So, we've been talking about strategic network formation. And we talked a little bit about the variations of positive and negative externality models. And, and why we might end up having inefficiencies in terms of the networks that form. And now I want to talk a little bit about. the possibility of transfers, so subsidies, payments across different individuals. And try to understand how that might rectify things, and in terms of, you know, what we've seen, we saw this conflict, you know we can talk about all the difference kinds of ways of modeling. And fitting such things, so here we're just going to talk a little bit about the transfers at this point. And then try and understand what we see there. There's a lot more that could be said on this subject, but I want you to give you some feeling for it and at least a little bit of a basic understanding. Okay, so what do we mean by transfers? so here we mean sort of outside intervention, somebody taxing or subsidizing relationships, say a government supporting R and D relationships. it could also be due to the fact that there's bargaining among the individuals involved. So, somebody says look, you know, it's worthwhile to form this link. I'm going to pay something to you to help you form this, favors exchanged among friends and so forth. So, the idea is that whatever those utilities numbers we're dealing with some of that could be moved from one node to another. Either by some outside entity saying I'm going to tax some people and, and subsidize others or by individuals bargaining and say look, I'll give you something if you're willing to do it. And, and certainly when, when countries form alliances, there can be payments made either explicitly or implicitly. In terms of the arrangements to make sure that these things are in both people interest's in forming relationships. Okay, so lets have a look at this in detail and so what we can think of in now we're changing the base utility to the utility plus some transfer. Where this could be either a positive or negative number depending on whether somebody is making net payments or getting that receipts as a function of the network. So, for instance it could be that the peripheral players say look, it's really beneficial to be connected in this star, I'm willing to do favors to the center. And then the center is willing to maintain these relationships because they get value form the other players. so if we just sort of thinking about transfers. One possibility in terms of, let's go back to the inefficiency that we had in the co-authorship setting. So, remember the co-authorship model that we just talked about in our last video. We've got a situation now where the problem was that people wanted to over connect. And we could imagine that in this situation one possibility is the government says okay, we're going to tax people who form extra links and then move that to to the other players. So, the raw paths you know from, from just forming just one relationship was three now if people form this extra relationship here, their payoffs went up to 2.35, and these people went down to 2. One possibility is now what we do is we actually charge these individuals, so we charge them a 0.625 each and then pay that to these individuals. Right, so we tax people saying look, if you're going to form relationships, you're going to have to pay something for that and then we reallocate those taxes. So, instead of these people getting the 2s, now they get 2.625. Okay, so those particular taxes and subsidies, are a way of equilibrating the payments in this model, right? And so now, when we look at the incentives, what happens is the individuals no longer have an incentive to form this extra link. Because they have to pay the tax involved, and so this now becomes pairwise stable, if they include the cost that they're going to to have to pay in terms of taxes and so forth. And so what we've done is we've aligned the interest of the individuals. Now everybody sees the, an equal fraction of the value of this network, and these people say, oh, that's not a good idea. I'm better off in this network and so they don't form this extra link and this turns out to be pairwise stable. So that's a situation where, you know, equalizing the payoffs by, by proper tax and subsidy and reallocating that. Now everybody gets an equal payoff. And they have incentives to form the right relationships. Okay, so one possibility is, is we just, you know, tax and subsidize in a completely Egalitarian way. So, set the transfers that are being made to any individual. What we do is, we just look at the total value of the network. And average that across all individuals, and if they were getting less than the actual amount, then they're going to get a positive transfer. If they were getting more than the average amount, then they'll have a negative transfer. So, we're just going to adjust the transfers to move everybody back to the center. So, now everybody, in terms of their net utility when we account for the transfers is just the average overall utility, okay? Now everybody in the society has exactly the same incentives as a utilitarian planner would have, because now everybody's utility is just one nth of the total utility in a society. So now, the utility anybody gets is, is exactly proportional to the efficiency of the network, so, ones that are more efficient everybody gets more value. Ones that are less efficient, everybody gets less value. Now the most efficient, so directly out of this we're going to get the, get the overall efficient network is going to be pairwise stable. Right? So now we, we've solved that problem of efficiency being pairwise stable, by just equilibrating things and making sure that everybody is an equal sharer in the pie. Okay, that's wonderful. it works well, but it, it, it could involve a lot of transfers. It could involve a lot of spreading money around. And in particular or spreading utility around, it could involve. making transfers that are going to violate some fairly basic conditions. And let's put some basic requirements on the types of transfers we allow and then see if we can still achieve full efficiency in that kind of setting, okay? So we're going to put in some very basic requirements on transfers and we're going to put in two, two requirements. one is that completely isolated nodes that generate absolutely no value get zero. And this is a condition which you can think of as one that's going to make sure that the society doesn't want to sort of split up and, and, and fragment and secede. So, if somebody's not generating value, other, other people don't subsidize that. So, this is a you know, somewhat controversial condition, but it's one that says it complete, people who aren't generating anything and are completely disconnected, don't get payments. Second condition two nodes that are completely interchangeable, meaning that they generate the same value, in different, in any time they're in the same kind of configuration or interchangeable in a configuration. So, people that are completely interchangeable should get exactly the same transfers. I'm not going to define this formally at this point but we'll see in an example that the idea is very intuitive. and you, you, so, so, let's just take a look at an example where that condition will become quite clear. and we won't have to go through a lot of notation. to, to talk about it. So, let's just do an example, and show that transfers can't always help. And this is, again, from the Jackson-Wolinsky paper. so here, what do we have? We've got a situation where basically we've got if everybody's connected, they each get a value of 4. Overall total value to society is 12. If we have people in a star configuration, the center gets 5, the outside nodes get 4. now we get a total value of 13. This is the maximizing value. Two nodes on their own each get a value of 6 value of 12. Okay? So, the right network in this kind of setting is one of these networks, these are the efficient networks. And what we can begin to see is that these networks are not going to be pairwise stable. Right? Why aren't they pairwise stable? Well this center of the star could benefit by deleting one of these, and they're going to get six instead. Right? So, so this is a situation where these are all efficient in that whoever involved with two links can get rid of them and move it to a one link network, and improve the values. And here, this should be a 5,4,4 as well, and, and basically no matter what configuration you're in, somebody can benefit by deleting the link and, and getting a higher value. So, we've got efficient networks having a star configuration, but none of them are pairwise stable. Okay. So what we want to do is see if we can do some transfers to try and help this. Okay, so in order to have the the, this is supposed to be want to make one of these networks pairwise stable. let's just take this one for instance well, first thing we know is that in order for this to be pairwise stable, this person's going to have to get at least 6. Right, so there have 5 here, they get a 6 here. They're going to have to increase this by at least 1. And let's just sort of first go through the fact that Another way we could have done it is reduce what we get down here. But this is where those conditions come in. The fact that somebody's completely disconnected, not generating any value, means they did nothing so things have to be split between these two individuals. They're completely symmetric, doing the same things so each one of them has to be 6. So this the value of how things are, are allocated here are tied down by those conditions on transfers. And so here in order for this person to be willing to maintain both links, they've got to get a transfer of at least one, okay? but in order for these two individuals not to want to form a new link, they have to stay at, at least 4. They're interchangeable. So, they have to get the same value. And appear everybody's going to get a value of four. And so in order for these people to be willing not to form a new link. You can't take anything away from them. So somehow, you have to give this person something. But you can't take anything away from these individuals. So the only way you could make the efficient thing stable, is by somehow infusing extra value into this. You'd have to take some extra value from the outside and pay this person. In which case now you've lost the value that you were trying to generate. So, in this situation there's no set of transfers that satisfy those conditions that treat equal people equally. And don't pay people that are completely disconnected to anything. if you put those two conditions in there's no way to have transfers such that this persons willing not to delete a link. And you still have these people not wanting to add a new link. So, there's different incentive constraints that you have to take care of at the same time. And there's now way to do that with one set of transfers so there's no way to arrange the transfers through to make this work. Okay so let's talk a little bit about the ideas behind this because all of the examples fairly simple I think that the point is, is more important. so there's something which is known as, as the Coase theorem in, in economics. And it goes back to a paper by Coase quite some years ago where Coase was talking about bargaining. And, and basically was making a point well in the paper's fairly complicated. But, but one What was taken out of that paper to become a Coase theorem was the idea that if people have complete information. And it's clear what the externalities in a situation is, there should exist some bargain that they can reach which would make sure that, they take efficient actions in the society. So, without frictions transfers can help solve these kinds of inefficiencies. Okay. And you know, Coase's paper was actually about frictions and, and why that might fail. But in, what it does say is, is here we're in a world where we've got complete information, we can see what the value of each one of these things are. And people can realize, look we need to make some payments. And the difficulty is that we're still not able to make the payments to make sure that the, the right network forms. And the difficulty is coming from the fact that we have to take care of multiple externalities all at once. And, we have to worry about, you know, making sure that center agent is still willing to keep both things. But also making sure that the other two agents don't want to form a new relationship. And so, the fact that we have to pay one agent and not take away from the other agents at the same time, is the combination of those externalities. Which is, is detrimental here. The, the combination of the incentive constraints that we have to take care of Not forming a new link which will harm the center, or not deleting a link which'll harmful to the, the outsides. Those are the combination of externality issues that are, are troublesome, and it's the combination of these things. That all have to be handled at once, which makes for the conflict between efficiency and stability and that that conflict can't be solved by reasonable transfers. So, that sort of tells us that transfers can be helpful sometimes but not necessarily always. And it depends on the circumstances so network setting introduces a, sort of an interesting problem. It's not necessarily entirely correctable with bargaining or transfers. It's going to depend on exactly what kinds of transfers we allow, and what situations and sometimes there'll be ways out of this and sometimes there won't, but there's an interesting issue there. Okay. So, summary so far. efficient networks can take some very simple forms in a a variety of, of models. Efficient networks and pairwise networks need not coincide. And transfers can help, but not always without violating some fairly basic conditions. Okay, so that sort of is, is a, you know, a quick look at some of the issues of. Strategic network formation. Now we can also look at some more advanced topics, where we've got some we, we, we work with slightly richer solution concepts. We think about dynamics. we can begin to talk about other, other kinds of of, of, of settings where we've got directed link formation and so forth. So there's a series of other topics. We'll take a brief look at some of those. There's, the literature here is, is grown quite large so it's hard to sort of, you know, give you a, a full understanding of all this in, in, in a few lectures. But hopefully this gives you an understanding of the basic issues and a lot of the reasons. Some, some of these are going to be interesting and there's an active area of research that we'll talk about a little bit more going forward. Which has to do with sort of bringing together these kinds of strategic formation models with some of the random network models. That we saw before to try and fit these together to data and understand what's going on. so that'll do it for, for now and then we'll take a look at some other formation models in just a moment.