3 Rose and Colin repeatedly play a zero-sum game. The pay-off matrix shows the number of points won by Rose for each combination of strategies.

1. What is the greatest number of points that Colin can win when Rose plays strategy \(A\) and which strategy does Colin need to play to achieve this?
2. Show that strategy \(B\) dominates strategy \(C\) and also that strategy \(Y\) dominates strategy \(Z\). Hence reduce the game to a \(3 \times 3\) pay-off matrix.
3. Find the play-safe strategy for each player on the reduced game. Is the game stable? Rose makes a random choice between the strategies, choosing strategy \(A\) with probability \(p _ { 1 }\), strategy \(B\) with probability \(p _ { 2 }\) and strategy \(D\) with probability \(p _ { 3 }\). She formulates the following LP problem to be solved using the Simplex algorithm: $$\begin{array} { l l } \text { maximise } & M = m - 5 ,
   \text { subject to } & m \leqslant 4 p _ { 1 } + 10 p _ { 2 } ,
   & m \leqslant 9 p _ { 1 } + 3 p _ { 2 } + 11 p _ { 3 } ,
   & m \leqslant 2 p _ { 1 } + 10 p _ { 2 } + p _ { 3 } ,
   & p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 ,
   \text { and } & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , m \geqslant 0 . \end{array}$$ (You are not required to solve this problem.)
4. Explain how \(9 p _ { 1 } + 3 p _ { 2 } + 11 p _ { 3 }\) was obtained. A computer gives the solution to the LP problem as \(p _ { 1 } = \frac { 7 } { 48 } , p _ { 2 } = \frac { 27 } { 48 } , p _ { 3 } = \frac { 14 } { 48 }\).
5. Calculate the value of \(M\) at this solution.