ECON3012 Strategic behaviour
Q1 (33 points) Suppose that, statistically, 90% of males in a country are faithful to their partners, while the remaining 10% are unfaithful. James is coupled with Betty, who has no information on James' faithfulness aside from the statistics. Aware of this, Betty thinks, \"If I could access James' phone, I would know for sure whether he is faithful to me.\" Then Betty argues, \"If you didn't have anything to hide you'd show me your phone," and threatens James with a breakup if he does not give her his phone. The game tree below represents the situation. James has to make a choice between giving Betty his phone (P) and not giving his phone (NP). Following James' action, Betty has to decide whether to break up (B) or not (NB). If James gives Betty his phone, Betty will learn whether he is faithful or not. If he does not give her his phone, Betty does not learn whether he is faithful, as represented by the information set (the dotted line) in the game tree. At each terminal node, the rst number represents the utility for James and the second number represents the utility for Betty. As these numbers suggest, James prefers not to break up and not to give Betty his phone. Betty prefers a breakup if and only if James is unfaithful; her utility is the highrst if she is in a relationship with a faithful .lames. Nature Unfalthful (a) {2 points] How many subgames [including the original game) does this game have? (b) (3 points) What will Betty do if James is faithful and gives her his phone (P)? What will Betty do if James is unfaithful and gives her his phone (P)? (c) (4 points) Suppose that. Betty's strategy is to break up (B) if James does not give her his phone. James' strategy is to give his phone (P) if he is faithful and not give his phone (NP) if he is unfaithful. What is Betty's expected utility? What is James' expected utility? (Hint: you should use your answer in (b) in your calculation.) (d) (4 points) Suppose that Betty's strategy is not to break up (NB) if James does not give her his phone. James' strategy is to give his phone (P) if he is faithful and not give his phone (NP) if he is unfaithful. What is Betty's expected utility? What is James' expected utility? (e) {10 points) Write down the normal form of this game (i.e., the game matrix). The rows of your matrix should be James' strategies, and the columns of your matrix should be Betty's strategies if James does not give her the phone. The entries of your matrix should include your answers to (e) and (d), as well as the expected utilities in the other scenarios. (f) (10 points) Find all pure-strategy perfect Bayesian equilibrium of this genre. Be sure to fully describe each player's complete strategy. In each equilibrium, do they break up if James is faithful? Do they break up if James is unfaithful
