| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Zero-sum game optimal mixed strategy |
| Difficulty | Moderate -0.8 This is a standard textbook exercise in game theory requiring routine application of well-defined algorithms: finding saddle points (maximin/minimax) and solving 2×2 mixed strategies using the standard formula. While it involves multiple steps, each step follows a mechanical procedure with no novel insight or problem-solving required, making it easier than average A-level questions. |
| 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{2}{*}{} | \multirow[b]{2}{*}{Strategy} | Col | ||
| X | Y | Z | ||
| \multirow{3}{*}{Ros} | I | -4 | -3 | 0 |
| II | 5 | -2 | 2 | |
| III | 1 | -1 | 3 | |
| \cline { 2 - 5 } \multicolumn{1}{c|}{} | Col | |||
| \cline { 2 - 5 } \multicolumn{1}{c|}{} | Strategy | \(\mathbf { C } _ { \mathbf { 1 } }\) | \(\mathbf { C } _ { \mathbf { 2 } }\) | \(\mathbf { C } _ { \mathbf { 3 } }\) |
| \multirow{2}{*}{Ros} | \(\mathbf { R } _ { \mathbf { 1 } }\) | 3 | 2 | 1 |
| \cline { 2 - 5 } | \(\mathbf { R } _ { \mathbf { 2 } }\) | - 2 | - 1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Row min: \(-4, -2, -1\) | M1 | Attempt at row minimum and column maximum |
| Col max: \(5, -1, 3\) | A1 | All figures correct |
| \(\min(\text{col max}) = \max(\text{row min}) \Rightarrow\) stable solution | E1 | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Ros plays III and Col plays Y | B1 | |
| Value of game \(= -1\) | B1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Ros plays \(R_1\) with probability \(p\) and \(R_2\) with probability \(1-p\) | ||
| \(C_1: 3p - 2(1-p) = 5p - 2\) | ||
| \(C_2: 2p - (1-p) = 3p - 1\) | M1 | Attempt at least 2 |
| \(C_3: p + 2(1-p) = 2 - p\) | A1 | Correct unsimplified |
| Plot expected gains against \(p\) for \(0 \leq p \leq 1\) | M1 | |
| Correct graph | A1 | Must see 0 or 1 on \(P\) axis; A0 if highest point of region not visible |
| Choose highest point of region below lines \(\Rightarrow 3p-1 = 2-p\) | M1 | Must be this pair of lines or their highest point |
| Leading to \(p = \dfrac{3}{4}\) | A1 | |
| Ros plays \(R_1\) with prob \(\dfrac{3}{4}\) and \(R_2\) with prob \(\dfrac{1}{4}\) | B1\(\checkmark\) | ft their \(p\) from any lines; Total: 7 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Value of game \(= 3 \times \dfrac{3}{4} - 1 = 1\dfrac{1}{4}\) or \(\left(2 - \dfrac{3}{4}\right) = 1\dfrac{1}{4}\) | B1 | Total: 1 |
## Question 4:
### Part (a)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Row min: $-4, -2, -1$ | M1 | Attempt at row minimum and column maximum |
| Col max: $5, -1, 3$ | A1 | All figures correct |
| $\min(\text{col max}) = \max(\text{row min}) \Rightarrow$ stable solution | E1 | Total: 3 |
### Part (a)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Ros plays III and Col plays Y | B1 | |
| Value of game $= -1$ | B1 | Total: 2 |
### Part (b)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Ros plays $R_1$ with probability $p$ and $R_2$ with probability $1-p$ | | |
| $C_1: 3p - 2(1-p) = 5p - 2$ | | |
| $C_2: 2p - (1-p) = 3p - 1$ | M1 | Attempt at least 2 |
| $C_3: p + 2(1-p) = 2 - p$ | A1 | Correct unsimplified |
| Plot expected gains against $p$ for $0 \leq p \leq 1$ | M1 | |
| Correct graph | A1 | Must see 0 or 1 on $P$ axis; A0 if highest point of region not visible |
| Choose highest point of region below lines $\Rightarrow 3p-1 = 2-p$ | M1 | Must be this pair of lines or their highest point |
| Leading to $p = \dfrac{3}{4}$ | A1 | |
| Ros plays $R_1$ with prob $\dfrac{3}{4}$ and $R_2$ with prob $\dfrac{1}{4}$ | B1$\checkmark$ | ft their $p$ from any lines; Total: 7 |
### Part (b)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Value of game $= 3 \times \dfrac{3}{4} - 1 = 1\dfrac{1}{4}$ or $\left(2 - \dfrac{3}{4}\right) = 1\dfrac{1}{4}$ | B1 | Total: 1 |
---4
\begin{enumerate}[label=(\alph*)]
\item Two people, Ros and Col, play a zero-sum game. The game is represented by the following pay-off matrix for Ros.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multirow{2}{*}{} & \multirow[b]{2}{*}{Strategy} & \multicolumn{3}{|c|}{Col} \\
\hline
& & X & Y & Z \\
\hline
\multirow{3}{*}{Ros} & I & -4 & -3 & 0 \\
\hline
& II & 5 & -2 & 2 \\
\hline
& III & 1 & -1 & 3 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\roman*)]
\item Show that this game has a stable solution.
\item Find the play-safe strategy for each player and state the value of the game.
\end{enumerate}\item Ros and Col play a different zero-sum game for which there is no stable solution. The game is represented by the following pay-off matrix for Ros.
\begin{center}
\begin{tabular}{ | c | c | r | r | c | }
\cline { 2 - 5 }
\multicolumn{1}{c|}{} & \multicolumn{3}{c|}{Col} & \\
\cline { 2 - 5 }
\multicolumn{1}{c|}{} & Strategy & $\mathbf { C } _ { \mathbf { 1 } }$ & $\mathbf { C } _ { \mathbf { 2 } }$ & $\mathbf { C } _ { \mathbf { 3 } }$ \\
\hline
\multirow{2}{*}{Ros} & $\mathbf { R } _ { \mathbf { 1 } }$ & 3 & 2 & 1 \\
\cline { 2 - 5 }
& $\mathbf { R } _ { \mathbf { 2 } }$ & - 2 & - 1 & 2 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\roman*)]
\item Find the optimal mixed strategy for Ros.
\item Calculate the value of the game.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA D2 2007 Q4 [13]}}