Midterm Exam
Date: March 4, 2009
Subject: Game Theory (ECO290E) Instructor: Yosuke YASUDA
1. Dominant Strategy (10 points, easy)
State the definition of the “(strictly) dominant strategy” (either by words or mathematically) within FOUR lines.
2. True or False (20 points, difficult)
Answer whether each of the following statements is true or false. You do NOT need to explain the reason.
a) If a game has finite number of players and strategies, there ALWAYS exists a pure strategy Nash equilibrium.
b) A Nash equilibrium outcome is NOT necessary Pareto efficient. c) A fact is called “COMMON knowledge” if everyone knows it.
d) If two different pure strategies are used (with positive probabilities) in a mixed strategy Nash equilibrium, then these strategies MUST yield the same expected payoff given the equilibrium strategies for other players.
3. Simple 2-2 games (25 points)
Consider the following 2-2 game.
P1 / P2 L R
U 2,0 0,1
D 0,1 1,0
a) Is there any strategy that is strictly dominated by other strategy? (5 points, easy) b) Find all pure-strategy Nash equilibria in this game. If there is no pure strategy equilibrium, explain why. (10 points, moderate)
c) Suppose player 1 takes U with probability q and D with probability (1-q). Likewise, player 2 takes L with probability p and R with probability (1-p). Find a combination of p and q which constitutes a mixed strategy Nash equilibrium. (10 points, moderate)
4. Spatial Competition (25 points)
Consider a spatial competition model discussed in the lecture: Two ice cream shops simultaneously chose the location between 0 and 100 on the beach, and the payoffs are given by the number of customers who are uniformly located on the beach and go to the nearest shop.
a) Remember the argument in the lecture. To obtain the Nash equilibrium, i.e., both shops choose 50, by iterated elimination of strictly dominated strategies, we assumed common knowledge of rationality. Explain why mutual knowledge alone is not enough for this elimination process to work. (5 points, moderate)
b) Suppose the number of ice cream shops increase, and now THREE (instead of two) shops chose the location simultaneously in the above game. Is there any pure strategy Nash equilibrium? If yes, find all such equilibria. If no, explain why. (10 points, difficult) c) How does your answer change if there are FOUR (instead of two or three) ice cream shops? (10 points, difficult)
