| Exam Board | OCR |
|---|---|
| Module | Further Discrete AS (Further Discrete AS) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Game theory LP formulation |
| Difficulty | Standard +0.3 This is a straightforward game theory question requiring construction of a 3×3 payoff matrix using given rules, then applying standard decision mathematics concepts (play-safe strategy, dominance). The calculations are arithmetic, and the concepts (maximin, dominated strategies, expected value) are core D1/Further Discrete content with no novel problem-solving required beyond following the rubric. |
| Spec | 7.08a Pay-off matrix: zero-sum games7.08c Pure strategies: play-safe strategies and stable solutions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct entries in cells where total is even: \((1,3)=2\), \((2,4)=4\), \((2,6)=4\), \((5,3)=10\) | B1 | May also see working for later parts |
| \((1,4)=(5,4)\) and \((1,6)=(5,6)\) | M1 | |
| Correct entries in cells where total is odd | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| If S chooses card 3, K should play card 5 | M1 FT | Any one correct, follow through their pay-off matrix |
| If S chooses card 4 or 6, K should play card 2 | A1 FT | All correct, follow through their pay-off matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Row maximin \(= \max\{-2, 1, -2\} = 1\) | M1 FT | Row minima seen, for their pay-offs (may be seen in part (a)) |
| Kareem's play-safe is card numbered 2 | A1 FT | Card 2, from correct working seen, for their pay-offs. Need 'card' or equivalent (e.g. strategy) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| (Card 1 is) (weakly) dominated by (card) 5 | B1 FT | Or explained in words or showing appropriate comparisons (all). i.e. Card 5 because [valid comparisons or explanation]. Allow "(Card 1 is) never chosen in part (b)" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| K plays 2: \(\frac{1}{3}(1+4+4)=3\) | M1 FT | Either expected score (3 or \(2\frac{2}{3}\)) for their pay-offs |
| K plays 5: \(\frac{1}{3}(10+0-2)=\frac{8}{3}\) or \(2\frac{2}{3}\) | A1 FT | Both expected scores, for their pay-offs |
# Question 4:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct entries in cells where total is even: $(1,3)=2$, $(2,4)=4$, $(2,6)=4$, $(5,3)=10$ | B1 | May also see working for later parts |
| $(1,4)=(5,4)$ and $(1,6)=(5,6)$ | M1 | |
| Correct entries in cells where total is odd | A1 | |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| If S chooses card 3, K should play card 5 | M1 FT | Any one correct, follow through their pay-off matrix |
| If S chooses card 4 or 6, K should play card 2 | A1 FT | All correct, follow through their pay-off matrix |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Row maximin $= \max\{-2, 1, -2\} = 1$ | M1 FT | Row minima seen, for their pay-offs (may be seen in part (a)) |
| Kareem's play-safe is card numbered 2 | A1 FT | Card 2, from correct working seen, for their pay-offs. Need 'card' or equivalent (e.g. strategy) |
## Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| (Card 1 is) (weakly) dominated by (card) 5 | B1 FT | Or explained in words or showing appropriate comparisons (all). i.e. Card 5 because [valid comparisons or explanation]. Allow "(Card 1 is) never chosen in part (b)" |
## Part (e)
| Answer | Marks | Guidance |
|--------|-------|----------|
| K plays 2: $\frac{1}{3}(1+4+4)=3$ | M1 FT | Either expected score (3 or $2\frac{2}{3}$) for their pay-offs |
| K plays 5: $\frac{1}{3}(10+0-2)=\frac{8}{3}$ or $2\frac{2}{3}$ | A1 FT | Both expected scores, for their pay-offs |
---4 Kareem and Sam play a game in which each holds a hand of three cards.
\begin{itemize}
\item Kareem's cards are numbered 1, 2 and 5.
\item Sam's cards are numbered 3, 4 and 6 .
\end{itemize}
In each round Kareem and Sam simultaneously choose a card from their hand, they show their chosen card to the other player and then return the card to their own hand.
\begin{itemize}
\item If the sum of the numbers on the cards shown is even then the number of points that Kareem scores is $2 k$, where $k$ is the number on Kareem's card.
\item If the sum of the numbers on the cards shown is odd then the number of points that Kareem scores is $4 - s$, where $s$ is the number on Sam's card.
\begin{enumerate}[label=(\alph*)]
\item Complete the pay-off matrix for this game, to show the points scored by Kareem.
\item Write down which card Kareem should play to maximise the number of points that he scores for each of Sam's choices.
\item Determine the play-safe strategy for Kareem.
\item Explain why Kareem should never play the card numbered 1.
\end{itemize}
Sam chooses a card at random, so each of Sam's three cards is equally likely.
\item Calculate Kareem's expected score for each of his remaining choices.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete AS 2022 Q4 [10]}}