Hello there. So I'm back again. And we are here now to talk about strategic network formation, and strategic meaning people are actually making choices. And the strategic aspect means that they have to worry about what other people are doing. And they care actually about things beyond just the, possibly the direct relationships they are involved in with. So we end up with a rich context in which to analyze network formation. So in terms of our outline of where we are in, in the course, we've gone through and we talked quite a bit about random network models. After we did our background and fundamentals. Now we're moving into the course where we're, we're dealing with, again, network formation. But choices of individuals and then, once we've got this behind us, we'll start looking at how networks influence behavior. So in order to understand it, network Economic game theoretic models of network formation, this idea that people are making choices. The, the basic techniques are, we're going to think of, of nodes now as actors that are actively making choices. And there's going to be costs and benefits. I'll, I'll often refer to these as players, so they could be nodes, they could be players, agents. And, it'll be fairly vague, because sometimes they might be countries, sometimes they might be people, choosing friends. Sometimes they might be researchers choosing collab or a, collaborators. Sometimes it might be firms choosing with whom to have a research and development agreement. you know, it could be a whole series of different entities. It might be an employee thinking about a company to work for. so the, the ideas here are that this could be fairly rich, but the main item is that there's actually choice. And what we'll be doing is contrasting the incentives of the individuals to form relationships with what might be best for society, what we'll call social efficiency. What's the overall optimum in terms of networks compared to what people will do when left to their own devices to form networks. So that's the, the basic underlying theme. Now when we, when we, once we go in this direction, we've got all kinds of modeling choices to make. So how do we model the incentives to form in several links. first of all there is consensus needed, is this directed or undirected, network. So, you know, does somebody have to say yes to need to be my friend? Or can I just be friends with somebody other wise. Often in things like citation networks, I can sight somebody else without having their permission. but forming a friendship, an alliance, a calibration agreement, we're going to have consensus. can people coordinate changes in the network, so can we, we at one time say, okay look, I'll form an alliance with you. But only if we can also ally with someone else at the same time. is the process dynamic or static? Is it sort of are we going to think of a lot of people coming together, and forming friendships at the same time or is this on going. How sophisticated are agents? Are these people who are calculating you know, in the case of international agreements, you know. There people calculating what's the value of these, how should they structured or are these. You know, people you know, kids in high school bumping into somebody and, and you know, forming a, a friendship. How much are they forward looking about this? what do they know when they're making a decision? Do they know a lot about the structure? Do they make errors? What's going on? What's generating value? I mean, where are people getting value from this? Where are the benefits coming from? What are the costs? Can people compensate each other? So you know, if, if, somebody's a very valuable friend can I do favors for them to make it more worthwhile? are there different intensities of links? Are we going to think of zero one? There's all kinds of modeling choices which come up and which are very interesting. A lot of these different things have been looked at in the literature. I'm not going to go through all of these different things in detail in the, in the short time that we have together But, instead I'm going to give you some basic feeling for how these kinds of things are modeled, and what the issues are. And then you can, you know, dig into the literature as you see fit. in, in seeing, you know, how people have dealt with a lot of these things. So, we'll do some examples of different variations of these things. But we're not going to go through all of the different issues that you can think of associated with this. Okay, now questions to keep in mind as we are going through this. Which networks are going to form, are some more stable than others, are ones that form going to be the right ones from society's perspective. If they're not, are they way off, or are they pretty close? if somebody wanted to come in and improve the networks, can they do so? So, for instance, if the government decides that there's not enough collaborations between firms in terms of research. And development, can they subsidize things to make more of these research and development alliances come true. can such models provide insight into observed characteristics? So these will be fairly stylized models so we can actually say something about them. Can we take these to data? Is there, are there ways of enriching these to the point that they begin to be, data friendly. Okay, so what I'm going to start with is just a basic approach in terms of, of how we would represent this stuff. And, and it comes out of a work I did with Asher Wolinsky in, in the mid 1990s, or early 90s but 96 by the time it was published. so what, what we're looking at is we're actually going to have a network now generate some payoff to given individuals. So we'll keep track of a utility that a person I or noda I gets from graph G. So an agent I is going to get a payoff, if the network turns out to be g. And for the simplest version of this, we'll think of undirected, but you, you can, you know can do directed, weighted networks and so forth. The, the, the, but, for illustration purposes we'll take this to be undirected network formation. Okay. So let's start with a simple example that came out of the work with Asher in the 90s. So this is just one of the examples that we use to sort of illustrate some of the ideas. So this is called the connections model. It's a very simple model that gives the idea that I get benefits from having friends. I also get benefits from having friends of friends. So the simplest possible network thing is there's values to having friends of friends and friends of friends of friends and so forth. And so we'll think of Sum Delta as a parameter which is going to capture how that decays. So if you remember decay centrality there was some decay, decay centrality is actually based on this model. So the benefit parameters for I for a connection between I and J is going to be some value of Delta. And there can be costs so the value of a, of a, link between I and J, is, could also have some costs associated with it. Now the general version of this model you could have the delta depend on I and j as well. But um,let's, let's, for now let's just take delta to be some parameter. costs between I and J. And when we're thinking about different individuals I could be directly connected to somebody. I could be indirectly connected to somebody. And what's going to happen is the benefits you get from being connected in a network are going to be proportional to the the deltas raised to the power of how far away you are. So for instance if delta is .5 then I get .5 from the direct friendship .25 from an indirect friendship and so forth. .125 from from a friendship of .3 whereas if its .9 then it falls off more gradually right? An indirect friendship is almost worth as much whereas here is worth half as much and so forth. Right, so, so we're, we're sort of moving at, at different rates depending on how big delta is. And then the costs are just going to be costs for maintaining direct friendships. So, if we take a syre-, symmetric version of this model, everybody has the same delta. and so the benefit's of a friend of a friend is dealt a square, and so forth cost's and seagrade of is zero. So if there's just one link in the world, then both of the individuals get a benefit of delta and a cost of c. So their net utility, the worth of this network to them, is delta minus c, in this case everybody else is just getting zero. They're not connected, they're not getting any cost's, they're not getting any benefit's. So this would be you know just one link just delta minus c. Okay so what happens is we add a link. So now we add a link between now one connects to two and one connects to three. one is now involved in two relationships. So they're getting two benefits two deltas paying two costs two c's. Let's look the more interesting utility let's say person two, who's now getting a delta from their direct connection between one and two. And they're also getting an indirect connection. So they're getting a delta squared from this indirect connection to person three.They're still only paying to maintain one relationship. But now, the fact that 1 has this other friend, is beneficial to 2. So, I might, 2 might be getting extra information from 3 through 1. Or getting favors from 3 through 1. Now, delta squared, given that delta is less than 1, is worth less than delta. So, 2 doesn't get as much value from the, the indirect value. Of the relationship to 3, but they still get some value. So the idea here is, you know, for any number configuration, we can assign values. Right? So now if we add another link here, then we can go through. Right now one has two direct connections, one indirect connection of length two Four has a connection to two. A connection to one, a connection to three, delta, delta squared, delta cubed minus c. And so for every network configuration, we can talk about how much value each individual in the network gets. And so, once we've got that then we can talk about, well, which is the most valuable network for society? Which is the one that each individual would like to be in? which is the ones they'll choose, which links will they choose to form if they can make these decisions? so that's the basic idea. All right? And you can just keep adding links, keep doing calculations. As you go through, for each different configuration you can do different calculations. Now here you knwo for instance three when counting the distance to four the way that we work this model is that three counts four as a delta squared. And doesnt account for the fact that they could also have other paths that are longer to get to four. So they have a delta cubed, to 5, and a delta squared to 4, and then the two deltas to the two people they're directly connected with. But they're not counting these extra paths that they have. Now you could enrich this model to have multiple path variations and so forth. The basic ideas here are already going to be captured in the fact that, you know, direct relationships are valuable. Indirect relationships are, are slightly less valuable. you could allow for multiple paths to matter. that something that would enrich this. Okay, so major questions we have. Which networks are best for society, and which ones are going to be formed by the people involved? So we'll, we'll take a look at both of those next. And the idea is going to be that we're going to have to first of all model how the network formation process works. And then also have some ways of evaluating the overall societal benefits from, from the network.