Along a stretch of a 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 a cluster, all 100 children in that cluster will buy an ice cream. For clusters without a vendor, 50 of the 100 children are willing to walk to a vendor who is one cluster away, only 20 are willing to walk to a vendor two clusters away, and no children are willing to walk the distance of three or more clusters. The ice cream melts quickly, so the walkers cannot buy for the non-walkers.
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 (locals or walkers) 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.
(a) Construct the five-by-five payoff table for the vendor location game; the entries stated here will give you a start and a check on your calculations: If both vendors choose to locate at A, each sells 85 units. If the first vendor chooses B and the second chooses C, the first sells 150 and the second sells 170. If the first vendor chooses E and the second chooses B, the first sells 150 and the second sells 200.
(b) Eliminate dominated strategies as far as possible.
(c) In the remaining table, locate all pure-strategy Nash equilibria.
(d) If the game is altered to one with sequential moves, where the first vendor chooses her location first and the second vendor follows, what are the locations and the sales that result from the sub game perfect equilibrium? How does the change in the timing of moves here help players resolve the coordination problem in part (c)?