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We've seen graphs involving
boys and girls and connections

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between them in the context
of our sexual demographics

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calculation and study.

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A similar problem
comes up in terms

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of what I call stable
matching, which is again

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the issue of matching up boys
and girls in a special way

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according to some constraints.

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It turns out to have a
lot of applications which

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we'll discuss toward the end.

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Let's just look at
what the problem is.

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The setup is that there's some
number of boys, in this case 5,

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1 through 5, and an equal number
of girls labeled A through E.

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And each of the boys has
a ranking of the girls.

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Different rankings,
because different boys

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have different preferences.

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And likewise, the girls
have rankings of the boys.

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Different girls have
different preferences.

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So here, girl A likes boy 3
best and boy 5 second best,

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and boy 1 likes girls C
best and girl D least.

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So the problem,
basically, is that we

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want to get all the boys
married to all the girls.

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We want to form 5 monogamous
bisexual marriages, a boy

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and a girl, and we'd
like, in some vague way,

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to acknowledge these preferences
and satisfy as many as we can.

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I'll be more specific
about that in a minute.

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But let's just
play with that idea

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of trying to accommodate
people's preferences.

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So one way to do it
is let's just decide,

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well, we'll favor
the boys this time.

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Let's try a greedy
strategy for the boys.

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Let's just look at
the boy preferences.

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And a greedy strategy
means I'm going

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to try to give each boy the
best possible choice that he

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can make.

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So I'm going to
start off by deciding

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that let's let boy 1 have
his first choice, girl

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C. I'm going to marry them off.

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And once I've married
them off, I'll just

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stop considering 1 and C. And
now I have a reduced problem.

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I have four remaining boys
and four remaining girls.

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And proceeding in this
way-- greedy way for boys--

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I'm going to now give boy 2
his next choice that remains,

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namely girl A. And
I'll marry them off.

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And again, now 2 and A can be
eliminated from consideration.

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I continue in this
way and I wind up

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with this set of five
marriages ending with boy 5

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married to girl E.

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OK.

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Now if we look at
this set of marriages,

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there's a problem
which motivates

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the whole stable marriage
problem that we're

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going to be examining.

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Namely, we've married off boy
1 to his first choice, girl C.

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He should be happy,
but she may not be.

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And we've also married
off boy 4 to girl B.

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Now a difficulty here is that
if you look at the preferences,

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girl C actually is more
desirable to boy 4 than girl B.

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Girl C, or boy 4, likes
somebody else's wife

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better than his own.

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And what makes it really bad is
that girl C, the other person's

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wife, likes boy 4
better than her husband.

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Each of them would be better
off if they ran off together.

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They are, whether
they do or not,

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they certainly are under
tremendous pressure.

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It makes this set of
marriages unstable.

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So they're called
a rogue couple.

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When you have in
a set of marriages

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a boy and a girl who
prefer each other

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over their current
spouses, they are

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said to be a rogue couple
and a source of instability.

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So the stable
marriage problem is

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let's see if we can get
everybody married off

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and have no rogue couples.

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It'll be a stable
set of marriages.

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Now people may not
be happy, but it

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doesn't matter
because they'll never

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find anybody else that is
unhappy in the same way that

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would be willing to run off
with them and make them happier.

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So it's stable.

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And it turns out
that there always

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is a way to find a
stable set of marriages.

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A couple of ways.

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But why don't we just
play with the idea.

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Here is display of
those preferences again

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and you can stop the video and
fiddle with a piece of paper

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and see if you can come up
with a stable set of marriages

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between the boys and girls.

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We used to do this in
class in real time.

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We would give five
different boys

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a chart of preferences
of girls, and we'd

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give the five different girls
a chart of preferences of boys.

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They were not supposed
to tell each other what

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their preferences
were, but the girls

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were supposed to be interviewing
the boys and the boys

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interviewing the girls
simultaneously in a parallel,

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and try to agree to
get married and see

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whether the set of marriages
that they wound up with

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were stable.

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Most of the time they
actually did wind up

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with a stable set of
marriages, but not always.

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Just an amusing exercise.

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And it does illustrate
something about the fact

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that the procedures that we're
going to be going through

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to find stable marriages
work very nicely if you

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choose to do them in parallel.

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Anyway, there are, it turns out,
two sets of stable marriages

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that we can find in this
particular set of preferences.

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The simplest one to
understand is all the girls

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get their first choice.

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It so happens, if you look at
that chart, all of the girls

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have different
first choice boys.

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If we simply give them
their first choice,

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no girl is going to be tempted
to be part of a rogue couple

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because she's got
her first choice.

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It's absolutely stable.

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But of course, that's a
very special circumstance.

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It would be unusual that all
the girls' first choices were

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different, or likewise,
it would be unusual

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if all the boys' first
choices were different.

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But if they were,
then you instantly

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get a stable set of marriages.

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There's another stable set
that's not quite so obvious.

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And you can check that
all of these pairs

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have no instability.

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There's no rogue couples in here
when I marry 5 to A and 1 to E.

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This is a so-called "boy
optimal" set of marriages.

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It turns out that in
this set of marriages,

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every boy gets the
best possible spouse

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that he could possibly get in
any set of stable marriages.

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There's no set of
stable marriages

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in which boy 5 gets a more
desirable girl than A. There's

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no set of stable marriages in
which boy 1 gets a girl that's

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more desirable to
him than girl E.

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The sad news is that
it's simultaneously

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pessimal for the girls.

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That is, each girl is getting
their worst possible spouse

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among all sets of
stable marriages.

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We'll examine that
further in a minute.

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But let me just point out that
this is more than a puzzle.

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I mean, it's fun, and
it's a nice puzzle,

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but it's more than a puzzle.

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Because the original case where
it was studied or published

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first was in a paper by
Gale and Shapley in 1962.

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You may remember the name
David Gale from the subset game

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that we played early in
the term when we were

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practicing with set relations.

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And what they were dealing
was with the problem

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of college admissions
where students

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have rankings of colleges
that they've applied to

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and their preferences
and colleges have

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rankings of students that
have applied to them.

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And we're trying to get matching
up between college offers

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and student preferences.

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And in a circumstance
where a college made

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an offer and a student sort
of accepted, but then later

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the student got another offer
from a place they preferred

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more, and they were
changing their mind,

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and withdrawing
acceptances and so on,

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it was making everybody
crazy, the administrators

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and the students themselves.

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And the desire was,
let's get some stable set

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of offers on the table.

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And Gale and Shapley
proposed a protocol

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to get stable
marriages that would

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apply to college admissions.

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It turns out,
interestingly enough,

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that although Gale and
Shapley are credited

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with the stable
marriage solution

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that we're going
to discuss, they

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did that because they were
the first to publish it.

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But, in fact, it had
been discovered and put

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into practice at
least 20 years earlier

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by a national board whose
job was to match interns

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in hospitals, that is
graduating medical students who

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were about to start their
further clinical training

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as interns, now called
residents in modern language,

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had to be matched
up with hospitals.

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And the residents had
preferred hospitals

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that they'd like to go
to, and the hospitals

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had rankings of residents
that met their criteria.

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And again, the issue was,
how do you assign residents

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to hospitals in a stable way?

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Before they had discovered
this stability algorithm,

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it was a mess.

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Again, there's a wonderful
story in the book by Gusfield

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and Irving that is an entire
book about the stable marriage

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problem published by MIT
Press, I think in '89.

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And as a matter of
fact, I was the editor

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of the series in
which it appears.

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This stable marriage
problem turns out

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to have a lot of structure.

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And they describe a wonderful
anecdote in their preface

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about the problems
that were happening

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between the hospitals
and the residents

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and the measures that were taken
to try to achieve stability

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before they discovered
this algorithm.

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Another genuine computer
science application

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is one that was described
by Tom Leighton who

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was a co-author of the text
and then now the CEO of Akamai,

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an internet infrastructure
plumbing company that

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has some large
number of servers,

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I think 65,000 servers
in 2010 around the world.

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And it basically is
providing cached web pages

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so that they can respond
more quickly to local calls.

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They have a problem
that they get something

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like 62,000, 65,000
servers and they

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get I think 200 billion
web requests a day.

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So the web requests
are kind of acting

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like the boys and
the servers are

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kind of acting like the
girls, or the hospitals

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and the residents.

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And the web requests
have preferences

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based on proximity
and speed of server,

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and the servers have preference
based on where they're located

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and the magnitude of the web
request that they're coming to.

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And the question is, how do you
assign web requests to servers

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so that things get
done expeditiously?

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And it turns out that
the stable marriage

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method gave a satisfactory
way to accomplish

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that kind of matching.

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And in particular,
because there's such

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large numbers involved that the
stable marriage ritual, which

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we'll describe shortly,
is very amenable to being

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run in parallel.

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Another application, it
turns out, to come up

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was in matching
dance partners when

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I was teaching this course ten
years ago with a co-instructor

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who was a member of
the Indian dance team.

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She said, we could use
this, because it turns out

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that, again, there are boy and
girl partners in the dance,

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and it was constantly the
case that one boy would

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like another boy's partner
better, and vice versa.

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And they would start
pairing up and leaving

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the other people hanging
and there were bad feelings,

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and it was a source of
disruption in the society.

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There's a picture of
that Indian dance group.

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My co-instructor's
not actually there,

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but it gives you some
sense of the reality

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of the problem here at MIT.

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And she told me
that it was actually

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being used by that group.