| Exam Board | OCR |
|---|---|
| Module | Further Discrete (Further Discrete) |
| Year | 2018 |
| Session | March |
| Marks | 8 |
| Topic | Dynamic Programming |
| Type | Zero-sum game stable solution |
| Difficulty | Challenging +1.2 This is a standard game theory question testing dominance, stability, and Nash equilibrium concepts from Further Discrete. While it requires systematic checking of multiple cases and understanding of these definitions, the techniques are routine for this module with no novel insight required. The multi-part structure and case analysis elevate 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 interpretation |
| Ben | |||
| \cline{2-4} \multicolumn{1}{c}{} | Card X | Card Y | Card Z |
| \cline{2-4} \multirow{3}{*}{Alix} | |||
| Card P | (4, 4) | (5, 9) | (1, 7) |
| \cline{2-4} Card Q | (3, 5) | (4, 1) | (8, 2) |
| \cline{2-4} Card R | \((x, y)\) | (2, 2) | (9, 4) |
| \cline{2-4} | |||
| Answer | Marks | Guidance |
|---|---|---|
| For Alix, if Ben plays Y: \(P > Q > R\) but if Ben plays Z: \(R > Q > P\). No row dominance. For Ben, if Alix plays P: \(Y > Z > X\) but if Alix plays Q: \(X > Y > Z\) | B1, B1 | Showing no row dominance; Showing no col dominance, using pay-offs given (not negatives) |
| Answer | Marks | Guidance |
|---|---|---|
| Row minima: P = 1, Q = 3, R = min(x, 2). Maximin for Alix = 3 for all values of \(x\). Col minima: X = min(4, y), Y = 1, Z = 2 | B1 | Calculating maximin for Alix and for Ben. |
| If the value of the game of Alix is 3, then the game is only stable is the value for Ben is 5. The value for Ben is min{4,y} or 2 so the game is unstable | E1 | Showing that the game is not stable |
| Answer | Marks | Guidance |
|---|---|---|
| If Alix plays P: max for Ben is (P, Y). If Alix plays Q: max for Ben is (Q, X). If Alix plays R: max for Ben is (R, X) if \(y \geq 4\) and (R, Z) if \(y \leq 4\). If Ben plays X: max for Alix is (P, X) if \(x \leq 4\) and (R, X) if \(x \geq 4\). If Ben plays Y: max for Alix is (P, Y). If Ben plays Z: max for Alix is (R, Z). Cell (P, Y) is a Nash equilibrium solution for all values of \(x\) and \(y\). If \(y \leq 4\) then (R, Z) is also a Nash equilibrium and if \(x\), \(y\) are both \(\geq 4\) then (R, X) is a Nash equilibrium | M1, A1, B1, E1 | Finding max for Ben in each row and max for Alix in each column; Identifying appropriate cells for Alix and for Ben, in terms of \(x\) and \(y\); (P, Y); Dealing with cases for (R, Z) and (R, X) correctly |
## (i)
| For Alix, if Ben plays Y: $P > Q > R$ but if Ben plays Z: $R > Q > P$. No row dominance. For Ben, if Alix plays P: $Y > Z > X$ but if Alix plays Q: $X > Y > Z$ | B1, B1 | Showing no row dominance; Showing no col dominance, using pay-offs given (not negatives) |
## (ii)
| Row minima: P = 1, Q = 3, R = min(x, 2). Maximin for Alix = 3 for all values of $x$. Col minima: X = min(4, y), Y = 1, Z = 2 | B1 | Calculating maximin for Alix and for Ben. |
| If the value of the game of Alix is 3, then the game is only stable is the value for Ben is 5. The value for Ben is min{4,y} or 2 so the game is unstable | E1 | Showing that the game is not stable |
## (iii)
| If Alix plays P: max for Ben is (P, Y). If Alix plays Q: max for Ben is (Q, X). If Alix plays R: max for Ben is (R, X) if $y \geq 4$ and (R, Z) if $y \leq 4$. If Ben plays X: max for Alix is (P, X) if $x \leq 4$ and (R, X) if $x \geq 4$. If Ben plays Y: max for Alix is (P, Y). If Ben plays Z: max for Alix is (R, Z). Cell (P, Y) is a Nash equilibrium solution for all values of $x$ and $y$. If $y \leq 4$ then (R, Z) is also a Nash equilibrium and if $x$, $y$ are both $\geq 4$ then (R, X) is a Nash equilibrium | M1, A1, B1, E1 | Finding max for Ben in each row and max for Alix in each column; Identifying appropriate cells for Alix and for Ben, in terms of $x$ and $y$; (P, Y); Dealing with cases for (R, Z) and (R, X) correctly |
---Each day Alix and Ben play a game. They each choose a card and use the table below to find the number of points they win. The table shows the cards available to each player. The entries in the cells are of the form $(a, b)$, where $a =$ points won by Alix and $b =$ points won by Ben. Each is trying to maximise the points they win.
\begin{center}
\begin{tabular}{c|c|c|c|}
\multicolumn{1}{c}{} & \multicolumn{3}{c}{Ben} \\
\cline{2-4}
\multicolumn{1}{c}{} & Card X & Card Y & Card Z \\
\cline{2-4}
\multirow{3}{*}{Alix} & \multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} \\
Card P & (4, 4) & (5, 9) & (1, 7) \\
\cline{2-4}
Card Q & (3, 5) & (4, 1) & (8, 2) \\
\cline{2-4}
Card R & $(x, y)$ & (2, 2) & (9, 4) \\
\cline{2-4}
\end{tabular}
\end{center}
\begin{enumerate}[label=(\roman*)]
\item Explain why the table cannot be reduced through dominance no matter what values $x$ and $y$ have. [2]
\item Show that the game is not stable no matter what values $x$ and $y$ have. [2]
\item Find the Nash equilibrium solutions for the various values that $x$ and $y$ can have. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete 2018 Q7 [8]}}