Okay, so we're back and we're talking about strategic network formation. And now we're 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 pair y stable. So, just to remind you of this model, we had this benefit parameter delta somewhere between 0 and 1. That gave the direct benefit and then this decays as you get further away. So, in terms of overall value, 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 length to those individuals. And then you might have a delta, which depends on those personal connection and then you raise that to that power. So if you are not connected at all you can make this infinite and then this will indicate a zero and then you subtract off costs for maintaining links. Okay and then the simplest version of this, in terms of the symmetric version, will get rid of the subscripts and 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 me just say one thing before we go on. So, one nice thing about this type of approach is that once we've specified utilities and welfare calculations for each individual. Then, we end up having the ability to evaluate networks and say which ones are good and which ones are bad, directly by doing calculations in terms of the welfare properties. And this is something that's very different from the random network. So, is a network formed by preferential attachment better than a network that is formulated 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 there are preferences and where you 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. And one other thing to say in terms of pair y stability and this idea of forming networks strategically. It's not necessarily true that each individual has to be Machiavellian or, calculating about all their friendships or I want to form this friendship because it's beneficial and that one because it's not. It's more just that what's necessary is that individuals will tend to form things which are beneficial and when things aren't working out, they tend to get rid of them. So as long as their 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 they'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 kind of networks. Okay, so let's have a look now at this particular setting. And let's look at the ones, the efficient networks, so the ones that are maximizing total utility in this setting. And they break into 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 then situations where there's a high cost to links, so the cost is above some level. And basically for very low cost of links, the complete network is the unique efficient network. And that's very intuitive. Links are so cheap and in particular, when c is less than delta- delta squared, that's going to be the situation where it's more beneficial to have a direct relationship, right? This is the value for direct relationship 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 cost of adding a link. So adding a link is always going to be beneficial, and the complete network is going to be uniquely efficient in that setting. And it will have to worry about externalities, but that's going to be the conclusion. In the middle 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 that one link to that person. And that's going to be the thing that maximizes total sum of utilizes necessity, and the only thing which maximizes the total sum of utilities. So that does it and its the only type of architecture that does it. And then once costs are so high it just makes sense that nobody should connect any links are just too expensive it 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, if 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 insight here is that you want to have the star networks form. Now, a couple of things to say is that this, is a fairly stark characterization. It's actually going to be true for a set of models beyond what 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 two distance, a value for three distance. And you'll see that everything we say in all the proofs and so forth, would go through if you just separated, put in, 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 and understand stars before we go into a formal proof of this. So letÂ´s think about, we start with one relationship that gives us 2 delta- 2C, and we think about adding a second one. So, there's two different ways we can add this second relationship. One, is that we could connect the person to somebody who's already connected, or they could form a new relationship. If they form a new relationship, then weÃ¨ve got four people connected, four benefits, four costs. Four minus delta four c. If we connect this person here, each one of these 2 still gets a benefit so we'd still end up with 4 deltas and 4c's. 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 it gives us a higher value than connecting in this separate way, okay? So that's the start of it. Now when we think, 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 can connect them back to the centre. And again, we're going to have 6 values and 6 costs coming from the direct links because there's 3 links in either network. And the question is why is it better to do it in the star form? Well, in a star form, all these indirect connections now are at a distance too. Whereas in this one some of the indirect connections are at a distance three. And so by doing it in a star form, we end up with a higher value for all the indirect connections. We get 6 delta squared as opposed to 4 deltas and 2 delta cubes. These longer distance relationships are worth less, they're worth a delta cubed. Which is less than a delta squared and so we get higher value by having more direct connections which come through a star. So those indirect connections are shorter and more valuable that way, okay? And then, if you think about adding a fourth person in. Well, if you add them in directly to the center, again, more value from the indirect connections, than if we added to the periphery. Where now a lot of these indirect things are going to be of distance three, as opposed to all of distance two here in the 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, let's suppose we've got the four individuals and we keep them in a star. Or else we could add extra links in, to connect these two people. Well, what we get here is now, before here when between these people, there 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 that. And so the question is, when is it that this gain in having shorter distances outweighs the cost? And it's precisely 1 delta minus delta squared is bigger than c, right? When is it better to have direct relationship than an indirect relationship? So 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 relationships. So that gives you the idea 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 a 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 quick peek at it. And the next thing I'll do is actually go through a form of proof of this. You can skip that if you like, or you can look through the details. So the formal proof is very 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 pair y stable networks and trying to understand the difference between the networks that are most efficient and the ones that are pair y stable.