Assignment:
Signaling game Caesar is at a cafe, trying to choose what to drink with breakfast: beer or orange juice. Brutus, sitting at a nearby table, is pondering whether or not to challenge Caesar to a duel after breakfast. Brutus does not know whether Caesar is brave or cowardly, and he will only dare to challenge Caesar if Caesar is cowardly. If he fights a cowardly opponent, he receives one unit of utility, and he receives the same single unit of utility if he avoids fighting a brave opponent. In contrast, he loses one unit of utility if he fights a brave opponent, and similarly loses one unit of utility if he dishonors himself by failing to fight a cowardly opponent. Brutus ascribes probability 0.9 to Caesar being brave, and probability 0.1 to Caesar being a coward. Caesar has no interest in fighting Brutus: he loses 2 units of utility if he fights Brutus, but loses nothing if there is no fight. Caesar knows whether he is brave or cowardly. He can use the drink he orders for breakfast to signal his type, because it is commonly known that brave types receive one unit of utility if they drink beer (and receive nothing if they drink orange juice), while cowards receive one unit of utility if they drink orange juice (and receive nothing if they drink beer). Assume that Caesar’s utility is additive; for example, he receives three units of utility if he is brave, drinks beer, and avoids fighting Brutus.
Answer the following questions:
(a) Describe this situation as an extensive-form game, where the root of the game tree is a chance move that determines whether Caesar is brave (with probability 0.9) or cowardly (with probability 0.1).
(b) Find all the Nash equilibria of the game.
(c) Find all the sequential equilibria of the game.
Provide complete and step by step solution for the question and show calculations and use formulas.