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Okay, so let's have a look at general forms of externalities in, in these

Â network formation models. We're modeling the payoffs to players.

Â And in particular we can differentiate between two types of, of externalities.

Â There can be mixed externalities. But we'll call, think of positive

Â externalities as a situation where if we add a link, ij, to a network g.

Â And we consider some other individual who's not one of the participants in that

Â link that they do weekly better, than they did before.

Â And you know, we can have them strictly positive.

Â We could have it be that people benefit who aren't directly in, involved.

Â But basically, what's happening is, is any spill overs that go to other

Â individuals from a relationship are net positives.

Â So, if I form a new friendship my current friends get value from information that

Â I'm getting or they can get indirect favors.

Â so they're not harmed and sometimes they might be helped.

Â The connections model had positive externalities.

Â Every time we add a link between two individuals, that either shortens paths

Â or keeps them the same for other individuals.

Â Nobody's hurt, and sometimes they're even helped.

Â So, that's positive externalities. Negative externalities is exactly the

Â opposite, and this is a situation where if two people add a link, the other

Â individuals are hurt by that. And this can come about in, you know,

Â some different settings where now I'm losing time with friends or if you're a,

Â a company and two other companies merge. Or form some sort of alliance that might

Â hurt your your ability to compete with them so we can think of situations where

Â ties among other individuals somehow is detrimental to a given individual.

Â and so these kinds of, of externalities often, in some situations they might both

Â be present. Some are going to have more positive,

Â some more negative, but it's useful to keep these in mind when we're thinking

Â about networks and thinking about different structures.

Â What's really going on in terms of the way paths are generated, how are the

Â externalities. Are they positive?

Â Are they negative? And whether they're positive or negative

Â will have different implications for which networks we might want to see.

Â And whether they're going to tend to be underconnected or overconnected.

Â what, what's missing in terms of the extra values that people might not be

Â considering. Okay.

Â So, inefficiency in the connections model was due to the fact that there were

Â positive externalities. And basically you know, the fact that

Â that you know, the star wasn't willing to maintain these external relationships was

Â coming from the fact that those weren't giving that the star, the center of the

Â star any value. And yet there were positive externalities

Â to the other players that the center was not taking into account.

Â So, this lead to the fact that, that there weren't any loose ends and it led

Â to either you know, complete failure or nothing formed.

Â Or overconnections in the sense that we ended up with, with people having to have

Â multiple indirect paths before they were willing to or indirect paths before they

Â were willing to link with somebody. so let's have a look at a different model

Â where we see negative externalities. And this is another example that came out

Â of the paper with Asher, the Jackson and Wolinsky 96 paper.

Â And it's what's known as the Coauthor model.

Â And it's a very simple, another simple variation on, on something you can

Â imagine generating value for individuals. And here people are going to be involved

Â in research collaborations. And the value from each relationship

Â depends on how much time people put into those relationships.

Â And we also get an interaction term which is going to capture the some sort of

Â synergies. That if I spend more time collaborating

Â with somebody, we have more time to brainstorm, get better ideas, and that's

Â going to be valuable. So, in particular, what we do is look at

Â all the friends that a given individual has in a network, so look at all the

Â ij's. And what is a given individual get for

Â each relationship in a network? Well, they get one over their degrees.

Â So, if I have four people that I'm involved with, I spend one fourth of my

Â time with each of them. So, for each one of them, I, I sum up one

Â over my degree, okay, so I split my time. I also get a fraction of their time, so

Â they're putting something into it. If somebody is, that I'm linked to has

Â five relationships, then I get one fifth from them.

Â 4:46

And then the value of this interaction term, this synergy is coming, is

Â proportional to 1 over di times 1 over dj.

Â So, if I spend all of my time with somebody, I'm going to get more synergy

Â than if I spend one quarter of my time with somebody, then I only get one

Â quarter of the synergy. And I get something proportional to how

Â much they, time they put into the relationship.

Â So, the more they drop it, the less productive it is.

Â So, this is going to be maximized when we limit these things to one, and then when

Â we put them as two, three, four and so forth, the value of the synergy is

Â going to decrease, okay? So, if you add these things up, then what

Â do you get? You get a value here which just the, the

Â sum of 1 over di is, is going to come out to be a 1.

Â So, basically what I'm getting is, I look across my coauthors, I get one fraction

Â of their time. And then the synergy term, which is

Â going to be proportional to how much time invest in different things.

Â Okay. So, that's a very simple value.

Â here we didn't, we, we, the benefits, the costs are implicit in this model.

Â We're not going to put in explicit c's or costs to links.

Â The costs from adding extra links come from the fact that you're diluting your

Â synergies with different collaborations. so here, you're just spreading your time

Â out and the more thinly you spread your time the lower the value from any

Â relationship you get, okay? So, we don't put explicit cost into this

Â model. Okay, so let's look at these well, if you

Â have two people together, what's the value?

Â You get 1 over 1 for your time, 1 over 1 for their time, 1 over 1 times 1, so you

Â get 3 as the total value. That's where in our earlier picture those

Â 3's were coming from. If I have a connection here, I split my

Â time between two individuals, I'm going to get a 3.25.

Â Where does that 3.25 come from? Well now, I'm splitting my time between

Â two individuals so I get a half on the first value, a half on the second.

Â 7:05

And then, what's the synergy? The synergy from the person I'm spending

Â with, time with who is spending all their time with me, I get a 1 over a half.

Â I'm spending half my time, they're spending 1.

Â And the other percent, I get a quarter. Add all these up, and you get your 3.25,

Â right? So, we get 1.

Â We get 2, 2.5, 3, 3 and a quarter. So, 3 and a quarter is what this gets.

Â This person is just getting half this person's time plus their own time, plus

Â half the synergy. So, they're getting 2, and so forth,

Â okay? So, you can go through and do these

Â calculations in this model. And that's exactly where these payoffs

Â came from, and again, the efficient thing to do is form pairs.

Â And yet, the unique pairwise stable network was that everybody overconnected

Â and formed a complete network. So, here is another situation.

Â Now, that the externalities are negative and we're seeing people form too many

Â relationships because they're not taking into account the fact that when they form

Â an additional relationship which increases their payoff, right?

Â So here, they went to a 3 to a 3.25, that they're actually decreasing the synergies

Â that they're giving to some other players, which lowers that person's

Â payoff. And so, they end up with lower payoffs

Â when they continue and everybody keeps forming extra relationships.

Â So, this is a situation where we get too many relationships and the overall

Â payoffs are diluted. Okay, so here, no direct costs to links.

Â You can go through and analyze pairwise stable and efficient networks in this

Â efficient networks, it's easy to check. They're always just going to be having

Â individual pairs. That's the best way to do things if n is

Â even. If n is odd and you've got three

Â individuals and so forth, it gets a little more complicated.

Â If you have a lot of people hanging around.

Â But, but generally efficient networks are going to be just splitting society into

Â pairs. pairwise stable networks people are

Â going to over connect in the same way we just saw in that example.

Â And you're going to see, they're going to consist of completely connected

Â components. But there could be some separate

Â components. And what's true is that the components,

Â if there are more than one component, they have to be of different sizes.

Â And each one has to have more than the square of the number of nodes in the

Â other in order to, to work. So, you have to have very different sized

Â components. and you know, but, but basically by

Â adding a link, you would dilute existing synergies and so you only want to add a

Â new coauthor if they bring in sort of comparable worth to, to your own values.

Â And that's what gives these the fact that, that pairwise stable networks, if

Â they have separate components, have to have very different sizes, so that one

Â isn't going to, to group with another. Okay.

Â So, you can go through and check those details it's a fun exercise to play with.

Â to check your thinking on these things. but basically what we're seeing now is

Â negative externalities. Again, there's a difference between

Â efficient networks and what people are going to form and now we're seeing people

Â forming too many relationships. Because they, they aren't taking into

Â account the harm they're doing to other people.

Â Before in the connections model, we saw possibly too few because the center of a

Â star might not be willing to maintain a relationship even though it's beneficial

Â to other individuals. Okay.

Â stable and efficient networks are only going to coincide in special cases.

Â And so, what we can begin to do is then ask, you know, can transfers, suppose we,

Â we start subsidizing the center will that help?

Â Can we say situations, things about when these kinds of conflicts occur?

Â can we say things about when transfers might help improving efficiency our

Â transfers going to be in, in players' interests?

Â So, we have a whole series of questions we can ask to try and rectify these

Â problems. So, those are some of the things we'll

Â take up next in, in looking at whether transfers can help avoid some of the

Â difficulties we have.

Â