Problem Set 2

PROBLEM SET 2: LESSON 3
ACADEMIC YEAR 2019/2020 GAME THEORY
1. Consider the following extensive form game. Suppose that players only play pure strategies:
a. Identify the players’ set of strategies.

b. Obtain all the Nash equilibria of the game.
c. Obtain the subgame perfect Nash equilibrium of the game.
2. Consider the following game. Suppose that players only play pure strategies:
a. Obtain all the Nash equilibria of the game.

b. Obtain the subgame perfect Nash equilibrium of the game.
3. Consider the following sequential game with two players and an initial pot with 5 dollars. In stage 1
player A can either claim the pot and get 80% of the money (player B receives the remainer), or
pass. If she passes, the pot is doubled and it is player B’s turn, who can either claim the pot and get
80% of the money (the remainder goes to player A), or pass. If she passes, the pot is doubled and it
is player A’s turn… (and so on).
a. Represent the game in extensive form with two repetitions for each player. Note: the
termination payoffs that correspond to player B passing in his second information set are 64
for player 1 and 16 for player 2.
b. Obtain the Nash equilibria in pure strategies of the game.
c. Obtain the subgame perfect Nash equilibrium of the game.
4. A challenger (player 1) is planning to enter an industry in which there is currently a monopolist (player
2). If player 1 does not enter, payoffs are (0,2) for players 1 and 2 respectively. If player 1 enters, then
both firms choose simultaneously whether to be aggressive (fight) or passive. If both fight, payoffs are
(‐3, ‐1). If firm 1 fights and firm 2 is passive, payoffs are (1,‐2). If firm 1 is passive and firm 2 fights,
payoffs are (‐2,‐1). If both firms are passive, then they play the following simultaneous game:
1\2 A B
A 3,1 0,0
B 0,0 𝑥 ,3
Suppose that players only play pure strategies. Then, you have to:
a. Represent the game in extensive form. How many subgames does the game have?
b. Identify the set of strategies of each player.
c. If 𝑥 ∈ 0,1 , obtain the subgame perfect Nash equilibria of the game.
5. Two consumers 𝐶 and 𝐶 have a common valuation 𝑣 ∈ 3/4,1 of an object that belongs to a seller.
At 𝑡 1, the seller offers the good to consumer 1 at price 𝑝 1. If 𝐶 buys the object, the game
ends. If he does not buy the object, then we move to 𝑡 2, where the seller offers the good to
consumer 2 at price 𝑝 3/4. If 𝐶 buys the object, the game ends. If he does not buy the object,
then we move to 𝑡 3, where the seller offers the good to consumer 1 at price 𝑝 2/3. If 𝐶 buys
the object, the game ends. If he does not buy the object, then the seller gives the object as a present
to either one of the consumers. Each consumer has ½ probability of receiving the object (in this case,
at 𝑝 0). Suppose that if a consumer buys the object, his utility is 𝑣 𝑝, and it is 0 otherwise.
Suppose also that players only play pure strategies:
a. Represent the game in extensive form (note that the only two players are 𝐶 and 𝐶 ).
b. Obtain all the Nash equilibria of the game.
c. Obtain the subgame perfect equilibrium of the game. Who buys the object?
6. Let us consider an industry with three firms: 1, 2 and 3. Firms compete in quantities and they all have
the same cost function, given by 𝑐 𝑥 2𝑥 , with 𝑖 ∈ 1,2,3 . The market demand is 𝑃 𝑥 102
𝑥, with 𝑥 𝑥 𝑥 𝑥 . Suppose that firm 1 is the leader in the industry and firms 2 and 3 are the
followers. That means that firm 1 chooses first its output level 𝑥 and upon observing it, firms 2 and 3
choose, simultaneously, their output levels 𝑥 and 𝑥 .
a. Obtain the subgame perfect Nash equilibrium of the game. How much output does each firm
produce in equilibrium? What are the profits of each firm in equilibrium?
b. Suppose that firms 2 and 3 threat firm 1 with the following strategy: If you produce 𝑥 50 we
will each produce . Otherwise, we will each produce 25. Is this a credible threat? Argue your
answer.

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PROBLEM SET 2: LESSON 3

ACADEMIC YEAR 2019/2020 GAME THEORY

a. Identify the players’ set of strategies.

a. Obtain all the Nash equilibria of the game.

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