3 Basil runs a luxury hotel. He advertises summer breaks at the hotel in several different magazines. Last summer he won the opportunity to place a full-page colour advertisement in one of four magazines for the price of the usual smaller advertisement. The table shows the expected additional weekly income, in $\pounds$, for each of the magazines for each possible type of weather. Basil wanted to maximise the additional income.

\begin{center}
\begin{tabular}{ l l | c | c | }
 &  & \multicolumn{2}{c}{Weather} \\
 &  & Rainy & Sunny \\
\cline { 2 - 4 }
 & Activity holidays & 4000 & 5000 \\
\cline { 2 - 4 }
Magazine & British beaches & 1000 & 7000 \\
\cline { 2 - 4 }
 & Country retreats & 3000 & 6000 \\
\cline { 2 - 4 }
 & Dining experiences & 5000 & 3000 \\
\cline { 2 - 4 }
\end{tabular}
\end{center}

(i) Explain carefully why no magazine choice can be rejected using a dominance argument.\\
(ii) Treating the choice of strategies as being a zero-sum game, find Basil's play-safe strategy and show that the game is unstable.\\
(iii) Calculate the expected additional weekly income for each magazine choice if the weather is rainy with probability 0.4 and sunny with probability 0.6 .

Suppose that the weather is rainy with probability $p$ and sunny with probability $1 - p$.\\
(iv) Which magazine should Basil choose if the weather is certain to be sunny ( $p = 0$ ), and which should he choose if the weather is certain to be rainy ( $p = 1$ )?\\
(v) Graph the expected additional weekly income against $p$. Hence advise Basil on which magazine he should choose for the different possible ranges of values of $p$.

\hfill \mbox{\textit{OCR D2 2011 Q3 [12]}}