Exercise 2. While this question has similarity with exercise 1, treat this as an independent ques- tion. Like the last part of Exercise 1, this

Exercise 2. While this question has similarity with exercise 1, treat this as an independent ques-
tion. Like the last part of Exercise 1, this one is also about incomplete information. However, here, not
only A does not know about B’s type, but also B does not know about A’s type. This is an example of
two-sided private information. There are two players, A and B. Each player i e {A, B} can be of one of two types:
I: E {1, 2] . The probability that a player is of type 2 equals p. When A and B meet, each can decide to fight or cave.
If both players fight, then player 1′ gets payoff
It
If + Ij —C where j at i and c>0. Rest of the payoffs are the same as in Exercise 1. (e) (2 marks) Assume that A never fights.
(i) If B is of type 2, what should B do?
(ii) If B is of type 1, what should B do?
(f) (2 marks) Is there a Bayesian Nash equilibrium in which no player ever fights? 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 < p < 1. Player A knows the distribution (p) but not the actual value
of t. Continue with the assumption that c = . Let A, OB1 and o B2 respectively denote the probabilities
that player A, B-type 1, and B-type 2 assign to F. We are looking for Bayesian Nash equilibrium in pure
strategies.
(i) (2 marks) Are there values of p E (0, 1) such that all player types choose to fight? In other words,
are there values of p such that (A, OBI, OB2) = (1, 1, 1) is a Bayesian Nash equilibrium (BNE)?
(ii)(2 marks) Are there values of p E (0, 1) such that player A chooses to fight in a pure strategy
Bayesian Nash equilibrium?