| Exam Board | OCR |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 20 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Difficulty | Moderate -0.5 This is a game theory problem from Decision Mathematics involving zero-sum games, play-safe strategies, and mixed strategies. While it requires multiple steps and understanding of game theory concepts, the techniques are standard for D2: finding row/column minima/maxima, checking stability, and solving for optimal mixed strategy probabilities using expected value equations. The calculations are straightforward once the method is understood, making it slightly easier than average for A-level. |
| 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 |
| Cricket club | ||||
| \cline { 2 - 5 }\cline { 2 - 5 } | Doug | Euan | Fiona | |
| \cline { 2 - 5 } Sanjeev | 0 | 4 | - 2 | |
| \cline { 2 - 5 } Rugby club | Tom | - 4 | 2 | - 4 |
| \cline { 2 - 5 } | Ursula | 2 | - 6 | 0 |
| \cline { 2 - 5 } | ||||
| \cline { 2 - 5 } | ||||
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| 5 | B1 | 5 |
| \((10-4)\div 2 = 3\) | M1, A1 | 3 or 7; 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Calculating row minima | M1 | |
| Calculating col maxima (or equivalent) | M1 | |
| Play-safe for rugby club (rows) is Sanjeev | A1 | Sanjeev or S (not just \(-2\) or identifying row) |
| Play-safe for cricket club (cols) is Fiona | A1 | Fiona or F (not just 0 or identifying column) |
| Not stable because \(-2 \neq 0\) | B1 | Any correct explanation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Fiona | B1 | F |
| Ursula | B1 | U |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sanjeev's row dominates Tom's row | B1 | This or any equivalent statement about Tom and Sanjeev |
| Doug | M1 | Doug |
| Fiona's column dominates Doug's (once Tom's row has been removed) | A1 | This or any equivalent statement about Doug and Fiona |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E: 4p - 6(1-p) = 10p - 6\); \(F: -2p\) | M1 | Both expressions seen in any form (note: \(D\) gives \(2(1-p)=2-2p\)) |
| \(10p - 6 = -2p \Rightarrow p = 0.5\) | A1 | \(p=0.5\) (cao) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Delete \(T\) row and multiply entries by \(-1\) | B1 | |
| Add 4 to make entries non-negative | B1 | |
| Identifying meaning of \(x\), \(y\), \(z\): Choose Doug with probability \(x\), Euan with probability \(y\) and Fiona with probability \(z\); If Sanjeev plays, expected score \(= 4x+6z\); If Ursula plays, expected score \(= 2x+10y+4z\) | B1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(z = \frac{5}{6}\), maximum value for \(m = 5\) | M1 | |
| Hence maximum value for \(M = 1\) | A1 | [2] |
# Question 5:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| 5 | B1 | 5 |
| $(10-4)\div 2 = 3$ | M1, A1 | 3 or 7; 3 | [3] |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Calculating row minima | M1 | |
| Calculating col maxima (or equivalent) | M1 | |
| Play-safe for rugby club (rows) is Sanjeev | A1 | Sanjeev or S (not just $-2$ or identifying row) |
| Play-safe for cricket club (cols) is Fiona | A1 | Fiona or F (not just 0 or identifying column) |
| Not stable because $-2 \neq 0$ | B1 | Any correct explanation | [5] |
## Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Fiona | B1 | F |
| Ursula | B1 | U | [2] |
## Part (iv)
| Answer | Mark | Guidance |
|--------|------|----------|
| Sanjeev's row dominates Tom's row | B1 | This or any equivalent statement about Tom and Sanjeev |
| Doug | M1 | Doug |
| Fiona's column dominates Doug's (once Tom's row has been removed) | A1 | This or any equivalent statement about Doug and Fiona | [3] |
## Part (v)
| Answer | Mark | Guidance |
|--------|------|----------|
| $E: 4p - 6(1-p) = 10p - 6$; $F: -2p$ | M1 | Both expressions seen in any form (note: $D$ gives $2(1-p)=2-2p$) |
| $10p - 6 = -2p \Rightarrow p = 0.5$ | A1 | $p=0.5$ (cao) | [2] |
## Part (vi)
| Answer | Mark | Guidance |
|--------|------|----------|
| Delete $T$ row and multiply entries by $-1$ | B1 | |
| Add 4 to make entries non-negative | B1 | |
| Identifying meaning of $x$, $y$, $z$: Choose Doug with probability $x$, Euan with probability $y$ and Fiona with probability $z$; If Sanjeev plays, expected score $= 4x+6z$; If Ursula plays, expected score $= 2x+10y+4z$ | B1 | [3] |
## Part (vii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $z = \frac{5}{6}$, maximum value for $m = 5$ | M1 | |
| Hence maximum value for $M = 1$ | A1 | [2] |5 The local rugby club has challenged the local cricket club to a chess match to raise money for charity. Each of the top three chess players from the rugby club has played 10 chess games against each of the top three chess players from the cricket club. There were no drawn games. The table shows, for each pairing, the number of games won by the player from the rugby club minus the number of games won by the player from the cricket club. This will be called the score; the scores make a zero-sum game.
\begin{center}
\begin{tabular}{ l | l | c | c | c | }
\multicolumn{2}{l}{} & \multicolumn{3}{c}{Cricket club} \\
\cline { 2 - 5 }\cline { 2 - 5 }
& & Doug & Euan & Fiona \\
\cline { 2 - 5 }
Sanjeev & 0 & 4 & - 2 & \\
\cline { 2 - 5 }
Rugby club & Tom & - 4 & 2 & - 4 \\
\cline { 2 - 5 }
& Ursula & 2 & - 6 & 0 \\
\cline { 2 - 5 }
& & & & \\
\cline { 2 - 5 }
\end{tabular}
\end{center}
(i) How many of the 10 games between Sanjeev and Doug did Sanjeev win? How many of the 10 games between Sanjeev and Euan did Euan win?
Each club must choose one person to play. They want to choose the person who will optimise the score.\\
(ii) Find the play-safe choice for each club, showing your working. Explain how you know that the game is not stable.\\
(iii) Which person should the cricket club choose if they know that the rugby club will play-safe and which person should the rugby club choose if they know that the cricket club will play-safe?\\
(iv) Explain why the rugby club should not choose Tom. Which player should the cricket club not choose, and why?
The rugby club chooses its player by using random numbers to choose between Sanjeev and Ursula, where the probability of choosing Sanjeev is $p$ and the probability of choosing Ursula is $1 - p$.\\
(v) Write down an expression for the expected score for the rugby club for each of the two remaining choices that can be made by the cricket club. Calculate the optimal value for $p$.
Doug is studying AS Mathematics. He removes the row representing Tom and then models the cricket club's problem as the following LP.
$$\begin{array} { l l }
\operatorname { maximise } & M = m - 4 \\
\text { subject to } & m \leqslant 4 x \quad + 6 z \\
& m \leqslant 2 x + 10 y + 4 z \\
& x + y + z \leqslant 1 \\
\text { and } & m \geqslant 0 , x \geqslant 0 , y \geqslant 0 , z \geqslant 0
\end{array}$$
(vi) Show how Doug used the values in the table to get the constraints $m \leqslant 4 x + 6 z$ and $m \leqslant 2 x + 10 y + 4 z$.
Doug uses the Simplex algorithm to solve the LP problem. His solution has $x = 0$ and $y = \frac { 1 } { 6 }$.\\
(vii) Calculate the optimal value of $M$.
\hfill \mbox{\textit{OCR D2 2009 Q5 [20]}}