程序代写代做代考 game C Question 1 (15 points): Consider the situation represented by the following bimatrix:

Question 1 (15 points): Consider the situation represented by the following bimatrix:
L
M
R
U
1,0
2,5
-2,-1
D
2,1
2,1
-3,2
1

(a) (5 points) Is there a strictly dominant strategy equilibrium of this game? Explain.
Answer: No. Neither player has a strictly dominant strategy. To see this, let us write down the best responses for each player. For player 1, we have
Therefore, player 1 does not have a strictly or weakly dominant strategy For player 2, we have
Note that M weakly dominates L but player 2 does not have a strictly or weakly domi- nant strategy. This follows because
u2(U,M) = 5 > u2(U,R) = 1 u2(D,M) = 1 3a)a<1and 21 3 > 2+2a)a<2 D is a strictly dominant strategy for player 2 if and only if: 1+a > 3a)a<1and 21 3 > 2+2a)a<2 These conditions are the same for the two players since the game is symmetric). Thus for a 2 [0, 1 ) the game is a Prisoners¡¯ Dilemma game. 2 When the game is not a Prisoners¡¯ Dilemma game, i.e., a 1 , there are two cases. (i) If a = 1, the game table is as follows: 2 In this case all the pure strategy profiles are Nash Equilibria, i.e., there are four pure strategy Nash Equilibria: (D,D), (D,C), (C,D), (C,C). (ii) If a > 1, C is now a strictly dominant strategy for both players, so there is only 2
D
C
D
1 + a, 1 + a
3, 3a
C
3a, 3
2+2a,2+2a
2
D
C
D
3, 3 22
3, 3
2
C
3,3 2
3,3
one Nash Equilibrium: (C,C).
7

Question 4 (5 points)
Find all the pure strategy Nash Equilibria of the following game.
Answer: The two pure strategy Nash Equilibria are (T,L) and (M,C).
First, B is strictly dominated by both T and M for Player 1. To see this note that
L
C
R
T
2,3
1,3
5,1
M
2,3
2,4
6,0
B
1,2
0,5
1,5
and
BR1(L) = T, M BR1(C) = M BR1(R) = M
BR2(T) = L, C BR2(M) = C BR2(B) = C, R
u1(T,L) = 2 > u1(B,L) = 1
u1(M,L) = 2 > u1(B,L) = 1
u1(T,C) = 2 > u1(B,C) = 0
u1(M,C) = 2 > u1(B,C) = 0
u1(T,R) = 5 > u1(B,R) = 1
u1(M,R) = 6 = u1(B,R) = 1
Now note that after eliminating B for player 1, R becomes strictly dominated for player 2 in the following game by both C and R
After eliminating R, we have
In the reduced 2×2 game it can be easily seen that there are two pure strategy Nash Equilibria: (T,L) and (M,C).
8
L
C
R
T
2,3
1,3
5,1
M
2,3
2,4
6,0
L
C
T
2,3
1,3
M
2,3
2,4