# Question 6:

## Part 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Row minima: $-1, -2, (-1)$; Column maxima: $4, 0, (1)$ | M1 | Identifies correctly all three row minima or all three (two) column maxima |
| $\max(\text{row min}) = -1$; $\min(\text{col max}) = 0$ | A1 | Determines correctly max(row min) and min(col max) |
| Victoria plays $R$ (or $P$); Albert plays $Y$ | A1 (CAO) | Correct play-safe strategies for each player |

## Part 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Albert should never play $Z$ because strategy $Y$ dominates strategy $Z$ | E1 | Identifies strategy and justifies with reference to dominance |

## Part 6(c)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Victoria should never play $P$ because strategy $R$ dominates strategy $P$ | B1 | States correctly that strategy $R$ dominates strategy $P$ |
| Let Victoria play $Q$ with probability $p$ and $R$ with probability $1-p$ | M1 | Introduces and defines a probability variable |
| $X$: expected gain for Victoria $= -2p + 4(1-p) = 4 - 6p$ | A1 | Finds correctly both expected gain expressions for Victoria |
| $Y$: expected gain for Victoria $= -(1-p) = p - 1$ | | |
| No stable solution, so optimal value of $p$: $4 - 6p = p - 1$, giving $p = \frac{5}{7}$ | A1F | States no stable solution; sets expressions equal; solves correctly |
| Victoria plays $Q$ with probability $\frac{5}{7}$ and $R$ with probability $\frac{2}{7}$ | E1 | Interprets solution giving optimal mixed strategy for Victoria |

## Part 6(c)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $4 - 6 \times \frac{5}{7} = -\frac{2}{7}$ or $\frac{5}{7} - 1 = -\frac{2}{7}$ | B1F | Substitutes value of $p$ into expected gain expression giving value of game |

## Part 6(c)(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Expected pay-off for Victoria will only be $-\frac{2}{7}$ if Albert plays an optimal mixed strategy between $X$ and $Y$ | E1 | Recognises Victoria can improve on value of game if Albert does not play optimal mixed strategy |

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