The A. J. Swim Team soon will have an important swim meet with the G. N. Swim

Question:

The A. J. Swim Team soon will have an important swim meet with the G. N. Swim Team. Each team has a star swimmer (John and Mark, respectively) who can swim very well in the 100- yard butterfly, backstroke, and breaststroke events. However, the rules prevent them from being used in more than two of these events. Therefore, their coaches now need to decide how to use them to maximum advantage.
Each team will enter three swimmers per event (the maximum allowed). For each event, the following table gives the best time previously achieved by John and Mark as well as the best time for each of the other swimmers who will definitely enter that event. (Whichever event John or Mark does not swim, his team€™s third entry for that event will be slower than the two shown in the table.)

The points awarded are 5 points for first place, 3 points for second place, 1 point for third place, and none for lower places. Both coaches believe that all swimmers will essentially equal their best times in this meet. Thus, John and Mark each will definitely be entered in two of these three events.
(a) The coaches must submit all their entries before the meet without knowing the entries for the other team, and no changes are permitted later. The outcome of the meet is very uncertain, so each additional point has equal value for the coaches. Formulate this problem as a two-person, zero-sum game. Eliminate dominated strategies, and then use the graphical procedure described in Sec. 15.4 to find the optimal mixed strategy for each team according to the minimax criterion.
(b) The situation and assignment are the same as in part (a), except that both coaches now believe that the A. J. team will win the swim meet if it can win 13 or more points in these three events, but will lose with less than 13 points. [Compare the resulting optimal mixed strategies with those obtained in part (a).]
(c) Now suppose that the coaches submit their entries during the meet one event at a time. When submitting his entries for an event, the coach does not know who will be swimming that event for the other team, but he does know who has swum in preceding events. The three key events just discussed are swum in the order listed in the table. Once again, the A. J. team needs 13 points in these events to win the swim meet. Formulate this problem as a two-person, zero-sum game. Then use the concept of dominated strategies to determine the best strategy for the G. N. team that actually €œguarantees€ it will win under the assumptions being made.
(d) The situation is the same as in part (c). However, now assume that the coach for the G. N. team does not know about game theory and so may, in fact, choose any of his available strategies that have Mark swimming two events. Use the concept of dominated strategies to determine the best strategies from which the coach for the A. J. team should choose. If this coach knows that the other coach has a tendency to enter Mark in the butterfly and the backstroke more often than in the breaststroke, which strategy should she choose?