There are two players, A and B. Each player i {A, B} can be of one of two types ti {0, 1}. The probability that a player is of type 1 equals . When A...

There are two players, A and B. Each player i ∈ {A, B} can be of one of two types ti ∈ {0, 1}. The probability that a player is of type 1 equals π. When A and B meet, each can decide to fight or cave. If both players fight, then player i gets payoff

ti − c, j ̸= i ti + tj

where c > 0. If player i fights, but player j does not, then player i gets payoff 1, while player j gets payoff 0. If both players do not fight, each gets payoff 1/2. 

1. 1.1 Draw the Bayesian normal form representation of this game.
2. 1.2 Recall that a strategy in a static Bayesian game is a function that specifies an action for each type of a player. Write down all the possible strategies for player i.
3. 1.3 Assume that A plays fight if tA = 1 and cave otherwise. IfBisoftype1,whatshouldshedo? Ifsheisoftype 0, what should she do?
4. 1.4 Is there a Bayesian Nash equilibrium in which each player fights if and only if she is of type 1? If so, what is the equilibrium probability of a fight?
5. 1.5 Assume that A never fights. If B is of type 1, what should she do? If she is of type 0, what should she do?
6. 1.6 Is there a Bayesian Nash equilibrium in which no player ever fights?