Homework # 1 - ISyE 6230 – Economic Decision Analysis II – Spring 2010 Due Thursday Jan 28, 2010 Grading: Up to 10 points per problem for completing each problem; the remaining 50 points will come from grading one randomly selected problem for correctness. 1. Give an example of a business situation that can be modeled as a game: Find an article from a reputable journalistic source (e.g., NY Times, Wall Street Journal, Forbes) detailing a business situation. Identify the players involved, the set of actions, and a description of the payoffs. If you can identify the classic game to which the situation belongs (e.g., Chicken, Prisoner’s Dilemma, etc.), then make this argument. Predict the outcome of this game; if applicable, describe how the outcome would be different if there were not multiple players acting strategically. Make sure to detail your reference for this article explicitly. Be prepared to describe your business example in class the day homework is due. 2. Consider the following payoff matrix: Player II Left Center Right Up 2,2 2,2 0,0 Down 0, 0 2,4 2,4 Find an example using this payoff matrix that shows the equilibrium that would result from eliminating weakly dominated actions may depend on the order in which actions are eliminated. 3. Consider the following payoff matrix for a game: Player II C D A 2.5, 2.5 - 4, 4 B -3.5, - 4 0,0 (i) Are there any dominated strategies for player I? For player II? (ii) Are there any dominated outcomes? (iii) Find all mixed-strategy Nash equilibrium in this game? (Recall that a pure strategy is also a mixed strategy). (iv) Assume now that the payoff for the outcome (A,D) is (-4,x). Describe all the equilibria (mixed -strategy and pure-strategy) achieved as a function of x. Player I Player I4. Consider the following game in normal form: Player II BLUE RED YELLOW WHITE blue 0, -2 5, 5 0, 0 2, 0 red 0, 3 0,0 5, 5 1, 0 yellow 6, 2 2, 0 1, 3 1, 3 white 5, 5 1, 0 0, 1 0, 6 (i) Iteratively eliminate all strictly dominated strategies; state the assumptions necessary for each elimination. (ii) What are the rationalizable strategies? (iii) What are the pure-strategy Nash equilibria? 5. In the following bargaining game, players A and B are bargaining over how to split one dollar. Both players simultaneously name shares they would like to have, sA and sB, where 0 ≤ sA; sB ≤1. If sA + sB ≤ 1, then the players receive the shares they named; if sA + sB > 1, then both players receive zero. (i) What are each player's strictly dominated strategies? (ii) What are each player's weakly dominated strategies? (iii) What are the pure strategy Nash equilibria of this game? Player
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