Answer to Question #63497 in Microeconomics for boset00

Question

Consider a two player game with payoff matrix
L R
X

Y

Z

3, θ 0, 0
2, 2θ 2, θ
0, 0 3, −θ
where θ ∈ {−1, 1} is a parameter known by Player 2. Player 1 believes that θ = −1
with probability 1/2 and θ = 1 with probability 1/2. Everything above is common
knowledge.
(a) Write this game formally as a Bayesian game.
(b) Compute the Bayesian Nash equilibrium of this game.
(c) What would be the Nash equilibria in pure strategies (i) if it were common
knowledge that θ = −1, or (ii) if it were common knowledge that θ = 1?