| Exam Board | OCR |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Difficulty | Standard +0.3 This is a standard game theory question from Decision Mathematics covering zero-sum games, dominance, and play-safe strategies. While it requires understanding multiple concepts (converting to zero-sum, finding saddle points, checking dominance), these are all routine algorithmic procedures taught directly in the D2 syllabus with no novel problem-solving required. The multi-part structure and computational steps place it slightly above average difficulty, but it remains a textbook-style question testing recall and application of standard techniques. |
| 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 |
| Collwen's choice | ||||||
| Fire | Ice | Gale | |||
| \cline { 2 - 5 } | Fire | 1 | 7 | 2 | ||
\cline { 2 - 5 }
| Ice | 6 | 2 | 4 | ||
| \cline { 2 - 5 } | Gale | 5 | 1 | 3 | ||
| \cline { 2 - 5 } | ||||||
| Collwen's choice | ||||||
| Fire | Ice | Gale | |||
| \cline { 2 - 5 } | Fire | 7 | 1 | 6 | ||
\cline { 2 - 5 }
| Ice | 2 | 6 | 4 | ||
| \cline { 2 - 5 } | Gale | 3 | 7 | 5 | ||
| \cline { 2 - 5 } | ||||||
| Collwen's choice | ||||||
| Fire | Ice | Gale | ||||
| \cline { 2 - 5 } | Fire | - 3 | 3 | - 2 | ||
\cline { 2 - 5 }
| Ice | 2 | - 2 | 0 | ||
| \cline { 2 - 5 } | Gale | 1 | - 3 | - 1 | ||
| \cline { 2 - 5 } | ||||||
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Each entry in zero-sum table = Ross's score \(-\) Collwen's score | M1 | Method explained |
| e.g. Fire/Fire: \(1-7=-6\)... checking given table: Fire/Fire \(= 1-7+(-3)\)... The zero-sum entries represent Ross's score minus Collwen's score (e.g. \(1-7=-6\), but table shows \(-3\)) — scores sum to constant 8, so zero-sum payoff \(=\) Ross score \(- \frac{8}{2}\) ... Each cell: Ross score \(+\) Collwen score \(=\) constant (e.g. \(1+7=8\)), so rewrite as difference by subtracting constant | A1 | Accept valid explanation showing scores sum to constant and conversion method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| If Ross chooses Ice, Collwen's scores (for Collwen) are: Fire\(=2\), Ice\(=-2\), Gale\(=0\); Collwen maximises so chooses Fire | B1 | Answer: Fire |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Ross's play-safe: find row maximin. Row minima: Fire\(=-3\), Ice\(=-2\), Gale\(=-3\); maximin \(= -2\), so Ross plays Ice | M1 A1 | |
| Collwen's play-safe: find column minimax. Column maxima: Fire\(=2\), Ice\(=3\), Gale\(=0\); minimax \(= 0\), so Collwen plays Gale | M1 A1 | |
| Maximin \((-2) \neq\) minimax \((0)\), so game is unstable | A1 | Must explicitly compare values |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Compare columns pairwise for Collwen (Collwen maximises — looks for larger values) | M1 | |
| Fire vs Ice: \(-3<3\), \(2>-2\), \(1>-3\) — neither dominates | A1 | |
| Fire vs Gale: \(-3>-2\)... not consistent domination | A1 | Conclusion: no strategy dominates any other, shown for all pairs |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Gale row: \(1, -3, -1\). Ice row: \(2, -2, 0\). Since \(1<2\), \(-3<-2\), \(-1<0\): Ice dominates Gale for Ross | B1 | Ross would never choose Gale as Ice always gives higher payoff |
| Reduced \(2\times3\) game (Fire and Ice rows): \(\begin{pmatrix}-3&3&-2\\2&-2&0\end{pmatrix}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Expected payoffs: vs Fire: \(-3p+2(1-p) = -5p+2\); vs Ice: \(3p-2(1-p)=5p-2\); vs Gale: \(-2p+0(1-p)=-2p\) | M1 | Lines correctly expressed |
| Graph drawn with three lines | A1 | |
| Optimal \(p\) at intersection of highest lower envelope: solve \(-5p+2 = 5p-2 \Rightarrow 4=10p \Rightarrow p=0.4\) | A1 | \(p = \frac{2}{5}\) |
# Question 4:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Each entry in zero-sum table = Ross's score $-$ Collwen's score | M1 | Method explained |
| e.g. Fire/Fire: $1-7=-6$... checking given table: Fire/Fire $= 1-7+(-3)$... The zero-sum entries represent Ross's score minus Collwen's score (e.g. $1-7=-6$, but table shows $-3$) — scores sum to constant 8, so zero-sum payoff $=$ Ross score $- \frac{8}{2}$ ... Each cell: Ross score $+$ Collwen score $=$ constant (e.g. $1+7=8$), so rewrite as difference by subtracting constant | A1 | Accept valid explanation showing scores sum to constant and conversion method |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| If Ross chooses Ice, Collwen's scores (for Collwen) are: Fire$=2$, Ice$=-2$, Gale$=0$; Collwen maximises so chooses **Fire** | B1 | Answer: Fire |
## Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Ross's play-safe: find row maximin. Row minima: Fire$=-3$, Ice$=-2$, Gale$=-3$; maximin $= -2$, so Ross plays **Ice** | M1 A1 | |
| Collwen's play-safe: find column minimax. Column maxima: Fire$=2$, Ice$=3$, Gale$=0$; minimax $= 0$, so Collwen plays **Gale** | M1 A1 | |
| Maximin $(-2) \neq$ minimax $(0)$, so game is unstable | A1 | Must explicitly compare values |
## Part (iv)
| Answer | Mark | Guidance |
|--------|------|----------|
| Compare columns pairwise for Collwen (Collwen maximises — looks for larger values) | M1 | |
| Fire vs Ice: $-3<3$, $2>-2$, $1>-3$ — neither dominates | A1 | |
| Fire vs Gale: $-3>-2$... not consistent domination | A1 | Conclusion: no strategy dominates any other, shown for all pairs |
## Part (v)
| Answer | Mark | Guidance |
|--------|------|----------|
| Gale row: $1, -3, -1$. Ice row: $2, -2, 0$. Since $1<2$, $-3<-2$, $-1<0$: Ice dominates Gale for Ross | B1 | Ross would never choose Gale as Ice always gives higher payoff |
| Reduced $2\times3$ game (Fire and Ice rows): $\begin{pmatrix}-3&3&-2\\2&-2&0\end{pmatrix}$ | B1 | |
## Part (vi)
| Answer | Mark | Guidance |
|--------|------|----------|
| Expected payoffs: vs Fire: $-3p+2(1-p) = -5p+2$; vs Ice: $3p-2(1-p)=5p-2$; vs Gale: $-2p+0(1-p)=-2p$ | M1 | Lines correctly expressed |
| Graph drawn with three lines | A1 | |
| Optimal $p$ at intersection of highest lower envelope: solve $-5p+2 = 5p-2 \Rightarrow 4=10p \Rightarrow p=0.4$ | A1 | $p = \frac{2}{5}$ |
---4 Ross and Collwen are playing a game in which each secretly chooses a magic spell. They then reveal their choices, and work out their scores using the tables below. Ross and Collwen are both trying to get as large a score as possible.
\begin{center}
\begin{tabular}{ l c | c | c | c }
& & \multicolumn{3}{|c}{Collwen's choice} \\
& \begin{tabular}{ r }
Score for \\
Ross \\
\end{tabular} & Fire & Ice & Gale \\
\cline { 2 - 5 }
& Fire & 1 & 7 & 2 \\
\cline { 2 - 5 }
\begin{tabular}{ l }
Ross's \\
choice \\
\end{tabular} & Ice & 6 & 2 & 4 \\
\cline { 2 - 5 }
& Gale & 5 & 1 & 3 \\
\cline { 2 - 5 }
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ l c | c | c | c }
& & \multicolumn{2}{c}{Collwen's choice} & \\
& \begin{tabular}{ r }
Score for \\
Collwen \\
\end{tabular} & Fire & Ice & Gale \\
\cline { 2 - 5 }
& Fire & 7 & 1 & 6 \\
\cline { 2 - 5 }
\begin{tabular}{ l }
Ross's \\
choice \\
\end{tabular} & Ice & 2 & 6 & 4 \\
\cline { 2 - 5 }
& Gale & 3 & 7 & 5 \\
\cline { 2 - 5 }
\end{tabular}
\end{center}
(i) Explain how this can be rewritten as the following zero-sum game.
\begin{center}
\begin{tabular}{ l c | c | c | c }
& & \multicolumn{2}{c}{Collwen's choice} & \\
& & Fire & Ice & Gale \\
\cline { 2 - 5 }
& Fire & - 3 & 3 & - 2 \\
\cline { 2 - 5 }
\begin{tabular}{ l }
Ross's \\
choice \\
\end{tabular} & Ice & 2 & - 2 & 0 \\
\cline { 2 - 5 }
& Gale & 1 & - 3 & - 1 \\
\cline { 2 - 5 }
\end{tabular}
\end{center}
(ii) If Ross chooses Ice what is Collwen's best choice?\\
(iii) Find the play-safe strategy for Ross and the play-safe strategy for Collwen, showing your working. Explain how you know that the game is unstable.\\
(iv) Show that none of Collwen's strategies dominates any other.\\
(v) Explain why Ross would never choose Gale, hence reduce the game to a $2 \times 3$ zero-sum game, showing the pay-offs for Ross.
Suppose that Ross uses random numbers to choose between Fire and Ice, choosing Fire with probability $p$ and Ice with probability $1 - p$.\\
(vi) Use a graphical method to find the optimal value of $p$ for Ross.
\hfill \mbox{\textit{OCR D2 2014 Q4 [16]}}