Very difficult game theory question..anyone up for it???

Question

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Consider a Cournot duopoly operating in a market with inverse demand P(Q) = a−q , where q = q1 +q2 is the aggregate quality on the market. Both firms have the total costs ci(qi) = q2i (where 2 is the SQUARE), but demand is uncertain – there is a positive probability that a new product will appear at the market and the demand falls drastically. The information about that is

asymmetric, while firm 1 receives information whether demand is high aH = 80, or small aL = 20, before its quantity decision, firm 2 knows just the probability of high demand (θ1 = 2/3 )

and the probability of low demand (θ2 = 1/3 ). All of this is common knowledge. The two firms simultaneously choose quantity. What is the Bayesian equilibrium of the game?

What is the difference between the expected profits of the particular firms (that can be understand as the value of the perfect information about demand)?

Answer 1

i)

q² = (θqh + (1-θ)al - c)) / 3

q¹h = ((3-θ)ah - (3-θ)al - 2c)) / 6

q¹l = ((2+θ)al - (3-θ)ah - 2c)) / 6

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