Final Exam
Date: July 22, 2008
Subject: Advanced Microeconomics II (ECO601E) Professor: Yosuke YASUDA
1. True or False (10 points, easy)
Answer whether each of the following statements is true or false. You DON’T need to explain the reason.
(a) The third-degree price discrimination always increases total sur- plus.
(b) There always exists at least one Nash equilibrium, possibly in mixed strategies, in any …nite games, i.e, games with …nite number of players and actions.
2. Uncertainty (10 points, easy)
Suppose you are on the admission committee of the CRIPS, and must decide the minimum acceptance score of the entrance examination. There are two kinds of students, excellent and geniuses. All students would like to be admitted to the CRIPS as long as their expected ben- e…ts (in monetary term) are non-negative, but the object of the admis- sion committee is to accept only geniuses. It is presumably easier for geniuses to obtain high scores on the exam. In particular, suppose that the cost of obtaining a score of x out of 100 is $1200x for an excellent student and $1000x for a genius. The value to each student of being admitted to the CRIPS is $90,000. Then, what range of (minimum) exam scores would meet the admission committee’s objective?
3. Repeated Game (10 points, moderate)
Cooperation in prisoner’s dilemma might be possible if the game will be played repeatedly. However, as we have seen in the class, there are several cases in which cooperation cannot be achieved in any sub- game perfect equilibrium. Mention one of those cases and explain why cooperation is impossible in such a situation.
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4. Simultaneous Game (20 points, moderate)
Suppose three cafe chain companies, i = 1; 2; 3, are considering to open new shops near the Roppongi cross (Each company opens at most one shop). They make the decision independently and simultaneously. A company receives 0 pro…t if it does not open a shop. If opens, then each …rm’s pro…ts depend on the number of shops which are given as follows:
Number of …rms 1 2 3 Each …rm’s pro…t 10 4 -2
(a) Derive all pure-strategy Nash equilibria.
(b) Is there any mixed-strategy Nash equilibrium in which companies decides to open a shop with the same probability p? If yes, solve such p.
(c) Is there any equilibrium in which one company opens a shop for sure while other two …rms open with equal probability q? If yes, solve such q.
5. Bayesian Game (20 points, hard)
Suppose a government auctions one block of radio spectrum to two risk neutral mobile phone companies, i = 1; 2. The companies submit bids simultaneously, and the company with higher bid receives a spectrum block. The loser pays nothing while the winner pays a weighted average of the two bids:
b+ (1 )b0
where b is the winner’s bid, b0 is the loser’s bid, and is some constant satisfying 0 1. (In case of ties, each company wins with equal probability.) Assume the valuation of the spectrum block for each company is independently and uniformly distributed between 0 and 1.
(a) Solve a Bayesian Nash equilibrium.
Hint: You can assume the equilibrium strategy is sym- metric and linear, i.e., bi = vi for i = 1; 2.
(b) Show that bi = vi is always (weakly) better than bi = 0:5vi for i when = 0.
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