Okay, so we are now going to talk about setting which will be analyzed quite extensively basically for the rest of the course, in terms of the types of mechanisms we're going to be looking at, and the alternative games we're looking at. And in particular what this is known as is a setting with transferable utility. And what's looked at in these settings are situations where people have what's known as quasilinear preferences. And the idea, before we get into the formal definitions, is that there's something like money, some transferable good, that we can move back and forth between agents and we know that, how much that's worth to them. And we can trade that off versus utility. And that gives us a nice sort of private good that we can move back and forth, sort of payments in an auction. Or payments, contributions to a public good. And we'll assume that translates directly into some utility numbers. And the importance of doing that is that it's going to give us a lot of power in terms of aligning incentives, by making people's payments encapsulate the externalities that they're imposing on others in terms of decisions. So once we have transferable utility, it'll give us a lot of power in designing mechanisms in terms of making sure. We basically can price everything, and we can figure out what kinds of prices people should be paying to change one decision to another decision. And that's going to be a very useful tool in designing mechanisms as we go forward. So what do the formal definitions look like instead of just having some abstract set of outcomes? Now the outcomes are going to have structure. Where there'll still be some basic sort of public aspect to it, some decision X. And then the other part is going to be a set of real numbers, where we give each individual some payment. Or it may have them make a payment. So there's some transfers going on between the different individuals. And so we will have some list Rn of what those payments are. So in this particular setting, when we have quasilinear preferences. So people have quasilinear, so here what we're going to have is things are going to be linear in this second dimension. So we can think of an outcome now as being a list of what's the public decision, some x and x and then also some p. Which is going to be a p1 through pn of what those payments are, and a given individual's utility for the outcome can be split into a utility function which describes how they like the x's and then also, they subtract off whatever payment they're making. And that payment could be positive or negative, so it could be that they're paying something into the society as part of the outcome or it could be that they're receiving payments. And these payments are going to be very important in designing efficient mechanisms, designing mechanisms to align people's incentives with what we'd like. So quasi-linearity gives us a lot of power, you can see where the quasi-linearity, where's the linearity part of this. Linearity part is that the preferences are always just moving linearly with whatever this payment scheme is, whatever the money dimension is. Okay, so when we start talking about mechanisms in this world, then we can split the mechanism into making a choice. So it's going to choose something, so I've got our outcomes are equal to X cross Rn. So what it's doing is first of all its going to make a choice out of the X. And then also have payments in Rn. So x and X is a non-monetary outcome and we use the term money here. It's not clear exactly what the transferable good is, but there's some way of moving something back and forth which people can equate with utility. So there's some nonmonetary outcome. Often these are called public decisions, the aspect of which is going to be common to all the agents. And then we've got these private payments where each person is making a payment into the mechanism. And if pi is negative, then that means they're actually getting a net payment to them. And the implications in terms of making this kind of assumption in terms of these preferences, first of all, the utility that somebody has for this outcome can be separated out from the utility that they get from the payment. So it's not influenced by the amount of money or wealth an agent has. And second, secondly, the people care, a given agent cares only about x, and their payment pi, they don't care about say, pj, where j is not equal to i. Right? So they don't care what payments other people are making. They just care about what's the overall decision. Which candidate do we pick or which public good do we pick or what decision are we making in terms of who gets what good? And then what payment do I have to make and I don't care what other people's payments are to the extent that it doesn't enter into my payment? Okay, so that's the setting. And then a direct mechanism in this world is going to be a combination of some choice, some x theta which comes out of x, and a payment scheme as a function of the thetas. So now we announce our thetas, and then society spits back at us a public decision, this non-monetary decision. And then a list of transfers or payments that we're each going to make. Okay, so that's a direct mechanism. One thing that's going to be very useful in these kinds of settings, and a lot of the analyses that we do going forward will be in a special case of quasi-linearity. And when we're thinking about the utility that individuals have, so we have, now we're writing people's utility overall as a utility of what the x is and the theta minus pi of theta, right. And this x can depend on theta, what we're going to do now is we're going to make a look at situations where there's a further assumption made. Where instead of having people's preferences depend on the full vector of types in the society, it's going to depend only on their own type. So we'll say that preferences have private values. Or satisfy conditional utility and dependence. If a particular person i's utility function, depends only on theta i. Okay, so the utility for this overall outcome does not depend on anybody else's type, it only depends on my own type. So that means once I know my own theta, I know everything about my preferences. And what this rules out is things like investing in a stock where I'm not quite sure what the value of that investment is. I don't know how well it's going to pay off. And other people might have information that could be very valuable to me. That's ruled out here. Once I know my information, I know everything I need to know about my preferences. And nobody else's information enters into that preference calculation. Okay. So if we're talking about a particular candidate, I know whether I like this candidate or not. Or if it's a public good, I know whether I want that public good go. So what's nice about the private good setting is now instead of just thinking about theta i's we can think of just people telling us what their utility function looks like. So we can think of the private information they have as just a valuation function, vi of x, which is equivalent to this ui of x of theta i. So the theta just becomes telling us what that function is, okay. So the agents have a valuation function which is basically the value that they have of any particular allocation x, okay? And so then when we start thinking of direct mechanisms, we can think of the space of the private information individuals have as these vi's. So in particular everybody can tell us, instead of a type, they tell us now their valuation function. And the standard notation we'll use is that people will tell us some v hat i, which might be a lie. So dominant strategy mechanisms might not always exist. It could be that people are going to tell us some alternative valuation function, instead of the true valuation function. So we can look at, when is it that they're going to want to tell us their true valuation function. So now we ask people, what, how do you value these different alternatives, and under this private value or conditional independent, conditional utility independence condition, they know their own preferences. And they can tell us what that preference function is and then we can look at mechanism design, and ask it when it is that possible and now this quasilinear world with these private values to get people to truthfully tell us what their preferences look like.
