# Question 4:

## Part (a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Row minima: $0, -2, -4$ — max is $0$ | M1 | 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) |
| Column maxima: $4, 5, 3$ — min is $3$ | A1 | Correct Row maximin and Column minimax (dependent on all row mins and column maxs correct) |
| Play safe for Eugene is $1$ and for Stephen is $3$ | A1 | Correct play safe for E (1) and S (3) – not dependent on the previous A mark |
| Row maximin $(0) \neq$ Column minimax $(3)$ so not stable | A1 **(4)** | CAO – 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 |

## Part (b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| 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)** | CAO – must mention option 3 and either gain 3 or equivalent in words |

## Part (c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Reverses signs in pay-off matrix followed by add 5 to each element. Condone one error. | B1 | Making all terms non-negative (any addition $\geq 5$ is acceptable) |
| Let $p_1, p_2, p_3$ be the probability of (S) playing 1, 2 and 3 respectively (where $p_1, p_2, p_3 \geq 0$) | B1 | Defining probability variables |
| Let $V =$ value of the game (to S) | B1 | Defining $V$ |
| Maximise $(P=)\ V$ | B1 | 'maximise' + function/expression |
| Subject to: $V - p_1 - 5p_3 + r = 0$ | M1 A1 | 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. CAO - the three constraints involving $V$ and $p_i$ expressed as equations with slack variables |
| $V - 7p_1 - 4p_2 - 4p_3 + s = 0$ | | |
| $V - 8p_1 - 9p_2 - 2p_3 + t = 0$ | | |
| $p_1 + p_2 + p_3 (+u) = 1$ | A1 **(7)** | Probability sum equation correct (allow presence of a slack variable in this equation) |
| $(r, s, t, u \geq 0)$ | | |

**Total: 12 marks**

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