3 The 'Rovers' and the 'Collies' are two teams of dog owners who compete in weekly dog shows. The top three dogs owned by members of the Rovers are Prince, Queenie and Rex. The top four dogs owned by the Collies are Woof, Xena, Yappie and Zulu.

In a show the Rovers choose one of their dogs to compete against one of the dogs owned by the Collies. There are 10 points available in total. Each of the 10 points is awarded either to the dog owned by the Rovers or to the dog owned by the Collies. There are no tied points. At the end of the competition, 5 points are subtracted from the number of points won by each dog to give the score for that dog.

The table shows the score for the dog owned by the Rovers for each combination of dogs.

\begin{center}
\begin{tabular}{ l | r | r | r | r | r | }
 & \multicolumn{5}{c}{Collies} \\
\cline { 2 - 6 }
 &  & $W$ & \multicolumn{1}{|c|}{$X$} & $Y$ & $Z$ \\
\cline { 2 - 6 }
 & $P$ & 1 & 2 & - 1 & 3 \\
\cline { 2 - 6 }
 & $Q$ & - 2 & 1 & - 3 & - 1 \\
\hline
$R$ & 2 & - 4 & 1 & 0 &  \\
\cline { 2 - 6 }
 &  &  &  &  &  \\
\cline { 2 - 6 }
\end{tabular}
\end{center}

(i) Explain why calculating the score by subtracting 5 from the number of points for each dog makes this a zero-sum game.\\
(ii) If the Rovers choose Prince and the Collies choose Woof, what score does Woof get, and how many points do Prince and Woof each get in the competition?\\
(iii) Show that column $W$ is dominated by one of the other columns, and state which column this is.\\
(iv) Delete the column for $W$ and find the play-safe strategy for the Rovers and the play-safe strategy for the Collies on the table that remains.

Queenie is ill one week, so the Rovers make a random choice between Prince and Rex, choosing Prince with probability $p$ and Rex with probability $1 - p$.\\
(v) Write down and simplify an expression for the expected score for the Rovers when the Collies choose Xena. Write down and simplify the corresponding expressions for when the Collies choose Yappie and for when they choose Zulu.\\
(vi) Using graph paper, draw a graph to show the expected score for the Rovers against $p$ for each of the choices that the Collies can make. Using your graph, find the optimal value of $p$ for the Rovers.

If Queenie had not been ill, the Rovers would have made a random choice between Prince, Queenie and Rex, choosing Prince with probability $p _ { 1 }$, Queenie with probability $p _ { 2 }$ and Rex with probability $p _ { 3 }$.

The problem of choosing the optimal values of $p _ { 1 } , p _ { 2 }$ and $p _ { 3 }$ can be formulated as the following linear programming problem:

$$\begin{array} { l l } 
\operatorname { maximise } & M = m - 4 \\
\text { subject to } & m \leqslant 6 p _ { 1 } + 5 p _ { 2 } , \\
& m \leqslant 3 p _ { 1 } + p _ { 2 } + 5 p _ { 3 } , \\
& m \leqslant 7 p _ { 1 } + 3 p _ { 2 } + 4 p _ { 3 } , \\
& p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 \\
\text { and } & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , m \geqslant 0 .
\end{array}$$

(vii) Explain how the expressions $6 p _ { 1 } + 5 p _ { 2 } , 3 p _ { 1 } + p _ { 2 } + 5 p _ { 3 }$ and $7 p _ { 1 } + 3 p _ { 2 } + 4 p _ { 3 }$ were obtained. Also explain how the linear programming formulation tells you that $M$ is a maximin solution.

The Simplex algorithm is used to find the optimal values of the probabilities. The optimal value of $p _ { 1 }$ is $\frac { 5 } { 8 }$ and the optimal value of $p _ { 2 }$ is 0 .\\
(viii) Calculate the optimal value of $p _ { 3 }$ and the corresponding value of $M$.

\hfill \mbox{\textit{OCR D2 2013 Q3 [19]}}