please answer question 1,2,3

Consider the following variant of the standard Hotelling spacial location model (NOTE: this is a similar environment as the one in a question in PS1). The set of the product characteristics is [0,1]. There are two firms, firms 1 and 2 , whose production costs are both zero. Fiach firm can produce only one type of product, and thus chooses exactly one characteristic. There is a continuum of consumers whose characteristics (tastes) are uniformly distributed in the product characteristic space. Each consumer buys exactly one unit of the product, and buys from a firm whose product characteristic is closer to his or her own characteristic (and buys with equal probability if s/ he is in an equal distance from both products). The price is fixed exogenously, say at p>0, and the firm's payoff is equal to its profit. Assume that, unlike in the standard Hotelling model, the timing of the game is as follows. Firm 1 chooses its product characteristic first, and then firm 2 observes 1's choice and chooses its product characteristic. 1. (5 points) What is the space of (pure) strategies for each firm? 2. (5 points) Identify a pure-strategy subgame perfect Nash equilibrium of this game if there exists any. If not, then prove nonexistence. 3. (5 points) Now, modify the environment and assume that there are an odd number of product characteristics, 1,2,,n, and consumers are distributed uniformly on {1,2,,n}1 In this setting, identify a purestrategy subgame perfect Nash equilibrium of this game if there exists any. If not, then prove nonexistence. Consider the following variant of the standard Hotelling spacial location model (NOTE: this is a similar environment as the one in a question in PS1). The set of the product characteristics is [0,1]. There are two firms, firms 1 and 2 , whose production costs are both zero. Fiach firm can produce only one type of product, and thus chooses exactly one characteristic. There is a continuum of consumers whose characteristics (tastes) are uniformly distributed in the product characteristic space. Each consumer buys exactly one unit of the product, and buys from a firm whose product characteristic is closer to his or her own characteristic (and buys with equal probability if s/ he is in an equal distance from both products). The price is fixed exogenously, say at p>0, and the firm's payoff is equal to its profit. Assume that, unlike in the standard Hotelling model, the timing of the game is as follows. Firm 1 chooses its product characteristic first, and then firm 2 observes 1's choice and chooses its product characteristic. 1. (5 points) What is the space of (pure) strategies for each firm? 2. (5 points) Identify a pure-strategy subgame perfect Nash equilibrium of this game if there exists any. If not, then prove nonexistence. 3. (5 points) Now, modify the environment and assume that there are an odd number of product characteristics, 1,2,,n, and consumers are distributed uniformly on {1,2,,n}1 In this setting, identify a purestrategy subgame perfect Nash equilibrium of this game if there exists any. If not, then prove nonexistence