Professor Ben Polak: So
last time we left things in the
middle of a model,
which was the candidate-voter
model.
What was--I don't want to go
over the whole model again,
but just to reiterate a little
bit--what was different about
that model from what we saw
before--the main thing that was
different was that the
candidates cannot choose their
positions.
If you like,
every voter is a potential
candidate but you know the
positions of the voters.
Let me just bring out two
lessons that we left hanging
last time.
I want to just put them on the
board to make sure they're in
your notes.
So the first lesson is--one
thing we saw already last
time--is there can be lots of
different Nash Equilibrium this
model.
There are multiple possible
Nash Equilibrium in this model
and more to the point,
not all of those equilibria
have the candidates crowded at
the center.
We saw early on in the classic
Downs or median-voter model that
that model predicted crowding
the center.
This one doesn't,
and we'll come back to that in
a second.
And a second thing we saw last
time was that entry can--if you
enter on the left one affect of
entering on the left can be to
make the candidate on the right
more likely to win.
Conversely, if you enter on the
right--you're a right-wing voter
candidate and you enter,
that can make it more likely
that the left wings are--can
lead to the winner being
more--being further from your
ideal position.
Just to revisit this a little
bit, let me go back and just
illustrate those two points
again.
So I'll take a row further back
this time and get a nice full
row that we can see the whole
of.
This way, I'm going to take a
row further back so that this
time we have no confusion about
what left wing and right wing
is,
at least almost no confusion.
So let me choose this row,
this row good okay.
I'm sorry, the people in the
balcony are going to have to
imagine this.
It's a penalty being in the
balcony.
So this row.
And here's my left wing of this
row (for everyone who's in front
of you which is almost
everybody), and here's my right
wing.
Let's try and illustrate some
equilibrium we saw last time.
So, in particular,
if I can get the guy in the
blue Yale shirt to stand up a
second and the guy with his
computer to stand up a second.
Sorry.
Let's assume all the seats are
filled for now so--just to help
me out a little bit since I'm
doing this on the fly.
This, I'm going to claim is an
equilibrium.
Notice that there are two
candidates standing and notice
that they are not particularly
close to the center.
We have our sort of
middle-of-the-democratic-party
left candidates.
So I'm tempted to give you a
name but perhaps I will not.
And here's a
middle-of-the-republican-party
candidate.
And this is the election.
They're going to split the
votes equally if I've actually
chosen correctly.
So, just to observe some things
here.
First, for this to be an
equilibrium, it better be the
case that they're symmetric on
left and right.
If they're not symmetric,
then it isn't an equilibrium
because one of those candidates
is going to lose for sure and
the way we set up the model that
means that one of them will drop
out: they will deviate to drop
out.
Is that correct?
Let's also illustrate this.
So, we've already illustrated
that they're not particularly
close to the center.
Let's illustrate what I meant
by somebody on the left causing
the right wing to win.
So, if this was the election,
here we are getting closer and
closer to the election,
and suddenly one of our
left-wing guys,
so Dennis Kucinich or something
decides to enter--so let's
suppose this is Dennis Kucinich
and he enters.
If Dennis Kucinich,
our left-wing guy enters,
there might be some sort of
moral victory in entering,
but the result of this will be
that our right-wing candidate
wins;
everyone sees that now?
If these three guys are
standing, Dennis Kucinich is
going to steal some votes from
our center-left candidate and
cause our center-right candidate
to win.
Now, that may or may not be Mr.
Kucinich's intention,
but we should at least be aware
of it.
So, this is a real effect.
Go back to the 2000 election
and think about what happened
when Nadar entered and "do the
math" as it were,
which people didn't do at the
time apparently.
If you were listening to the
newspapers or if you read The
New York Times this morning
or listened to the radio this
morning,
you'll find exactly the same
debate is going on now in the
republican party.
Some member--actually it's the
other way around,
it's the right wing this time;
I suppose we should switch it
around.
So the right--some people on
the right of the republican
party are saying that if it
turns out that Giuliani,
who currently is leading in the
polls, wins the republican
nomination they will run a
third-party candidate.
Of course the debate is
actually in these terms.
If they run a right-wing
third-party candidate--so that
would be our guy over here,
this might be--this might have
payoff in terms of other things.
But in terms of the election,
it's going to result in Hilary
winning.
So we've seen these two effects.
They're very real effects.
We're not necessarily getting
people crowding the center and
we have to worry about,
when we enter,
causing the other wing to win.
This isn't-- I'm not making a
left wing/right wing politics
argument here--this is true for
both wings symmetrically.
So, let's try and bring one
more idea in here,
which is where we ended up last
time, which is just how far away
from the center can we be?
So these guys sit down a second
and let's stand up,
Mr.
Kucinich , again;
I know that isn't your real
name but never mind.
Mr.
Kucinich out here on the left
and Mr.
crazy-right-wing guy whose name
I've-- who's the most crazy
right wing guy of these
candidates?
I'm not--I'll get in trouble
whoever I name--so whoever the
most crazy right-wing candidate
you can think of.
and now we have the full
spectrum represented with just
the extremes standing.
Here we have--they're symmetric
around the center,
but I claim this is not an
equilibrium.
So the people in the balcony,
I've got the extreme right and
the extreme left standing here.
Why is this not an equilibrium?
Yeah I should--it's my fault I
should have brought the mike.
Can I have the mike?
I'm sorry my fault.
Thanks.
This one.
So Katie, why is that not an
equilibrium?
Student: Because the
person in the center could stand
up and win the majority.
Professor Ben Polak:
Exactly, because the person in
the center could stand up and
win.
Actually, it doesn't only have
to be the person exactly in the
center.
A wide array of center
candidates could deviate and win
at this point.
So, if this was the two
candidates standing,
and you imagine a third
candidate standing,
who for example,
is this gentleman,
if he was to stand,
we could--he has to be a little
bit closer to the center,
let's say this guy,
the guy in gray--fairly clearly
he's going to end up winning.
So this is the third lesson.
If the candidates are too far
apart we're going to see some
center entry,
which is going to win.
Thank you guys.
So even though there isn't this
full Downsian effect of pushing
candidates towards the center,
even though we don't have the
median-voter theorem here,
we still have part of the
intuition surviving.
The part of the intuition
that's survives is,
if the candidates are too far
apart, then the center will
enter and win.
So there is something pulling
people to the center still.
So a reasonable question here
is just how far apart in
equilibrium can the candidates
be?
We've established that we can
have two candidates and they
needn't both be at the center.
We've established that they
can't be at the extremes.
How far apart can they be?
Well, this is really just
a--it's kind of a nerdy question
to get it precisely but let's
get it precisely nevertheless.
So let's have a look,
let's use the other board.
So, here's the full extent of
our political spectrum from 0 to
1, and let me just try and
illustrate how far apart these
can be.
What I'm going to do is I'm
going to divide this into sixths
- 1/6,2/6,1/2,4/6,5/6.
So I claim--and I'll show
afterwards--I'll claim that
provided the two candidates
aren't outside of 1/6 and 5/6
that that will be an
equilibrium.
So, in particular,
if the candidates are just
inside 1/6 and just inside 5/6,
or just more than 1/6 and just
less than 5/6--so here's one of
these candidates who's standing
and here's the other one--then
we'll be okay.
Now why?
Why is that the right answer?
Anyone want to try a guess?
This was the question I sent
you over the weekend.
I'm sure you were doing other
things over the weekend but
nevertheless,
why is this answer?
Well, let's see.
What are they vulnerable too?
They're vulnerable to deviation
by somebody entering at the
center.
And if somebody enters at the
center--what would make somebody
enter at the center?
They're going to enter at the
center if they can win
basically.
So if they enter at the center
in this case,
let's see how many votes
everyone gets.
So if we enter at the center
here--here's our new
candidate--who's thinking about
entering at the center.
So he's sort of thinking about
it.
So what's his or her
calculation going to be?
Well, let's look at what would
have happened.
So all of these voters are
going to vote for our left-wing
candidate.
So these are going to vote for
the left-wing candidate.
All of these voters are going
to vote for the right-wing
candidate.
They're all closest to the
right-wing candidate.
And the middle third--that's a
third of the voters,
another third--and the middle
third (these ones here) are
going to vote for the center
candidate.
Is that right?
So I've basically divided--the
reason I divided it into sixths
is I want to put everybody at
the middle of a third.
So here the left-wing candidate
is at the middle of the left
third;
the right wing candidate is at
the middle of the right third,
and the center candidate is at
the middle of the center third
not surprisingly.
So if the center candidate
enters, if they were exactly at
1/6 and 5/6, they would split
the vote and the center
candidate would win with
probably 1/3.
Is that right?
But without worrying about that
exact case suppose that this--as
I claimed that the left-wing
candidate is just slightly to
the right of 1/6 and the right
wing candidate is just slightly
to the left of 5/6.
Now this left-wing candidate
gets a few extra votes here and
the right wing candidate gets a
few extra votes here (let's put
them a bit harder),
and you can see now that the
center candidate isn't going to
win.
Is that right?
Because the left-wing candidate
is getting slightly more than
1/3 of the votes;
the right wing candidate's
going to get slightly more than
1/3 of the votes;
so the center candidate is
going to get squeezed out.
So just a little bit of
thinking about it,
we can tell in this very simple
model the furthest apart the
left and right wing candidates
can be is at 1/6 and 5/6.
They can't go to the extremes,
but they're not pulled all the
way to the center.
So that's just a little bit of
nerdy math to confirm it,
or nerdy thinking about it.
But coming back to our lesson,
which is what I guess we care
about-- coming back to our
lessons,
the third lesson here is if the
candidates--if the two
candidates are too
extreme--where too extreme in
this model meant beyond…
meant less than 1/6 and more
than 5/6--but are too extreme,
someone in the center will
enter.
Again, if you look back in both
American History and other
countries' history,
you'll see that when candidates
are perceived to be too far
apart there's been tremendous
temptation for center parties
with third parties to establish
themselves in the center.
So again, with some biased
towards England,
this is what happened in
English History during the
Thatcher Period for example.
The Thatcher government was
perceived to be quite far on the
right.
The Labor Party at times could
be quite far on the left.
And we saw a center party set
up in between them.
Okay, so these seem to be the
three main politics lessons of
this model.
Is everyone happy with that?
I'm rushing it slightly because
we said that already last time
and we're happy with how it
works?
There's a also a Game Theory
lesson that I want to just keep
in mind here without--well,
there's a Game Theory lesson
here.
And the Game Theory lesson is
that our method of finding
equilibrium in this model,
which was what--it's guess and
check--is actually pretty
effective.
You might think that guess and
check, since it doesn't sound
like advanced mathematics,
wouldn't be such a great way of
going about solving games and
thinking about them,
and thinking about the real
world.
But actually,
guess and check did pretty well
here.
We were able to make sensible
guesses pretty quickly.
We were pretty quickly able to
figure out what was going on.
And the key here is what?
Without writing it necessarily,
the key here is:
be systematic when you're
guessing.
Make sure you've looked
everywhere.
And second, be systematic,
be careful when you're
checking.
The big error is to ignore
certain types of deviation.
In this particular model,
people very quickly realized
that one possible deviation is
for someone else to enter.
But perhaps they're a little
bit less good at spotting that
another possible type of
deviation is for somebody to
drop out.
You want to look at all
possible types of deviation.
But if you're careful,
this is a very effective
method.
So, that's all I want to say
about this politics model,
and I want to move onto a model
that perhaps has more to do with
sociology.
We'll do a little tour of the
Social Sciences here in showing
how Game Theory can apply to
each in turn.
So to do that I'm going
to--I'll put this up so you can
still read it and work on this
board.
So, we're going to play another
game this morning,
and it's going to be completely
different from the games we've
played so far but it's still
going to have lessons in it.
It's going to be another
location model,
so that's a connection to what
we've done before.
But the idea of this game is as
follows.
We're going to imagine that
there are two towns,
two possible locations,
and we'll call them East Town
and West Town.
And we're going to assume that
there are two types of people in
the world.
So, there's two types of
people, and these types of
people are tall and short.
Deliberately,
this is arbitrary,
right?
East and West seems kind of
meaningless, almost meaningless,
in the nomenclature of the
towns.
I guess there's something about
where they are.
And tall and short is for
pretty meaningless nomenclature
for the people,
except it says something about
how tall they are.
The idea here is that people
are going to choose where they
live.
Let's assume that there's lots
of people.
there's 100,000 of each type of
person.
And let's assume that each town
holds 100,000 people.
These are fairly big towns.
So, the players in this game
are going to be the people,
the 200,000 people--200,000
tall people and 200,000 short
people--but in a minute I'm
going to tell you whether you're
tall or short.
So actually,
you're going to be the players
in this game.
The strategies are going to be
a choice of whether you choose
East or West.
So these are the players and
these are going to end up being
the strategies.
Each of you is going to choose
do I want to live in the East
Town or do I want to live in
West Town?
So, as usual,
what's missing is the payoffs.
To model the payoffs,
let me first of all draw a
picture and then come back and
explain it.
So, the picture is going to
look like this.
Here's the payoff picture,
and it's a little complicated
so just bear with me for a
second.
So, on the horizontal axis I'm
going to put the number of
people of your type in the town
you end up in.
So on the horizontal axis is
the number of your type in your
town.
So the most this could possibly
be is 100,000 because that would
say everybody is the same type
as you in the town and the
lowest it could possibly be is 0
because that would say that
everyone in the town except for
you is of the other type.
Is that right?
I'm going to draw this payoff
function.
This is going to be your payoff.
This is going to be
utility--the payoff of you.
Let's assume this is Type X,
it doesn't really matter.
Let's assume this is the tall
type payoff which is going to be
symmetric for the other type as
well.
So the payoffs going to look
like this.
Be careful.
Let me draw it and then explain
it.
That goes like this and then
like this.
So the idea here--the idea is
more important than the
picture--the idea is,
if you are a minority of 1 in
your town,
so everyone else is of the
other type, you get a payoff of
0.
If you are in the majority,
and in fact,
everyone in your town is of the
same type then you get a payoff
of a 1/2.
And if it's the case that
everyone in your town--sorry,
if it's the case that your town
is exactly mixed,
so half of your town is tall
and half of your town are short,
then you get a payoff of 1.
I put a half here.
That's really the wrong thing
to put here, it should be a 1/2
of 100,000 so I guess it's
50,000.
So if it's the case that 50,000
of the people in the town are
your type and 50,000 are the
other type, then you get the
highest possible payoff which is
1.
So this is--we're going to
assume that each and every one
of you has this payoff.
Does that make sense?
So these are people--let's just
try and get the idea across in
words--these are people who
would like to live in mixed
towns,
but if they're going to live in
a town that's not mixed,
they'd rather live in a town in
which they're the majority.
Does that make sense?
They'd like to live in mixed
towns, but if the town is
not--if they had a choice
between two towns and the towns
are not evenly mixed,
then theyd prefer to be in the
majority town-- the town in
which they're the majority.
Everyone understand this?
It's important;
since we're about to play this
game, it's important you're all
on board here.
So to play this game,
I need to put down a few more
rules.
So the first rule is going to
be that the choice is
simultaneous and that's a little
unrealistic because in practice,
of course, people choose their
towns sequentially,
wherever they happen to be
moving,
but for the moment let's leave
it as that.
Second, I need to just say what
happens if too many people
choose the same town.
So if there's no room in a
town, for example,
if too many people chose East
Town then we allocate the
surplus randomly.
We ration the places randomly.
So then--we then randomize to
ration;
people understand that?
So, for example,
there's 100,000 places in East
Town, so if 150,000 people chose
East Town,
you're going to have a 2/3
chance of getting into East Town
and the rest of you is going to
be allocated to West Town.
Does that make sense?
I just need something to make
things add up.
Okay, so to do this,
I first of all need to decide
who in the class is tall,
and who in the class is short
(and I'm going to grab that mike
again).
So let me just count backwards.
I guess we'll ignore this row.
So, 1,2, 3,4,
5,6, 7, possibly 8.
Well, maybe this will work,
so up to here this row and
forwards you are short people.
You understand that?
Everyone forward of where I am
now, this row included,
is short.
The rest of you guys and the
guys in the balcony too,
who can't see me anymore,
the guys in the balcony,
you are tall.
Short.
Tall.
Okay, so how many rows of these
do you have?
One, two, three, four, five.
Seven rows here, all right.
One, two, three,
four, five, six.
We have seven rows of short
people and the rest of you are
tall and I'm going to hope I've
split that evenly.
I'm going to cheat a little
bit--in a minute,
you're going to have to choose
which town you're going to live
in and we'll do it by show of
hands.
But before we do that I'm going
to cheat a little bit by giving
each of you an initial position
--;this is irrelevant but I just
want to sort of set things up.
So what we'll do is we'll put
this row--so this row,
one, two, three,
four--these rows,
your initial position is in
East Town.
It doesn't matter,
you can move,
but your initial position is in
East Town.
So East, East,
East, East--everyone understand
that?
These three rows,
this row, this row,
and this row,
you're in West Town.
And these next four rows--so
you guys, you guys,
you guys, and you guys--let's
do it again,
this row, this row,
this row and this row you're at
East Town and--oh,
I didn't notice this
row--you're in East Town as
well,
up to here is East Town.
And the rest of you is in West
Town.
Okay, everyone understand where
they are now?
Let's do a test with the camera
watching.
So if you are currently in East
Town raise your hands.
And if you're currently in West
Town raise your hands.
So I've set you up with a
stripy pattern and notice that
I've set things up so that
slightly more of the short
people were in East Town and
slightly more of the tall
people--if I did that
right--were in West Town.
If I did that correctly--I'm
not sure I did actually--but if
I did that correctly,
slightly more of my short
people started life in East
Town, and slightly more of my
tall people started life in West
Town.
Okay, so now think about it a
second.
Remember, these are your
payoffs.
That's a suspension of
disbelief.
These are your payoffs.
And in a minute I want all the
people who are choosing East
Town to raise their hands.
I'm going to do it at a count
of three and don't cheat.
Don't look around you and see
what's going on.
You all close your eyes.
Everyone in the room close
their eyes.
I can do anything now;
I could be stripping naked.
So, you've all got your eyes
closed.
On the count of three,
anyone who's choosing East Town
should raise their hands 1,2,
3--so this is my East Town.
People can open their eyes now
and look at the East Town
distribution.
So I've got this big block of
East Town here.
I've got a little bit going on
in the wings,
a little bit of spillage here.
Let's just make--let's just
check it so those of you who
didn't raise your hands just now
are West Town.
So, West Town people raise
their hands, so we've got a
little bit of spillage here but
basically I've got West Town
over here.
Everyone understood just what
happened?
Everyone saw what happened?
Let's do it again.
That's where you are now;
let's see what happens next
time.
So once again close your eyes,
don't communicate.
Everybody who's going to choose
East Town raise their hands now.
So I've got pretty much all of
these rows now are in East Town
and a little bit of spillage
here, but basically I've got
East Town in the front;
I can't see the balcony.
What do I got in the balcony up
there?
No hands up in the balcony;
that's typical.
They live in the balcony, right?
So I've more or less got East
Town here.
Where's my West Town?
Raise your hands if you're in
West Town now.
West Town: a little bit of
spillage here but basically I've
got West Town in the back.
Let's do it one more time,
everyone understands where
things were just now?
Am I leaving the hands up long
enough?
Are they up long enough?
Okay, so let's try it again.
So East Town on the count of
three 1,2, 3.
So here's my East Town.
I'm pretty much in the--among
the East Town dwellers right
now.
West Town raise your hands 1,2,
3.
And I'm pretty much in the West
Town dwellers right now and it's
not at all stripy.
We started up with a stripy
pattern and we ended up--we
ended up how?
We ended up with pretty much
the whole front of the room
choosing East Town,
and the whole back of the room,
maybe including the balcony,
choosing West Town.
The balcony were just flowing
over.
So everyone saw what happens.
Now, can anyone say what's
happened there?
Why did we end up like that?
How did we end up like that?
We started off with a stripy
pattern.
It was a little bit off from
being kind of perfect,
because all of you,
by these preferences would
prefer to be in a town that was
exactly 50/50.
It wasn't quite at 50/50 but it
wasn't far off actually and we
pretty quickly ended up with all
these short people,
you guys are short all living
in East Town and all of you all
tall people living in West Town.
So all of you pretty much ended
up with a payoff of 1/2.
Some of you didn't,
there are a few deviants.
Who are my deviants?
Who are my guys--who are my
tall guys who end up East Town?
So these guys.
It's fine, but they end up with
a payoff close to 0.
And who are my short deviants
who ended up in West Town?
There's a few.
Not many of them at all
actually, but a few of them,
but they end up with a payoff
of 0 as well.
But pretty much everyone else
was ending up with a payoff
close to a 1/2,
and splitting down the middle
of the room.
Now, why?
What do we call that?
What do we call that process?
What's the outcome here?
Student: Segregation.
Professor Ben Polak:
Yeah, so this--say that again.
Student: Segregation
Professor Ben Polak: So
this is segregation.
Right, we ended up getting the
class to segregate on tall and
short.
It wasn't that people wanted to
segregate.
I gave you the preferences.
I told you that you have to
have these preferences,
which may have actually been
the preferences that favored
being in a mixed town,
but we very quickly settled
down to a segregated
distribution of the class.
Why?
What led to that?
Let's just talk it through a
bit.
Who has an opinion of what it
was that kind of got us there.
Yes, so Patrick shout out for
the classroom.
This doesn't speak to the class.
Student: I mean,
I think what happened is
basically because it was the
first choice is simultaneous.
Everyone in the slightly bad
situation all changed to what
was perceived to be a better
situation.
So we sort of shot past the
equilibrium where we were 50/50.
Professor Ben Polak:
Right, so we--I cheated a little
bit by starting you off away
from the 50/50 point and you
kind of went further very,
very quickly.
Now, I want to say that I think
with a longer time period to
play with I could have started
you off just random,
just choose whatever,
and if we played enough times
my guess is we would have ended
up segregated anyway.
It might not have been
segregated with East here and
West there, might have been the
other way around,
but I'm pretty sure we'd have
ended up segregated anyway.
I can't prove that because I
can't--we wouldn't have enough
time to do it.
I cheated by pushing us that
way.
So what happened was--is this
right?
People who started off in the
minority but not badly in the
minority in--who were short guys
in West Town--they moved to East
Town.
That was largely the rows here;
you guys moved to West Town.
And guys who ended up--who I
started off in the minority--I'm
not even sure it was actually a
minority--who were tall guys in
East Town you guys drifted over
to West Town.
Is that right?
Okay, we'll talk about this
more in a second,
but let's just get some ideas
down on the board.
Well, no--I don't know.
Actually, while I'm down here
let's do a little bit more work.
So before we leave this,
I mean before we leave talking
about it, let's figure out what
are the equilibria here.
So what's an equilibrium of
this game?
It's a pretty simple game.
It has lots of people,
two choices per person.
We're going to be doing guess
and check here.
So what do people think the
equilibrium is?
Let me get someone who hasn't
spoken yet.
I surely shouldn't have to cold
call here.
This is an easy one.
What's an equilibrium here?
Somebody?
What's an equilibrium here?
How about the gentleman here.
You want to have a guess at
what the equilibrium is?
Student: Everyone being
segregated.
Professor Ben Polak:
Everyone being segregated.
Let's be a bit more precise.
So I claim there's two ways
they can be segregated.
So spell out the two ways.
Student: If all of the
tall people are in West Town and
all the short people are in East
Town,
or if all the tall people are
in the West Town and the other
way around.
Professor Ben Polak:
Good and your name is?
Student: Greg.
Professor Ben Polak: So
Greg is saying there's two ways
to be segregated and each of
them seems like an equilibrium.
Let's just spell it out.
So all the tall people being in
the East is one equilibrium,
and all the tall people being
in the West is the other
equilibrium.
Both of those,
I claim--or Greg claims a bit
more carefully--Greg claims both
of those are Nash Equilibrium.
How do we check that they're
Nash Equilibrium?
I mean they are,
you're right,
but how do we check that they
are in fact equilibrium?
How do we go about checking
that they're equilibrium?
The guessing part is easy,
checking requires a bit of
thought, so how do we check?
Yeah.
Student: You check for
profitable deviation.
Professor Ben Polak: You
check for profitable deviations.
So what's a deviation here?
If all the short people are at
East Town and all the tall
people are at West Town,
what's a deviation?
A deviation--let's ignore the
capacity constraints for a
minute--a deviation is for one
of the short guys to move to
West Town.
Is that right?
So at the equilibrium that
short guy was getting a payoff
of what?
What was his or her payoff in
equilibrium?
One half.
If he or she deviates and moves
to West Town,
again ignoring the capacity
constraints,
let's assume that they can,
if that short person moves to
West Town what's their payoff
going to be?
Zero.
So that's not a profitable
deviation.
Conversely, if we did the same
from the other side for the tall
people, we'd find the same
thing.
So we've just checked that that
is an equilibrium because nobody
can deviate profitably.
So we found two equilibria here
and as people pointed out they
are segregated.
Now, I claim that there's
another equilibrium here.
What's the other equilibrium
here?
I should get the mike right
there on the other side.
So let me try and lean in here,
lean my way.
Yeah can you--thanks.
Student: If you were to
put exactly 50/50 then that
would be an equilibrium.
Professor Ben Polak:
Good, so if the crowd had split
50/50 that would also be an
equilibrium.
Right, but the key word there
is what?
The key--50/50 is the key word.
What's the other key word there?
Exact;
it really has to be exact for
this to work.
And since I'm hand waving a bit
because equilibrium are always
exact statements,
but can people see what's going
on here?
So if people split exactly
50/50, right split the class
down the middle--I start off by
splitting it this way into tall
people and short people--if I
had split the town down the
middle into East and West,
then everyone would have been
happy and they would have stayed
put.
That's this gentleman's claim
and I think that's correct.
But what's suspicious if you
like, what's worrying about that
equilibrium?
That is an equilibrium.
What's worrying about that
equilibrium, that mixed
equilibrium, that integrated
equilibrium?
Can we get the guy here?
Student: It's a weak
Nash Equilibrium because while
there's no real incentive for
you to deviate,
there's no incentive for you to
not deviate either.
Professor Ben Polak:
Good, so one thing that
distinguishes--thank you that's
very good.
So one thing that distinguishes
that equilibrium from the
equilibria that we were just
looking at is it--sometimes it's
weak,
whereas, the other ones are
strict.
What do we mean by that?
If we deviate away from this
exactly mixed equilibrium,
roughly speaking,
a little bit of hand waving
here,
but roughly speaking we're
exactly indifferent.
It's true, that's not strictly
true because I guess by
ourselves moving we're changing
the balance a little bit,
but nevertheless we know if I
smooth out the top it'll we'll
be exactly right.
So at that mixed equilibrium--I
don't want to call it a mixed
equilibrium--at that integrated
equilibrium,
I'm exactly indifferent about
where I live,
both towns look the same to me.
They're called East and West
but they have the same mixture
of tall and short people in
them.
Whereas, at the segregated
equilibria, I strictly prefer to
go to the town in which I'm the
majority.
I'm doing a strictly higher
payoff, 1/2 versus 0,
by being in the town in which
I'm in the majority.
So there's this notion of
strictness, there's also a
notion of stability here,
so again, I don't want to be
too formal here.
I want to give you an informal
idea about why we might worry
about stability.
So what do I mean by stability
here?
Why do I think that that
integrated equilibrium in which
we divide the towns equally
might not be stable?
What do we mean by that?
For the physicists this is an
easy idea, but for everyone I
think it's a fairly intuitive
idea.
Why is that not likely to be
stable?
Anybody?
Yeah here.
What's your name?
Student: Chris.
Professor Ben Polak: So
why?
Student: If even one
person deviates and all of a
sudden then everyone would
prefer to deviate to the
completely segregated position.
Professor Ben Polak:
Right, good.
So at that integrated
equilibria, if we move away from
it a little bit,
if it turns out that,
let's say, one town has 5% more
short people and the other town
has 5% more tall people,
then in some sense we're in
trouble already.
We haven't gone very far from
this nice equilibrium but
already we're in trouble because
now all of the short people are
going to prefer East Town and
all of the tall people are going
to prefer West Town.
And in a few moves we're very
quickly going to be back at
segregation again.
It's not stable in the sense
that if we were a little bit off
we're going to go a long way
off.
Again, I'm not being formal
here but the informal idea I
think is important.
Conversely, those segregated
equilibria, because they were
strict equilibria,
as the gentleman over there was
pointing out,
they're already pretty stable.
If we start close to 100% short
people in East and close to 100%
tall people in West and then we
move it a little bit away,
so I shake you up a bit and I
force you to--I reallocate a few
people and then let you play,
I claim you'll go back to that
equilibrium.
Is that right?
So, the equilibrium that is
integrated here,
it is an equilibrium,
it's only a weak equilibrium
and it's not very stable in some
sense.
Whereas, the segregated
equilibria, they're clearly
strict equilibria and they are
stable.
Let's put some of this down on
the board now,
partly because it feels weird
for me being out there,
even if it doesn't for you.
So, we've got this several
equilibria here;
we've got at least three and
we'll come back and talk about
whether there are others in a
minute.
We've got at least three
equilibria, two of them are
segregated.
So the Nash Equilibrium in this
game--we have two segregated
Nash Equilibria and these
correspond to tall in East and
short in West and vice versa.
We had a separate one which was
an integrated one which was
roughly half of each in each
town.
So there's at least three
equilibria here,
although the two segregated
ones are kind of the same,
and we argued that these
segregated equilibria were in
some sense stable and they were
strict equilibria in the sense
that you strictly preferred not
to deviate.
Whereas, these integrated
equilibria, these were actually
perhaps not stable,
and again I'm not being formal
here, so I was being a bit
careful putting it in inverted
commas,
and again, a little bit
informal but a kind of weak
equilibrium.
Now, I want to bring up one
other concept here,
which is the idea of a "tipping
point."
So this is a game that was
introduced by a guy called
Schelling.
Schelling went on to win the
Nobel Prize largely for this;
certainly in large part for
this idea.
This is a game that has a
tipping point.
There are really two stable
equilibria, the segregation of
one way and the segregation the
other way,
and in between there's a
tipping point beyond which if
you--beyond which--if you go
beyond which you go to the other
equilibrium.
I said that very badly,
but do people get the idea?
There are two strict equilibria
and if we got beyond the points
of a 50/50 mix going the other
way, we can whiz off to the
other equilibrium.
We already saw a game that had
a tipping point when we played
the investment game,
where there were two
equilibria, all invest and no
one invest, there was a natural
tipping point in that game and
the tipping point was having
exactly 90% of you invest,
the point at which you actually
would want to flip over and go
to the other equilibrium.
So this is a game that has a
tipping point.
We can push people away from
the equilibrium and they'll go
on coming back and they'll go on
coming back and go on coming
back,
but if we just push them a
little bit beyond a half,
whoops, they'll go on off to
the other equilibrium;
very dramatic change.
That seems like a rather
important idea in,
for example,
sociology.
So I've got these segregated
equilibria and I've got these
integrated equilibria and let's
just make the other obvious
remark.
Which of these equilibrium is
preferred by the population?
Would they rather be in the
integrated equilibrium,
given that these are their
payoffs, or the segregated
equilibrium?
They'd rather be in the
integrated one.
So here's a world in which
they'd like to be in
the--everybody would like to be
in the integrated equilibrium.
This is not a Prisoner's
Dilemma.
This is not a case that we've
seen before but it turns out
that you're likely to end up in
these inefficient,
less preferred by everybody,
segregated equilibria.
However, if we're subtle about
this, you might notice there's
actually a third equilibrium in
this game.
If we're a little bit nerdy,
there's actually another
equilibrium in this game.
So, I'll need to give you your
mike back, I'm sorry.
There's a guy there in gray.
Student: If everyone
chooses one town or the other,
regardless of whether they're
tall or short,
then they would be
redistributed.
Professor Ben Polak:
Good, your name is?
Student: Nick.
Professor Ben Polak:
Nick;
so Nick's pointing out there's
actually hidden in here another
equilibrium.
It doesn't sound like anything
very realistic,
but let's just focus on it a
second because I think it's an
important lesson here.
There's going to turn out to be
important lessons.
The other equilibrium
is--actually there's two of
them--is if everybody,
everyone in the room chose East
Town, what would happen?
Well, we'd have to find a way
of assigning everybody,
so what we would do is we'd
essentially randomize over the
room and half the room would be
in East Town and half the room
would be in West Town.
So there's this kind of silly
equilibrium in a sense,
in which everyone does the same
thing,
and the way in which people
actually get allocated is not by
their choice but by this detail
of the original game,
which was if there was
overcrowding we were going to
randomize people.
So, there is actually a third
equilibrium which is
all--there's two of these of
course--so all choose the same
town and get randomized.
To check that is an
equilibrium, notice that if
everyone else is choosing--if
everyone else is playing this
strategy of all choosing East
Town and allowing the
randomization device to place
you,
then you're completely happy to
do that, completely happy to do
the same thing.
So it is in fact an equilibrium.
So, this is a slightly odd
equilibrium here,
and there's immediately a Game
Theory lesson here.
This equilibrium,
sounds like something we might
see in society,
and this one's certainly worth
talking about,
it seems a natural part of the
game.
This equilibrium seems to have
nothing to do with anything
that's really in the world.
It's just arising from a
particular detail I threw into
the model at the end to make
things add up.
Is that right?
When I was setting up this
model and describing reality,
I threw this in at the end to
say we better do something just
to make things add up,
otherwise towns are going to be
overcrowded.
It wasn't that's really out
there in the real world.
It was just to make it a game,
sort of define things
carefully.
This seemingly innocent detail
of my modeling technique threw
up another equilibrium.
So there's a sort of warning
lesson here.
The lesson is,
seemingly irrelevant details of
the game, things that aren't
really attempts to capture
reality,
they're just trying to get
things through in a hurry,
can end up mattering--can
matter.
They can lead your game to give
you a prediction you really
don't believe in.
Is that right?
Well, that's a very general
lesson for those of you who are
going to go out and model things
more widely.
There's a second lesson here,
however.
The second lesson here is if in
fact, if this randomization
process was available,
if in fact it was possible for
everyone in the town to chose
East Town and then have the
local government randomize you,
then if we use the law of large
numbers, if there's 100,000--I
guess 200,000 people in all and
they're all being randomized,
we're going to end up very,
very close in the limit exactly
at integration and everyone's
going to be better off.
Is that right?
So the other kind of strange
thing here is by
randomizing--wll,
I don't want to say
randomizing,
I want to say having society
randomize for you--ended up
being better than
choosing--ended up better than,
what you might want to call
"active choice."
Here's an example of a place
where by abdicating the right to
choose my town and simply by all
choosing East,
having society randomize for
me, we ended up better off:
a slightly surprising result.
Let's try and push this a
little harder.
Okay, so let's try and draw
some lessons.
We've drawn out some Game
Theory lessons already from this
about irrelevant details and
about stability and so on.
Let's try and draw out some
other lessons here.
So, one lesson might be a
lesson in sociology.
I'm not a sociologist so I want
to be careful.
I'll put it in inverted commas.
The sociological lesson is what?
In this game,
segregation is what resulted,
at least in the stable
equilibria,
and you might be tempted if
you're an empirical sociologist,
to go around the world and say,
look I see segregation
everywhere.
I've gone from country to
country, from society to
society, and wherever I go I see
segregation.
You might be tempted to
conclude that that's because
people prefer segregation and
that might be right.
Nothing in this model disproves
that.
It might be the facts that--it
might in fact be in the case
that the reason you see
segregation in virtually every
society is because segregation
is preferred.
I'm not ruling that out here
I'm just raising another
possibility.
What's the other possibility?
The other possibility is it
could be that preferences are
like this, roughly speaking.
People don't actually prefer
segregation, but when all people
act in their own interest you
end up with segregation anyway.
So, the fact that that we're
seeing segregation in this model
does not imply that there's a
preference for segregation.
It doesn't rule it out,
of course, but you can't
conclude, just because you see
segregation everywhere that
necessarily people want
segregation.
Let me just take that outside
of the context of segregation,
more generally.
If you see a social phenomenon
in society after society,
after society whether they are
anthropologists going across
societies or a historian going
through societies in historical
time,
and you see the same phenomenon
in each of these societies which
results from the choices of
thousands of different people,
you can't conclude from the
fact that you see it in all of
these societies that those
people prefer it.
All you know is that each of
their individual choices add up
to this social outcome that may
or may not be something they
prefer.
In this case it's not.
That was really Schelling's big
idea.
I don't want to write that all
out but it's kind of a--it's a
huge idea.
It got him the Nobel Prize.
So you don't want to conclude
from observation straight back
to preference in these strategic
settings.
Now, this matters a little bit,
obviously in our own society
because we live in a rather
segregated society.
I mean, we needn't be quite so
coy as to talk about tall people
and short people all the time,
what we're really talking about
mostly I guess in our society is
ethnic segregation,
and we see this very
dramatically in Connecticut,
for example.
So where we live in New Haven,
at least if you go a few feet
outside of the university,
and actually where people live
in New Haven,
we see that New Haven is a
fairly integrated town.
It's not a hundred-percent
integrated town but it has a
fairly wide array of
ethnicities.
But if you go upstate in
Connecticut, if you go north,
ignoring Hartford,
if you go in the rural areas of
Connecticut,
you find something dramatically
different.
How many of you have traveled
around in rural Connecticut?
Some of you have.
So it's quite shocking for me
as a foreigner when I came here.
So New Haven and the towns
along the coast are fairly
integrated but,
if you drive into rural
Connecticut, you see something
quite different.
So this weekend,
for example,
I was up at the Durham Fair.
How many of you know what the
Durham Fair is?
Some of you know.
The Durham Fair is actually
very good.
It's a place where you
can--it's the biggest
agricultural fair in Connecticut
and you can take your kids
there,
I have a four-year old and a
two-year old,
and they discover at this fair
that in fact it is the case that
cows say moo and sheep say baa.
Who knew?
Right, that's a good thing.
So I don't want to knock the
Durham Fair.
I think it's a good thing.
But if you're wandering around
the Durham Fair and you're a
social scientist as I am I
guess,
you can't help but be struck
by--how do I put this without
getting in trouble?
You would find more ethnic
diversity at a Klan rally.
This is not--I mean this
is--that's not going to get me
out of trouble.
There's got to be some way of
saying that it doesn't get me in
trouble.
Okay, but you know what I mean,
right?
It is strikingly,
strikingly white at the Durham
Fair.
We're only--what--it's about--I
think it's 18 miles from New
Haven.
It's a 20 minute drive from
here.
It's a public event,
20 miles from New Haven.
So this is a phenomenon one
might be tempted to say,
this is because people choose,
people want-- clearly,
people are choosing to go to
the Durham Fair;
they're choosing to some extent
where they live in Connecticut.
And you might think the fact
that we see this incredibly
dramatic segregation between
activities in New Haven and
activities 20 miles away is
evidence that people might
actually prefer segregation.
And just to repeat,
I can't prove that it isn't.
But we have to be at least
aware of the possibility that
it's got nothing to do with the
preference for segregation;
it could be the preferences
look like the preferences above.
It could be that simply
thousands of activities by
thousands of individuals,
who at least would prefer to be
in the majority than the--they'd
rather be in the majority and
not the minority,
but they'd like to be
integrated--leads to incredible
segregation in the aggregate.
So let's talk about policy now.
So as I go down here I'm
getting more and more in trouble
I'm sure as a foreigner.
This is a big policy question
in the U.S.
How many of you are unaware of
the fact this is a big policy
question in the U.S.?
So at least since the 1960s
this issue about segregation
versus integration has been a
hot-button issue,
particularly in what part of
social life?
Student: Schools bussing
for segregating schools.
Professor Ben Polak:
Schooling okay,
your name sorry?
Student: Jessica.
Professor Ben Polak: So
Jessica's pointing out correctly
that since the 1960s,
this has been an incredibly hot
issue in the U.S.,
and we're talking about the
bussing debate.
How many of you have not heard
of bussing?
How many of you have heard of
bussing?
So it's slightly a policy
debate of the 1960s so people
tend to forget it,
but in the 1960s and early 70s,
people were so worried about
segregation in schools that
they--that children were bussed
from one neighborhood to another
neighborhood to go to school.
This was an incredibly
controversial issue in the U.S..
And I'm not--I don't want to
take a position on it here.
I just want to point out that
you could arrive at that policy
by thinking about this kind of
model.
In fact, this is not just an
issue of history.
If you read the newspapers last
week, including the Yale
Daily, you'll see that it's
an issue today in Connecticut.
There's a worry in Connecticut
today, particularly around
Hartford, that the schools are
still illegally segregated.
They're not in compliance with
the law.
So this continues to be an
issue.
Let me take to a slightly less
contentious area,
since that's obviously so
contentious, and talk about--and
try and relate it back to the
model a bit.
So in the model what we saw was
if everyone chose to go to East
Town or if everyone chose to go
to West Town,
there was a policy that was a
little bit like bussing going
on.
If everyone chose to go East
Town, what we ended up with was
randomized placement across the
towns.
Is that right?
That's a little bit like the
bussing policy.
It says everybody--we take away
people who we--take away
people's active choice of their
school and in place we just
randomize everybody.
So leaving aside what happened
in Connecticut,
let's think about a school,
not a very good school,
but a school that's a little
bit north of here.
So there's this school called
Harvard and so that's--Harvard
is outside Boston and at least
when I was there as a graduate
student,
this was a hot issue in Harvard.
So whereas Yale has colleges,
Harvard has "houses," but
they're the same thing.
They are the same thing;
it's just the same name for
they're--maybe they're not the
same, maybe there's some subtle
difference--but they're roughly
speaking the same thing.
I'm still getting hissed.
So these houses at Harvard,
at least until about 1990 I
think, roughly,
certainly when I was there as a
graduate student,
the way in which you ended up
at a particular house was that
you chose which house to be in.
Harvard administrators started
worrying.
So these houses started
looking--started taking on
certain characteristics.
So, for example,
there was a house--I guess
there still is--called Elliot
House,
and if you went to,
as I used to occasionally as a
graduate student,
go and eat a dinner at Elliot
House,
you'd meet all these people.
And it was kind of an odd
experience because more than
half of them had names that just
happened to be the same as the
swimming pool that had just been
built at Harvard.
Then if you went to Adams
House, where I was actually a
tutor, it was a very different
mix of people.
And an extraordinary large
portion of them were daughters
of recently deposed Latin
American dictators.
It's not exactly what you
expect to meet at random.
And then to give you a third
example, there was a house
called--it's wrong now,
Kirkland House,
is that right?
Kirkland House.
And Kirkland House was known to
be the jock house,
the athlete house,
and of course this being
Harvard that meant that everyone
there was Canadian.
So this was--this worried the
Harvard administrators.
So they were worried about it
and presumably they were worried
that little Ms.
Pinochet wasn't going to have a
chance to meet Mr.
Master Gretzky or something,
whatever it was.
So they did something about it,
so what did they do?
What did the Harvard
administrator so to change this
outcome?
Anyone know?
Shout it out.
Student: They randomized
it.
So you go in as a group but you
don't get to choose which one
you're going to go into.
Professor Ben Polak:
Right, so what happened was they
imposed randomization.
They did exactly the policy
that we described up here.
This was essentially their
policy and notice actually the
policy that they adopted,
the policy in which house
allocation was random,
was basically going to what the
policy Yale had all along.
So as happens in other cases in
education, Harvard arrived at
the Yale solution eventually.
So randomization or more
dramatic things like bussing are
policies arrived at really
because we know of the existence
of these models.
I'm not saying these are right
or wrong.;
It's not my position to say
that they're right and wrong.
I'm saying if you were going to
argue for these policies,
this might be a way in which
you might argue for these
policies.
Now, I want to bring out the
third lesson here,
and the third lesson here is
once--we've drawn--I've already
said I won't repeat it,
there's a Game Theory lesson
here about irrelevant details;
irrelevant details mattering,
but I want to bring out one
more Game Theory lesson here.
So here we've been discussing
randomization in these settings
in the following form.
In the game,
the local governments
randomized where you lived.
In the bussing experiment it
was randomly chosen who went to
which school.
I guess it was done by social
security number,
I don't know.
In Harvard and in Yale too,
there's some randomization
about which college house you
live in.
That's one way to achieve
randomization:
you could have it done
centrally.
The central administration,
this local government,
the central government,
can randomly assign people.
But in principle,
there's another way to achieve
randomization.
What's the other way you could
achieve randomization?
Both in this experiment and
elsewhere.
There was a hand way in the
back;
can we get it?
Student: Each player in
the game could randomly choose.
Professor Ben Polak:
Right, so rather than having
centralized randomization,
in principle,
you could get to the same
outcome by having individual
randomization.
So there's another possibility
here, which is individual
randomization.
In principle,
you could have everybody in the
room decide that what they're
going to do is throw a coin,
a fair coin,
each of them separately,
and if it comes up heads
they'll go to East Town and if
it comes up tails they'll go to
West Town,
and if all of them do that and
they all stick with it,
again, by the law of large
numbers we'll get pretty close
to integrated towns.
But it won't have been some
central authority doing the
randomization.
it'll be each of you
individually doing the
randomization.
Is that right?
So that's a little bit
different, and notice that if
everyone else is doing this,
if everyone else is
randomizing, the towns will in
fact be, at least
asymptotically,
the towns will be equally mixed
and you'll be happy to
randomize.
Why?
Because you'll be indifferent
whether you end up in East Town
or West Town,
so you may as well toss a coin
anyway.
So this is actually another
Nash Equilibirum.
I'm being a bit loose here but
we'll be much more precise in a
moment.
There's another Nash
Equilibrium, yet another Nash
Equilibrium, in this model,
and it isn't as silly in some
sense as having everyone going
to East Town and then having the
government randomize.
It's each person on their own
tossing a coin and deciding
where to go.
It sounds a little bit less
like central planning and more
like choice.
Now, however,
as soon as we introduced the
idea of individual randomization
we've gone a little bit beyond
where we got to in the class.
Why?
Because so far in the class,
we've talked about strategies
as the choices that are
available to you.
The choices available to you
were go to the East Town,
go to the West Town.
Or, in the numbers game we
played, it was choose a number.
Or, in the ά
β game we played,
it was choose ά
or β.
Or deciding whether to invest
or not, it was invest or don't
invest, and so on and so forth.
What we're seeing now is a new
type of strategy.
And the new type of strategy is
to randomize over your existing
strategies.
So we're introducing here a new
notion that's going to occupy us
for the rest of the week,
and the new notion is a
randomized, or as we're going to
call them, "mixed strategies."
So what is--I'll be more formal
next time, but what is a mixed
strategy?
It is a randomization over your
pure strategies.
The strategies that we've dealt
with in the course up to now,
from now on,
we're going to refer to those
as pure strategies,
they are your choices.
And we're going to expand your
actual available choices to
include all randomizations over
those.
This may seem a little weird,
so to make it seem a little bit
less weird, let's move
immediately to an example.
So, the example I want to talk
about is a game that I think is
familiar to a number of you,
but I'll put up the payoffs and
see.
So the payoffs look like this;
each person has--there's two
players--each of them has three
strategies and the payoff matrix
looks as follows:
(0,0) is down the lead diagonal
and going around it's going to
be (1,-1), (-1,1),
(-1,1), (1,-1),
(1,-1), (-1,1).
Now, without me putting anymore
letters up there what is this
game?
Right, so somebody shouted it
out, this is "rock,
paper, scissors."
So I think, if I get this
right, this better be rock,
and this better be scissors
actually and this better be
paper.
This is rock, paper, scissors.
How many of you have not heard
of rock, paper,
scissors?
Good;
the other day I was flicking
channels and ESPN had a rock,
paper, scissors contest and I
just want to know who watches
that?
So this is quite a fun game and
it occurs all over the place.
It occurs even in episodes of
the Simpsons.
So there's I guess by now
famous episode of the Simpsons
in which the kids who are I
guess Bart and Lisa,
is that right?
Somebody help me out here,
Bart and Lisa is that what
they're called?
So Bart and Lisa are playing
rock, paper, scissors for some
purpose, to allocate some prize.
And Bart is seen to think (as
you can do it in cartoons) seed
to think, "Rock,
paper, scissors,
I love playing rock,
paper, scissors.
Rock rocks.
What could possibly beat rock?"
Lisa is seen to think,
"I love playing rock,
paper, scissors;
Bart always chooses rock."
That's really a hint of where
we're going here.
That's a hint of where we're
going.
This is a very simple game
"rock, paper,
scissors," but it's pretty
obvious,
I hope, that pure strategies in
this game are probably not
enough to model this guy.
Is that right?
So, in particular,
I claim and if necessary will
prove that there is no Nash
Equilibrium in pure strategies,
in the strategies that we've
been looking at up to now.
There's no Nash Equilibrium in
which people choose pure
strategies here.
So there's no Nash Equilibrium.
From now on I'm going to use
the term pure strategies to mean
the strategies we've been
looking at so far.
So by pure strategies here,
the set of pure strategies is
equal to the set rock,
paper, and scissors;
so what we've called strategies
up to now in the class.
So now, can everyone see why
there's not going to be a Nash
Equilibrium in pure strategies?
Let's just talk it through.
So, if one person plays rock,
the best response against rock
is?
Let's try that again;
you all know.
Best response to rock is?
Paper;
but if you played paper the
best response to paper is?
Scissors;
and the best response to
scissors is?
Rock;
so everyone knows how to play
this game.
So clearly, there's not going
to be a pure strategies Nash
Equilibrium, right?
Because any attempt to look for
best responses that are best
responses to each other can lead
to a cycle.
Everyone see that?
There's no hope of finding two
pure strategies that are best
responses to each other because
of that cycle.
It's also, I claim,
pretty easy to figure out what
must be the Nash Equilibrium in
this game.
We're going to prove it in a
second, more or less,
but nevertheless I think we
actually know it.
So what is, in fact,
the Nash Equilibrium strategy
in this game?
Let's get some--you could
probably cold call somebody;
just cold call somebody at
random.
I'm sure they all know it,
so just cold call somebody.
So what's the--yeah the
gentleman in yellow--what's the
Nash Equilibrium in this game?
I'm allowing you mixed
strategies now;
I'm allowing you to
randomize--if I allow you to
randomize over your pure
strategies,
I'm allowing you to play mixed
strategies, what's the mixed
strategy we think people are
going to play?
Student: No idea.
Professor Ben Polak: No
idea;
you should be on that--I want
to play against you.
Let's have someone else picked
out, yeah.
What do we think is likely to
be the mixed strategy people are
going to play in this game in
equilibrium?
Student: No idea.
Professor Ben Polak: No
idea?
Who here is regarded as a
champion of "rock,
paper, scissors" player?
Student: Playing each
choice with one-third
probability.
Professor Ben Polak:
Thank you.
I thought that would be okay.
So I'm--people didn't play this
with their siblings?
Maybe they did and they just
lost a lot.
So I claim as a guess that the
Nash Equilibrium is each player,
both players,
each player chooses--and I'm
going to call it 1/3,1/3,1/3--in
other words,
each player is playing the
mixed strategy 1/3,1/3,1/3 which
I'm sorry I didn't catch your
name,
your name is?
Student: Moses.
Professor Ben Polak:
Which is Moses' recommended
strategy.
So each player is going to play
1/3,1/3,1/3.
So I'm actually amazed that
people didn't realize that and
again I'm sort of puzzled as
what happened in your childhood,
but never mind.
Let's figure out what in fact
is the payoff to each of the
pure strategies against it.
So what I'm going to do is I'm
going to show you that this will
in fact be a Nash Equilibrium by
showing you payoffs.
So what's the expected
payoff--to start off,
what's the expected payoff of
rock against 1/3,1/3,1/3?
So the other person's
randomizing equally over rock,
paper, scissors and I'm going
to choose rock.
What will be my payoff,
my expected payoff?
Well, I claim that with
probability 1/3 I'll meet
another rock and get 0,
and with probability of 1/3
I'll meet a scissors and get -1
and with probability of 1/3 I'll
meet paper--I said that wrong,
let me start again.
The probability 1/3 I'll meet
rock and get 0,
probability 1/3 I'll meet
scissors and get +1,
and a probability 1/3 I'll meet
paper and get -1.
Is that right this time?
So my expected payoff is what?
It's 1/3 of 0,1/3 of +1,
and 1/3 of -1 for a total of 0.
It's not hard to check that the
same is true if I chose
scissors.
If I chose scissors and the
other person is randomizing
1/3,1/3,1/3 then with
probability 1/3 I will get -1,
with probability 1/3 I will
meet another scissors and get 0,
and with probability 1/3 I'll
meet paper and get 1,
and once again it nets out at 0.
Finally, if I played paper,
and once again I'm going to
meet somebody who is in fact
randomizing a 1/3,1/3,1/3 over
rock, paper, and scissors then
with probability 1/3 I'll meet a
rock and get 1,
with probability 1/3 I'll meet
scissors and get -1,
and with probability 1/3 I'll
meet paper and get 0.
Is that correct?
Everyone happy with that?
So it will net out at 0.
So each of the pure strategies
here rock, paper,
or scissors–actually,
I did rock, scissors,
and paper, each of them when
they play against 1/3,1/3,1/3
they yield an expected payoff of
0.
What about playing the mix
itself?
What's the expected payoff of
playing--myself playing
1/3,1/3,1/3 if I play against
somebody who's playing
1/3,1/3,1/3?
Well, a 1/3 of the time I'm
going to be playing rock,
so a 1/3 of the time I'm going
to be playing rock and then I'll
get the expected payoff from
playing rock against
1/3,1/3,1/3.
What was the expected payoff
from playing rock against
1/3,1/3,1/3?
Zero.
So, 1/3 of the time I'm going
to play rock and I'm going to
get this 0.
And 1/3 of the time I'm going
to be playing scissors,
and then I'm going to get the
expected payoff from scissors
against 1/3,1/3,1/3 and what was
the expected payoff of scissors
against 1/3,1/3,1/3?
Zero again.
So 1/3 of the time I'll play
scissors and I'll get this 0.
And 1/3 of the time I'll be
playing paper,
in which case I'll get the
expected payoff of paper against
1/3,1/3,1/3 which once again is
0.
So my total expected payoff is
1/3 of 0, plus 1/3 of 0,
plus 1/3 of 0 which,
for the math phobics in the
room comes, out as 0.
So notice what we've shown here;
we've shown that if I play what
I claim is the equilibrium
strategy 1/3,1/3,1/3 against
1/3,1/3,1/3 I get 0 and if I
played any other strategy I'd
still get 0,
is that right?
So therefore playing
1/3,1/3,1/3 is indeed a best
response, albeit weakly,
it is indeed a best response
and this is in fact a Nash
Equilibrium.
So what we've shown here is in
"rock, paper,
scissors" playing 1/3,1/3,1/3
against the other guy playing
1/3,1/3,1/3 is a best response,
anything would be a best
response, but in particular it's
a best response.
So if both people play this,
one person plays a 1/3,1/3,1/3
and the other person plays
1/3,1/3,1/3, this is a Nash
Equilibrium.
So we're belaboring a point
that we all knew already:
playing 1/3,1/3,1/3 against--if
everyone plays a 1/3,1/3,1/3
that is a Nash Equilibrium.
It's a little harder to show
but it's true,
that that's the only
equilibrium in the game.
Before you go,
if any of you know the people
that run ESPN,
this is why watching this game
on TV is unlikely to be
exciting.
We'll come back and look at
other mixed strategies and more
interesting games on Wednesday.
 
