Zero-sum games, play-safe strategies and saddle pointsEdexcel A-Level Further Maths: Subtopic test

1. 1
   Rose and Colin play a zero-sum game. Rose chooses a row and Colin chooses a column, and the pay-off matrix shows Rose's winnings: $\begin{pmatrix} 1 & 7 & 2 \\ 4 & 5 & 6 \\ 2 & 0 & 3 \end{pmatrix}$.

   (a)

   Which row is Rose's play-safe strategy?

   [1 mark]

   • ARow 1
   • BRow 2
   • CRow 3
   • DShe has none, because play-safe strategies only exist when a game has no stable solution

   (b)

   Which column is Colin's play-safe strategy?

   [1 mark]

   • AColumn 3
   • BColumn 2
   • CColumn 1
   • DHe has none, because the column maxima are all different

   (c)

   Show that the game has a stable solution and state its value.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Rose and Colin play a different zero-sum game, with Rose choosing the rows and Colin the columns. Rose's pay-off matrix is $\begin{pmatrix} -1 & 3 & -2 \\ 2 & -3 & 4 \end{pmatrix}$.

   (a)

   Rose plays row 2 and Colin plays column 2. What is Colin's pay-off?

   [1 mark]

   • A$3$
   • B$-3$
   • C$0$
   • D$2$

   (b)

   Both players use their play-safe strategies. What is Rose's pay-off?

   [1 mark]

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

   (c)

   Explain why this game has no stable solution.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two online retailers, Aria and Bolt, each choose one of three advertising strategies. The pay-off matrix shows Aria's gain in market share (percentage points) for each pair of strategies, where Aria chooses the rows and Bolt the columns, and $x$ is a constant: $\begin{pmatrix} 4 & 2 & 6 \\ 4 & x & 5 \\ 1 & 7 & 0 \end{pmatrix}$. Any gain for Aria is an equal loss for Bolt.

   (a)

   Given that $x=1$, find each retailer's play-safe strategy and show that the game has no stable solution.

   [3 marks]

   (b)

   Find the set of values of $x$ for which the game has a stable solution, and state the value of the game.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two political parties, Green and Blue, each choose one of three campaign strategies. The pay-off matrix shows the net change in the number of seats won by Green, where Green chooses the rows and Blue the columns: $\begin{pmatrix} 2 & -3 & 4 \\ -1 & 3 & 1 \\ 0 & 2 & -2 \end{pmatrix}$. Every seat Green gains is a seat Blue loses.

   (a)

   Find the play-safe strategy for each party, show that the game has no stable solution, and state the outcome if both parties play safe.

   [6 marks]

   (b)

   Green believes that Blue will play safe. Justify whether Green should also play safe, and explain what happens next if each party keeps reacting to the other. State what this shows about the game.

   [6 marks]

   Total for question 4: 12 marks

End of questions