AS June 2018 Q2

EdexcelCurrent spec15 marksGame Theory

(a) Explain what the term ‘zero-sum game’ means. (1)

Two teams, A and B, are to face each other as part of a quiz.

There will be several rounds to the quiz with 10 points available in each round.

For each round, the two teams will each choose a team member and these two people will compete against each other until all 10 points have been awarded. The number of points that Team A can expect to gain in each round is shown in the table below.

		Team B
		Paul	Qaasim	Rashid
Team A	Mischa	5	6	3
	Noel	4	1	7
	Olive	4	5	8

The teams are each trying to maximise their number of points.

(b) State the number of points that Team B will expect to gain each round if Team A chooses Noel and Team B chooses Rashid. (1)

(d)

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

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

At the last minute, Olive becomes unavailable for selection by Team A.
Team A decides to choose its player for each round so that the probability of choosing Mischa is \(p\) and the probability of choosing Noel is \(1 - p\).

(e) Use a graphical method to find the optimal value of \(p\) for Team A and hence find the best strategy for Team A. (6)

For this value of \(p\),

(f)

(i) find the expected number of points awarded, per round, to Team A,

(ii) find the expected number of points awarded, per round, to Team B. (2)

Mark scheme (a)

Scheme	Marks	AO
The gains (or losses) made by one player are exactly balanced by the losses (or gains) made by the other player.	B1	1.2
	(1)

Notes

B1: cao - indication that either the losses of one (player) are balanced by the gains of the other (player) or that the total points scored by both (players) is zero

Scheme	Marks	AO
3	B1	1.1b
	(1)

Mark scheme (c)

Scheme	Marks	AO
e.g. if a member of team \(A\) gains \(x\) points then a member of team \(B\) gains \(10 - x\) points. Subtracting 5 from both gives \(A\): \(x - 5\) and \((10 - x) - 5 = 5 - x\). The sum is \((x - 5) + (5 - x) = 0\)	B1	2.4
	(1)

Notes

B1: correct explanation – could explain their reasoning using a numerical example

Mark scheme (d)

Scheme	Marks	AO
(i) Row minima: \(-2, -4, -1\) max is \(-1\) Column maxima: \(0, 1, 3\) min is \(0\) Play safe for Team \(A\) is Olive and for Team \(B\) is Paul	M1 A1 A1	1.1b 1.1b 1.1b
(ii) Row maximin (\(-1\)) \(\neq\) Col minimax (0) so not stable	B1	2.4
	(4)

Notes

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

A1: max (row minima) and min (column maxima) correct – dependent on all correct values of row minimums and column maximums

A1: correct play safes for both teams (Olive (O) and Paul (P))

(ii) B1: row maximin (\(-1\)) \(\neq\) col minimax (0) (so not stable)

SC for (d) and (e) for those candidates who do not consider the zero-sum game

For (d) M1 only for finding row minimums (3, 1, 4) and column maximums (5, 6, 8) – condone one error. Then allow possibility of full marks in (e) – expressions should be \(p + 4\), \(5p + 1\) and \(-4p + 7\) and the graph should lead to \(5p + 1 = -4p + 7\)

graph of A’s gains against p from 0 to 1, with lines p − 1 (from −1 to 0), 5p − 4 (from −4 to 1) and 2 − 4p (from 2 to −2) — Mark scheme (e)

Scheme	Marks	AO
(i) 13/3	B1	1.1b
(ii) 17/3	B1ft	2.2a
	(2)
	(15 marks)