Mixed strategies: the graphical methodEdexcel A-Level Further Maths: Subtopic test

1. 1
   Rose and Colin play a zero-sum game with Rose's pay-off matrix $\begin{pmatrix} 3 & -1 \\ -2 & 4 \end{pmatrix}$, where Rose chooses the rows. Rose plays row 1 with probability $p$ and row 2 with probability $1-p$.

   (a)

   Which expression is Rose's expected pay-off when Colin plays column 1?

   [1 mark]

   • A$5p-2$
   • B$p+2$
   • C$3p-2$
   • D$4-5p$

   (b)

   What is the value of the game to Rose?

   [1 mark]

   • A$\frac35$
   • B$3$
   • C$-1$
   • D$1$

   (c)

   Find Colin's optimal strategy.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Rose and Colin play a zero-sum game with Rose's pay-off matrix $\begin{pmatrix} 2 & -1 & 4 \\ -3 & 3 & -2 \end{pmatrix}$, where Rose chooses the rows. Rose plays row 1 with probability $p$ and row 2 with probability $1-p$.

   (a)

   The expected pay-off lines for columns 1 and 2 intersect at which value of $p$?

   [1 mark]

   • A$\frac12$
   • B$\frac13$
   • C$\frac23$
   • D$\frac49$

   (b)

   Which column does Colin never play in an optimal strategy?

   [1 mark]

   • AColumn 1
   • BColumn 3
   • CColumn 2
   • DNone: he uses all three columns

   (c)

   Find Colin's optimal strategy.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A zero-sum game has Rose's pay-off matrix $\begin{pmatrix} 4 & -1 \\ 1 & 3 \\ -2 & 5 \end{pmatrix}$, where Rose chooses the rows. Colin plays column 1 with probability $q$ and column 2 with probability $1-q$.

   (a)

   Find Colin's optimal strategy.

   [3 marks]

   (b)

   Find Rose's optimal strategy and the value of the game.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two supermarkets, X and Y, each choose a promotion. The pay-off matrix shows the change in daily customers (in hundreds) for X, where X chooses between two promotions (the rows) and Y chooses between four (the columns): $\begin{pmatrix} 3 & -2 & 1 & 4 \\ -1 & 4 & 2 & -3 \end{pmatrix}$. Every customer gained by X is lost by Y. X plays row 1 with probability $p$ and row 2 with probability $1-p$.

   (a)

   Use a graphical method to find X's optimal strategy and the value of the game to X.

   [6 marks]

   (b)

   Hence find Y's optimal strategy and interpret the value of the game in context.

   [6 marks]

   Total for question 4: 12 marks

End of questions