1
Final Exam
Date: March 30, 2013
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) Any extensive form is uniquely translated into a normal form, while there can be multiple ways such that a normal form is translated into an extensive form.
B) A dynamic game may have multiple Nash equilibria, but its subgame perfect equilibrium is always unique.
C) In any finitely repeated games, repeated play of the stage game Nash equilibrium (playing an NE in every period) is a unique subgame perfect equilibrium.
D) Experimental studies sometimes show that the subjects play differently from theoretical prediction, even after they have played the same game for many times. E) A matching is called “stable” if it is Pareto efficient.
2. Dynamic Game (12 points, moderate)
Consider the dynamic game described by following game tree.
a) How many subgames does the game have (except for the entire game)? b) Find the subgame perfect Nash equilibrium.
c) Is there a Nash equilibrium different from the subgame Nash equilibrium (your answer in b)? If yes, derive the equilibrium. If not, explain why.
1 2 1
A
B D F
C E
(5, 1)
(4, 4)
(2, 2) (1, 3)
2
3. Repeated Game (12 points, moderate)
Consider the following two persons 2 x 2 game.1 / 2 L R
U 3, 3 0, 5
D 5, 0 1, 1
A) Find all pure‐strategy Nash equilibria.
B) Consider the two‐period repeated game in which the above stage game will be played twice. Then, can (U, L) be sustained as a subgame perfect Nash equilibrium? If yes, 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.
4. Incomplete Information (16 points, think carefully)
There are four different bills, $1, $5, $10, and $20. Two individuals randomly receive one bill each. The (ex ante) probability of an individual receiving each bill is therefore 1/4. An individual knows only her own bill, and is simultaneously given the option of exchanging her bill for the other individual’s bill. The bills will be exchanged if and only if both individuals wish to do so; otherwise no exchange occurs. That is, each individuals can choose either exchange (E) or not (N), and exchange occurs only when both choose E. We assume that individuals’ objective is to maximize their expected monetary payoff ($).
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.
Now suppose that the rule of the game is modified as follows. If exchange occurs, each individual receives k times as much money as the bill she has. For example, if individual 1 received $5 and 2 received $10 initially and both wish to exchange, then 1 will receive $10k and 2 will receive $5k. Nothing happens if they do not exchange.
c) If k = 2, is there any Bayesian Nash equilibrium in which trade occurs? Explain. d) If k = 4, explain if the following strategies become a Bayesian Nash equilibrium or
not: both players choose “exchange” (E) regardless of the bill they received.
