Consider a population of consumers uniformly distributed along the interval from left-hand (x = 0) to right-hand (x=1). The mass of consumers is 1. There are two firms (F1 and F2) that supply homogrnouse goods. The objective of each firm is to maximize its profit. The price is regulated at p. each of the riems simultaneously chooses its location (i.e. a point on the line between x=0 and x=1). The consumers observe the firm's choice, and then each consumer buys from the firm whose location is closest to the consumer's position on the line. If the two firms locate at a same point, they equally split the consumer demand.

For example, if Fi locates as in the following figure, the left-hand consumers buy from F1 and the right-hand consumers buy from F2.

1. In this case, what is a pure-strategy Nash equilibrium? You must explain how to derive it.

2. When there are three firms, no pure-strategy Nash equilibrium exists. Prove it.

3. When there are four firms, what is a pure-strategy Nash equilibrium? You must explain how to derive it.