Pay-off matrices and play-safe strategiesAQA A-Level Further Maths: Subtopic test

1. 1
   Two companies, Rowan Ltd and Colbert Ltd, each choose one of three advertising strategies at the same time. The pay-off matrix for Rowan, in percentage points of market share gained by Rowan (a negative entry is a loss for Rowan and a gain for Colbert), is $\begin{pmatrix} 3 & -1 & 2 \\ 1 & 0 & 4 \\ -2 & 5 & 1 \end{pmatrix}$. Rows are Rowan's strategies and columns are Colbert's.

   (a)

   Which is Rowan's play-safe strategy?

   [1 mark]

   • ARow 1
   • BRow 2
   • CRow 3
   • DNone, because the game is not stable

   (b)

   Which is Colbert's play-safe strategy?

   [1 mark]

   • AColumn 3
   • BColumn 2
   • CColumn 1 or column 3
   • DColumn 1

   (c)

   Show that this game does not have a stable solution.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A supermarket chain, Firm F, and its rival choose pricing strategies at the same time. The pay-off matrix for Firm F, in percentage points of market share gained by Firm F, is $\begin{pmatrix} 4 & 2 & 5 \\ 3 & 1 & 6 \\ 1 & 0 & 2 \end{pmatrix}$. Rows are Firm F's strategies and columns are the rival's. The game is zero-sum.

   (a)

   What is the value of the game?

   [1 mark]

   • A2
   • B3
   • C4
   • D1

   (b)

   Which pair of strategies gives the stable solution?

   [1 mark]

   • ARow 1 and column 1
   • BRow 1 and column 2
   • CRow 2 and column 2
   • DRow 3 and column 2

   (c)

   Interpret the value of the game in the context of the problem.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Xavier and Yara play a game. At the same time, Xavier chooses 1, 2 or 3 and Yara chooses 2 or 3. If the sum of the two numbers is even, Yara pays Xavier the product of the two numbers, in pounds. If the sum is odd, Xavier pays Yara the sum of the two numbers, in pounds.

   (a)

   Construct the pay-off matrix for Xavier, with a row for each of Xavier's choices and a column for each of Yara's choices.

   [3 marks]

   (b)

   Find each player's play-safe strategy and hence determine whether the game has a stable solution.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Rowan and Colin play a zero-sum game. The pay-off matrix for Rowan, in points, is $\begin{pmatrix} 2 & 3 & 1 \\ 5 & 4 & 6 \\ 7 & 2 & 0 \end{pmatrix}$. Rows are Rowan's strategies and columns are Colin's.

   (a)

   Prove that the game has a stable solution. State the play-safe strategy for each player and the value of the game.

   [6 marks]

   (b)

   Construct Colin's pay-off matrix, with a row for each of Colin's strategies, and use it to confirm Colin's play-safe strategy and the value of the game for Colin.

   [6 marks]

   Total for question 4: 12 marks

End of questions