1
00:00:00,336 --> 00:00:01,010
PROFESSOR: OK.

2
00:00:01,010 --> 00:00:03,520
So how do we find
these stable marriages?

3
00:00:03,520 --> 00:00:05,730
Well there's out
procedure for doing it,

4
00:00:05,730 --> 00:00:08,530
which is kind of elegantly
described as a day

5
00:00:08,530 --> 00:00:13,090
by day mating ritual that the
boys and girls cooperate in.

6
00:00:13,090 --> 00:00:18,030
So let's see what
happens on the first day.

7
00:00:18,030 --> 00:00:21,180
On the morning of the
first day, each boy

8
00:00:21,180 --> 00:00:24,000
looks at his list
of girls and picks

9
00:00:24,000 --> 00:00:27,190
the one that he likes the
best, at the top of this list,

10
00:00:27,190 --> 00:00:31,150
and he goes off and serenades
her or proposes to her.

11
00:00:31,150 --> 00:00:34,700
So here we have Billy Bob and
Brad proposing to Angelina.

12
00:00:34,700 --> 00:00:37,170
That means that on
the first day Angelina

13
00:00:37,170 --> 00:00:40,170
was at the top of both Brad's
list and Billy Bob's list.

14
00:00:40,170 --> 00:00:42,380
And they're both going to
be proposing and asking

15
00:00:42,380 --> 00:00:45,260
if she's willing to marry them.

16
00:00:45,260 --> 00:00:50,110
Well in the afternoon,
each girl rejects all but

17
00:00:50,110 --> 00:00:51,310
her favorite suitor.

18
00:00:51,310 --> 00:00:54,360
So in this case, Angelina
likes Brad best of all.

19
00:00:54,360 --> 00:00:58,980
So she says to everybody else,
if you're not Brad take a hike.

20
00:00:58,980 --> 00:01:00,739
And that's what
happens at that stage.

21
00:01:00,739 --> 00:01:02,280
And then in the
evening, here we look

22
00:01:02,280 --> 00:01:04,810
at rejected boy, Billy Bob.

23
00:01:04,810 --> 00:01:09,350
A boy who's rejected crosses
off the girl who rejected him.

24
00:01:09,350 --> 00:01:12,950
So Billy Bob is going to
cross Angelina off his list.

25
00:01:12,950 --> 00:01:14,670
And then the whole
ritual is going

26
00:01:14,670 --> 00:01:16,250
to repeat starting
the next morning.

27
00:01:16,250 --> 00:01:19,250
Except now Billy Bob will
have a different woman

28
00:01:19,250 --> 00:01:21,690
at the top of his list because
Angelina's gone forever.

29
00:01:25,100 --> 00:01:29,440
This mating ritual continues
until nothing changes.

30
00:01:29,440 --> 00:01:33,160
And that's going to happen
exactly when each girl has

31
00:01:33,160 --> 00:01:34,740
at most one suitor.

32
00:01:34,740 --> 00:01:36,620
Because if she has
more than one suitor,

33
00:01:36,620 --> 00:01:39,380
she's going to reject
the less favorite ones.

34
00:01:39,380 --> 00:01:42,740
So that is the definition
of the stopping condition.

35
00:01:42,740 --> 00:01:45,510
And on that day, by
definition, no girl

36
00:01:45,510 --> 00:01:47,280
can have more than one suitor.

37
00:01:47,280 --> 00:01:50,520
So she will marry the
one suitor she has.

38
00:01:50,520 --> 00:01:54,110
And that's the definition of
the marriages that result,

39
00:01:54,110 --> 00:01:58,620
if and when, the
mating ritual stops.

40
00:01:58,620 --> 00:02:04,010
And we claim that they are
going to be stable marriages.

41
00:02:04,010 --> 00:02:06,400
Now if we think
about this process,

42
00:02:06,400 --> 00:02:07,750
it's really a state machine.

43
00:02:07,750 --> 00:02:10,169
In fact, if you think about
it, what the states are

44
00:02:10,169 --> 00:02:14,270
is the set of girls on the
boys list on any given morning.

45
00:02:14,270 --> 00:02:17,890
And then those states
evolve to a new list

46
00:02:17,890 --> 00:02:20,560
after the crossing out
happens on the next morning.

47
00:02:20,560 --> 00:02:24,290
So this is kind
of a memorable way

48
00:02:24,290 --> 00:02:27,110
to tell a story about the
transitions of a state machine.

49
00:02:27,110 --> 00:02:29,470
And we can bring our state
machine concepts to bear.

50
00:02:29,470 --> 00:02:30,928
So the first thing
we want to prove

51
00:02:30,928 --> 00:02:33,170
is that this state
machine terminates.

52
00:02:33,170 --> 00:02:36,870
That is to say, there
exists a wedding day.

53
00:02:36,870 --> 00:02:39,880
Then we want to
prove that this state

54
00:02:39,880 --> 00:02:41,350
machine is partially correct.

55
00:02:41,350 --> 00:02:46,900
And what that means that
when the machine stops,

56
00:02:46,900 --> 00:02:51,250
everyone is married and that
the marriages are stable.

57
00:02:51,250 --> 00:02:53,250
So that's our task.

58
00:02:53,250 --> 00:02:54,530
Well termination's easy.

59
00:02:54,530 --> 00:02:56,650
Because if you
look at the state,

60
00:02:56,650 --> 00:03:00,910
the state is the boys
lists on a given morning.

61
00:03:00,910 --> 00:03:04,620
And things evolve
because boys get rejected

62
00:03:04,620 --> 00:03:06,480
and they cross
girls off the list.

63
00:03:06,480 --> 00:03:09,510
So what that means, if we
take the total number of names

64
00:03:09,510 --> 00:03:13,740
remaining on the boys lists
on any given morning, that

65
00:03:13,740 --> 00:03:17,480
is a strictly decreasing and
non-negative integer valued

66
00:03:17,480 --> 00:03:18,390
variable.

67
00:03:18,390 --> 00:03:20,410
So by the well-ordering
principle,

68
00:03:20,410 --> 00:03:22,090
that's strictly decreasing.

69
00:03:22,090 --> 00:03:28,440
Well-ordered derived variable
will reach a minimum value.

70
00:03:28,440 --> 00:03:31,290
And by definition,
that's when the algorithm

71
00:03:31,290 --> 00:03:33,220
has to stop because it's
strictly increasing.

72
00:03:33,220 --> 00:03:36,890
So it can't move once
it's reached its minimum.

73
00:03:36,890 --> 00:03:39,610
So there's a wedding day.

74
00:03:39,610 --> 00:03:40,110
All right.

75
00:03:40,110 --> 00:03:43,860
So now let's examine
correctness of this procedure

76
00:03:43,860 --> 00:03:47,600
and figure out what's supposed
to happen on the wedding day.

77
00:03:47,600 --> 00:03:49,810
We want to show that
everybody's married

78
00:03:49,810 --> 00:03:51,370
and that the
marriages are stable.

79
00:03:51,370 --> 00:03:52,870
And in order to do
that, we're going

80
00:03:52,870 --> 00:03:56,050
to look at some more derived
variables and an invariant that

81
00:03:56,050 --> 00:03:58,930
explains why the
mating ritual works.

82
00:03:58,930 --> 00:04:04,480
So the first remark is that
the girls improve day by day,

83
00:04:04,480 --> 00:04:06,370
or at least they
don't get any worse.

84
00:04:06,370 --> 00:04:09,050
Namely, a girl's
favorite tomorrow

85
00:04:09,050 --> 00:04:13,340
will be at least as desirable
to her as her favorite today.

86
00:04:13,340 --> 00:04:14,790
That's for any given day.

87
00:04:14,790 --> 00:04:16,769
If you look at a girl's
favorite on this day,

88
00:04:16,769 --> 00:04:19,970
and if there is a tomorrow,
then her favorite tomorrow

89
00:04:19,970 --> 00:04:23,010
is going to be at least as
good as the one she has.

90
00:04:23,010 --> 00:04:24,450
And why is that?

91
00:04:24,450 --> 00:04:28,670
Well because today's favorite
will keep serenading her

92
00:04:28,670 --> 00:04:30,300
until he gets rejected.

93
00:04:30,300 --> 00:04:34,390
And he only gets rejected when
the girl that he's serenading

94
00:04:34,390 --> 00:04:37,520
gets yet a better suitor.

95
00:04:37,520 --> 00:04:40,370
And so the girls are
always going to improve.

96
00:04:40,370 --> 00:04:43,680
We could reformulate this
in state machine language

97
00:04:43,680 --> 00:04:46,670
by saying that the rank
of a girl's favorite,

98
00:04:46,670 --> 00:04:50,180
where she rates the boy
that's serenading her

99
00:04:50,180 --> 00:04:54,320
on her own list of preferences,
is a weekly increasing

100
00:04:54,320 --> 00:04:55,300
variable.

101
00:04:55,300 --> 00:04:59,680
It never gets any worse
from one day to the next.

102
00:04:59,680 --> 00:05:01,800
By the same reasoning
or similar reasoning,

103
00:05:01,800 --> 00:05:07,371
a boy's favorite tomorrow is
going to be no better than

104
00:05:07,371 --> 00:05:07,870
today's.

105
00:05:07,870 --> 00:05:09,130
It might be worse.

106
00:05:09,130 --> 00:05:11,795
And so it's no more-- the
woman that he's going to say,

107
00:05:11,795 --> 00:05:14,420
tomorrow is no more desirable to
him than today's, is basically

108
00:05:14,420 --> 00:05:15,920
because he works
straight down his list.

109
00:05:15,920 --> 00:05:17,430
If he hasn't been
rejected, he'll

110
00:05:17,430 --> 00:05:19,420
keep serenading the
same woman tomorrow.

111
00:05:19,420 --> 00:05:22,550
If he has been
rejected, then he's

112
00:05:22,550 --> 00:05:24,150
going to be working
on somebody lower

113
00:05:24,150 --> 00:05:26,210
on his list that's
less desirable.

114
00:05:26,210 --> 00:05:30,510
So again, the rank of a girl
on a boy, that a boy serenades,

115
00:05:30,510 --> 00:05:35,670
is a weekly decreasing derived
variable of this process.

116
00:05:35,670 --> 00:05:39,530
And these observations lead
us to an invariant that

117
00:05:39,530 --> 00:05:41,270
holds for the mating ritual.

118
00:05:41,270 --> 00:05:45,770
And the invariant is that
if you look at any girl G,

119
00:05:45,770 --> 00:05:48,200
and if there's a
boy B and G is not

120
00:05:48,200 --> 00:05:51,250
on B's list, that is G
must've been crossed off

121
00:05:51,250 --> 00:05:57,120
by B at some point, then the
invariant is that the girl

122
00:05:57,120 --> 00:06:03,360
G has a favorite that
is better than B.

123
00:06:03,360 --> 00:06:06,480
And the reason is that
again, when G rejected B,

124
00:06:06,480 --> 00:06:09,240
she had a better suitor
than B. And her suitors keep

125
00:06:09,240 --> 00:06:12,010
getting better by
the weekly increasing

126
00:06:12,010 --> 00:06:13,720
property of her suitors.

127
00:06:13,720 --> 00:06:16,810
And therefore,
she's going to have

128
00:06:16,810 --> 00:06:21,160
a suitor that she likes better
than B, whose list she's not

129
00:06:21,160 --> 00:06:21,660
on.

130
00:06:21,660 --> 00:06:23,930
This holds for all G's and B's.

131
00:06:23,930 --> 00:06:28,890
And that is an invariant
from one day to the next.

132
00:06:28,890 --> 00:06:31,130
So let's look at what
happens on the wedding day

133
00:06:31,130 --> 00:06:34,770
and use the invariant to
prove that everybody's married

134
00:06:34,770 --> 00:06:37,360
and that stability holds.

135
00:06:37,360 --> 00:06:41,670
So the first remark is that each
girl has at most one suitor,

136
00:06:41,670 --> 00:06:44,990
and we've observed that's by
definition of a wedding day.

137
00:06:44,990 --> 00:06:51,750
And now what we want to prove
is that each boy gets married.

138
00:06:51,750 --> 00:06:56,370
Well what's going on with a
boy, a boy is either married

139
00:06:56,370 --> 00:06:59,970
because he's serenading
the top woman on his list,

140
00:06:59,970 --> 00:07:03,840
or maybe all the women on his
list have been crossed off,

141
00:07:03,840 --> 00:07:06,260
then he's not
serenading anybody.

142
00:07:06,260 --> 00:07:08,961
So that's the only way
he could be not married.

143
00:07:08,961 --> 00:07:10,460
Now the reason that
this is the case

144
00:07:10,460 --> 00:07:11,980
is that there's no bigamy here.

145
00:07:11,980 --> 00:07:15,770
So boys serenade only
one girl at a time.

146
00:07:15,770 --> 00:07:22,880
So if a boy is married,
there's only one possible woman

147
00:07:22,880 --> 00:07:24,120
that he can be married to.

148
00:07:24,120 --> 00:07:27,120
And a woman's married
to one possible boy.

149
00:07:27,120 --> 00:07:29,635
So let's now put these
facts together and argue

150
00:07:29,635 --> 00:07:31,510
that everybody is married
on the wedding day.

151
00:07:31,510 --> 00:07:33,640
And the proof is
by contradiction.

152
00:07:33,640 --> 00:07:36,270
Suppose there's some boy
B that's not married.

153
00:07:36,270 --> 00:07:39,700
Well that happens exactly
when his list is empty.

154
00:07:39,700 --> 00:07:43,170
Otherwise, he'd be serenading
somebody and be married to her.

155
00:07:43,170 --> 00:07:45,670
But if his list is
empty by the invariant,

156
00:07:45,670 --> 00:07:50,100
every girl has a
suitor that she likes

157
00:07:50,100 --> 00:07:53,290
better than B, which means she's
going to be married to somebody

158
00:07:53,290 --> 00:07:55,530
that she likes better than B.

159
00:07:55,530 --> 00:07:56,591
Every girl's married.

160
00:07:56,591 --> 00:07:58,590
But there are the same
number of boys and girls.

161
00:07:58,590 --> 00:08:01,560
So in fact, given that
there's no bigamy,

162
00:08:01,560 --> 00:08:03,710
all the boys have
to be married too.

163
00:08:03,710 --> 00:08:06,610
And that settles that one.

164
00:08:06,610 --> 00:08:09,340
So the next crucial property
that we're interested in

165
00:08:09,340 --> 00:08:11,360
is stability.

166
00:08:11,360 --> 00:08:13,250
That in fact this
set of marriages,

167
00:08:13,250 --> 00:08:16,400
which must come into
being on the wedding day,

168
00:08:16,400 --> 00:08:17,630
are all stable.

169
00:08:17,630 --> 00:08:21,410
And the argument for
stability has two cases,

170
00:08:21,410 --> 00:08:25,740
both of them trivial that follow
immediately from the invariant.

171
00:08:25,740 --> 00:08:28,720
First of all, let's look
at an arbitrary boy, Bob.

172
00:08:28,720 --> 00:08:31,210
I claim that he won't
be in a rogue couple

173
00:08:31,210 --> 00:08:36,840
with case one, any girl G
that's on his final list.

174
00:08:36,840 --> 00:08:39,740
Because if a girl is
on his final list,

175
00:08:39,740 --> 00:08:43,740
then he's already married
to the best of them.

176
00:08:43,740 --> 00:08:45,600
He marries the girl at
the top of his list.

177
00:08:45,600 --> 00:08:49,600
So he's not going to
be tempted to switch--

178
00:08:49,600 --> 00:08:53,770
to be part of a rogue couple
with some girl that's still

179
00:08:53,770 --> 00:08:54,920
on his list.

180
00:08:54,920 --> 00:08:57,920
Case two is, he's not going
to be in a rogue couple

181
00:08:57,920 --> 00:08:59,590
with a girl that's
not on his list.

182
00:08:59,590 --> 00:09:02,370
Because, by the invariant,
she's married to somebody

183
00:09:02,370 --> 00:09:04,360
she likes better than him.

184
00:09:04,360 --> 00:09:07,400
So there's no available
girl either way for Bob

185
00:09:07,400 --> 00:09:08,680
to be in a rogue couple with.

186
00:09:08,680 --> 00:09:11,840
Bob, of course, is
an arbitrary boy.

187
00:09:11,840 --> 00:09:14,190
And therefore, no boy
is in a rogue couple.

188
00:09:14,190 --> 00:09:18,450
And indeed, there
are no rogue couples.