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Midterm Exam: Solutions
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.
Answer:
There is no unique answer. Any subway station in Tokyo could potentially become the one. But, I expect many students would select “Roppongi”, since most of them use this station on the way to the GRIPS campus.
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
Answer:
(a) There is no dominated strategy. (b) (U, L) and (D, R).
(c) No, since there is no dominated strategy.
2
(2)
P1 ╲ P2 A B
X 1, 2 3, 1
Y 2, 1 0, 0
Z 0, 0 2, 2
Answer:
(a) Z (is strictly dominated by X). (b) (Y, A).
(c) Iterated process eliminates Z, B, and X and picks up (Y, A).
3. Mixed Strategy (16 points, difficult)
Consider a two player static game expressed by the following payoff matrix.
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.
Answer: (a) (C, F)
(b) Let p be the probability with which P2 chooses D and q be the probability with which P1 chooses A. By indifference condition for P1, i.e., choosing A must yield the same expected payoff as choosing B, we obtain
7p + 5(1-p) = 5p + 7(1-p) ⇒ p = 1/2.
Similarly, by indifference condition for P2, we obtain 6q + 8(1-q) = 8q + 6(1-q) ⇒ q = 1/2.
(c) Yes, there exists a mixed strategy Nash equilibrium. Let x, y be the probabilities with
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which P2 chooses D, E. And let a, b, be the probabilities with which P1 chooses A, B. By indifference condition for P1, i.e., choosing A, B or C yields the same expected payoff, we obtain
7x + 5y = 5x + 7y = 4(1-x-y) ⇒ x = y = 1/5. By indifference condition for P2, we obtain
6a + 8b = 8a + 6b = 4(1-a-b) ⇒ a = b = 2/11.
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.
Answer: (a) The table below is the payoff matrix:
1 ╲ 2 CE CF DE DF
A 1, 1 1, 1 4, 2 4, 2
B 3, 1 2, 5 3, 1 2, 5
1
2
2
A
B
C
D
E
F
(1, 1)
(4, 2)
(3, 1)
(2, 5)
Figure 1
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(b) There are Nash equilibria: (A, DE), (A, DF), (B, CF). (c) The backward induction solution is (A, DF).
