| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2008 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Zero-sum game optimal mixed strategy |
| Difficulty | Standard +0.3 This is a standard textbook exercise in zero-sum game theory requiring routine application of dominance elimination and the 2×2 mixed strategy formula. While it involves multiple parts, each step follows a well-established algorithm with no novel problem-solving required, making it slightly easier than average for A-level Further Maths. |
| 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}{*}{Rob} | Con | |||
| Strategy | \(\mathrm { C } _ { 1 }\) | \(\mathbf { C } _ { \mathbf { 2 } }\) | \(\mathrm { C } _ { 3 }\) | |
| \(\mathbf { R } _ { \mathbf { 1 } }\) | -2 | 5 | 3 | |
| \(\mathbf { R } _ { \mathbf { 2 } }\) | 3 | -3 | -1 | |
| \(\mathbf { R } _ { \mathbf { 3 } }\) | -3 | 3 | 2 | |
Question 3:
M1: Identify route SQZWT with flow value
A1: Correct flow value for SQZWT
M1: Identify route SPYXZVT with flow value
A1: Correct flow value for SPYXZVT3 Two people, Rob and Con, play a zero-sum game.
The game is represented by the following pay-off matrix for Rob.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multirow{5}{*}{Rob} & \multicolumn{4}{|c|}{Con} \\
\hline
& Strategy & $\mathrm { C } _ { 1 }$ & $\mathbf { C } _ { \mathbf { 2 } }$ & $\mathrm { C } _ { 3 }$ \\
\hline
& $\mathbf { R } _ { \mathbf { 1 } }$ & -2 & 5 & 3 \\
\hline
& $\mathbf { R } _ { \mathbf { 2 } }$ & 3 & -3 & -1 \\
\hline
& $\mathbf { R } _ { \mathbf { 3 } }$ & -3 & 3 & 2 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Explain what is meant by the term 'zero-sum game'.
\item Show that this game has no stable solution.
\item Explain why Rob should never play strategy $R _ { 3 }$.
\item \begin{enumerate}[label=(\roman*)]
\item Find the optimal mixed strategy for Rob.
\item Find the value of the game.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA D2 2008 Q3 [13]}}