Problem
Consider a two-period model with two firms, A and B. In the first period, they simultaneously choose one of two actions, Enter or Don't enter. Entry requires the expenditure of a fixed entry cost of 10. In the second period, whichever firms enter play a pricing game as follows. If no firm enters, the pricing game is trivial and profits are zero. If only one firm enters, it earns the monopoly profit of 50. If both firms enter, they engage in competition as in the Bertrand model with homogeneous products.
Using backward induction, fold the game back to the first period in which firms make their choice of Enter or Don't enter. Write down the normal form (a 2 x 2 "matrix") for this game. Denote p the probability that firm A plays Enter; and denote q the probability that firm B plays Enter.
I. Identify all the pure-strategy Nash equilibrium. (Multiple choice possible.)
II. Now focusing on the mixed-strategy Nash equilibrium, with what probability will firm A enter the market?
III. Still focusing on the mixed-strategy Nash equilibrium, with what probability will firm B enter the market?
IV. Now suppose that the two firms have different costs of entry. For firm A, the cost of entry is still 10; but for firm B, the cost is now 20. In the mixed-strategy Nash equilibrium, with what probability will firm A enter the market?
V. In the mixed-strategy Nash equilibrium, with what probability will firm B now enter the market?