| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2013 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Difficulty | Easy -2.5 This is a misclassified question - it's about game theory (zero-sum games, saddle points, play-safe strategies) from Decision Mathematics, not group theory. Even within its actual topic, it's a straightforward textbook exercise requiring only mechanical identification of row minima and column maxima with no problem-solving insight. |
| 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 interpretation |
| Will | |||||
| \cline { 2 - 6 } | Strategy | \(\boldsymbol { D }\) | \(\boldsymbol { E }\) | \(\boldsymbol { F }\) | \(\boldsymbol { G }\) |
| Harry | \(\boldsymbol { A }\) | - 1 | 2 | 3 | |
| \cline { 2 - 6 } | \(\boldsymbol { B }\) | 4 | 6 | 3 | 7 |
| \cline { 2 - 6 } | \(\boldsymbol { C }\) | 1 | 3 | - 2 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Row minima: \(A:-1\), \(B:3\), \(C:-2\); maximin \(= 3\) (row \(B\)) | M1 | Finding row minima |
| Column maxima: \(D:4\), \(E:6\), \(F:3\), \(G:7\); minimax \(= 3\) (column \(F\)) | M1 | Finding column maxima |
| Maximin \(=\) minimax \(= 3\), so stable solution exists | A1 | Both values equal with conclusion |
| Harry plays \(B\); Will plays \(F\) | A1 | Both strategies stated |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Saddle point at \((B, F)\) with value \(3\) | B1 |
# Question 2:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| Row minima: $A:-1$, $B:3$, $C:-2$; maximin $= 3$ (row $B$) | M1 | Finding row minima |
| Column maxima: $D:4$, $E:6$, $F:3$, $G:7$; minimax $= 3$ (column $F$) | M1 | Finding column maxima |
| Maximin $=$ minimax $= 3$, so stable solution exists | A1 | Both values equal with conclusion |
| Harry plays $B$; Will plays $F$ | A1 | Both strategies stated |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| Saddle point at $(B, F)$ with value $3$ | B1 | |2 Harry and Will play a zero-sum game. The game is represented by the following pay-off matrix for Harry.
\begin{center}
\begin{tabular}{ l | c | c | c | c | c | }
& \multicolumn{5}{c}{Will} \\
\cline { 2 - 6 }
& Strategy & $\boldsymbol { D }$ & $\boldsymbol { E }$ & $\boldsymbol { F }$ & $\boldsymbol { G }$ \\
\hline
Harry & $\boldsymbol { A }$ & - 1 & 2 & 3 & \\
\cline { 2 - 6 }
& $\boldsymbol { B }$ & 4 & 6 & 3 & 7 \\
\cline { 2 - 6 }
& $\boldsymbol { C }$ & 1 & 3 & - 2 & 4 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Show that this game has a stable solution and state the play-safe strategy for each player.
\item List any saddle points.
\end{enumerate}
\hfill \mbox{\textit{AQA D2 2013 Q2 [5]}}