| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Discrete (Further Paper 3 Discrete) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Zero-sum game stable solution |
| Difficulty | Standard +0.3 This is a standard game theory question testing basic concepts: dominance, stable solutions (saddle points), and play-safe strategies. Part (a) requires identifying dominated strategy D (dominated by B), part (b) involves checking row minima against column maxima (no saddle point exists), and part (c) applies the play-safe concept. All are routine applications of textbook algorithms with no novel insight required, making it slightly easier than average for 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 interpretation |
| Jadzia | |||
| Strategy | X | Y | Z |
| A | -3 | 2 | 3 |
| Ben B | 6 | 0 | -4 |
| C | 7 | -1 | 1 |
| D | 6 | -2 | 1 |
| Answer | Marks |
|---|---|
| 4(a) | Explains correctly that strategy |
| Answer | Marks | Guidance |
|---|---|---|
| strategy D should not be played | 3.5c | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Total | 1 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 4(b) | Translates the problem of |
| Answer | Marks | Guidance |
|---|---|---|
| Ignore inclusion of strategy D | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| maxima) | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| exist using correct terminology | 3.2a | R1 |
| Total | 3 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 4(c) | Explains that the play-safe | |
| strategy for Jadzia is Y | 2.4 | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| strategy A | 1.1b | B1 |
| Total | 2 | |
| Question total | 6 | |
| Q | Marking instructions | AO |
| 4 | 4 | 0 |
| Question total | 8 | |
| Q | Marking instructions | AO |
Question 4:
--- 4(a) ---
4(a) | Explains correctly that strategy
D is dominated by strategy C, so
strategy D should not be played | 3.5c | B1 | As 6 ≤ 7, –2 ≤ –1 and 1 ≤ 1,
strategy C dominates strategy D
Ben should never play strategy D
Total | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 4(b) ---
4(b) | Translates the problem of
finding a stable solution into
identifying all correct row
minima or all correct column
maxima
May be seen around table
Ignore inclusion of strategy D | 3.1a | M1 | row minima: –3, –4, –1
column maxima: 7, 2, 3
max(row minima) = –1
min(column maxima) = 2
max(row minima) = –1 ≠ 2 =
min(col maxima),
therefore, a stable solution does
not exist
Identifies the correct max(row
minima) and the correct min(col
maxima) | 1.1b | A1
Clearly shows that the correct
value max(row minima) and
correct value min(col maxima)
are not equal and concludes
that a stable solution does not
exist using correct terminology | 3.2a | R1
Total | 3
Q | Marking instructions | AO | Marks | Typical solution
--- 4(c) ---
4(c) | Explains that the play-safe
strategy for Jadzia is Y | 2.4 | E1 | Play-safe strategy for Jadzia = Y
Therefore, Ben should play
strategy A
States that Ben should play
strategy A | 1.1b | B1
Total | 2
Question total | 6
Q | Marking instructions | AO | Marks | Typical solution
4 | 4 | 0 | 1 | 2 | 3
Question total | 8
Q | Marking instructions | AO | Marks | Typical solutionBen and Jadzia play a zero-sum game.
The game is represented by the following pay-off matrix for Ben.
\begin{tabular}{|c|c|c|c|}
\hline
& \multicolumn{3}{|c|}{Jadzia} \\
\hline
Strategy & X & Y & Z \\
\hline
A & -3 & 2 & 3 \\
\hline
Ben B & 6 & 0 & -4 \\
\hline
C & 7 & -1 & 1 \\
\hline
D & 6 & -2 & 1 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\item State, with a reason, which strategy Ben should never play.
[1 mark]
\item Determine whether or not the game has a stable solution.
Fully justify your answer.
[3 marks]
\item Ben knows that Jadzia will always play her play-safe strategy.
Explain how Ben can maximise his expected pay-off.
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Discrete 2022 Q4 [6]}}