AS October 2020 Q3

EdexcelCurrent spec14 marksGame Theory

3. Two teams, A and B, each have three team members. One member of Team A will compete against one member of Team B for 10 rounds of a competition. None of the rounds can end in a draw.

Table 1 shows, for each pairing, the expected number of rounds that the member of Team A will win minus the expected number of rounds that the member of Team B will win. These numbers are the scores awarded to Team A. This competition between Teams A and B is a zero-sum game. Each team must choose one member to play. Each team wants to choose the member who will maximise its score.

		Team B
		Paul	Qaasim	Rashid
Team A	Mischa	\(4\)	\(-6\)	\(2\)
	Noel	\(0\)	\(-2\)	\(6\)
	Olive	\(-6\)	\(2\)	\(0\)

Table 1

(a)

(i) Find the number of rounds that Team A expects to win if Team A chooses Mischa and Team B chooses Paul.

(ii) Find the number of rounds that Team B expects to win if Team A chooses Noel and Team B chooses Qaasim. (2)

Table 1 models this zero-sum game.

(b)

(i) Find the play-safe strategies for the game.

(ii) Explain how you know that the game is not stable. (4)

(c) Determine which team member Team B should choose if Team B thinks that Team A will play safe. Give a reason for your answer. (1)

At the last minute, Rashid is ill and is therefore unavailable for selection by Team B.

(d) Find the best strategy for Team B, defining any variables you use. (7)

Scheme	Marks	AO
(i) 7	B1	3.4
(ii) 6	B1	3.4
	(2)

Mark scheme (b)

Scheme	Marks	AO
(i) Row minima: \(-6, -2, -6\) max is \(-2\) Column maxima: \(4, 2, 6\) min is \(2\)	M1 A1	1.1b 1.1b
Play-safe for Team A is Noel and play-safe for Team B is Qaasim	A1	1.1b
(ii) Row(maximin) \(\neq\) Col(minimax) therefore game is not stable	B1	2.4
	(4)

Notes

(i) M1: finding row minimums and column maximums – condone one error

A1: row minima and column maxima correct

A1: correct play safes for both teams

(ii) B1: row maximin (\(-2\)) \(\neq\) col minimax (2) so not stable

Mark scheme (c)

Scheme	Marks	AO
e.g. If Team A plays safe then Team B should also play their play-safe option which is Qaasim as by playing Qaasim they will gain 2 compared to gaining zero (if playing Paul) or losing 6 (if playing Rashid)	B1	2.4
	(1)

Notes

B1: cao (or equivalent – e.g. Qaasim because \(-2\) is the lowest value in Noel’s row) – explanation must involve consideration of values and not just (for example) a general statement that Qaasim will gain the most

graph of B’s expected gains against q from 0 to 1, with lines 6 − 10q, 2 − 2q and −2 + 8q — Mark scheme (d)

Scheme	Marks	AO
Let B play Paul with probability \(q\) and Qaasim with probability \(1 - q\)	B1	3.3
If \(A\) plays Mischa, \(B\)’s gains are \(-\left(4q + (-6)(1-q)\right) = 6 - 10q\) If \(A\) plays Noel, \(B\)’s gains are \(-\left(-2(1-q)\right) = 2 - 2q\) If \(A\) plays Olive, \(B\)’s gains are \(-\left(-6q + 2(1-q)\right) = -2 + 8q\)	M1 A1	1.1b 1.1b
	M1 A1	1.1b 1.1b
\(2 - 2q = -2 + 8q \Rightarrow q = 2/5\)	A1	1.1b
Team B should play Paul with probability 0.4 and play Qaasim with probability 0.6	A1ft	3.2a
	(7)
	(14 marks)