 So, very quick background, okay. I'm pretty old, I work on a lot of things, including mathematical areas that have nothing to do with computational, with social sciences. And I've been here since the beginning, given lots of talks, those have been to all of them, which will be nobody here would see some similarities, but there's also some new stuff. So, starting with the sort of general idea of voting or aggregation of preferences, let's say, there's a huge number of rules, which that word cloud on the left is supposed to show, that have been named in the literature, that's probably not all of them, and there's a huge number of axiomatic properties that they satisfy, and it's really difficult to navigate through this jungle of them all. So, one important unifying principle is something which Arkady Piotr and Edith Elkin have worked a lot on, the idea of distance rationalization. More generally, you think about having a profile of preferences in some format, that's the input, you have an outcome, which is the result of your rule, it could be a social choice, or it could be a multi-winner setup, you can have many different possibilities. And there are often, in many applications, there are some very easy outcomes that you can decide. If everyone has exactly the same preference order and you're voting for a president, then it's relatively easy to work out who you should choose. In general, what we do is come up with a distance measure to these so-called consensus sets, and we use this standard sort of Voronoi idea of looking at what is the closest consensus result, and those different colored regions there correspond to inputs, preferences, profiles, which have the same outcome under whatever your rule is. That's the general principle, so try to understand things. You decompose your rule somehow into a consensus notion and a distance notion, and then all you're arguing can be done at that level. And so I've been working a bit on this last year, well, for several years, but submitting last year, so a very systematic way of deriving properties of the rule from its components, and perhaps even more interestingly, in the case of anonymous rules, or rules that are homogeneous, which means that if you clone everyone the same number of times, you get the same result, you get to some interesting mathematical areas some buzzwords being Wasserstein or Kantorovich metric, optimal transport, finite-dimensional Bunnack spaces, things like that. So these are quite unusual mathematical ideas that haven't come up much in social choice before. So those are some of the things we're working at on, and the idea would be to move on next to different models, different output models. So the inputs are the same, usually there could be different inputs, you could have different types of ballots or preferences, but the same general unifying principle should be extended more generally. Not to say that no one has worked on this, but there's a lot more work that could be done. Right, next topic, I mentioned matching on the other slide. Another word for that might be resource allocation. You're trying to match items to agents, for example, lots of those things up there. Very important practical applications, school assignment, medical residence, these kinds of things. There's lots of, it's tricky because in the case of voting, if everyone has the same preferences, then it's easy because you know what's gonna happen. It's very easy to work out what to do. That's the hardest case in some sense for resource allocation. If everyone wants the same resource, it's not a single public good, it's the private one. It's more complicated. Recently, I've been working quite a bit and we've introduced some new algorithms for the so-called house allocation problem, which is also called one-sided matching. Those algorithms actually are quite competitive when you look at the welfare results even though their axiomatic properties are not fantastic. So here's an example from some simulation that did where the top one, higher is worse here, this is the welfare loss that you're getting under various assumptions. And the top one on top blue line is the standard random serial dictatorship which has lots of nice axiomatic properties. But it's welfare results are not very good and the purple one is the new one that we've introduced which actually was inspired by a Christmas party that I went to three years ago, interestingly enough. Now, electoral systems done quite a bit on electoral systems. These pictures are intended to talk about prediction and how hard it is. That's the left-hand picture and the right-hand picture is talking about evaluation of electoral systems in terms of multiple criteria. So there are two criteria here. You want to be kind of at the bottom left. You want to be down here to be a really perfect system and the color here indicates a certain parameter that you can try to optimize in your electoral system. And the question is, which is better? Which is the best value of the parameter? Are we doing some kind of optimization by criterion optimization? So you want to find the Pareto Frontier that you have a whole lot of scatter in these points. So you've got distributions rather than single points. So I still haven't worked out exactly the methodology we're gonna use here but that's really the kind of picture we're talking about, give you a rough idea of the kind of things we're talking about. So we've done all sorts of stuff involving simulations, practical submissions to government, currently working on interesting models for artificial preference data using earned models and some predictions, some other political science applications involving swing and then we'll continue to talk about optimizing systems. So that's sort of that stream and the final stream I wanted to talk about was networks, starting a lot of stuff on networks because my PhD students are working in that area. For example, this network here, question is, is it balanced? Balance is a technical term which is not difficult to work out. Each edge is either plus or minus and a network is balanced if every cycle has an even number of negative edges in it. Friend or foe? Yeah, friend or foe, yeah, this one. This here would be an ally or an enemy. And you can have, so it's not hard to tell whether something is balanced or not because you really can apply a little breadth-first search basically, so well-known graph algorithms to do that but what's hard is if it's not balanced working out how to make it balanced with the minimum amount of change to the network, removing like flipping a minimum number of edges in order to make it balanced. That's a very well-known NP hard problem. I don't know. I haven't looked at everything there but these are all, so it's created by some. Are they both vertices? They're both vertices because they are all actors in some sense. So Germany was part, I guess, of this P5 plus one or whatever, wasn't it? It was one of those deals with Iran, for example. The interesting thing here is that, yeah, if you go through and trace everything through you see that Iran and the US, for example, here have quite positive relations in many ways. I guess the other thing that's clear is that everyone hates ISIS pretty much, yeah. It's based on, well, it's here. It's come from some source but it's based on formal alliances and so there are formal treaties and alliances and also news reports. So there's some judgment required there. Someone is making a judgment here. Yeah, this is not a complete graph but it's a graph that shows the various relations. There are some non-relations like Russia and China don't have a direct relation in this sense, right? Okay. But there might, there's no formal alliance, there's probably no formal alliance or enmity, yeah. At some point you get to the boundary of a network, right? So you have to decide where you stop. You could get rid of these external players if you like but they could be relevant. Some of them are not states either, right? Like here, for example. This is a few years old as well so it's not gonna be the same as now. This is an example of a signed network where you have plus or minus on the edges, right? So that's, I'm just giving you that as a sort of a taste of a problem. I'll zoom down to the second point because this idea of working out whether something is balanced or not, if it is or isn't, it's easy to determine but measuring partial balance, how close you are to being balanced is not something that's been properly analyzed so we've tried to axiomatize that and looked at it, performance on various real world networks. So that was one paper there. It might be a bad thing to have but in certain contexts it's a good thing. Social settings usually it's considered to be good. Otherwise you have a lot of tension. You have friends of friends who are enemies and things like that. But here it may or may not be a good thing because the perfect balance here would in fact be equivalent to a bipartisan of the network into two sort of Cold War alliances which may not necessarily be a good thing. But it is something which is used a lot in a lot of applications, not just here but in physics and it's a very basic concept, yeah. So whether we want to actually, yeah, so first of all we're trying to improve the theory, then we, if you look under the next there, we're trying to actually work out how to compute what we think of as the best performing overall measure of performance that unfortunately is, it's an NP hard problem to compute this thing, exactly. We also have some other problems in the networks area so Addison mentioned decide, we'd use that in some at length to experiment on undergraduates a few years ago and finally have submitted something based on that. A diffusion model in the network, sorry, a diffusion study, experimental study from which we hope to develop interesting models. We already have enough to know that the original model is not very good so we're trying to improve on that. Also looking at a completely different network which is a citation network of all New Zealand laws, laws cite other laws, usually for the purposes of definitions but sometimes for amendment for various other reasons. Trying to understand the structure and evolution of that as a complex network. Yeah, it's still, there's a lot of very interesting questions involved there. We've tried really hard to find let's say legal historians, people who are quantitative political scientists in New Zealand and we haven't done too well. We've had one, we've talked to a lot of people. We're intending to publish in the New Zealand law journal which will be your first thing first. And just to ask the legal community if there's any interesting questions that they can think of, we can use this network for. We have, there's a lot you can say from the network science point of view whether it has relevance in the application area is gonna be an interesting. So we're gonna work on that. One thing is very clear is that the law has become extremely more, much more complex. Not only are there more laws, there are more interrelations between the density has been increasing over time. It's hard to understand exactly why that is, but immediately, but you can look at various hypotheses. So those are the things there, very quick overview. And as I said, they're always looking for some new people to work with and I'm gonna be talking at least once this year in our seminar, so be based to some extent on feedback. Thanks. We can then, on demand we can ask Mark to talk more about one particular. Yeah, you can think about it. Have a think about it, it's all there and the slide will be up there. Yeah, catch up.