Perfect Bayesian Equilibrium in a two stage game with incomplete information

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, $A,B$ 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 lowest price.

Both firms initially are able to produce the product at cost $C$, but each firm $i in {A,B}$ may decrease it productions costs by $c_i$ in private by investing $d_i (c_i)^2$.

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 game stages is:

1. The two firms simultaneously privately invest into cost reductions $c_A,c_B$.
2. The firms set prices $p_A,p_B$ for the 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?