ECON 383
Practice Problems from Chapter 6
10, 11, 12(a), 13(a), 15
H. K. Chen (SFU)
ECON 383
1 / 14
Chapter 6 — Ex.10(a)
10(a) Find all pure strategy Nash equilibria for the game below:
Player B
L
R
U 3,3 1,2
Player A
D 2,1 3,0
H. K. Chen (SFU)
ECON 383
2 / 14
Chapter 6 — Ex.10(a)
10(a) Find all pure strategy Nash equilibria for the game below:
Player B
L
R
U 3,3 1,2
Player A
D 2,1 3,0
BRA (L) = U
H. K. Chen (SFU)
ECON 383
2 / 14
Chapter 6 — Ex.10(a)
10(a) Find all pure strategy Nash equilibria for the game below:
Player B
L
R
U 3,3 1,2
Player A
D 2,1 3,0
BRA (L) = U
BRA (R) = D
H. K. Chen (SFU)
ECON 383
2 / 14
Chapter 6 — Ex.10(a)
10(a) Find all pure strategy Nash equilibria for the game below:
Player B
L
R
U 3,3 1,2
Player A
D 2,1 3,0
BRA (L) = U
BRA (R) = D
BRB (U ) = L
H. K. Chen (SFU)
ECON 383
2 / 14
Chapter 6 — Ex.10(a)
10(a) Find all pure strategy Nash equilibria for the game below:
Player B
L
R
U 3,3 1,2
Player A
D 2,1 3,0
BRA (L) = U
BRA (R) = D
BRB (U ) = L
BRB (D) = L
H. K. Chen (SFU)
ECON 383
2 / 14
Chapter 6 — Ex.10(a)
10(a) Find all pure strategy Nash equilibria for the game below:
Player B
L
R
U 3,3 1,2
Player A
D 2,1 3,0
BRA (L) = U
BRA (R) = D
BRB (U ) = L
BRB (D) = L
Therefore the only NE is (U, L)
H. K. Chen (SFU)
ECON 383
2 / 14
Chapter 6 — Ex.10(a)
10(a) Find all pure strategy Nash equilibria for the game below:
Player B
L
R
U 3,3 1,2
Player A
D 2,1 3,0
BRA (L) = U
BRA (R) = D
BRB (U ) = L
BRB (D) = L
Therefore the only NE is (U, L)
Can this game be solved using ISD?
H. K. Chen (SFU)
ECON 383
2 / 14
Chapter 6 — Ex.10(a)
10(a) Find all pure strategy Nash equilibria for the game below:
Player B
L
R
U 3,3 1,2
Player A
D 2,1 3,0
BRA (L) = U
BRA (R) = D
BRB (U ) = L
BRB (D) = L
Therefore the only NE is (U, L)
Can this game be solved using ISD?
Yes. First delete R for Player B, and then delete D for Player A.
H. K. Chen (SFU)
ECON 383
2 / 14
Chapter 6 — Ex.10(b)
10(b) Note that PA (U, L) = 3. Can you change this to a non-negative
number so that the resulting game has no pure strategy NE?
L
R
U 3,3 1,2
D 2,1 3,0
H. K. Chen (SFU)
ECON 383
3 / 14
Chapter 6 — Ex.10(b)
10(b) Note that PA (U, L) = 3. Can you change this to a non-negative
number so that the resulting game has no pure strategy NE?
L
R
U x,3 1,2
D 2,1 3,0
Need to change x to some non-negative number so that no pure
strategy NE exists
H. K. Chen (SFU)
ECON 383
3 / 14
Chapter 6 — Ex.10(b)
10(b) Note that PA (U, L) = 3. Can you change this to a non-negative
number so that the resulting game has no pure strategy NE?
L
R
U x,3 1,2
D 2,1 3,0
Need to change x to some non-negative number so that no pure
strategy NE exists
Recall that L is a strictly dominant strategy for Player B, so Player B
will always choose L
H. K. Chen (SFU)
ECON 383
3 / 14
Chapter 6 — Ex.10(b)
10(b) Note that PA (U, L) = 3. Can you change this to a non-negative
number so that the resulting game has no pure strategy NE?
L
R
U x,3 1,2
D 2,1 3,0
Need to change x to some non-negative number so that no pure
strategy NE exists
Recall that L is a strictly dominant strategy for Player B, so Player B
will always choose L
Changing the value of x is only going to affect Player A’s best
response to L
H. K. Chen (SFU)
ECON 383
3 / 14
Chapter 6 — Ex.10(b)
10(b) Note that PA (U, L) = 3. Can you change this to a non-negative
number so that the resulting game has no pure strategy NE?
L
R
U x,3 1,2
D 2,1 3,0
Need to change x to some non-negative number so that no pure
strategy NE exists
Recall that L is a strictly dominant strategy for Player B, so Player B
will always choose L
Changing the value of x is only going to affect Player A’s best
response to L
If x > 2, then BRA (L) = U; if x < 2, then BRA (L) = D; and if
x = 2, then BRA (L) = {U, D}
H. K. Chen (SFU)
ECON 383
3 / 14
Chapter 6 — Ex.10(b)
10(b) Note that PA (U, L) = 3. Can you change this to a non-negative
number so that the resulting game has no pure strategy NE?
L
R
U x,3 1,2
D 2,1 3,0
Need to change x to some non-negative number so that no pure
strategy NE exists
Recall that L is a strictly dominant strategy for Player B, so Player B
will always choose L
Changing the value of x is only going to affect Player A’s best
response to L
If x > 2, then BRA (L) = U; if x < 2, then BRA (L) = D; and if
x = 2, then BRA (L) = {U, D}
For all values of x, either (U, L) or (D, L) will always be a pure NE.
H. K. Chen (SFU)
ECON 383
3 / 14
Chapter 6 — Ex.10(b)
10(b) Note that PA (U, L) = 3. Can you change this to a non-negative
number so that the resulting game has no pure strategy NE?
L
R
U x,3 1,2
D 2,1 3,0
Need to change x to some non-negative number so that no pure
strategy NE exists
Recall that L is a strictly dominant strategy for Player B, so Player B
will always choose L
Changing the value of x is only going to affect Player A’s best
response to L
If x > 2, then BRA (L) = U; if x < 2, then BRA (L) = D; and if
x = 2, then BRA (L) = {U, D}
For all values of x, either (U, L) or (D, L) will always be a pure NE.
Therefore, it is impossible to achieve the requirement.
H. K. Chen (SFU)
ECON 383
3 / 14
Chapter 6 — Ex.10(c)
10(c) Note that PB (U, L) = 3. Can you change this to a non-negative
number so that the resulting game has no pure strategy NE?
L
R
U 3,3 1,2
D 2,1 3,0
H. K. Chen (SFU)
ECON 383
4 / 14
Chapter 6 — Ex.10(c)
10(c) Note that PB (U, L) = 3. Can you change this to a non-negative
number so that the resulting game has no pure strategy NE?
L
R
U 3,x 1,2
D 2,1 3,0
Need to change x to some non-negative number so that no pure
strategy NE exists
H. K. Chen (SFU)
ECON 383
4 / 14
Chapter 6 — Ex.10(c)
10(c) Note that PB (U, L) = 3. Can you change this to a non-negative
number so that the resulting game has no pure strategy NE?
L
R
U 3,x 1,2
D 2,1 3,0
Need to change x to some non-negative number so that no pure
strategy NE exists
Since U is a best response to L, we need to make L not a best
response to U; that is, to make R a strict best response to U
H. K. Chen (SFU)
ECON 383
4 / 14
Chapter 6 — Ex.10(c)
10(c) Note that PB (U, L) = 3. Can you change this to a non-negative
number so that the resulting game has no pure strategy NE?
L
R
U 3,1 1,2
D 2,1 3,0
Need to change x to some non-negative number so that no pure
strategy NE exists
Since U is a best response to L, we need to make L not a best
response to U; that is, to make R a strict best response to U
This happens when x < 2, for example, take x = 1
H. K. Chen (SFU)
ECON 383
4 / 14
Chapter 6 — Ex.10(c)
10(c) Note that PB (U, L) = 3. Can you change this to a non-negative
number so that the resulting game has no pure strategy NE?
L
R
U 3,1 1,2
D 2,1 3,0
Need to change x to some non-negative number so that no pure
strategy NE exists
Since U is a best response to L, we need to make L not a best
response to U; that is, to make R a strict best response to U
This happens when x < 2, for example, take x = 1
Since the best response to R is D, not U, the game now has no pure
strategy NE, as desired.
H. K. Chen (SFU)
ECON 383
4 / 14
Chapter 6 — Ex.11
11. Consider any two player game which has at least one pure strategy
NE. Explain why the strategies used in an equilibrium of this game will not
be (strictly) dominated strategies.
H. K. Chen (SFU)
ECON 383
5 / 14
Chapter 6 — Ex.11
11. Consider any two player game which has at least one pure strategy
NE. Explain why the strategies used in an equilibrium of this game will not
be (strictly) dominated strategies.
Suppose (s1∗ , s2∗ ) is a NE. Then by definition, s1∗ is a best response to
s2∗
H. K. Chen (SFU)
ECON 383
5 / 14
Chapter 6 — Ex.11
11. Consider any two player game which has at least one pure strategy
NE. Explain why the strategies used in an equilibrium of this game will not
be (strictly) dominated strategies.
Suppose (s1∗ , s2∗ ) is a NE. Then by definition, s1∗ is a best response to
s2∗
If s1∗ is a strictly dominated strategy, then there must exist another
strategy s10 such that
P1 (s10 , s2∗ ) > P1 (s1∗ , s2∗ )
H. K. Chen (SFU)
ECON 383
5 / 14
Chapter 6 — Ex.11
11. Consider any two player game which has at least one pure strategy
NE. Explain why the strategies used in an equilibrium of this game will not
be (strictly) dominated strategies.
Suppose (s1∗ , s2∗ ) is a NE. Then by definition, s1∗ is a best response to
s2∗
If s1∗ is a strictly dominated strategy, then there must exist another
strategy s10 such that
P1 (s10 , s2∗ ) > P1 (s1∗ , s2∗ )
But this contradicts the supposition that s1∗ is a best response to s2∗
H. K. Chen (SFU)
ECON 383
5 / 14
Chapter 6 — Ex.11
11. Consider any two player game which has at least one pure strategy
NE. Explain why the strategies used in an equilibrium of this game will not
be (strictly) dominated strategies.
Suppose (s1∗ , s2∗ ) is a NE. Then by definition, s1∗ is a best response to
s2∗
If s1∗ is a strictly dominated strategy, then there must exist another
strategy s10 such that
P1 (s10 , s2∗ ) > P1 (s1∗ , s2∗ )
But this contradicts the supposition that s1∗ is a best response to s2∗
Therefore, strategies in a NE cannot be strictly dominated.
H. K. Chen (SFU)
ECON 383
5 / 14
Chapter 6 — Ex.11
11. Consider any two player game which has at least one pure strategy
NE. Explain why the strategies used in an equilibrium of this game will not
be (strictly) dominated strategies.
Suppose (s1∗ , s2∗ ) is a NE. Then by definition, s1∗ is a best response to
s2∗
If s1∗ is a strictly dominated strategy, then there must exist another
strategy s10 such that
P1 (s10 , s2∗ ) > P1 (s1∗ , s2∗ )
But this contradicts the supposition that s1∗ is a best response to s2∗
Therefore, strategies in a NE cannot be strictly dominated.
Can NE contain weakly dominated strategies?
H. K. Chen (SFU)
ECON 383
5 / 14
Chapter 6 — Ex.12(a)
12(a) Find all pure NE for the game below. Do any of them use weakly
dominated strategies?
Player B
L
R
U 1,1 1,1
Player A
D 0,0 2,1
H. K. Chen (SFU)
ECON 383
6 / 14
Chapter 6 — Ex.12(a)
12(a) Find all pure NE for the game below. Do any of them use weakly
dominated strategies?
Player B
L
R
U 1,1 1,1
Player A
D 0,0 2,1
BRA (L) = U, BRA (R) = D
H. K. Chen (SFU)
ECON 383
6 / 14
Chapter 6 — Ex.12(a)
12(a) Find all pure NE for the game below. Do any of them use weakly
dominated strategies?
Player B
L
R
U 1,1 1,1
Player A
D 0,0 2,1
BRA (L) = U, BRA (R) = D
BRB (U ) = {L, R}, BRB (D) = R
H. K. Chen (SFU)
ECON 383
6 / 14
Chapter 6 — Ex.12(a)
12(a) Find all pure NE for the game below. Do any of them use weakly
dominated strategies?
Player B
L
R
U 1,1 1,1
Player A
D 0,0 2,1
BRA (L) = U, BRA (R) = D
BRB (U ) = {L, R}, BRB (D) = R
There are two NEs: (U, L) and (D, R)
H. K. Chen (SFU)
ECON 383
6 / 14
Chapter 6 — Ex.12(a)
12(a) Find all pure NE for the game below. Do any of them use weakly
dominated strategies?
Player B
L
R
U 1,1 1,1
Player A
D 0,0 2,1
BRA (L) = U, BRA (R) = D
BRB (U ) = {L, R}, BRB (D) = R
There are two NEs: (U, L) and (D, R)
Does either of the players have any dominated strategies?
H. K. Chen (SFU)
ECON 383
6 / 14
Chapter 6 — Ex.12(a)
12(a) Find all pure NE for the game below. Do any of them use weakly
dominated strategies?
Player B
L
R
U 1,1 1,1
Player A
D 0,0 2,1
BRA (L) = U, BRA (R) = D
BRB (U ) = {L, R}, BRB (D) = R
There are two NEs: (U, L) and (D, R)
Does either of the players have any dominated strategies?
L is a weakly dominated strategy for Player B
H. K. Chen (SFU)
ECON 383
6 / 14
Chapter 6 — Ex.12(a)
12(a) Find all pure NE for the game below. Do any of them use weakly
dominated strategies?
Player B
L
R
U 1,1 1,1
Player A
D 0,0 2,1
BRA (L) = U, BRA (R) = D
BRB (U ) = {L, R}, BRB (D) = R
There are two NEs: (U, L) and (D, R)
Does either of the players have any dominated strategies?
L is a weakly dominated strategy for Player B
So the NE (U, L) contains a weakly dominated strategy
H. K. Chen (SFU)
ECON 383
6 / 14
Chapter 6 — Ex.13(a)
13(a) Find all pure NEs of this three player (simultaneous-move) game:
Player 2
Player 2
L
R
L
R
U 4,4,4 0,0,1
U 2,0,0 1,1,1
Player 1
D 0,2,1 2,1,0
D 1,1,1 2,2,2
l
r
Player 3
H. K. Chen (SFU)
ECON 383
7 / 14
Chapter 6 — Ex.13(a)
13(a) Find all pure NEs of this three player (simultaneous-move) game:
Player 2
Player 2
L
R
L
R
U 4,4,4 0,0,1
U 2,0,0 1,1,1
Player 1
D 0,2,1 2,1,0
D 1,1,1 2,2,2
l
r
Player 3
Nash equilibrium in an n-player game
A strategy profile (s1 , . . . , si−1 , si , si+1 , . . . , sn ) is a Nash equilibrium if
every player i’s strategy si is a best response to the other players’
strategies (s1 , . . . , si−1 , si+1 , . . . , sn ). That is, for all i = 1, . . . , n,
Pi (s1 , . . . , si , . . . , sn ) ≥ Pi (s1 , . . . , si0 , . . . , sn )
holds for all si0 ∈ STRATi .
H. K. Chen (SFU)
ECON 383
7 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
l
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
r
Player 3
H. K. Chen (SFU)
ECON 383
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) =
BR1 (L, r) =
BR1 (R, l) =
BR1 (R, r) =
The notation BR1 (L, l) means “Player 1’s best response(s) when Player
2 chooses L and Player 3 chooses l”
H. K. Chen (SFU)
ECON 383
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
H. K. Chen (SFU)
BR1 (L, r) =
ECON 383
BR1 (R, l) =
BR1 (R, r) =
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
H. K. Chen (SFU)
BR1 (L, r) =
ECON 383
BR1 (R, l) =
BR1 (R, r) =
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
H. K. Chen (SFU)
BR1 (L, r) = U
ECON 383
BR1 (R, l) =
BR1 (R, r) =
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
H. K. Chen (SFU)
BR1 (L, r) = U
ECON 383
BR1 (R, l) =
BR1 (R, r) =
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
H. K. Chen (SFU)
BR1 (L, r) = U
ECON 383
BR1 (R, l) = D
BR1 (R, r) =
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
H. K. Chen (SFU)
BR1 (L, r) = U
ECON 383
BR1 (R, l) = D
BR1 (R, r) =
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
H. K. Chen (SFU)
BR1 (L, r) = U
ECON 383
BR1 (R, l) = D
BR1 (R, r) = D
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) =
BR2 (U, r) =
BR2 (D, l) =
BR2 (D, r) =
H. K. Chen (SFU)
ECON 383
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) =
BR2 (D, l) =
BR2 (D, r) =
H. K. Chen (SFU)
ECON 383
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) =
BR2 (D, l) =
BR2 (D, r) =
H. K. Chen (SFU)
ECON 383
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) = R
BR2 (D, l) =
BR2 (D, r) =
H. K. Chen (SFU)
ECON 383
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) = R
BR2 (D, l) =
BR2 (D, r) =
H. K. Chen (SFU)
ECON 383
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) = R
BR2 (D, l) = L
BR2 (D, r) =
H. K. Chen (SFU)
ECON 383
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) = R
BR2 (D, l) = L
BR2 (D, r) =
H. K. Chen (SFU)
ECON 383
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) = R
BR2 (D, l) = L
BR2 (D, r) = R
H. K. Chen (SFU)
ECON 383
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) = R
BR2 (D, l) = L
BR2 (D, r) = R
BR3 (U, L) = BR3 (U, R) =
H. K. Chen (SFU)
BR3 (D, L) =
ECON 383
BR3 (D, R) =
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) = R
BR2 (D, l) = L
BR2 (D, r) = R
BR3 (U, L) = l BR3 (U, R) =
H. K. Chen (SFU)
BR3 (D, L) =
ECON 383
BR3 (D, R) =
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) = R
BR2 (D, l) = L
BR2 (D, r) = R
BR3 (U, L) = l BR3 (U, R) =
H. K. Chen (SFU)
BR3 (D, L) =
ECON 383
BR3 (D, R) =
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) = R
BR2 (D, l) = L
BR2 (D, r) = R
BR3 (U, L) = l BR3 (U, R) = {l, r} BR3 (D, L) =
H. K. Chen (SFU)
ECON 383
BR3 (D, R) =
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) = R
BR2 (D, l) = L
BR2 (D, r) = R
BR3 (U, L) = l BR3 (U, R) = {l, r} BR3 (D, L) =
H. K. Chen (SFU)
ECON 383
BR3 (D, R) =
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) = R
BR2 (D, l) = L
BR2 (D, r) = R
BR3 (U, L) = l BR3 (U, R) = {l, r} BR3 (D, L) = {l, r} BR3 (D, R) =
H. K. Chen (SFU)
ECON 383
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) = R
BR2 (D, l) = L
BR2 (D, r) = R
BR3 (U, L) = l BR3 (U, R) = {l, r} BR3 (D, L) = {l, r} BR3 (D, R) =
H. K. Chen (SFU)
ECON 383
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) = R
BR2 (D, l) = L
BR2 (D, r) = R
BR3 (U, L) = l BR3 (U, R) = {l, r} BR3 (D, L) = {l, r} BR3 (D, R) = r
H. K. Chen (SFU)
ECON 383
8 / 14
Chapter 6 — Ex.13(a)
Player 1
U
D
Player 2
L
R
4,4,4 0,0,1
0,2,1 2,1,0
U
D
Player 2
L
R
2,0,0 1,1,1
1,1,1 2,2,2
l
r
Player 3
BR1 (L, l) = U
BR1 (L, r) = U
BR1 (R, l) = D
BR1 (R, r) = D
BR2 (U, l) = L
BR2 (U, r) = R
BR2 (D, l) = L
BR2 (D, r) = R
BR3 (U, L) = l BR3 (U, R) = {l, r} BR3 (D, L) = {l, r} BR3 (D, R) = r
There are two pure NEs: (U, L, l) and (D, R, r)
H. K. Chen (SFU)
ECON 383
8 / 14
Chapter 6 — Ex.15
15. Two identical firms, 1 and 2, must decide simultaneously and
independently whether to enter a new market and which product, A or B,
to produce.
If both enter and produce A, each loses 10 million
If both enter and produce B, each eanrs 5 million
If both enter but produce differen products, each earns 10 million
If a firm doesn’t enter, it earns zero profit
If only a single firm enters and produces A, it earns 15 million
If only a single firm enters and produces B, it earns 30 million
You are the manager of firm 1 and you have to choose a strategy for your
firm.
H. K. Chen (SFU)
ECON 383
9 / 14
Chapter 6 — Ex.15(a)
15(a) Use a two-player game to model this situation. Each firm has three
strategies, not enter (N), enter and produce A (A), enter and produce B
(B). Recall that the payoffs are...
If both enter and produce A, each loses 10 million
If both enter and produce B, each eanrs 5 million
If both enter but produce differen products, each earns 10 million
If a firm doesn’t enter, it earns zero profit
If only a single firm enters and produces A, it earns 15 million
If only a single firm enters and produces B, it earns 30 million
N
A
B
H. K. Chen (SFU)
N
0,0
A
ECON 383
B
10 / 14
Chapter 6 — Ex.15(a)
15(a) Use a two-player game to model this situation. Each firm has three
strategies, not enter (N), enter and produce A (A), enter and produce B
(B). Recall that the payoffs are...
If both enter and produce A, each loses 10 million
If both enter and produce B, each eanrs 5 million
If both enter but produce differen products, each earns 10 million
If a firm doesn’t enter, it earns zero profit
If only a single firm enters and produces A, it earns 15 million
If only a single firm enters and produces B, it earns 30 million
N
A
B
H. K. Chen (SFU)
N
0,0
A
0,15
ECON 383
B
10 / 14
Chapter 6 — Ex.15(a)
15(a) Use a two-player game to model this situation. Each firm has three
strategies, not enter (N), enter and produce A (A), enter and produce B
(B). Recall that the payoffs are...
If both enter and produce A, each loses 10 million
If both enter and produce B, each eanrs 5 million
If both enter but produce differen products, each earns 10 million
If a firm doesn’t enter, it earns zero profit
If only a single firm enters and produces A, it earns 15 million
If only a single firm enters and produces B, it earns 30 million
N
A
B
H. K. Chen (SFU)
N
0,0
A
0,15
ECON 383
B
0,30
10 / 14
Chapter 6 — Ex.15(a)
15(a) Use a two-player game to model this situation. Each firm has three
strategies, not enter (N), enter and produce A (A), enter and produce B
(B). Recall that the payoffs are...
If both enter and produce A, each loses 10 million
If both enter and produce B, each eanrs 5 million
If both enter but produce differen products, each earns 10 million
If a firm doesn’t enter, it earns zero profit
If only a single firm enters and produces A, it earns 15 million
If only a single firm enters and produces B, it earns 30 million
N
A
B
H. K. Chen (SFU)
N
0,0
15,0
A
0,15
ECON 383
B
0,30
10 / 14
Chapter 6 — Ex.15(a)
15(a) Use a two-player game to model this situation. Each firm has three
strategies, not enter (N), enter and produce A (A), enter and produce B
(B). Recall that the payoffs are...
If both enter and produce A, each loses 10 million
If both enter and produce B, each eanrs 5 million
If both enter but produce differen products, each earns 10 million
If a firm doesn’t enter, it earns zero profit
If only a single firm enters and produces A, it earns 15 million
If only a single firm enters and produces B, it earns 30 million
N
A
B
H. K. Chen (SFU)
N
0,0
15,0
A
0,15
-10,-10
ECON 383
B
0,30
10 / 14
Chapter 6 — Ex.15(a)
15(a) Use a two-player game to model this situation. Each firm has three
strategies, not enter (N), enter and produce A (A), enter and produce B
(B). Recall that the payoffs are...
If both enter and produce A, each loses 10 million
If both enter and produce B, each eanrs 5 million
If both enter but produce differen products, each earns 10 million
If a firm doesn’t enter, it earns zero profit
If only a single firm enters and produces A, it earns 15 million
If only a single firm enters and produces B, it earns 30 million
N
A
B
H. K. Chen (SFU)
N
0,0
15,0
A
0,15
-10,-10
ECON 383
B
0,30
10,10
10 / 14
Chapter 6 — Ex.15(a)
15(a) Use a two-player game to model this situation. Each firm has three
strategies, not enter (N), enter and produce A (A), enter and produce B
(B). Recall that the payoffs are...
If both enter and produce A, each loses 10 million
If both enter and produce B, each eanrs 5 million
If both enter but produce differen products, each earns 10 million
If a firm doesn’t enter, it earns zero profit
If only a single firm enters and produces A, it earns 15 million
If only a single firm enters and produces B, it earns 30 million
N
A
B
H. K. Chen (SFU)
N
0,0
15,0
30,0
A
0,15
-10,-10
ECON 383
B
0,30
10,10
10 / 14
Chapter 6 — Ex.15(a)
15(a) Use a two-player game to model this situation. Each firm has three
strategies, not enter (N), enter and produce A (A), enter and produce B
(B). Recall that the payoffs are...
If both enter and produce A, each loses 10 million
If both enter and produce B, each eanrs 5 million
If both enter but produce differen products, each earns 10 million
If a firm doesn’t enter, it earns zero profit
If only a single firm enters and produces A, it earns 15 million
If only a single firm enters and produces B, it earns 30 million
N
A
B
H. K. Chen (SFU)
N
0,0
15,0
30,0
A
0,15
-10,-10
10,10
ECON 383
B
0,30
10,10
10 / 14
Chapter 6 — Ex.15(a)
15(a) Use a two-player game to model this situation. Each firm has three
strategies, not enter (N), enter and produce A (A), enter and produce B
(B). Recall that the payoffs are...
If both enter and produce A, each loses 10 million
If both enter and produce B, each eanrs 5 million
If both enter but produce differen products, each earns 10 million
If a firm doesn’t enter, it earns zero profit
If only a single firm enters and produces A, it earns 15 million
If only a single firm enters and produces B, it earns 30 million
N
A
B
H. K. Chen (SFU)
N
0,0
15,0
30,0
A
0,15
-10,-10
10,10
ECON 383
B
0,30
10,10
5,5
10 / 14
Chapter 6 — Ex.15(b)
15(b) One of your employees argues that you should enter the market
(although he is not sure what product you should produce) because no
matter what firm 2 does, entering and producing product B is better than
not entering. Evaluate this argument.
N
A
B
H. K. Chen (SFU)
N
0,0
15,0
30,0
A
0,15
-10,-10
10,10
ECON 383
B
0,30
10,10
5,5
11 / 14
Chapter 6 — Ex.15(b)
15(b) One of your employees argues that you should enter the market
(although he is not sure what product you should produce) because no
matter what firm 2 does, entering and producing product B is better than
not entering. Evaluate this argument.
The employee is right. Firm 1 should not stay out of the market
because N is a strictly dominated strategy.
N
A
B
H. K. Chen (SFU)
N
0,0
15,0
30,0
A
0,15
-10,-10
10,10
ECON 383
B
0,30
10,10
5,5
11 / 14
Chapter 6 — Ex.15(c)
15(c) Another employee agrees with the person in part (b) and argues that
as strategy A could result in a loss (if the other firm also produces A) you
should enter and produce B. If both firms reason this way, and thus enter
and produce product B, will their play of the game form a Nash
equilibrium? Explain.
N
A
B
H. K. Chen (SFU)
N
0,0
15,0
30,0
A
0,15
-10,-10
10,10
ECON 383
B
0,30
10,10
5,5
12 / 14
Chapter 6 — Ex.15(c)
15(c) Another employee agrees with the person in part (b) and argues that
as strategy A could result in a loss (if the other firm also produces A) you
should enter and produce B. If both firms reason this way, and thus enter
and produce product B, will their play of the game form a Nash
equilibrium? Explain.
If firm 2 chooses B, what is firm 1’s best response?
N
A
B
H. K. Chen (SFU)
N
0,0
15,0
30,0
A
0,15
-10,-10
10,10
ECON 383
B
0,30
10,10
5,5
12 / 14
Chapter 6 — Ex.15(c)
15(c) Another employee agrees with the person in part (b) and argues that
as strategy A could result in a loss (if the other firm also produces A) you
should enter and produce B. If both firms reason this way, and thus enter
and produce product B, will their play of the game form a Nash
equilibrium? Explain.
If firm 2 chooses B, what is firm 1’s best response?
N
A
B
H. K. Chen (SFU)
N
0,0
15,0
30,0
A
0,15
-10,-10
10,10
ECON 383
A
B
0,30
10,10
5,5
12 / 14
Chapter 6 — Ex.15(c)
15(c) Another employee agrees with the person in part (b) and argues that
as strategy A could result in a loss (if the other firm also produces A) you
should enter and produce B. If both firms reason this way, and thus enter
and produce product B, will their play of the game form a Nash
equilibrium? Explain.
If firm 2 chooses B, what is firm 1’s best response?
A
Therefore, (B, B) is not a pair of mutual best responses, hence not
NE.
N
A
B
H. K. Chen (SFU)
N
0,0
15,0
30,0
A
0,15
-10,-10
10,10
ECON 383
B
0,30
10,10
5,5
12 / 14
Chapter 6 — Ex.15(d)
15(d) Find all the pure strategy Nash equilibria
N
A
B
N
0,0
0,15
0,30
A
15,0
-10,-10 10,10
B
30,0
10,10
5,5
H. K. Chen (SFU)
ECON 383
13 / 14
Chapter 6 — Ex.15(d)
15(d) Find all the pure strategy Nash equilibria
N
A
B
N
0,0
0,15
0,30
A
15,0
-10,-10 10,10
B
30,0
10,10
5,5
H. K. Chen (SFU)
ECON 383
13 / 14
Chapter 6 — Ex.15(d)
15(d) Find all the pure strategy Nash equilibria
N
A
B
N
0,0
0,15
0,30
A
15,0
-10,-10 10,10
B
30,0
10,10
5,5
H. K. Chen (SFU)
ECON 383
13 / 14
Chapter 6 — Ex.15(d)
15(d) Find all the pure strategy Nash equilibria
N
A
B
N
0,0
0,15
0,30
A
15,0
-10,-10 10,10
B
30,0
10,10
5,5
The two pure NEs are (B, A) and (A, B)
H. K. Chen (SFU)
ECON 383
13 / 14
Chapter 6 — Ex.15(e)
15(e) Another employee of your firm suggests merging the two firms and
deciding cooperatively on strategies so as to maximize the sum of profits.
Ignoring whether this merger would be allowed by the regulators, do you
think it’s a good idea? Explain.
H. K. Chen (SFU)
ECON 383
14 / 14
Chapter 6 — Ex.15(e)
15(e) Another employee of your firm suggests merging the two firms and
deciding cooperatively on strategies so as to maximize the sum of profits.
Ignoring whether this merger would be allowed by the regulators, do you
think it’s a good idea? Explain.
Without merger, joint profit of the two firms is 20
H. K. Chen (SFU)
ECON 383
14 / 14
Chapter 6 — Ex.15(e)
15(e) Another employee of your firm suggests merging the two firms and
deciding cooperatively on strategies so as to maximize the sum of profits.
Ignoring whether this merger would be allowed by the regulators, do you
think it’s a good idea? Explain.
Without merger, joint profit of the two firms is 20
With merger, two firms becomes two production plants of the merged
firm.
H. K. Chen (SFU)
ECON 383
14 / 14
Chapter 6 — Ex.15(e)
15(e) Another employee of your firm suggests merging the two firms and
deciding cooperatively on strategies so as to maximize the sum of profits.
Ignoring whether this merger would be allowed by the regulators, do you
think it’s a good idea? Explain.
Without merger, joint profit of the two firms is 20
With merger, two firms becomes two production plants of the merged
firm.
The management can decide to let one produce B while shutting the
other firm down.
H. K. Chen (SFU)
ECON 383
14 / 14
Chapter 6 — Ex.15(e)
15(e) Another employee of your firm suggests merging the two firms and
deciding cooperatively on strategies so as to maximize the sum of profits.
Ignoring whether this merger would be allowed by the regulators, do you
think it’s a good idea? Explain.
Without merger, joint profit of the two firms is 20
With merger, two firms becomes two production plants of the merged
firm.
The management can decide to let one produce B while shutting the
other firm down.
Doing so will result in a total profit of 30
H. K. Chen (SFU)
ECON 383
14 / 14
Chapter 6 — Ex.15(e)
15(e) Another employee of your firm suggests merging the two firms and
deciding cooperatively on strategies so as to maximize the sum of profits.
Ignoring whether this merger would be allowed by the regulators, do you
think it’s a good idea? Explain.
Without merger, joint profit of the two firms is 20
With merger, two firms becomes two production plants of the merged
firm.
The management can decide to let one produce B while shutting the
other firm down.
Doing so will result in a total profit of 30
Therefore merger is a good idea in terms of profit maximization.
H. K. Chen (SFU)
ECON 383
14 / 14