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.

\begin{center}
\begin{tabular}{ c | c | c c c | }
 & \multicolumn{3}{c}{Computer} &  \\
\cline { 2 - 5 }
 &  & $X$ & $Y$ & $Z$ \\
\cline { 2 - 5 }
Rose & $A$ & 1 & 3 & 4 \\
$B$ & 4 & 3 & 2 &  \\
$C$ & 3 & 2 & 1 &  \\
\cline { 2 - 5 }
\end{tabular}
\end{center}

(i) 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?\\
(ii) Explain why Rose should never choose row $C$ and hence reduce the game to a $2 \times 3$ pay-off matrix.\\
(iii) 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.\\
(iv) Graham produces the following pay-off matrix.

\begin{center}
\begin{tabular}{ | l | l | l | }
\hline
3 & 1 & 0 \\
\hline
0 & 1 & 2 \\
\hline
\end{tabular}
\end{center}

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.\\
(v) 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.

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\hfill \mbox{\textit{OCR D2  Q6 [17]}}