Consider the following description of an extensive form game with incomplete information. A sender is either a strong type or a weak type with equal probability. The sender can select either action A or action B. A receiver observes the sender's action and can either respond with action not brawl or action brawl. If the sender is strong and the receiver chooses action not brawl, then sender receives 1 and receiver gets 0, regardless of sender's action. If the sender is strong and the receiver chooses action brawl, the sender gets 0 and receiver gets -2, regardless of sender's action. If the sender is weak and selects action A and the receiver responds with action not brawl, sender gets 1 while the receiver gets 0. If the sender is weak and chooses action A and the receiver responds with action fight, the sender receives -1 while the receiver gets 1. If the sender is weak and chooses action B and the receiver responds with action not brawl, sender gets 1 c and receiver gets 0. If the sender is weak and chooses action B and the receiver responds with action brawl, sender gets 1 c while the receiver gets 1.

a. Draw the extensive form game tree.

b. Explain what it means for the sender to have a pooling strategy in this game. What are the pooling strategies for this game?

c. Explain what it means for the sender to have a separating strategy? What are the separating strategies for this game?

d. What conditions on c guarantee the existence of a separating Perfect Bayesian equilibrium? Characterize such an equilibrium. (Characterize conditions on c such that player 1 doesn't have incentives to deviate from the equilibrium strategy.)

e. What conditions on c guarantee the existence of a pooling Perfect Bayesian equilibrium? Characterize such an equilibrium. (Also specify what off-path beliefs are to fully characterize the equilibrium.)