4) HOTELLING LOCATIONS: MOVIE RATINGS Profit-seeking firms F1, F2, and F3 serve a market in which 300 movie fans have heterogeneous preferences about kid- friendliness. We could think of this in terms of moving ratings, such as G, PG, PG-13, R, etc. E.g., the 60 fans near the low end of the spectrum prefer something very kid-friendly, whereas the 30 fans near the high end of the spectrum prefer to see a very kid-unfriendly movie! Suppose travel costs are not an issue here: all 300 fans will attend a movie at the nearest firm, and if two or more firms are equidistant, then those firms will split the customers equally (if there are 2 movies to choose from, they flip a coin). NOTE: Kids are located at odd-numbered locations (1,3,5,7); firms can locate only at even-numbered locations (0,2,4,6,8). a) Suppose F2 has already located at 4/8 and F3 has already located at 6/8. Your job, as manager of F1, is to determine the customer-maximizing "location" on this spectrum. The last row of the table says that if you locate at 8/8, you will split the 30 customers at 7/8 with rival firm F3, and won't get any other customers farther to the left, so you will attract only 15 customers. Complete the rest of the table. b) In general terms, describe a Nash Equilibrium in this type of game, assuming that all three firms can choose their locations. TIP: You don't have to use any specific numbers from this game in your answer. Is (L1=2/8, L2 = 4/8, L3 = 6/8) a Nash Equilibrium? Explain. c) d) Is (L1 e) f) L2 = L3 = 4/8) a Nash Equilibrium? Explain. Is (L1 L2 = L3 = 6/8) a Nash Equilibrium? Explain. Is (L1 = L2 = 4/8, L3 = 6/8) a Nash Equilibrium? Explain. 60 fans at 1/8 90 fans at 3/8 120 fans at 5/8 30 fans at 7/8 2/8 4/8 6/8 L2=4/8 L3=6/8 8/8 # of Fl's Customers a) Signposts at → 0 2/8 4/8 6/8 8/8=1 Group of kids at → 1/8 3/8 5/8 7/8 If L1 = 0 L1 L2 L3 If L1 = 2/8 LI L2 L3 If L1 = 4/8 L2, L1 L3 If L1 = 6/8 L2 L3, L1 If L1 = 8/8 0 0 L2 0 L3 15 LI 15 0 2/8 4/8 For the other parts of the problem, this table may be useful in checking whether a firm would benefit from a unilateral relocation: Signposts at → 6/8 8/8=1 Group of kids at → 1/8 3/8 5/8 7/8 # of Fl's Customers If L1 = 0 L1 If L1 = 2/8 If L1 = 4/8 If L1 = 6/8 If L1 = 8/8 L1 L1 L1 LI