| Exam Board | Edexcel |
|---|---|
| Module | FD2 (Further Decision 2) |
| Year | 2021 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Zero-sum game LP formulation |
| Difficulty | Challenging +1.2 This is a standard Further Maths D2 game theory question requiring LP formulation interpretation, graphical solution, and dual problem solving. While it involves multiple parts and techniques (reconstructing payoff matrix, graphical LP, finding dual strategy), these are all routine procedures taught explicitly in the syllabus with no novel insight required. The 'complete the payoff matrix' scaffolding makes it more accessible than if students had to formulate from scratch. |
| Spec | 7.08a Pay-off matrix: zero-sum games7.08e Mixed strategies: optimal strategy using equations or graphical method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Either one correct row or column of pay-off matrix | M1 | SC M1 A0 for \(\begin{pmatrix}6&0&4\\1&8&2\end{pmatrix}\) |
| Full correct matrix: Q row: 3, \(-3\), 1; R row: \(-2\), 5, \(-1\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| If B plays X: \(6p_1+p_2=6p_1+(1-p_1)=5p_1+1\); If B plays Y: \(8p_2=8(1-p_1)=-8p_1+8\); If B plays Z: \(4p_1+2p_2=4p_1+2(1-p_1)=2p_1+2\) | M1 | Setting up three expressions in terms of \(p_1\) or \(p_2\) |
| All three expressions correct (equiv: \(5p_1-2\), \(-8p_1+5\), \(2p_1-1\)) | A1 | |
| Axes correct, at least one line correctly drawn | M1 | |
| Correct graph | A1 | |
| \(2+2p_1=8-8p_1 \Rightarrow p_1=0.6\) | A1 | Using correct graph to obtain correct value of \(p_1\) or \(p_2\) |
| Alexis should play option Q with probability 0.6 and option R with probability 0.4 | A1ft | Interpret values in context — must refer to play and associated probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Value of game \(= 2+2\!\left(\dfrac{3}{5}\right)-3=\dfrac{1}{5}\) | B1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(4q_3=\dfrac{16}{5}\), \(8q_2+2q_3=\dfrac{16}{5}\) or \(-3q_2+q_3=\dfrac{1}{5}\), \(5q_2-q_3=\dfrac{1}{5}\) | M1 | Setting up two equations in \(q_2\) and \(q_3\) with value of game |
| Correct two equations | A1 | |
| \(q_2=\dfrac{1}{5}\), \(q_3=\dfrac{4}{5}\) \(\Rightarrow\) Becky should play option X never, option Y with probability 0.2 and option Z with probability 0.8 | A1 | Interpret values in context — must refer to play and associated probabilities |
# Question 7:
## Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Either one correct row or column of pay-off matrix | M1 | SC M1 A0 for $\begin{pmatrix}6&0&4\\1&8&2\end{pmatrix}$ |
| Full correct matrix: Q row: 3, $-3$, 1; R row: $-2$, 5, $-1$ | A1 | cao |
## Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| If B plays X: $6p_1+p_2=6p_1+(1-p_1)=5p_1+1$; If B plays Y: $8p_2=8(1-p_1)=-8p_1+8$; If B plays Z: $4p_1+2p_2=4p_1+2(1-p_1)=2p_1+2$ | M1 | Setting up three expressions in terms of $p_1$ or $p_2$ |
| All three expressions correct (equiv: $5p_1-2$, $-8p_1+5$, $2p_1-1$) | A1 | |
| Axes correct, at least one line correctly drawn | M1 | |
| Correct graph | A1 | |
| $2+2p_1=8-8p_1 \Rightarrow p_1=0.6$ | A1 | Using correct graph to obtain correct value of $p_1$ or $p_2$ |
| Alexis should play option Q with probability 0.6 and option R with probability 0.4 | A1ft | Interpret values in context — must refer to play and associated probabilities |
## Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Value of game $= 2+2\!\left(\dfrac{3}{5}\right)-3=\dfrac{1}{5}$ | B1 | cao |
## Part (d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $4q_3=\dfrac{16}{5}$, $8q_2+2q_3=\dfrac{16}{5}$ or $-3q_2+q_3=\dfrac{1}{5}$, $5q_2-q_3=\dfrac{1}{5}$ | M1 | Setting up two equations in $q_2$ and $q_3$ with value of game |
| Correct two equations | A1 | |
| $q_2=\dfrac{1}{5}$, $q_3=\dfrac{4}{5}$ $\Rightarrow$ Becky should play option X never, option Y with probability 0.2 and option Z with probability 0.8 | A1 | Interpret values in context — must refer to play and associated probabilities |7. Alexis and Becky are playing a zero-sum game.
Alexis has two options, Q and R . Becky has three options, $\mathrm { X } , \mathrm { Y }$ and Z .\\
Alexis intends to make a random choice between options Q and R , choosing option Q with probability $p _ { 1 }$ and option R with probability $p _ { 2 }$
Alexis wants to find the optimal values of $p _ { 1 }$ and $p _ { 2 }$ and formulates the following linear programme, writing the constraints as inequalities.
$$\begin{aligned}
& \text { Maximise } P = V \\
& \text { where } V = 3 + \text { the value of the gan } \\
& \text { subject to } V \leqslant 6 p _ { 1 } + p _ { 2 } \\
& \qquad \begin{aligned}
& V \leqslant 8 p _ { 2 } \\
& V \leqslant 4 p _ { 1 } + 2 p _ { 2 } \\
& p _ { 1 } + p _ { 2 } \leqslant 1 \\
& p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , V \geqslant 0
\end{aligned}
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Complete the pay-off matrix for Alexis in the answer book.
\item Use a graphical method to find the best strategy for Alexis.
\item Calculate the value of the game to Alexis.
Becky intends to make a random choice between options $\mathrm { X } , \mathrm { Y }$ and Z , choosing option X with probability $q _ { 1 }$, option Y with probability $q _ { 2 }$ and option Z with probability $q _ { 3 }$
\item Determine the best strategy for Becky, making your method and working clear.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FD2 2021 Q7 [12]}}