Question

	w	x	y	z
a	3,2	4,1	2,3	0,4
b	4,4	2,5	1,2	0,4
c	1,3	3,1	3,1	4,2
d	5,1	3,1	2,3	1,4

a) for this game, use iterated elimination of strictly dominated strategies. explain each step of your work.

b) what strategy profiles survive IESDS? what are the Nash equilibrium of this game?

Answer 1

a) Given that player 1 choose a, player 2's best response is z(4).
Given that player 1 choose b, player 2's best response is x(5).
Given that player 1 choose c, player 2's best response is w(3).
Given that player 1 choose d, player 2's best response is z(4).
So, we can see that player 2 never choose strategy y. Thus, IESDS eliminates strategy y. So, the game reduces to:

	w	x		z
a	3,2	4,1		0,4
b	4,4	2,5		0,4
c	1,3	3,1		4,2
d	5,1	3,1		1,4

Now, given that player 2 choose w, player 1's best response is d(5).
Given that player 2 choose x, player 1's best response is a(4).
Given that player 2 choose z, player 1's best response is c(4).
So, we can see that player 1 never choose strategy b. Thus, IESDS eliminates strategy b. So, the game reduces to:

	w	x		z
a	3,2	4,1		0,4
c	1,3	3,1		4,2
d	5,1	3,1		1,4

Again, given that player 1 choose a, player 2's best response is z(4).
Given that player 1 choose c, player 2's best response is w(3).
Given that player 1 choose d, player 2's best response is z(4).
So, we can see that player 2 never choose strategy x. Thus, IESDS eliminates strategy x. So, the game reduces to:

	w			z
a	3,2			0,4
c	1,3			4,2
d	5,1			1,4

Now, given that player 2 choose w, player 1's best response is d(5).
Given that player 2 choose z, player 1's best response is c(4).
So, we can see that player 1 never choose strategy a. Thus, IESDS eliminates strategy a. So, the game reduces to:

	w			z
c	1,3			4,2
d	5,1			1,4

Again, given that player 1 choose c, player 2's best response is w(3).
Given that player 1 choose d, player 2's best response is z(4).
So, we can see that player 2 never choose any single strategy. Thus, IESDS does not eliminate any strategy for player 2.
Similarly, given that player 2 choose w, player 1's best response is d(5).
Given that player 2 choose z, player 1's best response is c(4).
So, we can see that player 1 never choose any single strategy. Thus, IESDS does not eliminate any strategy for player 1.

b) Thus, the strategy profiles that survive IESDS are (c, w), (c, z), (d, w), and (d, z).

	w	z
c	1, 3	4, 2
d	5, 1	1, 4

Given that player 1 choose c, player 2's best response is w(3).
Given that player 1 choose d, player 2's best response is z(4).
Given that player 2 choose w, player 1's best response is d(5).
Given that player 2 choose z, player 1's best response is c(4).
Thus, there are no Nash equilibrium as best response of both players never occur simultaneously.