Game Theory Question

• Nov 23rd 2008, 11:01 PM
Phyxius117
Game Theory 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 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.)

A.J. Swim Team
Entry
1
Butterfly stroke = 1:01.6
Backstroke = 1:06.8
Breaststroke = 1:13.9
2
Butterfly stroke = 59.1
Backstroke = 1:05.6
Breaststroke = 1:12.5
John
Butterfly stroke = 57.5
Backstroke = 1:03.3
Breaststroke = 1:04.7

G. N. Swim Team
Entry
1
Butterfly stroke = 1:03.2
Backstroke = 1:04.9
Breaststroke = 1:15.3
2
Butterfly stroke = 59.8
Backstroke = 1:04.1
Breaststroke = 1:11.8
Mark
Butterfly stroke = 58.4
Backstroke = 1:02.6
Breaststroke = 1:06.1

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 time in this meet. Thus, John and Mark each will definately be entered in two of these three events

Question:
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 to find the optimal mixed strategy for each team according to the minimax criterion.

Thank you very much for the help!