 OK, we've seen some results on the difficulties associated with aggregating preferences coherently. And what I want to do is talk a little bit about a setting where things are nicely behaved. And these are known as single-peaked preferences. And sometimes voters' preferences have nice properties. And a very prominent case of that is when candidates can actually be ordered from left to right. So we can think of leftmost candidates, rightmost candidates, and candidates can be nicely ordered. And in situations where people have a most preferred outcome or candidate on this spectrum, then candidates were going to have nice properties in terms of having an existing, say, winner, having nicely defined voting schemes, and so forth. And so the idea here is people have a most preferred point, and they prefer candidates closest to that and don't prefer ones that are further away. Now, we can think of this as a political spectrum. You could also think of how much tax would you like to have? Well, people might have an optimal tax rate and not like to have higher tax rates or lower tax rates, or what's the extent of some regulation? We can think of a series of different settings where we have a nice ordering over the alternatives, and people's preferences are going to have this property. So in particular, when we think about, so our set of outcomes, let's say our set of outcomes here is A, B, C, D, E. And we look at a person who has single peak preferences. We look at, say, the utility. So here, the utility, the payoff they get from a given alternative, and their most preferred alternative in this case is B. They get the highest. And then when we look at the other alternatives, in this case, C would be second highest. A would be third highest. Then D and then E is way down here. So their preferences would look like B, C, A, D, E in terms of a ranking. But what's important in terms of single peakedness is that the further you move this way, the lower the preferences, the lower the utility is. So the voter always prefers something which is closer to B than something further away on the same side. And similarly, the more you move in this direction, the voter prefers the alternative B to A or anything further to the left of A. So that's what's known as a single peaked preference. And the fact that you've got this nice ordering, and it can be ordered on a line with the most preferred alternative and things getting worse when you move away, that kind of preference means that you've got nice voting procedures and the existence of condesce winners. So let's have a peek at that, just to understand why things are going to be better behaved in this kind of world. So imagine we had three voters that have single peaked preferences, and we had our A, B, C, D, E. So we'll keep an odd number of voters to make sure we don't have to worry about ties. What's true, then, is if we look at the median most preferred outcome, that's going to be a condesce winner. So in this case, that's B. So B's most preferred by this middle person. So let's call this person one, person two, person three. Person three's most preferred alternative is D. Person one's most preferred alternative is A. And let's verify that this outcome B is actually going to beat everything in a pairwise comparison. So suppose we compare B to C. Well, both persons one and two like B better than C. They also like it better than D and E, right? So given the fact that when we look at the median voters, all the voters to the left of the median are going to prefer the median's peak to any alternative which lies to the right. Because going to the right further to the right is bad. And similarly, we're going to have a majority of people on the right preferring the median's alternative to anything that's to the left of the median's alternative. The median's most preferred alternative. So the median outcome here, median outcome is a Condorcet winner, median meaning median peak here, right? So we look at the peak, we look at the median of the peaks that's a Condorcet winner. And so we're going to have a well-defined alternative which is going to beat every other one. And this gives us a nice way of choosing an alternative. It's known as median voting. And it gives us an idea of why we always talk about median voters and so forth because then we've got a majority that prefers that position to anything on the other side and vice versa. We have a Condorcet winner. Moreover, this is going to have nice properties in terms of people misrepresenting their preferences, right? So could two improve by changing their ordering? No, their alternative is picked, right? B, their favorite ordering. Let's think about person A. How could you affect this by, if you're median voting, we're picking the median of the three alternatives. So if we think about picking the median alternative of these three alternatives, then the only way that person one could affect the outcome would be to change the median. How could they change the median? They would have to announce a peak over here, C, D, or E. And all of those alternatives are worse than B. So there's, by misrepresenting their preferences, this person one would actually do worse. So you can verify that that's also two for three or any individual. The only way you're gonna be able to make a difference is to change the median, which means you have to flip to the other side, which is something you're generally not gonna wanna do. So nice thing about single peak preference worlds, median most preferred point is a Condorcet winner and individuals are gonna have incentives to be truthful and to announce their preferences in a truthful manner. In fact, it's gonna be a dominant strategy if we ask people for their preferences to announce their truthful preference if we're using majority voting, sorry, median voting in this case. So there are settings where a lot of the negative results of impossibilities of aggregating preferences are overturned and those are gonna be ones where we've got a lot of structure on the problem and that structure of the problem means we have nice orderings and there's no voting cycles. So this kind of setting, there's no voting cycles. We don't have the difficulties that underlie some of the issues like aero's theorem and so forth. Once we go into the setting, we have a nicer set of preferences in terms of the restrictions that they satisfy and that allows us to aggregate preferences in ways that are gonna be nicely efficient strategy proof, having dominant strategies and so forth. So there are settings which are quite natural where things are gonna work out more effectively than they do when we allow for all preference orderings over alternatives. Now, when that happens depends on the setting and there can be situations where you don't have this nice left-right ordering. So it is a fairly restrictive setting but one that at least captures a good number of applications.