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

Exam questions on Zero-sum games, play-safe strategies and saddle points

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: $\begin{pmatrix} 1 & 7 & 2 \\ 4 & 5 & 6 \\ 2 & 0 & 3 \end{pmatrix}$.
   Show that the game has a stable solution and state its value.2 marks
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 $\begin{pmatrix} -1 & 3 & -2 \\ 2 & -3 & 4 \end{pmatrix}$.
   Explain why this game has no stable solution.2 marks
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.
   Given that $x=1$, find each retailer's play-safe strategy and show that the game has no stable solution.3 marks