Decision Mathematics 2: Game theoryEdexcel A-Level Further Maths: Topic test

1. 1
   Two rival bakeries, Crust and Crumb, each choose one of three pricing strategies. The game is zero-sum and the pay-off matrix shows Crust's gain in daily sales (in hundreds of pounds). Crust chooses the rows $\mathrm{R}_1, \mathrm{R}_2, \mathrm{R}_3$ and Crumb chooses the columns $\mathrm{C}_1, \mathrm{C}_2, \mathrm{C}_3$. The matrix is $\begin{pmatrix} 3 & 6 & 4 \\ 1 & 7 & 2 \\ -2 & 4 & 0 \end{pmatrix}$.

   (a)

   What is Crust's play-safe strategy?

   [1 mark]

   • A$\mathrm{R}_1$
   • B$\mathrm{R}_2$
   • C$\mathrm{R}_3$
   • D$\mathrm{C}_1$

   (b)

   What is the value of the game to Crust?

   [1 mark]

   • A$7$
   • B$4$
   • C$3$
   • D$1$

   (c)

   Show that the game has a stable solution.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two ice-cream sellers, Ayla and Bran, each choose one of three pitches on a beach. The game is zero-sum and the pay-off matrix shows Ayla's change in daily takings (in pounds). Ayla chooses the rows $\mathrm{A}_1, \mathrm{A}_2, \mathrm{A}_3$ and Bran chooses the columns $\mathrm{B}_1, \mathrm{B}_2, \mathrm{B}_3$. The matrix is $\begin{pmatrix} 1 & -2 & 3 \\ -1 & 2 & -4 \\ 0 & 1 & -1 \end{pmatrix}$.

   (a)

   What is the row maximin?

   [1 mark]

   • A$-4$
   • B$1$
   • C$-2$
   • D$-1$

   (b)

   Which pitch is Bran's play-safe strategy?

   [1 mark]

   • A$\mathrm{B}_2$
   • B$\mathrm{B}_1$
   • C$\mathrm{B}_3$
   • D$\mathrm{A}_3$

   (c)

   Explain why this game has no stable solution.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two bike-hire firms, Pedal and Quick, each choose a pricing policy. Pedal chooses one of two policies $\mathrm{P}_1, \mathrm{P}_2$ (rows) and Quick chooses one of three policies $\mathrm{Q}_1, \mathrm{Q}_2, \mathrm{Q}_3$ (columns). The game is zero-sum and the pay-off matrix shows Pedal's gain in weekly bookings: $\begin{pmatrix} 4 & -1 & 2 \\ -2 & 3 & 1 \end{pmatrix}$.

   (a)

   Pedal plays $\mathrm{P}_1$ with probability $p$ and $\mathrm{P}_2$ with probability $1-p$. Find the value of $p$ that maximises Pedal's smallest expected gain, and find the value of the game.

   [3 marks]

   (b)

   Find Quick's optimal mixed strategy, and explain why Quick should never play $\mathrm{Q}_3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two delivery apps, Dash and Zoom, compete for customers. Each month Dash chooses one of three promotions $\mathrm{D}_1, \mathrm{D}_2, \mathrm{D}_3$ (rows) and Zoom chooses one of three promotions $\mathrm{Z}_1, \mathrm{Z}_2, \mathrm{Z}_3$ (columns). The game is zero-sum and the pay-off matrix shows Dash's gain in market share (percentage points): $\begin{pmatrix} 3 & -1 & 4 \\ -2 & 4 & 3 \\ 1 & 2 & 2 \end{pmatrix}$.

   (a)

   Use a dominance argument to reduce the game, and then find Zoom's optimal mixed strategy and the value of the game to Dash.

   [6 marks]

   (b)

   Dash wants to find its own optimal strategy using the Simplex algorithm. Dash plays $\mathrm{D}_i$ with probability $p_i$. Add a suitable constant to every entry of the matrix and let $V$ be the value of the new game. Formulate Dash's problem as a linear programming problem in which $P=V$ is maximised, stating the objective, all the constraints and the non-negativity conditions.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Two railway companies, Alto and Brisk, each choose a timetable strategy. The game is zero-sum and the pay-off matrix shows Alto's gain in daily passengers (in hundreds). Alto chooses the rows $\mathrm{A}_1, \mathrm{A}_2, \mathrm{A}_3$ and Brisk chooses the columns $\mathrm{B}_1, \mathrm{B}_2, \mathrm{B}_3$. The matrix is $\begin{pmatrix} 3 & 1 & 4 \\ 2 & 4 & 3 \\ 0 & 2 & 1 \end{pmatrix}$.

   (a)

   Which of Brisk's strategies can be removed by dominance?

   [1 mark]

   • A$\mathrm{B}_1$
   • B$\mathrm{B}_2$
   • C$\mathrm{B}_3$
   • DNo column can be removed

   (b)

   Which matrix remains when all dominated strategies have been removed?

   [1 mark]

   • A$\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$
   • B$\begin{pmatrix} 3 & 1 & 4 \\ 2 & 4 & 3 \end{pmatrix}$
   • C$\begin{pmatrix} 3 & 1 \\ 0 & 2 \end{pmatrix}$
   • D$\begin{pmatrix} 1 & 4 \\ 4 & 3 \end{pmatrix}$

   (c)

   Alto plays $\mathrm{A}_1$ with probability $p$ in the reduced game. Find the optimal value of $p$ and the value of the game.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Ria and Cal play a zero-sum game in which Ria chooses the rows and Cal chooses the columns. The pay-off matrix shows Ria's gain and is $\begin{pmatrix} -2 & 3 \\ 4 & -1 \end{pmatrix}$. Cal plays column $j$ with probability $q_j$, and the game is to be solved as a linear programming problem using the Simplex algorithm.

   (a)

   What is the smallest whole number that can be added to every entry so that no entry is negative?

   [1 mark]

   • A$1$
   • B$2$
   • C$3$
   • D$4$

   (b)

   After the constant is added, let $V$ be the value of the new game and $y_j=\dfrac{q_j}{V}$. Cal's problem is to maximise $P=y_1+y_2$. Which constraint comes from the first row of the new matrix?

   [1 mark]

   • A$-2y_1+3y_2\le1$
   • B$5y_1\le1$
   • C$y_1+y_2\le1$
   • D$5y_2\le1$

   (c)

   The optimal solution of this linear programme is $y_1=\frac{2}{15}$ and $y_2=\frac15$. Find Cal's optimal mixed strategy and the value of the original game.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Two farm shops, Hill and Dale, compete for customers. Hill chooses one of two offers $\mathrm{H}_1, \mathrm{H}_2$ (rows) and Dale chooses one of four offers $\mathrm{D}_1, \mathrm{D}_2, \mathrm{D}_3, \mathrm{D}_4$ (columns). The game is zero-sum and the pay-off matrix shows Hill's gain in weekly customers (in tens): $\begin{pmatrix} 5 & 1 & 4 & 2 \\ 1 & 4 & 0 & 4 \end{pmatrix}$.

   (a)

   Hill plays $\mathrm{H}_1$ with probability $p$ and $\mathrm{H}_2$ with probability $1-p$. Find Hill's optimal value of $p$ and the value of the game.

   [3 marks]

   (b)

   Find Dale's optimal mixed strategy, explaining why Dale never plays $\mathrm{D}_1$ or $\mathrm{D}_4$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Two gyms, Atlas and Beam, compete for new members. Atlas chooses one of three campaigns $\mathrm{A}, \mathrm{B}, \mathrm{C}$ (rows) and Beam chooses one of three campaigns $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$ (columns). The game is zero-sum and the pay-off matrix shows Atlas's gain in new members per month: $\begin{pmatrix} 5 & 2 & 3 \\ 1 & 4 & 5 \\ 2 & 1 & 3 \end{pmatrix}$.

   (a)

   Use dominance to reduce the game to a $2\times2$ game, and hence find the optimal mixed strategy for each gym and the value of the game.

   [6 marks]

   (b)

   Beam plays $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$ with probabilities $q_1, q_2, q_3$ and $V$ is the value of the game. Let $y_j=\dfrac{q_j}{V}$. Formulate Beam's problem as a linear programme in $y_1, y_2, y_3$ that can be solved by the Simplex algorithm, and use your answer to (a) to state the optimal value of $P$ and of $y_1, y_2, y_3$.

   [6 marks]

   Total for question 8: 12 marks

End of questions