
Title:
24. Asymmetric information: auctions and the winner's curse

Description:
Game Theory (ECON 159)
We discuss auctions. We first distinguish two extremes: common values and private values. We hold a common value auction in class and discover the winner's curse, the winner tends to overpay. We discuss why this occurs and how to avoid it: you should bid as if you knew that your bid would win; that is, as if you knew your initial estimate of the common value was the highest. This leads you to bid much below your initial estimate. Then we discuss four forms of auction: firstprice sealedbid, secondprice sealedbid, open ascending, and open descending auctions. We discuss bidding strategies in each auction form for the case when values are private. Finally, we start to discuss which auction forms generate higher revenues for the seller, but a proper analysis of this will have to await the next course.
00:00  Chapter 1. Auctions: Common versus Private Values
08:16  Chapter 2. Auctions: Winner's Curse in the FirstPrice SealedBid Auction
42:38  Chapter 3. Auctions: Other Types of Auction
58:35  Chapter 4. Auctions: Revenue Generation in Different Types of Auctions
Complete course materials are available at the Open Yale Courses website: http://open.yale.edu/courses
This course was recorded in Fall 2007.

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 notebookjust
write down,

but don't show it to your
neighborwrite 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 auctionthis is
what we're going to study

todayis 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
notepadsyou've already written

down how many coins you think
are in the jar.

For the large jarwe'll do the
large jar firstwrite 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
isif I can just borrow Ale a

secondhere'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

theWhere'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 youI should say

half the people we
sampledoverestimated 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
coinsthe number of pennies in

the jarminus 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 estimatethis

is not a mistakepeople 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

estimateeach person forms an
estimate, which we could call

Y_iand this
Y_i we could think of

as being the truth plus noise.
So let's call it

ε_i.
Let's even put a tilde on it to

make it clearer that it's a
random term.

So for some people
ε_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
ε_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 lifethere's a true V and
then people make some estimate

of it which is essentially V
plus noisewho'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.,

ε_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
thatsomeone can do an

empirical test of thisyou
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
questionnot to bidbut 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 150it
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

wasactually it wasn't 150,
it was $5.00so 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 elsethere were
probably what,

30 other people therehas an
estimate lower than

mineeveryone elsewhat 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
valuessorry not their

valuesif 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 ideait

can't be an equilibriumfor
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

expost 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 youlet'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

accountpeople 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
yoube honest nowhow 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

twentyfourth 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 firstprice,
sealedbid 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 secondRobert

in the second one.
So a firstpriced sealedbid

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

secondprice sealedbid auction.
So what happens in a

secondprice,
sealedbid 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

auctionSo 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 isit'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 headit's
useful to think of it this

wayyou'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 secondprice

sealedbid 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
minusit'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

secondpriced,
sealedbid 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

youryou're only going to pay
the second priceso 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 showwe haven't

got time nowbut 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 firstprice
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 firstprice

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 secondprice 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 firstprice 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 secondprice 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

secondprice 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
itWho 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

saidpure private value,
absolutely independent,

my value is completely
statistically independent of

your value, and we're all
basically similarindependent,

symmetric, private valueboth
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 auctionor
any other silly old auction you

come up with,
at least it has the property

that in equilibrium,
the highest value winsall 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.