D2 June 2016 Q6
6. 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 | 5 | –3 | 1 |
| A plays 2 | 2 | 5 | 0 |
| A plays 3 | –4 | –1 | 4 |
| Scheme | Marks |
|---|---|
| Row mins {−3, 0, −4} Column max {5, 5, 4} | M1 |
| Row maximin (0) \(\neq\) column minimax (4) (so not stable) | A1 |
| (2) |
Notes
a1M1: Finding row minimums and column maximums – condone one error
a1A1: CAO must state that \(0 \neq 4\) (oe) – if \(0 \neq 4\) stated with no working then award M1A0 only
| Scheme | Marks |
|---|---|
| E.g. add 5 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: \(V - 10p_1 - 7p_2 - p_3 + r = 0\) \(V - 2p_1 - 10p_2 - 4p_3 + s = 0\) \(V - 6p_1 - 5p_2 - 9p_3 + t = 0\) \(p_1 + p_2 + p_3 (+u) = 1\) \((r, s, t, u \geqslant 0)\) | M1 A1 A1 |
| (7) |
Notes
b1B1: Making all terms non-negative (any addition \(\geqslant 4\) is acceptable)
b2B1: Defining probability variables
b3B1: Defining \(V\)
b4B1: ‘maximise’ + function/expression
b1M1: At least three (of the four) equations or inequalities in \(V\), \(p_1\), \(p_2\), \(p_3\) (with all \(p_i\) terms in the first three constraint equations having correct signs for the coefficients) – condone no slack variables for this mark
b1A1: CAO - the three constraints involving \(V\) and \(p_i\) expressed as equations with slack variables
b2A1: Probability sum equation correct (allow presence of a slack variable in this equation)
| Scheme | Marks | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
e.g. (adding 5 to each element)
| B1 M1 A1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (3) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 12 marks |
Notes
c1B1: All row and column labels correct for Simplex tableau
c1M1: Any two (numerical in nature) rows correct following from their constraints or a ‘correct’ answer (no follow through) with either one column or one row or one of both (so both a row and column) missing
c1A1: CAO – candidates may not label columns or rows in the order as given above – please check these carefully. Furthermore, candidates may add any value \(\geqslant 4\) which will change the nine bolded values above (so if +4 has been used this will increase each of the bolded values by +1). If all these bolded values are different then check the candidate’s original constraints in (b) to see if consistent with equations seen earlier