# Game Theory

Question 1 [10 points]:

Consider the game given above

a. Find the Nash equilibria of this game.

b. If you eliminate weakly dominated strategies, what will be the prediction of this game.

How is it different from part (a) above?

Question 2 [20 points]:

Consider a matrix game with two players, where the players can play only pure strategies.

In this context do you think the following statement is true?

The set of rationalizable strategies (strategies that survive iterated elimination of strategies

that are never best responses) is identical to the set of strategies that survive IESDS.

L M R

U 2, 3 4, 1 1, 2

C 2, 1 3, 2 0, 0

D 0, 2 3, 1 1, 2

If ‘yes’, give a proof. If ‘no’ give an example to prove your point. Also, if your answer is ‘no’,

please discuss how you can resolve this so that the two sets become identical.

Question 3 [30 points]:

Consider the following game:

a) Write down the strategy sets of players 1, 2, and 3.

b) What is the Subgame-perfect Nash Equilibrium of this game?

Question 4 [20 points]:

Consider two competing firms in a declining industry that cannot support both firms

profitably. Each firm has three possible choices, as it must decide whether or not to exit the

industry immediately, at the end of this quarter, or at the end of the next quarter. If a firm

chooses to exit then its payoff is 0 from that point onward. Each quarter that both firms

operate yields each a loss equal to −1, and each quarter that a firm operates alone yields it a

payoff of 2. For example, if firm 1 plans to exit at the end of this quarter while firm 2 plans

to exit at the end of the next quarter then the payoffs are (−1, 1) because both firms lose −1

in the first quarter and firm 2 gains 2 in the second. The payoff for each firm is the sum of

its quarterly payoffs.

a. Write down this game in matrix form.

b. Are there any strictly dominated strategies? Are there any weakly dominated

strategies?

c. Find all the Nash equilibria of this game

Question 5 [20 points}

Consider the following game that proceeds in two stages: In the first stage one brother

(player 2) has two $10 bills and can choose one of two options: he can give his younger

brother (player 1) $20 or give him one of the $10 bills. This money will then be used to buy

snacks at the show they will see, and each $1 of snacks purchased yields one unit of payoff

for a player who uses it. In the second stage the show they will see is determined by the

following game:

a. Draw the game tree for the above interaction.

b. Write out the set of pure strategies for the two brothers.

c. Present the entire game in one matrix.

d. Find all the Nash equilibria of this game. How many of them survive sequential

rationality?