| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Zero-sum game optimal mixed strategy |
| Difficulty | Moderate -0.3 This is a standard textbook exercise in zero-sum games requiring routine application of dominance and mixed strategy formulas. Parts (a)-(c) are definitional/observational, and part (d) involves straightforward calculation of a 2×2 mixed strategy after eliminating the dominated row—slightly easier than average A-level difficulty. |
| Spec | 7.08a Pay-off matrix: zero-sum games7.08b Dominance: reduce pay-off matrix7.08c Pure strategies: play-safe strategies and stable solutions7.08d Nash equilibrium: identification and interpretation7.08e Mixed strategies: optimal strategy using equations or graphical method |
| \multirow{5}{*}{Rowena} | Colin | |||
| Strategy | \(\mathrm { C } _ { 1 }\) | \(\mathbf { C } _ { \mathbf { 2 } }\) | \(\mathrm { C } _ { 3 }\) | |
| \(\mathbf { R } _ { \mathbf { 1 } }\) | -4 | 5 | 4 | |
| \(\mathbf { R } _ { \mathbf { 2 } }\) | 2 | -3 | -1 | |
| \(\mathbf { R } _ { \mathbf { 3 } }\) | -5 | 4 | 3 | |
| Answer | Marks | Guidance |
|---|---|---|
| \(2\) | \(H\) | \(K\) |
| \(I\) | \(K\) |
| Answer | Marks |
|---|---|
| \(J\) | \(L\) |
Question 2:
$2$ | $H$ | $K$ | $(cid:1)2 + 7 = 5$
$I$ | $K$
$L$
$J$ | $L$2 Two people, Rowena and Colin, play a zero-sum game.\\
The game is represented by the following pay-off matrix for Rowena.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multirow{5}{*}{Rowena} & \multicolumn{4}{|c|}{Colin} \\
\hline
& Strategy & $\mathrm { C } _ { 1 }$ & $\mathbf { C } _ { \mathbf { 2 } }$ & $\mathrm { C } _ { 3 }$ \\
\hline
& $\mathbf { R } _ { \mathbf { 1 } }$ & -4 & 5 & 4 \\
\hline
& $\mathbf { R } _ { \mathbf { 2 } }$ & 2 & -3 & -1 \\
\hline
& $\mathbf { R } _ { \mathbf { 3 } }$ & -5 & 4 & 3 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Explain what is meant by the term 'zero-sum game'.
\item Determine the play-safe strategy for Colin, giving a reason for your answer.
\item Explain why Rowena should never play strategy $R _ { 3 }$.
\item Find the optimal mixed strategy for Rowena.
\end{enumerate}
\hfill \mbox{\textit{AQA D2 2009 Q2 [11]}}