| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) |
| Year | 2020 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Non-group structures |
| Difficulty | Moderate -0.5 This is a straightforward game theory question requiring identification of a saddle point in a pay-off matrix. Students need to find row minima and column maxima, then check if they coincide—a mechanical procedure taught explicitly in the syllabus. While it requires careful checking of values, it involves no problem-solving insight or novel reasoning, making it easier than average. |
| 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 |
| Strategy | \(\mathbf { H } _ { \mathbf { 1 } }\) | \(\mathbf { H } _ { \mathbf { 2 } }\) | \(\mathbf { H } _ { \mathbf { 3 } }\) | |
| Summer | \(\mathbf { S } _ { \mathbf { 1 } }\) | 4 | - 4 | 0 |
| \cline { 2 - 5 } | \(\mathbf { S } _ { \mathbf { 2 } }\) | - 12 | 0 | 10 |
| \cline { 2 - 5 } | \(\mathbf { S } _ { \mathbf { 3 } }\) | 10 | 4 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Row minima: \((-4, -12, 4)\); Column maxima: \((10, 4, 10)\) | M1 | Identifies correctly the row minima or column maxima, OR identifies at least one dominated strategy |
| \(\max(\text{row minima}) = 4 = \min(\text{column maxima})\) | A1 | Finds correctly max(row minima) and min(column maxima), OR finds a \(2 \times 2\) pay-off matrix by removing dominated strategies |
| Therefore a stable solution exists | R1 | Completes a reasoned argument to show that the game has a stable solution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Value of the game for Summer \(= 4\) | B1 | States the value of the game for Summer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Play safe strategy for Summer: \(S_3\); Play safe strategy for Haf: \(H_2\) | B1 | States the play-safe strategy for each player |
## Question 3(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Row minima: $(-4, -12, 4)$; Column maxima: $(10, 4, 10)$ | M1 | Identifies correctly the row minima or column maxima, OR identifies at least one dominated strategy |
| $\max(\text{row minima}) = 4 = \min(\text{column maxima})$ | A1 | Finds correctly max(row minima) and min(column maxima), OR finds a $2 \times 2$ pay-off matrix by removing dominated strategies |
| Therefore a stable solution exists | R1 | Completes a reasoned argument to show that the game has a stable solution |
## Question 3(b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Value of the game for Summer $= 4$ | B1 | States the value of the game for Summer |
## Question 3(b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Play safe strategy for Summer: $S_3$; Play safe strategy for Haf: $H_2$ | B1 | States the play-safe strategy for each player |
---3 Summer and Haf play a zero-sum game.
The pay-off matrix for the game is shown below.
Haf
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
& Strategy & $\mathbf { H } _ { \mathbf { 1 } }$ & $\mathbf { H } _ { \mathbf { 2 } }$ & $\mathbf { H } _ { \mathbf { 3 } }$ \\
\hline
Summer & $\mathbf { S } _ { \mathbf { 1 } }$ & 4 & - 4 & 0 \\
\cline { 2 - 5 }
& $\mathbf { S } _ { \mathbf { 2 } }$ & - 12 & 0 & 10 \\
\cline { 2 - 5 }
& $\mathbf { S } _ { \mathbf { 3 } }$ & 10 & 4 & 6 \\
\hline
\end{tabular}
\end{center}
3
\begin{enumerate}[label=(\alph*)]
\item Show that the game has a stable solution.\\
3
\item (i) State the value of the game for Summer.
3 (b) (ii) State the play-safe strategy for each player.
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete 2020 Q3 [5]}}