Simon ECON 482 – Problem Set 4

Simon Fraser UniversityECON 482 Fall 2016Problem Set 4Due on November 28, at 2:30pm Note: In the likely event of presentations, you will want to have these questions in front of you asyour peers present. Therefore, bring a copy of this questionnaire to class on the due date, anddo not attach it to your answers.1. Consider the repeated prisoner’s dilemma from class and section 17.1 of Tadelis, where1player 1 is a “grim-trigger type” with probability ? (0, 3). There are three periods labeled1, 2 and 3, and there is no discounting.a) Let q be the probability that player 1 cooperates in period 1 conditional on being thestrategic type. For each ? [0,1], assume that:- both players best respond in period 2 and 3, and that- player 2’s beliefs in periods 2 and 3 is derived from Bayes’ rule whenever possible.Find player 2’s expected* utilities in periods 2 and 3 from each of the following:i) defecting in period 1, andii) cooperating in period 1.*The expectation is taken before player 1’s action in period 1 is known.Hints: 1. Can the argument from class describing behaviour in the periods 2 and 3 be used here? 2. For ii, you may need to consider the cases ? 1? and ? 1? separately.b) Use part a to show that, in any PBE, player 2 defects with probability 1 in period 1.Hint: Use p < 1/3, and evaluate the cost and benefit of cooperation for player 2 in period1. There is an easy one-line argument.c) Use part b to show that every PBE has the same on-path play, and find it.1 1 2. Consider the repeated prisoner’s dilemma from problem 1, but now suppose that ? (3 , 2).a) Show that there is now no PBE where player 2 defects with probability 1 in period 1.Hint: Use the reasoning from question 1c to pin down what q (as defined in problem 1)would have to be in such a PBE, and then use p>1/3 and question 1a to study player 2’sincentives in period 1.b) Use question 1a to show that, if cooperating is a best response for player 2 in period 1,then cooperating is the unique best response for player 2 in period 2 conditional on bothplayers having cooperated in period 1.Hint: Start by finding the range of q (as a function of p) for which the hypothesis holds,and use what you found in question 1a about what would happen in period 2 for such q. c) Use part b to show that there is no PBE where player 2 cooperates with probability 1 inperiod 1.Hint: What would player 2 cooperating with probability 1 in period 1 imply about player1’s best response in period 1? Given your work on previous questions, is this possible?d) Use parts a-c to show that there is a unique PBE strategy profile, and find it.Note: Unique PBE strategy profile ? unique PBE, since off-path beliefs might differ!