Solved: Alice and Bob Case Study-Probability Matrix
Alice is visiting New York. She calls her friend Bob, who lives in New Haven, from a pay phone, and they decide that Bob comes to New York and they meet at the train station. Before they can clarify which train station, Bob's cell phone's battery dies, and they can no longer communicate. Unfortunately, there are two train stations from New Haven to New York.: Amtrak, which arrives at the Penn Station, and MetroLiner, which arrives at the Grand Central, both arriving at noon. Clearly, Bob may take Amtrak, MetroLiner, or give up and stay at home. Alice may check either of the stations (but not both). It is commonly known among them that they are both expected utility maximizers and that the following about their preferences are true. They are indifferent between Bob taking Amtrak while Alice checking the Grand Central and Bob taking MetroLiner while Alice waiting at the Penn Station. That is, missing each other is equally bad. Alice is also indifferent between where they meet. For Alice, waiting at the Penn Station while Bob stays home is as bad as missing each other. But she would feel worse if she waits at the Grand Central and Bob stays home. In particular, Alice's preferences are such that, if she assigns probabilities p, q, r to Amtrak, MetroLiner, and staying home respectively, then she would prefer the Penn Station to the Grand Central if and only if p is greater than q minus r/2. If Bob stays home, he does not care which station Alice checks. He prefers Amtrak to MetroLiner if and only if the probability of Alice waiting at the Penn Station is greater than 1/3. He prefers Amtrak to staying home if the probability of Alice waiting at the Penn Station is greater than 2/3.- What is the payoff matrix for the above (Alice is Row, Bob is Column).
- Find all the possible outcomes given that it is common knowledge that both players are expected utility maximizers with the above preferences.