In this problem set, we will solve a game of incomplete information.

Adrian got Bailey's number! Now Adrian has to decide whether to call Bailey immediately or wait three days. After Bailey gets the call, Bailey will be able to decide whether to go on a date or not. Suppose there are two types of Adrian: desperate or cool. Let's say that they are cool .6 of the time, and desperate .4 of the time. If Adrian is desperate, Bailey doesn't want to go on the date; if Adrian is cool, Bailey does. Bailey gets 1 point of utility from dating a cool Adrian, 0 from not going on a date, and -1 from dating a desperate Adrian.

Adrian always prefers to call immediately, but desperate types like waiting less. Suppose it

costs 1 point for a cool Adrian to wait, and 3 points for a desperate Adrian to wait. The date itself, if it happens, is worth C points.

1. (10 pts) Solve this game as a signaling game.
2. (15 pts) First suppose C is 2. Is there a seperating equilibrium in which only the cool type waits, while the desperate type calls immediately?
3. (15 pts) Is there a pooling equilibrium where they both call right away?
4. (15 pts) Is there a pooling equilibrium where they both wait to call?
5. (45 pts) Now suppose C is 4. Repeat questions 2-4 with this assumption. How does this change the results.