Submitted

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

MANAGERIAL SCIENCE:- 2nd

trimester{2017}

ASSIGNMENT ON GAME THEORY

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