Hi folks. It's Matt again and what we're going to do now is, is actually sort of wrap up our discussion of, of coalitional games and allocating value on 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, and see exactly what's going on. so let's, let's look at a very interesting one. so think about the UN, UN Security Council. So the United Nations has a security council which makes a whole series of. Passing off in resolutions among doing other things. which can, can be very, very important in international politics.And in particular there are 15 members of the UN security council at, at present. How does this. There is 5 permanent members of the council so those are China, France, Russia, UK and US and so they, they are always in the security council. Hmm, there are 10 temporary members so beyond these 5 there are other members rotate in and out of the security council other chosen, a general from the UN. And the, the um.one, one sort of very important aspect of this is that the five permanent members can veto resolutions, okay? 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 a abstention you basically have to have them on board. If anyone of them says no to something, it won't pass in the security council. but the ten other members do not have vetos. Okay? So, if we start thinking about a cooperative game to capture this. and, and the security counsel can use different rules for voting. Sometimes they use majority rule, sometimes they use 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, you know labeling say China France Russia UK a nd US as, as individuals or players 1 2 3 4, 5 then what's true about the value of the coalition so a, a value of a coalition. A coalition can pass legislation or pass a resolution and in this case. and get a value of 1, so let's say 1 is success you, you pass a resolution. you can do that if all 5 are on board, so you need the 5 permanent members, plus you need a majority so you need at least half of the 15 so you need at least 8 members in total to vote yes. to overwhelm the, the other 7 that might be voting no, but all 5 of this, these have to be present. And if you have a coalition that wants to, to pass a resolution that does not include some of these members or fall short of the 8 Then you get 0, okay? So this is a cooperative game, it's a very particular one, and we can analyze then, what are the core al, allocations for this, what's the Shapley value and so forth. So in order to do that, let's start with a, a, a simple 3 player game that has a similar structure Okay. And what's the structure of this, this is sort of a simplified version of the UN Security Counsel. say 1 permanent member with a veto and 2 temporary members. And we still operate by majority rule. So what's true is The value of a coalition is 1 if you got 1 as a member and, you've got at least 2 members on board. So if these two agree and person 1 is one of them you get, you can pass something otherwise you can't. Okay. So that's just a simplified version, but it has the same kind of structure as the UN Security Counsel. so what happens, let's start with the core and try and analyze it. So we've got our game that v gets 1 if, if you've got 1 in there and at least 2 members. Otherwise you get 0. So now what does 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 1 and 2 to get they can, generate a value of 1. So they have to be getting at least 1. 1 and 3 together have to be getting at least 1. 1, 2, an 3 together hav e to be getting one. Right? So we're dividing up the total value among the 3 members. And it has to be that, that everybody gets at least zero. Since you could generate, zero, But, you, you, you can't be, forced, in this case, to, participate. Okay, so now when we think about what the core is going to do,um, when we want to look at this, the fact that, 1 and 2 have to be getting at least 1, and the total of all 3 have to be equal to 1, and nobody can get a negative value. That these together. Imply that x of 3. Sorry, x sub 3 has to be equal to zero. Right? So, there's no way of giving, giving one and two at least 1. And all 3, a total of 1. And, except by giving 3, 0. Okay. So, then we can do the same thing here. That means that x2 is equal to zero. if we've got x2 equals zero, and x3 equal to zero. That's going to imply that x1 has to be 1. So, in this case, the fact that 1. Is a vital player, an essential player this means that, that the core actually gives 1 the full value here. Now if you do the core for the, the security council, it was a full 15 members. you can work through that, what are you going to get? your going to get that essentially the division of the full value, is going to end up going completely to the 5 permanent numbers. So you are going to get the 5 permanent numbers X 1 through X 5, getting a value of 1, and then everyone else getting a value of 0. But you can have many different ways of allocating that amount in between those numbers 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 it we're looking at the Shapley Value for this game, what are we going to end up with? Well we can do our, our calculations from the shapley value, we know that the value of I is given according to this formula and in this, in particular, you know, we can sort of just build this up we could build it up by first putting in 1 and 1,2 then 1,2,3 1, 1,3 1,2,3 2 first, then 1 3 first, or 2 first, then oops, then, Then, 3, 3, then 1, 3, then 2, etcetera. And in this case, whe-, when 1 comes in, these 2, they, they add nothing. and in every other case, no matter where 1 comes in, They, 1 comes in when there's at least 1 other player there. in this, 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 3rds because 2 3rds of the time, they're adding a value of 1. And 1 3rd of the time, they're adding a value of zero. Okay? If you go through these kinds of calculations, you can, you know? You've got, This weighted by 2 6th. This weighted by 1 6th This is evaluated by 1 6th and so forth. So, what you're going to end with is here. 2/3 Then 2 is going to get 1/6. 3 is going to get 1/6, and so forth. And so what we end up with is, is Shapley value of 2/3 for 1. 1 6th for each of the other players. So the Core and the Shapley value in this case are both unique and they are giving as different predictions, one, the core 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, in this case 1 is more important so they get more between 3 are still valuable members in this society and the Shapley value were. Reflects that, but these are very different logics you might think of the core in a situation where people might secede and that one could walk away and say, you know, without me you get nothing whereas the Shapley value is, is doing calculations based on marginal contributions. Okay, in terms of, of cooperative games then, what have we done? We've looked at modeling fairly complicated multilateral bargaining settings; you know, something like, say the UN Security Council, something like that And we, you know, the idea, part of the idea behind cooperative game theory is that it, you know, we could do everything as a non cooperative game. We could have written down a normal form game for bar gaining or we could have written down a, a giant extensive form for how the security counsel al, operates and who can bring in a resolution and then who has to vote yes and how it all works. And then calculate what a Nash equilibrium of that game is, the so the sub game perfect equilibrium, and then trying to figure out what the payoffs are. And 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. And this is a different way of approaching things which takes an axiomatic approach, a very simple approach and ends up you know, generating a predictions. And there's a number of different solutions that people have used. so, you know thi, thi, 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, on different approaches, to characterizing what fair kinds of values are.
