Question 2:
2 | (a) | (i) | R S T Row min for A
K (7, 3) (2, 6) (5, 3) 2
L (1, 5) (8, 2) (2, 5) 1
M (3, 2) (1, 5) (4, 6) 1
Play-safe strategy for Annie is K | B1
B1
[2] | May reduce every a value by a constant and/or every b
value by a constant, but not e.g. reducing rows and not
calculating differences, a – b
Row minima 2, 1, 1 (o.e.) seen, on table or written
But not just stating row maximin = 2
K identified as play-safe
2 | (a) | (ii) | R S T
K (7, 3) (2, 6) (5, 3)
L (1, 5) (8, 2) (2, 5)
M (3, 2) (1, 5) (4, 6)
Col min for B 2 2 3
T has largest minimum
Maximin = max {2, 2, 3} = 3 (so play-safe for Brett is T, as given) | M1
A1
[2] | May reduce every a value by a constant and/or every b
value by a constant, but not e.g. reducing rows and not
calculating differences, a – b
Col minina 2, 2, 3 (o.e.) seen, on table or written
Col maximin = 3
‘max’ or ‘largest’
(7, 3) | (2, 6) | (5, 3) | 2
(1, 5) | (8, 2) | (2, 5) | 1
(3, 2) | (1, 5) | (4, 6) | 1
(7, 3) | (2, 6) | (5, 3)
(1, 5) | (8, 2) | (2, 5)
(3, 2) | (1, 5) | (4, 6)
2 | 2 | 3
2 | (b) | (i) | K | B1
[1] | K
2 | (b) | (ii) | S | B1 FT
[1] | Follow through to col with max b value in row from (b)(i)
2 | (c) | (Column R is weakly dominated by) column T
3 = 3, 5 = 5 and 2 < 6 | M1
A1
[2] | T (only)
3, 3 5, 5 and 2 < 6
2 | (d) | Let Brett play S with probability p and T with probability 1 – p
If Annie plays X, Brett expects to win 6p + 3(1 – p) = 3 + 3p
If Annie plays Y, Brett expects to win 2p + 5(1 – p) = 5 – 3p
If Annie plays Z, Brett expects to win 5p + 6(1 – p) = 6 – p
Optimum at p =
1
Play S with probability and T with probability
3
1 2
3 3 | B1
B1
M1
A1FT
[4] | Three correct expressions (in any form) for expected winnings
in terms of a single variable (which need not be called p)
Graph showing lines (0, 3) to (1, 6)
(0, 5) to (1, 2)
and (0, 6) to (1, 5)
Horizontal axis from 0 to 1 using at least half width of grid
(values 0 and 1 may be implied from lines)
Vertical axis with a scale (at least 2 values marked on axis, or
origin and at least 1 value marked)
or prob corresponding to max point of their lower boundary
1
Interpretation of their p in context, or implied if meaning of p
3
(or 1 – p) was given when setting up equations
2 | (e) | Best outcome in each col for Annie: (K, R), (L, S), (K, T)
Best outcome in each row for Brett: (K, S), (L, R), (L, T), (M, T)
No common cell so no Nash equilibrium | B1
B1
[2] | (K, R), (L, S) and (K, T) identified as a set in any way
(K, S), (L, R), (L, T) and (M, T) identified as a set in any way
Or one set and verify none of these are optimal for other player
R S T R S T
K (7, 3) (2, 6) (5, 3) K (7, 3) (2, 6) (5, 3)
L (1, 5) (8, 2) (2, 5) L (1, 5) (8, 2) (2, 5)
M (3, 2) (1, 5) (4, 6) M (3, 2) (1, 5) (4, 6)
(7, 3) | (2, 6) | (5, 3)
(1, 5) | (8, 2) | (2, 5)
(3, 2) | (1, 5) | (4, 6)
(7, 3) | (2, 6) | (5, 3)
(1, 5) | (8, 2) | (2, 5)
(3, 2) | (1, 5) | (4, 6)