Kira and Julian play a zero-sum game that does not have a stable solution.

Kira has three strategies to choose from: $\mathbf{K_1}$, $\mathbf{K_2}$ and $\mathbf{K_3}$

To determine her optimal mixed strategy, Kira begins by defining the following variables:

$v =$ value of the game for Kira

$p_1 =$ probability of Kira playing strategy $\mathbf{K_1}$

$p_2 =$ probability of Kira playing strategy $\mathbf{K_2}$

$p_3 =$ probability of Kira playing strategy $\mathbf{K_3}$

Kira then formulates the following linear programming problem.

Maximise        $v$

subject to      $7p_1 + p_2 + 8p_3 \geq v$
                $3p_1 + 7p_2 + 2p_3 \geq v$
                $9p_1 + 2p_2 + 4p_3 \geq v$

and             $p_1 + p_2 + p_3 \leq 1$
                $p_1, p_2, p_3 \geq 0$

\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Explain why the condition $p_1 + p_2 + p_3 \leq 1$ is necessary in Kira's linear programming problem.
[1 mark]

\item Explain why the condition $p_1, p_2, p_3 \geq 0$ is necessary in Kira's linear programming problem.
[1 mark]
\end{enumerate}

\item Julian has three strategies to choose from: $\mathbf{J_1}$, $\mathbf{J_2}$ and $\mathbf{J_3}$

Complete the following pay-off matrix which represents the game for Kira.
[3 marks]

\begin{tabular}{|c|c|c|c|}
\hline
 & \multicolumn{3}{|c|}{Julian} \\
\hline
Strategy & $\mathbf{J_1}$ & $\mathbf{J_2}$ & $\mathbf{J_3}$ \\
\hline
$\mathbf{K_1}$ & 7 & & \\
\hline
Kira $\mathbf{K_2}$ & & & \\
\hline
$\mathbf{K_3}$ & & & \\
\hline
\end{tabular}
\end{enumerate}

\hfill \mbox{\textit{AQA Further Paper 3 Discrete 2022 Q10 [5]}}