3. Two teams, A and B , each have three team members. One member of Team A will compete against one member of Team B for 10 rounds of a competition. None of the rounds can end in a draw.
Table 1 shows, for each pairing, the expected number of rounds that the member of Team A will win minus the expected number of rounds that the member of Team B will win. These numbers are the scores awarded to Team A. This competition between Teams A and B is a zero-sum game. Each team must choose one member to play. Each team wants to choose the member who will maximise its score.
\begin{table}[h]
| \cline { 3 - 5 }
\multicolumn{2}{c|}{} | Team B |
| \cline { 3 - 5 }
\multicolumn{2}{c|}{} | Paul | Qaasim | Rashid |
| \multirow{3}{*}{Team A} | Mischa | 4 | - 6 | 2 |
| \cline { 2 - 5 } | Noel | 0 | - 2 | 6 |
| \cline { 2 - 5 } | Olive | - 6 | 2 | 0 |
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{table}
- Find the number of rounds that Team A expects to win if Team A chooses Mischa and Team B chooses Paul.
- Find the number of rounds that Team B expects to win if Team A chooses Noel and Team B chooses Qaasim.
Table 1 models this zero-sum game.
- Find the play-safe strategies for the game.
- Explain how you know that the game is not stable.
- Determine which team member Team B should choose if Team B thinks that Team A will play safe. Give a reason for your answer.
At the last minute, Rashid is ill and is therefore unavailable for selection by Team B.
- Find the best strategy for Team B, defining any variables you use.