3 Two people, Rhona and Colleen, play a zero-sum game. The game is represented by the following pay-off matrix for Rhona.
|
| \cline { 2 - 5 } | Colleen |
| \cline { 2 - 5 }
Strategy | \(\mathbf { C } _ { \mathbf { 1 } }\) | \(\mathbf { C } _ { \mathbf { 2 } }\) | \(\mathbf { C } _ { \mathbf { 3 } }\) | |
| \cline { 2 - 5 }
Rhona | \(\mathbf { R } _ { \mathbf { 1 } }\) | 2 | 6 | 4 |
| \cline { 2 - 5 } | \(\mathbf { R } _ { \mathbf { 2 } }\) | 3 | - 3 | - 1 |
| \cline { 2 - 5 } | \(\mathbf { R } _ { \mathbf { 3 } }\) | \(x\) | \(x + 3\) | 3 |
| \cline { 2 - 5 } | | | | |
| \cline { 2 - 5 } |
It is given that \(x < 2\).
- Write down the three row minima.
- Show that there is no stable solution.
- Explain why Rhona should never play strategy \(R _ { 3 }\).
- Find the optimal mixed strategy for Rhona.
- Find the value of the game.