| Exam Board | OCR |
|---|---|
| Module | Further Discrete (Further Discrete) |
| Year | 2018 |
| Session | September |
| Marks | 9 |
| Topic | Dynamic Programming |
| Type | Zero-sum game optimal mixed strategy |
| Difficulty | Challenging +1.2 This is a standard Further Maths decision mathematics question on zero-sum games requiring three routine procedures: checking for saddle points, graphical solution for column player (plotting 3 lines and finding intersection), and setting up a simplex tableau. While it involves multiple steps and Further Maths content, each part follows a well-defined algorithm taught directly in the syllabus with no novel problem-solving required. |
| Spec | 7.07f Algebraic interpretation: explain simplex calculations7.08a Pay-off matrix: zero-sum games7.08c Pure strategies: play-safe strategies and stable solutions7.08d Nash equilibrium: identification and interpretation7.08e Mixed strategies: optimal strategy using equations or graphical method |
| X | Y | Z | |
| \cline { 2 - 4 } A | - 2 | 1 | 0 |
| \cline { 2 - 4 } B | 3 | 5 | - 3 |
| \cline { 2 - 4 } C | - 4 | - 2 | 2 |
| \cline { 2 - 4 } D | 0 | 2 | - 1 |
| \cline { 2 - 4 } | |||
| \cline { 2 - 4 } |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | 5 | -3 |
| -4 | -2 | 2 |
Question 3:
3 | 5 | -3
-4 | -2 | 23 The pay-off matrix for a zero-sum game is
\begin{center}
\begin{tabular}{ l | c | c | c | }
& \multicolumn{1}{c}{X} & \multicolumn{1}{c}{Y} & \multicolumn{1}{c}{Z} \\
\cline { 2 - 4 }
A & - 2 & 1 & 0 \\
\cline { 2 - 4 }
B & 3 & 5 & - 3 \\
\cline { 2 - 4 }
C & - 4 & - 2 & 2 \\
\cline { 2 - 4 }
D & 0 & 2 & - 1 \\
\cline { 2 - 4 }
& & & \\
\cline { 2 - 4 }
\end{tabular}
\end{center}
(i) Show that the game does not have a stable solution.\\
(ii) Use a graphical technique to find the optimal mixed strategy for the player on columns.\\
(iii) Formulate an initial simplex tableau for the problem of finding the optimal mixed strategy for the player on rows.
\hfill \mbox{\textit{OCR Further Discrete 2018 Q3 [9]}}