4 The table shows the pay-off matrix for player \(A\) in a two-person zero-sum game between \(A\) and \(B\).
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Player \(A\)}
| Strategy \(X\) | Strategy \(Y\) | Strategy \(Z\) |
| Strategy \(P\) | 4 | 5 | - 4 |
| Strategy \(Q\) | 3 | - 1 | 2 |
| Strategy \(R\) | 4 | 0 | 2 |
\end{table}
- Find the play-safe strategy for player \(A\) and the play-safe strategy for player \(B\). Use the values of the play-safe strategies to determine whether the game is stable or unstable.
- If player \(B\) knows that player \(A\) will use their play-safe strategy, which strategy should player \(B\) use?
- Suppose that the value in the cell where both players use their play-safe strategies can be changed, but all other entries are unchanged. Show that there is no way to change this value that would make the game stable.
- Suppose, instead, that the value in one cell can be changed, but all other entries are unchanged, so that the game becomes stable. Identify a suitable cell and write down a new pay-off value for that cell which would make the game stable.
- Show that the zero-sum game with the new pay-off value found in part (iv) has a Nash equilibrium and explain what this means for the players.