6 Rose is playing a game against a computer. Rose aims a laser beam along a row, \(A , B\) or \(C\), and, at the same time, the computer aims a laser beam down a column, \(X , Y\) or \(Z\). The number of points won by Rose is determined by where the two laser beams cross. These values are given in the table. The computer loses whatever Rose wins.

1. Find Rose's play-safe strategy and show that the computer's play-safe strategy is \(Y\). How do you know that the game does not have a stable solution?
2. Explain why Rose should never choose row \(C\) and hence reduce the game to a \(2 \times 3\) pay-off matrix.
3. Rose intends to play the game a large number of times. She decides to use a standard six-sided die to choose between row \(A\) and row \(B\), so that row \(A\) is chosen with probability \(a\) and row \(B\) is chosen with probability \(1 - a\). Show that the expected pay-off for Rose when the computer chooses column \(X\) is \(4 - 3 a\), and find the corresponding expressions for when the computer chooses column \(Y\) and when it chooses column \(Z\). Sketch a graph showing the expected pay-offs against \(a\), and hence decide on Rose's optimal choice for \(a\). Describe how Rose could use the die to decide whether to play \(A\) or \(B\). The computer is to choose \(X , Y\) and \(Z\) with probabilities \(x , y\) and \(z\) respectively, where \(x + y + z = 1\). Graham is an AS student studying the D1 module. He wants to find the optimal choices for \(x , y\) and \(z\) and starts off by producing a pay-off matrix for the computer.
4. Graham produces the following pay-off matrix.

   3	1	0
   0	1	2

   Write down the pay-off matrix for the computer and explain what Graham did to its entries to get the values in his pay-off matrix.
5. Graham then sets up the linear programming problem: $$\begin{array} { l l } \text { maximise } & P = p - 4 ,
   \text { subject to } & p - 3 x - y \leqslant 0 ,
   & p - y - 2 z \leqslant 0 ,
   & x + y + z \leqslant 1 ,
   \text { and } & p \geqslant 0 , x \geqslant 0 , y \geqslant 0 , z \geqslant 0 . \end{array}$$ The Simplex algorithm is applied to the problem and gives \(x = 0.4\) and \(y = 0\). Find the values of \(z , p\) and \(P\) and interpret the solution in the context of the game. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}