| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Zero-sum game optimal mixed strategy |
| Difficulty | Moderate -0.5 This is a standard textbook exercise in game theory requiring routine application of dominance and mixed strategy formulas. While it involves multiple parts, each step follows a well-defined algorithm taught in D2 with no novel insight required—easier than average A-level questions but not trivial due to the algebraic manipulation involved. |
| 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 |
| \multirow{4}{*}{Rowan} | Strategy | \(\mathrm { C } _ { 1 }\) | \(\mathrm { C } _ { 2 }\) | \(\mathrm { C } _ { 3 }\) |
| \(\mathrm { R } _ { 1 }\) | -3 | -4 | 1 | |
| \(\mathbf { R } _ { \mathbf { 2 } }\) | 1 | 5 | -1 | |
| \(\mathbf { R } _ { \mathbf { 3 } }\) | -2 | -3 | 4 |
| Surname | Other Names | |||||||||||
| Centre Number | Candidate Number | |||||||||||
| Candidate Signature | ||||||||||||
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Gain for Rowan + gain for Colleen in each strategy \(= 0\) | E1 | Gain for one \(=\) loss of other |
| Total: 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\begin{array}{cccc} & & & \text{min} \\ -3 & -4 & 1 & \boxed{-4} \\ 1 & 5 & -1 & \boxed{-1} \\ -2 & -3 & 4 & \boxed{-3} \\ \text{Max} & \boxed{1} & 5 & 4 \end{array}\) | M1 | Minimum of rows & max of columns, or maximum of minima or minimax |
| All values correct | A1 | All values correct (seen) or words maximin and minimax highlighted |
| \(1 \neq -1 \Rightarrow\) no stable solution | E1 | |
| Total: 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(R_3\) dominates \(R_1\) | ||
| \((-3,-4,1) < (-2,-3,4)\) so never play \(R_1\) | E1 | |
| Total: 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| R chooses \(R_2\) with prob \(p \Rightarrow\) choose \(R_3\) with prob \(1-p\) | M1 | Attempt at one expression |
| Expected gain when C plays: \(C_1: p - 2(1-p) = 3p - 2\) \(C_2: 5p - 3(1-p) = 8p - 3\) \(C_3: -p + 4(1-p) = 4 - 5p\) | A1 | All correct unsimplified |
| Plot expected gains for \(0 \leq p \leq 1\) | M1 | |
| Correct graph plotted | A1 | Condone mirror image |
| Choosing their "highest" point: \(C_1\) & \(C_3\) intersect \(\Rightarrow 3p - 2 = 4 - 5p\) | M1 | Any 2 lines |
| \(\Rightarrow p = \dfrac{3}{4}\) | A1 | |
| \(\Rightarrow\) play \(R_2\) with prob \(\dfrac{3}{4}\) and \(R_3\) with prob \(\dfrac{1}{4}\) | E1\(\checkmark\) | Statement of strategy |
| Total: 7 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Value of game is \(3 \times \dfrac{3}{4} - 2 = \dfrac{1}{4}\) | B1 | CSO or equivalent, e.g. \(0.25\) |
| Total: 1 mark |
# Question 6:
## Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Gain for Rowan + gain for Colleen in each strategy $= 0$ | E1 | Gain for one $=$ loss of other |
| **Total: 1 mark** | | |
## Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\begin{array}{cccc} & & & \text{min} \\ -3 & -4 & 1 & \boxed{-4} \\ 1 & 5 & -1 & \boxed{-1} \\ -2 & -3 & 4 & \boxed{-3} \\ \text{Max} & \boxed{1} & 5 & 4 \end{array}$ | M1 | Minimum of rows & max of columns, or maximum of minima or minimax |
| All values correct | A1 | All values correct (seen) or words maximin and minimax highlighted |
| $1 \neq -1 \Rightarrow$ no stable solution | E1 | |
| **Total: 3 marks** | | |
## Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $R_3$ dominates $R_1$ | | |
| $(-3,-4,1) < (-2,-3,4)$ so never play $R_1$ | E1 | |
| **Total: 1 mark** | | |
## Part (d)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| R chooses $R_2$ with prob $p \Rightarrow$ choose $R_3$ with prob $1-p$ | M1 | Attempt at one expression |
| Expected gain when C plays: $C_1: p - 2(1-p) = 3p - 2$ $C_2: 5p - 3(1-p) = 8p - 3$ $C_3: -p + 4(1-p) = 4 - 5p$ | A1 | All correct unsimplified |
| Plot expected gains for $0 \leq p \leq 1$ | M1 | |
| Correct graph plotted | A1 | Condone mirror image |
| Choosing their "highest" point: $C_1$ & $C_3$ intersect $\Rightarrow 3p - 2 = 4 - 5p$ | M1 | Any 2 lines |
| $\Rightarrow p = \dfrac{3}{4}$ | A1 | |
| $\Rightarrow$ play $R_2$ with prob $\dfrac{3}{4}$ and $R_3$ with prob $\dfrac{1}{4}$ | E1$\checkmark$ | Statement of strategy |
| **Total: 7 marks** | | |
## Part (d)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Value of game is $3 \times \dfrac{3}{4} - 2 = \dfrac{1}{4}$ | B1 | CSO or equivalent, e.g. $0.25$ |
| **Total: 1 mark** | | |
**Section Total: 13 marks**
**Paper Total: 75 marks**6 Two people, Rowan and Colleen, play a zero-sum game. The game is represented by the following pay-off matrix for Rowan.
Colleen
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multirow{4}{*}{Rowan} & Strategy & $\mathrm { C } _ { 1 }$ & $\mathrm { C } _ { 2 }$ & $\mathrm { C } _ { 3 }$ \\
\hline
& $\mathrm { R } _ { 1 }$ & -3 & -4 & 1 \\
\hline
& $\mathbf { R } _ { \mathbf { 2 } }$ & 1 & 5 & -1 \\
\hline
& $\mathbf { R } _ { \mathbf { 3 } }$ & -2 & -3 & 4 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Explain the meaning of the term 'zero-sum game'.
\item Show that this game has no stable solution.
\item Explain why Rowan should never play strategy $R _ { 1 }$.
\item \begin{enumerate}[label=(\roman*)]
\item Find the optimal mixed strategy for Rowan.
\item Find the value of the game.
\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*{MATHEMATICS \\
Unit Decision 2}
\section*{Insert}
Thursday 8 June 2006 9.00 am to 10.30 am
Insert for use in Questions 1, 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 [13]}}