14.Maths - Ijmcar - Some Remarkable .Full

7/29/2019 14.Maths - Ijmcar - Some Remarkable .Full

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SOME REMARKABLE RESULTS IN ROW AND COLUMN BOTH DOMINANCE

GAME WITH BROWNS ALGORITHM

K.V.L.N.ACHARYULU1 &MADDI.N.MURALI KRISHNA21Faculty 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

The paper aims to investigate thoroughly a row and column both dominance game with the help of

Browns Algorithm. The dominant strategy is applied with a criterion in each row and each column for investigation. Some

fruitful results are ascertained by considering maximum number of possible iterations in the classical Java program .The

graphs are illustrated whenever necessary. The relations between the Lower bounds and/or Upper bounds of this game are

identified. The errors are found and their nature is established.

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

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

INTRODUCTION

In general, it is very difficult to find solutions for the games which have the size more than two. But,Brown's

algorithm is an efficient and effective method to find the optimum mixed strategies for any game irrespective of the size.

The phenomenon of this method is that the past is the best guide to the future. Considering the experiences occurred in the

past which will guide us to gain to move towards best optimum strategies of the players in future. Brown investigated this

method for any size of the game problem.

Billy E.Gillett[1] discussed this algorithm to compute any complicated problem of game theory in introduction to

optimum research in 1979. Levin and Desjardins[2] explained about the theory of games and its applications in

1970.Rapoport[3], Dresher[4],Raiffa[5], McKinsey[6] etc explicated with a detailed study on the strategies and

applications of game theory and afforded innovative ideas in the filed of operations research.

The paper is aimed to look into a 10x10 game with row and column both dominance properties. The investigation

based on the criteria with a small increment of one unit in a possible action of Player A with respect to the possible actions

of Player B,similarly largest increment ten units in a possible action of Player B with respect to the possible actions of

Player A. The maximum possible iterations have been computed to reach the best optimum mixed strategy. In this

connection, the iterations are started from minimum 50 th iteration and terminated with maximum 500th iteration. The

authors obtained some fruitful results in this special case of 10x10 game. The lower and upper bounds at each iteration are

found. The relation and correlation between these bounds are established at each iteration. The authors employed

the classified method of Brown's algorithm with the help of programming language of Java for this investigation. The

graphs are illustrated whenever necessary and feasible. The errors are found in each iteration and the nature of the

considered 10x10 game is identified. The relation between the actions of Player A and the actions of Player B are

discussed. The optimum mixed strategies for player A and player B are determined from 50th iteration to 500th iterations.

The correlations among the optimum mixed strategies have been obtained as maximum value.

International Journal of Mathematics and

Computer Applications Research (IJMCAR)

ISSN 2249-6955Vol. 3, Issue 1, Mar 2013, 139-150

TJPRC Pvt Ltd.


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

BASIC FORMATION OF 10X10 GAME

The game is formed with 10 rows and 10 columns according to player A and Player B. One player selects only

one single action from his/her set possible actions. It consists of ten possible actions of A i.e

A1,A2,A3,A4,A5,A6,A7,A8,A9,A10 which will influence on the other ten possible actions of player B i.e

B1,B2,B3,B4,B5,B6,B7,B8,B9,B10. It is better to consider the influence of Player B on the components of Player A with

maximum possible extent ten units.In this connection, It is considered step by step increment with one unit in Actions of

Player B and systematic increase with quantity ten units in actions of player A to meet the required criteria of row and

column dominance in the game.

The pay off matrix of constituted game having the size 10x10 is given below.

100999897969594939291

90898887868584838281

80797877767574737271

70696867666564636261

60595857565554535251

5049484746454443424140393837363534333231

30292827262524232221

20191817161514131211

10987654321

MATERIAL AND METHODS

The authors adopted Browns algorithm to solve this special case of 10x10 game in which row and columns both

dominated. Browns Algorithm:

Step 1: Player A chooses one of the possible actions(Ai1) from A1-A10 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 - A10 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 Bj 2

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 (Ai 3)

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 = =


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Some Remarkable Results in Row and Column both Dominance Game with Browns Algorithm 141

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 are where are calculated in step 5. The Process completes.

DISCUSSIONS AND RESULTS

This case is investigated with the above Browns Algorithm to get best strategies by considering 50th iteration to

500th iterations by Java program.The influences of player B on Player A are presented in the tabular forms from Table (1)

to Table (10) at each iteration.

Table 1: Player A vs Player B at 50th

Iteration

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10

50 4460 4470 4480 4490 4500 4510 4520 4530 4540 4550

550 4510 4520 4530 4540 4550 4560 4570 4580 4590 4600

1050 4560 4570 4580 4590 4600 4610 4620 4630 4640 4650

1550 4610 4620 4630 4640 4650 4660 4670 4680 4690 47002050 4660 4670 4680 4690 4700 4710 4720 4730 4740 4750

2550 4710 4720 4730 4740 4750 4760 4770 4780 4790 4800

3050 4760 4770 4780 4790 4800 4810 4820 4830 4840 4850

3550 4810 4820 4830 4840 4850 4860 4870 4880 4890 4900

4050 4860 4870 4880 4890 4900 4910 4920 4930 4940 4950

4550 4910 4920 4930 4940 4950 4960 4970 4980 4990 5000


Iteration

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10

100 9010 9020 9030 9040 9050 9060 9070 9080 9090 9100

1100 9110 9120 9130 9140 9150 9160 9170 9180 9190 9200

2100 9210 9220 9230 9240 9250 9260 9270 9280 9290 9300

3100 9310 9320 9330 9340 9350 9360 9370 9380 9390 9400

4100 9410 9420 9430 9440 9450 9460 9470 9480 9490 9500

5100 9510 9520 9530 9540 9550 9560 9570 9580 9590 9600

6100 9610 9620 9630 9640 9650 9660 9670 9680 9690 9700

7100 9710 9720 9730 9740 9750 9760 9770 9780 9790 9800

8100 9810 9820 9830 9840 9850 9860 9870 9880 9890 9900

9100 9910 9920 9930 9940 9950 9960 9970 9980 9990 10000


Iteration

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10

150 13560 13570 13580 13590 13600 13610 13620 13630 13640 136501650 13710 13720 13730 13740 13750 13760 13770 13780 13790 13800

3150 13860 13870 13880 13890 13900 13910 13920 13930 13940 13950

4650 14010 14020 14030 14040 14050 14060 14070 14080 14090 14100

6150 14160 14170 14180 14190 14200 14210 14220 14230 14240 14250

7650 14310 14320 14330 14340 14350 14360 14370 14380 14390 14400

9150 14460 14470 14480 14490 14500 14510 14520 14530 14540 14550

10650 14610 14620 14630 14640 14650 14660 14670 14680 14690 14700

12150 14760 14770 14780 14790 14800 14810 14820 14830 14840 14850

13650 14910 14920 14930 14940 14950 14960 14970 14980 14990 15000


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Iteration

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10

200 18110 18120 18130 18140 18150 18160 18170 18180 18190 18200

2200 18310 18320 18330 18340 18350 18360 18370 18380 18390 18400

4200 18510 18520 18530 18540 18550 18560 18570 18580 18590 18600

6200 18710 18720 18730 18740 18750 18760 18770 18780 18790 18800

8200 18910 18920 18930 18940 18950 18960 18970 18980 18990 19000

10200 19110 19120 19130 19140 19150 19160 19170 19180 19190 19200

12200 19310 19320 19330 19340 19350 19360 19370 19380 19390 19400

14200 19510 19520 19530 19540 19550 19560 19570 19580 19590 19600

16200 19710 19720 19730 19740 19750 19760 19770 19780 19790 19800

18200 19910 19920 19930 19940 19950 19960 19970 19980 19990 20000


Iteration

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10250 22660 22670 22680 22690 22700 22710 22720 22730 22740 22750

2750 22910 22920 22930 22940 22950 22960 22970 22980 22990 23000

5250 23160 23170 23180 23190 23200 23210 23220 23230 23240 23250

7750 23410 23420 23430 23440 23450 23460 23470 23480 23490 23500

10250 23660 23670 23680 23690 23700 23710 23720 23730 23740 23750

12750 23910 23920 23930 23940 23950 23960 23970 23980 23990 24000

15250 24160 24170 24180 24190 24200 24210 24220 24230 24240 24250

17750 24410 24420 24430 24440 24450 24460 24470 24480 24490 24500

20250 24660 24670 24680 24690 24700 24710 24720 24730 24740 24750

22750 24910 24920 24930 24940 24950 24960 24970 24980 24990 25000


Iteration

Player A Player BA1 A2 A3 A4 A5 A6 A7 A8 A9 A10

300 27210 27220 27230 27240 27250 27260 27270 27280 27290 27300

3300 27510 27520 27530 27540 27550 27560 27570 27580 27590 27600

6300 27810 27820 27830 27840 27850 27860 27870 27880 27890 27900

9300 28110 28120 28130 28140 28150 28160 28170 28180 28190 28200

12300 28410 28420 28430 28440 28450 28460 28470 28480 28490 28500

15300 28710 28720 28730 28740 28750 28760 28770 28780 28790 28800

18300 29010 29020 29030 29040 29050 29060 29070 29080 29090 29100

21300 29310 29320 29330 29340 29350 29360 29370 29380 29390 29400

24300 29610 29620 29630 29640 29650 29660 29670 29680 29690 29700

27300 29910 29920 29930 29940 29950 29960 29970 29980 29990 30000


Iteration

Player A Player BA1 A2 A3 A4 A5 A6 A7 A8 A9 A10

350 31760 31770 31780 31790 31800 31810 31820 31830 31840 31850

3850 32110 32120 32130 32140 32150 32160 32170 32180 32190 32200

7350 32460 32470 32480 32490 32500 32510 32520 32530 32540 32550

10850 32810 32820 32830 32840 32850 32860 32870 32880 32890 32900

14350 33160 33170 33180 33190 33200 33210 33220 33230 33240 33250

17850 33510 33520 33530 33540 33550 33560 33570 33580 33590 33600

21350 33860 33870 33880 33890 33900 33910 33920 33930 33940 33950

24850 34210 34220 34230 34240 34250 34260 34270 34280 34290 34300

28350 34560 34570 34580 34590 34600 34610 34620 34630 34640 34650

31850 34910 34920 34930 34940 34950 34960 34970 34980 34990 35000


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Iteration

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10

400 36310 36320 36330 36340 36350 36360 36370 36380 36390 36400

4400 36710 36720 36730 36740 36750 36760 36770 36780 36790 36800

8400 37110 37120 37130 37140 37150 37160 37170 37180 37190 3720012400 37510 37520 37530 37540 37550 37560 37570 37580 37590 37600

16400 37910 37920 37930 37940 37950 37960 37970 37980 37990 38000

20400 38310 38320 38330 38340 38350 38360 38370 38380 38390 38400

24400 38710 38720 38730 38740 38750 38760 38770 38780 38790 38800

28400 39110 39120 39130 39140 39150 39160 39170 39180 39190 39200

32400 39510 39520 39530 39540 39550 39560 39570 39580 39590 39600

36400 39910 39920 39930 39940 39950 39960 39970 39980 39990 40000


Iteration

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10

450 40860 40870 40880 40890 40900 40910 40920 40930 40940 40950

4950 41310 41320 41330 41340 41350 41360 41370 41380 41390 41400

9450 41760 41770 41780 41790 41800 41810 41820 41830 41840 41850

13950 42210 42220 42230 42240 42250 42260 42270 42280 42290 42300

18450 42660 42670 42680 42690 42700 42710 42720 42730 42740 42750

22950 43110 43120 43130 43140 43150 43160 43170 43180 43190 43200

27450 43560 43570 43580 43590 43600 43610 43620 43630 43640 43650

31950 44010 44020 44030 44040 44050 44060 44070 44080 44090 44100

36450 44460 44470 44480 44490 44500 44510 44520 44530 44540 44550

40950 44910 44920 44930 44940 44950 44960 44970 44980 44990 45000


Iteration

Player A Player B

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10500 45410 45420 45430 45440 45450 45460 45470 45480 45490 45500

5500 45910 45920 45930 45940 45950 45960 45970 45980 45990 46000

10500 46410 46420 46430 46440 46450 46460 46470 46480 46490 46500

15500 46910 46920 46930 46940 46950 46960 46970 46980 46990 47000

20500 47410 47420 47430 47440 47450 47460 47470 47480 47490 47500

25500 47910 47920 47930 47940 47950 47960 47970 47980 47990 48000

30500 48410 48420 48430 48440 48450 48460 48470 48480 48490 48500

35500 48910 48920 48930 48940 48950 48960 48970 48980 48990 49000

40500 49410 49420 49430 49440 49450 49460 49470 49480 49490 49500

45500 49910 49920 49930 49940 49950 49960 49970 49980 49990 50000

CONCLUSIONS

It is observed that the correlation will be maxmium among at a possible action of Player B and possible actions ofPlayer of A .

A consistent and constant increase has been identified from iteration to iteration. The influence of Player B uniformly effects on the possible action of PlayerA in each iteration.


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EFFECTS OF PLAYER B ON THE POSSIBLE ACTIONS OF PLAYER A

The influence of Player B on the possible actions of Player A are depicted from Fig.1 to Fig.10 at each Iteration as

shown below.

0 1000 2000 3000 4000 5000

4400

4500

4600

4700

4800

4900

5000

Player A

P

la

y

e

r

B

Fig.1 : Player A vs Player B - 50th Iterative values

Lines:Red(Action B1) to Green(Action B10)

1500 2000 2500 3000 3500 4000 4500

9200

9250

9300

9350

9400

9450

9500

9550

Player A

P

la

y

e

r

B


Lines: Red(Action B1) to Green(Action B10)

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

4500 5000 5500 6000 6500 7000 75001.405

1.41

1.415

1.42

1.425

1.43

1.435

1.44x 10

4

Player A

P

la

y

e

r

B



0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35

x 104

1.91

1.92

1.93

1.94

1.95x 10

4 Fig.4 : Player A vs Player B - 200th Iterative values

P

la

y

e

r

B

Player A



1 1.1 1.2 1.3 1.4 1.5

x 104

2.37

2.38

2.39

2.4

2.41

2.42

x 104 Fig.5 : Player A vs Player B - 250th Iterative values

Player A

P

la

y

e

r

B


1.3 1.35 1.4 1.45 1.5 1.55 1.6

x 104

2.85

2.855

2.86

2.865

2.87

2.875

2.88

x 104 Fig.6 : Player A vs Player B - 300th Iterat ive values

Player A

P

la

y

e

r

B




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1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45

x 104

3.29

3.295

3.3

3.305

3.31

3.315

3.32

3.325

x 104 Fig.7 : Player A vs Player B -350th Iterative values

Player A

P

la

y

e

r

B


1.85 1.9 1.95 2 2.05 2.1 2.15 2.2

x 104

3.81

3.82

3.83

3.84

3.85

x 104 Fig.8 : Player A vs Player B - 400th Iterative values

Player A

P

la

y

e

r

B



1.3 1.4 1.5 1.6 1.7 1.8

x 104

4.21

4.22

4.23

4.24

4.25

4.26

x 104

Player A


P

la

y

e

r

B


1.8 1.9 2 2.1 2.2 2.3 2.4x 10

4

4.72

4.73

4.74

4.75

4.76

4.77

4.78 x 104 Fig.10 : Player A vs Player B - 500th Iterative values

Player A

P

la

y

e

r

B



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

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0

1

2

3

4

5

x 104

Player A

P

la

y

e

r

B

Action A1 of Player A vs Player B

50

100

150

200

250

300

350

400

450

500

Constant difference areidentified for both Players(A1 Vs B)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0

1

2

3

4

5x 10

4

Player A

P

la

y

e

r

B


50

100

150

200

250

300

350

400

450

500


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


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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0

1

2

3

4

5x 10

4

Player A

P

la

y

e

r

B


50

100

150

200

250

300

350

400

450

500


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0

1

2

3

4

5x 10

4

Player A

P

la

y

e

r

B


50

100

150200

250

300

350

400

450

500

Constant difference areidentified for bothPlayers(A4 Vs B)


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0

1

2

3

4

5x 10

4

Player A

P

la

y

e

r

B


50

100

150

200

250

300

350

400

450

500

Constant difference are

identified for both Players(A5 Vs B)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0

1

2

3

4

5x 10

4

Pla er A

P

la

y

e

r

B


50

100

150

200

250

300

350

400

450

500

Constant difference are

identified for both Players(A6 Vs B)


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0

1

2

3

4

5x 10

4

Player A

P

la

y

e

r

B


50

100

150

200

250300

350

400

450

500


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0

1

2

3

4

5x 10

4

Player A

P

la

y

e

r

B


50

100

150

200

250

300

350

400

450

500




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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0

1

2

3

4

5x 10

4

Player A

P

la

y

e

r

B


50

100

150

200250

300

350

400

450

500


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0

1

2

3

4

5x 10

4

Player A

P

la

y

e

r

B


50

100

150

200250

300

350

400

450

500



CONCLUSIONS

From Figure 1-Figure 10

The variations among the all iterations are uniform. No fluctuations are identified. Periodic developments have been established. The rate of change between any two iterations is identical

From Figure 11-Figure 20

Constant differences between the values of possible actions of player A at any two consequent iterationshave been determined

Constant differences between the values of possible actions of player B at any two consequent iterationshave been ascertained.

Taxonomical improvement between the players is discovered.OPTIMUM MIXIED STRATEGIES OF PLAYER A AND PLAYER B

The optimum mixed strategies of Player A and Player B are illustrated and shown in table 11

Table 11

Opitimum Mixted Strategies of Player A and Player (Iteration Wise)

50 100 150 200 250 300 350 400 450 500

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

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

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

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

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


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Table-11;Contd.,

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

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

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

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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

UPPER BOUNDS AND LOWER BOUNDS AT ALL ITERATIONS

At each play of the game the minimum 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 maximum 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 10 x 10 game are tabulated in Table-12.

Table 12

U.BLower Bounds

Iterations 50 100 150 200 250 300 350 400 450 500

50-500

91 89.2 90.1 90.4 90.55 90.64 90.7 90.742 90.775 90.8 90.82

91 89.4 90.2 90.46 90.6 90.68 90.73 90.771 90.8 90.82 90.84

91 89.6 90.3 90.53 90.65 90.72 90.767 90.8 90.825 90.845 90.86

91 89.8 90.4 90.6 90.7 90.76 90.8 90.82 90.85 90.86 90.88

91 90 90.5 90.67 90.75 90.8 90.83 90.85 90.875 90.89 90.9

91 90.2 90.6 90.73 90.8 90.84 90.86 90.88 90.9 90.91 90.92

91 90.4 90.7 90.8 90.85 90.88 90.9 90.91 90.925 90.93 90.94

91 90.6 90.8 90.86 90.9 90.92 90.934 90.94 90.95 90.955 90.96

91 90.8 90.9 90.934 90.95 90.96 90.967 90.97 90.975 90.977 90.98

91 91 91 91 91 91 91 91 91 91 91

CONCLUSIONS

The optimum mixed strategies of player A and player B are unique in any iteration. The value of the game is 91.since maximum of row minima coincides with minimum of column

maxima.

The value of upper bound is as same as the value of the game at any level of iteration. The value of lower bound of this game is initially not equal to the value of the game but it tends towards

to become the value of the game at final stage in any iteration.

The error has occurred initially with 0.8 and it will be minimised step by step.


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With the maximum possible computed iterations, the error will be vanished.OVER ALL CONCLUSIONS

The optimum mixed strategies of the players in the conceived game are the best strategies. The row and column both dominance game will become a strictly determinable game because lower

bound and upper bound are same as value of the game.

No vacillations are obtained. The errors are gradually denigrated. Conservative strategies between opponent Players are accomplished. The revolution of the conflict is obtained in this game.

ACKNOWLEGDEMENTS

The authors are grateful to HOD and the Faculty members of Dept. of M.C.A, Bapatla Engineering College for

their affectionate cooperation and constant encouragement.The authors are also thankful to Sri.J.Ravi Kiran for his support

and suggestions.

REFERENCES

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

Company, Scranton, Pa.

3. Rapoport, A.(1966).Two Person Game Theory,The Essential Ideas, University of Michigan Press, AnnArbor, Mich.

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

5. Raiffa, R. D.(1958). Games and Decisions,John Wiley & Sons, Inc., NewYork.6. McKinsey,J.C.C.(1952).Introduction of the Theory of Games,McGraw-Hill Book Company, NewYork.


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14.Maths - Ijmcar - Some Remarkable .Full