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Hi folks. It's Matt again and what we're going to do now is, is actually sort of

Â wrap up our discussion of, of coalitional games and allocating value on a set of

Â individuals and we're going to do so by looking at a particular example which we

Â can do some comparison, say, of the Core and the Shapley Value and, and see exactly

Â what's going on. so let's, let's look at a very interesting one. so think about the

Â UN, UN Security Council. So the United Nations has a security council which makes

Â a whole series of. Passing off in resolutions among doing other things.

Â which can, can be very, very important in international politics.And in particular

Â there are 15 members of the UN security council at, at present. How does this.

Â There is 5 permanent members of the council so those are China, France,

Â Russia, UK and US and so they, they are always in the security council. Hmm, there

Â are 10 temporary members so beyond these 5 there are other members rotate in and out

Â of the security council other chosen, a general from the UN. And the, the um.one,

Â one sort of very important aspect of this is that the five permanent members can

Â veto resolutions, okay? So, basically, if you want to pass something within the

Â security council. The five permanent members actually have to agree to it.

Â there's some subtleties in this, it can be that they can abstain, but ignoring a

Â abstention you basically have to have them on board. If anyone of them says no to

Â something, it won't pass in the security council. but the ten other members do not

Â have vetos. Okay? So, if we start thinking about a cooperative game to capture this.

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and, and the security counsel can use different rules for voting. Sometimes they

Â use majority rule, sometimes they use two thirds rule and so forth. But we'll look

Â at a situation where they're using majority rule. So, if we want to represent

Â that as a cooperative game in order to come to a resolution. what's going to have

Â to be true. so if we think about the, you know labeling say China France Russia UK a

Â nd US as, as individuals or players 1 2 3 4, 5 then what's true about the value of

Â the coalition so a, a value of a coalition. A coalition can pass

Â legislation or pass a resolution and in this case. and get a value of 1, so let's

Â say 1 is success you, you pass a resolution. you can do that if all 5 are

Â on board, so you need the 5 permanent members, plus you need a majority so you

Â need at least half of the 15 so you need at least 8 members in total to vote yes.

Â to overwhelm the, the other 7 that might be voting no, but all 5 of this, these

Â have to be present. And if you have a coalition that wants to, to pass a

Â resolution that does not include some of these members or fall short of the 8 Then

Â you get 0, okay? So this is a cooperative game, it's a very particular one, and we

Â can analyze then, what are the core al, allocations for this, what's the Shapley

Â value and so forth. So in order to do that, let's start with a, a, a simple 3

Â player game that has a similar structure Okay.

Â And what's the structure of this, this is sort of a simplified version of the UN

Â Security Counsel. say 1 permanent member with a veto and 2 temporary members. And

Â we still operate by majority rule. So what's true is The value of a coalition is

Â 1 if you got 1 as a member and, you've got at least 2 members on board. So if these

Â two agree and person 1 is one of them you get, you can pass something otherwise you

Â can't. Okay. So that's just a simplified version, but

Â it has the same kind of structure as the UN Security Counsel. so what happens,

Â let's start with the core and try and analyze it. So we've got our game that v

Â gets 1 if, if you've got 1 in there and at least 2 members. Otherwise you get 0. So

Â now what does the core has to satisfy? Remember the core has to be allocating

Â each coalition, a total that's at least what it gets. So that means that if you

Â put what 1 and 2 to get they can, generate a value of 1. So they have to be getting

Â at least 1. 1 and 3 together have to be getting at least 1. 1, 2, an 3 together

Â hav e to be getting one. Right? So we're dividing up the total value among the 3

Â members. And it has to be that, that everybody gets at least zero. Since you

Â could generate, zero, But, you, you, you can't be, forced, in this case, to,

Â participate. Okay, so now when we think about what the core is going to do,um,

Â when we want to look at this, the fact that, 1 and 2 have to be getting at least

Â 1, and the total of all 3 have to be equal to 1, and nobody can get a negative value.

Â That these together. Imply that x of 3. Sorry, x sub 3 has to be equal to zero.

Â Right? So, there's no way of giving, giving one and two at least 1. And all 3,

Â a total of 1. And, except by giving 3, 0. Okay.

Â So, then we can do the same thing here. That means that x2 is equal to zero. if

Â we've got x2 equals zero, and x3 equal to zero. That's going to imply that x1 has to

Â be 1. So, in this case, the fact that 1. Is a vital player, an essential player

Â this means that, that the core actually gives 1 the full value here. Now if you do

Â the core for the, the security council, it was a full 15 members. you can work

Â through that, what are you going to get? your going to get that essentially the

Â division of the full value, is going to end up going completely to the 5 permanent

Â numbers. So you are going to get the 5 permanent numbers X 1 through X 5, getting

Â a value of 1, and then everyone else getting a value of 0. But you can have

Â many different ways of allocating that amount in between those numbers and still

Â be in the Core. Okay so simple idea of what the core is in

Â this game. Okay now let's stick with the same game and do the Shapley Value. so it

Â we're looking at the Shapley Value for this game, what are we going to end up

Â with? Well we can do our, our calculations from the shapley value, we know that the

Â value of I is given according to this formula and in this, in particular, you

Â know, we can sort of just build this up we could build it up by first putting in 1

Â and 1,2 then 1,2,3 1, 1,3 1,2,3 2 first, then 1 3 first, or 2 first, then oops,

Â then, Then, 3, 3, then 1, 3, then 2, etcetera. And in this case, whe-, when 1

Â comes in, these 2, they, they add nothing. and in every other case, no matter where 1

Â comes in, They, 1 comes in when there's at least 1 other player there. in this, in

Â all these other cases they're adding a value of 1. So this is going to tell us

Â that the value to 1 should be equal to 2 3rds because 2 3rds of the time, they're

Â adding a value of 1. And 1 3rd of the time, they're adding a value of zero.

Â Okay? If you go through these kinds of calculations, you can, you know? You've

Â got, This weighted by 2 6th. This weighted by 1 6th This is evaluated by 1 6th and so

Â forth. So, what you're going to end with is here. 2/3 Then 2 is going to get 1/6. 3

Â is going to get 1/6, and so forth. And so what we end up with is, is Shapley value

Â of 2/3 for 1. 1 6th for each of the other players. So the Core and the Shapley value

Â in this case are both unique and they are giving as different predictions, one, the

Â core saying everything should go to person 1 the Shapley value says well 2 and 3

Â actually do generate some value and we should be giving them some of the fruits

Â of their production and in, in this case 1 is more important so they get more between

Â 3 are still valuable members in this society and the Shapley value were.

Â Reflects that, but these are very different logics you might think of the

Â core in a situation where people might secede and that one could walk away and

Â say, you know, without me you get nothing whereas the Shapley value is, is doing

Â calculations based on marginal contributions.

Â Okay, in terms of, of cooperative games then, what have we done? We've looked at

Â modeling fairly complicated multilateral bargaining settings; you know, something

Â like, say the UN Security Council, something like that And we, you know, the

Â idea, part of the idea behind cooperative game theory is that it, you know, we could

Â do everything as a non cooperative game. We could have written down a normal form

Â game for bar gaining or we could have written down a, a giant extensive form for

Â how the security counsel al, operates and who can bring in a resolution and then who

Â has to vote yes and how it all works. And then calculate what a Nash equilibrium of

Â that game is, the so the sub game perfect equilibrium, and then trying to figure out

Â what the payoffs are. And the idea of cooperative game theory is sometimes you

Â want to model things in a more compact way, and actually trying to model an

Â extensive Form for that bargaining process would be overwhelming. And this is a

Â different way of approaching things which takes an axiomatic approach, a very simple

Â approach and ends up you know, generating a predictions. And there's a number of

Â different solutions that people have used. so, you know thi, thi, you can do core

Â based ideas Shapley value. There's other solutions as well. So there's a fairly

Â rich literature on cooperative game theory that's based on, on different approaches,

Â to characterizing what fair kinds of values are.

Â