4 The table gives the pay-off matrix for a zero-sum game between two players, Rowan and Colin.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Colin}
| \cline { 2 - 5 } | | Strategy \(X\) | Strategy \(Y\) | Strategy \(Z\) |
| \cline { 2 - 5 }
Rowan | Strategy \(P\) | 5 | - 3 | - 2 |
| \cline { 2 - 5 } | Strategy \(Q\) | - 4 | 3 | 1 |
| \cline { 2 - 5 } | | | | |
| \cline { 2 - 5 } |
\end{table}
Rowan makes a random choice between strategies \(P\) and \(Q\), choosing strategy \(P\) with probability \(p\) and strategy \(Q\) with probability \(1 - p\).
- Write down and simplify an expression for the expected pay-off for Rowan when Colin chooses strategy \(X\).
- Using graph paper, draw a graph to show Rowan's expected pay-off against \(p\) for each of Colin's choices of strategy.
- Using your graph, find the optimal value of \(p\) for Rowan.
- Rowan plays using the optimal value of \(p\). Explain why, in the long run, Colin cannot expect to win more than 0.25 per game.