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So, we've been talking about strategic network formation.

Â And we talked a little bit about the variations of positive and negative

Â externality models. And, and why we might end up having

Â inefficiencies in terms of the networks that form.

Â And now I want to talk a little bit about.

Â the possibility of transfers, so subsidies, payments across different

Â individuals. And try to understand how that might

Â rectify things, and in terms of, you know, what we've seen, we saw this

Â conflict, you know we can talk about all the difference kinds of ways of modeling.

Â And fitting such things, so here we're just going to talk a little bit about the

Â transfers at this point. And then try and understand what we see

Â there. There's a lot more that could be said on

Â this subject, but I want you to give you some feeling for it and at least a little

Â bit of a basic understanding. Okay, so what do we mean by transfers?

Â so here we mean sort of outside intervention, somebody taxing or

Â subsidizing relationships, say a government supporting R and D

Â relationships. it could also be due to the fact that

Â there's bargaining among the individuals involved.

Â So, somebody says look, you know, it's worthwhile to form this link.

Â I'm going to pay something to you to help you form this, favors exchanged among

Â friends and so forth. So, the idea is that whatever those

Â utilities numbers we're dealing with some of that could be moved from one node to

Â another. Either by some outside entity saying I'm

Â going to tax some people and, and subsidize others or by individuals

Â bargaining and say look, I'll give you something if you're willing to do it.

Â And, and certainly when, when countries form alliances, there can be payments

Â made either explicitly or implicitly. In terms of the arrangements to make sure

Â that these things are in both people interest's in forming relationships.

Â Okay, so lets have a look at this in detail and so what we can think of in now

Â we're changing the base utility to the utility plus some transfer.

Â Where this could be either a positive or negative number depending on whether

Â somebody is making net payments or getting that receipts as a function of

Â the network. So, for instance it could be that the

Â peripheral players say look, it's really beneficial to be connected in this star,

Â I'm willing to do favors to the center. And then the center is willing to

Â maintain these relationships because they get value form the other players.

Â so if we just sort of thinking about transfers.

Â One possibility in terms of, let's go back to the inefficiency that we had in

Â the co-authorship setting. So, remember the co-authorship model that

Â we just talked about in our last video. We've got a situation now where the

Â problem was that people wanted to over connect.

Â And we could imagine that in this situation one possibility is the

Â government says okay, we're going to tax people who form extra links and then move

Â that to to the other players. So, the raw paths you know from, from

Â just forming just one relationship was three now if people form this extra

Â relationship here, their payoffs went up to 2.35, and these people went down to 2.

Â One possibility is now what we do is we actually charge these individuals, so we

Â charge them a 0.625 each and then pay that to these individuals.

Â Right, so we tax people saying look, if you're going to form relationships,

Â you're going to have to pay something for that and then we reallocate those taxes.

Â So, instead of these people getting the 2s, now they get 2.625.

Â Okay, so those particular taxes and subsidies, are a way of equilibrating the

Â payments in this model, right? And so now, when we look at the

Â incentives, what happens is the individuals no longer have an incentive

Â to form this extra link. Because they have to pay the tax

Â involved, and so this now becomes pairwise stable, if they include the cost

Â that they're going to to have to pay in terms of taxes and so forth.

Â And so what we've done is we've aligned the interest of the individuals.

Â Now everybody sees the, an equal fraction of the value of this network, and these

Â people say, oh, that's not a good idea. I'm better off in this network and so

Â they don't form this extra link and this turns out to be pairwise stable.

Â So that's a situation where, you know, equalizing the payoffs by, by proper tax

Â and subsidy and reallocating that. Now everybody gets an equal payoff.

Â And they have incentives to form the right relationships.

Â Okay, so one possibility is, is we just, you know, tax and subsidize in a

Â completely Egalitarian way. So, set the transfers that are being made

Â to any individual. What we do is, we just look at the total

Â value of the network. And average that across all individuals,

Â and if they were getting less than the actual amount, then they're going to get

Â a positive transfer. If they were getting more than the

Â average amount, then they'll have a negative transfer.

Â So, we're just going to adjust the transfers to move everybody back to the

Â center. So, now everybody, in terms of their net

Â utility when we account for the transfers is just the average overall utility,

Â okay? Now everybody in the society has exactly

Â the same incentives as a utilitarian planner would have, because now

Â everybody's utility is just one nth of the total utility in a society.

Â So now, the utility anybody gets is, is exactly proportional to the efficiency of

Â the network, so, ones that are more efficient everybody gets more value.

Â Ones that are less efficient, everybody gets less value.

Â Now the most efficient, so directly out of this we're going to get the, get the

Â overall efficient network is going to be pairwise stable.

Â Right? So now we, we've solved that problem of

Â efficiency being pairwise stable, by just equilibrating things and making sure that

Â everybody is an equal sharer in the pie. Okay, that's wonderful.

Â it works well, but it, it, it could involve a lot of transfers.

Â It could involve a lot of spreading money around.

Â And in particular or spreading utility around, it could involve.

Â making transfers that are going to violate some fairly basic conditions.

Â 6:34

And let's put some basic requirements on the types of transfers we allow and then

Â see if we can still achieve full efficiency in that kind of setting, okay?

Â So we're going to put in some very basic requirements on transfers and we're

Â going to put in two, two requirements. one is that completely isolated nodes

Â that generate absolutely no value get zero.

Â And this is a condition which you can think of as one that's going to make sure

Â that the society doesn't want to sort of split up and, and, and fragment and

Â secede. So, if somebody's not generating value,

Â other, other people don't subsidize that. So, this is a you know, somewhat

Â controversial condition, but it's one that says it complete, people who aren't

Â generating anything and are completely disconnected, don't get payments.

Â Second condition two nodes that are completely interchangeable, meaning that

Â they generate the same value, in different, in any time they're in the

Â same kind of configuration or interchangeable in a configuration.

Â So, people that are completely interchangeable should get exactly the

Â same transfers. I'm not going to define this formally at

Â this point but we'll see in an example that the idea is very intuitive.

Â and you, you, so, so, let's just take a look at an example where that condition

Â will become quite clear. and we won't have to go through a lot of

Â notation. to, to talk about it.

Â So, let's just do an example, and show that transfers can't always help.

Â And this is, again, from the Jackson-Wolinsky paper.

Â so here, what do we have? We've got a situation where basically

Â we've got if everybody's connected, they each get a value of 4.

Â Overall total value to society is 12. If we have people in a star

Â configuration, the center gets 5, the outside nodes get 4.

Â now we get a total value of 13. This is the maximizing value.

Â Two nodes on their own each get a value of 6 value of 12.

Â Okay? So, the right network in this kind of

Â setting is one of these networks, these are the efficient networks.

Â 8:48

And what we can begin to see is that these networks are not going to be

Â pairwise stable. Right?

Â Why aren't they pairwise stable? Well this center of the star could

Â benefit by deleting one of these, and they're going to get six instead.

Â Right? So, so this is a situation where these

Â are all efficient in that whoever involved with two links can get rid of

Â them and move it to a one link network, and improve the values.

Â And here, this should be a 5,4,4 as well, and, and basically no matter what

Â configuration you're in, somebody can benefit by deleting the link and, and

Â getting a higher value. So, we've got efficient networks having a

Â star configuration, but none of them are pairwise stable.

Â Okay. So what we want to do is see if we can do

Â some transfers to try and help this. Okay, so in order to have the the, this

Â is supposed to be want to make one of these networks pairwise stable.

Â let's just take this one for instance well, first thing we know is that in

Â order for this to be pairwise stable, this person's going to have to get at

Â least 6. Right, so there have 5 here, they get a 6

Â here. They're going to have to increase this by

Â at least 1. And let's just sort of first go through

Â the fact that Another way we could have done it is

Â reduce what we get down here. But this is where those conditions come

Â in. The fact that somebody's completely

Â disconnected, not generating any value, means they did nothing so things have to

Â be split between these two individuals. They're completely symmetric, doing the

Â same things so each one of them has to be 6.

Â So this the value of how things are, are allocated here are tied down by those

Â conditions on transfers. And so here in order for this person to

Â be willing to maintain both links, they've got to get a transfer of at least

Â one, okay? but in order for these two individuals

Â not to want to form a new link, they have to stay at, at least 4.

Â 11:42

So, in this situation there's no set of transfers that satisfy those conditions

Â that treat equal people equally. And don't pay people that are completely

Â disconnected to anything. if you put those two conditions in

Â there's no way to have transfers such that this persons willing not to delete a

Â link. And you still have these people not

Â wanting to add a new link. So, there's different incentive

Â constraints that you have to take care of at the same time.

Â And there's now way to do that with one set of transfers so there's no way to

Â arrange the transfers through to make this work.

Â Okay so let's talk a little bit about the ideas behind this because all of the

Â examples fairly simple I think that the point is, is more important.

Â so there's something which is known as, as the Coase theorem in, in economics.

Â And it goes back to a paper by Coase quite some years ago where Coase was

Â talking about bargaining. And, and basically was making a point

Â well in the paper's fairly complicated. But, but one What was taken out of that

Â paper to become a Coase theorem was the idea that if people have complete

Â information. And it's clear what the externalities in

Â a situation is, there should exist some bargain that they can reach which would

Â make sure that, they take efficient actions in the society.

Â So, without frictions transfers can help solve these kinds of inefficiencies.

Â Okay. And you know, Coase's paper was actually

Â about frictions and, and why that might fail.

Â But in, what it does say is, is here we're in a world where we've got complete

Â information, we can see what the value of each one of these things are.

Â And people can realize, look we need to make some payments.

Â And the difficulty is that we're still not able to make the payments to make

Â sure that the, the right network forms. And the difficulty is coming from the

Â fact that we have to take care of multiple externalities all at once.

Â And, we have to worry about, you know, making sure that center agent is still

Â willing to keep both things. But also making sure that the other two

Â agents don't want to form a new relationship.

Â And so, the fact that we have to pay one agent and not take away from the other

Â agents at the same time, is the combination of those externalities.

Â Which is, is detrimental here. The, the combination of the incentive

Â constraints that we have to take care of Not forming a new link which will harm

Â the center, or not deleting a link which'll harmful to the, the outsides.

Â Those are the combination of externality issues that are, are troublesome, and

Â it's the combination of these things. That all have to be handled at once,

Â which makes for the conflict between efficiency and stability and that that

Â conflict can't be solved by reasonable transfers.

Â 14:35

So, that sort of tells us that transfers can be helpful sometimes but not

Â necessarily always. And it depends on the circumstances so

Â network setting introduces a, sort of an interesting problem.

Â It's not necessarily entirely correctable with bargaining or transfers.

Â It's going to depend on exactly what kinds of transfers we allow, and what

Â situations and sometimes there'll be ways out of this and sometimes there won't,

Â but there's an interesting issue there. Okay.

Â So, summary so far. efficient networks can take some very

Â simple forms in a a variety of, of models.

Â Efficient networks and pairwise networks need not coincide.

Â And transfers can help, but not always without violating some fairly basic

Â conditions. Okay, so that sort of is, is a, you know,

Â a quick look at some of the issues of. Strategic network formation.

Â Now we can also look at some more advanced topics, where we've got some we,

Â we, we work with slightly richer solution concepts.

Â We think about dynamics. we can begin to talk about other, other

Â kinds of of, of, of settings where we've got directed link formation and so forth.

Â So there's a series of other topics. We'll take a brief look at some of those.

Â There's, the literature here is, is grown quite large so it's hard to sort of, you

Â know, give you a, a full understanding of all this in, in, in a few lectures.

Â But hopefully this gives you an understanding of the basic issues and a

Â lot of the reasons. Some, some of these are going to be

Â interesting and there's an active area of research that we'll talk about a little

Â bit more going forward. Which has to do with sort of bringing

Â together these kinds of strategic formation models with some of the random

Â network models. That we saw before to try and fit these

Â together to data and understand what's going on.

Â so that'll do it for, for now and then we'll take a look at some other formation

Â models in just a moment.

Â