| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Zero-sum game optimal mixed strategy |
| Difficulty | Moderate -0.3 This is a standard textbook application of game theory requiring dominance reduction and solving a 2×3 game using mixed strategies. While it involves multiple steps (identifying dominance, setting up probability equations, solving), the techniques are routine for D2 students with no novel problem-solving required. Slightly easier than average A-level due to being a direct application of learned algorithms. |
| Spec | 7.08b Dominance: reduce pay-off matrix7.08d Nash equilibrium: identification and interpretation7.08e Mixed strategies: optimal strategy using equations or graphical method |
| Fiendish uses plan 1 | Fiendish uses plan 2 | Fiendish uses plan 3 | |
| Air | 0 | 4 | 5 |
| Sea | 2 | -3 | 1 |
| Land | -2 | 3 | -2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (a) Row 1 (air) dominates row 3(land), (so Row 3 can be deleted) | B1 | 1 |
| (b) Let Goodie play row 1 with probability \(p\), and row 2 with probability \(1 - p\). | 1M1 1A1 | |
| If F plays 1 G's expected winnings are \(0 + 2(1-p) = 2 - 2p\) | ||
| If F plays 2 G's expected winnings are \(4p - 3(1-p) = 7p - 3\) | ||
| If F plays 3 G's expected winnings are \(5p + (1-p) = 4p + 1\) | ||
| \(7p - 3 = 2 - 2p\) | 2M1 2A1 | |
| \(9p = 5\) | ||
| \(p = \frac{5}{9}\) | 3A1 | |
| Goodie should play Row 1 (air) with probability \(\frac{5}{9}\), row 2 (sea) with probability \(\frac{4}{9}\) and never row 3 (land). | 4A1ft | 7 |
| (c) The value of the game to Goodie is \(\frac{8}{9}\). | B1 | 1 |
| Total 9 |
| Answer/Working | Marks | Guidance |
|---|---|---|
| **(a)** Row 1 (air) dominates row 3(land), (so Row 3 can be deleted) | B1 | **1** |
| **(b)** Let Goodie play row 1 with probability $p$, and row 2 with probability $1 - p$. | 1M1 1A1 | |
| If F plays 1 G's expected winnings are $0 + 2(1-p) = 2 - 2p$ | | |
| If F plays 2 G's expected winnings are $4p - 3(1-p) = 7p - 3$ | | |
| If F plays 3 G's expected winnings are $5p + (1-p) = 4p + 1$ | | |
| $7p - 3 = 2 - 2p$ | 2M1 2A1 | |
| $9p = 5$ | | |
| $p = \frac{5}{9}$ | 3A1 | |
| Goodie should play Row 1 (air) with probability $\frac{5}{9}$, row 2 (sea) with probability $\frac{4}{9}$ and never row 3 (land). | 4A1ft | **7** |
| **(c)** The value of the game to Goodie is $\frac{8}{9}$. | B1 | **1** |
| | | **Total 9** |
**Notes for question 5:**
- a1B1: CAO. Accept 'air dominates land' etc. Must have a named row dominating a named row
- b1M1: Setting up three probability equations, implicit definition of p.
- b1A1: CAO
- b2M1: Three lines drawn, accept $p > 1$ or $p < 0$ here. Must be functions of p.
- b2A1: CAO 0 ≤ p ≤ 1, scale clear (or 1 line = 1), condone lack of labels. Rulers used.
- b3DM1: Must have drawn 3 lines. Finding their correct optimal point, must have three lines and set up an equation to find 0 ≤ p ≤ 1. If solving each pair of SE's must clearly select the correct one or M0, but allow recovery if their choice is clear from (c).
- b3A1: CAO 5/9
- b4A1ft: All three options listed must fit from their p, check page 1, no negatives.
- c1B1: CAO
---5. Agent Goodie is planning to break into Evil Doctor Fiendish's secret base.
He uses game theory to determine whether to approach the base from air, sea or land.\\
Evil Doctor Fiendish decides each day which of three possible plans he should use to protect his base.
Agent Goodie evaluates the situation. He assigns numbers, negative indicating he fails in his mission, positive indicating success, to create a pay-off matrix. The numbers range from - 3 (he fails in his mission and is captured) to 5 (he successfully achieves his mission and escapes uninjured) and the pay-off matrix is shown below.
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
& Fiendish uses plan 1 & Fiendish uses plan 2 & Fiendish uses plan 3 \\
\hline
Air & 0 & 4 & 5 \\
\hline
Sea & 2 & -3 & 1 \\
\hline
Land & -2 & 3 & -2 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Reduce the game so that Agent Goodie has only two choices, explaining your reasoning.
\item Use game theory to determine Agent Goodie's best strategy.
\item Find the value of the game to Agent Goodie.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D2 2012 Q5 [9]}}