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Midterm Exam
Date: February 29, 2012
Subject: Game Theory (ECO290E) Instructor: Yosuke YASUDA
1. Focal Point (4 points, think strategically!)
Select a subway station in Tokyo, and write down its name. You should NOT write more than one name. If you will successfully choose the most popular answer, you would get 4 points. Otherwise, you will receive 0. You do NOT need to explain any reason why you choose your answer.
2. Static games (18 points, easy)
Consider the following two player static games expressed by (1), (2). For each game, answer the following questions:
a) Find a dominated strategy (if any).
b) Find all pure strategy Nash equilibria (if any).
c) Explain whether the game can be solved by iterated elimination of strictly dominated strategies.
(1)
P1 ╲ P2 L R
U 4, 4 0, 3
D 3, 0 2, 2
(2)
P1 ╲ P2 A B
X 1, 2 3, 1
Y 2, 1 0, 0
Z 0, 0 2, 2
3. Mixed Strategy (16 points, difficult)
Consider a two player static game expressed by the following payoff matrix.
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P1 ╲ P2 D E F
A 7, 6 5, 8 0, 0
B 5, 8 7, 6 0, 0
C 0, 0 0, 0 4, 4
a) Find all pure strategy Nash equilibria.
b) Find the mixed strategy Nash equilibrium in which each player randomizes over just the first two actions, i.e., A, B for P1 and D, E for P2, respectively.
c) Is there a mixed strategy Nash equilibrium in which both players randomize over all three strategies? If yes, derive the equilibrium. If not, explain why.
4. Dynamic Game (12 points, moderate)
Consider a dynamic game depicted by Figure 1.
a) Express this game into normal-form (strategic-form) by drawing the payoff matrix. b) Find all Nash equilibria.
c) Solve the game by backward induction. You do not need to worry about the way to write the answer. As long as the path which survives in backward induction process is apparent, you will receive the full score.
