3 Two people, Roseanne and Collette, play a zero-sum game. The game is represented by the following pay-off matrix for Roseanne.
| \multirow{2}{*}{} | | Collette |
| Strategy | \(\mathrm { C } _ { 1 }\) | \(\mathbf { C } _ { \mathbf { 2 } }\) | \(\mathrm { C } _ { 3 }\) |
| \multirow{2}{*}{Roseanne} | \(\mathrm { R } _ { 1 }\) | -3 | 2 | 3 |
| \(\mathbf { R } _ { \mathbf { 2 } }\) | 2 | -1 | -4 |
- Find the optimal mixed strategy for Roseanne.
- Show that the value of the game is - 0.5 .
- Collette plays strategy \(\mathrm { C } _ { 1 }\) with probability \(p\) and strategy \(\mathrm { C } _ { 2 }\) with probability \(q\). Write down, in terms of \(p\) and \(q\), the probability that she plays strategy \(\mathrm { C } _ { 3 }\).
- Hence, given that the value of the game is - 0.5 , find the optimal mixed strategy for Collette.