The two towers: Ann and Bob stay in different hotels in San Francisco and plan to share a cab visit their friend Carol. They agree to meet by “the tower”, but they forget to specify which tower. They both realize that there are two natural choices: Sutro Tower or Coit Tower. Not having cell phones, each must choose independently which tower to go to. Sutro tower is closer to their destination, i.e., if they pick a cab from Sutro they are going to pay $20 each; whereas if they pick a cab from Coit they are going to pay $30 each. They share the same preferences, which look as follows: 

(i) they both prefer to meet over not meeting, 

(ii) they both prefer to pay the lower fare, and 

(iii) if Ann believes that the probability of Bob going to Sutro is 1/4 (and of course, the probability of him going to Coit is 3/4), then she is indifferent between the two towers. 

(a) Model this as a normal-form game and write down the matrix form of the game. 

(b) Suppose that Ann believes that Bob will go to Sutro with probability 1/3. Then is it rational for her to toss a (fair) coin in order to decide which tower she is going to? 

(c) (challenging). Suppose now that Ann has a third choice, namely to stay at her hotel. Her preferences are such that: 

(i) if she stays at the hotel then she does not care where Bob goes, 

(ii) if Bob goes to Sutro then she is indifferent between staying at the hotel and tossing a coin to decide which tower she will go to. Model this new situation as a normal form game and write down the matrix form.