Exercise 1 is only a reference, please only answer Exercise 2

Exercise 1. The environment described in this question applies to many situations including advertising, innovation, political campaign among others. In case of advertising and innovation, the game can be interpreted as fighting for market shares. In political context, it can be a fight for vote shares. There are two players, A and B. When A and B meet, each can decide to fight (F) or cave (C). Fighting is costly. It requires resources. If both choose to fight, fight occurs and each gets A gets 14 - c while B gets 147 - c. If only one, A or B, chooses to fight, then the one who chooses to fight gets 1 while the one who caves gets 0. In case both decide not to fight, i.e. both choose cave (C) each gets ?. (a) (3 marks) Suppose t = 1 and c = 3. (i) (1 mark) Write down the payoff matrix for the game and state all Nash equilibria in pure strategies. (ii)(2 marks) Find the unique Nash equilibrium in mixed strategies. (b) (3 marks) Suppose t = 2 and c = 3. (i) (1 mark) Write down the payoff matrix for the game and state all Nash equilibria in pure strategies. (ii) (2 marks) Consider the mixed strategy Nash equilibrium (MSNE) that you found in (a). Is that still a Nash equilibrium? (c) (4 marks) Suppose t denotes the type of player B. Player B's type is t = 1 with probability p and t = 2 with probability 1-p where 0 0. Rest of the payoffs are the same as in Exercise 1. (c)(2 marks) Assume that A plays fight if tA = 2 and cave otherwise. (i) If B is of type 2, what should B do? (ii) If B is of type 1, what should B do? (d) (2 marks) Is 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? [Note: Fight occurs in equilibrium when both player chooses to fight]