| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Zero-sum game stable solution |
| Difficulty | Easy -1.2 This is a straightforward application of the standard algorithm for finding stable solutions in zero-sum games: identify row minima, column maxima, and check if maximin equals minimax. The question requires only mechanical application of a well-defined procedure with no problem-solving insight or multi-step reasoning beyond the basic algorithm taught in D2. |
| 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 interpretation |
| \multirow{5}{*}{Alex Strategy} | D | E | F | G | |
| A | 5 | - 4 | - 1 | 1 | |
| B | 4 | 3 | 0 | 1 | |
| C | - 3 | 0 | - 5 | - 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Row minima found: \(A=-4\), \(B=0\), \(C=-5\) | M1 | Correct row minima identified |
| Maximin \(= 0\) (Row \(B\)) | A1 | Play-safe for Alex is strategy \(B\) |
| Column maxima found: \(D=5\), \(E=3\), \(F=0\), \(G=1\) | M1 | Correct column maxima identified |
| Minimax \(= 0\) (Column \(F\)) | A1 | Play-safe for Roberto is strategy \(F\); since maximin \(=\) minimax \(= 0\), stable solution exists |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((B, F)\) with value \(0\) | B1 | Must identify position and value |
# Question 2:
## Part (a) - Stable Solution [4 marks]
| Answer | Mark | Guidance |
|--------|------|----------|
| Row minima found: $A=-4$, $B=0$, $C=-5$ | M1 | Correct row minima identified |
| Maximin $= 0$ (Row $B$) | A1 | Play-safe for Alex is strategy $B$ |
| Column maxima found: $D=5$, $E=3$, $F=0$, $G=1$ | M1 | Correct column maxima identified |
| Minimax $= 0$ (Column $F$) | A1 | Play-safe for Roberto is strategy $F$; since maximin $=$ minimax $= 0$, stable solution exists |
## Part (b) - Saddle Points [1 mark]
| Answer | Mark | Guidance |
|--------|------|----------|
| $(B, F)$ with value $0$ | B1 | Must identify position and value |2 Alex and Roberto play a zero-sum game. The game is represented by the following pay-off matrix for Alex.
\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Roberto}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
\multirow{5}{*}{Alex Strategy} & D & E & F & G & \\
\hline
& A & 5 & - 4 & - 1 & 1 \\
\hline
& B & 4 & 3 & 0 & 1 \\
\hline
& C & - 3 & 0 & - 5 & - 2 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Show that this game has a stable solution and state the play-safe strategy for each player.
\item List any saddle points.
\end{enumerate}
\hfill \mbox{\textit{AQA D2 2014 Q2 [5]}}