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Final Exam: Solution
Date: March 26, 2010
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) In Stackelberg models, the FOLLOWER gets benefit because she can observe the leader’s output and take the best reply against it.
B) Each person’s equilibrium share in alternating offer games increases as her discount factor DECREASES.
C) In ANY repeated games, playing a Nash equilibrium of the stage game in every period ALWAYS constitutes a subgame perfect Nash equilibrium.
Answer: (A) F (B) F (C) T
2. Dynamic Game (8 points, moderate)
Consider a dynamic game expressed by the following game tree.
A) How many information sets (containing two or more decision node) does this game have?
Answer: 5
B) How many subgames (including the entire game) does this game have? Answer: 3
C) Find all pure‐strategy subgame perfect Nash equilibria. Answer: There are 4 SPNE:
(U, AA’, XY’), (U, AB’, XX’), (D, BA’, YY’), (O, BB’, YX’)
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3. Bargaining (8 points, easy)
Players 1 (proposer) and 2 (receiver) are bargaining over how to split the ice‐cream of size 1. In the first stage, player 1 proposes a share (x, 1‐x) to player 2 where 0 x 1. Player 2 can decide whether accepting the offer or reject it. If player 2 accepts, then the game finishes and players get their shares. If player 2 rejects, the game move to the second stage, in which the size of the ice‐cream becomes 80% of the original size due to melting. In the second stage, by flipping a coin, the ice‐cream is randomly assigned to one of the players. Suppose each player maximizes expected size of the ice‐cream that she can get. Derive a subgame perfect Nash equilibrium of this game.
Answer: x = 0.6. Player 2 accepts the offer if and only if x 0.6.
Note that each player will receive 0.4 as his/her expected payoff in the second stage. Taking this into account, the proposer must give at least 0.4 to the receiver.
4. Repeated Game (12 points, think carefully)
Consider the following two persons 3 x 2 game. Assume x > 0.
1 / 2 L R
U 3, 3 0, 1
M 2, 0 x, 2
D 5, 1 0, 0
A) Find all pure‐strategy Nash equilibrium. Answer: (DL), (MR)
B) Consider the two‐period repeated game in which the above stage game will be played twice. Suppose the payoff for each player is simply the sum of the payoffs in the stage games, i.e., there is no discounting. Then, for what value of x does there exist a subgame perfect Nash equilibrium achieving (U, L) in the first period? Answer: 0 < x 3 and x 7.
In the former case, players select (D, L) on the equilibrium path while choosing (M, R) as a punishment in the second stage. In the latter case, they play (M, R) on the equilibrium path while choosing (D, L) as a punishment in the second stage.
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5. Incomplete Information (16 points, moderate)
Two countries are involved in a conflict. Country 1 does not know whether country 2 is strong or weak; it assigns probability x to country 2 being strong, and 1‐x to weak. Country 2 is fully informed. Under this asymmetric information, countries simultaneously decide either “fight” (F) or “yield” (Y). Each country receives payoff of 0 if it yields (irrespective of other country’s action), and obtains 1 if it fights and the other yields. When both countries fight, their payoffs become (‐1, 1) if country 2 is strong and (1, ‐1) if country 2 is weak. Then, answer the following questions.
A) Describe the strategies for each country.
Answer: 1’s strategy is {F, Y} and 2’s strategy is {FF, FY, YF, YY}.
B) What is an optimal action for country 2 when it is strong? Answer: F
C) Derive a Bayesian Nash equilibrium when x > 0.5. Answer: (Y, FF)
D) Derive a Bayesian Nash equilibrium when x < 0.5. Answer: (F, FY)
