1
Midterm: Solutions
Date: March 9, 2011
Subject: Game Theory (ECO290E) Instructor: Yosuke YASUDA
1. Dominant Strategy (6 points, easy)
State the definition of dominant strategy, either by words or by mathematically.
Answer: A dominant strategy is the strategy that is best for the player regardless of the strategies chosen by other players.
2. True or False (12 points, think carefully)
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) The normal form representation of games must specify 1) players, 2) strategies, and 3) outcomes.
B) When strategy x is dominated by strategy y, and y is dominated by z, then x MUST by dominated by z.
C) Nash equilibrium is ALWAYS Pareto efficient.
D) For ANY finite game with 2 players and 2 strategies each, there exists at least one Nash equilibrium, possibly in mixed strategies.
Answer: (A) F (B) T (C) F (D) T.
3. Finite Game (12 points, moderate)
For each of the static games A, B, and C, whose payoff matrices are given below, answer the following questions:
i. Find all pure strategy Nash equilibria.
ii. Find all outcomes which are Pareto efficient.
iii. Can the game be solved by iterated elimination of strictly dominated strategies? If yes, describe the elimination process. If not, explain why.
Note) In each cell, the number on the left (/right) shows a payoff for P1 (/P2).
2
A)
P1 / P2 L R
U 4, 4 1, 5
D 5, 1 0, 0
Answer: i) (U, R), (D, L), ii) (U, L), (U, R), (D, L), iii) No, since no strategy is dominated.
B)
P1 / P2 L R
U 4, 2 1, 1
D 0, 5 5, 3
Answer: i) (U, L), ii) (D, L), (D, R), iii) Yes; First erase R and second erase D.
C)
P1 / P2 L M R
U 4, 4 1, 100 0, 5
D 0, 0 2, 1 1, 2
Answer: i) (D, R), ii) (U, L), (U, M), iii) Yes; First erase L (dominated by R), second erase U, and third erase M.
4. Continuous Game (12 points, difficult)
Two neighboring homeowners, i = 1, 2, simultaneously choose how many hours (denoted by li ) to spend maintaining a beautiful lawn. Since the appearance of one’s property depends in part on the beauty of the surrounding neighborhood, homeowner i’s benefit is increasing in the hours that neighbor j spends on his own lawn. Suppose that i’s benefit (or payoff) is expressed by
(10 – li + lj/3) li where i ≠ j. Then, answer the following questions.
a) Derive the best response function, BRi (lj ), for homeowner i.
Hint: Recall how we have derived the best response functions in Cournot model. You can solve this question in a similar way.
3
Answer: BRi (lj ) = 5 + lj /6.
b) Graph the best response functions of both players (taking l1 on X‐axis and l2 on Y‐axis), and show the Nash equilibrium on your figure.
Answer: Skipped. (Note that BR functions are both upward‐sloping and the intersection coincides with the Nash equilibrium.)
c) Compute the Nash equilibrium.
Answer: li = lj = 6.
5. Mixed Strategy (8 points, moderate)
Compute the mixed strategy Nash equilibrium of the following game. That is, calculate the probabilities of choosing each strategy for each player under equilibrium.
P1 / P2 L R
U 4, 2 2, 4
D 0, 7 8, 1
Answer: Let p be the probability that player 1 chooses L and q be the probability that player 2 chooses U under equilibrium. Using the indifference property, we obtain, p = 3/4 and q = 3/5.
