| Mia | | | X | Y | Z | | | \multirow{3}{*}{Li} | X | 5 | - 6 | 0 | | \cline { 2 - 5 } | Y | - 2 | 3 | 4 | | \cline { 2 - 5 } | Z | - 1 | 4 | 8 | | \cline { 2 - 5 } |
| Mia | | X | Y | Z | | | \multirow{2}{*}{Li} | X | 4 | | | | \cline { 2 - 5 } | Y | 11 | | 5 | | \cline { 2 - 5 } | Z | 10 | 5 | 1 | | \cline { 2 - 5 } |
The game can be converted into a zero-sum game, this means that the total number of points won by Li and Mia is the same for each combination of strategies.- Complete the table in the Printed Answer Booklet to show the points won by Mia.
- Convert the game into a zero-sum game, giving the pay-offs for Li .
Use dominance to reduce the pay-off matrix for the game to a \(2 \times 2\) table. You do not need to explain the dominance.
Mia knows that Li will choose his play-safe strategy.Determine which strategy Mia should choose to maximise her points.
5 A linear programming problem is formulated as below.
Maximise \(\quad \mathrm { P } = 4 \mathrm { x } - \mathrm { y }\)
subject to \(2 x + 3 y \geqslant 12\)
\(x + y \leqslant 10\)
\(5 x + 2 y \leqslant 30\)
\(x \geqslant 0 , y \geqslant 0\)- Identify the feasible region by representing the constraints graphically and shading the regions where the inequalities are not satisfied.
- Hence determine the maximum value of the objective.
The constraint \(x + y \leqslant 10\) is changed to \(x + y \leqslant k\), the other constraints are unchanged.
Determine, algebraically, the value of \(k\) for which the maximum value of \(P\) is 3 .
Do not draw on the graph from part (a) and do not use the spare grid.Determine, algebraically, the other value of \(k\) for which there is a (non-optimal) vertex of the feasible region at which \(P = 3\).
Do not draw on the graph from part (a) and do not use the spare grid.
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