3. A two-person zero-sum game is represented by the following pay-off matrix for player \(A\).
| \(B\) plays X | \(B\) plays Y |
| \(A\) plays Q | 2 | -2 |
| \(A\) plays R | -1 | 5 |
| A plays S | 3 | 4 |
| \(A\) plays T | 0 | 2 |
- Show that this game is stable.
- State the value of this game to player \(B\).
Option S is removed from player A's choices and the reduced game, with option S removed, is no longer stable.
- Write down the reduced pay-off matrix for player \(B\).
Let \(B\) play option X with probability \(p\) and option Y with probability \(1 - p\).
- Use a graphical method to find the optimal value of \(p\) and hence find the best strategy for player \(B\) in the reduced game.
- Find the value of the reduced game to player \(A\).
- State which option player \(A\) should never play in the reduced game.
- Hence find the best strategy for player \(A\) in the reduced game.