# Question 2:

## Part (a)

| Answer/Working | Marks | Guidance |
|---|---|---|
| The gains (or losses) made by one player are exactly balanced by the losses (or gains) made by the other player. | B1 | Either: losses of one player balanced by gains of the other, **or** total points scored by both is zero |

## Part (b)

| Answer/Working | Marks | Guidance |
|---|---|---|
| $5$ | B1 | CAO |

## Part (c)

| Answer/Working | Marks | Guidance |
|---|---|---|
| Row minimum $\{-3, 0, -5\}$, Row maximin $= 0$ | M1 | Clear attempt to find Row maximin and Column minimax (row minimums or column maximums correct, **or** at least four of six values stated correctly) |
| Column maximum $\{2, 4, 2\}$, Column minimax $= 2$ | A1 | Correct Row maximin **and** Column minimax (dependent on all row mins and column maxs correct) |
| $0 \neq 2$ so no stable solution | A1 | CAO – states $0 \neq 2$ and draws correct conclusion |

## Part (d)

| Answer/Working | Marks | Guidance |
|---|---|---|
| Column 1 dominates column 2 so remove column 2 | B1 | CAO (accept reduced matrix or 'column 1 dominates column 2' or column crossed out) |
| $\begin{pmatrix} 3 & 0 & -2 \\ -2 & -1 & 5 \end{pmatrix}$ | B1ft, B1 | Either $3\times2$ matrix with correct values for G (all signs changed correctly) or $2\times3$ matrix with correct values for G; d3B1: CAO |

## Part (e)

| Answer/Working | Marks | Guidance |
|---|---|---|
| Let $p$ = probability that Greg plays new row 1. If R plays 1: G's expected winnings $= 3p - 2(1-p)\ (= 5p-2)$; If R plays 2: $= 0p - 1(1-p)\ (= p-1)$; If R plays 3: $= -2p + 5(1-p)\ (= -7p+5)$ | M1, A1 | Setting up all three probability expressions (allow $p-1$); CAO (condone incorrect simplification) |
| Graph with three lines, correct slant direction and relative intersection | B2, 1ft, 0 | Attempt at three lines; CAO $0\leq p\leq1$, scaling correct and clear |
| $p - 1 = -7p + 5 \Rightarrow 8p = 6 \Rightarrow p = \frac{3}{4}$ | DM1, A1 | Finding correct optimal point; must have three lines and set up equation for $0\leq p\leq1$; CSO |
| G should play 1 with probability $\frac{3}{4}$, 2 never, and play 3 with probability $\frac{1}{4}$ | A1ft | All three options listed must fit from their $p$; dependent on both previous M marks |
| Value of game to G is $-\frac{1}{4}$ | A1 | CAO $\left(-\frac{1}{4}\right)$ |

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