0:00

Okay, so we are now going to talk about setting which will be

Â analyzed quite extensively basically for the rest of the course,

Â in terms of the types of mechanisms we're going to be looking at, and

Â the alternative games we're looking at.

Â And in particular what this is known as is a setting with transferable utility.

Â And what's looked at in these settings

Â are situations where people have what's known as quasilinear preferences.

Â And the idea, before we get into the formal definitions,

Â is that there's something like money, some transferable good, that we can move

Â back and forth between agents and we know that, how much that's worth to them.

Â And we can trade that off versus utility.

Â And that gives us a nice sort of private good that we can move back and

Â forth, sort of payments in an auction.

Â Or payments, contributions to a public good.

Â And we'll assume that translates directly into some utility numbers.

Â And the importance of doing that is that it's going to give us a lot of power

Â in terms of aligning incentives, by making people's payments

Â 1:15

encapsulate the externalities that they're imposing on others in terms of decisions.

Â So once we have transferable utility,

Â it'll give us a lot of power in designing mechanisms in terms of making sure.

Â We basically can price everything, and we can figure out what kinds of prices people

Â should be paying to change one decision to another decision.

Â And that's going to be a very useful tool in designing mechanisms as we go forward.

Â So what do the formal definitions look like instead of just having some

Â abstract set of outcomes?

Â Now the outcomes are going to have structure.

Â Where there'll still be some basic sort of public aspect to it, some decision X.

Â And then the other part is going to be a set of real numbers,

Â where we give each individual some payment.

Â 2:02

Or it may have them make a payment.

Â So there's some transfers going on between the different individuals.

Â And so we will have some list Rn of what those payments are.

Â So in this particular setting, when we have quasilinear preferences.

Â So people have quasilinear, so here what

Â we're going to have is things are going to be linear in this second dimension.

Â 2:26

So we can think of an outcome now as being a list of what's the public decision,

Â some x and x and then also some p.

Â Which is going to be a p1 through pn of what those payments are, and

Â a given individual's utility for the outcome can be split

Â into a utility function which describes how they like the x's and

Â then also, they subtract off whatever payment they're making.

Â And that payment could be positive or negative, so

Â it could be that they're paying something into the society

Â 2:59

as part of the outcome or it could be that they're receiving payments.

Â And these payments are going to be very important in designing efficient

Â mechanisms, designing mechanisms to align people's incentives with what we'd like.

Â So quasi-linearity gives us a lot of power,

Â you can see where the quasi-linearity, where's the linearity part of this.

Â Linearity part is that the preferences are always just moving

Â linearly with whatever this payment scheme is, whatever the money dimension is.

Â Okay, so when we start talking about mechanisms in this world,

Â then we can split the mechanism into making a choice.

Â So it's going to choose something, so I've got our outcomes are equal to X cross Rn.

Â So what it's doing is first of all its going to make a choice out of the X.

Â And then also have payments in Rn.

Â So x and X is a non-monetary outcome and we use the term money here.

Â It's not clear exactly what the transferable good is, but there's some way

Â of moving something back and forth which people can equate with utility.

Â So there's some nonmonetary outcome.

Â Often these are called public decisions,

Â the aspect of which is going to be common to all the agents.

Â And then we've got these private payments

Â where each person is making a payment into the mechanism.

Â And if pi is negative,

Â then that means they're actually getting a net payment to them.

Â 4:25

And the implications in terms of making this kind of assumption in terms of these

Â preferences, first of all, the utility that somebody has for

Â this outcome can be separated out from the utility that they get from the payment.

Â So it's not influenced by the amount of money or wealth an agent has.

Â And second, secondly, the people care, a given agent cares only about x,

Â and their payment pi, they don't care about say, pj, where j is not equal to i.

Â Right? So

Â they don't care what payments other people are making.

Â They just care about what's the overall decision.

Â Which candidate do we pick or which public good do we pick or

Â what decision are we making in terms of who gets what good?

Â And then what payment do I have to make and I don't care what

Â other people's payments are to the extent that it doesn't enter into my payment?

Â 5:11

Okay, so that's the setting.

Â And then a direct mechanism in this world is going to be a combination of some

Â choice, some x theta which comes out of x, and

Â a payment scheme as a function of the thetas.

Â So now we announce our thetas, and

Â then society spits back at us a public decision, this non-monetary decision.

Â And then a list of transfers or payments that we're each going to make.

Â Okay, so that's a direct mechanism.

Â One thing that's going to be very useful in these kinds of settings,

Â and a lot of the analyses that we do going forward will be

Â in a special case of quasi-linearity.

Â And when we're thinking about the utility that individuals have, so we have,

Â now we're writing people's utility overall as a utility of what the x is and

Â the theta minus pi of theta, right.

Â And this x can depend on theta, what we're going to do now is we're going to

Â make a look at situations where there's a further assumption made.

Â Where instead of having people's preferences depend on the full vector of

Â types in the society, it's going to depend only on their own type.

Â So we'll say that preferences have private values.

Â Or satisfy conditional utility and dependence.

Â If a particular person i's utility function, depends only on theta i.

Â 6:45

Okay, so the utility for this overall outcome

Â does not depend on anybody else's type, it only depends on my own type.

Â So that means once I know my own theta, I know everything about my preferences.

Â And what this rules out is things like investing in a stock

Â where I'm not quite sure what the value of that investment is.

Â I don't know how well it's going to pay off.

Â 7:06

And other people might have information that could be very valuable to me.

Â That's ruled out here.

Â Once I know my information,

Â I know everything I need to know about my preferences.

Â And nobody else's information enters into that preference calculation.

Â Okay. So if we're talking about a particular

Â candidate, I know whether I like this candidate or not.

Â Or if it's a public good, I know whether I want that public good go.

Â So what's nice about the private good setting

Â is now instead of just thinking about theta i's we can think of just people

Â telling us what their utility function looks like.

Â So we can think of the private information they have as just a valuation function,

Â vi of x, which is equivalent to this ui of x of theta i.

Â So the theta just becomes telling us what that function is, okay.

Â So the agents have a valuation function which is basically

Â the value that they have of any particular allocation x, okay?

Â And so then when we start thinking of direct mechanisms, we can think of

Â the space of the private information individuals have as these vi's.

Â So in particular everybody can tell us,

Â instead of a type, they tell us now their valuation function.

Â And the standard notation we'll use is that people will tell us some v hat i,

Â which might be a lie.

Â So dominant strategy mechanisms might not always exist.

Â It could be that people are going to tell us some alternative valuation function,

Â instead of the true valuation function.

Â So we can look at,

Â when is it that they're going to want to tell us their true valuation function.

Â So now we ask people, what, how do you value these different alternatives, and

Â under this private value or conditional independent, conditional utility

Â independence condition, they know their own preferences.

Â And they can tell us what that preference function is and

Â then we can look at mechanism design, and ask it when it is that possible and

Â now this quasilinear world with these private values to

Â get people to truthfully tell us what their preferences look like.

Â