Consider the following game. The players are 1 and 2. Nature determines that player 1's type is A with probability .5 and B with probability .5. Then player 1 observes her type and chooses Left or Right. Then player 2 observes player 1's move (but not type) and chooses Up or Down. Payoffs are as follows for players 1 and 2 respectively:

For type A :

Left, Up: 3,3

Left, Down: 4,0

Right, Up: 6,8

Right, Down: 2,2

For type B :

• Left, Up: 2,2
• Left, Down: 2,4
• Right, Up: 4,2
• Right, Down: 4,4

(a) Draw the extensive form. Be sure to include the information sets for each player.

(b) What are player 1's pure strategies? What are player 2's pure strategies?

(c) What are the possible separating pure strategies for player 1? What are the possible pooling pure strategies for player 1?

(d) Is there a separating perfect Bayesian equilibrium? Be sure to include player 2's beliefs at every information set.

(e) Is there a pooling perfect Bayesian equilibrium? If so, what is it? Be sure to include player 2's beliefs at every information set.

(f) Is this a signaling game? Why or why not?