2. An incumbent can commit to producing any quantity of output before the potential rival decides whether to enter. The incumbent chooses whether to commit to produce a small quantity or a large quantity. The rival then decides whether to enter.

- If the incumbent commits to the small output level and the rival does not enter, the rival makes $0 and the incumbent makes $900.
- If the rival does enter, the rival makes $125 and the incumbent earns $450.
- If the incumbent commits to producing the large quantity and the potential rival stays out of the market, the potential rival makes $0 and the incumbent makes $800.
- If the rival enters, it loses $20 and the incumbent earns $400.

a. Draw the extensive form of the game.

b. Solve for the Nash equilibrium.

The Nash equilibrium of the game is where each player is choosing its best response to the other player's choice. In this game, there are two strategies for the incumbent:

(i) Commit to a small output level

(ii) Commit to a large output level. For the potential rival, there are two strategies:

(i) Enter the market

(ii) Stay out of the market.

The Nash equilibrium of this game is found by working backward, starting with the rival's decision nodes.

1. If the incumbent produces a small quantity, the rival's best choice is to enter the market, earning a payoff of $125. If the rival stays out, its payoff will be $0.2. If the incumbent produces a large quantity, the rival's best choice is to stay out of the market, earning a payoff of $50.

If the rival enters, its payoff will be -$20.3. Given the rival's best response, the incumbent's best choice is to produce a small quantity, earning a payoff of $900. If the incumbent chooses to produce a large quantity, its payoff will be $800.

Hence, the Nash equilibrium of the game is (Small Quantity, Enter), where the incumbent produces a small quantity, and the potential rival enters the market.

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