| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Zero-sum game optimal mixed strategy |
| Difficulty | Standard +0.3 This is a standard D2 game theory question covering routine procedures: defining zero-sum games, checking for saddle points, finding optimal mixed strategies using the 2×3 formula, and formulating the dual LP. All techniques are textbook exercises with no novel insight required, though the multi-part structure and LP formulation push it slightly above average difficulty. |
| Spec | 7.08a 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 |
| I | II | III | |
| I | 5 | 2 | 3 |
| II | 3 | 5 | 4 |
7. (a) Explain briefly what is meant by a zero-sum game.
A two person zero-sum game is represented by the following pay-off matrix for player $A$.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
& I & II & III \\
\hline
I & 5 & 2 & 3 \\
\hline
II & 3 & 5 & 4 \\
\hline
\end{tabular}
\end{center}
(b) Verify that there is no stable solution to this game.\\
(c) Find the best strategy for player $A$ and the value of the game to her.\\
(d) Formulate the game as a linear programming problem for player $B$. Write the constraints as inequalities and define your variables clearly.\\
(Total 17 marks)\\
\hfill \mbox{\textit{Edexcel D2 2005 Q7 [17]}}