 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 what people have what's known as quasi-linear 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 that translates directly into some utility numbers. And the importance of doing that is 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 will give us a lot of power in designing mechanisms in terms of making sure we basically 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 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 quasi-linear preferences so people have quasi-linear 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, the linearity part is that the preferences are always just moving linearly with whatever this payment scheme is, whatever the money dimension is. 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 we've got our outcomes are equal to X cross Rn so what it's doing is it's first of all 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 non-monetary outcomes so often these are called public decisions, the aspect 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 your wealth an agent has and secondly 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 Theta's so now we announce our Theta's 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 analysis 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's and the Theta minus PI of Theta right so in 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 independence 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 that 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 would 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 though so what that's what's nice about the private good setting is now instead of just thinking about Theta I's we can think of just people just telling us what their utility function looks like so we can think of the the private information they have is 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 agents have a valuation function which is basically they out the value that they have of any particular allocation X okay and so then when we start thinking about direct mechanisms we can think of the space of individuals that 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 and so dominant strategy mechanisms they might not always exist it could be that people are going to tell us some alternative valuation function instead of the true valuation function and 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 you 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 is it that possible and now this quasi linear world with these private values to get people to truthfully tell us what their preferences look like