Ice Cream Vendors at the Beach

Warning: This is a difficult problem set.  Print it out, read it, and think about it.  Allow yourself enough time to get it done.

Along a stretch of beach 'down the shore' there are 500 children in five clusters of 100 each.  They are clustered in this fashion due to the location of the lifeguard stands.  The stands are labeled A, B, C, D, E.  Each child buys one ice cream cone per day.

There are two rival ice cream vendors named Ben and Jerry.  Each morning they roll there wagons down to the beach for business at 10 AM.  They must locate their wagon at the exact location of one of the clusters. The children always go to the nearest vendor. If there is a vendor in a cluster then all 100 children buy from that vendor.  For clusters without a vendor, 50 of the children will walk to an adjacent cluster with a vendor for ice cream, only 20 will walk as far as two clusters, none will walk three or more clusters for ice cream.  If both Ben and Jerry are located in the same cluster then they each get 50% of the demand for ice cream at that cluster.  If Ben and Jerry locate in different clusters then children that are equidistant from both of them will be equally split.

Last Name

First Name

Social Security Number

 Ben A B C D E Jerry A 85, 85 100, 170 125, 195 150, 200 160, 160 B 170, 100 110, 110 150, 170 200, 150 C 195, 125 170, 150 195, 125 D 200, 150 See B, D See C, D 110, 110 170, 100 E 160, 160 150, 200 125, 195 100, 170 85, 85

1.  Fill in the missing cells in the payoff matrix.  To get you started:  If Ben chooses B and Jerry chooses C, then Ben will sell 150 cones (100 from his own cluster and 50 walking over from A) and Jerry will sell 170 cones (100 from his own cluster plus 50 walkers from D and 20 walkers from E).

2. Suppose that Jerry plays B.  What is Ben's best response? Enter the letter that Ben will play:

3. How many dominated strategies does Ben have?
Enter the number:

4. How many Nash equilibria are there? Enter the number:

5. Will Ben ever use his dominated strategies in constructing the support for his mixed strategy?  Yes  No

6. Solve for the probabilities with which Ben will play each of the pure strategies in the support for his mixed strategy:

 Strategy A B C D E Probability

Hint: Print out the payoff table.  Add a final column.  Cross out the dominated strategies for Ben.   Choose a symbol for Ben's probability of playing the different strategies.   Compute the expected payoff for Ben for each pure strategy played by Jerry.   Now use the expected payoffs to solve for the probabilities in Ben's mixed strategy.

Closing remark: Give some thought to the setup of this problem and the recent history of Hechinger, Loews, and Home Depot.  Also think about the proliferation by the big manufacturers of 'models' of ready to eat cereal.   After the due date we'll talk about these cases.