Along a stretch of beach are 500 children in five clusters of 100 each. (Label the clusters A, B, C, D and E in that order.) Two ice-cream vendors are deciding simultaneously where to locate. They must choose the exact location of one of the clusters. If there is a vendor in that cluster, all 100 children in that cluster will buy an ice cream. For clusters without a vendor, 60 of the 100 children are willing to walk to a vendor who is one cluster away, only 20 are willing to walk to a cluster that is two clusters away, and no children are willing to walk a distance of three or more clusters. The ice cream melts quickly so the only way to get any is to purchase directly from a vendor. If the two vendors choose the same cluster, each will get a 50% share of the total demand for ice cream. If they choose different clusters, then those children for whom one vendor is closer than the other will go to the closer one and those for whom the two are equidistant will split 50% each. Each vendor seeks to maximize her sales. For example, if both vendors choose E, then there will be 180 total customers (100 from E, 60 from D and 20 from C) which will be divided equally between them. Payoffs are (90, 90) As a second example, if vendor 1 chooses A and vendor 2 chooses B, then vendor 1 gets 100 customers from A and vendor 2 gets 100 customers from B 180. Payoffs are (100, 180). , 60 from C and 20 from D for a total of:. (a) Construct the 5x5 payoff table for the vendor location game.
(b) Are there any dominated strategies in the game, and if so, what are they?
(c) Find all Nash equilibria
(d) If the game is altered to one with sequential moves, where one vendor chooses her location first and the other vendor follows, what are the location choices that result from the subgame perfect equilibrium? Note: you do not need to write the game tree; just state what will happen In the sequential game. How does the change in the timing of moves help the vendors resolve the coordination problem in part (c)?