| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Difficulty | Moderate -0.5 This is a standard game theory problem requiring the minimax/maximin method to find optimal strategies and game value. While it involves multiple steps (finding row minima, column maxima, checking for saddle point, possibly mixed strategies), it's a routine algorithmic procedure from Decision Mathematics with no novel insight required. Easier than average A-level maths due to being a straightforward application of a taught algorithm. |
| Spec | 7.08a Pay-off matrix: zero-sum games7.08c Pure strategies: play-safe strategies and stable solutions7.08e Mixed strategies: optimal strategy using equations or graphical method |
| B plays 1 | B plays 2 | B plays 3 | |
| A plays 1 | - 2 | 4 | 3 |
| A plays 2 | 4 | - 1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| If B plays 1: A's expected gain \(= 4 - 6p\) | M1 A1 | |
| If B plays 2: A's expected gain \(= 5p - 1\) | ||
| If B plays 3: A's expected gain \(= p + 2\) | ||
| Graph plotted with correct lines | M1 A1 | |
| \(4 - 6p = 5p - 1 \Rightarrow 11p = 5 \Rightarrow p = \dfrac{5}{11}\) | A1 | |
| A should play 1 with probability \(\dfrac{5}{11}\) and play 2 with probability \(\dfrac{6}{11}\) | B1 | |
| The game has value \(\dfrac{14}{11}\) to A | B1(7) |
## Question 3:
| Answer/Working | Marks | Guidance |
|---|---|---|
| If B plays 1: A's expected gain $= 4 - 6p$ | M1 A1 | |
| If B plays 2: A's expected gain $= 5p - 1$ | | |
| If B plays 3: A's expected gain $= p + 2$ | | |
| Graph plotted with correct lines | M1 A1 | |
| $4 - 6p = 5p - 1 \Rightarrow 11p = 5 \Rightarrow p = \dfrac{5}{11}$ | A1 | |
| A should play 1 with probability $\dfrac{5}{11}$ and play 2 with probability $\dfrac{6}{11}$ | B1 | |
| The game has value $\dfrac{14}{11}$ to A | B1(7) | |
---3. A two-person zero-sum game is represented by the following pay-off matrix for player A.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\hline
& B plays 1 & B plays 2 & B plays 3 \\
\hline
A plays 1 & - 2 & 4 & 3 \\
\hline
A plays 2 & 4 & - 1 & 2 \\
\hline
\end{tabular}
\end{center}
Find the best strategy for player A and the value of the game.\\
(Total 7 marks)\\
\hfill \mbox{\textit{Edexcel D2 Q3 [7]}}