How to Solve Bayesian Game Equations?

[Poisonous]

Homework Statement

My Textbook gives the following example problem:

Cournot Duopoly with incomplete information.

The profit functions are given by:
u_i = q_i(θ_¡ - q_i - q_j)

Firm 1 has one type θ_1 = 1, but firm 2 has private information about its type θ_2. Firm 1
believes that θ_2= 3/4 with probability 1/2 and θ _2 = 5/4 with probability 1/2, and this belief is common knowledge.

We will look for a pure strategy equilibrium of this game. Firm 2 of type θ_2’s decision
problem is to

max q_2: q_2(θ_2 - q_1 - q_2)

which is solved at
q_'2(θ_2) = (θ_2-q_1)/2


Firm 1’s decision problem, on the other hand, is
max q_1: 1/2 * q_1 (1 - q_1 - q_'2(3/4)) + 1/2 * q_1(1-q_1-q_'2(5/4))

which is solved at
q_'1 = (2-q_'2 (3/4)-q_'2(5/4))/4

Solving yields,
q_'1 = 1/3
q_'2(3/4) = 11/24
q_'2(5/4) = 5/24

Atempt at a solution

I can't figure out how they got from the equations q_'1 and q_'2 to the answers given in the "solving yields" part. It seems like simultaneous equations, but I'm not sure what to do with the q_1 or really where to begin.

 

[cris(c)]

Recall that in any bayesian game, strategies are mappings from types to quantities. This means that firm 2 'quantity' must specify what the the type \theta_{2}=3/4 will produce and how much the type \theta_{2}=5/4 will produce. Now, since firm 2 knows his own type, it can maximize using this information: this is why you obtain two best response functions, one for each type. Next, to compute the Nash equilibrium you also need to derive a best response for firm 1. Firm 1 only know the probability of each type and hence, incorporates two different quantities (one for each type) together with the probabilities of meeting each type in his maximization problem. This will give you a third best response that must be a function of the quantity type \theta_2=3/4 is expected to produce, and the quantity that type \theta_2=5/4 is expected to produce. All this gives you 3 equations (2 best responses for firm 2 and one for firm 1) and 3 unknowns (quanties for each type of firm 2 and the quantity for firm 1). You can solve this system of equations the way you like (substitution, reduction, etc). This will give you the solutions you state in the last part of your question.