Okay, so let's have a look at general forms of externalities in, in these network formation models. We're modeling the payoffs to players. And in particular we can differentiate between two types of, of externalities. There can be mixed externalities. But we'll call, think of positive externalities as a situation where if we add a link, ij, to a network g. And we consider some other individual who's not one of the participants in that link that they do weekly better, than they did before. And you know, we can have them strictly positive. We could have it be that people benefit who aren't directly in, involved. But basically, what's happening is, is any spill overs that go to other individuals from a relationship are net positives. So, if I form a new friendship my current friends get value from information that I'm getting or they can get indirect favors. so they're not harmed and sometimes they might be helped. The connections model had positive externalities. Every time we add a link between two individuals, that either shortens paths or keeps them the same for other individuals. Nobody's hurt, and sometimes they're even helped. So, that's positive externalities. Negative externalities is exactly the opposite, and this is a situation where if two people add a link, the other individuals are hurt by that. And this can come about in, you know, some different settings where now I'm losing time with friends or if you're a, a company and two other companies merge. Or form some sort of alliance that might hurt your your ability to compete with them so we can think of situations where ties among other individuals somehow is detrimental to a given individual. and so these kinds of, of externalities often, in some situations they might both be present. Some are going to have more positive, some more negative, but it's useful to keep these in mind when we're thinking about networks and thinking about different structures. What's really going on in terms of the way paths are generated, how are the externalities. Are they positive? Are they negative? And whether they're positive or negative will have different implications for which networks we might want to see. And whether they're going to tend to be underconnected or overconnected. what, what's missing in terms of the extra values that people might not be considering. Okay. So, inefficiency in the connections model was due to the fact that there were positive externalities. And basically you know, the fact that that you know, the star wasn't willing to maintain these external relationships was coming from the fact that those weren't giving that the star, the center of the star any value. And yet there were positive externalities to the other players that the center was not taking into account. So, this lead to the fact that, that there weren't any loose ends and it led to either you know, complete failure or nothing formed. Or overconnections in the sense that we ended up with, with people having to have multiple indirect paths before they were willing to or indirect paths before they were willing to link with somebody. so let's have a look at a different model where we see negative externalities. And this is another example that came out of the paper with Asher, the Jackson and Wolinsky 96 paper. And it's what's known as the Coauthor model. And it's a very simple, another simple variation on, on something you can imagine generating value for individuals. And here people are going to be involved in research collaborations. And the value from each relationship depends on how much time people put into those relationships. And we also get an interaction term which is going to capture the some sort of synergies. That if I spend more time collaborating with somebody, we have more time to brainstorm, get better ideas, and that's going to be valuable. So, in particular, what we do is look at all the friends that a given individual has in a network, so look at all the ij's. And what is a given individual get for each relationship in a network? Well, they get one over their degrees. So, if I have four people that I'm involved with, I spend one fourth of my time with each of them. So, for each one of them, I, I sum up one over my degree, okay, so I split my time. I also get a fraction of their time, so they're putting something into it. If somebody is, that I'm linked to has five relationships, then I get one fifth from them. And then the value of this interaction term, this synergy is coming, is proportional to 1 over di times 1 over dj. So, if I spend all of my time with somebody, I'm going to get more synergy than if I spend one quarter of my time with somebody, then I only get one quarter of the synergy. And I get something proportional to how much they, time they put into the relationship. So, the more they drop it, the less productive it is. So, this is going to be maximized when we limit these things to one, and then when we put them as two, three, four and so forth, the value of the synergy is going to decrease, okay? So, if you add these things up, then what do you get? You get a value here which just the, the sum of 1 over di is, is going to come out to be a 1. So, basically what I'm getting is, I look across my coauthors, I get one fraction of their time. And then the synergy term, which is going to be proportional to how much time invest in different things. Okay. So, that's a very simple value. here we didn't, we, we, the benefits, the costs are implicit in this model. We're not going to put in explicit c's or costs to links. The costs from adding extra links come from the fact that you're diluting your synergies with different collaborations. so here, you're just spreading your time out and the more thinly you spread your time the lower the value from any relationship you get, okay? So, we don't put explicit cost into this model. Okay, so let's look at these well, if you have two people together, what's the value? You get 1 over 1 for your time, 1 over 1 for their time, 1 over 1 times 1, so you get 3 as the total value. That's where in our earlier picture those 3's were coming from. If I have a connection here, I split my time between two individuals, I'm going to get a 3.25. Where does that 3.25 come from? Well now, I'm splitting my time between two individuals so I get a half on the first value, a half on the second. One of the people I'm getting, they're spending all of their time with me, the other person is spending half their time with me. And then, what's the synergy? The synergy from the person I'm spending with, time with who is spending all their time with me, I get a 1 over a half. I'm spending half my time, they're spending 1. And the other percent, I get a quarter. Add all these up, and you get your 3.25, right? So, we get 1. We get 2, 2.5, 3, 3 and a quarter. So, 3 and a quarter is what this gets. This person is just getting half this person's time plus their own time, plus half the synergy. So, they're getting 2, and so forth, okay? So, you can go through and do these calculations in this model. And that's exactly where these payoffs came from, and again, the efficient thing to do is form pairs. And yet, the unique pairwise stable network was that everybody overconnected and formed a complete network. So, here is another situation. Now, that the externalities are negative and we're seeing people form too many relationships because they're not taking into account the fact that when they form an additional relationship which increases their payoff, right? So here, they went to a 3 to a 3.25, that they're actually decreasing the synergies that they're giving to some other players, which lowers that person's payoff. And so, they end up with lower payoffs when they continue and everybody keeps forming extra relationships. So, this is a situation where we get too many relationships and the overall payoffs are diluted. Okay, so here, no direct costs to links. You can go through and analyze pairwise stable and efficient networks in this efficient networks, it's easy to check. They're always just going to be having individual pairs. That's the best way to do things if n is even. If n is odd and you've got three individuals and so forth, it gets a little more complicated. If you have a lot of people hanging around. But, but generally efficient networks are going to be just splitting society into pairs. pairwise stable networks people are going to over connect in the same way we just saw in that example. And you're going to see, they're going to consist of completely connected components. But there could be some separate components. And what's true is that the components, if there are more than one component, they have to be of different sizes. And each one has to have more than the square of the number of nodes in the other in order to, to work. So, you have to have very different sized components. and you know, but, but basically by adding a link, you would dilute existing synergies and so you only want to add a new coauthor if they bring in sort of comparable worth to, to your own values. And that's what gives these the fact that, that pairwise stable networks, if they have separate components, have to have very different sizes, so that one isn't going to, to group with another. Okay. So, you can go through and check those details it's a fun exercise to play with. to check your thinking on these things. but basically what we're seeing now is negative externalities. Again, there's a difference between efficient networks and what people are going to form and now we're seeing people forming too many relationships. Because they, they aren't taking into account the harm they're doing to other people. Before in the connections model, we saw possibly too few because the center of a star might not be willing to maintain a relationship even though it's beneficial to other individuals. Okay. stable and efficient networks are only going to coincide in special cases. And so, what we can begin to do is then ask, you know, can transfers, suppose we, we start subsidizing the center will that help? Can we say situations, things about when these kinds of conflicts occur? can we say things about when transfers might help improving efficiency our transfers going to be in, in players' interests? So, we have a whole series of questions we can ask to try and rectify these problems. So, those are some of the things we'll take up next in, in looking at whether transfers can help avoid some of the difficulties we have.
