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