| Exam Board | OCR |
|---|---|
| Module | Further Discrete (Further Discrete) |
| Year | 2019 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Difficulty | Standard +0.8 This is a game theory question from Further Maths requiring understanding of Nash equilibrium, maximin strategies, and zero-sum games. While the concepts are A-level appropriate, game theory is a specialized topic that requires careful logical reasoning about strategic choices. Part (a) requires inequality manipulation, parts (b-c) test conceptual understanding, and part (d) involves converting to zero-sum and finding optimal mixed strategies. The multi-step nature and specialized vocabulary place this moderately above average difficulty. |
| Spec | 7.08c Pure strategies: play-safe strategies and stable solutions7.08d Nash equilibrium: identification and interpretation7.08e Mixed strategies: optimal strategy using equations or graphical method |
| Vlad | |||
| \cline { 2 - 4 } \multicolumn{1}{c}{} | \(X\) | \(Y\) | \(Z\) |
| A | \(( x , 1 )\) | \(( 4 , - 2 )\) | \(( 2,0 )\) |
| B | \(( 3 , - 1 )\) | \(( 6 , - 4 )\) | \(( - 1,3 )\) |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (a) | x > 3 |
| (b) | If Vlad plays X, Sumi’s highest score is by |
| Answer | Marks |
|---|---|
| playing X | E1 |
| E1 | 2.4 |
| 2.4 | Max{x, 3} = x |
| Max {1, -2, 0} = 1 | Two separate statements, not |
| Answer | Marks |
|---|---|
| (iii) | X Y Z Min pay- |
| Answer | Marks |
|---|---|
| for Vlad. | M1 |
| Answer | Marks |
|---|---|
| A1 | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | Finding play-safe or maximin Sumi |
| Answer | Marks |
|---|---|
| (A, Z) = (2, 0) | A or 2, -1 (allow ‘2 or x’, 1) |
| Answer | Marks | Guidance |
|---|---|---|
| X | Y | Z |
| A | (x, 1) | (4, -2) |
| B | (3, -1) | (6, -4) |
| (d) | x = 1 |
| Answer | Marks |
|---|---|
| and B with probability 0.2 | B1 |
| Answer | Marks |
|---|---|
| A1 | 3.1a |
| Answer | Marks |
|---|---|
| 3.2a | Seen or implied from zero-sum pay- |
| Answer | Marks |
|---|---|
| following correct working | Each cell must have the |
| Answer | Marks | Guidance |
|---|---|---|
| X | Y | Z |
| A | 0 | 3 |
| B | 2 | 5 |
| col max | 2 | 5 |
Question 6:
6 | (a) | x > 3 | B1 | 2.1 | Allow x > 3
(b) | If Vlad plays X, Sumi’s highest score is by
playing A
If Sumi plays A, Vlad’s highest score is by
playing X | E1
E1 | 2.4
2.4 | Max{x, 3} = x
Max {1, -2, 0} = 1 | Two separate statements, not
merged into one
(iii) | X Y Z Min pay-
off Sumi
A (x, 1) (4, -2) (2, 0) 2 *
B (3, -1) (6, -4) (-1, 3) -1
Min pay-off -1 -4 0
Vlad *
Play-safe for Sumi is A, maximin pay-off = 2
Play-safe for Vlad is Z, maximin pay-off = 0.
Maximin pay-off for Sumi is 2 and maximin
pay-off for Vlad is 0.
Cell (A, Z) has pay-off 2 for Sumi and pay-off 0
for Vlad. | M1
M1
A1 | 1.1
1.1
1.1 | Finding play-safe or maximin Sumi
Finding play-safe or maximin Vlad
(A, Z) = (2, 0) | A or 2, -1 (allow ‘2 or x’, 1)
Z or -1, -4, 0 or 1, 4, 0
Cell (A, Z) as well as 2 and
0 from correct working
X | Y | Z
A | (x, 1) | (4, -2) | (2, 0)
B | (3, -1) | (6, -4) | (-1, 3)
(d) | x = 1
X Y Z row min
A 0 3 1 0
B 2 5 -2 -2
col max 2 5 1
Game is unstable (0 ≠ 1)
Sumi chooses randomly, P(A) = p
Vlad plays X: 0(p) + 2(1 – p) or 2 – 2p
(Vlad plays Y: 3(p) + 5(1 – p) or 5 – 2p)
Vlad plays X: 1(p) – 2(1 – p) or 3p – 2
2 – 2p = 3p – 2
p = 0.8
Choose randomly between rows,
so that A is played with probability 0.8
and B with probability 0.2 | B1
M1
A1
B1ft
M1ft
depM1
A1 | 3.1a
3.1a
2.4
2.4
1.1
1.1
3.2a | Seen or implied from zero-sum pay-
off = 0
2÷2 = 1, subtract 1 from each score
to get pay-off for Sumi (and
negative of pay-off for Vladimir)
Pay-off’s for Sumi correct, or a
positive multiple
Verifying that game is unstable
Expected winnings for Sumi when
Vlad plays X and Z in terms of one
parameter (may still have constant x)
Solving their X = their Z,
algebraically, or using a graph
p = 0.8, interpreted in context,
following correct working | Each cell must have the
same sum
These entries for cells apart
from (A, X), or a non-zero
multiple
With 0 for (A, X)
(Y is dominated by X so
may be excluded, or not)
Allow their Y = their X or Z
or max of their lower
boundary (shown)
Or interpret for B with 0.2
[13]
X | Y | Z | row min
A | 0 | 3 | 1 | 0
B | 2 | 5 | -2 | -2
col max | 2 | 5 | 16 The pay-off matrix for a game between two players, Sumi and Vlad, is shown below. If Sumi plays A and Vlad plays X then Sumi gets X points and Vlad gets 1 point.
Sumi
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\multicolumn{1}{c}{} & Vlad & & \\
\cline { 2 - 4 }
\multicolumn{1}{c}{} & $X$ & $Y$ & $Z$ \\
\hline
A & $( x , 1 )$ & $( 4 , - 2 )$ & $( 2,0 )$ \\
\hline
B & $( 3 , - 1 )$ & $( 6 , - 4 )$ & $( - 1,3 )$ \\
\hline
\end{tabular}
\end{center}
You are given that cell ( $\mathrm { A } , \mathrm { X }$ ) is a Nash Equilibrium solution.
\begin{enumerate}[label=(\alph*)]
\item Find the range of possible values of X .
\item Explain what the statement 'cell (A, X) is a Nash Equilibrium solution' means for each player.
\item Find a cell where each player gets their maximin pay-off.
Suppose, instead, that the game can be converted to a zero-sum game.
\item Determine the optimal strategy for Sumi for the zero-sum game.
\begin{itemize}
\item Record the pay-offs for Sumi when the game is converted to a zero-sum game.
\item Describe how Sumi should play using this strategy.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete 2019 Q6 [13]}}