A chevalier contests a marquis to a duel to seize his land. The chevalier is strong with probability 1/2 and weak with probability 1/2. He knows his own type but the marquis does not. The chevalier may either get ready for the duel or remain unready (he does not have the option to not fight). Marquis observes the chevaliers readiness, but not his type, and chooses whether to accept the duel & fight or surrender. (The marquis updates his beliefs about the marquis type upon observing his action.) An unready chevaliers payoff is 10 if the marquis surrenders. Preparations cost a strong chevalier 3 unit of payoff and a weak one 6 units, and fighting entails a loss of 4 units for the chevalier regardless of his type. The marquis prefers to fight with (payoff 3) rather than surrender to (payoff 0) a weak chevalier, and prefers to surrender to (payoff 5)rather than to fight with (payoff -1) a strong one.

a.(9 points)Draw this game in extensive form. Make sure to include information sets where relevant. Write down chevaliers payoff first and marquis payoff second.

b.(6 points)Write down all possible strategies for chevalier and for marquis. (Hint 1: Be careful how you define strategies. I am not asking what their possible actionsare. A strategy is a complete set of actions for every possible scenario a player can find himself in. Hint 2:Chevaliers strategy should include a component for what to do if hes strong and a component for what to do if hes weak. Marquisstrategy should include a component for what to do if chevalier gets ready and a component for what to do if he remains unready.)

c.(10 points)Find a separating PBNE.(Hint: PBNE must include strategies for both players and belief of Marquis about Chevaliers type.) (Warning:Do not simply guess here. I need you to show your work proving why this is a PBNE. Show why no player has an incentive to deviate given the Marquis belief.)

d.(10 points)Find a pooling PBNE.(Hint: PBNE must include strategies for both players and belief of Marquis about Chevaliers type.) (Warning:Do not simply guess here. I need you to show your work proving why this is a PBNE. Show why no player has an incentive to deviate given the Marquis belief.)