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Midterm Exam: Solutions
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.
Answer: A dominant strategy is a strategy which yields strictly higher payoff than any other strategies do irrespective of other players’ strategies.
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.
Answer: False. A finite game always has a mixed strategy Nash equilibrium, but may not have a pure strategy Nash equilibrium. Remember the “matching penny” game.
b) A Nash equilibrium outcome is NOT necessary Pareto efficient.
Answer: True. Nash equilibrium is not linked with Pareto efficiency. There are many games, e.g., Prisoner’s Dilemma, whose Nash equilibrium is less efficient than other outcomes.
c) A fact is called “COMMON knowledge” if everyone knows it.
Answer: False. This is a definition of “mutual knowledge.” The definition of common knowledge requires higher order knowledge such that everyone knows that everyone knows that everyone knows that … that 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.
Answer: True. If not, then the player prefers not to assign positive probability on the strategy yielding strictly lower expected payoff, which contradicts to the assumption.
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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) Answer: No. It can easily be checked.
b) Find all pure‐strategy Nash equilibria in this game. If there is no pure strategy equilibrium, explain why. (10 points, moderate)
Answer: There is no pure strategy Nash equilibrium. For all strategy combinations, exactly one player has an incentive to switch her strategy.
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) Answer: (p,q) = (1/3,1/2) constitutes a mixed strategy Nash equilibrium.
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)
Answer: Although mutual knowledge of rationality is enough to conclude that the rival never chooses the end point, it is not sufficient to conclude that the rival never chooses next to the end point. Without further assumption on knowledge, the elimination process stops after one round of elimination (eliminating only end points).
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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) Answer: There is no pure strategy Nash equilibrium. Suppose there exist an equilibrium. Then, it must be the case that all three shops locate at the same place. Otherwise, at least one firm has an incentive to move. However, if the location of all the shops is identical, then again each of them has an incentive to change the location. Thus, there can be no equilibrium when we have three shops.
c) How does your answer change if there are FOUR (instead of two or three) ice cream shops? (10 points, difficult)
Answer: There is a unique pure strategy Nash equilibrium in which two shops choose 25 and other two shops choose 75. Try to verify by yourself that no one has an incentive to deviate.
