Question

Game Theory Econ

Imagine a market setting with three firms. Firms 2 and 3 are already operating as monopolists in two different industries (they are not competitors). Firm 1 must decide whether to enter
firm 2’s industry and thus compete with firm 2 or enter firm 3’s industry and thus compete with firm 3. Production in firm 2’s industry occurs at zero cost, whereas the cost of production in firm 3’s industry is 2 per unit. Demand in firm 2’s industry is given by p = 9-Q, whereas demand in firm 3’s industry is given by p' = 14-Q', where p and Q denote the price and total quantity in firm 2’s industry and p' and Q' denote the price and total quantity in firm 3’s industry. The game runs as follows: First, firm 1 chooses between E² and E³. (E² means "enter firm 2’s industry" and E³ means "enter firm 3’s industry.") This choice is observed by firms 2 and 3. Then, if firm 1 chooses E², firms 1 and 2 compete as Cournot duopolists, where they select quantities q₁ and q₂ simultaneously. In this case, firm 3 automatically gets the monopoly profit of 36 in its own industry. In contrast, if firm 1 chooses E³, then firms 1 and 3 compete as Cournot duopolists, where they select quantities q₁ and q₃ simultaneously; and in this case, firm 2 automatically gets its monopoly profit of 81/4.

(a) Calculate and report the subgame perfect Nash equilibrium of this game. In the equilibrium, does firm 1 enter firm 2’s industry or firm 3’s industry?

(b) Is there a Nash equilibrium (not necessarily subgame perfect) in which firm 1 selects E²? If so, describe it. If not, briefly explain why

Answer 1

a. We need to find out how the sub games starting after E3 and E2 are played.

E2: We have Cournot competition between 1 and 2 in firm 2’s industry, where p= 9-Q and production can be seen at zero cost. This subgame has a unique Nash equilibrium, where 1 and 2 choose quantity of q1=q2= 3, for a profit of 9, and 3 chooses q3= 6 for a profit of 36.

E3: We now have Cournot competition between 1 and 3 in firm 3’s industry, where p= 14-Q and the marginal cost stands at 2. This subgame has a unique Nash equilibrium, where 1 and 3 choose quantity of q1=q3= 4, for a profit of 16, and 2 chooses q2= 9/2 for a profit of 81/4.

So, we can see that there is a unique subgame perfect equilibrium.

Firm 1 chooses E3, produces q1= 3 if it chooses E2, and produces q1= 4

Firm 2 produces q2= 9/2 if firm 1 chooses E3 and produces q2= 3

Firm 3 produces q3= 6 if firm 1 chooses E2 and produces q3= 4

b. Yes there is a Nash equilibrium in which firm 1 selects E2.

Firm 1 chooses E2, produces q1= 3 if it chooses E2 and produces zero

•Firm 2 produces q2= 9/2 if firm 1 chooses E3 and produces q2= 3

•Firm 3 produces q3= 6 if firm 1 chooses E2 and produces q3= 12

Firm 3 can adopt a strategy where q3= 12 if it enters the industry when it is booming, lowering firm 1's profit from entering to 0. At equilibrium, this restricts firm 1 from entering against firm 3, this way sub optimal behavior following entry does not lower firm3's payoff. This game is Nash equilibrium but not subgame-perfect.