3 Two people, Rose and Callum, play a zero-sum game. The game is represented by the following pay-off matrix for Rose.
| Callum |
| \cline { 2 - 5 } | | \(\mathbf { C } _ { \mathbf { 1 } }\) | \(\mathbf { C } _ { \mathbf { 2 } }\) | \(\mathbf { C } _ { \mathbf { 3 } }\) |
| \cline { 2 - 5 } | \(\mathbf { R } _ { \mathbf { 1 } }\) | 5 | 2 | - 1 |
| \cline { 2 - 5 }
Rose | \(\mathbf { R } _ { \mathbf { 2 } }\) | - 3 | - 1 | 5 |
| \cline { 2 - 5 } | \(\mathbf { R } _ { \mathbf { 3 } }\) | 4 | 1 | - 2 |
| \cline { 2 - 5 } | | | | |
| \cline { 2 - 5 } |
- State the play-safe strategy for Rose and give a reason for your answer.
- Show that there is no stable solution for this game.
- Explain why Rose should never play strategy \(\mathbf { R } _ { \mathbf { 3 } }\).
- Rose adopts a mixed strategy, choosing \(\mathbf { R } _ { \mathbf { 1 } }\) with probability \(p\) and \(\mathbf { R } _ { \mathbf { 2 } }\) with probability \(1 - p\).
- Find expressions for the expected gain for Rose when Callum chooses each of his three possible strategies. Simplify your expressions.
- Illustrate graphically these expected gains for \(0 \leqslant p \leqslant 1\).
- Hence determine the optimal mixed strategy for Rose.
- Find the value of the game.