| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) |
| Year | 2018 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Zero-sum game stable solution |
| Difficulty | Moderate -0.5 This is a straightforward zero-sum game question requiring identification of a stable solution (saddle point) and play-safe strategies. Students need to find row minima and column maxima, which is a standard algorithmic procedure taught in Further Maths Decision modules. The 3×3 matrix is small and the saddle point is easily identifiable at value 2, making this easier than average for Further Maths content. |
| 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 |
| Sam | ||||
| \cline { 2 - 5 } | Strategy | |||
| \cline { 2 - 5 } | \(\mathbf { S } _ { \mathbf { 1 } }\) | \(\mathbf { S } _ { \mathbf { 2 } }\) | \(\mathbf { S } _ { \mathbf { 3 } }\) | |
| \(\mathbf { A } _ { \mathbf { 1 } }\) | 2 | 2 | 3 | |
| \cline { 2 - 5 } | \(\mathbf { A } _ { \mathbf { 2 } }\) | 0 | 3 | 5 |
| \(\mathbf { A } _ { \mathbf { 3 } }\) | - 1 | 2 | - 2 | |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Row minima \(= (2, 0, -2)\), Col maxima \(= (2, 3, 5)\) | M1 | Finds 4 or more correct row minima/col maxima |
| \(\max(\text{row minima}) = 2\), \(\min(\text{col maxima}) = 2\) | A1 | Correctly finds all row minima/col maxima |
| As \(\max(\text{row minima}) = 2 = \min(\text{col maxima})\), a stable solution exists and the value of the game is \(2\) | E1 | Allow 'Maximin' and 'Minimax' |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(A_1\) (or \(A_2\)) dominates \(A_3\), so remove \(A_3\) | M1 | Uses dominance to reduce pay-off matrix to \(2 \times 3\) or \(3 \times 2\) |
| \(S_1\) dominates \(S_2\) and \(S_3\), so remove \(S_2\) and \(S_3\); then \(A_1\) dominates \(A_2\), remove \(A_2\) | A1 | Uses dominance to reduce to \(1 \times 1\) OR correctly finds all remaining row minima/col maxima |
| Alex will only ever play \(A_1\) and Sam will only ever play \(S_1\), resulting in Alex gaining 2 each game. Value of game is \(2\) | E1 | Allow Maximin/Minimax explanation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Alex strategy \(A_1\), Sam strategy \(S_1\) | B1 | Condone stating '\(A_1\)' and '\(S_1\)' |
## Question 3(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Row minima $= (2, 0, -2)$, Col maxima $= (2, 3, 5)$ | M1 | Finds 4 or more correct row minima/col maxima |
| $\max(\text{row minima}) = 2$, $\min(\text{col maxima}) = 2$ | A1 | Correctly finds all row minima/col maxima |
| As $\max(\text{row minima}) = 2 = \min(\text{col maxima})$, a stable solution exists and the value of the game is $2$ | E1 | Allow 'Maximin' and 'Minimax' |
## Question 3(a) ALT:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $A_1$ (or $A_2$) dominates $A_3$, so remove $A_3$ | M1 | Uses dominance to reduce pay-off matrix to $2 \times 3$ or $3 \times 2$ |
| $S_1$ dominates $S_2$ and $S_3$, so remove $S_2$ and $S_3$; then $A_1$ dominates $A_2$, remove $A_2$ | A1 | Uses dominance to reduce to $1 \times 1$ OR correctly finds all remaining row minima/col maxima |
| Alex will only ever play $A_1$ and Sam will only ever play $S_1$, resulting in Alex gaining 2 each game. Value of game is $2$ | E1 | Allow Maximin/Minimax explanation |
## Question 3(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Alex strategy $A_1$, Sam strategy $S_1$ | B1 | Condone stating '$A_1$' and '$S_1$' |
---3 Alex and Sam are playing a zero-sum game.
The game is represented by the pay-off matrix for Alex.
\begin{center}
\begin{tabular}{ l | c | c | c | c | }
\multicolumn{4}{c}{Sam} & \\
\cline { 2 - 5 }
& \multicolumn{4}{c}{Strategy} \\
\cline { 2 - 5 }
& $\mathbf { S } _ { \mathbf { 1 } }$ & $\mathbf { S } _ { \mathbf { 2 } }$ & $\mathbf { S } _ { \mathbf { 3 } }$ & \\
\hline
$\mathbf { A } _ { \mathbf { 1 } }$ & 2 & 2 & 3 & \\
\cline { 2 - 5 }
& $\mathbf { A } _ { \mathbf { 2 } }$ & 0 & 3 & 5 \\
\hline
$\mathbf { A } _ { \mathbf { 3 } }$ & - 1 & 2 & - 2 & \\
\hline
\end{tabular}
\end{center}
3
\begin{enumerate}[label=(\alph*)]
\item Explain why the value of the game is 2\\
3
\item Identify the play-safe strategy for each player.\\
Each pipe is labelled with its upper capacity in $\mathrm { cm } ^ { 3 } \mathrm {~s} ^ { - 1 }$\\
\includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-04_620_940_450_550}
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete 2018 Q3 [4]}}