Firm A supplies parts for Firm 2's products. They value being located close to each other. But Firm A also likes to be located towards the west (a lower number) and Firm B likes to locate towards the east (a higher number). With x1 being the location of Firm A and x2 being the location of Firm B. Assume that decisions are made simultaneously. The payoffs to the two firms are:
π1 (x1, x2) = 12 - x1^2 - (x2-x1)^2
π2 (x1, x2) = 12 - (1 - x2)^2 - (x2-x1)^2
Derive best response functions for the two firms and find the nash equilibrium.