3. (32 pts) Consider a modified Hotelling game. Two food vendors, A and B, decide where to set-up their stands on a linear beach that is broken into five segments. There are two customers per segment. The two vendors dislike each other; if they locate at the same segment each will get a profit of $0. If the vendors locate at different segments, each customer will go to the closest vendor. If any customers are equal distance from the two vendors, half will go to one and half will go to the other. The vendors make $5 profit per customer. For instance, if vendor A locates in 1 and vendor B locates in 3,A will have 3 customers ( 2 from 1 and 1 from 2 ) and B will have 7 customers ( 1 from 2 and 2 each from 3,4 and 5.) A 's payout will be $15 and B 's payout will be $35. If vendor A and vendor B both locate in 4 , each of their payoffs will be $0. a. If the players choose their locations simultaneously and independently, identify all pure strategy Nash equilibria (if any exist). Show your work/logic. For the rest of the problem assume that vendor A chooses her location first. Vendor B observes her choice and then chooses his location. (This assumption makes it a sequential game.) b. Is [A=1,B={2AifA=1fA=2,3,4,5] a Nash equilibrium? If so, is it subgame perfect? Briefly explain. In words, the strategy says that vendor A chooses to locate in 1. Vendor B chooses to locate in 2 if vendor A locates in 1. Vendor B chooses to locate in the same location as Vendor A if vendor A locates in either 2, 3, 4 or 5 . c. Identify all subgame perfect Nash equilibria (if any exist). Show enough of the extensive form to demonstrate your logic. You do NOT need to graph the entire tree