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Final Exam
Date: March 29, 2014
Subject: Game Theory (ECO290E) Instructor: Yosuke YASUDA
1. True or False (10 points)
Answer whether each of the following statements is true (T) or false (F). You do NOT need to explain the reason. Please just indicate T or F.
A) A dynamic game may have multiple Nash equilibria, but its subgame perfect equilibrium is ALWAYS unique.
B) A strategy in dynamic games specifies what the player will do in her decision node or information set reached ONLY on the equilibrium path.
C) In the Stackelberg models, the FOLLOWER gets higher profit than the leader because it can observe the leader’s output and take the best reply against it. D) In finitely repeated games, a stage game Nash equilibrium must be played in the
LAST period in any subgame perfect equilibria.
E) For ANY dynamic games of bargaining, there exists a subgame perfect equilibrium that realizes equal surplus division among players.
2. Game Tree (15 points)
See the following game tree.
1
2
2
1 1
1
A F
C
K
L
M
N
D E
G B
H
I H
I
(2, 2) (3, 4)
(1, 3)
(1, 3) (0, 0) (0, 0) (3, 1) (1, 5) (0, 2)
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A) How many information sets (containing two or more decision nodes) does this game have?
B) How many subgames (including the entire game) does this game have? C) Find all subgame perfect Nash equilibria.
Remark: You should make it clear which action each player will take in every decision node (or information set). Writing a resulting payoff alone is not enough.
3. Dynamic Game (16 points)
A manufacturer of automobile tires produces at a cost of $20 per tire. It sells units to a retailer who in turn sells the tires to consumers. Imagine that the retailer faces the (inverse) demand function, p = 200 – q/100. That is, if the retailer brings q tires to the market, these tires will be sold at this price. The retailer has no cost of production, other than whatever it must pay to the manufacturer for the tires. Suppose that the manufacturer and retailer interact as follows. First, the manufacturer sets a price x that the retailer must pay for each tire. Then, the retailer decides how many tires q to purchase from the manufacturer and sell to consumers.
A) What is the profit function of the manufacturer? B) What is the profit function of the retailer?
C) Derive the retailer’s best reply function, i.e., the optimal reaction to the manufacturer’s strategy, x.
D) Solve for the subgame perfect equilibrium of this game.
4. Repeated Games (12 points)
Consider the following two persons 2 x 2 game.
1 / 2 L R
U 2, 3 0, 6
D 4, 0 1, 1
A) Find all pure‐strategy Nash equilibria.
B) Consider the repeated game in which the above stage game will be played 5 times. Then, can (U, L) be sustained as a subgame perfect Nash equilibrium? If yes,
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derive the equilibrium. If not, explain why.
C) Now suppose that the game will be played infinitely many times, and each player tries to maximize the discounted sum of payoffs with the discount factor δ (< 1). For what value of δ, can (U, L) be sustained as a subgame perfect Nash equilibrium? You can focus on the trigger strategy.
5. Bayesian Nash Equilibrium (12 points)
There are three different bills, $5, $10, and $20. Two individuals randomly receive one bill each. The (ex ante) probability of an individual receiving each bill is therefore 1/3. Each individual knows only her own bill, and is simultaneously given the option of exchanging her bill for the other individual’s bill. The bills will be exchanged if and only if both individuals wish to do so; otherwise no exchange occurs. That is, each individuals can choose either exchange (E) or not (N), and exchange occurs only when both choose E. We assume that individuals’ objective is to maximize their expected monetary payoff ($).
A) Consider the above situation as a Bayesian game. How many strategies does each individual have? Recall that a strategy is the complete plan of actions.
B) Prove that no trade will occur in any Bayesian Nash equilibrium.
C) Now suppose that the rule of the game is modified as follows. If exchange occurs, each individual receives 3 times as much amount as the bill she will have. For example, if individual 1 receives $5 and 2 receives $10 initially and both wish to exchange, then 1 will receive $30 (= $10 x 3) and 2 will receive $15 (= $5 x 3). Nothing happens if they do not exchange. Then, does trade occur in a Bayesian Nash equilibrium? Explain.
