Game Theory

Assignment I

Q1.      Solve the following Game                 

Player A		Player B
	A1	20	80
	A2	40	30
	A3	50	60

Ans :   Value of Game = 50

Q2.      Solve the following Game

Firms X		Firm Y
		Y1	Y2	Y3
	X1	4	20	6
	X2	18	12	10

Ans :   Value of Game = 10

Q3.      A company management and labour union are negotiating a new three years settlement. Each party has four strategies these are

I           Hard and aggressive bargaining approach

II         Reasoning and Logical approach

III        Legalistic approach

IV        Conciliator approach

Union Strategy		Company Strategy
		I	II	III	IV
	I	20	15	12	35
	II	25	14	8	10
	III	40	02	10	05
	IV	-5	04	11	00

Ans :   Value of Game = 12

Q4.      Solve the following game matrix

Firm A		Firm B
		I	II	III	IV
	I	-2	-1	-2	8
	II	1	0	-1	-1
	III	-3	1	-3	1

Ans :   Value of Game = -1, Pure Strategy A-II & B-III

Q5.      Solve the following game whose Pay off matrix is

Firm A		Firm B
		I	II	III
	I	-2	-1	-2
	II	1	0	-1
	III	-3	1	-3

Ans :   Value of Game = 0, Pure Strategy A-II & B-II

Q6.      Shruti Ltd has developed a Sales forecasting function for its products and the products of its competitor, Purmima Ltd. These are four strategies S1,S2,S3 & S4 available to shruti Ltd and three strategies P1, P2 & P3 to Purnima Ltd. The Pay off corresponding to all the twelve combination of the strategies are given below. From the table we can see that for example, if strategy S1 is employed by Shruti Ltd & Strategy P1 by Purnima Ltd. Then these shall be gain of Rs 30,000 in quarterly sales to the former. Other entries can be similarly interpreted. Considering this information, State what would be the optimal strategy for shruti Ltd? Purima Ltd? What is the value of the game? Is the game Fair?

Shruti Strategy		Purnima Ltd. Strategy
		P1	P2	P3
	S1	30,000	-21,000	1,000
	S2	18,000	14,000	12,000
	S3	-6,000	28,000	4,000
	S4	18,000	6,000	2,000

Ans :   Value of Game =12,000, Pure Strategy Shruti –S2 & Purnima –P3

Q7.      Find the optimum strategies for two stores from the following Payoff matrix gain or loss of customers for StoreS

Action of Store I		Action of Store II
		A	B	C
	I	0	20	-60
	II	30	-10	-20
	III	70	-80	-30

Q8.      The following is the Payoff matrix table is given fine the value of game and optimal strategy for both of the firms

Firm A		Firm B
		B1	B2	B3	B4	B5
	A1	3	-1	4	6	7
	A2	-1	8	2	4	12
	A3	16	8	6	14	12
	A4	1	11	-4	2	1

Ans :   Value of Game =6

Q9       Using the dominance property obtain the optimal strategies for both the players and determine the value of the game.

Player A	Player B
		B1	B2	B3	B4	B5
	A1	2	4	3	8	4
	A2	5	6	3	7	8
	A3	6	7	9	8	7
	A4	4	2	8	4	3

Ans :   Value of Game = 6, Pure Strategy A-III & B-I

Q10     Using the dominance property obtain the optimal strategies for both the players and determine the value of the game.

Firm A		Firm B
		I	II	III	IV
	I	35	65	25	05
	II	30	20	15	00
	III	40	50	00	10
	IV	55	60	10	15

Ans :   Value of Game = 13

Q11     Using the dominance property obtain the optimal strategies for both the players and determine the value of the game.

Firm X		Firm Y
		I	II	III
	I	2	8	3
	II	6	2	8
	III	4	1	6

Ans :   Value of Game = 22/5

Q12     Solve the following Game

Firm P		Firm Q
		I	II	III	IV
	I	6	4	8	0
	II	6	8	4	8
	III	8	4	8	0
	IV	0	8	0	16

Q13.    Solve the game whose payoff matrix is given below:

Firm A		Firm B
		I	II	III	IV
	I	3	2	4	0
	II	3	4	2	4
	III	4	2	4	0
	IV	0	4	0	8

Ans :   Value of Game = 8/3

Q14.    Solve the game whose Payoff matrix is given below

Strategy of ABC		Strategy of XYZ
		Newspaper	Radio	Television
	Newspaper	30	40	-80
	Radio	0	15	-20
	Television	90	20	50

Ans :   Value of Game = 24,  p1=1/5, 1-p1=4/5, q1=13/15, 1-q2=2/15

Q15.    Two breakfast food manufactures ABC and XYZ are competing for an increased market share. The Pay off matrix shown in the following table, describes the increase in market share for ABC and decrease in market share of XYZ

Firm ABC		Firm XYZ
		Give Coupons	Decrease Prices	Maintain present strategy	Increase Advertising
	Give Coupons	2	-2	4	1
	Decrease Prices	6	1	12	3
	Maintain present strategy	-3	2	0	6
	Increase Advertising	2	-3	7	1

Determine the optimal strategy for both the manufacturers & value of the Game

Ans :   Value of Game = 3/2,  p1=1/2, 1-p1=1/2, q1=1/10, 1-q2=9/10

Q16.    Two competitors are competing for the market share of the similar product. The pay off matrix in terms of their advertising plan is show below

Company A		Company B
		No advertising	Medium Advertising	Heavy Advertising
	No advertising	10	5	-2
	Medium Advertising	13	12	15
	Heavy Advertising	16	14	10

Ans :   Value of Game = 90/7,  p1=4/7, 1-p1=3/7, q1=5/7, 1-q2=2/7

Q17.    Use Graphical method in solving the following game & fine the value of game

Firm A		Firm B
		I	II	III	IV
	I	2	2	3	-2
	II	4	3	2	6

Q18.    Obtain the optimal strategies for both persons and the value of the game for two person zero sum game whose payoff matrix is as follow:

Player A		Player B
		B1	B2
	A1	1	-3
	A2	3	5
	A3	-1	6
	A4	4	1
	A5	2	2
	A6	-5	0

Q19     Solve the following (2 x 3) game graphically

           

Firm A		Firm B
		I	II	III
	I	1	3	11
	II	8	5	2

Ans :   Value of Game = 49/11,  p1=3/11, p2=8/11, q1=0, q2=2/11, q3=9/11

Q20.    Obtain the optimal strategies for both persons and the value of the game for two person zero sum game whose payoff matrix is as follow:

Player A		Player B
		B1	B2
	A1	2	4
	A2	2	3
	A3	3	2
	A4	-2	6

Ans :   Value of Game = 8/3,  p1=1/3, p2=0 p3=2/3, p4=0, q1=0, q1=2/3, q3=1/3

Q21     Solve the following (2 x 4) game graphically

Firm A		Firm B
		I	II	III	IV
	I	2	2	3	-1
	II	4	3	2	6

Ans :   Value of Game = 5/2

Q22.    Obtain the optimal strategies for both persons and the value of the game for two person zero sum game whose payoff matrix is as follow:

Player A		Player B
		B1	B2
	A1	-6	7
	A2	4	-5
	A3	-1	-2
	A4	-2	5
	A5	7	-6

            Ans :   Value of Game = 23/20

Q23.    Obtain the optimal strategies for both persons and the value of the game for two person zero sum game whose payoff matrix is as follow:

Player A		Player B
		B1	B2
	A1	1	2
	A2	5	4
	A3	-7	9
	A4	-4	-3
	A5	2	1

Ans :   Value of Game = 73/17 A’s Strategy (0, 16/17,1/17,0,0)

  B’s Strategy (5/17, 12/17)

Q24     Solve the following (2 x 3) game graphically

Firm A		Firm B
		I	II	III
	I	6	4	3
	II	2	4	8

Ans :   Value of Game = 4

Q25.    Solve the following game graphically. Payoff matrix for player A is given

Firm b		Firm a
		I	II	III	IV
	I	-7	7	-4	8
	II	6	-4	-2	-6

Ans :   Value of Game = 7/12 A’s Strategy (11/24, 13/24, 0, 0)

  B’s Strategy (5/12, 7/12)

Q26     solve the following game by Graphic method where payoff matrix has been prepared for player A

Player B		Player A
		A1	A2
	B1	45	40
	B2	35	50
	B3	70	35
	B4	50	30

Ans :   Value of Game = 290/7 A’s Strategy (4/7, 3/7)

  B’s Strategy (0, 4/7, 0, 3/7)

Q27.    Solve the following Game

Firms X		Firm Y
		Y1	Y2	Y3
	X1	1	9	2
	X2	8	5	4

Ans :   Value of Game = 4

Q27.    Solve the following Game whose pay off matrix is

Firms X		Firm Y
		Y1	Y2	Y3
	X1	-3	-2	6
	X2	2	0	4
	X3	5	-2	-4

Ans :   Value of Game = 0

Q28.    Solve the following Game whose pay off matrix is

Firms X		Firm Y
		Y1	Y2
	X1	4	6
	X2	3	5

Ans :   Value of Game = 4

Q29.    Find the solution of the game whose Pay off matrix is given below

Firm A	Firm B
		I	II	III	IV	V
	I	-4	-2	-2	3	1
	II	1	0	-1	0	0
	III	-6	-5	-2	-4	4
	IV	3	1	-6	0	-8

Ans :   Value of Game = -1

Q30     Apply Dominance Property the following Payoff Matrix

Firm A		Firm B
		I	II	III	IV
	I	2	-4	-3	4
	II	4	-3	-4	2

Q31     Solve the game whose pay off matrix is

Firms X		Firm Y
		Y1	Y2	Y3
	X1	1	7	2
	X2	6	2	7
	X3	5	1	6

Ans :   Value of Game = 4     x’s Strategy (2/5, 3/5, 0)

  y’s Strategy (0, 1/2, 1/2)

Q32     Use the concept of dominance to solve the game

Firm A		Firm B
		I	II	III	IV
	I	3	2	4	0
	II	3	4	2	4
	III	4	2	4	0
	IV	0	4	0	8

Ans :   Value of Game = 4     A’s Strategy (0 0, 2/3,1/3)

B’s Strategy (0, 0, 2/3, 1/3)

Q33.       Solve the optimum strategy for P& Q and Value of the Game

Firms P		Firm Q
		Q1	Q2	Q3
	P1	0	-2	7
	P2	2	5	6
	P3	3	-3	8

Ans :       Value of Game = 21/9        P’s Strategy (0, 2/3, 1/3)

                 Q’s Strategy (8/9, 1/9,0)

Q34        Find the solution of the game whose Pay off matrix is given below

Firm A	Firm B
		I	II	III	IV	V
	I	1	3	2	7	4
	II	3	4	1	5	6
	III	6	5	7	6	5
	IV	2	0	6	3	1

Ans :       Value of Game = 5

Q35     Find the solution of the game whose Pay off matrix is given below

Firm A				Firm B
		I	II	III	IV	V	VI
	I	4	2	0	2	1	1
	II	4	3	1	3	2	2
	III	4	3	7	-5	1	2
	IV	4	3	4	-1	2	2
	V	4	3	3	-2	2	2

Ans :   Value of Game = 13/7            A’s Strategy (0, 6/7, 1/7, 0, 0)

             B’s Strategy (0, 0, 2/7, 5/7, 0)

Q36.    Solve the game whose payoff matrix

Firms A		Firm B
		B1	B2	B3
	A1	-4	3	-1
	A2	6	-4	-2

Ans :   Value of Game =-14/11          A’s Strategy (8/11, 3/11)

             B’s Strategy (1/11, 0, 10/11)

Q37.    Solve the following sub game

Player A		Player B
		B1	B2
	A1	1	8
	A2	3	5
	A3	11	2

Ans :   Value of Game =85/16           A’s Strategy (9/16, 0, 7/16)

             B’s Strategy (6/16, 10/16)

Q37.    Solve the following game by method of sub games whose Payoff matrix is

Player A		Player B
		B1	B2
	A1	6	5
	A2	3	6
	A3	8	4

Ans :   Value of Game =36/7             A’s Strategy (3/4, ¼, 0)

            B’s Strategy (1/4, ¾, 0)

Q38.    Solve the following game by method of sub games whose Payoff matrix is

Firms A		Firm B
		B1	B2	B3
	A1	-5	5	0
	A2	8	-4	-1

Ans :   Value of Game = -5/14           A’s Strategy (9/14, 5/14)

             B’s Strategy (1/15, 0, 14/15)

Q39.    Solve the following game by method of Graphically

Firm B		Firm A
		I	II	III	IV
	I	1	4	-2	-3
	II	2	1	4	5

Ans :   Value of Game = 7/4              A’s Strategy (1/4, 3/4)

             B’s Strategy (3/4, ¼, 0, 0)

Q40.    Solve the following game by method of Graphically

Firm A			Firm B
		I	II	III	IV	V
	I	-5	5	0	-1	8
	II	8	-4	-1	6	-5

Ans :   Value of Game = 5/4              A’s Strategy (9/14, 5/14)

             B’s Strategy (1/4, 0, 13/14, 0, 0)

Q41.    Solve the following game by method of Graphically

Player A		Player B
		B1	B2
	A1	-6	7
	A2	4	-5
	A3	-1	-2
	A4	-2	5
	A5	7	-6

Ans :   Value of Game = 23/20          A’s Strategy (0, 0, 0, 13/20, 7/20) 

             B’s Strategy (11/20, 9/20)

Q42.    Solve the following game by method of Graphically

Firm A		Firm A
		I	II	III	IV
	I	3	4	10	12
	II	8	4	3	2

Ans :   Value of Game = 4                 A’s Strategy (1, 0)

             B’s Strategy (0, 1, 0, 0)