| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Game theory LP formulation |
| Difficulty | Challenging +1.2 This is a standard D2 game theory question requiring conversion of a pay-off matrix to LP form using the textbook method. While it involves multiple constraints and careful bookkeeping (7 marks suggests substantial work), it's a direct application of a taught algorithm with no novel problem-solving or insight required. Slightly above average difficulty due to the mechanical complexity and potential for algebraic errors, but well within the scope of routine D2 questions. |
| Spec | 7.08f Mixed strategies via LP: reformulate as linear programming problem |
| B plays 1 | B plays 2 | B plays 3 | |
| A plays 1 | 1 | - 3 | 2 |
| A plays 2 | - 2 | 3 | - 1 |
| A plays 3 | 5 | - 1 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Add 4 to each element | B1 | 1B1: Making all terms non-negative. |
| Let \(p_1, p_2, p_3\) be the probability of (A) playing 1, 2 and 3 respectively (where \(p_1, p_2, p_3 \geq 0\)) | B1 | 2B1: Defining probability variables |
| let \(V =\) value of the game (to player A) | B1 | 3B1: Defining \(V\) |
| maximise \(P = V\) | B1 | 4B1: 'maximise' + function/expression |
| subject to: | ||
| \(5p_1 + 2p_2 + 9p_3 \geq V\) | M1, A1 | 1M1: At least three equations/inequalities in \((V)\), \(p_1\), \(p_2\) and \(p_3\) |
| \(p_1 + 7p_2 + 3p_3 \geq V\) | 1A1: The three inequalities in \(V\), \(p_1\), \(p_2\) and \(p_3\) CAO | |
| \(6p_1 + 3p_2 + 4p_3 \geq V\) | ||
| \(p_1 + p_2 + p_3 \leq 1\) | A1 | |
| (7) | ||
| Total | 7 marks |
| Add 4 to each element | B1 | 1B1: Making all terms non-negative. |
|---|---|---|---|
| | | |
| Let $p_1, p_2, p_3$ be the probability of (A) playing 1, 2 and 3 respectively (where $p_1, p_2, p_3 \geq 0$) | B1 | 2B1: Defining probability variables |
| let $V =$ value of the game (to player A) | B1 | 3B1: Defining $V$ |
| | | |
| maximise $P = V$ | B1 | 4B1: 'maximise' + function/expression |
| subject to: | | |
| $5p_1 + 2p_2 + 9p_3 \geq V$ | M1, A1 | 1M1: At least three equations/inequalities in $(V)$, $p_1$, $p_2$ and $p_3$ |
| $p_1 + 7p_2 + 3p_3 \geq V$ | | 1A1: The three inequalities in $V$, $p_1$, $p_2$ and $p_3$ CAO |
| $6p_1 + 3p_2 + 4p_3 \geq V$ | | |
| $p_1 + p_2 + p_3 \leq 1$ | A1 | |
| | **(7)** | |
| **Total** | **7 marks** | |
---7. 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 & 1 & - 3 & 2 \\
\hline
A plays 2 & - 2 & 3 & - 1 \\
\hline
A plays 3 & 5 & - 1 & 0 \\
\hline
\end{tabular}
\end{center}
Formulate the game as a linear programming problem for player A. Write the constraints as inequalities. Define your variables clearly.\\
(Total 7 marks)\\
\hfill \mbox{\textit{Edexcel D2 2013 Q7 [7]}}