6 Lucy and Maria repeatedly play a zero-sum game. The pay-off matrix shows the number of points won by Lucy, who is playing rows, for each combination of strategies.
| | | \cline { 2 - 5 } | \(X\) | \(Y\) | \(Z\) | | | \(A\) | 2 | - 3 | 4 | | | \cline { 2 - 5 }
Lucy's | \(B\) | - 3 | 5 | 1 | | \cline { 2 - 5 }
strategyy | \(C\) | 4 | 2 | - 3 |
- Show that strategy \(A\) does not dominate strategy \(B\) and also that strategy \(B\) does not dominate strategy \(A\).
- Show that Maria will not choose strategy \(Y\) if she plays safe.
- Give a reason why Lucy might choose to play strategy \(B\).
Lucy decides to play strategy \(A\) with probability \(p _ { 1 }\), strategy \(B\) with probability \(p _ { 2 }\) and strategy \(C\) with probability \(p _ { 3 }\). She formulates the following LP problem to be solved using the Simplex algorithm:
$$\begin{array} { l l }
\text { maximise } & M = m - 3 ,
\text { subject to } & m \leqslant 5 p _ { 1 } + 7 p _ { 3 } ,
& m \leqslant 8 p _ { 2 } + 5 p _ { 3 } ,
& m \leqslant 7 p _ { 1 } + 4 p _ { 2 } ,
& p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 ,
\text { and } & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , m \geqslant 0 .
\end{array}$$
[You are not required to solve this problem.] - Explain why 3 has to be subtracted from \(m\) in the objective row.
- Explain how \(5 p _ { 1 } + 7 p _ { 3 } , 8 p _ { 2 } + 5 p _ { 3 }\) and \(7 p _ { 1 } + 4 p _ { 2 }\) were obtained.
- Explain why \(m\) has to be less than or equal to each of the expressions in part (v).
Lucy discovers that Maria does not intend ever to choose strategy \(Y\). Because of this she decides that she will never choose strategy \(B\). This means that \(p _ { 2 } = 0\).
- Show that the expected number of points won by Lucy when Maria chooses strategy \(X\) is \(4 - 2 p _ { 1 }\) and find a similar expression for the number of points won by Lucy when Maria chooses strategy \(Z\).
- Set your two expressions from part (vii) equal to each other and solve for \(p _ { 1 }\). Calculate the expected number of points won by Lucy with this value of \(p _ { 1 }\) and also when \(p _ { 1 } = 0\) and when \(p _ { 1 } = 1\). Use these values to decide how Lucy should choose between strategies \(A\) and \(C\) to maximise the expected number of points that she wins.
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