| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Discrete (Further Paper 3 Discrete) |
| Year | 2024 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Zero-sum game stable solution |
| Difficulty | Standard +0.8 This is a Further Maths game theory question requiring systematic analysis of a 4×4 payoff matrix. Part (a) demands proving no stable solution exists by checking all 16 entries for saddle points (requiring understanding of maximin/minimax), and part (b) requires finding play-safe strategies. While methodical rather than conceptually deep, it involves multiple calculations and formal justification beyond standard A-level content. |
| Spec | 7.08a Pay-off matrix: zero-sum games7.08c Pure strategies: play-safe strategies and stable solutions7.08d Nash equilibrium: identification and interpretation |
| Strategy | W | X | Y | Z | |
| \multirow{4}{*}{Daniel} | A | 3 | \(-2\) | 1 | 4 |
| B | 5 | 1 | \(-4\) | 1 | |
| C | 2 | \(-1\) | 1 | 2 | |
| D | \(-3\) | 0 | 2 | \(-1\) |
| Answer | Marks |
|---|---|
| 4(a) | Identifies the four correct row |
| Answer | Marks | Guidance |
|---|---|---|
| maxima | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| and min(column maxima) = 1 | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| stable solution does not exist | 3.2a | R1 |
| Subtotal | 3 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 4(b) | Deduces C for Daniel | |
| and X for Jackson | 2.2a | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Subtotal | 1 | |
| Question total | 4 | |
| Q | Marking instructions | AO |
Question 4:
--- 4(a) ---
4(a) | Identifies the four correct row
minima or four correct column
maxima | 3.1a | M1 | row minima: –2, –4, –1, –3
column maxima: 5, 1, 2, 4
max(row minima) = –1
min(column maxima) = 1
As
max(row minima) = –1 ≠ 1
= min(col maxima),
therefore, a stable solution does
not exist.
States max(row minima) = –1
and min(column maxima) = 1 | 1.1b | A1
Completes a reasoned
argument to show that the
max(row minima) and
min(column maxima) are not
equal and concludes that a
stable solution does not exist | 3.2a | R1
Subtotal | 3
Q | Marking instructions | AO | Marks | Typical solution
--- 4(b) ---
4(b) | Deduces C for Daniel
and X for Jackson | 2.2a | B1 | Play-safe strategy for Daniel = C
Play-safe strategy for Jackson = X
Subtotal | 1
Question total | 4
Q | Marking instructions | AO | Marks | Typical solutionDaniel and Jackson play a zero-sum game.
The game is represented by the following pay-off matrix for Daniel.
Jackson
\begin{tabular}{c|c|c|c|c}
& Strategy & W & X & Y & Z \\
\hline
\multirow{4}{*}{Daniel} & A & 3 & $-2$ & 1 & 4 \\
& B & 5 & 1 & $-4$ & 1 \\
& C & 2 & $-1$ & 1 & 2 \\
& D & $-3$ & 0 & 2 & $-1$ \\
\end{tabular}
Neither player has any strategies which can be ignored due to dominance.
\begin{enumerate}[label=(\alph*)]
\item Prove that the game does not have a stable solution.
Fully justify your answer.
[3 marks]
\item Determine the play-safe strategy for each player.
[1 mark]
Play-safe strategy for Daniel _______________________________________________
Play-safe strategy for Jackson ______________________________________________
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Discrete 2024 Q4 [4]}}