## 24. Asymmetric information: auctions and the winner’s curse

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 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 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., 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

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

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