I would like to solve a game where firms have private information about their own type, but only know the distribution of the other firm's type. They interact in two stages, where the strategies depend on each other. The first period is an optimization problem, and in the second period Nash Equilibria have to be identified.
Example: Consider two firms who are trying to sell their product to one customer. The customer just needs one indivisible unit of the product, which has value V for the customer, and will buy the product from the firm that offers the cheaper price. Both firms initially are able to produce the costs at costs C, but each firm i may invest d_i (c_i)^2 to decrease it productions costs by c_1. Each firm i has private knowledge about its own d_i, but only knows the distribution of d_j. (If the solution needs a specific distribution just pick one which fits your needs.) The sequence of the game is: First, the two asymmetric firms invest into cost reductions. Second, the firms price their product.
Straightforward backward induction does not work in my trials, due to the asymmetry of cost reduction investments.
Question: Is there a good textbook or paper where sth. similar is done or do you have any technical knack which I have missed?