Dominance and mixed strategiesAQA A-Level Further Maths: Subtopic test

1. 1
   Rowan and Colin play a zero-sum game. The pay-off matrix for Rowan is $\begin{pmatrix} 3 & 2 & 4 \\ 1 & 5 & 2 \\ 4 & 3 & 5 \end{pmatrix}$. Rows are Rowan's strategies and columns are Colin's. Rowan wants to maximise his pay-off and Colin wants to minimise it.

   (a)

   Which statement about dominance in this game is correct?

   [1 mark]

   • ARow 1 dominates row 3
   • BRow 2 dominates row 1
   • CRow 3 dominates row 1
   • DRow 2 dominates row 3

   (b)

   All dominated rows and columns are removed. What is the reduced matrix?

   [1 mark]

   • A$\begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}$
   • B$\begin{pmatrix} 1 & 2 \\ 4 & 5 \end{pmatrix}$
   • C$\begin{pmatrix} 3 & 2 & 4 \\ 4 & 3 & 5 \end{pmatrix}$
   • D$\begin{pmatrix} 1 & 5 \\ 4 & 3 \end{pmatrix}$

   (c)

   Explain why Colin will never play column 3.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Rowan and Colin play a zero-sum game with pay-off matrix for Rowan $\begin{pmatrix} 5 & 1 \\ 2 & 4 \end{pmatrix}$. The game has no stable solution. Rowan chooses row 1 with probability $p$ and row 2 with probability $1-p$, to maximise his smallest expected pay-off.

   (a)

   What is the optimal value of $p$?

   [1 mark]

   • A$\frac12$
   • B$\frac13$
   • C$\frac23$
   • D$\frac14$

   (b)

   What is the value of the game?

   [1 mark]

   • A$3$
   • B$\frac72$
   • C$2$
   • D$\frac52$

   (c)

   Find Colin's optimal strategy.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Rowan and Colin play a zero-sum game with pay-off matrix for Rowan $\begin{pmatrix} 2 & 5 & 1 \\ 4 & 1 & 6 \end{pmatrix}$. The game has no stable solution. Rowan chooses row 1 with probability $p$ and row 2 with probability $1-p$. Colin has three strategies, columns 1, 2 and 3.

   (a)

   Write down Rowan's expected pay-off, in terms of $p$, when Colin plays each of columns 1, 2 and 3.

   [3 marks]

   (b)

   Use these expressions to find Rowan's optimal strategy and the value of the game, and show that Colin should never play column 3.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Rowan and Colin play a zero-sum game with pay-off matrix for Rowan $\begin{pmatrix} 3 & 6 & 4 \\ 7 & 2 & 8 \\ 2 & 5 & 3 \end{pmatrix}$. Rows are Rowan's strategies and columns are Colin's.

   (a)

   Use dominance to reduce the game to a $2\times2$ game, and hence find Rowan's optimal mixed strategy and the value of the game.

   [6 marks]

   (b)

   Find Colin's optimal mixed strategy for the reduced game. Show that if Rowan uses the strategy found in part (a), Colin does worse by playing column 3.

   [6 marks]

   Total for question 4: 12 marks

End of questions