Final Exam
Date: July 27, 2010
Subject: Advanced Microeconomics II (ECO601E) Professor: Yosuke YASUDA
1. True or False (9 points)
Answer whether each of the following statements is true (T) or false (F). You do NOT need to explain the reason.
(a) The backward induction solution coincides with the subgame perfect Nash equilibrium for ANY perfect information game.
(b) Revenue equivalence theorem claims that the equilibrium bidding strategy under the …rst-price auction is ALWAYS identical to the one under the second- price auction.
(c) EVERY perfect Bayesian equilibrium is a weak perfect Bayesian equilibrium. 2. Dynamic Game (14 points)
Consider the following two-person dynamic game. In the …rst period, game A is played; after observing each player’s actions, they play game B in the second period. Assume that the payo¤s are simply the sum of the payo¤s of two games (i.e., there is no discounting).
1 2 L R
U 3; 3 0; 4 D 4; 0 1; 1
Game A
1 2 L0 R0 U0 4; 4 0; 0 D0 0; 0 2; 2
Game B
(a) Suppose above games are played separately, that is, each game is played only once. Then, derive all Nash equilibria for each game.
(b) Now consider the dynamic game. How many subgames (including the entire game) does this game have?
(c) Is there a subgame perfect Nash equilibrium that can achieve (U; L) is the …rst period? If so, describe the equilibrium strategies. If not, explain why.
3. Repeated Game (12 points)
Find conditions on the discount factor under which cooperation (=(C; C)) can be supported in the in…nitely repeated games with the following stage games.
1
1 2 C D C 3; 3 0; 4 D 4; 0 1; 1
Game 1
1 2 C D
C 3; 4 0; 7 D 5; 0 1; 2
Game 2
1 2 C D
C 3; 2 0; 4 D 7; 0 2; 1
Game 3
Hint: Note that every stage game above is a prisoner’s dilemma. You can focus on the trigger strategy, i.e., players choose a stage game Nash equilibrium (D; D) as a punishment whenever someone has once deviated from (C; C).
4. Auction (15 points)
Suppose a seller auctions one object to two risk neutral buyers, = 1; 2. The buyers submit bids simultaneously, and the buyer with higher bid receives the object. The loser pays nothing while the winner pays a weighted average of the two bids b+b20 where b is the winner’s bid, b0 is the loser’s bid. Assume that the valuation of the object for each buyer is independently and uniformly distributed between 0 and 1. (a) Suppose that buyer 2 takes a linear strategy, b2 = v2. Then, derive the
probability such that buyer 1 wins as a function of b1. (b) Solve a Bayesian Nash equilibrium.
Hint: You can assume that equilibrium bidding strategy is symmetric and linear, bi = vi for i = 1; 2.
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