And so the idea now is that the types represent somebody's
private value in this particular example for the two possible options.
So this is a valuation function.
So they basically have utilities for doing nothing, and utilities for
undertaking the project.
And undertaking the project is going to involve, possibly, some taxes and
other kinds of things, and maybe benefits.
So each person has utility of doing nothing and then the utility for
what happens if they undertake this investment.
Okay, so for
instance, one possibility is that somebody has a utility of 0 for doing nothing.
We can normalize it to 0.
And then if they did, undertook the project, this is a person who
likes the project, they would get some 4 utils for doing this.
So given that vi(0) is 0 and vi(1) is 4,
we can just represent that by a vector (0, 4).
So we could say that person has a vector of utility (0, 4), okay?
Another possible utility, maybe there's somebody who doesn't like the project.
That type could be represented by a 0 and a -2.
So what this does is it keeps track of how much each person values
not doing the project, how much they value doing the project.
And so now a person's type is just represented by these valuations.
And these valuations can be thought of also as transferable in the sense that we
can just pay them to compensate them.
So if you wanted to compensate this person down here for
the pain that they're enduring for building this thing, you would
have to pay them 2 utils in order to make them happy if you undertake this.
This person may be willing to pay up to 4 utils to have the project built.
Okay, so
in this kind of mechanism now we can think of this as kind of a direct mechanism.
People are going to announce their type.
So for instance if we keep things simple and they're either types who like it and
get 4 utils for it or don't like it and get -2, then there would be two possible
type vectors, the (0, 4) type and the( 0, -2) types.
In general you could have lots of types, but for this example,
let's just keep it simple.
And one possible mechanism would be to say, okay, look, if a majority of people
announce (0,4), if a majority of people like the project, then we'll build it.
If a majority of people don't like the project, we won't build it.
So we'll just take a majority vote, and then we'll build it or not.
So in terms of our notation of a mechanism, if a majority of people
announced the (0,4) types, then x, the outcome is going to be 1.
You undertake the project.
And if we're not doing any compensation, then everybody gets a pi of 0.
So basically, we just go ahead, we undertake the project, and
there's no transfers or subsidies back and forth in between individuals.
In contrast if the majority of people announce (0,
-2), then we don't undertake the project and
everybody ends up again, with a payment of 0.
So this is a simple example, just a straight majority vote over the project.
What's interesting about this mechanism?
Well it's actually a dominant strategy in this particular example
to announce your type, and better off voting to not
get the project built if I don't like it and voting for it if I do.
So truth is an equilibrium.
But sometimes what are the downfalls of this?
Sometimes some of the people get negative utility.
So sometimes the project is built, and there's a bunch of people getting -2s.
And also the choice doesn't maximize the total utility in this society, right?
So this mechanism isn't good in terms of maximizing the total utility.
It's not accounting for
the fact that these people like it twice as much as these people dislike it.
So one possibility would be to say, okay look, let's try a different mechanism.
Why don't we try, instead of having a majority,
if a third of the population likes it, then we'll go ahead and build it.
So now we're trying to do something where, instead what we take into account is these
people like it twice as much as these people.
So we'll weight their votes twice as much.
And we only need a third of the population to announce the v hat types, sorry,
the (0, 4) types, in order to have the project undertaken.
So we could change that mechanism, right?
So now, again, people announce their types.
And now if the number of people,
let's let m be the number of people who voted for it, so the (0, 4) types.
If that's bigger than a third of the population,
then we go ahead and undertake the project.
But now what we're going to do is we're going to try and
also compensate the people who don't like it, right?
So, there's a bunch of people suffering for it.
