4 Euan and Wai Mai play a zero-sum game. Each is trying to maximise the total number of points that they score in many repeats of the game. The table shows the number of points that Euan scores for each combination of strategies.
| Wai Mai |
| \cline { 2 - 5 } | \(X\) | \(Y\) | \(Z\) | |
| \(A\) | 2 | - 5 | 3 | |
| \cline { 2 - 5 }
\(E u a n\) | - 1 | - 3 | 4 | |
| \cline { 1 - 5 }
\(C\) | 3 | - 5 | 2 | |
| \(D\) | 3 | - 2 | - 1 | |
- Explain what the term 'zero-sum game' means.
- How many points does Wai Mai score if she chooses \(X\) and Euan chooses \(A\) ?
- Why should Wai Mai never choose strategy \(Z\) ?
- Delete the column for \(Z\) and find the play-safe strategy for Euan and the play-safe strategy for Wai Mai on the table that remains. Is the resulting game stable or not? State how you know.
The value 3 in the cell corresponding to Euan choosing \(D\) and Wai Mai choosing \(X\) is changed to - 5 ; otherwise the table is unchanged.
Wai Mai decides that she will choose her strategy by making a random choice between \(X\) and \(Y\), choosing \(X\) with probability \(p\) and \(Y\) with probability \(1 - p\).
- Write down and simplify an expression for the expected score for Wai Mai when Euan chooses each of his four strategies.
- Using graph paper, draw a graph showing Wai Mai's expected score against \(p\) for each of Euan's four strategies and hence calculate the optimum value of \(p\).