1
Midterm Exam: Solution
Date: February 24, 2010
Subject: Game Theory (ECO290E) Instructor: Yosuke YASUDA
1. True or False (15 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) A strategy which is strictly dominated by some other strategy can NEVER be a dominant strategy.
B) The (unique) Nash equilibrium of the Prisoners’ Dilemma game is Pareto effieicnt. C) ANY finite game with 2 players and 2 strategies (for each player) has a pure
strategy Nash equilibrium.
D) There is a duopoly model in which each firm’s profit under Nash equilibrium becomes 0.
E) To exclude non‐credible Nash equilibria in dynamic games, we should solve a game forwardly, that is, from the first stage to the last stage.
Answer: (A) T (B) F (C) F (D) T (E) F
2. Simple 2‐2 games (12 points, moderate)
For the 2‐2 games A, B, and C below, answer the following questions: i. Find (all) pure strategy Nash equilibrium.
ii. Find (all) outcomes which are NOT Pareto efficient. iii. Explain whether there exists a dominant strategy.
iv. Explain whether the game can be solved by iterated elimination of strictly dominated strategies.
Note) In each cell, the number on the left (/right) shows a payoff for P1 (/P2).
2
A)
P1 / P2 L R
U 2, 2 0, 3
D 3, 0 1, 1
Answer: (i) (D, R) (ii) (D, R) (iii) D and R (iv) Yes
B)
P1 / P2 L R
U 2, 2 1, 3
D 3, 1 0, 0
Answer: (i) (D, L) (U, R) (ii) (D, R) (iii) None (iv) No
C)
P1 / P2 L R
U 2, 2 3, 0
D 1, 1 0, 3
Answer: (i) (U, L) (ii) (D, L) (iii) U (iv) Yes
3. Mixed Strategy (15 points)
Consider the modified version of the “Rock‐Paper‐Scissors” game. The basic rule is the same; Rock wins Scissors, Scissors wins Paper, and Paper wins Rock. A winner gets the payoff of 1 while a loser receives ‐1. If the players choose the same action, both receive 0. In our modified version, an available set of actions is limited in the following way. Player 1 can choose an action from {Rock, Paper, Scissors}, but Player 2 chooses an action only from {Rock, Scissors}. Suppose the game is played only once, and answer the following questions.
A) Draw a payoff matrix.
B) Show that there is no pure strategy Nash equilibrium. C) Explain whether there exists a strictly dominated strategy. D) Solve the mixed strategy Nash equilibrium.
3
Hint: You can use the property that a player will not assign positive probability to strictly dominated strategies in any mixed strategy Nash equilibrium.
Answer: (A)
1 / 2 Rock Scissors
Rock 0, 0 1, ‐1
Paper 1, ‐1 ‐1, 1
Scissors ‐1, 1 0, 0
(B) Skipped. (C) Scissors is strictly dominated by Rock for player 1. (D) p = q= 2/3
4. Dynamic Game (8 points, easy)
Solve each of the following games 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) (AC, CG) (B) (U, AD)
5. Focal Point (3 points, bonus)
Choose a 4 digits number (for example, 1984), and clearly write it down. Do NOT write more than one number. If you will successfully choose the most popular answer, you would get additional 3 points. Otherwise, you will receive 0. You need NOT explain any reason why you choose your answer.
Answer: The majority of the students chose “2010.”
Although there are many potential Nash equilibria, it is reasonable that the current calendar year becomes a “focal point.”