AQA Further Paper 3 Discrete 2021 June — Question 7 14 marks

Exam BoardAQA
ModuleFurther Paper 3 Discrete (Further Paper 3 Discrete)
Year2021
SessionJune
Marks14
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicDynamic Programming
TypeZero-sum game optimal mixed strategy
DifficultyStandard +0.3 This is a standard Further Maths game theory question requiring routine application of the mixed strategy algorithm (equalizing expected payoffs) and straightforward optimization when opponent's strategy is known. The calculations are mechanical with no novel insight required, making it easier than average even for Further Maths.
Spec7.08c Pure strategies: play-safe strategies and stable solutions7.08d Nash equilibrium: identification and interpretation7.08e Mixed strategies: optimal strategy using equations or graphical method
7 Avon and Roj play a zero-sum game. The game is represented by the following pay-off matrix for Avon. 7 (c) (i) Find the optimal mixed strategy for Avon.
7 (c) (ii) Find the value of the game for Avon.
7 (d) Roj thinks that his best outcome from the game is to play strategy \(\mathbf { R } _ { \mathbf { 2 } }\) each time. Avon notices that Roj always plays strategy \(\mathbf { R } _ { \mathbf { 2 } }\) and Avon wants to use this knowledge to maximise his expected pay-off from the game. Explain how your answer to part (c)(i) should change and find Avon's maximum expected pay-off from the game. \includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-16_2490_1735_219_139}
Question 7(a):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(3p_1 + 2p_2 + 6p_3 \geq v\)M1 (AO3.1a) Translates pay-off matrix into at least one correct expression
\(p_1 + 3p_2 + 2p_3 \geq v\), \(4p_1 + 2p_2 + p_3 \geq v\), \(p_1 + p_2 + p_3 \leq 1\)A1 (AO1.1b) All three correct inequalities
Maximise \(P = v\)B1 (AO1.1b) Correct condition for sum of probability variables
Subject to: \(3p_1 + 2p_2 + 6p_3 \geq v\), \(p_1 + 3p_2 + 2p_3 \geq v\), \(4p_1 + 2p_2 + p_3 \geq v\), \(p_1 + p_2 + p_3 \leq 1\), \(p_1, p_2, p_3 \geq 0\)A1 (AO1.1b) Full LP formulation including 'maximise', 'subject to', non-negativity conditions
Total: 4 marks
Question 7(b)(i):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Introduces four slack variables \(s_1, s_2, s_3, s_4\) in column headingsM1 (AO3.1a) Translates LP by introducing four slack variables
Initial simplex tableau (two correct rows):A1 (AO1.1b)
\(\begin{array}{c\ccccccccc\ c} P & v & p_1 & p_2 & p_3 & s_1 & s_2 & s_3 & s_4 & \text{val} \\ \hline 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & -3 & -2 & -6 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & -1 & -3 & -2 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & -4 & -2 & -1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 \end{array}\)
Total: 3 marks
Question 7(b)(ii):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Applies simplex algorithm to modify at least one non-pivot row correctlyM1 (AO3.1a)
At least three rows correct after iterationA1 (AO1.1b)
Final tableau: \(\begin{array}{c\ccccccccc\ c} P & v & p_1 & p_2 & p_3 & s_1 & s_2 & s_3 & s_4 & \text{val} \\ \hline 1 & 0 & -3 & -2 & -6 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & -3 & -2 & -6 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & -1 & 4 & -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 5 & -1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 \end{array}\)
Total: 3 marks
Question 7(c)(i):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
The optimal mixed strategy for Avon is: play \(A_1\) with probability \(0.25\), play \(A_2\) with probability \(0.70\), play \(A_3\) with probability \(0.05\)B1 (AO3.2a) Interprets final simplex tableau to find optimal mixed strategy
Total: 1 mark
Question 7(c)(ii):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(3 \times 0.25 + 2 \times 0.70 + 6 \times 0.05 = 2.45\)B1 (AO2.2a) Deduces value of the game for Avon
Total: 1 mark
Question 7(d):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
To improve their outcome, Avon should play \(A_2\) each timeE1 (AO2.4) Explains correctly that Avon should play \(A_2\) each time
Under these conditions, Avon's maximum expected pay-off will be \(3\)E1 (AO3.2a) Determines correctly that Avon's maximum expected pay-off will be \(3\)
Total: 2 marks
AnswerMarks
Question total: 14 marksPaper total: 50 marks 
## Question 7(a):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $3p_1 + 2p_2 + 6p_3 \geq v$ | M1 (AO3.1a) | Translates pay-off matrix into at least one correct expression |
| $p_1 + 3p_2 + 2p_3 \geq v$, $4p_1 + 2p_2 + p_3 \geq v$, $p_1 + p_2 + p_3 \leq 1$ | A1 (AO1.1b) | All three correct inequalities |
| Maximise $P = v$ | B1 (AO1.1b) | Correct condition for sum of probability variables |
| Subject to: $3p_1 + 2p_2 + 6p_3 \geq v$, $p_1 + 3p_2 + 2p_3 \geq v$, $4p_1 + 2p_2 + p_3 \geq v$, $p_1 + p_2 + p_3 \leq 1$, $p_1, p_2, p_3 \geq 0$ | A1 (AO1.1b) | Full LP formulation including 'maximise', 'subject to', non-negativity conditions |

**Total: 4 marks**

---

## Question 7(b)(i):

| Answer/Working | Marks | Guidance |
|---|---|---|
| Introduces four slack variables $s_1, s_2, s_3, s_4$ in column headings | M1 (AO3.1a) | Translates LP by introducing four slack variables |
| Initial simplex tableau (two correct rows): | A1 (AO1.1b) | |
| $\begin{array}{c\|ccccccccc\|c} P & v & p_1 & p_2 & p_3 & s_1 & s_2 & s_3 & s_4 & \text{val} \\ \hline 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & -3 & -2 & -6 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & -1 & -3 & -2 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & -4 & -2 & -1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 \end{array}$ | A1 (AO1.1b) | All rows correct |

**Total: 3 marks**

---

## Question 7(b)(ii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| Applies simplex algorithm to modify at least one non-pivot row correctly | M1 (AO3.1a) | |
| At least three rows correct after iteration | A1 (AO1.1b) | |
| Final tableau: $\begin{array}{c\|ccccccccc\|c} P & v & p_1 & p_2 & p_3 & s_1 & s_2 & s_3 & s_4 & \text{val} \\ \hline 1 & 0 & -3 & -2 & -6 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & -3 & -2 & -6 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & -1 & 4 & -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 5 & -1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 \end{array}$ | A1 (AO1.1b) | All rows correct |

**Total: 3 marks**

---

## Question 7(c)(i):

| Answer/Working | Marks | Guidance |
|---|---|---|
| The optimal mixed strategy for Avon is: play $A_1$ with probability $0.25$, play $A_2$ with probability $0.70$, play $A_3$ with probability $0.05$ | B1 (AO3.2a) | Interprets final simplex tableau to find optimal mixed strategy |

**Total: 1 mark**

---

## Question 7(c)(ii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $3 \times 0.25 + 2 \times 0.70 + 6 \times 0.05 = 2.45$ | B1 (AO2.2a) | Deduces value of the game for Avon |

**Total: 1 mark**

---

## Question 7(d):

| Answer/Working | Marks | Guidance |
|---|---|---|
| To improve their outcome, Avon should play $A_2$ each time | E1 (AO2.4) | Explains correctly that Avon should play $A_2$ each time |
| Under these conditions, Avon's maximum expected pay-off will be $3$ | E1 (AO3.2a) | Determines correctly that Avon's maximum expected pay-off will be $3$ |

**Total: 2 marks**

**Question total: 14 marks | Paper total: 50 marks**
7 Avon and Roj play a zero-sum game.

The game is represented by the following pay-off matrix for Avon.

7 (c) (i) Find the optimal mixed strategy for Avon.\\

7 (c) (ii) Find the value of the game for Avon.\\

7 (d) Roj thinks that his best outcome from the game is to play strategy $\mathbf { R } _ { \mathbf { 2 } }$ each time.

Avon notices that Roj always plays strategy $\mathbf { R } _ { \mathbf { 2 } }$ and Avon wants to use this knowledge to maximise his expected pay-off from the game.

Explain how your answer to part (c)(i) should change and find Avon's maximum expected pay-off from the game.\\

\includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-16_2490_1735_219_139}

\hfill \mbox{\textit{AQA Further Paper 3 Discrete 2021 Q7 [14]}}
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