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 chevalier's 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

chevalier's 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. Draw this game in extensive form. Make sure to include information

sets where relevant. Write down chevalier's payoff first and marquis' payoff second.

b. Write down all possible strategies for chevalier and for marquis. (Hint1:

Be careful how you define strategies. I am not asking what their possible actions

are. A strategy is a complete set of actions for every possible scenario a player

can find himself in. Hint2: Chevalier's strategy should include a component for

what to do if he's strong and a component for what to do if he's weak. Marquis'

strategy should include a component for what to do if chevalier gets ready and a

component for what to do if he remains unready.)

c. Find a separating PBNE. (Hint: PBNE must include strategies for

both players and belief of Marquis about Chevalier's 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. Find a pooling PBNE. (Hint: PBNE must include strategies for both

players and belief of Marquis about Chevalier's 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.)