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                                               Submitted
to:-PROF.SONALI SINGH                                         Submitted by:AHMER HASSAN (JN170173)

                                                                              

                                       MANAGERIAL SCIENCE:- 2nd
trimester{2017}                                                ASSIGNMENT ON GAME THEORY   

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Q. Two companies are competing for the same
product. To improve its market share, company A decides to launch the following
strategies.

A1 = give discount coupons
A2 = home delivery services
A3 = free gifts

The company B decides to use media
advertising to promote its product.

B1 = internet
B2 = newspaper
B3 = magazine

 

 

                                 Company B

 

Company A

 

B1

B2

B3

A1

6

-5

3

A2

2

-3

-7

A3

    -2

      7            

     4

Use
linear programming to determine the best strategies for both the companies.

 

solution.

 

 
 
 
 
Company B

 
 
 
 
 
 
Minimum

Company A

 

B1

B2

B3

 

A1

6

-5

3

-5

A2

2

-3

-7

-7

A3

-2

7

4

-2

Maximum

 

6

7

4

 

 

Minimax=-2
Maximin = 4

This game has no saddle point. So
the value of the game lies between –2 and +4. It is possible that the value of
game may be negative or zero. Thus, a constant k is added to all the elements
of pay-off matrix. Let k = 4, then the given pay-off matrix becomes:

 

Company B

Company A

 

B1

B2

B3

A1

10

-1

7

A2

6

1

-3

A3

2

11

8

Let
V = value of the game
p1, p2 & p3 = probabilities of selecting
strategies A1, A2 & A3 respectively.
q1, q2 & q3 = probabilities of selecting
strategies B1, B2 & B3 respectively.

 

  

 

 
 
 
 
Company
B

 
 
 
 
 
Probability

Company A

 

B1

B2

B3

 

A1

10

-1

7

p1

A2

6

1

-3

p2

A3

2

11

8

p3

Probability

 

q1

q2

q3

 

 

Company A’s objective is to maximize the
expected gains, which can be achieved by maximizing V, i.e., it might gain more
than V if company B adopts a poor strategy. Hence, the expected gain for company
A will be as follows:

 

10p1+6p2+2p3?V
-p1+p2+11p3?V
7p1-3p2+8p3?V
p1+p2+p3=1
and p1, p2, p3 ? 0

Dividing the above constraints by V, we get

10p1/V + 6p2/V + 2p3/V ? 1
-p1/V + p2/V + 11p3/V ? 1
7p1/V – 3p2/V + 8p3/V ? 1
p1/V + p2/V + p3/V = 1/V

To simplify the problem, we put 

p1/V = x1, p2/V =
x2, p3/V = x3

In order to maximize V, company A can

Minimize 1/V = x1+ x2+ x3

subject to;

10×1 +
6×2 + 2×3 ? 1

-x1 +x2 + 11×3 ? 1

7×1 -3×2 + 8×3 ? 1

and x1, x2, x3 ? 0         

          Company B’s
objective is to minimize its expected losses, which can be reduced by
minimizing V, i.e., company A adopts a poor strategy. Hence, the expected loss
for company B will be as follows:

10q1 – q2 + 7q3 ? V

6q1 +q2 -3q3 ? V

2q1 + 11q2 + 8q3 ? V

q1 + q2 + q3 = 1

               and q1,
q2, q3 ? 0

Dividing the above constraints by V, we get

10q1/V – q2/V + 7q3/V ? 1

6q1/V +q2/V – 3q3/V ? 1

2q1/V +11q2/V + 8q3/V ? 1

q1/V + q2/V + q3/V = 1/V

 

To simplify the problem, we put 

q1/V = y1, q2/V =
y2, q3/V = y3

 

In order to minimize V, company B can 

 

Maximize 1/V = y1+ y2+ y3

subject to

10y1 – y2 + 7y3 ? 1

6y1 +y2 – 3y3 ? 1

2y1 +
11y2 + 8y3 ? 1

      and y1, y2, y3 ? 0

 

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