| Exam Board | Edexcel |
|---|---|
| Module | FD2 AS (Further Decision 2 AS) |
| Year | 2020 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Zero-sum game stable solution |
| Difficulty | Standard +0.3 This is a standard textbook zero-sum game question requiring routine application of play-safe strategies (maximin/minimax), stability checking, and basic mixed strategy calculation. While it has multiple parts, each step follows a well-defined algorithm with no novel insight required—slightly easier than average for Further Maths. |
| Spec | 7.08a Pay-off matrix: zero-sum games7.08b Dominance: reduce pay-off matrix7.08c Pure strategies: play-safe strategies and stable solutions7.08d Nash equilibrium: identification and interpretation7.08e Mixed strategies: optimal strategy using equations or graphical method |
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | Team B | |||
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | Paul | Qaasim | Rashid | |
| \multirow{3}{*}{Team A} | Mischa | 4 | - 6 | 2 |
| \cline { 2 - 5 } | Noel | 0 | - 2 | 6 |
| \cline { 2 - 5 } | Olive | - 6 | 2 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| (i) 7 | B1 | cao |
| (ii) 6 | B1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| (i) Row minima: \(-6, -2, -6\), max is \(-2\); Column maxima: \(4, 2, 6\), min is \(2\) | M1 | Finding row minimums and column maximums – condone one error |
| Row minima and column maxima correct | A1 | cao |
| Play-safe for Team A is Noel and play-safe for Team B is Qaasim | A1 | Correct play safes for both teams |
| (ii) Row(maximin) \(\neq\) Col(minimax) therefore game is not stable | B1 | Row maximin \((-2) \neq\) col minimax \((2)\) so not stable |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| e.g. If Team A plays safe then Team B should also play their play-safe option which is Qaasim; by playing Qaasim they will gain 2 compared to gaining zero (if playing Paul) or losing 6 (if playing Rashid) | B1 | cao (or equivalent); explanation must involve consideration of values and not just a general statement that Qaasim will gain the most |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Let B play Paul with probability \(q\) and Qaasim with probability \(1-q\) | B1 | Defining variable \(q\) |
| If \(A\) plays Mischa, \(B\)'s gains are \(-(4q+(-6)(1-q))=6-10q\) | M1 | Setting up three expressions in terms of \(q\) |
| If \(A\) plays Noel, \(B\)'s gains are \(-(-2(1-q))=2-2q\) | A1 | All three expressions correct – allow correct unsimplified expressions |
| If \(A\) plays Olive, \(B\)'s gains are \(-(-6q+2(1-q))=-2+8q\) | ||
| Correct graph with axes correct, at least one line correctly drawn | M1 | Axes correct, at least one line correctly drawn for their expressions |
| Correct graph | A1 | cao |
| \(2-2q=-2+8q \Rightarrow q=2/5\) | A1 | Using graph to obtain correct probability expressions leading to correct value of \(q\) |
| Team B should play Paul with probability 0.4 and play Qaasim with probability 0.6 | A1ft | Interpret their value of \(q\) in context – must refer to play/choose and the two players |
## Question 3:
### Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| (i) 7 | B1 | cao |
| (ii) 6 | B1 | cao |
### Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| (i) Row minima: $-6, -2, -6$, max is $-2$; Column maxima: $4, 2, 6$, min is $2$ | M1 | Finding row minimums and column maximums – condone one error |
| Row minima and column maxima correct | A1 | cao |
| Play-safe for Team A is Noel and play-safe for Team B is Qaasim | A1 | Correct play safes for both teams |
| (ii) Row(maximin) $\neq$ Col(minimax) therefore game is not stable | B1 | Row maximin $(-2) \neq$ col minimax $(2)$ so not stable |
### Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| e.g. If Team A plays safe then Team B should also play their play-safe option which is Qaasim; by playing Qaasim they will gain 2 compared to gaining zero (if playing Paul) or losing 6 (if playing Rashid) | B1 | cao (or equivalent); explanation must involve consideration of values and not just a general statement that Qaasim will gain the most |
### Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| Let B play Paul with probability $q$ and Qaasim with probability $1-q$ | B1 | Defining variable $q$ |
| If $A$ plays Mischa, $B$'s gains are $-(4q+(-6)(1-q))=6-10q$ | M1 | Setting up three expressions in terms of $q$ |
| If $A$ plays Noel, $B$'s gains are $-(-2(1-q))=2-2q$ | A1 | All three expressions correct – allow correct unsimplified expressions |
| If $A$ plays Olive, $B$'s gains are $-(-6q+2(1-q))=-2+8q$ | | |
| Correct graph with axes correct, at least one line correctly drawn | M1 | Axes correct, at least one line correctly drawn for their expressions |
| Correct graph | A1 | cao |
| $2-2q=-2+8q \Rightarrow q=2/5$ | A1 | Using graph to obtain correct probability expressions leading to correct value of $q$ |
| Team B should play Paul with probability 0.4 and play Qaasim with probability 0.6 | A1ft | Interpret their value of $q$ in context – must refer to play/choose and the two players |
---3. Two teams, A and B , each have three team members. One member of Team A will compete against one member of Team B for 10 rounds of a competition. None of the rounds can end in a draw.
Table 1 shows, for each pairing, the expected number of rounds that the member of Team A will win minus the expected number of rounds that the member of Team B will win. These numbers are the scores awarded to Team A. This competition between Teams A and B is a zero-sum game. Each team must choose one member to play. Each team wants to choose the member who will maximise its score.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & \multicolumn{3}{c|}{Team B} \\
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & Paul & Qaasim & Rashid \\
\hline
\multirow{3}{*}{Team A} & Mischa & 4 & - 6 & 2 \\
\cline { 2 - 5 }
& Noel & 0 & - 2 & 6 \\
\cline { 2 - 5 }
& Olive & - 6 & 2 & 0 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the number of rounds that Team A expects to win if Team A chooses Mischa and Team B chooses Paul.
\item Find the number of rounds that Team B expects to win if Team A chooses Noel and Team B chooses Qaasim.
Table 1 models this zero-sum game.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Find the play-safe strategies for the game.
\item Explain how you know that the game is not stable.
\end{enumerate}\item Determine which team member Team B should choose if Team B thinks that Team A will play safe. Give a reason for your answer.
At the last minute, Rashid is ill and is therefore unavailable for selection by Team B.
\item Find the best strategy for Team B, defining any variables you use.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FD2 AS 2020 Q3 [14]}}