| Exam Board | OCR |
|---|---|
| Module | Further Discrete (Further Discrete) |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Non-group structures |
| Difficulty | Standard +0.8 This is a multi-part game theory question requiring understanding of play-safe strategies, stability conditions, Nash equilibrium, and strategic reasoning. While the individual concepts are A-level accessible, the question demands careful analysis across five parts including a proof element (part iii) and strategic modification (part iv), making it moderately challenging but within reach for well-prepared Further Maths students. |
| Spec | 7.08a Pay-off matrix: zero-sum games7.08c Pure strategies: play-safe strategies and stable solutions7.08d Nash equilibrium: identification and interpretation |
| Strategy \(X\) | Strategy \(Y\) | Strategy \(Z\) | |
| Strategy \(P\) | 4 | 5 | - 4 |
| Strategy \(Q\) | 3 | - 1 | 2 |
| Strategy \(R\) | 4 | 0 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (i) | row maxi |
| Answer | Marks |
|---|---|
| Unstable, since 0(cid:122)2 | M1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 2.2a | n |
| Answer | Marks | Guidance |
|---|---|---|
| oe | Play-safe values not equal | |
| 4 | (ii) | If player A uses strategy R, player B would get |
| the best pay-off by changing from Z to Y | B1 | |
| [1] | i2.2a | m |
| 4 | (iii) | Increasing entry in (R, Z ) has no effect on row |
| Answer | Marks |
|---|---|
| equal and the game is still unstable. | e |
| Answer | Marks |
|---|---|
| [2] | c |
| Answer | Marks |
|---|---|
| 2.4 | Describing what happens if entry is |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (iv) | e.g. row R column Y is increased to 2 (or more) |
| Answer | Marks |
|---|---|
| [2] | 2.1 |
| 2.3 | Identifying a suitable cell |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | 5 | –4 |
| 4 | 0 | 2 |
| 4 | (v) | Player A: (P, Y ), (Q, X ), (R, X ) |
| Answer | Marks |
|---|---|
| the other changed as well | M1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 2.4 | Identifying cells where row maxima |
Question 4:
4 | (i) | row maxi
X Y Z
min min
P 4 5 –4 –4
Q 3 –1 2 –1
R 4 0 2 0 *
col max 4 5 2
minimax *
Play-safe for player A is strategy R
Play-safe for player B is strategy Z
Unstable, since 0(cid:122)2 | M1
M1
A1
[3] | 1.1
1.1
2.2a | n
e
oe | Play-safe values not equal
4 | (ii) | If player A uses strategy R, player B would get
the best pay-off by changing from Z to Y | B1
[1] | i2.2a | m
4 | (iii) | Increasing entry in (R, Z ) has no effect on row
min and increases col max. Row maximin is
still 0 and col minimax is greater thapn 2, they
are not equal and the game is still unstable.
Decreasing entry has no effeSct on col max, as
(cid:11)Q, Z(cid:12)(cid:32)2, and may or may not decrease row
min. Row maximin is either still 0 or is less
than 0 and col minimax is still 2, they are not
equal and the game is still unstable. | e
E1
E1
[2] | c
2.4
2.4 | Describing what happens if entry is
increased
Describing what happens if entry is
decreased
4 | (iv) | e.g. row R column Y is increased to 2 (or more) | M1
A1
[2] | 2.1
2.3 | Identifying a suitable cell
Giving a valid new value
4 | 5 | –4
4 | 0 | 2
4 | (v) | Player A: (P, Y ), (Q, X ), (R, X )
Player B: (Q, X ), (R, Y ), (P, Z )
When player A chooses row Q and player B
choses column X it is a Nash equilibrium
Neither player would want to change, unless
the other changed as well | M1
A1
E1
[3] | 1.1
2.5
2.4 | Identifying cells where row maxima
occur
(Q, X ) is a Nash equilibrium
Explaining Nash equilibrium in context4 The table shows the pay-off matrix for player $A$ in a two-person zero-sum game between $A$ and $B$.
Player $A$\\
Player $B$
\begin{center}
\begin{tabular}{ l | c | c | c }
& Strategy $X$ & Strategy $Y$ & Strategy $Z$ \\
\hline
Strategy $P$ & 4 & 5 & - 4 \\
\hline
Strategy $Q$ & 3 & - 1 & 2 \\
\hline
Strategy $R$ & 4 & 0 & 2 \\
\hline
\end{tabular}
\end{center}
(i) 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.\\
(ii) If player $B$ knows that player $A$ will use their play-safe strategy, which strategy should player $B$ use?\\
(iii) 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.\\
(iv) 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.\\
(v) Show that the zero-sum game in the table above has a Nash equilibrium and explain what this means for the players.
\hfill \mbox{\textit{OCR Further Discrete Q4 [11]}}