| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Zero-sum game optimal mixed strategy |
| Difficulty | Moderate -0.8 This is a standard D2 game theory question testing routine procedures: dominance definition (recall), checking for saddle points (mechanical), and finding optimal mixed strategy via graphical method (algorithmic). Part (d) requires recognizing when graphical methods fail, but this is textbook knowledge. No novel problem-solving or insight required—purely procedural application of learned techniques. |
| Spec | 7.08a Pay-off matrix: zero-sum games7.08c Pure strategies: play-safe strategies and stable solutions7.08e Mixed strategies: optimal strategy using equations or graphical method |
| Mark plays 1 | Mark plays 2 | Mark plays 3 | |
| Liz plays 1 | 5 | 3 | 2 |
| Liz plays 2 | 4 | 5 | 6 |
| Liz plays 3 | 6 | 4 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | B2,1,01 | 2 |
| M1 | 3 | |
| A1 | 3 | |
| B1 | ||
| B2,1,0, M1A1 | 2 | |
| A1ftA1 | 4 | |
| B1 | ||
| B1 | 2 |
**(a)** | B2,1,01 | 2 |
| M1 | 3 |
| A1 | 3 |
| B1 | |
| B2,1,0, M1A1 | 2 |
| A1ftA1 | 4 |
| B1 | |
| B1 | 2 |
[16]5. (a) In game theory, explain the circumstances under which column $( x )$ dominates column $( y )$ in a two-person zero-sum game.
Liz and Mark play a zero-sum game. This game is represented by the following pay-off matrix for Liz.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\hline
& Mark plays 1 & Mark plays 2 & Mark plays 3 \\
\hline
Liz plays 1 & 5 & 3 & 2 \\
\hline
Liz plays 2 & 4 & 5 & 6 \\
\hline
Liz plays 3 & 6 & 4 & 3 \\
\hline
\end{tabular}
\end{center}
(b) Verify that there is no stable solution to this game.\\
(c) Find the best strategy for Liz and the value of the game to her.
The game now changes so that when Liz plays 1 and Mark plays 3 the pay-off to Liz changes from 2 to\\
4. All other pay-offs for this zero-sum game remain the same.\\
(d) Explain why a graphical approach is no longer possible and briefly describe the method Liz should use to determine her best strategy.\\
(2) (Total 16 marks)\\
\hfill \mbox{\textit{Edexcel D2 2008 Q5 [16]}}