 Hi, folks. It's Matt again. And what we're going to do now is actually sort of wrap up our discussion of coalitional games and allocating value among a set of individuals. And we're going to do so by looking at a particular example, which we can do some comparison say of the core and the Shapley value and see exactly what's going on. So let's look at a very interesting one. So think about the UN Security Council. So the United Nations has a Security Council, which makes a whole series of passes off in resolutions among doing other things, which can be very important in international politics. And in particular, there are 15 members of the UN Security Council at present. How does this work? There's five permanent members of the Council. So those are China, France, Russia, the UK, and the US. And so they're always in the Security Council. There are 10 temporary members. So beyond these five, there are other members that rotate in and out of the Security Council that are chosen in general from the UN. And one sort of very important aspect of this is that the five permanent members can veto resolutions. So basically, if you want to pass something within the Security Council, the five permanent members actually have to agree to it. There's some subtleties in this. It can be that they can abstain, but ignoring abstention, you basically have to have them on board. If any one of them says no to something, it won't pass in the Security Council. But the 10 other members do not have vetoes. So if we start thinking about a cooperative game to capture this, and the Security Council can use different rules for voting. Sometimes they use majority rules, sometimes there's a two-thirds rule and so forth. But we'll look at a situation where they're using majority rule. So if we want to represent that as a cooperative game, in order to come to a resolution, what's going to have to be true? So if we think about the labeling, say China, France, Russia, UK, and US as individuals or players, one, two, three, four, five, then what's true about the value of the coalition? So a value of a coalition, a coalition can pass legislation or pass a resolution in this case and get a value of one. So let's say one is success, you pass a resolution. You can do that if all five are on board. So you need the five permanent members. Plus you need a majority, so you need at least half of the 15, so you need at least eight members in total to vote yes, to overwhelm the other seven that might be voting no. But all five of these have to be present. And if you have a coalition that wants to pass a resolution that does not include some of these members or falls short of the eight, then you get zero. So this is a cooperative game. It's a very particular one. And we can analyze then what are the core allocations for this, what's the Shapley value and so forth. So in order to do that, let's start with a simple three-player game that has a similar structure. And what's the structure of this? This is sort of a simplified version of the UN Security Council, say one permanent member with a veto and two temporary members. And we still operate by majority rule. So what's true is the value of a coalition is one if you've got one as a member and you've got at least two members on board. So if at least two agree and one person, one player, one is one of them, you can pass something otherwise you can't. So that's a simplified version, but it has the same kind of structure as the UN Security Council. So what happens? Let's start with the core and try to analyze this. So we've got our game that V is one if you've got one in there and at least two members, otherwise you get zero. So now what is the core has to satisfy? Remember, the core has to be allocating each coalition a total that's at least what it gets. So that means that if you put what one and two to get, they can generate a value of one. So they have to be getting at least one. One and three together have to be getting at least one. One, two, and three together have to be getting one, right, so we're dividing up the total value among the three members. And it has to be that everybody gets at least zero since you could generate zero, but you can't be forced in this case to participate. Okay, so now when we think about what the core is going to do, when we want to look at this, the fact that one and two have to be getting at least one, and the total of all three have to be equal to one, and nobody can get a negative value, that these together imply that X of three, sorry, X sub three has to be equal to zero, right? So there's no way to be giving one and two at least one and all three a total of one, except by giving three to zero, okay. So then we can do the same thing here. That means that X two is equal to zero. If we've got X two equals zero and X three equal to zero, that's going to imply that X one has to be one. So in this case, the fact that one is a vital player, an essential player, this means that the core actually gives one the full value here. Now if you do the core for the Security Council with the full 15 members, you can work through that. What are you going to get? You're going to get that essentially the division of the full value is going to end up going completely to the five permanent members, so you're going to get the five permanent members X one through X five getting a value of one, and then everybody else getting a value of zero, but you could have many different ways of allocating that amount in between those members and still be in the core, okay. So simple idea of what the core is in this game. Okay, now let's stick with the same game and do the Shapley value. So if we're looking at the Shapley value for this game, what are we going to end up with? Well, we can do our calculations from the Shapley value. We know that the value of I is given according to this formula, and in this in particular, we can sort of just build this up. We could build it up by first putting in 1, then 1, 2, then 1, 2, 3, 1, 1, 3, 1, 2, 3, 2 first, then 1, 3 first, or 2 first, then 3, 3, then 1, 3, then 2, et cetera. In this case, when 1 comes in, these two, they add nothing, and in every other case, no matter where 1 comes in, 1 comes in when there's at least one other player there. In all these other cases, they're adding a value of 1, so this is going to tell us that the value to 1 should be equal to 2 thirds, because 2 thirds of the time they're adding a value of 1 and 1 third of the time they're adding a value of 0. If you go through these kinds of calculations, you've got this weighted by 2, 6, this weighted by 1, 6, this by-to-weighted by 1, 6, and so forth, so what you're going to end up with is here 2 thirds, then 2 is going to get 1, 6, 3 is going to get 1, 6, then so forth, and so what we end up with is a Shapley value of 2 thirds for 1, 1, 6 for each of the other players. The core and the Shapley value, in this case, are both unique, and they're giving us different predictions. The core is saying everything should go to person 1. The Shapley value says, well, 2 and 3 actually do generate some value, and we should be giving them some of the fruits of their production, and in this case, 1 is more important, so they get more, but 2 and 3 are still valuable members in this society, and the Shapley value reflects that, but these are very different logics. You might think of the core in a situation where people might secede, and one could walk away and say, without me you get nothing, whereas the Shapley value is doing calculations based on marginal contributions. In terms of cooperative games, then, what have we done? We've looked at modeling fairly complicated multilateral bargaining settings, something like the UN Security Council, something like that, and part of the idea behind cooperative game theory is that we could do everything as a non-cooperative game. We could have written down a normal form game for bargaining, or we could have written down a giant extensive form for how the Security Council operates and who can bring in a resolution and who has to vote yes and how it all works, and then calculate what a Nash equilibrium of that game is, or a sub-game perfect equilibrium, and then try and figure out what the payoffs are. The idea of cooperative game theory is sometimes you want to model things in a more compact way, and actually trying to model an extensive form for that bargaining process would be overwhelming. This is a different way of approaching things, which takes an axiomatic approach, a very simple approach, and ends up generating predictions. There's a number of different solutions that people have used. You can do core-based ideas, Shapley value, there's other solutions as well, so there's a fairly rich literature on cooperative game theory that's based on different approaches to characterizing what fair kinds of values are.