Consider a road which is represented by the interval [0, 1]. Let a be a number such that 0 < a < 1. Vendor 1 can locate at any point on the interval [0, a] (that is, he can locate at any point x such that 0 x a). Vendor 2 can locate at any point on the interval [a, 1]. A unit mass of onsumers are uniformly distributed on [0, 1] and each consumer buys one unit of the good from the vendor who is closest to him. If the two vendors locate at the same point a, then each gets one-half of the consumers. The game is as follows. Vendors choose locations simultaneiously, and a vendor's payoff is given by the number of consumers who purchase from him. (a) Write down the strategy sets and payoff functions in this game. (b) Suppose a = 0.5. Show that this game has a unique Nash equilibrium in pure strategies. That is, you need to show (i) there is a Nash equilibrium, and (ii) there is no other Nash equilibrium. (c) Suppose a < 0.5. Show that the game does not have a Nash equilibrium in pure strategies