| Exam Board | OCR |
|---|---|
| Module | Further Discrete (Further Discrete) |
| Year | 2018 |
| Session | December |
| Marks | 7 |
| Topic | Dynamic Programming |
| Type | Zero-sum game stable solution |
| Difficulty | Standard +0.8 This is a Further Maths game theory question requiring systematic calculation of 16 outcomes, understanding of zero-sum games, and construction of a 4×4 pay-off matrix. While the concepts are standard for Further Discrete, the computational burden and need to carefully evaluate powers (e.g., 2^5 vs 5^2) across all cases makes it moderately challenging but still within typical Further Maths scope. |
| Spec | 7.08a Pay-off matrix: zero-sum games7.08b Dominance: reduce pay-off matrix |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | 0 | 0 |
| 7 | 2 | |
| 7 | 4 |
Question 1:
1 | 0 | 0 | 0 | 0 | 0 | 3
7 | 2
7 | 4
28
71 Arif and Bindiya play a game as follows.
\begin{itemize}
\item They each secretly choose a positive integer from $\{ 2,3,4,5 \}$.
\item They then reveal their choices. Let Arif's choice be $A$ and Bindiya's choice be $B$.
\item If $A ^ { B } \geqslant B ^ { A }$, Arif wins $B$ points and Bindiya wins $- 4 - B$ points.
\item If $A ^ { B } < B ^ { A }$, Arif wins $- 4 - A$ points and Bindiya wins $A$ points.
\begin{enumerate}[label=(\alph*)]
\item Assuming that each of the 16 possible outcomes is equally likely to be chosen, show that the average amount won by Arif is 0 .
\item \begin{enumerate}[label=(\roman*)]
\item Describe how to convert this game to a zero-sum game.
\item Construct the pay-off matrix for this zero-sum game, with Arif on rows.
\end{itemize}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete 2018 Q1 [7]}}