The row maximin is 4, so A to play it safe A should play strategy 2, and the column minimax is 4, so B should play strategy 4. This results in a win of 4 for A and a loss of 4 for B. We now consider if it is worthwhile for A or B to change strategy. If A assumes B will play strategy 4, then if A plays strategy 1, they will win 3 and if A plays strategy 3 they will win 2, but will win 8 by staying with strategy 2. If B assumes A will play strategy 2, B will lose 5 if they play strategy 1, 6 if they play strategy 2, 5 if they play strategy 3 but only 4 if they stick with strategy 4. It is not sensible for either player to change strategy, if they are both playing safe. The game has a stable equilibrium or saddle point, at (A2, B4) and the game has value 4 to A and -4 to B.
Suppose we have the following payoff matrix.
|
5 |
-2 |
3 |
3 |
|
5 |
6 |
5 |
4 |
|
2 |
3 |
-1 |
2 |
We find the play safe strategies.
|
|
|
|
|
|
Row Minimum |
|
|
5 |
-2 |
3 |
3 |
-2 |
|
|
5 |
6 |
5 |
4 |
4 |
|
|
2 |
3 |
-1 |
2 |
-1 |
|
Column Maximum |
5 |
6 |
5 |
4 |
|
The row maximin is 4, so A to play it safe A should play strategy 2, and the column minimax is 4, so B should play strategy 4. This results in a win of 4 for A and a loss of 4 for B.
We now consider if it is worthwhile for A or B to change strategy.
If A assumes B will play strategy 4, then if A plays strategy 1, they will win 3 and if A plays strategy 3 they will win 2, but will win 8 by staying with strategy 2.
If B assumes A will play strategy 2, B will lose 5 if they play strategy 1, 6 if they play strategy 2, 5 if they play strategy 3 but only 4 if they stick with strategy 4.
It is not sensible for either player to change strategy, if they are both playing safe. The game has a stable equilibrium or saddle point, at (A2, B4) and the game has value 4 to A and -4 to B.