Question: Consider the following market game: An incumbent firm, called firm 3, is already in an industry. Two potential entrants, called firms 1 and 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 of 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. Assume 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.)
(a) Compute the subgame perfect equilibrium of this market game. Do so by first finding the equilibrium quantities and profits in the Cournot subgames. Show your answer by designating optimal actions on the tree and writing the complete subgame perfect equilibrium strategy profile. [Hint: In an n-firm Cournot oligopoly with demand p = 12 - Q and 0 costs, the Nash equilibrium entails each firm producing the quantity q = 12>(n + 1).]
(b) In the subgame perfect equilibrium, which firms (if any) enter the industry?