6002230 6003610 6005215

2831500 6053495 6053765 6054260 6054980 6055925 6057095 6058490 6060110

3424000 6108535 6108820 6109330 6110065 6111025 6112210 6113620 6115255

4072750 6163825 6164125 6164650 6165400 6166375 6167575 6169000 6170650

4777750 6219365 6219680 6220220 6220985 6221975 6223190 6224630 6226295

5539000 6275155 6275485 6276040 6276820 6277825 6279055 6280510 6282190

Table 10: Player A vs Player B at 250th Iteration from Action A9 to A15

Player A Player B

A9 A10 A11 A12 A13 A14 A15

250 5524105 5526025 5528170 5530540 5533135 5535955 5539000

30250 5576765 5578700 5580860 5583245 5585855 5588690 5591750

116500 5629675 5631625 5633800 5636200 5638825 5641675 5644750

259000 5682835 5684800 5686990 5689405 5692045 5694910 5698000

457750 5736245 5738225 5740430 5742860 5745515 5748395 5751500

712750 5789905 5791900 5794120 5796565 5799235 5802130 5805250

1024000 5843815 5845825 5848060 5850520 5853205 5856115 5859250

1391500 5897975 5900000 5902250 5904725 5907425 5910350 5913500

1815250 5952385 5954425 5956690 5959180 5961895 5964835 5968000

2295250 6007045 6009100 6011380 6013885 6016615 6019570 6022750

2831500 6061955 6064025 6066320 6068840 6071585 6074555 6077750

3424000 6117115 6119200 6121510 6124045 6126805 6129790 6133000

4072750 6172525 6174625 6176950 6179500 6182275 6185275 6188500

4777750 6228185 6230300 6232640 6235205 6237995 6241010 6244250

5539000 6284095 6286225 6288580 6291160 6293965 6296995 6300250

Table 11: Player A vs Player B at 300th Iteration from Action A1 to A8

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8

300 6624645 6624765 6625110 6625680 6626475 6627495 6628740 6630210

36300 6687735 6687870 6688230 6688815 6689625 6690660 6691920 6693405

139800 6751125 6751275 6751650 6752250 6753075 6754125 6755400 6756900

310800 6814815 6814980 6815370 6815985 6816825 6817890 6819180 6820695

549300 6878805 6878985 6879390 6880020 6880875 6881955 6883260 6884790

855300 6943095 6943290 6943710 6944355 6945225 6946320 6947640 6949185

1228800 7007685 7007895 7008330 7008990 7009875 7010985 7012320 7013880

1669800 7072575 7072800 7073250 7073925 7074825 7075950 7077300 7078875

2178300 7137765 7138005 7138470 7139160 7140075 7141215 7142580 7144170

2754300 7203255 7203510 7203990 7204695 7205625 7206780 7208160 7209765

3397800 7269045 7269315 7269810 7270530 7271475 7272645 7274040 7275660

4108800 7335135 7335420 7335930 7336665 7337625 7338810 7340220 7341855

4887300 7401525 7401825 7402350 7403100 7404075 7405275 7406700 7408350

5733300 7468215 7468530 7469070 7469835 7470825 7472040 7473480 7475145

6646800 7535205 7535535 7536090 7536870 7537875 7539105 7540560 7542240

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Table 12: Player A vs Player B at 300th Iteration from Action A9 to A15

Player A Player B

A9 A10 A11 A12 A13 A14 A15

300 6631905 6633825 6635970 6638340 6640935 6643755 6646800

36300 6695115 6697050 6699210 6701595 6704205 6707040 6710100

139800 6758625 6760575 6762750 6765150 6767775 6770625 6773700310800 6822435 6824400 6826590 6829005 6831645 6834510 6837600

549300 6886545 6888525 6890730 6893160 6895815 6898695 6901800

855300 6950955 6952950 6955170 6957615 6960285 6963180 6966300

1228800 7015665 7017675 7019910 7022370 7025055 7027965 7031100

1669800 7080675 7082700 7084950 7087425 7090125 7093050 7096200

2178300 7145985 7148025 7150290 7152780 7155495 7158435 7161600

2754300 7211595 7213650 7215930 7218435 7221165 7224120 7227300

3397800 7277505 7279575 7281870 7284390 7287135 7290105 7293300

4108800 7343715 7345800 7348110 7350645 7353405 7356390 7359600

4887300 7410225 7412325 7414650 7417200 7419975 7422975 7426200

5733300 7477035 7479150 7481490 7484055 7486845 7489860 7493100

6646800 7544145 7546275 7548630 7551210 7554015 7557045 7560300

Table 13: Player A vs Player B at 350th Iteration from Action A1 to A8

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8

350 7732445 7732565 7732910 7733480 7734275 7735295 7736540 7738010

42350 7806085 7806220 7806580 7807165 7807975 7809010 7810270 7811755

163100 7880075 7880225 7880600 7881200 7882025 7883075 7884350 7885850

362600 7954415 7954580 7954970 7955585 7956425 7957490 7958780 7960295

640850 8029105 8029285 8029690 8030320 8031175 8032255 8033560 8035090

997850 8104145 8104340 8104760 8105405 8106275 8107370 8108690 8110235

1433600 8179535 8179745 8180180 8180840 8181725 8182835 8184170 8185730

1948100 8255275 8255500 8255950 8256625 8257525 8258650 8260000 8261575

2541350 8331365 8331605 8332070 8332760 8333675 8334815 8336180 8337770

3213350 8407805 8408060 8408540 8409245 8410175 8411330 8412710 84143153964100 8484595 8484865 8485360 8486080 8487025 8488195 8489590 8491210

4793600 8561735 8562020 8562530 8563265 8564225 8565410 8566820 8568455

5701850 8639225 8639525 8640050 8640800 8641775 8642975 8644400 8646050

6688850 8717065 8717380 8717920 8718685 8719675 8720890 8722330 8723995

7754600 8795255 8795585 8796140 8796920 8797925 8799155 8800610 8802290

Table 14: Player A vs Player B at 350th Iteration from Action A9 to A15

Player A Player B

A9 A10 A11 A12 A13 A14 A15

350 7739705 7741625 7743770 7746140 7748735 7751555 7754600

42350 7813465 7815400 7817560 7819945 7822555 7825390 7828450

163100 7887575 7889525 7891700 7894100 7896725 7899575 7902650

362600 7962035 7964000 7966190 7968605 7971245 7974110 7977200

640850 8036845 8038825 8041030 8043460 8046115 8048995 8052100

997850 8112005 8114000 8116220 8118665 8121335 8124230 8127350

1433600 8187515 8189525 8191760 8194220 8196905 8199815 8202950

1948100 8263375 8265400 8267650 8270125 8272825 8275750 8278900

2541350 8339585 8341625 8343890 8346380 8349095 8352035 8355200

3213350 8416145 8418200 8420480 8422985 8425715 8428670 8431850

3964100 8493055 8495125 8497420 8499940 8502685 8505655 8508850

4793600 8570315 8572400 8574710 8577245 8580005 8582990 8586200

5701850 8647925 8650025 8652350 8654900 8657675 8660675 8663900

6688850 8725885 8728000 8730340 8732905 8735695 8738710 8741950

7754600 8804195 8806325 8808680 8811260 8814065 8817095 8820350

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Table 15: Player A vs Player B at 400th Iteration from Action A1 to A8

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8

400 8840245 8840365 8840710 8841280 8842075 8843095 8844340 8845810

48400 8924435 8924570 8924930 8925515 8926325 8927360 8928620 8930105

186400 9009025 9009175 9009550 9010150 9010975 9012025 9013300 9014800414400 9094015 9094180 9094570 9095185 9096025 9097090 9098380 9099895

732400 9179405 9179585 9179990 9180620 9181475 9182555 9183860 9185390

1140400 9265195 9265390 9265810 9266455 9267325 9268420 9269740 9271285

1638400 9351385 9351595 9352030 9352690 9353575 9354685 9356020 9357580

2226400 9437975 9438200 9438650 9439325 9440225 9441350 9442700 9444275

2904400 9524965 9525205 9525670 9526360 9527275 9528415 9529780 9531370

3672400 9612355 9612610 9613090 9613795 9614725 9615880 9617260 9618865

4530400 9700145 9700415 9700910 9701630 9702575 9703745 9705140 9706760

5478400 9788335 9788620 9789130 9789865 9790825 9792010 9793420 9795055

6516400 9876925 9877225 9877750 9878500 9879475 9880675 9882100 9883750

7644400 9965915 9966230 9966770 9967535 9968525 9969740 9971180 9972845

8862400 10055305 10055635 10056190 10056970 10057975 10059205 10060660 10062340

Table 16: Player A vs Player B at 400th Iteration from Action A9 to A15

Player A Player B

A9 A10 A11 A12 A13 A14 A15

400 8847505 8849425 8851570 8853940 8856535 8859355 8862400

48400 8931815 8933750 8935910 8938295 8940905 8943740 8946800

186400 9016525 9018475 9020650 9023050 9025675 9028525 9031600

414400 9101635 9103600 9105790 9108205 9110845 9113710 9116800

732400 9187145 9189125 9191330 9193760 9196415 9199295 9202400

1140400 9273055 9275050 9277270 9279715 9282385 9285280 9288400

1638400 9359365 9361375 9363610 9366070 9368755 9371665 9374800

2226400 9446075 9448100 9450350 9452825 9455525 9458450 9461600

2904400 9533185 9535225 9537490 9539980 9542695 9545635 9548800

3672400 9620695 9622750 9625030 9627535 9630265 9633220 9636400

4530400 9708605 9710675 9712970 9715490 9718235 9721205 9724400

5478400 9796915 9799000 9801310 9803845 9806605 9809590 9812800

6516400 9885625 9887725 9890050 9892600 9895375 9898375 9901600

7644400 9974735 9976850 9979190 9981755 9984545 9987560 9990800

8862400 10064245 10066375 10068730 10071310 10074115 10077145 10080400

Table 17: Player A vs Player B at 450th Iteration from Action A1 to A8

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8

450 9948045 9948165 9948510 9949080 9949875 9950895 9952140 9953610

54450 10042785 10042920 10043280 10043865 10044675 10045710 10046970 10048455

209700 10137975 10138125 10138500 10139100 10139925 10140975 10142250 10143750

466200 10233615 10233780 10234170 10234785 10235625 10236690 10237980 10239495

823950 10329705 10329885 10330290 10330920 10331775 10332855 10334160 10335690

1282950 10426245 10426440 10426860 10427505 10428375 10429470 10430790 10432335

1843200 10523235 10523445 10523880 10524540 10525425 10526535 10527870 10529430

2504700 10620675 10620900 10621350 10622025 10622925 10624050 10625400 10626975

3267450 10718565 10718805 10719270 10719960 10720875 10722015 10723380 10724970

4131450 10816905 10817160 10817640 10818345 10819275 10820430 10821810 10823415

5096700 10915695 10915965 10916460 10917180 10918125 10919295 10920690 10922310

6163200 11014935 11015220 11015730 11016465 11017425 11018610 11020020 11021655

7330950 11114625 11114925 11115450 11116200 11117175 11118375 11119800 11121450

8599950 11214765 11215080 11215620 11216385 11217375 11218590 11220030 11221695

9970200 11315355 11315685 11316240 11317020 11318025 11319255 11320710 11322390

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Table 18: Player A vs Player B at 450th Iteration from Action A9 to A15

Player A Player B

A9 A10 A11 A12 A13 A14 A15

450 9955305 9957225 9959370 9961740 9964335 9967155 9970200

54450 10050165 10052100 10054260 10056645 10059255 10062090 10065150

209700 10145475 10147425 10149600 10152000 10154625 10157475 10160550466200 10241235 10243200 10245390 10247805 10250445 10253310 10256400

823950 10337445 10339425 10341630 10344060 10346715 10349595 10352700

1282950 10434105 10436100 10438320 10440765 10443435 10446330 10449450

1843200 10531215 10533225 10535460 10537920 10540605 10543515 10546650

2504700 10628775 10630800 10633050 10635525 10638225 10641150 10644300

3267450 10726785 10728825 10731090 10733580 10736295 10739235 10742400

4131450 10825245 10827300 10829580 10832085 10834815 10837770 10840950

5096700 10924155 10926225 10928520 10931040 10933785 10936755 10939950

6163200 11023515 11025600 11027910 11030445 11033205 11036190 11039400

7330950 11123325 11125425 11127750 11130300 11133075 11136075 11139300

8599950 11223585 11225700 11228040 11230605 11233395 11236410 11239650

9970200 11324295 11326425 11328780 11331360 11334165 11337195 11340450

Table 19: Player A vs Player B at 500th Iteration from Action A1 to A8

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8

500 11055845 11055965 11056310 11056880 11057675 11058695 11059940 11061410

60500 11161135 11161270 11161630 11162215 11163025 11164060 11165320 11166805

233000 11266925 11267075 11267450 11268050 11268875 11269925 11271200 11272700

518000 11373215 11373380 11373770 11374385 11375225 11376290 11377580 11379095

915500 11480005 11480185 11480590 11481220 11482075 11483155 11484460 11485990

1425500 11587295 11587490 11587910 11588555 11589425 11590520 11591840 11593385

2048000 11695085 11695295 11695730 11696390 11697275 11698385 11699720 11701280

2783000 11803375 11803600 11804050 11804725 11805625 11806750 11808100 11809675

3630500 11912165 11912405 11912870 11913560 11914475 11915615 11916980 11918570

4590500 12021455 12021710 12022190 12022895 12023825 12024980 12026360 120279655663000 12131245 12131515 12132010 12132730 12133675 12134845 12136240 12137860

6848000 12241535 12241820 12242330 12243065 12244025 12245210 12246620 12248255

8145500 12352325 12352625 12353150 12353900 12354875 12356075 12357500 12359150

9555500 12463615 12463930 12464470 12465235 12466225 12467440 12468880 12470545

11078000 12575405 12575735 12576290 12577070 12578075 12579305 12580760 12582440

Table 20: Player A vs Player B at 500th Iteration from Action A9 to A15

Player A Player B

A9 A10 A11 A12 A13 A14 A15

500 11063105 11065025 11067170 11069540 11072135 11074955 11078000

60500 11168515 11170450 11172610 11174995 11177605 11180440 11183500

233000 11274425 11276375 11278550 11280950 11283575 11286425 11289500

518000 11380835 11382800 11384990 11387405 11390045 11392910 11396000915500 11487745 11489725 11491930 11494360 11497015 11499895 11503000

1425500 11595155 11597150 11599370 11601815 11604485 11607380 11610500

2048000 11703065 11705075 11707310 11709770 11712455 11715365 11718500

2783000 11811475 11813500 11815750 11818225 11820925 11823850 11827000

3630500 11920385 11922425 11924690 11927180 11929895 11932835 11936000

4590500 12029795 12031850 12034130 12036635 12039365 12042320 12045500

5663000 12139705 12141775 12144070 12146590 12149335 12152305 12155500

6848000 12250115 12252200 12254510 12257045 12259805 12262790 12266000

8145500 12361025 12363125 12365450 12368000 12370775 12373775 12377000

9555500 12472435 12474550 12476890 12479455 12482245 12485260 12488500

11078000 12584345 12586475 12588830 12591410 12594215 12597245 12600500

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184 K. V. L. N. Acharyulu& Maddi. N. Murali Krishna

CONCLUSIONS

The competitor of player A i.e player B influences on all available actions of player A in each computation. It has utmost level of correlation among scientific computations. There is a step by step accuracy obtained from each iteration.

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.10 at each computation.

05

10x 10

4

1.115 1.12 1.125 1.13 1.135 1.14

x 106

-1

-0.5

0

0.5

1

Fig.1:Actions of Player B vs Player A:50th Iteration

Player APlayer B

Lines:Action B1

(Red)to

ActionB15

(Green)

1

2

3 x 105

2.26 2.27 2.28 2.29 2.3 2.31

x 106

-1

-0.5

0

0.5

1

Fig.2:Actions of Player B vs Player A:100th Iteration

Player APlayer B

Lines:

ActionB2

(Red)

to

ActionB15

(Green)

Figure 1: Influence of Player B on Player A at 50th Iteration Figure 2: Influence of Player B on Player A at 100th Iteration

23

45

6x 10

5

3.443.45

3.463.47

3.483.49

x 106

-1

-0.5

0

0.5

1

Fig.3:Actions of Player B vs Player A:150th Iteration

Player APlayer B

Lines:

Action B3(Red)

to

Action B15

(Green)

24

6 x 105

4.564.58 4.6

4.62

x 106

-1

-0.5

0

0.5

1

Fig.4:Actions of Player B vs Player A:200th Iteration

Player APlayer B

Lines:

Action B4

(Red)

to

Action B15

(Green)

Figure 3: Influence of Player B on Player A at 150th Iteration Figure 4: Influence of Player B on Player A at 200th Iteration

24

68

x 105

5.685.7

5.725.74

5.765.78

x 106

-1

-0.5

0

0.5

1

Fig.5:Actions of Player B vs Player A:250th Iteration

Player APlayer B

Lines:

Action B5

(Red)

to

Action B15(Green)

02

46

8 x 105

6.86.82

6.846.86

6.886.9

x 106

-1

-0.5

0

0.5

1

Fig.6:Actions of Player B vs Player A:300th Iteration

Player APlayer B

Lines:

Action B6(Red)to

Action B15(Green)

Figure 5: Influence of Player B on Player A at 250th Iteration Figure 6: Influence of Player B on Player A at 300th Iteration

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0

5

10x 10

5

7.97.95

88.05

x 106

-1

-0.5

0

0.5

1

Fig.7:Actions of Player B vs Player A:350th Iteration

Player APlayer B

Lines:ActionB7

(Red)

toActionB15

(Green)

24

68

1012

x 105

9.19.15

9.29.25

x 106

-1

-0.5

0

0.5

1

Fig.8:Actions of Player B vs Player A:400th Iteration

Player A

Player B

Lines:Action B8(Red)

toAction B15

(Green)

Figure 7: Influence of Player B on Player A at 350th Iteration Figure 8: Influence of Player B on Player A at 400th Iteration

24

68

1012

14

x 1051.025

1.031.035

1.041.045

x 107

-1

-0.5

0

0.5

1

Fig.9:Actions of Player B vs Player A:450th Iteration

Player APlayer B

Lines:Action B9(Red)to

Action B15(Green)

0.5

1

1.5 x 106

1.1451.15

1.1551.16

1.165x 10

7

-1

-0.5

0

0.5

1

Fig.10:Actions of Player B vs Player A:500th Iteration

Player APlayer B

Lines:Action B10

(Red)to

Action B15(Green)

Figure 9: Influence of Player B on Player A at 450th Iteration Figure 10: Influence of Player B on Player A at 500th Iteration

Each Action of Player A with the Competition of the Player B at all Iterations

0

5

10

15

x 106

0

5

10

15

x 106

-1

0

1

Player A Player B

Fig.11:Action A1 vs Player B at all Iterations

50

100

150

200

250300

350

400

450

500

Constant differences areidentified forboth Players(A1 Vs B)

0

5

10

15

x 106

05

1015

x 106

-1

0

1

Player A Player B

Fig.12:Action A2 vs Player B at all Iterations50

100

150

200

250

300350

400

450

500

Constant differences areidentified for both Players(A2 Vs B)

Figure 11: Effect of A1 on Player B at all Iterations Figure 12: Effect A2 on Player B at all Iterations

0

5

10

15

x 106

05

1015

x 106

-1

0

1

Player BPlayer A

Fig.13:Action A3 vs Player B at all Iterations50

100150

200

250

300

350

400

450

500

Constant differencesareidentifiedfor bothPlayers(A3Vs B)

0

5

10

15

x 106

0

5

10

15

x 106

-1

0

1

Player A Player B

Fig.14:Action A4 vs Player B at all Iterations

50

100

150

200

250

300

350

400

450

500

Constant differencesareidentifiedforbothPlayers(A4VsB)

Figure 13: Effect A3 on Player B at all Iterations Figure 14: Effect A4 on Player B at all Iterations

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0

5

10

15

x 106

0

5

10

15

x 106

-1

0

1

Player A Player B

Fig.15:Action A5 vs Player B at all Iterations

50

100

150

200

250

300

350

400

450

500

Constantdifferencesare

identifiedforbothPlayers(A5VsB)

0

5

10

15

x 106

0

5

10

15

x 106

-1

0

1

Player BPlayer A

Fig.16:Action A6 vs Player B at all Iterations

50

100

150

200

250

300

350

400

450

500

Constantdifferences areidentifiedfor bothPlayers(A6VsB)

Figure 15: Effect A5 on Player B at all Iterations Figure 16: Effect A6 on Player B at all Iterations

0

5

10

15

x 106

0

5

10

15

x 106

-1

0

1

Player A Player B

Fig.17:Action A7 vs Player B at all Iterations

50

250

150

200

250

300

350

400

450

500

ConstantdifferencesareidentifiedforbothPlayers(A7VsB)

0

5

10

15

x 106

0

5

10

15

x 106

-1

0

1

Player A Player B

Fig.18:Action A8 vs Player B at all Iterations50

150

200

250

250

300

350

400

450

500

Constant differencesareidentified forbothPlayers(A8Vs B)

Figure 17: Effect A7 on Player B at all Iterations Figure 18: Effect A8 on Player B at all Iterations

0

5

10

15

x 106

0

5

10

15

x 106

-1

0

1

Player A Player B

Fig.19:Action A9 vs Player B at all Iterations50

100

150

200

250

300

350

400

450

500

Constant differences areidentified for both Players(A9 VsB)

0

5

10

15

x 106

0

5

10

15

x 106

-1

0

1

Player A Player B

Fig.20:Action A10 vs Player B at all Iterations50

100

150

200

250

300350

400

450

500

Constant differences areidentified forbothPlayers(A10 VsB)

Figure 19: Effect A9 on Player B at all Iterations Figure 20: Effect A10 on Player B at all Iterations

0

5

10

15

x 106

0

5

10

15

x 106

-1

0

1

Player A Player B

Fig.21:Action A11 vs Player B at all Iterations

50

100

150

200

250

300

350

400

450

500

Constantdifferencesareidentified forbothPlayers(A11VsB)

0

5

10

15

x 106

0

5

10

15

x 106

-1

0

1

Player BPlayer A

Fig.22:Action A12 vs Player B at all Iterations

50

100

150200

250

300

350

400

450

500

Constantdifferencesareidentifiedfor bothPlayers(A12VsB)

Figure 21: Effect A11 on Player B at all Iterations Figure 22: Effect A12 on Player B at all Iterations

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0

5

10

15

x 106

0

5

10

15

x 106

-1

0

1

Player BPlayer A

Fig.23:Action A13 vs Player B at all Iterations50

100

150

200

250

300

350

400

450

500

Constant differences areidentifiedfor bothPlayers(A13 Vs B)

0

5

10

15

x 106

0

5

10

15

x 106

-1

0

1

Player BPlayer A

Fig.24:Action A14 vs Player B at all Iterations50

100

150

200

250

300

350

400

450

500

Constant differences areidentifiedforbothPlayers

(A14 Vs B)

Figure 23: Effect A13 on Player B at all Iterations Figure 24: Effect A14 on Player B at all Iterations

0

5

10

15

x 106

05

1015

x 106

-1

0

1

Fig.25:Action A15 of Player A vs Player B

Player BPlayer A

50

100

150

200

250

300

350400

450

500

Constant differences areidentified for both Players(A15 Vs B)

Figure 25: Effect A15 on Player B at all Iterations

CONCLUSIONS

From Fig.1-Fig.10:

Undifferentiated deviations occurred among the iterations. Ameliorate improvements have been discovered. Invariant rate of change exited in any two computations. Unwavering fluctuations are obtained.

From Fig.11- Fig.25:

Each available action of player A at any two successive computations has fixed contraventions. Systematic developments have been identified in both the players. Any two possible actions of player B have fixate variations.

OPTIMUM MIXIED STRATEGIES OF PLAYER A AND PLAYER B

The optimum mixed strategies of the playerA from the iteration 50-500 are uniquely obtained as

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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Similarly the optimum mixed strategies of the player B from the iteration 50-500

are also uniquely obtained as

B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12 B13 B14 B15

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

UPPER BOUNDS, LOWER BOUNDS AND ERRORS 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 and error

estimations are shown in the tables from Table (21) to Table (22).

Table 21

U.B Lower Bound

Iterations

50-50050 100 150 200 250 300 350 400 450 500

22156 21712.9 21934.45 22008.3 22045.23 22067.38 22082.15 22092.7 22100.61 22106.76 22111.69

22156 21715.3 21935.65 22009.1 22045.83 22067.86 22082.55 22093.04 22100.91 22107.03 22111.93

22156 21722.2 21939.1 22011.4 22047.55 22069.24 22083.7 22094.03 22101.78 22107.8 22112.62

22156 21733.6 21944.8 22015.2 22050.4 22071.52 22085.6 22095.65 22103.2 22109.06 22113.76

22156 21749.5 21952.75 22020.5 22054.38 22074.7 22088.25 22097.92 22105.18 22110.83 22115.35

22156 21769.9 21962.95 22027.3 22059.48 22078.78 22091.65 22100.84 22107.74 22113.1 22117.39

22156 21794.8 21975.4 22035.6 22065.7 22083.76 22095.8 22104.4 22110.85 22115.86 22119.88

22156 21824.2 21990.1 22045.4 22073.05 22089.64 22100.7 22108.6 22114.53 22119.13 22122.82

22156 21858.1 22007.05 22056.7 22081.53 22096.42 22106.35 22113.44 22118.76 22122.9 22126.21

22156 21896.5 22026.25 22069.5 22091.13 22104.1 22112.75 22118.92 22123.56 22127.16 22130.05

22156 21939.4 22047.7 22083.8 22101.85 22112.68 22119.9 22125.05 22128.93 22131.93 22134.34

22156 21986.8 22071.4 22099.6 22113.7 22122.16 22127.8 22131.82 22134.85 22137.2 22139.08

22156 22038.7 22097.35 22116.9 22126.68 22132.54 22136.45 22139.24 22141.34 22142.96 22144.2722156 22095.1 22125.55 22135.7 22140.78 22143.82 22145.85 22147.3 22148.39 22149.23 22149.91

22156 22156 22156 22156 22156 22156 22156 22156 22156 22156 22156

Table 22

Iterations

50 100 150 200 250 300 350 400 450 500

443.1 221.55 147.7 110.77 88.62 73.85 63.3 55.39 49.24 44.31

440.7 220.35 146.9 110.17 88.14 73.45 62.96 55.09 48.97 44.07

433.8 216.9 144.6 108.45 86.76 72.3 61.97 54.22 48.2 43.38

422.4 211.2 140.8 105.6 84.48 70.4 60.35 52.8 46.94 42.24

406.5 203.25 135.5 101.62 81.3 67.75 58.08 50.82 45.17 40.65

386.1 193.05 128.7 96.52 77.22 64.35 55.16 48.26 42.9 38.61

361.2 180.6 120.4 90.3 72.24 60.2 51.6 45.15 40.14 36.12

331.8 165.9 110.6 82.95 66.36 55.3 47.4 41.47 36.87 33.18

297.9 148.95 99.3 74.47 59.58 49.65 42.56 37.24 33.1 29.79

259.5 129.75 86.5 64.87 51.9 43.25 37.08 32.44 28.84 25.95

216.6 108.3 72.2 54.15 43.32 36.1 30.95 27.07 24.07 21.66

169.2 84.6 56.4 42.3 33.84 28.2 24.18 21.15 18.8 16.92

117.3 58.65 39.1 29.32 23.46 19.55 16.76 14.66 13.04 11.73

60.9 30.45 20.3 15.22 12.18 10.15 8.7 7.61 6.77 6.09

0 0 0 0 0 0 0 0 0 0

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A Scientific Computation on a Peculiar Case of Game Theory in Operations Research 189

CONCLUSIONS

The same optimum mixed pure strategies are endured for player A and player B in each computation. The value of the game is 22156.Because the optimal mini max strategies concur with the value at saddle position. The value of upper bound is unique and same as the value of the game at any stage of computation. At beginning,the value of lower bound of this game is not adequate to the value of the game.But it inclines to

turn to the value of the game at last the stage in the computation.

The error has found initially with 443.1 and it converges gradually to zero at final .The error value is 1/10 timesminimized from 50th iteration to 500th iteration at each possible action.

OVER ALL CONCLUSIONS

Indistinguishable optimum pure mixed strategies for the both players are incurred.

Moderate strategies for the players have been achieved adversely.

The game is investigated as a strictly determinable game. Since lower bound and upper bound are equal to valueof the game.

No disinclination is found. The obtained errors are declined step by step from iteration to iteration. Gyration of conflicts are observed in this special case.

ACKNOWLEGDEMENTS

The authors are thankful to the principal, HOD & the Faculty members of Dept. of M.C.A, Bapatla Engineering

College for their constant encouragement.

REFERENCES

1. K.V.L.N.Acharyulu and Maddi.N.Murali Krishna,(2013). Some Remarkable Results In Row And ColumnBoth Dominance Game With Rowns Algorithm, International Journal of Mathematics and Computer

Applications Research (IJMCAR),Vol.3,Issue.1,Mar 2013,pp.139-150 .

2. Billy E. Gillett (1979). Introduction to operations Research,Tata McGraw-Hill Edition.3. Levin, R. L., and R.B. Desjardins (1970).Theory of Games and Strategies, International Textbook Company,

Scranton, Pa.

4. Rapoport, A.(1966).Two Person Game Theory,The Essential Ideas, University of Michigan Press, Ann Arbor,Mich.

5. Dresher, M., Games of Strategy (1961). Theory and Applications,Prentice-Hall, Inc., Englewood Cliffs,N.J.6. Raiffa, R. D.(1958). Games and Decisions,John Wiley & Sons, Inc., NewYork.7. McKinsey,J.C.C.(1952).Introduction of the Theory of Games,McGraw-Hill Book Company, NewYork.

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