There are two players, A and B. Each player i {A,B} can be of one of two types t_i {1,2}. The probability that a player is of type 2 is . When A and B meet, each can decide to fight or cave. If both players fight, then player i gets payoff (t_i/(t_i + t_j)) - c, j != i, where c > 0. If player i fights, but player j does not, then player i gets payoff 1 and j gets payoff 0. If both players do not fight, each gets payoff 1/2.

1.1 Draw the Bayesian normal form representation of this game.

1.2 Recall that a strategy in a static Bayesian game is a function that specifies an action for each type of player. Write down all possible strategies for player i.

1.3 Assume that A players fight if t_A = 2 and cave otherwise. If B is of type 2, what should they do? If they are of type 1, what should they do?

1.4 If there a Bayesian Nash equilibrium in which each player fights if and only if she is of type 2? If so, what is the equilibrium probability of a fight?

1.5 Assume that A never fights. If B is of type 2, what should she do? If she is of type 1, what should she do?

1.6 Is there a Bayesian Nash equilibrium in which no player ever fights?