| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Difficulty | Moderate -0.5 This is a standard game theory question requiring routine application of dominance principles and LP formulation. Part (a) involves checking for saddle points (straightforward max-min comparison), part (b) requires identifying dominated strategies (systematic comparison of rows/columns), and part (c) is a textbook LP setup for mixed strategy games. While multi-part, each component follows well-established algorithms with no novel insight required, making it easier than average A-level material. |
| 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 method7.08f Mixed strategies via LP: reformulate as linear programming problem |
| B plays 1 | B plays 2 | B plays 3 | B plays 4 | |
| A plays 1 | - 2 | 1 | 3 | - 1 |
| A plays 2 | - 1 | 3 | 2 | 1 |
| A plays 3 | - 4 | 2 | 0 | - 1 |
| A plays 4 | 1 | - 2 | - 1 | 3 |
| - 2 | 1 | 3 |
| - 1 | 3 | 2 |
| 1 | - 2 | - 1 |
5. 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 | c | }
\hline
& B plays 1 & B plays 2 & B plays 3 & B plays 4 \\
\hline
A plays 1 & - 2 & 1 & 3 & - 1 \\
\hline
A plays 2 & - 1 & 3 & 2 & 1 \\
\hline
A plays 3 & - 4 & 2 & 0 & - 1 \\
\hline
A plays 4 & 1 & - 2 & - 1 & 3 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Verify that there is no stable solution to this game.
\item Explain why the $4 \times 4$ game above may be reduced to the following $3 \times 3$ game.
\item Formulate the $3 \times 3$ game as a linear programming problem for player A. Write the
\begin{center}
\begin{tabular}{ | c | c | c | }
\hline
- 2 & 1 & 3 \\
\hline
- 1 & 3 & 2 \\
\hline
1 & - 2 & - 1 \\
\hline
\end{tabular}
\end{center}
constraints as inequalities. Define your variables clearly.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D2 2006 Q5 [13]}}