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Final Exam
Date: April 3, 2011
Subject: Game Theory (ECO290E) Instructor: Yosuke YASUDA
1. True or False (6 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 strategy in dynamic games is the plan of actions, which specifies what the player will do in the information set reached ONLY on the equilibrium path.
B) For ANY dynamic games of bargaining, the subgame perfect equilibrium outcome ALWAYS realizes equal surplus division among players.
C) In finitely repeated games, players cannot sustain (as a subgame perfect Nash equilibrium) ANY action profiles that are different from the stage game Nash equilibrium in the LAST period.
2. Dynamic Game (12 points, moderate)
Consider the dynamic game described by following game tree.
a) Translate this game into normal-form by drawing the payoff bi-matrix. b) Find all pure-strategy Nash equilibria. How many are there?
c) How many subgames (except for the entire game) does this game have?
d) Solve this game by backward induction, and derive the subgame perfect Nash equilibrium. You may state the strategy profile or show it on the game tree.
1 2 1
A
B D F
C E
(10, 3)
(8, 8)
(0, 5) (1, 1)
2
3. Bargaining (8 points, easy)
Players 1, 2 and 3 are bargaining over how to split the ice-cream of size 1. In the first period, player 1 proposes a share (x, y, 1-x-y) to players 2 and 3 where each share must be between 0 and 1. Players 2 and 3 can decide whether accepting the offer or reject it. If both players accept, then the game finishes and each player gets the proposed shares. If either one of them rejects, the game moves to the second period, in which the size of the ice-cream becomes 60% of the original size due to melting. In the second stage, by flipping a coin, the ice-cream is randomly assigned to player 2 or 3 with equal probability, i.e., 50% each. Suppose that each player maximizes expected size of the ice-cream that she can get. Derive a subgame perfect Nash equilibrium of this game.
Hint: You can focus on the equilibrium in which both players 2 and 3 accept the offer in the first period.
4. Repeated Game (12 points, moderate)
Consider the following two persons 2 x 2 game.
1 / 2 L R
U 3, 4 0, 6
D 4, 0 1, 2
A) Find all pure-strategy Nash equilibria.
B) Consider the two-period repeated game in which the above stage game will be played twice. Suppose that the payoff for each player is simply the sum of the payoffs in the stage games. 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?
Hint: You can focus on the trigger strategy. Consider both players incentive constraints, since the game is not symmetric.
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5. Incomplete Information (12 points, think carefully)
Consider a game of election with asymmetric information among voters (citizens). Whether candidate A or candidate B is elected depends on the votes of two citizens. The social situation may be in one of two states, a and b. The citizens agree that candidate A is best if the state is a, and candidate B is best if the state is b. The payoff for each citizen is symmetric and given as follows: 1 if the best candidate wins, 0 if the other candidate wins, and 0.5 if the candidates tie. Suppose that citizen 1 knows the true state, whereas citizen 2 believes that the state is a with probability 0.8 and b with probability 0.2. Each citizen takes either one of the three actions: vote for candidate A, vote for candidate B, and not vote.
A) Consider the corresponding Bayesian game. What is the strategy for each player? B) Derive the pure strategy Bayesian Nash equilibria.
Hint: There are two equilibria, and the one of them involves weakly dominated strategy.
