| Exam Board | OCR |
|---|---|
| Module | Further Discrete (Further Discrete) |
| Year | 2017 |
| Session | Specimen |
| Marks | 11 |
| Topic | Dynamic Programming |
| Type | Zero-sum game stable solution |
| Difficulty | Standard +0.8 This is a multi-part Further Maths game theory question requiring understanding of play-safe strategies, stability conditions, and Nash equilibrium. While the individual concepts are standard for Further Discrete, the question requires careful analysis across multiple parts, including showing why certain modifications won't work (part iii) and finding suitable modifications (part iv). The Nash equilibrium explanation adds theoretical depth. This is moderately challenging for Further Maths students but follows predictable game theory procedures. |
| Spec | 7.08a Pay-off matrix: zero-sum games7.08c Pure strategies: play-safe strategies and stable solutions7.08d Nash equilibrium: identification and interpretation |
| Player \(B\) | |||
| Strategy \(X\) | Strategy \(Y\) | Strategy \(Z\) | |
| Player \(A\) Strategy \(P\) | 4 | 5 | \(-4\) |
| Player \(A\) Strategy \(Q\) | 3 | \(-1\) | 2 |
| Player \(A\) Strategy \(R\) | 4 | 0 | 2 |
The table shows the pay-off matrix for player $A$ in a two-person zero-sum game between $A$ and $B$.
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
& \multicolumn{3}{|c|}{Player $B$} \\
\hline
& Strategy $X$ & Strategy $Y$ & Strategy $Z$ \\
\hline
Player $A$ Strategy $P$ & 4 & 5 & $-4$ \\
\hline
Player $A$ Strategy $Q$ & 3 & $-1$ & 2 \\
\hline
Player $A$ Strategy $R$ & 4 & 0 & 2 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\roman*)]
\item Find the play-safe strategy for player $A$ and the play-safe strategy for player $B$. Use the values of the play-safe strategies to determine whether the game is stable or unstable. [3]
\item If player $B$ knows that player $A$ will use their play-safe strategy, which strategy should player $B$ use? [1]
\item Suppose that the value in the cell where both players use their play-safe strategies can be changed, but all other entries are unchanged. Show that there is no way to change this value that would make the game stable. [2]
\item Suppose, instead, that the value in one cell can be changed, but all other entries are unchanged, so that the game becomes stable. Identify a suitable cell and write down a new pay-off value for that cell which would make the game stable. [2]
\item Show that the zero-sum game with the new pay-off value found in part (iv) has a Nash equilibrium and explain what this means for the players. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete 2017 Q4 [11]}}