7. Alexis and Becky are playing a zero-sum game. Alexis has two options, Q and R . Becky has three options, \(\mathrm { X } , \mathrm { Y }\) and Z .
Alexis intends to make a random choice between options Q and R , choosing option Q with probability \(p _ { 1 }\) and option R with probability \(p _ { 2 }\) Alexis wants to find the optimal values of \(p _ { 1 }\) and \(p _ { 2 }\) and formulates the following linear programme, writing the constraints as inequalities. $$\begin{aligned} & \text { Maximise } P = V
& \text { where } V = 3 + \text { the value of the gan }
& \text { subject to } V \leqslant 6 p _ { 1 } + p _ { 2 }
& \qquad \begin{aligned} & V \leqslant 8 p _ { 2 }
& V \leqslant 4 p _ { 1 } + 2 p _ { 2 }
& p _ { 1 } + p _ { 2 } \leqslant 1
& p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , V \geqslant 0 \end{aligned} \end{aligned}$$

1. Complete the pay-off matrix for Alexis in the answer book.
2. Use a graphical method to find the best strategy for Alexis.
3. Calculate the value of the game to Alexis. Becky intends to make a random choice between options \(\mathrm { X } , \mathrm { Y }\) and Z , choosing option X with probability \(q _ { 1 }\), option Y with probability \(q _ { 2 }\) and option Z with probability \(q _ { 3 }\)
4. Determine the best strategy for Becky, making your method and working clear.