Hi, Welcome back, In this set of lectures
we are talking about mechanism design. And
the idea here, is that you want to use
models to help us design institutions, and
to also how to think about which
institution we might use. In this
particular lecture, we're going to talk
about auctions, and how we auction things
off. Now auctions are used in a lot of
settings, they're used to auction off air
waves, oil leases, there's even things
like, wine auctions. And you go to these
auctions, sometimes they have ascending
bids, where people call out prices, and
you keep bidding, until no one can bid
anymore. Other times there's sealed bid
auctions, where you just write down an
amount. There's a third type of auction,
that I'm going to talk about as well,
called a. And price auction, which has a
slightly more complicated set of rules.
And when you auction something off, your
objective is to get as much money as you
can possibly get. And so that's what we'll
talk about here, we'll talk about auctions
from the perspective, of the person who's
selling the thing off. And if you're
selling the thing off, you want to think
about, how can I make as much money as I
possibly can. So, we're gonna talk about
these three types of auctions, Ascending
bid, second price and sealed price. Now
again, an ascending bid is an auction
where, we just keep calling out prices
until an want, no one wants to stay in
anymore. A second price auction is a
sealed auction, where each person writes
down an amount. The highest bid gets it,
but they get it with the second highest
bid. Okay, So it's sorta complicated. So,
the highest bidder wins the good, but they
only pay the second highest bid. And then
finally a sealed bid auction, is everybody
submits a bid and the highest bidder gets
it, but they pay it at the amount that
they bid. 'Kay, so let's start with
ascending-bid auctions. Ascending-big
auctions are pretty simple. Individuals
keep calling out bids or there's an
auctioneer, until no one's willing to go
above a price, and whoever's bid the
highest price gets it. So let's think
about how that would work. You gotta think
about how that would work, you gotta think
about different types of behavioral
models. [inaudible] said there's, sort of,
three ways we can think about modeling
people. One is, we can think of people
being rational. The other is, we can
thinking of people following psychological
rules, having maybe some biases. And the
third thing we can think about people
being rule following. Having some sort of
heuristic or rule of thumb that they use
in different situations. So if we think
about an ascending bid auction where
somebody keeps calling out price. If I'm
rational, I'm gonna [inaudible], if I've
got some value, if it's worth $100 to me,
I'm gonna keep bidding Until it gets up to
$100. You know if, so if it's gonna sell
for 90, that's where the 100 to be, then
I'll bid 91. So I'll bid pretty much up to
my value. What about these other two? What
about psychological or rule following in
this setting? Let's first do rule
following. In this setting, a rule
following bidder might have some rule
like, I'm going to start off at half my
value, and then I'm going to go up by five
dollars, or two dollars. And there are
some that are you know, fairly sub-stated
[inaudible] about how they raise their
bids. But at the end of the day, it seems
like they're probably only going to bid up
to their value. They wouldn't bid above
their value, because then they're paying
more then it's worth to them, and they
probably wouldn't stop bidding at less
then their value. However, their rule
could determine how much they raise bids
by, and things like that. Now for
psychological bidder, here, a bunch of
things could come. It could be that their
initial bid is based on how much the
previous goods sold for, or all sorts of
things. But again, it's hard to imagine in
this ascending bid auction, someone not
bidding for something if it was going to
sell for less than they wanted it for.
It's also hard to imagine someone bidding
more than they value something for. Well,
maybe that's not that hard of it. Because
you can image, in an ascending good
action, that people could get, you know,
frenzy. They could really want to win. So,
even though something is valued at $100,
to them, it might be that if they're
winning it at 95 and then somebody else
get 105 that they did 110 just for, you
know, just for the thrill of winning. Once
I was, I was talking to a real auctioneer
recently at a charity auction. He said
that he feels he can raise the amount of
money, increase the amount of money that
you get because he gets people all
excited, and they get excited about
winning. And they forget that even though
they only want that vase for $100, they'll
pay 150, just for the thrill of winning,
for the thrill of the chase. So
psychological models in this setting could
actually lead to higher values. But let's
start out by assuming that people are
rational. So what's the outcome in a
rescinding bid auction. The good just goes
to whoever bid the most. So, fairly
straight forward, how much is that person
gonna pay? Well, they're probably gonna
pay the value of the second highest
bidder. Why's that? Because let's suppose
that one person buys at 100 and another
person buys at 70. So it starts out the
bidding is at 40, and then 50, and then
60, and then 70. And at 70, this person is
gonna drop out. So this person who buys at
100 shouldn't pay any more than 70. Yeah,
they could make a mistake, if they, if
their rule is to keep [inaudible] by ten,
they could pay 80, but most of the time,
you'd expect them to pay only a little bit
over 70, only a little bit over the value
of the second highest bidder. Okay, now
let's look a second price auction. Totally
different auction mechanism and we can
compare the two. In a second price
auction, everybody writes down a bid,
whoever is the highest bid gets it but you
pay the second highest price. So let's
suppose there's three bidders. One bidder
values it at, one puts in 90, one puts in
60, one puts in 70. So the winner is the
one that bids 90 but they only pay 70, be
cause they pay the second highest price.
Totally straightforward Well, let's think
about how you'd bid in this setting. You
could be a rational bidder, a
psychological bidder, or a rule-following
bidder. Let's focus on the rational bidder
to start with. Let's think about, how
would you rationally bid in this setting?
So let's suppose your value's 80. And
let's for a moment suppose you bid your
true value, you bid 80. But all we care
about is the highest other bid. So if the
highest other bid is 60, and you bid 80,
you're gonna get it. Right? You're going
to pay 60, so you're going to end up
winning, in a sense, $twenty. Because you
paid 60 and it was worth 80. Suppose the
highest other bid is 75. If you bid 80,
you get it, you pay 75, and so your net
gain is going to be five. So let's suppose
the highest other bid is 85. So you're no
longer the highest bid. That means
somebody else is going to get it. They're
going to pay 80. You don't get it. So your
value is zero. So your values are 25 and
zero. Well let's suppose you think, maybe
I should bid a little bit more, maybe I
should bid 90. Well if you bid 90, and the
highest other bid is 60, you're gonna get
it for 60, and so your net is gonna be
twenty. Now, why twenty? Twenty, because
you valued it at 80, and you paid 60. So
bidding 90 didn't hurt you anyway. You
know, the fact, you did ten over your bid,
your real value didn't cost you at all.
Well, suppose the highest other bid is 75.
Again you bid 90, but you only pay 75, and
so your net is five just as it was before.
But suppose the second the highest other
bid is 85 and now you bid 90. Well, now
you're gonna pay 85. You only value it at
80, so you're gonna lose five. Notice
you're worse off than you were before. Cuz
before in that case, you didn't lose
anything, and here you lose five. So it's
fairly [inaudible] in a second price
auction you don't wanna overbid. But do
you want to under bid. Suppose you bid 70.
Highest other bid is 60. You're gonna get
it for 60. So your net is gonna be twenty.
But, if the s econd highest bid is 75. And
you bid 70. And you're not gonna get it,
because they're gonna get it and they're
only gonna pay 70. And you're gonna say,
oh I wish I'da bid 80, or at least 76. So,
you're only gonna get zero, whereas if you
woulda bid 80, you'd have gotten it for 75
and you would've made $five. And finally
if the other highest other bid is 85,
you're not gonna get it anyway and your
payoff is zero. So what we see in the
second price auction, is if you tell the
truth you get 25 zero, if you over bid you
get 25 minus five, and if you underbid you
get twenty zero, zero. So the rational
bidder, In this case, should be your true
value. What about the other types of
bitters? What if you're a rule following
bitter? Well, the rule following bitter
here. Could do a lot of things. The
rule-following bidder could've. Say, will
maybe I'm shade my bid, ten percent or
they could over bid, it's hard to tell, so
rule following bidder, may not play the
optimal rule. They could, play it, over
bid or under bid. The psychological bidder
as well just going to be more variation we
don't know if people are going to tell the
truth or not but the interesting thing
about this option is, is that weather or
not the other people are irrational or not
it's still optimal for you. If your
rational to bid your true value. So the
interesting thing here is, there's no sort
of ratcheting up. Remember when you did
that race to the bottom game, if other
people were rational, then you wanted to
start taking into account their
irrationality. The interesting thing here
in the second price auction is, even if
other people are psychological, or other
people are role based, you should still
bid your true value. So what that's going
to mean is that's going to lead a general
tendency towards people being more
rational. It doesn't mean we have to
abandon the psychological and we're
following Rules for thinking about how
people behave. But it does mean that
there's probability this general
tendencies for people, over time at least
to make ration al bids. So, let's think
about this for a second. What happens in
this auction? The outcome goes to the
highest-valued bidder, and that person
pays the second-highest price. That's the
exact same thing we got in the
ascending-bid auction. Okay, now let's go
the, the sealed bid auction. This is, in
some ways, even though the simplest, it's
the most complicated. So now everybody
puts in a bid. They're all sealed, and the
highest bidder gets it, but they pay the
highest price. So if there's three
bidders, bidder one bids 90, bidder two
bids 60, bidder three bids 70. Bidder one
gets it at 90, but they, but she pays 90.
She doesn't pay 70 she pays 90. So in this
setting, it makes sense to do what? To
shade, to bid a little bit less. So if we
think about what a rational bidder should
do, that person should shade a little bit.
If we think about a psychological bidder,
that person might also shade but they
might, think, well other people are going
to bid even numbers like 75 some are going
to bid $75 and one cent. Now a rule
following bidder in this case might shade
by some fixed percentage. So think about a
rational bidder, how they should bid, it's
gonna depend on a bunch of things
including the number of other people in
the auction. Let's look at a simple case
where there's just two. And one thing we
know right away is the higher you bid, the
more likely it is you're gonna win. So you
wanna [inaudible] you wanna go under your
value. But you also wanna get, you know,
somewhat higher bids, 'cause then you're
more likely to win. So we wanna think
through how this logic plays out. So let's
do a two bidder model. And let's suppose
the value of the other bidder is a uniform
distribution between zero and one. So
remember, in a uniform distribution, it's
equally likely to be any value between
zero and one. Let's suppose the other
bidder bids her true value. So if she bids
her true value. What are the odds that you
win if you bid 60 cents? We are gonna win
if our value happens to be less than 60
cents, and that's gonna happen 60 percent
of the time. We can generalize this.
Suppose you bid some amount, B, which is
between zero and one. What are the odds
that you win? Well again, you're probably,
the odds that you win is just going to be
B. So let's formalize this. Let's suppose
the other person is bidding virtually
badly, and think about what you should do.
So V is your value, B is your bid. V minus
B is your surplus. That's how much you'd
win, if you win. So if your value is 90.
And your bid was 30, and you won, you'd
get 60. Right, that's how much you sort
of, it's the difference between your value
for the object and how much you bid, In
this case though, these values are going
to be at the interval 0,1, As are your
bids. Now B, if you bid point six, is also
your probability of winning. So if you bid
a half, the probability of winning is a
half. If you bid a quarter, the
probability of winning is a quarter. So
your expected winnings are just the
probability of winning times your surplus.
So that's just B times V minus D. All you
want to do is maximize D times V minus B.
Well if I multiply that out at B V minus B
squared. Now, if you've had calculus, all
you have to do is take the derivative to
this, with respect to D, and that's gonna
give you V minus two B equals zero, which
we've got right here, and your optimal bid
is to bid half your value. So if you think
the other person's bidding her true value,
you should bid half your value. And so
let's think about this one again. So
you're a rational bidder and you think, if
the other person's bidding her true value,
I should bid half my value. But that means
the other person should probably also be
bidding half of her value. If I'm bidding
half my value and I'm rational, she should
be bidding half her value. So let's think
of, let's suppose she's bidding half her
value, what should you do? If the other
bidder bids half her value, now if I bid B
the probability that I win is going to be
2B. Why is that? Let's think about it. So
suppose that I bid .25, If I did .25 I'm
gonna win as long a s her value is less
than .25 times two, because she's bidding
half her value. So that means I'm gonna l,
win half the time. So what we get is, the
probability that I win if I get B is gonna
be 2B, given that she's bidding half her
value. So now we just do the same
calculation. V is my value, B is my bid.
So the difference between those two is how
much I win. And now 2B is my probability
of winning. So my winnings are gonna be 2B
times VB. So if I write down that, and set
the derivative again equal to zero, what
I'm gonna get is that I again should bid V
over two. Now if you don't know how to
take derivatives, don't worry about it.
All we're doing here is we're just using a
little math to show that my [inaudible]
bid again is gonna be V over two. Well
this is great, because it says, if I bid
half my value. And she bids half her
value, we're both doing the optimal thing.
So the optimal thing for each of us in
this case, the rational thing to do, would
be to bid half her value. So what's gonna
happen is the highest value bidder is
gonna get it. And they're gonna get it at
half their value. Great, So let's look
through all three of our auctions. In the
sealed bit auction, the highest bidder
gets it at half her value. In the
ascending big auction, the highest value
bidder gets it at the second highest
value. And at the second price auction,
the highest value bidder gets it, also at
the second highest value. Notice this,
though. Half of the highest bidder's
value. If the highest bidder's value is
60, half of their value is 30, and that's
the expected value of the second highest
bidder. Why's that, remember, let's think
about it, we got this distribution of
values and their uniform. So if I bid.6
and I win, that means that the other
person's bid is somewhere in here. So
what's, if it's somewhere in there, the
expected value that would be halfway in
between, which would be my bid over two.
So half of my value, if I've got a uniform
distribution, is excatly equal to the
expected value of the second highest
bidder. So wha t we get is, all three of
these auctions seem to work about the same
way. The highest value bidder gets it, and
they get it at either the exact value of
the second highest bidder, or at the
expected value of the second highest
bidder. If since, if you're auctioning it
off, you don't know the exact value of the
second Hindspitter. All you can expect to
get is the expected value of the second
Hindspitter, so it looks like all three of
these things are the same. And in fact
they are. So there's a theorem proven by
Roger Myerson, and he, incidentally, he
won a Nobel Prize for this work. So it's a
fairly sophisticated theorem. That says if
you have rational bidders, there's a Y
class of auction mechanisms that includes
sealed bids, second price, and ascending
bid auctions, such that they get identical
expected outcomes. So the expected
outcomes in all three of those cases were
highest bidder gets it at the expected
value of the second highest bidder. And
that's what we got and it's called the
revenue equivalents here. So what the
model tells us is, it doesn't matter how
we auction things off, if voters are
rational. So this is a really powerful
theory, and here we see the value, one of
the values of models. Because we might sit
around and think, oh boy, ascending bid
auctions are better. Seal bid auctions are
better. Second price auctions are better.
What this tells us is, if we have rational
bidders, all three are equally good. But
we may not have rational bidders. We could
have psychological bidders, we could have
rule following bidders, so here's where we
take. Our model results, the revenue
equivalence term, and we then try to bring
our experience in and think something
about the bidders in the auction. So let's
suppose that we've got a bunch of really
sophisticated bidders, so these are
multinational firms bidding on oil leases.
Well in that case, we can imagine they are
probably fairly close to rational. And now
we know that. Pretty much any auction
method is going to give us the same
revenue. And so we could s ay, well maybe
it doesn't matter. Well now we may care
about things like transparency. So for
instance maybe we decide to have it be a
sealed bid auction so we can actually see
exactly how much people bid. And we know
that, none of the, since the, all of the
bidders are highly rational it's going to
be okay, we're going to get the same
revenue, and that way we'll see it. Let's
suppose instead of having some charity
auction we are auctioning off Something in
the community, just for fun, and now, we
know people maybe haven't participated in
auction before, and it's somewhat
confusing to them. And then, they may be
suffering from some psychological biases,
or they may just be following some simple
rules. Well, in those settings, when
you've got unsophisticated bidders. Let's
think about the three auctions. In the
sealed bid auction, they've gotta think
about what are the distribution of other
people's values. Well that could be really
hard for them to do and they may make all
sorts of mistakes. They may follow rules
that don't make sense. They may suffer
from psychological biases. What about the
second price auction? Where you say that
the highest bidder gets it at the second
highest price. That may be confusing to
people and they might not have any idea
how it works. So what about. The ascending
bid auction. This makes a lot of sense in
that setting, because even if people are
biased or if they are rule-following, it's
still probably going to be the case that
if the bid is lower than what they value
it, that they'll probably bid. So that
way, no one is going to. Do some silly
thing and underbid, and not get something
they want. And in addition, if we're
trying to make as much money as we can,
given that there could be some
psychological bias in that people could
just wanna win, we might even make more
money by having an ascending bid auction.
We're not gonna make more money in the
sealed bid case. So what we see is if you
have highly sophisticated people, maybe we
go with sealed bid. Or maybe we go to
second price cuz they can figure it out.
If you got unsophisticated people maybe we
go to the sending bid. For a couple
reasons. One it's easier, and the other is
maybe we get them sort of in a
psychological frenzy, and we make more
money. We have a powerful theorem, the
revenue equivalence theorem, and that
tells us it doesn't matter which auction
mechanism we do, we use, If people are
rational. But if we think about how people
actually behave, we could then start to
make some distinctions about what
institution to auction things off might
work best. And in some cases we might want
a sealed bid. And in some cases we might
want ascending. And in other cases we
might want second price. So what we've
seen here is we can write down models of
auctions and we can develop some really
profound results saying that it doesn't
matter how you auction things off provided
some conditions are met. So that's really
nice. It sort of frees us up to think
about other things. And it frees us up to
think about how are people are actually
going to behave. How much information do
they have? How sophisticated are they? How
many of them are there? And that can then,
then we can use those criteria to decide
which auctions we're going to use. As
opposed to spending our time thinking
about, well this auction is better than
this auction on purely rational grounds.
So we talked about what, why do we model.
But why do we assume even rational actors?
Remember, I said, benchmarks are good
things. Remember I said Roger Myerson
says, the one who's got the revenue
equivalent theorum, that, assuming
rational behavior's often a very good
benchmark. Well, we saw that was the case
here in options, because we see. If people
are rational, doesn't matter what
mechanism you use. Once we relax that
assumption, then the mechanism may matter.
But, now we know what criteria to use to
think about choosing among auction
mechanisms. So it's really useful. Models
are really helpful. All right. Thank you.