Okay. So were back. And were talking about strategic network formation. And now were going to have a look at the Connections model. The symmetric version of the connections model and see what is efficient in that setting and then what's[UNKNOWN] stable . So just to remind you of this model. We had this benefit perimeter delta, less than, somewhere between 0 and 1. I gave the direct benefit and then this decays as you get further away. So, in terms of overall value of the utility that a given individual gets is represented then by this utility form where you sum over different individuals. And you look at the path lengths to those individuals and then you might have a delta which depends on the, those that personal connection. And then you raise that to that power, so if you're not connected at all you can make this infinite. And then this would decay to 0, and then you subtract off costs for maintaining links. Okay. And then the simplest version of this in terms of the symmetric version. We'll get rid of the subscripts in all these things and everybody will bear the same types of costs and have the same types of benefits. So, when we looked at that, then for different networks we have utilities that each individual gets. And as a function of those utilities then we can end up figuring out what the value to society is. Okay let, let me just say one thing before we go on. you know, so, so one nice thing about this type of approach is that once we specify utilities. And welfare calculations for each individual then we end up having the ability to evaluate networks and see which ones are good. And which ones are bad directly by doing calculations in terms of the, the welfare property. And this something that's very different from the random network. So you know, is a network formed by preferential attachment better than a network that is formed at random. Well we have no idea because we didn't have any specification of what that did for the individuals involved. And so here we've got some idea that their preferences and you know, where we can value friendships at a certain level. We value indirect friendships at a certain level, and once we've done that now we can assign values to these things. And then make welfare evaluations. So, it's not only that by doing this we get predictions about which ones are going to form but we also can evaluate them and so that's an important aspect of this. One other thing to say in terms of pairway stability and this idea of forming networks strategically. It's not necessarily true that each individual has to be [UNKNOWN] or calculating about all their friendships. I want to form this friendship because it's beneficial and not one because it's not. It's it's more just that what's necessary is that individuals will tend to form things which are are beneficial. And when things aren't working out they tend to to get rid of them. So as long as there's pressures in those directions then we can talk about dynamics and so forth afterwards. But that'll give some push towards these kinds of networks. And these are ones that are going to be stable, in the sense that nobody then has an incentive to move away from these. So, there'll be rest points of different processes, where people don't have to be so calculating, but at least pushed in the right directions. And eventually reach these kinds of networks. Okay, so let's have a look now at this, this particular setting. And let's look at the ones, the efficient networks are the ones that are maximizing total utility in this setting. And they break in to three different categories. So we're going to deal with situations where there's very low cost to links. Situations where there's a medium cost to links, and situations where there's a high cost to links, so the cost is above some level. And basically for basic low cost of links the complete network, is a unique efficient network. And that's very intuitive, links are so cheap. And in particular, when c is less than delta minus delta squared. That's going to bet he situation where it's more beneficial to have a direct relationship. Right? This is the value for a direct relationships compared to an indirect relationship of distance 2, so shortening anything of distance 2 or even further. The gain in changing that is bigger than the loss in terms of adding the link. So adding a link is always going to be beneficial and the complete network is going to be uniquely efficient in that setting. you have to worry about externalities but thats going to be the conclusion. In a middlke range, then star networks are going to be uniquely efficient. So the only architecture which is going to be efficient is going to be have somebody in the center and then everybody else have one link to that person. And that's going to be the thing that maximizes the total sum of utilities in a society, and the only thing which maximizes the total sum of utilities. So it's, it's that does it, and it's the only type of architecture that does it. And then once costs are so high, then it just makes sense that nobody should connect to anybody. Links are just too expensive. Doesn't make sense to have anybody talking to anybody. So for very high costs, the empty network is uniquely efficient, okay? So the meat of understanding this, is really going to be understanding this middle part. Because the other extremes are going to be fairly, you know, if the links are so cheap you might as well just add them all. If links are so expensive it doesn't make sense to add any. And in the middle what's really the incite here is that, you want to have the star networks for. Now, a couple things to say is that, that this, is a fairly stark characterization. It's actually going to be true for a set of models beyond what, is, is true here. As long as you get a higher benefit from a direct connection and a lower benefit from an indirect connection and a lower benefit from things. There won't be anything special about it being delta, delta squared, delta cubed. It just has to be some value, some value for 2 distance, a value for 3 distance. And you'll see that everything we say and all the proofs and so forth would go through if you just substituted something else for the deltas and so forth. Okay, so it's not special to the functional form. Okay so let's first try to understand stars before we go into a, a formal proof of this. so lets think about, you know, we, we start with one relationship. That gives us 2 delta minus 2 c. And we think about adding a second one. So there's two different ways we can add this second relationship. One is we could connect these, the person to somebody whose already connected or they could form a new relationship. They form a new relationship then we've got four people connected, four benefits, four costs, 4 delta minus 4 c. If we connect this person here, each one of these two still get's a benefit so we, we'd still end up with 4 deltas and 4 cs . But now we're also benefiting from this indirect connection which is present in this society but not in this one. So, here what's important is the indirect benefits that flow through the network, generate extra value. And so connecting in this way gives us a higher value than connecting in this separate way. Okay? So, that's the, that's the start of it. Now, when we think, you know, let's connect this third person in somehow. We want to connect them as well. well, we could connect them to say, one of the agents, one of these peripheral agents or we could connect them back to the center. And again, we're going to have six , you know, six values, and six costs coming from the direct links because there's three links in either network. And the question is, wh, why is it better to do it in a star form? Well, in the star form all of these indirect connections now are of a distance 2, where as in this one some of the indirect connections are at a distance 3. And so by doing it in a star form, we end up with a higher value for all of the indirect connections. We get 6 delta squared. As opposed to four deltas and two delta cubes. These longer distance relationships are worth less. They're worth a delta cubed, which is less than a delta squared. And so this, we get higher value by having more direct connections which come through a star. So these indirect connections are shorter and more valuable that way. Okay, and then you know if you think about adding a, a fourth person in. Well, if you add them in directly to the center again more value from the indirect connections than if we add it to the periphary. Where now a lot of these indirect things are going to be of distance 3, as opposed to all of distance 2 here in any other network. So the stars are coming out because they're the most efficient way to connect people with a given number of links with the least distance between them. So it's a very efficient way to do connections. when is it that you want to keep connecting? So, you know, let's suppose we've got four individuals and we can think of, will we keep them in a star? Or else we could add extra lengths in. You know, so if we connect these two people. Well what we get here is now you know before here here, when, between these people that was a distance 2. Now there's a distance 1. So what we've done is we've moved some of the distance 2 things over to direct connections. But we've paid a couple of extra costs of maintaining links to do so. So we have more relationships but more benefits because of, of that. And so the question is, when is it that this gain in having shorter distances outweighs the cost and it's precisely when delta minus delta squared is bigger than c. Right, when is it better to have direct relationship than an indirect relationship. So if delta minus delta squared is bigger than c, then you don't want to stop at a star, you want to just keep adding links and shortening all those indirect relation. So that gives you the idea of, of when you'd want the complete network, versus instead stopping at a star. So a star is the most efficient way if you wanted to use the minimal number of links. And then there's the question of whether you want to keep adding links, and that'll happen if the links are cheap enough, and the benefits outweigh the costs. Okay, so that's just a, a quick peak at it. in the, the next thing I'll do is actually go through a formal proof of this, so if, you can skip that if you like or you can look through the details. So the formal proof is fairly straightforward but we'll just go through verifying that the star is actually the most efficient way to do this. Do a formal proof of this proposition. And then we're going to come back and look at the pairway stable networks. And try to understand what's different between the networks that are the most efficient and the ones that are paralyzed stable.