Hi. In our final lecture on mechanism
design, we're gonna talk about how we can
use mechanisms to decide whether or not we
should fund public projects. Now to think
about a public project, what we're
thinking about is some outlay of funds to
buy something from which everybody is
gonna benefit from it. So, a private good
would be something like this sweater, or a
chair, or a pen, it's something that just
one individual is gonna use. If we think
about a public project, you think about
something like a road or a park, where
everybody can use it. The example I'm
gonna use for this particular lecture is
the example of getting a coffee machine
for an office, because if you get a coffee
machine, everybody can use it. So one
coffee machi, machine can serve a whole
bunch of people. So here's the idea. Let's
suppose there's three people and person
one would be willing to contribute $40.
It's worth $40 to her to have a coffee
machine. Person two would be willing to
contribute $50. And person three would be
willing to contribute $twenty. So if you
add all these things up. You get a $110,
so you'd be collecting [inaudible] between
$110, and the machine only cost 80, which
is great. But here's the problem. The
problem is these values, the 40, the 50,
the twenty, they're private information.
No one knows those before the fact. And
so, what you've got is, you've got a
classic case of hidden information. People
value the coffee mach--, machine some
amount. But we'd like to get them to
reveal that information so we can decide
whether or not to buy the coffee machine.
Because it could be that they do not value
it enough. That person one values it at
twenty that person two values it at
twenty, and person three values it at
twenty. So the total value is only 60, and
the coffee machine costs 80, so it?s not
worth doing. So the challenge from a
design standpoint is to figure out how we
get people to reveal their hidden
information. So again it comes down to a
question of [inaudible] compatibility. So
what we're going to talk about her e is a
mechanism called a Pivot Mechanism. And
this was designed really by three people,
Clark, Groves, and Vicry. So this is
sometimes called the Clark-Groves-Vicry
Pivot Mechanism. We'll just sort of call
it the Pivot Mechanism. He's how it works;
you only have to pay the marginal amount.
So you pay the minimal amount that you
would have to pay given other people's
bids in order to make the project viable.
So if without you, people are willing to
pay 60 dollars and it costs 80. Then you
only have to pay twenty. If without you,
people are willing to pay $70 and it cost
80, then you only have to pay ten. So it
is, you only have to pay that amount that
would make it viable if you threw in that
much money. And this creates an incentive
for you to tell the truth, because it
doesn't hurt you to say how much you
really valued the project, cuz you're only
going to pay your marginal amount, whether
it's $twenty or whether it's $ten. So,
here's formally how the Clark Groves Vicry
mechanism works. Each person claims a
value. Now this, the real value is hidden
information, but they're gonna claim a
value. And the whole idea behind mechanism
design is to get them to claim their true
value. So each person claims a value, and
if the sum of those values is bigger than
the cost, if everybody's claiming values
that add up to more than the cost, since
we're gonna share this coffee machine,
then we go ahead and do the project and
buy the coffee machine. Now, how much do
people pay? Well, person one is just gonna
pay the cost, minus the bids of the other
two people, the values of the other two
people. Or if the [inaudible] are bigger
than the cost, Person one is going to pay
nothing because the project should be done
without Person one contributing anything.
So, that's it. That's all there is to the
mechanism. Let's see how it works. So
let's again go to our example where
there's three people. Person one buys at
40, person two at 50 and person three at
twenty and the cost is 80 dollars. So
again if we add these three things up we
are gonna get 110 dollars. So clearly you
wanna buy the coffee machine because they
value at 100 and it only cost 80. But how
do we decide how much each person should
pay? Again, the rule is, each person's
just gonna pay that cost, which is $80,
minus the values of the other two people.
So if we look at person one, the cost is
$80. The other two people added, added
together, 50+20 is 70. So that means
person one is just gonna pay $ten. What we
want to figure out is, does it make sense
for person one to tell the truth. So if
this is the mechanism, should person one
say 40 dollars or should they say 50
dollars or should they say twenty dollars.
Well let's think, let's suppose the cost
is 80 and person one values it at 40, and
let's suppose that they claim a value of
40, so in this trial we're gonna submit
they're actually claiming the truth. So
let's think through all the possibilities,
so one possibility is that some of the
other people's is only 30. So, if some of
the other people's 30, and they claim 40,
that only adds up to 70, so they're not
gonna pay anything. >> Cuz the project's
not gonna get done, and so they're not
going to make anything. Now, if the other
people, some of their bids is 40, and our
person claims 40, then they're going to
pay 40, and they're going to break even.
And they're going to pay $40, and they're
going to get a project for $40, so, you
know, whether you do it or not really
doesn't matter. Let's suppose that some of
the others is $45 though. So the others
bids, or values, add up to 45. This person
claims 40. They're only going to have to
pay 35, right? Because 45 plus 35 adds up
to 80, which is our cost, And, so on net
they're going to come out five ahead. >>
And then finally, let's look at some of
the others as ninety. Well now, their
claim value is forty, they're not gonna
pay anything because they're not gonna pay
anything and they're gonna come up forty
ahead. So if they claim that you value,
sometimes the project doesn't get done,
sometimes it gets done and they're
indifferent, sometimes it gets done and
you make a little money, sometimes you
don't have to pay anything. Now let's
suppose they decide to cheat a little bit,
and to claim a value of only thirty. So
you're gonna say, oh, no, I want forty, in
the shade. Well now if the sum of the
others is 30 the project doesn't get done
and then it'll pay anything Because the
project isn't done. If someone else is 40
and they did 30, again that adds up to 70.
So that's less than the cost and the
project doesn't get done, And because some
of the others is 45 and they'd done 30,
that's only 75. So it's a third time the
project doesn't get done. But in the case
where, that some of the others was 90,
then the project gets done and they don't
pay anything. So if you compare that to
the case where they tell the truth,
they're worse off in these two cases
because when the sum of the others is 45,
if they'd have told the truth, they only
would have paid 35 and they'd have come
out five ahead. But when the sum of the
others is 45 and they. A 30 that only adds
up to 75, And the product doesn't get done
and they're worse off. They are getting
nothing when they could've gotten a plus
five. So it seems pretty clear you don't
wanna under claim, because if you under
claim, the project could not get done in
cases where you'd like to get done. What
about over claiming? Well, let's think.
Suppose the value is 40, now I claim 50.
Well, if the sum of the others is 30, and
I claim 50, 50+30 adds up to 80, which is
the cost. That means I'm gonna have to pay
50, and I'm out ten. Now in the other
cases I'm going to only pay 40 or less, so
I'm just the same as if I told the truth,
but in this one case I'm worse off. So
what we see is if I look at the three
cases, do I tell the truth where I get
zero, zero, five and 40. Where I
understate, I get 00040 and when I
overstate I get minus 100540. What you see
is, I'm best off telling the truth. What
the Clark-iii mechanism does, which is
fantastic, it's a way to; induce people to
tell the truth. So, this is again what we
call the 'Incentive Compatibility'. Each
person has an incentive under the
mechanism we had provided the rational, to
give their true value. So that's great,
totally great, and we think well, this is
a fantastic mechanism. And let's see how
it plays out. Let's say there's a case
where person one has a value of thirty,
person two has a value of thirty, and
person three has a value of thirty. So the
other two people. Have values of 60. The
total cost is 90. So that means person one
is gonna pay 30, person two is gonna pay
30 and person three is gonna pay 30 for a
total revenue of 90, which is great. It
costs 90 and we're gonna get revenue of
90. So this mechanism not only gives
people the truth to reveal, but in this
case it actually raises enough money to
pay for the project. Now, notice that's
not always gonna be true. Let's go back to
our example of 40, 50, and twenty. In this
case, how much is person one gonna pay?
Person one is gonna pay 80 minus the sum
of the other two people's bids, which is
just gonna be $ten. >> Person two is going
to pay 80 minus the sum of the other two
people's bid, which is one and three,
which is $60. So they're going to pay
$twenty. And person three is going to pay
80 minus the sum of the other people's
bids, but that's 90. So the most you can
make see you can't pay, can't make someone
pay negative amount. So they're not going
to pay anything. So in this case the total
revenue is only $30. So now we have a
problem, we got people to truthfully
reveal, everybody gives their own value,
but. You don't raise enough money to pay
for the project. So to pay for the project
we have to have some other way of paying
for the project. Possible taxing people,
But if we tax people, that could cause
them to change their value assessment.
Because the thing is now they have less
money than they had before. So what we see
is, the [inaudible] mechanism work sorta,
[inaudible] some of the [inaudible]
funding the project. But it doesn't work
perfectly because it's not balanced, in
the sense that you ca n't fully pay for
the project. See cause what we we'd like a
mechanism to do, when there's public
project setting, we'd like it to be
efficient by that I mean we'd like it to
be the case that you always get. The
project if it's worth it. So you always,
you only do the project if it's
worthwhile, so you can have an efficient
outcome. You'd like it to be the case that
it makes sense for people to always join,
so that you are not coercing people into
doing it, that you have to join if you
don't want to. You'd like it to be
incentive compatible, so that people tell
the truth. And then finally, you'd like it
to be balanced, like to be able to pay for
the project. What you can show is that
that's impossible. So there's no mechanism
that'll do this. So what you can do is you
can sort of do second best. You can
sacrifice one of these things. And the
thing you can sacrifice in the [inaudible]
victory mechanism is that it's balanced.
Now, there's other mechanisms that
sacrifice other restrictions in order to
get this to work. So it'd be balanced, but
they won't necessarily satisfy the
condition that no one's coerced to join.
It could be that you have to join the
project, and sometimes you end up worse
off than if you said, I'm not even
participating at all. So what we learn
from mechanism design by writing models
where we say okay here's the values that
people have and think about how we induce
them to tell the truth, you learn what is
possible to accomplish and what is not
possible to accomplish. Okay. So when we
think about mechanism design, which we
should in some sense, two fundamental
issues we're trying to overcome. One is
hidden action. We may not see what people
do. So what we'd like to do is write down
incentive structures and information
structures so that we induce people to
take the right actions. We can't see what
they're going to do, so what we'd like
them to do is somehow incentives so we
know they're going to take the actions we
want. Second thing we can use mechanisms
to overcome is hidden information, Ju st
like we saw in this public project case,
or in the auction case. We don't know
people's values, so what we'd like to do
is we'd like to create a mechanism where
people truthfully reveal their values, and
where we get outcomes that are desired
outcomes. So in the case of an auction, we
sell it to the person who values it the
most. And in the case of a public project,
we undertake the public project if it's
worth doing. All right, so that's
mechanism design and it's really
interesting, it's a way to use models to
try and design institutions so they're
more efficient. Now one thing we saw now,
and I wanna be very careful, is most
mechanism design assumes people are
rational, but we talk about auctions, some
[inaudible] we saw that, if we don't
assume people rational, we maybe get
slightly different implications of what we
should do. So, for example, we have the
revenue equivalents theorem that said, it
doesn't matter what auction you use,
you're gonna get the same outcome in each
case. But, we saw that if we have a
different model of human behavior, some
auction mechanisms might be preferred over
others. So, depending on how sophisticated
people are, we might wanna use a different
auction mechanism. The same goes for these
public project mechanisms. We can think
about mechanism design but then we can
start around as a benchmark but setting
some rational actors or something. If
anybody can fully figure this out what we
can get then you can use your judgment and
say lets now try for more some more
realistic models on how people behave and
how these same mechanisms hold up. Again
having model O is going to be really
useful here because it actually helps to
think through all the implications and
decisions you are making when you are
design those institutions. All right.
Thank you.