Question 3: An ice-cream vendor game On a hot day at the beach just outside the city of Spilrup, Denmark, two ice-cream vendors (1 and 2) compete with each other. We can think of the beach as being one-dimensional, and as having the length one. Moreover, a continuum of people are located along the beach; their distribution is uniform and their total mass is one. During the day, each one of the beach visitors will buy one ice-cream, and he or she makes the purchase from the vendor that is closest to him/her. This is because the mayor of Spilrup has decided that any ice-cream sold at the beach must cost exactly 20 kroner, and the ice-cream quality and all other aspects of the ice-creams are the same. Consequently, the vendors can compete only in locations. Vendor i's location is denoted by s;. It is possible for the vendors to locate at the same spot (s; = s2), in which case they get half of the customers each. The normal-form representation of this game can be written as G = (N, S, ), where N = {1,2},S =[0,1] x [0,1], and 7w = (71, 712) with 51_552 if$1 at2 jfs) > sp. (a) Given the verbal description of the game, explain why the mathematical expression for Vendor 1's payoff, as stated in (*), should look like that. That is, where from do the terms % and %, etcetera, come from? (b) Derive Vendor 1's best-response correspondence. (c) Solve for all (pure strategy) Nash equilibria of the game