Consider a game between Joe and Snake. This game takes place in the morning after Joe has had breakfast. Snake can do two things: fight Joe or not. He likes fighting with people who are feeling cowardly and gets a payoff of 1 if he does. Snake, however, does not like to fight people who are feeling brave. He gets a payoff of -1 if he does. Regardless of how the other person feels Snake gets a payoff of 0 if he doesn't fight. There is a 75% chance that Joe is feeling cowardly and a 25% chance that he is feeling brave. Joe knows whether he is feeling brave or not, but Snake does not. However, Snake does observe what Joe eats for breakfast. Hence, Joe can use his breakfast choice as a signal to Snake. Joe can choose to have either Quiche or Beer for breakfast. Joe prefers to have Quiche for breakfast and he prefers not to fight. Regardless of whether he is feeling brave or cowardly, Joe gets a payoff of 2 by having Quiche and not fighting, a payoff of 1 from having Beer and not fighting, a payoff of 0 from having Quiche and fighting, and a payoff of -1 from having Beer and fighting.
Draw the extensive-form version of this game.
Find all pure-strategy sub-game perfect Bayes' Nash equilibrium.