Problem Set 4: Due in class on Tuesday July 28. Solutions to this homework will be posted right after

class hence no late submissions will be accepted. Test 4 on the content of this homework will be given

on August 4 at 9:00am sharp.

Problem 1 (4p)

Consider the following game:

(a) Suppose that the Column player announces that he will play X with probability 0.5 and Y

with probability 0.5 i.e., ½ X ½ Y. Identify all best response strategies of the Row

player, i.e., BR(½ X ½ Y) ?

(b) Identify all best response strategies of the Column player to Row playing ½ A ½ B,

i.e. BR(½ A ½ B)?

(c) What is BR(1/5 X 1/5 Y 3/5 Z)?

(d) What is BR(1/5 A 1/5 B 3/5 C)?

X

Y

Z

A

2

1

1

3

5

-2

B

4

-1

2

1

1

2

C

0

4

3

0

2

1

Page 2 of 4

Problem 2 (4p) Here comes the Two-Finger Morra game again:

C1

C2

C3

C4

R1

0

0

-2

2

3

-3

0

0

R2

2

-2

0

0

0

0

-3

3

R3

-3

3

0

0

0

0

4

-4 R4

0

0

3

-3

-4

4

0

0

To exercise notation and concepts involved in calculating payoffs to mixed strategies, calculate

the following (uR, uC stand for the payoffs to Row and Column respectively):

(a) uR(0.4 R1 0.6 R2, C2) =

(b) uC(0.4 C1 0.6 C2, R3) =

(c) uR(0.3 R2 0.7 R3, 0.2 C1 0.3 C2 0.5 C4 ) =

(d) uC(0.7 C2 0.3 C4, 0.7 R1 0.2 R2 0.1 R3) =

Problem 3 (4p)

X

Y

A

1

6

3

1

B

2

3

0

4

For the game above:

(1) Draw the best response function for each player using the coordinate system below.

Mark Nash equilibria on the diagram.

Page 3 of 4

(2) List the pair of mixed strategies in Nash equilibrium.

(3) Calculate each player’s payoffs in Nash equilibrium.

Problem 4 (4p)

C1 C2

C3

C4

R1

0

0

-2

2

3

-3

0

0

R2

2

-2

0

0

0

0

-3

3

R3

-3

3

0

0

0

0

4

-4 R4

0

0

3

-3

-4

4

0

0

In the Two-Finger morra game above suppose Row decided to play a mix of R1 and R2 and

Column decided to play a mix of C1 and C3. In other words, assume that the original 44 game

is reduced to the 22 game with R1 and R2 and C1 and C3. Using our customary coordinate

system:

(a) Draw the best response functions of both players in the coordinate system as above.

(b) List all Nash equilibria in the game.

(c) Calculate each player’s payoff in Nash equilibrium.

p=1

p=0

q=1 q=0

Page 4 of 4

Problem 5 (4p)

Lucy offers to play the following game with Charlie: “let us show pennies to each other, each

choosing either heads or tails. If we both show heads, I pay you $3. If we both show tails, I pay

you $1. If the two don’t match, you pay me $x.” For what values of x is it profitable for Charlie

to play this game?

Problem 6 (4p)

(a) Represent this game in normal form (payoff matrix).

(b) Identify all pure strategy Nash equilibria. Which equilibrium is the subgame perfect Nash

equilibrium?

Important: In game theory people often use the same name to identify actions in different

information nodes. This is the case above. In extensive form games, however, these actions are

formally and conceptually different. You need to keep this distinction in mind when solving this

problem. An easy way not to make a mistake is by using your own naming convention, e.g., X

and X. -----------------------------

Problem 7 (2 extra credit points)

Represent the following game in normal form and find its Nash equilibria.

B

A

1

2

C

X

X

Y

Y

0,0

8,6

0,0

6,8

4,4

B

A

1

2

C

X

X'

Y

Y'

0,0

8,8

2,2

6,6

5,5 2