D2 June 2013 (R) Q7
7. A two-person zero-sum game is represented by the following pay-off matrix for player A.
| B plays 1 | B plays 2 | B plays 3 | |
|---|---|---|---|
| A plays 1 | 1 | –3 | 2 |
| A plays 2 | –2 | 3 | –1 |
| A plays 3 | 5 | –1 | 0 |
Formulate the game as a linear programming problem for player A. Write the constraints as inequalities. Define your variables clearly. (7)
| Scheme | Marks |
|---|---|
| E.g. Add 4 to each element | B1 |
| Let \(p_1\), \(p_2\), \(p_3\) be the probability of (A) playing 1, 2 and 3 respectively (where \(p_1, p_2, p_3 \geqslant 0\)) | B1 |
| let \(V\) = value of the game (to player A) | B1 |
| maximise \(P = V\) | B1 |
| subject to: \(5p_1 + 2p_2 + 9p_3 \geqslant V\) \(p_1 + 7p_2 + 3p_3 \geqslant V\) \(6p_1 + 3p_2 + 4p_3 \geqslant V\) | M1 A1 |
| \(p_1 + p_2 + p_3 \leqslant 1\) | A1 |
| (7) | |
| 7 marks |
Notes
1B1: Making all terms non-negative.
2B1: Defining probability variables
3B1: Defining \(V\)
4B1: ‘maximise’ + function/expression
1M1: At least three equations/inequations in (\(V\)), \(p_1\), \(p_2\) and \(p_3\)
1A1: The three inequalities in \(V\), \(p_1\), \(p_2\) and \(p_3\) CAO
1A1: probability sum inequality (or equation) correct.