Consider the following discrete version of Bayesian Cournot duopoly with asymmetric information. Firm 1 has only one type but Firm 2 has has either low cost or high cost with equal probability. Firm 2 knows his cost level but Firm 1 does not know Firm 2’s cost level. Each firm (of each type) chooses whether to produce high (Qh) or low output (Ql). The game is illustrated as follows.

Solve for all Bayesian-Nash equilibria of this game (in pure and mixed strategies).
 

1\2 Qh Qe Qh -2, -2 0,2 Qe 2,0 0,0 H Cost Probability: 0.5 1\2 Qe Qh 2, 2 0,3 Qe 2, 1 6, 6 L Cost Probability: 0.5