Question

1\2 A B с D W X 5,6 4,20 2, 11 8,8 5,10 18,2 12,0 5,8 Y 7,3 10,2 4,6 3, 16 Z 7, 11 5, 10 6,1 10,6 QUESTION 2. (35 pts) Using the process of iterated dominance, find the rationalizable set for this game. Carefully (justifying fully each step you take) the process of arriving at the rationalizable set. If I cannot follow what you are doing, you will not get full credit on this question.

Answer 1

Answer:

There is no pure strategy Nash equilibrium in this game:

The initial matrix of payoffs:

		2
		W	X	Y	Z
1	A	5, 6	4, 20	7, 3	7, 11
	B	2, 11	8, 8	10, 2	5, 6
	C	5, 10	18, 2	4, 6	6, 1
	D	12, 0	5, 8	3, 16	10, 6

We see that Player 1 will never play A. So we can eliminate the first row, A:

		2
		W	X	Y	Z
1	A	5, 6	4, 20	7, 3	7, 11
	B	2, 11	8, 8	10, 2	5, 6
	C	5, 10	18, 2	4, 6	6, 1
	D	12, 0	5, 8	3, 16	10, 6

Now, Player 2 will never play Z, it can be eliminated:

		2
		W	X	Y	Z
1	A	5, 6	4, 20	7, 3	7, 11
	B	2, 11	8, 8	10, 2	5, 6
	C	5, 10	18, 2	4, 6	6, 1
	D	12, 0	5, 8	3, 16	10, 6

Out of the remaining cells. Player 1 will never play C. Let us eliminate C:

		2
		W	X	Y	Z
1	A	5, 6	4, 20	7, 3	7, 11
	B	2, 11	8, 8	10, 2	5, 6
	C	5, 10	18, 2	4, 6	6, 1
	D	12, 0	5, 8	3, 16	10, 6

Next, Player 2 will not play X. It can be eliminated:

		2
		W	X	Y	Z
1	A	5, 6	4, 20	7, 3	7, 11
	B	2, 11	8, 8	10, 2	5, 6
	C	5, 10	18, 2	4, 6	6, 1
	D	12, 0	5, 8	3, 16	10, 6

Now, Player 1 will not play B for Player 2's W, and D for Player 2's Y. They can be eliminated:

		2
		W	X	Y	Z
1	A	5, 6	4, 20	7, 3	7, 11
	B	2, 11	8, 8	10, 2	5, 6
	C	5, 10	18, 2	4, 6	6, 1
	D	12, 0	5, 8	3, 16	10, 6

Player 2 will never play Y for Player 1's B and W for Player 1's D. These can also be eliminated:

		2
		W	X	Y	Z
1	A	5, 6	4, 20	7, 3	7, 11
	B	2, 11	8, 8	10, 2	5, 6
	C	5, 10	18, 2	4, 6	6, 1
	D	12, 0	5, 8	3, 16	10, 6

Now there is nothing left for the players to check for elimination.

There is no pure strategy Nash equilibrium to this game.