| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Difficulty | Easy -1.8 This is a standard game theory problem from Decision Mathematics involving finding optimal strategies in a 3×3 zero-sum game using minimax/maximin methods. Despite being labeled 'Groups', this is actually a routine D2 topic requiring only algorithmic application of dominance and saddle point identification—significantly easier than typical Further Maths content and well below average A-level difficulty. |
| 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 |
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | \(B\) | |||
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | I | II | III | |
| \multirow{3}{*}{\(A\)} | I | - 3 | 4 | 0 |
| \cline { 2 - 5 } | II | 2 | 2 | 1 |
| \cline { 2 - 5 } | III | 3 | - 2 | - 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Row minima: \(-3, 1, -2\); maximin \(= 1\) (Row II) | M1 | Finding row minima and selecting maximum |
| Column maxima: \(3, 4, 1\); minimax \(= 1\) (Column III) | A1 | Finding column maxima and selecting minimum |
| Stable solution / saddle point since maximin \(=\) minimax \(= 1\) | A1 | Must state reason |
| Player \(A\) plays Row II (strategy II) | A1 | |
| Player \(B\) plays Column III (strategy III), value of game \(= 1\) | A1 |
# Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Row minima: $-3, 1, -2$; maximin $= 1$ (Row II) | M1 | Finding row minima and selecting maximum |
| Column maxima: $3, 4, 1$; minimax $= 1$ (Column III) | A1 | Finding column maxima and selecting minimum |
| Stable solution / saddle point since maximin $=$ minimax $= 1$ | A1 | Must state reason |
| Player $A$ plays Row II (strategy II) | A1 | |
| Player $B$ plays Column III (strategy III), value of game $= 1$ | A1 | |
---\begin{enumerate}
\item The payoff matrix for player $A$ in a two-person zero-sum game is shown below.
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & \multicolumn{3}{|c|}{$B$} \\
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & I & II & III \\
\hline
\multirow{3}{*}{$A$} & I & - 3 & 4 & 0 \\
\cline { 2 - 5 }
& II & 2 & 2 & 1 \\
\cline { 2 - 5 }
& III & 3 & - 2 & - 1 \\
\hline
\end{tabular}
\end{center}
Find the optimal strategy for each player and the value of the game.\\
\hfill \mbox{\textit{Edexcel D2 Q1 [5]}}