Mathematical Finance II (6CCM338A) spring 2026
Homework Problems 6CCM338A- Week 10
Problem 1. Consider the following zero-sum matrix game:
0 −1 2
A=
1 0 −3
−2 3 0
Find the set of all NE of the game. Find the value of the game.
Example solution. From the theorem in class, we have:
3 2 1
( , , )A ≥ v 1 1 · · · 1
6 6 6
⊤ ⊤
A 3 2 1 ≤ v 1 ··· 1 ,
6 6 6
⊤
∗ ∗ ⊤
Now, in this case x A = 0 0 0 and A(y ) = 0 0 0 So the value of the game is 0.
Problem 2. Consider the following zero-sum matrix game.
2 0 2 2
4 2 3 3
A=
0 −2 −1 −1
2 1 2 1
1. Calculate the payoff/utility of both player 1 and player 2 under the strategy ( 36 , 26 , 61 , 0), ( 12 , 61 , 16 , 16 )
2. By repeated row and column elimination (of only dominated strategy, not weakly dominated
strategy) reduce this matrix game to a 1 × 1 matrix game. (Hint: you may change this game
to a general non-zero sum game setup)
3. Show that {(0, 1, 0, 0), (0, 1, 0, 0)} is the only NE.
4. Compute the value of the game.
, Example solution. 1. Player 1’s payoff:
2 0 2 2 12
4 2 3 3 1
⊤ 3 2 1 6
A(x, y) = xAy = ( , , , 0)
6 6 6
0 −2 −1 −1 1
6
1
2 1 2 1 6
Player2’s payoff is −A(x, y).
2. Eliminate row 1,3,4 in any order. Also, eliminate columns 1, 3, 4 in any order.
ii
3. The previous game reduces to Ā = . So the only NE of this game is a pure NE and
ii 2,-2
given by σ ∗ = (x∗ , y ∗ ) = {(0, 1, 0, 0), (0, 1, 0, 0)}
4. From the previous section we have the value of the game to be 2. One can also see this in a
similar computation to problem 1.
Problem 3. Consider the following slightly modified Matching pennies:
H T
H 1,-1 -1,1
T -1,1 1,-1
1. Is this a zero-sum game?
2. Do we have a pure NE in this game?
3. Do we have any NE in this game?
4. If the value of this game is 0, find all the NE for this game.
Example solution.
Yes, in this case u1 = −u2 .
No. This can be seen by showing every pure strategy profile has a profitable deviation.
This is a finite game (the number of players is finite, and each player has finitely many strategies).
So Nash’s theorem says this game will have at least one (mixed/pure) NE.
Homework Problems 6CCM338A- Week 10
Problem 1. Consider the following zero-sum matrix game:
0 −1 2
A=
1 0 −3
−2 3 0
Find the set of all NE of the game. Find the value of the game.
Example solution. From the theorem in class, we have:
3 2 1
( , , )A ≥ v 1 1 · · · 1
6 6 6
⊤ ⊤
A 3 2 1 ≤ v 1 ··· 1 ,
6 6 6
⊤
∗ ∗ ⊤
Now, in this case x A = 0 0 0 and A(y ) = 0 0 0 So the value of the game is 0.
Problem 2. Consider the following zero-sum matrix game.
2 0 2 2
4 2 3 3
A=
0 −2 −1 −1
2 1 2 1
1. Calculate the payoff/utility of both player 1 and player 2 under the strategy ( 36 , 26 , 61 , 0), ( 12 , 61 , 16 , 16 )
2. By repeated row and column elimination (of only dominated strategy, not weakly dominated
strategy) reduce this matrix game to a 1 × 1 matrix game. (Hint: you may change this game
to a general non-zero sum game setup)
3. Show that {(0, 1, 0, 0), (0, 1, 0, 0)} is the only NE.
4. Compute the value of the game.
, Example solution. 1. Player 1’s payoff:
2 0 2 2 12
4 2 3 3 1
⊤ 3 2 1 6
A(x, y) = xAy = ( , , , 0)
6 6 6
0 −2 −1 −1 1
6
1
2 1 2 1 6
Player2’s payoff is −A(x, y).
2. Eliminate row 1,3,4 in any order. Also, eliminate columns 1, 3, 4 in any order.
ii
3. The previous game reduces to Ā = . So the only NE of this game is a pure NE and
ii 2,-2
given by σ ∗ = (x∗ , y ∗ ) = {(0, 1, 0, 0), (0, 1, 0, 0)}
4. From the previous section we have the value of the game to be 2. One can also see this in a
similar computation to problem 1.
Problem 3. Consider the following slightly modified Matching pennies:
H T
H 1,-1 -1,1
T -1,1 1,-1
1. Is this a zero-sum game?
2. Do we have a pure NE in this game?
3. Do we have any NE in this game?
4. If the value of this game is 0, find all the NE for this game.
Example solution.
Yes, in this case u1 = −u2 .
No. This can be seen by showing every pure strategy profile has a profitable deviation.
This is a finite game (the number of players is finite, and each player has finitely many strategies).
So Nash’s theorem says this game will have at least one (mixed/pure) NE.
