Professor Ben Polak:
Where have my jars of coins
got to?
That isn't very far.
They've only been on one row?
Well whiz them along this row
as fast as you can.
Just shake and pass now.
Everybody who has had access to
those jars, can you please write
down on your notebook--just
write down,
but don't show it to your
neighbor--write down for each of
those two jars how many coins
you think are in the small jar
and how many coins you think are
in the large jar.
How many coins you think are in
the small jar and how many coins
you think are in the large jar?
Keep passing it along.
All right, so today I want to
talk about auctions.
And just to put this in the
context of the whole class,
way back on the very first day
of the class,
we talked about different types
of people playing games.
We talked about evil gits
versus indignant angels,
and then for most of the
course,
really until this week,
we've been assuming that you
knew who it was you were
playing.
You knew your own payoffs but
you also knew whom it was you
were playing against or with.
But the feature,
the new feature of this week,
has been that we're looking at
settings where you don't
necessarily know what are the
payoffs of the other people
involved in the game or
strategic situation.
So in the signaling model we
looked at last time,
the different types of worker
had different types of payoffs
from going to get an MBA,
from going to business school,
and they yielded different
payoffs to you if you hired
them.
So we had to model the game
where you didn't quite know the
payoffs of the people you were
playing against.
Similarly, an auction--this is
what we're going to study
today--is such a setting.
Typically, in an auction,
you are competing or playing
with or against the other
bidders.
But typically you don't know
something crucial about those
other bidders.
You don't know how much they
value the good in question.
So there's a good up for sale,
and you don't know how much
they value that good.
So I want to start off by
thinking about a little bit of
the informational structure of
auctions and then we'll get into
more detail as we go along.
The first thing I wanted to
distinguish are two extremes.
At one extreme I want to talk
about "common values" and at the
other extreme I want to talk
about "private values."
So the idea of a common value
auction is that the good that is
for sale ultimately has the same
value for whoever buys it.
Now that doesn't mean they're
all going to be prepared to bid
the same amount because they may
not know what that value is.
For example,
imagine an oil well.
So there's an oil well out
there.
There's an oil reserve out
there, and different companies
are trying to estimate how much
they want to bid for the right
to draw oil out of this oil
field.
Each of them is going to make a
little practice well and get
some estimate of how much oil
there is in the well,
so they're going to bid
different amounts.
But at the end of the day what
comes out of that well is the
same for everybody.
There is just one amount of oil
in that well,
and that oil is just worth one
amount at the market price.
So that's a classic example of
a common value auction.
The value of the good for sale,
the true value if you like,
is the same for all.
We'll use the notation V to
denote this common value that
this object has.
Now, the other extreme is
private value and it's really
such an extreme it's hard to
think of good examples.
But the idea is that the value
of the good at hand,
not only is it different for
everybody,
but my valuation of this good
has no bearing whatsoever on
your value for the good,
and your value for the good has
no bearing whatsoever on my
value for the good.
So here's a case where the
value of the good,
the ultimate value of the good
in question,
not only is it different for
all, but, moreover,
it's completely idiosyncratic
and my value is irrelevant to
you.
So if you happen to buy this
good and you learn that in fact,
I valued it a lot,
that makes no difference to how
happy you feel at having bought
the good.
Now, these are extremes and
most of reality lies between.
I should give you the notation.
Let's use V_i to be
the private values where i
denotes the player in question.
These are extremes and most
things lie in between.
So we already mentioned that on
this extreme,
close to this extreme,
you could think about the oil
wells.
Oil wells are pretty much
common value goods.
There's a certain amount of oil
there and that's all there is to
it.
However, even there you could
imagine that the different firms
have different costs on
extracting that oil or these
different firms have their
machinery occupied to different
extents in other wells that
they're digging.
So even in that pure case,
that seemingly perfect example
of a common value,
it probably isn't literally a
common value.
Or these different firms have
different distances between the
wells and their refineries.
So the oil well is a good
example of something that's
close to common value but it
isn't really literally common
value, probably,
in reality.
One's tempted to say that homes
are private value,
after all, my valuation,
my happiness from living in my
house is not really affected by
how happy you would feel living
in my house.
I don't really care if you
would like to live in my house
or if you wouldn't like to live
in my house because I'm the
person living in it.
Is that right?
But there's a catch here.
What's the catch which makes
homes not literally private
value?
What's the catch?
The catch is that at some point
in time I may want to resell my
home.
The home is a durable good.
It's a consumption good,
my living in it,
that's a private value.
But it's also an investment
good, I'm going to want to
resell that home at some point
when I'm kicked out of Yale or
whatever,
and then at that point at which
I sell it, I'm going to care a
lot about how much you value it
because that's going to affect
the price that I'm going to get
at the end of the day.
So in the case of a home,
it's somewhere between a
private value and a common
value.
It's true that the consumption
part might be private value,
but the investment component
introduces common values.
So really for private values,
for pure private values,
we need to think about pure
consumption goods.
Goods that I consume,
they have no investment value,
they have no resale value.
So think about some good being
sold on eBay.
It's a cake, say.
So if I buy it,
once I've eaten it,
I can't resell it.
I can't have my cake and resell
it.
So think about pure consumption
goods over here.
And even in these pure
consumption goods I mustn't get
any psychological value out of
thinking I managed to get that
cake and you didn't.
So the private value case is
really an extreme thing,
but it turns out to be a useful
abstraction when we come to
consider things further.
Now, where have my jars got to?
So I've got certainly two rows
I can play with here.
Let's talk about this auction
for the jars.
So what we're going to do is
we're going to have people bid
for the value in the jar.
They're going to put forward a
bid.
The highest bidder is going to
win, and what they're going to
win is the amount of money in
the jar, but what they're going
to pay is their bid.
So what is that?
Is that a common value or a
private value?
That's a common value.
There is a certain amount of
money in that jar.
You don't know what it is,
but there is a certain amount
of money in that jar and that's
the common value.
So pretty much our jars of
coins lie over here.
They're probably even a purer
example than the oil well.
All right, now let me get the
first two rows of the class,
so this row here and this row
here.
All of you have now had a
chance to have a look at the
jars.
Let me just get you to write
down, without looking at each
other, write down on your
notepads--you've already written
down how many coins you think
are in the jar.
For the large jar--we'll do the
large jar first--write down your
bid.
Just so you can't cheat later
on, write down your bid.
We're playing this for real
cash, so if you win you're going
to have to pay me.
So write down what you're going
to bid.
Well I might not hold it to you
if it's too crazy,
we'll see, but at least in
principle we're playing for real
cash.
So write down your bid,
without changing your bid show
your neighbor your bid.
Now, what I'm going to do
is--if I can just borrow Ale a
second--here's some chalk.
Let me go along the row and
find out what those bids were.
Ale you want to record the bids?
So we're going to record
everybody's bid and we'll come
back and talk about it
afterwards.
Where are those jars by the way?
Let's have a look at
the--Where's the jar gone?
Whose got the large jar there?
Yeah the woman in the corner.
Hold up that large jar so that
everyone can see it.
That's the bid.
It's coins in a Sainsbury's
pesto jar.
Sainsbury's pesto turns out to
be quite good.
All right, so I won't bother
with names today.
I'm just going to get your bids.
Everyone's written down a bid.
No one's going to cheat?
So what is your bid?
Student: $4.50.
Professor Ben Polak:
$4.50.
Student: $3.00.
Professor Ben Polak:
Shout it out so everyone can
hear.
Student: $3.00.
Student: $4.00.
Student: $.99.
Professor Ben Polak: All
right, I'm going to pass this
along so?
Student: $.80.
Student: $3.80.
Professor Ben Polak:
Shout louder than that,
what was it?
Student: $3.80.
Professor Ben Polak:
$3.80, go on.
Student: $4.00.
Professor Ben Polak:
$4.00 again.
Student: $2.09.
Professor Ben Polak:
$2.09.
Student: $3.00.
Student: $1.60.
Student: $2.01.
Professor Ben Polak:
Sorry, the last one was what?
Student: $2.01.
Professor Ben Polak:
$2.01 here, that was after $1.60
though.
Student: $.89.
Professor Ben Polak:
$.89.
Student: This is for the
big jar?
Professor Ben Polak: The
big jar.
Student: $1.40.
Professor Ben Polak:
$1.40, all right.
Now we get a second row's worth
of people.
Student: $1.41.
Professor Ben Polak:
$1.41.
Student: $1.50.
Professor Ben Polak:
$1.50.
Student: $3.00.
Professor Ben Polak:
$3.00.
Student: $2.00.
Professor Ben Polak:
$2.00.
Student: $4.50.
Professor Ben Polak:
$4.50.
Student: $5.00
Professor Ben Polak:
$5.00, we're getting some high
ones now.
Student: $.01.
Professor Ben Polak:
$.01, okay.
What's wrong with my jar?
Okay.
All right, pass that along.
Student: $.80.
Professor Ben Polak:
That was an $.80.
Student: $1.50.
Professor Ben Polak:
$1.50.
Student: $1.59.
Student: $1.00.
Professor Ben Polak:
$1.00.
Student: $1.20.
Professor Ben Polak:
$1.20 and three more.
Student: $1.50.
Student: $1.50.
Student: $2.00.
Professor Ben Polak: All
right, so we have lots of bids
and the winner is?
The last one was $2.00.
I've actually forgotten how
many coins were in here.
Let me just remind myself.
This was the large jar right?
Okay now I know again.
All right, so who is our winner
there?
We've got a $4.50 here,
there's a $5.00.
Okay so here's our winner,
who's our winner?
Let's have our winner stand up
a second.
So now a round of applause for
our winner.
Now, let's talk about how
people bid, and why they bid
that amount.
Okay, so let's start with our
winner.
So why did you bid $5.00?
Student: It looked like
there could about $5.00 in
there.
Professor Ben Polak: All
right, so I've forgotten your
name, your name is?
Student: Ashley.
Professor Ben Polak: So
Ashley is saying she bid roughly
$5.00 because it looked like
there was about $5.00 in there.
Student: Plus you get
the jar.
Professor Ben Polak:
Plus you get the jar,
I'm not sure I'm throwing in
the jar.
Let's just sample a few other
people and see what they say.
What did you say again?
Student: I said $1.60
because I didn't want to over
estimate it because then I'd
have to pay you more than I'd
get.
Professor Ben Polak: All
right, so what was your
estimate?
Student: My estimate was
about $1.80 to $2.00 so I bid
under that.
Professor Ben Polak: So
your estimate was $1.80 to $2.00
and you bid around $1.60.
Person next to you?
Student: Well I guess
$3.00 and same reasoning.
I thought there would probably
be about $4.00 and then I valued
it at like $1.00.
Professor Ben Polak: All
right, so you thought there was
about $4.00 worth of coin and
you actually bid?
Student: $3.00.
Professor Ben Polak:
$3.00 all right,
so all of you actually wrote
down initially how many coins
you thought were in there,
right?
Is that right?
Let's just get some idea of the
distribution of those.
So how many people thought
there was less than $1.00 in
there?
Raise your hand:
no shame here,
just raise your hands.
How many people thought there
was between $1.00 and $1.50?
How many people thought there
was between $1.50 and $2.00?
How many people thought there
was between $2.00 and $2.50?
How many people thought there
was between $2.50 and $3.00?
How about more than $3.50?
Clearly the people who bid high
did.
So we have a whole range of
estimates there,
a wide range of estimates,
a wide range of bids.
And people are saying things
like: well, I thought there was
this many coins in there.
Maybe I shaded down a little
bit from the number of coins I
thought was in there because I
want to make some profit on
this, is that right?
That's kind of the explanations
I'm hearing from people.
What I want to suggest is
that's not a very good way to
bid in this auction.
So let's just repeat what I
think people did,
and people can contradict me if
this is wrong.
I think most people,
they shook this thing.
They weighed it a bit.
They figured out there was,
let's say, $3.50 worth of
pennies in there.
And then they said,
okay $3.50, so I'll bid $3.40,
$3.30 something like that.
So what's wrong with that?
Well first of all,
to reveal that there's
something wrong with it,
let me tell you how many coins
were in there.
In the larger jar there was
$2.07.
How many of you bid more than
$2.07?
Just raise your hands.
Quite a few of you, all right.
So what we see here is a number
of people, including our winner,
bid a lot more than the number
of coins in the jar.
What we find,
by a lot, is that the winning
bid was much,
much greater than the true
value.
This is a common phenomenon in
common value auctions.
It's such a common phenomenon
that it has a name.
The name is the "winner's
curse."
It's the winner's curse.
And the main lesson of the
first half of today is going to
be: let's figure out why there
exists a winner's curse;
let's try and avoid falling
into a winner's curse;
and maybe let's even figure out
how to do better.
So let's try and think through
why it is we fall into a
winner's curse.
So one way to think about this
is to think about naïve
bidding in this context.
So suppose people's strategy
was actually to bid their
estimate.
I know that isn't what people
did.
Most people shaded their
estimate a little bit.
But most people bid pretty
close to their estimate.
What's going to happen in that
instance is what?
People are going to bid
essentially what they think it's
worth, and we just saw that
fully half of you--I should say
half the people we
sampled--overestimated the
number of coins in there.
Is that right?
Let's just have that show of
hands.
How many people,
raise your hands again,
let's be honest,
if you thought there was more
than $2.07 in there.
So maybe roughly a half,
maybe a little less than a half
of you overestimated the number
of coins in there.
Now, what's that going to mean?
It's going to mean all of those
people who have this
overestimate are going to
overbid.
But we can be a little bit more
general and a little bit more
rigorous about this.
So let's try and be a little
bit more general.
So first of all,
let's just make sure we
understand what the payoffs are
in this auction.
The payoff in this auction is
what?
You get the true value,
you get the number of
coins--the number of pennies in
the jar--minus your bid,
if you are the highest;
and you get 0 otherwise.
I think it's straightforward,
we all understand that's what
the value is.
And what do people do?
People tried to estimate--this
is not a mistake--people tried
to estimate how many coins were
in the jar.
Now, in fact,
the true number of coins in the
jar was V which turned out be
$2.07.
But when people estimate it,
they don't get it exactly
right, neither here where you're
shaking the jar,
nor in the case of these oil
samples.
So what they actually
estimate--each person forms an
estimate, which we could call
Y_i--and this
Y_i we could think of
as being the truth plus noise.
So let's call it
Îµ_i.
Let's even put a tilde on it to
make it clearer that it's a
random term.
So for some people
Îµ_i is going to
be a positive amount,
which means they're going to
overestimate the number of coins
in the jar.
And for some people
Îµ_i is going to
be a negative amount,
which means they're going to
underestimate the number of
coins in the jar.
Everyone agree with that?
That's not a controversial
statement, everyone okay with
that?
So let's think about the
distribution of these
Y_i's.
Let's draw a picture that has
on the horizontal axis all the
different estimates that people
could form of the number of
coins in the jar.
And let's anchor this by V,
so here's V.
And here is going to be,
if you like,
the probability of getting that
estimate: so the frequency or
probability of estimating
Y_i given that
V_i is there.
So I don't know what the shape
of this distribution is but my
guess is it's kind of bell
shaped.
Is that right?
So it probably looks something
like this.
That seem plausible?
We could actually test this if
we had time.
We could actually go around all
of you and get you to report
what your estimates were,
and we could plot that
distribution and see if it is
bell shaped.
But my guess is,
it's reasonable to assume its
bell shaped.
There's some central tendency
to estimate something close to
the truth.
If I'd drawn this correctly I'd
have its highest point at V.
I haven't quite drawn it
correctly.
It's probably roughly symmetric.
Okay, so now suppose that
people's bidding strategy is
pretty much what they reported.
People are going to bid roughly
their estimate of the number of
coins in the jar.
So suppose people bid
B_i roughly equal to
Y_i.
So I know people are going to
shade a little bit,
but let's ignore that for now.
So people are bidding roughly
equal to Y_i.
So what's going to happen here?
Who's going to win?
If this is the way in which the
Y_i's emerge naturally
in life--there's a true V and
then people make some estimate
of it which is essentially V
plus noise--who's going to end
up being the winner,
the winner of the auction?
It's going to be the person who
has the highest estimate.
So if there's really a lot of
people the winner isn't going to
be the person who estimated it
correctly at V.
The winner's going to be way
out here somewhere.
The winner is going to be way
up in the right hand tail.
Why?
Because the winner will then be
the i who's Y_i is the
biggest, the maximum.
The problem with this is the
person whose Y_i is
the biggest has what?
They have the biggest error:
the person whose Y_i
is the max, i.e.,
Îµ_i is the max.
And that's exactly what
happened.
When we did estimates just now
the person who won was the
person, Ashley,
who had estimated there to be
roughly (maybe a little bit more
than) $5.00 worth of coins in
there.
So I'm guessing,
is this right,
that no one else estimated more
than $5.00 in these two rows,
is that correct?
No one estimated more than
$5.00.
So the person who had the
highest estimate bid the most,
which is pretty close to her
estimate, and that caused her to
lose money.
She ends up owing me whatever
it is, $1.93,
which I will collect
afterwards.
So the winner's curse is caused
by this.
It's caused by:
if people bid taking into
account their own estimate and
only their own estimates of the
number of coins in the jar or
the amount of oil in the oil
well,
then the winner ends up being
the person with the highest
estimate, which means the person
with the highest error.
So notice what this leads to.
On average, the winning bid is
going to be much,
much bigger than the truth.
Is that right?
The biggest error is typically
going to be way out in this
right tail and that's going to
mean people are going to lose
money.
All right, so this phenomenon
is very general because common
value auctions are very general.
I already mentioned the oil
fields.
In the early period after World
War II when the U.S.
Government started auctioning
out the rights to drill oil in
the gulf, in the Gulf of Mexico,
early on, it was observed that
these companies,
the winning companies,
the companies who won the bid
each time was losing money.
It was great for the
government, but these companies
were consistently losing money,
they were consistently
overbidding.
Be careful, it wasn't that the
companies as a whole were
overbidding.
It was that the winning bid was
over bidding:
it was the winner's curse.
Over time, companies figured
this out and they've figured out
that they shouldn't bid as much,
and this in fact went away.
But you also see this effect in
other places where naïve
bidders are involved.
So for example,
those people who have been
following the baseball free
agent market,
I think you could argue
that--someone can do an
empirical test of this--you
could argue that the winning
bids on free agents in the
baseball free agent market end
up being horrible overbids for
the same reason.
The team who has the highest
idiosyncratic estimate of the
person's value ends up hiring
that player,
but the highest idiosyncratic
value tends to be too high.
Similarly, perhaps more
importantly, if you look at
IPO's, initial public offerings
of companies,
they tend to sell too high.
The baseball one I haven't got
the data, but the IPO's we know
that there's a tendency for
IPO's,
initial public offerings of
companies, to have too high a
share price and for those shares
to fall back after awhile.
There may be a little bit of
initial enthusiasm,
but then they fall back.
Why?
Again, the people with the
highest estimates of the value
of the company end up winning
the company,
and if they're not
sophisticated about the way they
bid then they overbid.
So this is a serious problem
out there and it raises the
issue: well, how should I
correct this?
I might, in life,
be involved in an auction as a
bidder for something that has a
common value element.
How should I think about how I
should bid?
We've learned how we shouldn't
bid.
We shouldn't just bid my
estimate minus a little.
So how should we think about it?
Now, to walk us towards that
let me try and think about a
little bit more about the
information that's out there.
Let's go back to our oil well
example.
Each of these oil companies
drills a test well in the oil
field, and from this test well
each of them gets an estimate of
Y_i.
So you can imagine someone
doing a test drill into my jar
of coins, and when they do this
test drill into this jar of
coins they form an estimate
Y_i.
And suppose that your estimate
of the number of coins in the
jar or the amount of oil in the
oil well,
suppose that your particular
one is equal to 150.
Then, if I then asked you the
question--not to bid--but I
asked you the question how many
coins do you think are in the
jar.
Your answer would be 150.
That would be your best
estimate.
But suppose I then told you
that your neighbor,
let's go back to Ashley again.
So Ashley's estimate was,
let's say, it was 150--it
wasn't, but let's say it was
150.
And suppose I went to her
neighbor and asked her neighbor.
And her neighbor said:
actually, I think there's only
130 in there.
So suppose Ashley now knows
that she did a little test,
she thinks there's 150.
But she now knows that her
neighbor has done a similar test
and he thinks there's only 130.
Now what should be Ashley's
estimate of the number of coins
in the jar?
Somewhere in between;
so probably somewhere between
150 and 130, maybe about 140,
but certainly lower than 150.
Is that right?
So if I told you that someone
else had an estimate that was 20
lower than yours that would
cause you to lower your belief
about how many coins was in the
jar.
Now let's push this a little
harder.
Suppose I told you not that
your neighbor had an estimate of
130, but just that your neighbor
had an estimate that was lower
than 150.
I'm not going to tell you
exactly what your neighbor
estimates, I'm just going to
tell you that his estimate is
lower than yours.
So your initial belief was
there was 150 coins in this jar,
but now I know that my neighbor
thinks there's fewer than 150.
Do you think your belief is
still 150 or is it lower?
Who thinks it's gone up?
It hasn't gone up.
Who thinks it's gone down?
It's gone down.
I don't know exactly by how
much to pull it down,
but the fact that I know that
my neighbor has a lower estimate
than me suggests that I should
have a lower estimate than 150.
Now I'm going to tell you
something more dramatic.
Suppose I go to Ashley and say
your initial estimate
was--actually it wasn't 150,
it was $5.00--so let's do it.
So your initial estimate was
500 pennies.
And I'm not going to tell you
what your neighbor's estimate
was.
I'm not going to tell you what
your neighbor's,
neighbors estimate was.
But I'm going to tell you that
every single person in the row,
in the two rows other than you,
had an estimate lower than
$5.00.
So Ashley's estimate was $5.00,
but I'm now going to tell her
that every single person in the
room had a lower estimate than
hers.
So what I'm going to tell her
is that Y_j <
Y_i for all_
j,_ for all the
other people.
But I claim that if you tell me
that everybody else--there were
probably what,
30 other people there--has an
estimate lower than
mine--everyone else--what should
I now estimate?
What should happen to my
estimate?
It's going to come down a lot,
is that right?
If my estimate was $5.00 but I
know everybody else,
not just one person,
but everybody else had a lower
estimate,
then my guess of the number of
coins in the jar has come down a
whole lot.
But what?
But that's exactly what Ashley
knows as soon as she found out
that she's won the auction.
If people are bidding their
values--sorry not their
values--if people are bidding
their estimates,
then as soon as Ashley
discovers she's won,
she's going to say:
oh bother, I now know that my
estimate was too high.
She may say something more
extravagant than bother,
but at the very least,
she is going to say:
oh bother,
I now know that everyone else
had a lower estimate than me,
and therefore my estimate of
$5.00 is too high.
So what's going to happen is if
people start bidding their
estimates or close to their
estimates,
then once they've won,
they're going to learn exactly
this.
They're going to learn that
everyone else's estimate was
lower than theirs and they're
going to regret their choices.
It can't be a good idea--it
can't be an equilibrium--for
people to make choices which
they're going to regret if they
win.
That's crazy.
So we need to think about how
to correct for that.
So how do we correct for it?
Now let's talk about this a
little bit harder.
Each of you,
in your bidding for this jar of
coins, I claim you only really
care about how many coins were
in the jar, in one circumstance.
What's the only circumstance in
which you care at all how many
coins are in the jar?
If you win.
I claim you only care how many
coins are in the jar,
or how much oil is in the well,
if you win, if your bid is the
winning bid.
If your bid is the winning bid
what do you know?
You know that your estimate was
the highest estimate in the room
(at least if this equilibrium
has the property that bids are
increasing in estimates per se,
which is not much to expect).
So you know in this case,
you would have an estimate
Y_i that was at least
as big as Y_j for all
the other people in the room.
So where are we?
You only care how many coins
are in the jar if you win,
and if you win you know your
estimate was the highest.
So what's the relevant estimate?
The relevant estimate of the
number of coins in the jar for
you when you're bidding,
the relevant estimate is not
how many coins do I think is in
this jar, that's the naïve
thing.
The relevant estimate is:
how many coins do I think is in
this jar given my shaking of it
and given the fact that I
have won the auction,
given the supposition that I
might win the auction.
So the relevant estimate when
bidding is how many coins do I
think are there given my initial
guess,
Y_i,
and given that Y_i is
bigger than Y_j.
Now, notice this is kind of a
weird thing.
It's a counter factual thing.
I don't know at the time at
which I'm bidding,
I don't know that I'm going to
win.
But nevertheless,
I should bid as if I knew I was
going to win,
because I only care in the
circumstance in which I win.
So the way in which I should
estimate the number of coins in
the jar, and indeed,
the way in which I should bid
is, I should bid the number of
coins I would think were in the
jar if I won [correction:
less a few].
Say that again,
I should bid [correction:
fewer than]
the number of coins I would
think were in the jar if my bid
ends up being the winning bid.
So the lesson here is,
bid as if you know you win.
Now why is that a good idea?
Let's go back to this case of
now you discover you've won.
Provided you bid as if you know
you won, when you win you're not
going to be disappointed because
you already took that
information into account.
But if you bid not as if you
won, you failed to take into
account the possibility of
winning,
then winning's going to come as
a shock to you and cause regret.
So the only way to prevent this
ex-post regret,
the only way to bid optimally,
is to bid as if you know you're
going to win.
Estimate the number of coins
not on your own sample but on
the belief that your sample is
the biggest sample.
Question?
Student: I don't
understand what the difference
is between bidding,
sorry,
I don't understand what the
difference is between bidding as
if you know you win and what if
you won right?
Because if you bid,
like whenever you bid,
you're bidding the number that
you think,
oh well, I think there are this
many coins in the jar,
so if I win I don't want to bid
too many so that I don't lose,
right?
Professor Ben Polak:
Good.
Student: So how is that
different from bidding as if you
know you win versus if you won?
Professor Ben Polak:
Good question.
So how is it different to say
bidding as if I know I win?
Let me try and say it again.
So what you're going to do is
you're going to think of the
following thought experiment.
Suppose you told me I won,
now how many coins do I think
are in the jar?
Let me bid that amount.
So before we even do the bid,
let's do the following thought
experiment.
You're figuring out how many
coins you think are there.
Now I say, suppose it turns out
that your estimate's the highest
estimate, now how many coins do
you think are there?
That's what you should bid
[correction: minus a little].
I'm arguing that being told
that your estimate's the highest
is going to drag down that
estimate a long way.
But the key idea is if you bid
as if you know you win then you
won't regret winning and that's
what you want to avoid.
You want to avoid the winner's
curse.
Now let's see how that goes on,
let me swap places with Ale
again.
And let's see if we can
actually very quickly just do
one row on the second jar.
So this is the yellow mic.
So same group of people,
let me get people to shout
these out fairly quickly so that
we move on.
So these two rows,
write down your bid on the
smaller jar now.
This is your bid,
not your estimate.
Write down your bid on the
smaller jar.
As fast as we can go,
just shout out a number.
Student: $.40.
Professor Ben Polak:
$.40.
Student: $1.00.
Professor Ben Polak:
$1.00.
Student: $1.20.
Professor Ben Polak:
$1.20.
Student: $1.50.
Professor Ben Polak:
$1.50.
Student: $.98.
Professor Ben Polak:
$.98.
Student: $1.00.
Professor Ben Polak:
$1.00.
Student: $.90.
Professor Ben Polak:
$.90.
Student: $.75.
Professor Ben Polak:
$.75.
Student: $1.60.
Professor Ben Polak:
$1.60.
Student: $1.50.
Professor Ben Polak:
$1.50.
Student: $1.40.
Professor Ben Polak:
$1.40, okay I'm going to pass
this in, so I'll go to the other
side.
Keep shouting them out.
Student: $.95.
Student: $.80.
Professor Ben Polak:
$.80.
Student: $.50.
Professor Ben Polak:
$.50.
Student: $.75.
Student: $1.25.
Professor Ben Polak:
$1.25.
Student: $1.50.
Student: $1.30.
Professor Ben Polak:
$1.30.
Student: $1.25.
Professor Ben Polak:
$1.25.
Student: $1.09.
Professor Ben Polak:
$1.09.
Student: $1.15.
Professor Ben Polak:
$1.15 and?
Student: $.80.
Professor Ben Polak:
$.80, did I get everybody?
$.80 was the last one.
So okay the bids came down
partly because the jar was
smaller of course,
which is cheating.
But let me just tell you--let's
find out who the winner was
first of all.
So $1.60 is the winner and it
turns out that the number of
coins in this jar was $1.48.
So what happened here?
I think people took into
account--people did lower their
bids below the estimates.
Let's just check actually,
so who was my $1.60 person?
Stand up my $1.60 person,
so how many coins did you think
were in there?
Student: $1.60.
Professor Ben Polak:
$1.60, your name is?
Student: Robert.
Professor Ben Polak: So
Robert thought there was,
hang on, you bid $1.60,
how many coins did you think
was in there?
Student: $1.60.
Professor Ben Polak:
$1.60, I'm not doing well here
am I?
So what am I trying to convince
you to do here?
Let me find some other bidders,
so what did you bid?
Student: $1.00.
Professor Ben Polak:
$1.00, how many coins do you
think were in there?
Student: $1.25.
Professor Ben Polak:
$1.25, okay, so how many of
you--be honest now--how many of
you bid significantly below your
estimate?
So raise your hands,
how many of you bid below your
estimate?
Good, so we're learning
something.
I feel like this is the
twenty-fourth lecture,
I should be able to teach you
something.
Okay, so the idea here is:
in a common value auction you
need to shade your bid
considerably.
In fact, most of these bids,
even though you were shading
your bid considerably,
most of you didn't shade it
enough.
So even taking this into
account, even taking into
account the lesson of the day,
even with that into account,
a number of you are still
overbidding.
So what's the take away lesson?
If you're in a common value
auction, you need to bid as if
you have been told that your
estimate is the highest
estimate.
That means you need to shade
your estimate a lot.
If you don't do this you'll win
a lot of auctions,
and you'll be very unhappy.
All right, now we're not done
here, let me move forward by
moving away now just from common
value auctions.
So far we've focused on common
value auctions and we've focused
on a particular structure of
auction.
But I also want to talk about
different types of auctions
themselves because one
phenomenon you're going to be
seeing out there a lot these
days is that people run
different structures of
auctions.
And auctions are getting more
and more important in the U.S.
economy.
It used to be that auctions
were something you thought of as
a pretty rare event.
You'd see them when people were
selling cattle,
and you'd see them people were
selling art but that was pretty
much it.
But now you see auctions
everywhere.
We see auctions on eBay.
We see auctions for the
spectrum.
Pretty much everything is
auctioned these days.
So auctions are becoming
important.
In fact, at Yale this year,
we had a class solely devoted
to auctions.
We're having one day of this
class devoted to auctions,
but they had a whole 24
lectures on auctions,
it's that important.
One thing we should realize is
that there are lots of different
types of auctions.
So let's talk about four
different types of auctions.
Let's call them A, B, C, and D.
So the first type of auction is
a first-price,
sealed-bid auction.
And that's what we just did.
Everybody wrote down their bid
on a piece of paper.
And the winner was the person
with the highest bid,
and they paid their bid:
Ashley in the first case,
Jonathon in the second--Robert
in the second one.
So a first-priced sealed-bid
auction is what we just did,
and that's a typical auction
you might see,
for example,
in house sales.
Here's another type of auction
though.
So this sounds crazy,
but let me write it up anyway.
You could imagine a
second-price sealed-bid auction.
So what happens in a
second-price,
sealed-bid auction?
Everybody writes down their
bid, each player writes down
their bid.
The highest bidder gets the
goods so that's the same as
before.
But now instead of paying the
bid that they wrote down,
they pay the second highest
bid.
So the idea is the winner pays
the second bid.
The winner is the person with
the highest bid,
but they pay the second bid.
So that seems crazy doesn't it?
It seems a bit crazy.
These are sometimes called
Vickrey auctions.
And Vickrey won the Nobel
Prize, so it can't be that
crazy.
We'll come back and talk about
it.
Here's two other kinds of
auctions.
We can think about an ascending
open auction.
So this is what you all think
of when we say auction.
This is what happens at a
cattle auction or an art
auction, in which people are
shouting out bids.
One way to think about this is,
if we were auctioning off
something in the class today,
is that everybody who is still
in the auction would raise their
hands, and,
as the bid got higher and
higher, some of you would start
dropping out.
And eventually,
when there's only one hand
left, that person would win the
auction.
Is that right?
So that's a version of an
ascending open auction.
Everybody raises their hands
when the price is 0 and,
as the price goes up,
hands go down until there's
only one hand left.
But we can also imagine another
crazy auction,
which is a descending open
auction.
So how does a descending open
auction--So I should just say
this open ascending one is what
you typically see on eBay.
What's a descending open
auction?
A descending open auction is
the same idea,
almost, except instead of
starting the prices at 0 and
going up,
I'll start the price at
infinity and go down.
So now, if I start the price at
infinity, none of you have your
hands up I hope.
And then as the price goes
down, eventually one of you is
going to raise their hands,
and then you get that good and
pay that amount.
So this auction happens in the
real world.
It used to happen in a place
called Filene's Basement.
When I was a graduate student
struggling to have enough money,
there was a place in Boston
called Filene's Basement,
which would sell clothing in
this way.
So you go and pick out the
suits you needed for your
horrible job interview,
and the price would come down
each week, and you'd hope that
no one bought it before you did.
Sometimes it's called a Dutch
auction.
So here's four kinds of
auctions.
Two of them seem pretty
familiar.
So A and C sound like familiar
kind of auctions that you're
used to seeing,
and B and D seem a little
weird.
So the first thing I want to
convince you of is that B and D
are not as weird as they seem.
So let's look at these crazy
auctions.
The first thing I want to claim
is that auction D is really the
same as auction A.
So let's just remind ourselves
what those two auctions are.
Auction A is--it's exactly the
auction we just did for the
coins.
Everybody writes down their bid.
We open all the envelopes.
And the winner is the person
with the highest bid written
down, and they pay that.
The descending open auction:
nobody bids,
nobody bids,
nobody bids,
nobody bids,
suddenly somebody bids,
and then they pay that amount.
Why are those two auctions the
same?
Well, think about that
descending open auction.
During that descending open
auction, each of you may have
written down in your head--it's
useful to think of it this
way--you've written down in your
head the number at which you're
going to raise your hand.
Is that right?
So when I was a graduate
student waiting for my suit to
come down in price so I can
afford it,
and I know what the number is,
and from your point of view,
that's a sealed bid for me.
At the end of the day whoever
has that highest intended bid,
that highest sealed bid,
will end up winning the suit
and they'll pay their bid.
So from a strategic point of
view the descending open auction
and the sealed first price
auction are the same thing.
The person who has the highest
bid, the highest strategy,
wins.
And they pay their bid.
You don't get to see anyone
else's strategy until it's too
late.
So D is equal to A.
What about B and D?
So I claim that B and D are not
the same but they're very
closely related.
Sorry B and C.
So C is what?
C is our eBay auctions,
our classic auction you're all
used to playing on eBay.
And B is this slightly crazy
thing where we all write down
bids and the winner is the
person who has the highest bid
but they only pay the second
amount.
Now why am I saying that's the
same as our eBay auction in some
ways?
Well let's think about the eBay
auction.
Here you are playing the eBay
auction, so all of you have your
hands up meaning you're still
in, and the price is going up.
The price is going up over
time, and all of you have your
hands up, you're still in.
Each of you has some strategy
in mind which is what?
The strategy is when am I going
to lower my hand?
What's the highest price I'm
going to pay for this object?
So your strategy in the classic
eBay auction is:
the price at which I lower my
hand.
Who wins in that open ascending
auction?
The person who has the highest
intended bid.
The person whose hand is up at
the end is the person whose
intended bid is the highest.
But what amount do they pay?
When does the auction stop?
It stops when the second to
last hand goes down.
So if I'm the winning bid in an
open ascending auction my hand
is still up.
You don't know what I was
wiling to pay.
What I'm actually going to pay
is the price of the last person
whose hand went down.
I'm going to pay the bid of the
person whose amount that they
were going to pay is the second
highest amount.
Does that make sense?
So in an ascending open auction
the winner, the person who has
the highest intended bid,
actually pays the highest
intended bid of the second
highest player.
So an ascending open auction is
structurally very similar to
these sealed bid auctions,
which is really why the sealed
bid auctions are interesting.
Now, having said that,
they're not exactly the same,
and the reason they're not
exactly the same is that if in
fact the good for sale has
common value then we might learn
something by the fact that the
hands are up.
So the fact that people's hands
are still up in the open
auction, whereas you can't see
what people are doing in the
sealed auction,
makes these not identical,
but there's clearly a close
similarity between them.
Now, let's ask,
I guess, the question you've
been wondering about,
which is how should I bid at
eBay?
We figured out that for the
common value auction on eBay,
for example,
if the good you're buying is a
good you're later on going to
want to resell,
in that case we already know
that you should shade your
estimate of the value
considerably.
So let's go to the other
extreme.
Let's consider a private value
auction.
There's no common value here at
all and let's assume that this
auction is either second-price
sealed-bid or open ascending.
To summarize,
it's either what we call B or
what we call C.
So there you are,
you're bidding on eBay,
and it's a private value good:
there's nothing interesting
about how much anyone else
values this thing.
So what's your value?
Your value is V_i.
You might bid B_i.
So this is your value.
Your bid is B_i.
And what's your payoff?
Your payoff is V_i
minus--it's not minus
B_i.
What's it minus?
It's going to be minus
B_jbar,
and I'll say what it is in a
minute.
So your payoff is:
the value of the good to you
minus this thing I'm going to
call B_jbar (which
I'll say what it is in a minute)
if you win,
so if B_i is highest.
And it's 0 otherwise.
Is that right?
So what's B_jbar?
B_jbar is the highest
other bid.
So if my bid is the highest,
my final payoff is the value of
the good to me minus the highest
other bid, the second highest
bid, in other words.
So question,
how should I bid either on eBay
here or for that matter in a
second-priced,
sealed-bid auction:
what's the right way to bid in
these auctions?
Should I bid my value?
Should I bid under my value?
Let's have a poll.
Who thinks you should bid over
your--you're only going to pay
the second price--so who thinks
you should pay over your value?
Who thinks you should bid over
your value?
Who thinks you should bid your
value?
Who thinks you should bid under
your value?
Everyone knows this that's
good, okay good,
that's correct.
So the optimal thing to do is
bid your value.
Actually we can do better than
that, we can show--we haven't
got time now--but we can show
that bidding your value in a
second price auction is a weakly
dominant strategy.
So setting B_i =
V_i is weakly
dominant.
It's a weakly dominant
strategy:.
so it's really a very good idea.
So there's nothing subtle about
bidding on eBay if it's truly a
private value auction.
You're going to stay in until
it hits your value and then
you're going to drop out.
I'll leave proving that as an
exercise.
What about if we switch from
the second price auction,
or eBay, to a first-price
auction?
So now your payoff is
V_i minus your own bid
if you win and 0 otherwise.
So the first price auction
you're going to get your value
minus your own bid if you win,
and 0 otherwise.
Now how should you bid?
Remember this is a private
value auction,
so you don't have to worry
anymore about the winner's
curse.
But nevertheless,
how should you bid?
Should you bid more than your
value?
Should you bid the same as your
value?
Or should you bid less than
your value?
Who thinks you should bid more
than your value?
Let's have a poll.
Who thinks you should bid your
value?
Who thinks you should bid less
than your value?
Yeah, the answer is:
here you should bid less than
your value.
Let's see why.
Bid less than V_i.
Why?
Because if you bid
V_i,
even if you win the auction,
what's going to be your payoff?
0.
If you lose the auction you get
0, if you win the auction you
get 0.
If you bid less than
V_i,
if you shade your bid a bit,
then, if you win,
which will happen with some
probability, you'll make some
surplus.
So here it's flipped around.
Here it turns out that bidding
your value in the first-price
auction is weakly dominated.
All right, where are we.
We haven't gotten much time.
We want to get one more thing
out of the class,
so where are we here?
What we've argued is:
in a second-price auction
you're going to bid your value,
but the winner's only going to
pay the second price.
In a first price auction you're
going to shade your bid under
your value.
You're going to trade off two
things.
The two things you're going to
trade off are:
as you raise your bid,
you'll increase your chance of
winning the auction,
but you'll get less surplus if
you win.
So the first-price auction is a
classic trade off:
marginal benefit and marginal
cost.
The marginal benefit of raising
your bid is you increase the
probability of winning.
The marginal cost is you'll get
less surplus if you win.
But in summary,
in the second-price auction I
bid "truthfully my value," but
if I win I only pay the second
price.
In the first price auction I
bid less than my value but I pay
what I bid if I win.
That leads us to the natural
question.
Which of these two auctions,
at least in expectation,
is going to raise more money?
Let's make some assumptions.
Let's assume that it's a purely
private value environment.
And let's also assume that
these values are completely
independent, that my value is
statistically completely
independent of your value:
they're just completely
idiosyncratic.
Let's assume that we're all
kind of basically similar except
for that.
So I'm going to assume
independence,
symmetry, private values:
most simple thing you can
imagine.
Let's ask the question again.
So I'm selling the good now,
would I rather sell this as a
second-price auction in which at
least you'll bid your values but
the winner will only pay the
second value;
or would I rather sell it as a
first price auction in which
you're all going to shade your
values because of this effect of
trying to get some surplus,
but at least the winner will
actually pay you what they bid.
Which is going to generate more
revenue for me?
Let's have a poll.
Who thinks I should sell
it--Who thinks I'll get more
revenue from a second price
auction?
Who thinks I'll get more
revenue from a first price
auction?
This is the last poll of the
class.
We can surely get no
abstentions here.
Let's try it again,
no abstentions:
last poll of the class,
last poll of the whole course.
Who thinks I can expect more
revenue from a second price
auction in which people will bid
their values but I only get the
second price?
Who thinks I get more revenue
from a first price auction in
which people pay what they bid,
but they all shade their bids?
There's a slight majority of
the second price.
So here's a great theorem.
Provided we're in the setting I
said--pure private value,
absolutely independent,
my value is completely
statistically independent of
your value, and we're all
basically similar--independent,
symmetric, private value--both
of those type of auctions we
mentioned, the first price
auction and the second price
auction,
and indeed, any other kind of
auction which has the property
that in equilibrium,
the person with the highest
value ends up winning the good.
Any such auction in expectation
yields exactly the same revenue,
in expectation.
The first price auction,
the second price auction--or
any other silly old auction you
come up with,
at least it has the property
that in equilibrium,
the highest value wins--all of
them generate the same revenue
in expectation.
But to find out why,
you're going to have to take
another class in Game Theory.
We're done and I will see you
at the review session.
 
