4 A group of rowers have challenged some cyclists to see which team is fitter. There will be several rounds to the challenge. In each round, the rowers and the cyclists each choose a team member and these two compete in a series of gym exercises. The time by which the winner finishes ahead of the loser is converted into points. These points are added to the score for the winner's team and taken off the score for the loser's team.
The table shows the expected number of points added to the score for the rowers for each combination of competitors.
Rowers
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Cyclists}
| Chris | Jamie | Wendy |
| Andy | - 3 | 2 | - 4 |
| Kath | 5 | 4 | - 6 |
| Zac | 1 | - 4 | - 5 |
\end{table}
- Regarding this as a zero-sum game, find the play-safe strategy for the rowers and the play-safe strategy for the cyclists. Show that the game is stable.
Unfortunately, Wendy and Kath are needed by their coaches and cannot compete.
- Show that the resulting reduced game is unstable.
- Suppose that the cyclists are equally likely to choose Chris or Jamie. Calculate the expected number of points added to the score for the rowers when they choose Andy and when they choose Zac.
Suppose that the cyclists use random numbers to choose between Chris and Jamie, so that Chris is chosen with probability \(p\) and Jamie with probability \(1 - p\).
- Showing all your working, calculate the optimum value of \(p\) for the cyclists.
- The rowers use random numbers in a similar way to choose between Andy and Zac, so that Andy is chosen with probability \(q\) and Zac with probability \(1 - q\). Calculate the optimum value of \(q\).