| Exam Board | Edexcel |
|---|---|
| Module | FD2 (Further Decision 2) |
| Session | Specimen |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Zero-sum game LP formulation |
| Difficulty | Challenging +1.2 This is a structured Further Maths question on game theory requiring matrix reduction via dominance, play-safe strategy identification, LP constraint formulation, and value calculation. While it involves multiple steps and Further Maths content (inherently harder), each part follows standard algorithmic procedures taught in FD2 with no novel insight required. The LP formulation is template-based once the concept is understood. |
| Spec | 7.08a Pay-off matrix: zero-sum games7.08b Dominance: reduce pay-off matrix7.08c Pure strategies: play-safe strategies and stable solutions |
| B plays 1 | B plays 2 | B plays 3 | B plays 4 | |
| A plays 1 | 4 | -2 | 3 | 2 |
| A plays 2 | 3 | -1 | 2 | 0 |
| A plays 3 | -1 | 2 | 0 | 3 |
5.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
& B plays 1 & B plays 2 & B plays 3 & B plays 4 \\
\hline
A plays 1 & 4 & -2 & 3 & 2 \\
\hline
A plays 2 & 3 & -1 & 2 & 0 \\
\hline
A plays 3 & -1 & 2 & 0 & 3 \\
\hline
\end{tabular}
\end{center}
A two person zero-sum game is represented by the pay-off matrix for player A given above.
\begin{enumerate}[label=(\alph*)]
\item Explain, with justification, how this matrix may be reduced to a $3 \times 3$ matrix.
\item Find the play-safe strategy for each player and verify that there is no stable solution to this game.
The game is formulated as a linear programming problem for player A .\\
The objective is to maximise $P = V$, where $V$ is the value of the game to player A.\\
One of the constraints is that $p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1$, where $p _ { 1 } , p _ { 2 } , p _ { 3 }$ are the probabilities that player A plays 1, 2, 3 respectively.
\item Formulate the remaining constraints for this problem. Write these constraints as inequalities.
The Simplex algorithm is used to solve the linear programming problem.\\
The solution obtained is $p _ { 1 } = 0 , p _ { 2 } = \frac { 3 } { 7 } , p _ { 3 } = \frac { 4 } { 7 }$
\item Calculate the value of the game to player A.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FD2 Q5 [12]}}