D2 June 2019 Q4
4. Eugene and Stephen play a zero-sum game. The pay-off matrix shows the number of points that Eugene scores for each combination of strategies.
| Stephen plays 1 | Stephen plays 2 | Stephen plays 3 | |
|---|---|---|---|
| Eugene plays 1 | 4 | 5 | 0 |
| Eugene plays 2 | –2 | 1 | 1 |
| Eugene plays 3 | –3 | –4 | 3 |
| Scheme | Marks |
|---|---|
| Row minima: 0, −2, −4 max is 0 | M1 |
| Column maxima: 4, 5, 3 min is 3 | A1 |
| Play safe for Eugene is 1 and for Stephen is 3 | A1 |
| Row maximin (0) \(\neq\) Column minimax (3) so not stable | A1 |
| (4) |
Notes
a1M1: Clear attempt to find the Row maximin and Column minimax (either the Row minimums or Column maximums correct or at least four (of the six) values stated correctly)
a1A1: Correct Row maximin and Column minimax (dependent on all row mins and column maxs correct) – these could either be stated or clearly shown
a2A1: Correct play safe for E (1) and S (3) – not dependent on the previous A mark
a3A1: CAO (dependent on all rowmins and colmaxs correct) states \(0 \neq 3\) (or row (maximin) \(\neq\) col (minimax) as long as 0 is clearly identified as the row maximin and 3 as the column minimax)
| Scheme | Marks |
|---|---|
| If Stephen plays safe then Eugene should change from their play safe of option 1 to their option 3 as they will win more against Stephen’s play-safe (3 rather than 0) | B1 |
| (1) |
Notes
b1B1: CAO – must mention option 3 and either gain 3 or equivalent in words
| Scheme | Marks |
|---|---|
| Reverses signs in pay-off matrix followed by add 5 to each element. Condone one error. | B1 |
| Let \(p_1, p_2, p_3\) be the probability of (S) playing 1, 2 and 3 respectively (where \(p_1, p_2, p_3 \geqslant 0\)) | B1 |
| Let \(V\) = value of the game (to S) | B1 |
| Maximise \((P =)\, V\) | B1 |
| Subject to: \[\begin{aligned} V - p_1 - 5p_3 + r &= 0 \\ V - 7p_1 - 4p_2 - 4p_3 + s &= 0 \\ V - 8p_1 - 9p_2 - 2p_3 + t &= 0 \\ p_1 + p_2 + p_3 (+u) &= 1 \end{aligned}\] | M1 A1 A1 |
| \((r, s, t, u \geqslant 0)\) | |
| (7) | |
| 12 marks |
Notes
c1B1: Making all terms non-negative (any addition \(\geqslant 5\) is acceptable)
c2B1: Defining probability variables
c3B1: Defining V
c4B1: ‘maximise’ + function/expression
c1M1: 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
c1A1: CAO - the three constraints involving V and \(p_i\) expressed as equations with slack variables
c2A1: Probability sum equation correct (allow presence of a slack variable in this equation)