I've been struggling with this problem, if you could show work that would be appreciated

2. Consider a Cournot game between two rms, called 1 and 2, in which the rms simultaneously choose quantities of output to produce. The price of the rms' output is determined by the demand relationship P : (a1 + a2) Q, where Q is the sum of the rms' choices of output, q1 + q2, and a1 and 0:2 are random factors that affect demand. In particular, rm 1 gets to observe the realization of 0:1 before choosing its output (11, but not the realization of 0:2, while rm 2 gets to observe the realization of 052 before choosing its output (12, but not the realization of a1. a1 and cm are determined by the following joint probability distribution: Pr(a1 = a2 = 30) = Pr(a1 = 02 = 9) = 1/3; P'r(a1 = 30,052 = 9) = Pr(a1 = 9, 0:2 2 30) = 1/6. That is, 0:1 and 0:2 both can only equal either 9 or 30, and every combination of 9 and 30 for their values has positive probability. Note that a1 and 02 are not independently distributed. Each rm's cost of producing q units of output is 12q (i.e., the constant cost per unit is 12). (a) How many choices of output does a pure strategy for a given rm specify? Explain. (2 points) (b) Use Bayes's rule to calculate the probability that 052 is 30 given that 0:1 is 30 (i.e., Pr(o:2 = 30|a1 = 30)) and the probability that 112 is 30 given that a1 is 9 (i.e., 1971052 = 30|a1 = 9)). Note: because the probability distribution satises a property called symmetry, it will be the case that P'r(o:2 = 30|a1 = 30) = Pr(o:1 = 30|a2 = 30), and Pr(o:2 = 30|a1 = 9) = P'r(0z1 = 30|a2 : 9). (4 points) (c) Write down the expected payoff to a rm 1 when its observation of a1 is 30, as a function of its choice of output (11 and as a function of a strategy for rm 2. Do the same for rm 1 when it observes 0:1 = 9. (2 points) (c) Find the best-response function for each type of rm 1. Note: because the game is symmetric, a given type of rm 2 has the same best response, as a function of a strategy for 1, as the same type of rm 1. (4 points) (d) Solve for a Bayes Nash equilibrium of this game in which each rm uses the same strategy, i.e., in which what rm 1 does for a particular value of a1 is the same as what rm 2 does for that same value of 0:2. You may leave your answers as unsimplied fractions (i.e., fractions of fractions) if you wish. (8 points)