5.
| B plays 1 | B plays 2 | B plays 3 | B plays 4 |
| A plays 1 | 4 | -2 | 3 | 2 |
| A plays 2 | 3 | -1 | 2 | 0 |
| A plays 3 | -1 | 2 | 0 | 3 |
A two person zero-sum game is represented by the pay-off matrix for player A given above.
- Explain, with justification, how this matrix may be reduced to a \(3 \times 3\) matrix.
- Find the play-safe strategy for each player and verify that there is no stable solution to this game.
The game is formulated as a linear programming problem for player A .
The objective is to maximise \(P = V\), where \(V\) is the value of the game to player A.
One of the constraints is that \(p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1\), where \(p _ { 1 } , p _ { 2 } , p _ { 3 }\) are the probabilities that player A plays 1, 2, 3 respectively. - Formulate the remaining constraints for this problem. Write these constraints as inequalities.
The Simplex algorithm is used to solve the linear programming problem.
The solution obtained is \(p _ { 1 } = 0 , p _ { 2 } = \frac { 3 } { 7 } , p _ { 3 } = \frac { 4 } { 7 }\) - Calculate the value of the game to player A.