Question

Two students in a Game Theory course plan to take an exam tomorrow. The professor seeks to create incentives for students to study, so he tells them that the student with the highest score will receive a grade of A and the one with the lower score will receive a B. Student 1’s score equals x1 + 1.5, where x1 denotes the amount of hours studying. Student 2’s score equals x2, where x2 is the hours she studies. Note that these score functions imply that, if both students study the same number of hours, x1 = x2, student 1 obtains the higher score, i.e., she is “the smarter of the two”. Assume, for simplicity, that the hours of studying for the game theory course is an integer number, and that they cannot exceed 5. The payoff to student i is 10 – xi if she gets an A and 8 – xi if she gets a B.

a) Which outcome(s) survive iterated deletion of strictly dominated strategies?

b) Which outcomes survive iterated deletion of weakly dominated strategies?

Answer 1

Solution:

As Xi (No of hours of study) is an integer <= 5. It can be 0,1,2,3,4,5

a) The payof matrix has 5 options. First we will reduce it to more simpler form i.e in 3x3 matrix.

For this we will find out the minimum payoff for any player if he doesn't put any effort. that means Xi=0.

The minimum payoff will be 8-Xi =8-0=8

Now, If he puts effort on study the maximum payoff will be 10-Xi ( Because maximum payoff when he get A)

Now if Xi = 3,4 or 5 the payoff will be 7,6 and 5 which is below the payoff of 8. So a student will better not to put any effort if Xi>2. So, no student will put effort Xi = 3,4 and 5. So we are left with Xi= 0,1 and 2

Now, as in the question the player one score is X1 + 1.5 and player 2 score is X2. If player 1 put X1=0 and player 2 also put X2=0 then score of player 1 will be 1.5 and player 2 will be 0. So, player 1 will get A and player B will get B. the payoff for player 1 will be 10-Xi= 10-0=10 and for player 2 will be 8-X2= 8-0=8. Like wise we can calculate for each posibility and the 3x3 payoff matrix will be as follow

		Player 2
		0	1	2
Player 1	0	10,8	10,7	8,8
	1	9,8	9,7	9,6
	2	8,8	8,7	8,6

Now in this matrix we can see for player 1, X1= 1 is strictly dominating X1=2 becase player 1 is getting payoff 9 which is better than payoff of 8 regardless of what player 2 effort is . So we an delete option 2 for player 1. Now for player 2, X2= 0 is strictly dominating X2=1 becase player 2 is getting payoff 8 which is better than payoff of 7 regardless of what player 1 effort is . So we an delete option 1 for player 2.

The new matrix will be

		Player 2
		0		2
Player 1	0	10,8		8,8
	1	9,8		9,6

Now, no startegy of any player is dominating another player startegy.

So we can say for player 1 X1=(0,1) will survive iterated deletion of strictly dominated strategies and for player 2 X2=(0,2) will survive iterated deletion of strictly dominated strategies.

b) Now as we know if a strategy is strictly dominating another strategy it will dominate it weakly also. So from part a we can directly reach into the following matrix

		Player 2
		0		2
Player 1	0	10,8		8,8
	1	9,8		9,6

Now in this matrix player 2, X2=0 is weakly dominating X2=1 because in X2=0, it is getting payoff of 8 regardless of what player 1 is doing but in X2=1 it is getting 6 if player 1 will choose X1= 1. So we can delete option 2 for player 2. Now player 1 is geeting payoff of 10 if he choose X1=0. So, player 1 will not choose X1=1 and will go for option X1=0. So, the new matrix will be

		Player 2
		0
Player 1	0	10,8

So we can say for player 1 X1=(0) will survive iterated deletion of weakly dominated strategies and for player 2 also X2=(0) will survive iterated deletion of weakly dominated strategies.