| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Difficulty | Moderate -1.0 This is a standard game theory question from Decision Mathematics involving routine application of graphical methods to find mixed strategies in a 2×3 zero-sum game. The procedures are algorithmic and well-practiced, requiring no novel insight. Note: This question is incorrectly tagged as 'Groups (UFM Additional Further Pure)' - it's actually basic D2 game theory, making it easier than typical Further Maths content. |
| 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|}{} | \(Y\) | |||
| \cline { 2 - 5 } \multicolumn{2}{c|}{} | \(Y _ { 1 }\) | \(Y _ { 2 }\) | \(Y _ { 3 }\) | |
| \multirow{2}{*}{\(X\)} | \(X _ { 1 }\) | 10 | 4 | 3 |
| \cline { 2 - 5 } | \(X _ { 2 }\) | \({ } ^ { - } 4\) | \({ } ^ { - } 1\) | 9 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Let \(p\) = probability \(X\) plays \(X_1\) | M1 | Setting up expressions |
| \(E(\text{vs }Y_1) = 10p + (-4)(1-p) = 14p - 4\) | A1 | |
| \(E(\text{vs }Y_2) = 4p + (-1)(1-p) = 5p - 1\) | A1 | |
| \(E(\text{vs }Y_3) = 3p + 9(1-p) = 9 - 6p\) | A1 | |
| Graph lines and find highest lower envelope | M1 | |
| Optimal intersection: \(Y_1\) and \(Y_3\): \(14p-4 = 9-6p \Rightarrow 20p = 13 \Rightarrow p = \frac{13}{20}\) | M1 A1 | |
| \(X\) plays \(X_1\) with probability \(\frac{13}{20}\), \(X_2\) with probability \(\frac{7}{20}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(Y\) should only mix strategies that are active at optimum, i.e. \(Y_1\) and \(Y_3\) | M1 | |
| Let \(q\) = probability \(Y\) plays \(Y_1\): \(10q + 3(1-q) = -4q + 9(1-q)\) | M1 A1 | |
| \(7q + 3 = 9 - 13q \Rightarrow 20q = 6 \Rightarrow q = \frac{3}{10}\) | A1 | |
| \(Y\) plays \(Y_1\) with prob \(\frac{3}{10}\), \(Y_3\) with prob \(\frac{7}{10}\), \(Y_2\) with prob \(0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Value \(= 14\left(\frac{13}{20}\right) - 4 = \frac{182}{20} - 4 = \frac{102}{20} = \frac{51}{10}\) | M1 A1 | Or substitute into either active line |
# Question 5:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Let $p$ = probability $X$ plays $X_1$ | M1 | Setting up expressions |
| $E(\text{vs }Y_1) = 10p + (-4)(1-p) = 14p - 4$ | A1 | |
| $E(\text{vs }Y_2) = 4p + (-1)(1-p) = 5p - 1$ | A1 | |
| $E(\text{vs }Y_3) = 3p + 9(1-p) = 9 - 6p$ | A1 | |
| Graph lines and find highest lower envelope | M1 | |
| Optimal intersection: $Y_1$ and $Y_3$: $14p-4 = 9-6p \Rightarrow 20p = 13 \Rightarrow p = \frac{13}{20}$ | M1 A1 | |
| $X$ plays $X_1$ with probability $\frac{13}{20}$, $X_2$ with probability $\frac{7}{20}$ | A1 | |
## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $Y$ should only mix strategies that are active at optimum, i.e. $Y_1$ and $Y_3$ | M1 | |
| Let $q$ = probability $Y$ plays $Y_1$: $10q + 3(1-q) = -4q + 9(1-q)$ | M1 A1 | |
| $7q + 3 = 9 - 13q \Rightarrow 20q = 6 \Rightarrow q = \frac{3}{10}$ | A1 | |
| $Y$ plays $Y_1$ with prob $\frac{3}{10}$, $Y_3$ with prob $\frac{7}{10}$, $Y_2$ with prob $0$ | | |
## Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Value $= 14\left(\frac{13}{20}\right) - 4 = \frac{182}{20} - 4 = \frac{102}{20} = \frac{51}{10}$ | M1 A1 | Or substitute into either active line |
---5. The payoff matrix for player $X$ in a two-person zero-sum game is shown below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & \multicolumn{3}{|c|}{$Y$} \\
\cline { 2 - 5 }
\multicolumn{2}{c|}{} & $Y _ { 1 }$ & $Y _ { 2 }$ & $Y _ { 3 }$ \\
\hline
\multirow{2}{*}{$X$} & $X _ { 1 }$ & 10 & 4 & 3 \\
\cline { 2 - 5 }
& $X _ { 2 }$ & ${ } ^ { - } 4$ & ${ } ^ { - } 1$ & 9 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Using a graphical method, find the optimal strategy for player $X$.
\item Find the optimal strategy for player $Y$.
\item Find the value of the game.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D2 Q5 [13]}}