Hi folks. So we're back and we want to talk about enriching some of the strategic formation models to be have a little more heterogeneity in them. So that we can help expalin some observed fact and data. And so we're still in the part of wrapping up the strategic network formation. And in particular what we're going to do is enrich things basically just in terms of cost structure. so cost of forming relationships can depend on geography and characteristics of nodes. So, it's easier to be friends with somebody who lives very close by. It's easier to relate to people with similar backgrounds. you could also imagine that the benefits would depend on the characteristics of knowledge. So that people with similar characteristics find it easier to work together or share, share risk and so forth. there could also be complementarities in benefits from diversity. There's a lot of different ways we can enrich these models. We're going to do this in a very simple way just to get some ideas out. and the idea here is, is that we can get the so called small worlds. observations out from cost benefits, so we want to get simultaneous effects networks tend to have short average path length. And at the same time have high clustering. And so we want to look and see whether we can explain that with strategic model. And let me just give you the, the, the basic intuitions before we get into the details. the ideas here are going to be that effectively, the fact that there might be very low costs to linking to people. Who are very similar or very close by is going to give high clustering. So we'll get very dense networks on the local level, just because those relationships are easy to have. high value to distant connections means that then we'll have a low diameter. So if I'm not connected to somebody at, at a great distance. I'm not accessing part of a network that's far away from me and forming relationships with people. Who, who are distant can give me access to a lot of information or people I don't have access to before. So that tends to give high benefits to those which help string the diameter and the high cost of distant connections means you're not going to have too many distant links. So you'll have high density on a low level, a local level. a few long distance links so that's not to diminish the clustering too much but you'll still have a lower diameter. Because people will connect far away if you're not already connected. So that's the basic idea, and let's just go through the logic in a little more detail. So there's a whole series of models that have basically looked at the variations on the connections model, where geography is added in some way. And what I'll do is just take you through one version of that model, where we have people living on an island. And people that live on the same island can connect to each other very easily and there's different islands. So there's, so there's a, a cost little c for connecting to somebody who's on your same island, and a cost big C to linking to somebody on another island. But then the benefits that deltas and so forth are exactly as they were in the original connections model. And what this will do is give us high clustering within islands, few links across islands. But we'll still have enough links across islands to have small distances at least for some perimeter values. Okay, now the, the island here are metaphors. For it could be geography but it also could be characteristics so people with very similar characteristics find it very easy to link to each other. People with different characteristics find it more costly so the islands are, are metaphor but a fairly obvious one. Okay so let's have a, a peak at, at some versions of this. So imagine that we look at a given node here, in, in network, like this, where we have a, here the five individuals. in each group here on islands, so these are different islands. So the J the number J here is equals five and we also have five islands. So we have five islands and five individuals per island, and we can go through and look at the, the value to a given individual from their links. So for instance if this individual is in this particular network, what's their payoff? Well, there, they have four little c's because they're connecting to the people on their own island. They're also getting four direct delta's. they've got a delta squared which comes from their connection to somebody to away. They've got seven delta cubes from people three away and 12 delta to the fourth from people at a distance of four. Okay so, this is the similar to the connections model but now what we've done is enriched the cost structure, to have this geography involved. Okay so what, what depending on whether you have only distant connections or only close by connection or some combination. The pay outs are going to differ. So in this case, we can see that this individual here is maintaining a connection to somebody on another island. They're paying a large cost for that but their seeing additional benefits. And lower distances than an individual who's not connected across the islands. So there can be incentives for somebody to connect and also if that person was not connected then nobody in that island could access anybody on another island. So, so, if that person was to sever that link, they would lose connections with the other islands. Okay, so basically what happens here, low cost to an island. means that you want to connect within your island. High cost across islands means that you only want to have limited number of connections across islands. So here is a situation where for instance if the little c is below 0.04 the big C is bigger than 1 and still less then 4.5 so you still want to have some outside connections. Delta is reasonably large, 0.95, then this is a pairwise stable network. So you can go through and check that nobody wants to delete a link. And no two individuals who were not linked would want to add a link, so you can go through and do all those checks for these parameter values. hopefully I didn't make an error on that, but you can go ahead and check that[COUGH] and here what we end up with high clustering and low diameter. So we end up with high clustering given that many individuals have all of their friends talking to each other. and, we end up with low diameters because the greatest distance from somebody to somebody else in this network is 4, right? So the diameter here, is, is 4. And, you know, so, so, if you, if you kept enriching this mar to have more and more islands. you would end up with, you know, very large number of nodes, relatively low diameter. so in this case, we get high clustering, low diameter. you know, obviously, this is, is not, is still a very stark model. It ends up having very particular stable networks that are going to have certain kinds of regularities and degree and so forth. Which won't end up matching reality. But what it does do, is it gives us a different explanation and reasoning behind why you might see small worlds. And we can begin to, to enrich this kind of model with some random formation to begin to try and fit things to data. so you can go through and, you know, prove things about this model, so in the paper Jackson Rogers 04. We proved some things about this model so here, this is a, a. First of all, you can truncate the connections model. So, you only get value as long as you're within some distance, maximum distance of people. So you don't get infinite if, if I'm at distance of 50 from somebody, I don't get any value from that. So, you can put in some cap d, say for instance a value friends out of a distance three or a distance four et cetera. Then you can go through this Islands model and basically you can show that if the little c is small enough. And the big C isn't too large, then players on each island form a clique. You get a bound on the diameter. So here you're going to get clustering from the, within island connections. You're going to get a bound on the diameter, and if the big C is large enough. Then you won't have too many inter island connections, and you can get a lower bound on clustering. which, is depends on the number of islands and the size of each island. So basically what you can get is, is you know, some proof that this is a process, uh,[INAUDIBLE] . A set of properties which'll hold four parameters values within this kind of geographics connections model. The important thing to take away from this is, is now we see that, we're getting clustering because it's cheap to connect to people who are close by. And we're getting low diameter because it's, there's a high value to connecting to people who, to whom you have only in very long indirect paths. And so, the diameter's going to be limited just to the fact if, if there was too many missing connections then it would pay for somebody to add them. Okay, so in terms of the summary of the strategic formation we've gone through so far we've got efficient and stable networks need not coincide. even when some transfers are possible and with complete information. The details of this depend on the setting, which kinds of transfers we might make. we didn't talk too much about forward looking, but that's something that you can add to these models. and you can match and explain some of, observables, with these kinds of models. so in terms of the strengths of the, of this kind of approach. a big part is that the payoffs allow us to have a welfare analysis. So, we can say something about which are good networks, which are bad ones. And we can identify trade offs between individual incentives to form relationships and societal goals. And so we, we, we end up with is, is a real understanding of whether or not, the process of, of, networking formation that's out there is leading to good ones or bad ones. And we're also tying the nature of the externalities to network formation. So are there positive extrenalities, are there negative ones, how does this depend on context. and so when we also end up accounting for some observable facts like clustering and the low diameter. there's an answer of, of why this might happen because of, of certain features rather than sort of a, a mechanical model which imitates it. We have an idea of, of, what kinds of fundamental assumptions about human behavior would lead to this. Now of course the, the problems with the economic approach that we've talked about so far. is that the, a lot of the models we've, we've looked at in order to solve them analytically, have been very very simple. So they tend to be very stark over the regular lots of symmetry. And we want to add hydrogeneity to those. if, if we want to enrich those models we're going to have to add something, add some heterogeneity. simulations can help then If you want to take these to data. You can enrich the models, simulate them and see what happens. there's also a question of whether or not, you know, things are basically currently at random or whether people are really making determined choices. And I think depending on the application, thing might be swayed very much toward individual stategic choices. And in other applications it might be very much at random. And so depending on the application you, you might want some mixture of those two. in, in terms of, of applying these things, one challenge is figuring out what the payoffs are. So what are the payoffs? how do we relate network structure and outcomes to payoffs? How do we identify that? That's not an easy question, and it depends very much on the context. So you have to really begin to think about what is it, what is it that motivates people to form relationships? Why do they maintain certain kinds of relationships? What are they, what are they getting from those things? What's influencing their behavior. So that's an important element that needs to go into these kinds of models. So, models that start to marry the strategic random network models that we've seen before. And the, you know, put these two types of models together, are, are really needed. the strengths of the random network models, the, in terms of being able to match data or, or fit to data. are, are, are to some extent the weaknesses of the economic approach, and vice versa. So we've got basically two sets of models with very complimentary types of, of properties. And mixing these together would then allow for us to do this kind of welfare and efficiency analysis. Understand why things are happening, take the model to data and do so across a wide range of applications. these kinds of models are being developed. we'll talk about some of those in, in some other videos.