Econ 4550 – Game Theory with Economic Applications Final Exam
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Econ 4550 – Game Theory with Economic Applications
Final Exam
May 31, 2022
Question 1. (18 points)
Consider the following game
Player 1
A
B
C
Player 2
W X Y Z
|
5, 10 |
10, 20 |
15, 0 |
10, 10 |
|
10, 15 |
15, 8 |
20, 5 |
12, 9 |
|
12, 5 |
0, 7 |
0, 15 |
15, 8 |
a. (8 marks) Use IEDS to simplify the game as much as possible
b. (10 marks) In the simplified game, find all Nash equilibria, including mixed-strategy Nash equilibria.
Question 2. (30 points)
Two profit-maximizing firms, 1 and 2, produce the same product and compete in the same market. Together they face a demand:
P = 24 − q1 − q2
The two firms choose output to maximize own profits, and they choose q1 and q2 sequentially : firm 1 sets q1 first, and firm 2 observes this decision then sets q2 .
a. (10 marks) Suppose that both firms have zero cost. Find the subgame-perfect Nash equilibrium.
Now suppose that, firm 2’s cost is still zero, but firm 1’s marginal cost is zero with probability 0.5, and is $6 per unit with probability 0.5. Firm 1 knows its own cost, but firm 2 does NOT observe firm 1’s cost. Firm 2 knows this probability distribution.
b. (4 marks) Draw the game tree.
c. (4 marks) Is this game a perfect or imperfect information game? Use one or two sentences to explain briefly. Can you use backward induction to solve this game?
d. (12 marks) Find the subgame-perfect Nash equilibrium and the two firms’ output.
Question 3. (24 points)
A
Player 1
Player 2
C D
|
3, 5 |
x, 16 |
|
6, 9 |
6, 0 |
In the above normal form game, x is a random variable uniformly distributed between 0 and 12. Player 1 knows the true value of x, but player 2 does not know the value of x . Player 2 only knows the probability distribution of x .
a. (2 marks) When player 1 chooses A, what is player 2’s best response?
b. (2 marks) When player 1 chooses B, what is player 2’s best response?
c. (2 marks) When player 2 chooses C, what is player 1’s best response?
d. (4 marks) When player 2 chooses D, what is player 1’s best response?
e. (6 marks) Find the Nash equilibrium where player 2 chooses C. Explain briefly why it is a Nash equilibrium.
f. (8 marks) Is there a Nash equilibrium where player 2 chooses D? Explain why it is or it is not a Nash equilibrium.
Question 4. (17 points)
Consider a 1st-price common-value auction with two bidders. The true value of the object is Y = 
y1 + 
y2, where yi is bidder i’s estimate of Y . The two bidders are not sure about the value Y, because yi is bidder i’s private information. y1 and y2 are independently and uniformly distributed on [0, 20], and a bidder knows her own estimate, but does not know the other bidder’s estimate.
She only knows that the other bidder’s estimate is uniformly distributed on [0, 20]. The two bidders simultaneously submit bids. The one who bids higher wins and pays her own bid. Suppose bidder 2 uses the bidding strategy b2 (y2) = ay2, where a is a constant number which is publicly known.
a. What is bidder 1’s optimal bidding strategy?
2022-12-08