 Okay, and I apologize that the title of my talk has changed, but that's my fault. Because I've just recently visited Clemens Poupe in Karlsruhe. And we did significant progress on this paper and had great fun. And I think that our perspective on the whole thing has changed. So we will be talking, I will be talking about Condorcet domains. So first, what's Condorcet paradox? So everybody knows what important figure in French Revolution Condorcet was. And how important he was as a scientist. He was first to try to apply probability theory to social choice. Well, he noticed that sometimes majority relation can be intransitive. For, there are two situations here. We have three persons, three alternatives. So three persons choosing among three alternatives. And they have preferences, A preferred to B preferred to C, B preferred to C preferred to A, C preferred to A to B. So these are cyclic preferences and majority relations behave badly. For example, if you ask them to compare A and B, two of them will vote for A, first and third. So A will win. And if you ask them to compare B and C, then first and second will vote for B. So B will win over C. And if you ask them to compare C and A, two of them will vote for C. And so C wins over A. So majority relation is intransitive. You cannot decide what's better for this society. Okay, so, and this is a similar situation. So what is a condorsed domain? So you can try, you can think of finding a condorsed domain is kind of agreement in society. So to avoid coordination, certain coordination to avoid undesirable outcomes so that majority can take intransitive decisions. For example, you can, and of course you have to prohibit one of these orders and one of these orders. At least two orders you have to prohibit to get transitive majority. So for example, you can choose these four linear orders and society can agree on using only these. Or you can choose these four orders. So the difference between them is not quite clear from this, but if we put it on a permutahedron, what is permutahedron? Permutahedron, that's a polytope whose vertices are linear orders or elements of symmetric group if you are an algebraic, has an algebraic background. So and in this polytope, polytope edges are drawn between any pair of orders which differ by a single switch. For example, here you switch from A and B. That's a single switch and they are neighbors in the permutahedron. So the first domain was graphically described by the domain D1 graphically described like this. So it is connected in this graph and the second is disconnected. So and in a long literature existed on Condorcet domains. So Monjardet was one of the important figures because these two guys, Hemini, Nimboy and Bella wrote PhD thesis under Monjardet. Then at some point Peter Fishburne got strong interest in Condorcet domains. And Gallambos and Rayner did some important work too. And all of them, most of these work was about connected domains. So that's domains of the first type but not the second type. So those domains which are subgraphs of the permutahedron graph. Recently very important work have done Russian from Central Institute of Mathematics and Economics, Daniela from Koshyvoy and I will mention them slightly later. So our innovation is that we make all domains connected. How do we do that? So let's say that order r is between two other orders r prime and r2 prime. If r agrees with r prime and r2 prime whenever they agree. Or mathematically it can be written in this way. So an interval of between two orders. That's all linear orders that are between r1 and r2. So for example here, these all orders are everything between CBA and ABC. And BAC is between ABC and BCA. And I will give them more examples later. And that's important that we now declare two linear orders from the domain neighbors if there's nothing in between them. So like in permutahedron, two neighbors have nothing in between them. And so we get the associated graph gamma d after that. So now in the associated graph both domains are connected and there is no real reason for making a difference between them. That's one of our innovations. So condorsed domain is closed. If majority relation corresponding to any opinion of the society is also an element of this domain. So you do not go anywhere outside. And maximal condorsed domain, if it's not a proper superset of any other condorsed domain. So the two little domains in the first slide were both maximal. So condorsed domain, if you have a majority relation on some profile with odd number of waters and if it's not in the condorsed domain you can add it and it still will be a condorsed domain. So in particular this means that every condorsed domain is contained in a closed condorsed domain. And closed condorsed domain contains in the maximal condorsed domain and maximal condorsed domain is necessarily closed. So I apologize, I have to introduce another object. Eventually it will appear that all these are the same thing. But it's not clear now that they... So let's introduce median domains. So what is median domains? For every three linear order there is a median linear order in this domain. Which is between any pair of these, between R1 and R2, R1 and R3, R2 and R3. And if such a median order exists it's always unique. For example let's see here ABCD is a median of these three orders. Let's check that it's between the first two. So the first two they agree on A ranking A, C and D. They rank it in this order and this also agrees A, C, D. Now another pair they agree on B and D and ABCD also agrees with them. So you can check any pair here and you will see that indeed this is a median order. And the key lemma, very difficult, surprisingly difficult, is that betweenness, if we have a median domain, or alternatively another good case where you have the whole pyramid of hydrant, then betweenness relation on D is the same as geodesic betweenness in the associated graph. So let me show you, it's on this example. This is not a median domain and betweenness relation on D will not coincide with geodesic betweenness. Okay, so if we took A, B, A, C, B and B, C, A, then these two will be between them because this is just the reverse of that, so anything is between them. But what is geodesic betweenness? It's on the shortest path. So B, A, C is on the shortest path between these two. So this is the only thing that is geodesically between, but these are two orders that are between in the sense of domains. So in this case geodesic betweenness and betweenness do not coincide. And that is a key lemma that allows us to do whatever we can do. Now the third concept, I'm sorry, there's yet another new object. It's a median graph. It's a connected graph such that for every three distinct vertices, there is a unique vertex which is M, A, B, C. Here now I have to assume unities which is geodesically between every pair. So here M, A, B, C is between A and B because it's on the shortest path from A to B. It's between A and C because it's on the shortest path from A to C and it's between B and C because it's on the shortest path from B to C and it's unique, it's easy to check. Okay, the main theorem connects everything now together. So what it says is the main is median, the main if and only if it is closed condorsed domain. So you see condorsed domain was in terms of majority relation but the median domain was in terms of betweenness, but in fact it's appeared that it's the same thing. And for every closed condorsed domain, the associated graph is median graph. So now these two things come together. Conversely, if you have a median graph with a set of vertices V, there exists a closed condorsed domain on a set of alternatives and you need no more than vertices in V to construct such a condorsed domain and this condorsed domain will have associated graph exactly G. This is the main result but let me show you some... Now this is a contribution of Adam. Adam showed that you see in the previous result we have these restrictions that we need no more than the number of vertices for alternatives to realize this median graph. And that's the worst case scenario is when you have a star graph. So if you have a star graph with n vertices, you cannot... And I have to say also that graphs are median... Any tree is a median graph. So this is a tree but it's a median graph and you can realize it with four alternatives but you cannot realize it with three and that's what Adam proved that with a star graph you cannot realize it with less than an alternative if star graph is in vertices. Okay, so I'll show you also some beautiful pictures. It's not our invention but... So this is what you see here. It's a maximal condorsed domain for four alternatives. It's not only maximal but the largest. It has nine linear orders and it's a tiling domain. Tiling domain, that's the invention of Danilov and Karshevoy. So you have here the lowest point and the upper point in this zonogon and every snake here is a linear order in this picture. For example, you have B, D, A, C, so that's here. But you have also D, B, A, C and this is here. So these are tiling domain. That's another maximal condorsed domain. It's also tiling domain. It has one little square. It has only eight orders. This is a classical single crossing profile. So that's a corresponding graph is just a path, a line. So this is how permutahedron in the case of four alternative looks. It's already 24 vertices, this polytope. And here is what I show here is not connected in a classical sense maximal condorsed domain. So these two parts of it. But in our associated graph is a cube. So it is connected. And this is just for curiosity that's largest condorsed domain for five vertices. That's what associated graph, how its associated graph looks like. So this is distributive lattice. And Shemini and Nibwa and Danilov and Koshivoy, they showed that maximal condorsed domain is always a distributive lattice. We slightly generalized it to closed condorsed domain. But there is a condition. You have to have two completely reverse orders. Otherwise, this is not true. So this maximal condorsed domain does not have two mutually reverse orders. It's easy to check because B never goes below second position. And it's not a distributive lattice. Okay, well, interesting thing that we started to look at single crossing condition on trees. But eventually we were disappointed. We were totally now feel about trees not so good because it appears that median graph of a maximal condorsed domain is never a tree unless it's a chain. So now we feel that the more natural object is a median graph than a tree. So what are the further questions? Well, I haven't seen a complete account of maximal condorsed domains for even four alternatives. That wouldn't be too difficult, but that would be a good project for a student. And permutahedron is just a calligraph of a symmetric group relative to the set of generators, permutations 1, 2, 2, 3, and minus 1 in. And I think if we... But this is not the only set of generators for a symmetric group. So if we change the set of generators, then calligraph will change and probably disconnected domains in the first case will become connected. So it's probably connectedness or disconnectedness. It's only a matter of choosing the right set of generators for the symmetric group. Another interesting, very competitive question is that let's say mn is a maximal number of orders in a condorsed domain with n alternatives. So what's known? That was... most efforts went into this. And four values are known. So for three, we have four, maximal number. For four, we have nine, maximal number. For five, we have 20. I showed you that latest. Fishburn did enormous amount of work and I showed that for six, the maximal number is 45. So actually, Montjarde had this example and Fishburn managed to prove that it's maximal. However, we don't know m7 and with magma computing package, I believe it's possible to extend this sequence. So if you are a student and thinking about a project or for dissertation, that would be, I think, a very good set of questions. Thank you. Yeah, yeah, yeah, it grows exponentially. Oh, there are some bounds. Yes, but my talk was not about numerical values and bounds. Well, a lot is known about tiling domains. So about this tiling domains and there are some numbers for maximal values of tiling domains as they go quite far. But disconnected domains, we know nothing about cardinality of disconnected domains. And there are observations that at least by n equals 16, these tiling domains will lag behind. Other domains will be better disconnected ones. I have to be restricting almost all of the, if you're severely restricting the preference or the preference that they can be expressed by your population, it doesn't seem to have an enormous amount of probability to the original problem. I mean, it solves the problem, but it's sort of very, very restrictive. And then you realize there's almost an old opinion that people are allowed to have on all the values if you want to have a specific preference. But single peak domains are an example where you get natural restrictions because you put some ordering on the alternative straight. And so you can imagine that you end up with this sort of cutting because somehow people agree that there's some natural ordering over all the alternatives. So a good peak domain is one of the maximal domains, yes. There's some logic behind the order. Is there a natural ordering? So I guess there's an distributive lattice that comes out of one of the theorems. I'm trying to think of whether there's a natural way in which the alternatives can be ordered in order to always sustain the maximum domain. Or single peak ones. I'm assuming that no way there's some structure to the alternatives which then give you the logic behind the preferences that come out towards maximum domain. Well, if you... There are several identical notions where Danilo Kashiwoye calls this casting and Fishburne calls this never conditions. So they specify for each triple which four combinations are available. And there are four different never conditions according to Fishburne. And Fishburne found that especially the first one and the second one, if he alternates these, he can get quite large domains. All these never conditions kind of analogs of what you are saying. That single peakiness can be seen as one of these never conditions. That's a good question. Is it about the way the one was zero? Well, firstly, you mean these domains that means can there say closed under... Or maximal can there say domains? That's a good question. I haven't thought about this. Now, just one combination so you have to, instead of combining four different never conditions, you have to apply the same one all the time. We'll think about it.