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PROFESSOR: So stability
has some value,

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but it doesn't mean
that everybody's happy.

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In fact, it just
means that nobody

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can find anybody else who's
equally unhappy that they

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would want to run off with.

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So let's examine the
question of how well people

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do using the mating ritual,
or in other possible ways

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of finding stable marriages.

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So basically, we want to begin
with the question of who does

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better, the boys or the girls?

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Maybe it's a mixture.

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Maybe the boys do better,
the girls do better,

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or maybe some boys do
better than others,

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and some girls do
better than others.

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One thing to notice is we know
that the girl's suitors are

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getting better day by day, and
that sounds like the mating

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ritual might be
slanted towards them.

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Likewise the boy's
sweethearts, the ones

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that they're serenading, are
getting worse day by day,

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and that sounds like it might
be an argument for the girls

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to do better, but
that's not true.

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And the reason it's not is
that, if you think about it,

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the boys are starting off
with their first choice.

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They begin by serenading the
girl at the top of their list,

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and it's true that
day by day they

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keep going down, or staying
the same or going down,

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but they're only
sinking to, in fact,

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the best possible woman that
they could be married to.

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Let's examine that.

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So I need a definition,
which is that we'll

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say that a woman, Nicole,
is called optimal for Keith

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when she is the highest ranked
girl he can stably marry.

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So let's think about
that for a minute.

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So Keith has his preference for
different girls that he likes,

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to different degrees, and there
may be some that he likes.

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Like Keith thinks that
Angelina is terrific,

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but there's just
no way that she's

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going to wind up with
him, because she just

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ranks him very lowly, so there's
no stable set of marriages

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in which Keith can
wind up with this very

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desirable woman, Angelina.

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But if you look at all of
the sets of marriages that

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are stable, that Keith
can be involved in,

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among them Nicole is the
woman that he most likes,

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so that's what we mean by
Nicole is optimal for Keith.

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She's optimal among
the feasible women

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that he could stably
be married to.

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The claim that we're making
is that the mating ritual

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yields a set of stable
marriages, which

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is simultaneously optimal for
Keith and all the other boys

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at once.

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Now, that's a kind
of unusual thing.

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Usually when you're
optimizing, you

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figure you're
optimized for one boy,

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and it sacrifices the
optimality for the other boys,

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but that's not what happens
in the mating ritual.

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All of the boys get their
absolutely optimal spouse

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in the mating
ritual, and dually it

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turns out that all of the girls
get the worst possible spouse

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that they can get,
a pessimal spouse,

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among all possible
stable marriages.

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Well, with that
claim understood,

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let's go about
proving it, and we're

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going to prove that
the mating ritual leads

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to boy-optimal marriages
by contradiction.

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So let's suppose that Nicole
is optimal for Keith among all

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the women that
Keith could possibly

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be married to in a stable way.

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Nicole is the best.

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Suppose that Keith does
not wind up marrying

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Nicole in the mating ritual.

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So he doesn't marry Nicole
in the mating ritual.

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That means that since the
Nicole is optimal for Keith,

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he must be married
to somebody that's

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less desirable to
him than Nicole,

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so he must have crossed
Nicole off on some day.

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Let's call that his bad day.

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So on his bad day, Keith is
rejected by his optimal spouse.

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OK

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If this ever happens,
there's going

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to be some boy who has the
earliest bad day-- we may

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as well assume that it's Keith.

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So let's assume that Keith was
the earliest among the boys

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to have a bad day, that is,
a day on which he crosses off

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his optimal spouse, because
he was rejected by her.

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Well, on this bad day when
Keith crosses off Nicole,

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it's because Nicole
rejected him,

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which meant that Nicole had a
suitor that she liked better

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than Keith.

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Let's call that suitor Tom.

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So what we know is that Nicole
prefers this guy Tom to Keith

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on the day that she rejected
Keith, and he crossed her off,

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and we also know since
this is the earliest

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bad day that anybody
has, Tom has not yet

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crossed off his optimal girl.

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So what that means is that
since he's serenading Nicole,

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and she's going to wind up
rejecting Keith in favor

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of Tom, it must be the case
that Nicole is at least as

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desirable to Tom as
his optimal spouse,

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because he hasn't gotten
to his optimal spouse yet.

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He's working his
way down the list,

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and he hasn't had a bad day yet.

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So let's put these
two pieces together.

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Nicole is at least as desirable
to Tom as Tom's optimal spouse,

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and Nicole prefers
Tom to Keith, but what

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that tells us is that if I
had a set of stable marriages,

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with Nicole married
to Keith, then

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in the stable set of
marriages, of course,

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Tom is going to be married
to somebody that's,

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at best, optimal for him,
so he's married to somebody

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that he likes less than Nicole.

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And Nicole is married
to Keith, and she

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likes Tom better than
who she's married to.

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What that tells us is
that Nicole and Tom are

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a rogue couple in any stable set
of marriages where Nicole was

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married to Keith,
but that contradicts

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the fact that Nicole is supposed
to be optimal for Keith.

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There's supposed to be a
stable set of marriages where

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Nicole is married to Keith.

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So a similar argument-- it's
actually slightly easier--

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is that the mating ritual
yields a set of stable marriages

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in which all of the
girls get the worst

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possible spouse
that they can have

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in any set of stable marriages.

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So this leads to a whole
bunch of questions,

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and it turns out that
there's a very rich theory

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of stable marriages,
as I mentioned.

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First question to ask
is, well, are there

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other possible stable marriages?

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Well, one thing that
you can obviously do

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is you could switch the roles
of the boys and the girls.

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So if you switch the roles
of the boys and the girls,

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you'll get a set
of stable marriages

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that are optimal for the girls
and pessimal for the boys.

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Maybe that's fair or
you rather do that.

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So that's at least a possibility
of using the mating ritual

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to get two different
stable sets of marriages,

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unless the two happen
to be the same.

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And the question
arises, are there

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others that could exist that
the mating ritual doesn't find,

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either by choosing the boys
to act as boys or the boys

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to act as girls, and the answer
is that, in general, there

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can be many.

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As a matter of fact, if
there are N boys and girls,

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it's possible that there
could be an exponential number

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of stable marriages in N, and
that leads to the question of,

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well, which is one that
might be a better one

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to choose compared to the one
that completely favors the boys

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or completely favors girls.

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Another interesting
question that

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comes up that's an
issue that comes up

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with general protocols of
negotiation and optimization

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among multiple parties is,
does it serve anybody to lie?

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That is, instead of
following the protocol

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and always going to the-- and
the boys always serenading

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the girls that they like best,
and the girls always rejecting

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anybody that's less desirable
than their favorite suitor,

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suppose they violate
the convention and lie.

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Can they do better?

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Well, it turns out
that, of course,

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the boys in the mating
ritual aren't doing optimal,

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so they don't gain
anything by trying to lie,

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but the girls it turns
out-- it's almost the case--

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that girls can do
better by lying.

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If they conspire
together to lie,

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they can actually
force the mating ritual

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to wind up turning into
a stable set of marriages

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that's girl optimal.

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So that raises
another issue about,

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are there protocols which
are resistant to lying?

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We're not going to go
into these questions.

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We mainly wanted to
understand the stable marriage

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problem and its applications
and how to find them.

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Again, if you want to
learn more about this,

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you can look at the book
by Gusfield and Irving

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that I mentioned in
a previous video.