| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Difficulty | Easy -1.8 This is a standard game theory problem requiring routine application of the minimax/maximin algorithm to find optimal strategies in a zero-sum game. Despite being labeled as 'Groups', this is actually a straightforward Decision Mathematics question involving mechanical checking of row minima and column maxima—no abstract reasoning or proof required, just algorithmic application of a well-defined procedure. |
| Spec | 7.08e Mixed strategies: optimal strategy using equations or graphical method |
| S plays 1 | S plays 2 | S plays 3 | |
| L plays 1 | - 4 | - 1 | 1 |
| L plays 2 | 3 | - 1 | - 2 |
| L plays 3 | - 3 | 0 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Check for saddle point — none exists | B1 | |
| Eliminate dominated strategies if possible | M1 | |
| L plays 1 dominated — remove row 1 | A1 | |
| Remaining \(2\times 3\) matrix; check further dominance | M1 | |
| Set up probability equations: let \(p\) = prob L plays 2 | M1 | |
| \(E(\text{S plays 1}) = 3p - 3(1-p)\); \(E(\text{S plays 3}) = -2p + 2(1-p)\) | A1 | |
| Solve: equate expected values, \(p = \frac{1}{2}\) | A1 | |
| Value of game \(= 0\) | A1 | |
| Best strategy: L plays 2 and 3 each with probability \(\frac{1}{2}\) | A1 |
# Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Check for saddle point — none exists | B1 | |
| Eliminate dominated strategies if possible | M1 | |
| L plays 1 dominated — remove row 1 | A1 | |
| Remaining $2\times 3$ matrix; check further dominance | M1 | |
| Set up probability equations: let $p$ = prob L plays 2 | M1 | |
| $E(\text{S plays 1}) = 3p - 3(1-p)$; $E(\text{S plays 3}) = -2p + 2(1-p)$ | A1 | |
| Solve: equate expected values, $p = \frac{1}{2}$ | A1 | |
| Value of game $= 0$ | A1 | |
| Best strategy: L plays 2 and 3 each with probability $\frac{1}{2}$ | A1 | |
---4. Laura and Sam play a zero-sum game. This game is represented by the following pay-off matrix for Laura.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\hline
& S plays 1 & S plays 2 & S plays 3 \\
\hline
L plays 1 & - 4 & - 1 & 1 \\
\hline
L plays 2 & 3 & - 1 & - 2 \\
\hline
L plays 3 & - 3 & 0 & 2 \\
\hline
\end{tabular}
\end{center}
Find the best strategy for Laura and the value of the game to her.\\
\hfill \mbox{\textit{Edexcel D2 2011 Q4 [9]}}