## Subgame Perfect Nash Equilibrium in Cournot oligopoly

I have some questions about how I’m supposed to find the SPNE in this particular Cournot oligopoly:

Consider the following market game: An incumbent firm, called firm 3, is already in an industry. Two potential entrants, called firm 1 and firm 2, can each enter the industry by paying the entry cost of 10. First, firm 1 decides whether to enter or not. Then, after observing firm 1’s choice, firm 2 decides whether to enter or not. Every firm, including firm 3, observes the choices of firms 1 and 2. After this, all off the firms in the industry (including firm 3) compete in a Cournot oligopoly, where they simultaneously and independently select quantities. The price is determined by the inverse demand curve p = 12 – Q, where Q is the total quantity produced in the industry. Asssume that the firms produce at no cost in this Cournot game. Thus, if firm i is in the industry and produces qi, then it earns a gross profit of (12 – Q)qi, in the Cournot phase. (Remember that firms 1 and 2 have to pay the fixed cost 10 to enter.)

Now, I’ve drawn the tree to determine the exact pay offs and I know I’m supposed to use the backward induction in each subgame but how am I supposed to do that with knowing the quantities produced by each firm? Or am I supposed to make assumptions using the information that the observe each other’s actions?