| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Zero-sum game stable solution |
| Difficulty | Moderate -0.8 This is a standard textbook game theory question testing dominance elimination and mixed strategy Nash equilibrium. The steps are routine: identify dominated strategies, verify no saddle point, set up expected value equations, and solve for optimal probabilities. While it requires multiple steps, each follows a mechanical procedure taught directly in D2 with no novel insight required. |
| 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 interpretation7.08e Mixed strategies: optimal strategy using equations or graphical method |
| Road Conditions | ||||
| \cline { 2 - 5 } | \(\boldsymbol { C } _ { \mathbf { 1 } }\) | \(\boldsymbol { C } _ { \mathbf { 2 } }\) | \(\boldsymbol { C } _ { \mathbf { 3 } }\) | |
| \cline { 2 - 5 } | \(\boldsymbol { S } _ { \mathbf { 1 } }\) | - 2 | 2 | 4 |
| \cline { 2 - 5 } Sam's Car | \(\boldsymbol { S } _ { \mathbf { 2 } }\) | 2 | 4 | 5 |
| \cline { 2 - 5 } | \(\boldsymbol { S } _ { \mathbf { 3 } }\) | 5 | 1 | 2 |
| \cline { 2 - 5 } | ||||
| \cline { 2 - 5 } | ||||
| Surname | Other Names | |||||||||||
| Centre Number | Candidate Number | |||||||||||
| Candidate Signature | ||||||||||||
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((-2,2,4) < (2,4,5)\) so \(S_1\) dominated by \(S_2\) | E1 | |
| \(\begin{pmatrix}4\\5\\2\end{pmatrix} > \begin{pmatrix}2\\4\\1\end{pmatrix}\) so \(C_3\) dominated by \(C_2\) | E1 | 2 marks total; note \(>\) sign |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Reduced \(2\times2\) game: \(S_2\): \(2\ 4\); \(S_3\): \(5\ 1\) | ||
| Minimum of rows: \(\min(2,4)=2\), \(\min(5,1)=1\) | M1 | correct method for either S or C |
| Choose maximum \(= 2\) | A1 | play safe for Sam is \(S_2\) |
| Max of columns: \(\max(2,5)=5\), \(\max(4,1)=4\); choose minimum \(=4\) | A1 | play safe for computer is \(C_2\) |
| Since \(2 \neq 4 \Rightarrow\) not stable solution | E1 | 4 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Computer picks \(C_1\): Expected game \(= 2p + 5(1-p) = 5-3p\) | M1, A1 | |
| Computer picks \(C_2\): Expected gain \(= 4p + (1-p) = 1+3p\) | A1 | 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Best mixed strategy: \(5-3p = 1+3p\) | M1 | |
| \(\Rightarrow p = \dfrac{2}{3}\) | A1 | 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Expected points gain \(= 5 - 3\times\dfrac{2}{3} = 3\) | B1 | 1 mark total; or \(1+3\left(\dfrac{2}{3}\right)\) |
## Question 6:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(-2,2,4) < (2,4,5)$ so $S_1$ dominated by $S_2$ | E1 | |
| $\begin{pmatrix}4\\5\\2\end{pmatrix} > \begin{pmatrix}2\\4\\1\end{pmatrix}$ so $C_3$ dominated by $C_2$ | E1 | 2 marks total; note $>$ sign |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Reduced $2\times2$ game: $S_2$: $2\ 4$; $S_3$: $5\ 1$ | | |
| Minimum of rows: $\min(2,4)=2$, $\min(5,1)=1$ | M1 | correct method for either S or C |
| Choose maximum $= 2$ | A1 | play safe for Sam is $S_2$ |
| Max of columns: $\max(2,5)=5$, $\max(4,1)=4$; choose minimum $=4$ | A1 | play safe for computer is $C_2$ |
| Since $2 \neq 4 \Rightarrow$ not stable solution | E1 | 4 marks total |
### Part (c)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Computer picks $C_1$: Expected game $= 2p + 5(1-p) = 5-3p$ | M1, A1 | |
| Computer picks $C_2$: Expected gain $= 4p + (1-p) = 1+3p$ | A1 | 3 marks total |
### Part (c)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Best mixed strategy: $5-3p = 1+3p$ | M1 | |
| $\Rightarrow p = \dfrac{2}{3}$ | A1 | 2 marks total |
### Part (c)(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Expected points gain $= 5 - 3\times\dfrac{2}{3} = 3$ | B1 | 1 mark total; or $1+3\left(\dfrac{2}{3}\right)$ |6 Sam is playing a computer game in which he is trying to drive a car in different road conditions. He chooses a car and the computer decides the road conditions. The points scored by Sam are shown in the table.
\begin{center}
\begin{tabular}{ l | c | r | c | c | }
\multicolumn{2}{c}{} & \multicolumn{3}{c|}{Road Conditions} \\
\cline { 2 - 5 }
& & $\boldsymbol { C } _ { \mathbf { 1 } }$ & $\boldsymbol { C } _ { \mathbf { 2 } }$ & $\boldsymbol { C } _ { \mathbf { 3 } }$ \\
\cline { 2 - 5 }
& $\boldsymbol { S } _ { \mathbf { 1 } }$ & - 2 & 2 & 4 \\
\cline { 2 - 5 }
Sam's Car & $\boldsymbol { S } _ { \mathbf { 2 } }$ & 2 & 4 & 5 \\
\cline { 2 - 5 }
& $\boldsymbol { S } _ { \mathbf { 3 } }$ & 5 & 1 & 2 \\
\cline { 2 - 5 }
& & & & \\
\cline { 2 - 5 }
\end{tabular}
\end{center}
Sam is trying to maximise his total points and the computer is trying to stop him.
\begin{enumerate}[label=(\alph*)]
\item Explain why Sam should never choose $S _ { 1 }$ and why the computer should not choose $C _ { 3 }$.
\item Find the play-safe strategies for the reduced 2 by 2 game for Sam and the computer, and hence show that this game does not have a stable solution.
\item Sam uses random numbers to choose $S _ { 2 }$ with probability $p$ and $S _ { 3 }$ with probability $1 - p$.
\begin{enumerate}[label=(\roman*)]
\item Find expressions for the expected gain for Sam when the computer chooses each of its two remaining strategies.
\item Calculate the value of $p$ for Sam to maximise his total points.
\item Hence find the expected points gain for Sam.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
Surname & & & & & & \multicolumn{2}{|c|}{Other Names} & & & & & \\
\hline
\multicolumn{2}{|c|}{Centre Number} & & & & & & \multicolumn{2}{|l|}{Candidate Number} & & & & \\
\hline
\multicolumn{3}{|l|}{Candidate Signature} & & & & & & & & & & \\
\hline
\end{tabular}
\end{center}
\section*{General Certificate of Education January 2006 \\
Advanced Level Examination}
\section*{MATHEMATICS \\
Unit Decision 2}
MD02
\section*{Insert}
Wednesday 18 January 20061.30 pm to 3.00 pm
Insert for use in Questions 3 and 4.\\
Fill in the boxes at the top of this page.\\
Fasten this insert securely to your answer book.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA D2 2006 Q6 [11]}}