| Exam Board | OCR |
|---|---|
| Module | Further Discrete AS (Further Discrete AS) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Game theory LP formulation |
| Difficulty | Moderate -0.5 This is a straightforward game theory question requiring basic concepts: finding row minima for play-safe strategy (part a), applying dominance arguments to eliminate columns (part b), which are standard textbook procedures in Decision Mathematics. While it involves multiple steps and careful comparison of matrix entries, these are routine algorithmic tasks rather than requiring problem-solving insight or novel reasoning. |
| Spec | 7.08a Pay-off matrix: zero-sum games7.08c Pure strategies: play-safe strategies and stable solutions7.08d Nash equilibrium: identification and interpretation |
| C | ||||||
| \cline { 3 - 7 } \multirow{4}{*}{Alayer 1} | D | E | H | I | ||
| \cline { 3 - 7 } | A | 4 | 1 | 3 | 2 | 2 |
| \cline { 3 - 7 } | B | 0 | 2 | 1 | 2 | 1 |
| \cline { 3 - 7 } | F | 0 | 1 | 1 | 2 | 3 |
| \cline { 2 - 7 } | G | 2 | 0 | 3 | 3 | 3 |
| \cline { 3 - 7 } | J | 1 | 2 | 3 | 0 | 2 |
| \cline { 3 - 7 } | ||||||
| \cline { 3 - 7 } | ||||||
| C | D | E | H | ||
| \cline { 2 - 6 } A | 2 | 2 | 0 | 1 | I |
| \cline { 2 - 6 } B | 3 | 1 | 2 | 1 | 2 |
| \cline { 2 - 6 } F | 3 | 2 | 2 | 1 | 0 |
| \cline { 2 - 6 } G | 1 | 3 | 0 | 0 | 0 |
| \cline { 2 - 6 } J | 2 | 1 | 0 | 3 | 1 |
| \cline { 2 - 6 } | |||||
| \cline { 2 - 6 } |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (a) | C D E H I row min |
| Answer | Marks |
|---|---|
| Play-safe strategy for player 1 is A | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | Finding row minima for table of pay-offs for player 1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (b) | 2 > 0 and 2 > 1 |
| Answer | Marks |
|---|---|
| J 2 1 0 3 1 | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | These comparisons seen |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (c) | When they play A and C the total pay-off is 6 |
| but for (all the) other cells total pay-off is 3 | B1 | |
| [1] | 2.2a | Identifying cell (A, C) in any form and an appropriate comparison |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | 0 | 3 |
| 2 | 2 | 0 |
| 2 | 1 | 0 |
| 2 | 0 | 0 |
Question 2:
2 | (a) | C D E H I row min
A 4 1 3 2 2 1
B 0 2 1 2 1 0
F 0 1 1 2 3 0
G 2 0 3 3 3 0
J 1 2 3 0 2 0
Play-safe strategy for player 1 is A | M1
A1
[2] | 1.1
1.1 | Finding row minima for table of pay-offs for player 1
Accept 1, 0, 0, 0, 0 or an equivalent written description
A (from correct row minima)
2 | (b) | 2 > 0 and 2 > 1
3 > 2 3 > 2
3 > 2 3 > 0
1 > 0 1 > 0
2 > 0 2 > 1
C D E H I
A 2 2 0 1 1
B 3 1 2 1 2
F 3 2 2 1 0
G 1 3 0 0 0
J 2 1 0 3 1 | B1
B1
[2] | 1.1
1.1 | These comparisons seen
(may omit repeated 3 > 2 and 2 > 0 for E and repeated 2 > 1 for I,
i.e. 2 > 0, 3 > 2, 1 > 0 and 2 > 1, 3 > 2, 3 > 0, 1 > 0 or similarly)
Ignore extras
Deleting columns E and I (both)
Or saying that both E and I are dominated
Or indicating both E and I
2 | (c) | When they play A and C the total pay-off is 6
but for (all the) other cells total pay-off is 3 | B1
[1] | 2.2a | Identifying cell (A, C) in any form and an appropriate comparison
(of total or mean) with at least one other cell
2 | 0 | 3 | 3 | 3
2 | 2 | 0 | 1 | 1
2 | 1 | 0 | 3 | 1
2 | 0 | 0 | 1 | 0 | 12 In a game two players are each dealt five cards from a set of ten different cards.\\
Player 1 is dealt cards A, B, F, G and J.\\
Player 2 is dealt cards C, D, E, H and I.
Each player chooses a card to play.\\
The players reveal their choices simultaneously.
The pay-off matrix below shows the points scored by player 1 for each combination of cards.
Pay-off for player 1
\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Player 2}
\begin{tabular}{ l l | l | l | l | l | l | }
& & \multicolumn{5}{c}{C} \\
\cline { 3 - 7 }
\multirow{4}{*}{Alayer 1} & \multicolumn{1}{c}{D} & \multicolumn{1}{c}{E} & \multicolumn{1}{c}{H} & \multicolumn{1}{c}{I} & & \\
\cline { 3 - 7 }
& A & 4 & 1 & 3 & 2 & 2 \\
\cline { 3 - 7 }
& B & 0 & 2 & 1 & 2 & 1 \\
\cline { 3 - 7 }
& F & 0 & 1 & 1 & 2 & 3 \\
\cline { 2 - 7 }
& G & 2 & 0 & 3 & 3 & 3 \\
\cline { 3 - 7 }
& J & 1 & 2 & 3 & 0 & 2 \\
\cline { 3 - 7 }
& & & & & & \\
\cline { 3 - 7 }
\end{tabular}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Determine the play-safe strategy for player 1, ignoring any effect on player 2.
The pay-off matrix below shows the points scored by player 2 for each combination of cards.\\
Pay-off for player 2
Player 1
\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Player 2}
\begin{tabular}{ l | l | l | l | l | l | }
& \multicolumn{1}{l}{} & \multicolumn{1}{l}{C} & \multicolumn{1}{l}{D} & \multicolumn{1}{l}{E} & \multicolumn{1}{l}{H} \\
\cline { 2 - 6 }
A & 2 & 2 & 0 & 1 & \multicolumn{1}{l}{I} \\
\cline { 2 - 6 }
B & 3 & 1 & 2 & 1 & 2 \\
\cline { 2 - 6 }
F & 3 & 2 & 2 & 1 & 0 \\
\cline { 2 - 6 }
G & 1 & 3 & 0 & 0 & 0 \\
\cline { 2 - 6 }
J & 2 & 1 & 0 & 3 & 1 \\
\cline { 2 - 6 }
& & & & & \\
\cline { 2 - 6 }
\end{tabular}
\end{center}
\end{table}
\item Use a dominance argument to delete two columns from the pay-off matrix. You must show all relevant comparisons.
\item Explain, with reference to specific combinations of cards, why the game cannot be converted to a zero-sum game.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete AS 2024 Q2 [5]}}