Problem Set 2

Degree in Economics and Finance Prof. Arguedas

AvancedMicroeconomics

Problem set #2: Game Theory

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1. Consider the following normal-form game:

PLAYER 2

PLAYER 1

D E F

A 7, 6 5, 8 0, 0

B 5, 8 7, 6 1, 1

C 0, 0 1, 1 4, 4

a) Find all the pure-strategy Nash equilibria (if any)

b) Find the mixed-strategy Nash equilibrium in which each player randomizes just over the

first two actions.

c) Compare players’ expected payoffs in the equilibria found in parts a) and b).

2. In the following normal-form game:

PLAYER 2

PLAYER 1

L M R

U 2, 0 1, 1 4, 2

M 3, 4 1, 2 2, 3

D 1, 3 0, 2 3, 0

a) What strategies survive after the elimination of strictly dominated strategies?

b) Find all the Nash equilibria (both in pure and mixed strategies).

3. The mixed-strategy Nash equilibrium in the Battle of the Sexes may depend on the numerical

values for the payoffs. To generalize this solution, assume that the payoffs matrix for the game

is given by:

PLAYER 2

(husband)

PLAYER 1

(wife)

Ballet Boxing

Ballet K, 1 0, 0

Boxing 0, 0 1, K

where � � 1. Show how the mixed-strategy Nash equilibrium depends on the value of �.




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4. The game of Chicken is played by two macho teens who speed toward each other on a single-

lane road. The first to veer is branded the chicken, whereas the one who does not veer gains per-

group esteem. Of course, if neither veers, both die in the resulting crash. Payoffs to the Chicken

game are provided in the following table:

TEEN 2

TEEN 1

Veer Not veer

Veer 2, 2 1, 3

Not veer 3, 1 0, 0

a) Find all the pure strategy Nash equilibria.

b) Find the mixed strategy Nash equilibrium. What is the probability that both teenagers

will survive?

c) Suppose the game is played sequentially with teen 1 moving first and commiting to this

action by throwing away the steering wheel. Write down the extensive form game and

find the corresponding subgame perfect equilibrium.

d) Does the equilibrium found in part c) coincide with any of the equilibria found in parts

a) or b)? Why?

5. Two neighbouring homeowners, � � 1, 2, simultaneously choose how many hours � to spend

maintaining a beautiful garden. The average benefit per hour is 10 � � � �

�, with � � 1, 2 and

� � �, and the (opportunity) cost per hour is 4. Homeowner �’s average benefit is increasing in

the hours neighbour � spends on his own garden because the appearance of one’s property

depends on part on the beauty of the surrounding neighbourhood.

a) Compute the Nash equilibrium of this game.

b) Graph the best response functions and indicate the Nash equilibrium found in part a) in

the figure.

c) Show graphically how the Nash equilibrium would change if the opportunity cost of

neighbour 1 increases.

d) Compute the Subgame Perfect Equilibrium of this game (assume for example that

neighbour � went on holidays while neighbour � worked on his garden in the meantime).

e) Explain the differences between the results obtained in parts a) and d), both verbally

and graphically.

6. Consider the following normal-form game:




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PLAYER 2

PLAYER 1

A B

A 0, 0 3, -1

B -1, 3 1, 1

a) Verify that the Nash equilibrium is the usual one for the Prisoners’ Dilemma and that

both players have dominant strategies.

b) Suppose the stage game is repeated infinitely many times. Compute the discount factor

required for the players to sustain the cooperative strategy (B,B). Outline the trigger

strategies you are considering.

7. The simultaneous-move game (below) is played twice, with the outcome of the first stage

observed before the second stage begins. There is no discounting. Can the payoff (4,4) be

achieved in the first stage in a subgame-perfect equilibrium? If so, give strategies that do so. If

not, prove why not.

PLAYER 2

PLAYER 1

L C R

T 3, 1 0, 0 5, 0

M 2, 1 1, 2 3, 1

B 1, 2 0, 1 4, 4

8. Consider the following normal-form game of incomplete information:

PLAYER 2

PLAYER 1

C D

A 7, 6 0, 0

B 5, 8 1, 1

C 0, 0 t, 4

where t can take two possible values (t=4 or t=0) with equal probability. Assume player 1

knows his/her own type while player 2 only knows the probability distribution of player 1’

types.

a) Find all the pure-strategy Bayesian Nash equilibria of this game.




___________________________________________________________________ b) Now suppose that player 1 moves first and player 2 can observe player 1’s action.

Calculate the Perfect Bayesian equilibrium of the sequential game. Is the equilibrium

pooling or separating?

c) Solve parts a) and b) assuming that the normal-form game is now the following:

PLAYER 2

PLAYER 1

C D

A t, 6 0, 0

B 5, 8 1, 1

C 0, 0 t, 4

9. Consider the following normal-form game of incomplete information:

PLAYER 2

PLAYER 1

C D

A 7, h 0, 0

C 0, 0 t, 4

a) Find all the pure-strategy Bayesian Nash equilibria assuming that t and h are

independent and can take two possible values each with equal probability (t=4 or t=0

and h=6 or h=1). Assume each player knows his/her own type and the probability

distribution of the other player’ types.

b) Suppose player 1 moves first and player 2 can observe player 1’s action. Draw the

extensive form of the game.

c) Assume h = 1 and calculate the Perfect Bayesian equilibrium of the sequential game. Is

the equilibrium pooling or separating?

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Problem Set 2