| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Difficulty | Moderate -0.5 This is a standard game theory question requiring systematic application of dominance reduction to find optimal strategy. While it requires careful checking of multiple comparisons, it follows a well-defined algorithmic procedure with no novel insight needed. The topic is specialized but the execution is routine for students who have learned the method. |
| Spec | 7.08b Dominance: reduce pay-off matrix7.08e Mixed strategies: optimal strategy using equations or graphical method |
| B plays 1 | B plays 2 | B plays 3 | |
| A plays 1 | - 2 | 2 | - 3 |
| A plays 2 | 1 | 1 | - 1 |
| A plays 3 | 2 | - 1 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(B1\) | — |
| \(B1\) | — | Define \(p\); allow those who define B play 2 with prob. \(p\) but no incorrect statements |
| \(M1 A1\) | — | Setting up three probability equations, implicit definition of \(p\) |
| \(A1\) | — | CAO (condone incorrect simplification) |
| \(M1 A1\) | — | Three lines drawn, accept \(p > 1\) or \(p < 0\) here. Must be functions of \(p\) |
| \(A1\) | — | CAO \(0 \leq p \leq 1\), scale correct and clear (or 1 line = 1), condone lack of labels. Rulers used |
| \(DM1\) | — | Must have drawn 3 lines. Finding their correct optimal point, must have three lines and set up an equation to find \(0 \leq p \leq 1\). Dependent on previous M mark. Must have three intersection points. If solving each pair of SE's must clearly select correct one or M0, but allow recovery if their choice is clear |
| \(A1\) | — | CAO; dependent on all, but a2B1, being awarded in this part |
| \(A1\) | — | CAO |
| \(B1\) | (1) | CAO; dependent on all previous M marks in (a) |
| (b) | 10 marks |
**(a)** | $B1$ | — | CAO Col 3 dominates Col 1 |
| $B1$ | — | Define $p$; allow those who define B play 2 with prob. $p$ but no incorrect statements |
| $M1 A1$ | — | Setting up three probability equations, implicit definition of $p$ |
| $A1$ | — | CAO (condone incorrect simplification) |
| $M1 A1$ | — | Three lines drawn, accept $p > 1$ or $p < 0$ here. Must be functions of $p$ |
| $A1$ | — | CAO $0 \leq p \leq 1$, scale correct and clear (or 1 line = 1), condone lack of labels. Rulers used |
| $DM1$ | — | Must have drawn 3 lines. Finding their correct optimal point, must have three lines and set up an equation to find $0 \leq p \leq 1$. Dependent on previous M mark. Must have three intersection points. If solving each pair of SE's must clearly select correct one or M0, but allow recovery if their choice is clear |
| $A1$ | — | CAO; dependent on all, but a2B1, being awarded in this part |
| $A1$ | — | CAO |
| $B1$ | (1) | CAO; dependent on all previous M marks in (a) |
**(b)** | | 10 marks | |
---3. A two-person zero-sum game is represented by the following pay-off matrix for player A.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
& B plays 1 & B plays 2 & B plays 3 \\
\hline
A plays 1 & - 2 & 2 & - 3 \\
\hline
A plays 2 & 1 & 1 & - 1 \\
\hline
A plays 3 & 2 & - 1 & 1 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Starting by reducing player B's options, find the best strategy for player B.
\item State the value of the game to player B.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D2 2014 Q3 [10]}}