0:40

The difference is really going to be in the application.

Â So this has to be a setting where I can really connect to somebody without them

Â having to want to, to allow me to connect to somebody.

Â So it does work in settings like citing somebody's article.

Â I can do that without their permission, or linking to a webpage.

Â I can follow somebody on Twitter and so forth without them having to follow me.

Â So there are setting where you can have this kind of diretic, directed setting.

Â And so what we'll do is we'll model that as a very simple situation where people

Â just announce their preferred set of neighbors.

Â And then a network forms if and he based on which ever links people want to form.

Â And here we're going to keep track of ordered pairs.

Â Now, so the fact that ij is in this network, ij is now different from ji, so

Â this means i is following j. this means j is following i, or, or i

Â formed a direct link to j, and, and so forth.

Â And then just look at the Nash set of networks, where each person is forming

Â some links to some the links that they want.

Â Given the links that everyone else is forming.

Â [BLANK_AUDIO] Okay now when we think about these kinds of settings, we have to

Â think about the flow of payoffs too. So one situation is sort of a one way

Â flow. If I follow somebody on Twitter and they

Â don't follow me, then I get to hear what they say but they don't hear what I say.

Â so that's a setting where the person who pays the cost then hears information from

Â the, the node that they're accessing. Now two way flow could be that one player

Â forms a link and bears the cost but both benefit from it.

Â So it could be that for instance if I add a link to another page on the Internet,

Â then that's good for me because people can get from my page from the other.

Â It also helps the other page get new traffic that I direct there.

Â phone calls, you know, they, they, the, there's you could think of this, as

Â somebody bears the cost. two people end up but phone calls

Â actually are across to both people involved if you think about time and so

Â forth. So there's some situations where we could

Â think about two way flow. Now the two way flow, there's a paper by

Â Bala and Goyal in 2000 basically did a directed version of the connections

Â model. So the model we can think of as being the

Â same as what we saw from the Jackson Lewinsky version.

Â But now what's going to be different is instead of having consent to form a

Â relationship. People can form a relationship

Â unilateraly. And I can just direct my link to somebody

Â else. We'll both end up benefiting, so the

Â benefits will look just like the the connections model we had originally, but

Â now people can form a link unilaterally. So it's a different formation process but

Â the same payoff structure. so the person, the benefits structure,

Â the cost structure is going to be different.

Â It's only the person forming the link that bears the cost.

Â So if you just want to go through in terms of efficiency.

Â The efficiency's going to be exactly the same as before.

Â Except now that we have half as much costs being born, so instead of having 2c

Â per link, we're going to have 1c per link.

Â And then in terms of what that does for the efficient calculations, is it just

Â factors everything down by a, a factor of 2 in terms of the costs.

Â So the efficiency in this model is exactly as it was before except for we've

Â just divided through by 2. and so you get complete networks if costs

Â are low enough, star networks in the middle range and the empty network when

Â costs are high enough. One thing here is I put complete and star

Â in quotes, because complete doesn't mean that every link in both directions is

Â present. It just means that every two nodes are

Â connected, and in this case, they're connected by only one link.

Â It doesn't make sense to have links in both directions because you bear twice

Â the cost and there's no added information flow in this situation.

Â So, if you, if you go through this, it's, it's similar architectures.

Â slightly different cost structure. Somebody has to bear the cost.

Â But you only want one person bearing the cost between any two individuals, because

Â things flow in both directions once that's formed.

Â So efficiency's exactly the same as it was before.

Â when we begin to look at the Nash stable networks, I'll sort of list through what

Â you, what you can find. You can check this.

Â so if, if cost is very low, then the sort of complete network will be Nash stable.

Â So basically it makes sense to cut an indirect relationship to a direct

Â relationship. And somebody, if, if the other person's

Â not doing anything, the other, you know. If one person's not doing it, the second

Â person will have an incentive to do it. Medium cost range, medium low, all star

Â networks are Nash stable, plus some other networks.

Â Whereas a star now could be formed in, in different directions, so it could be that

Â the centre is forming some of the links. And peripheral agents are forming some of

Â the others, so you can have a star with multiple directions on the links.

Â again, you know, it doesn't make sense to form links in both directions because it

Â doesn't add any benefits, and adds to cost.

Â so you'll see stars that are going to be Nash stable, and depending on, the

Â configuration. it, it could be that center's bearing

Â more cost or the periphery is bearing more cost, there can be combinations of

Â different types of stars. So any star in that network is going to

Â be Nash stable. when you get to a higher cost so that c

Â is bigger than delta, now let's think a little bit about comparing two stars.

Â So if we look at a situation where we've got a, let's look at two extremes.

Â One extreme is where the periphery formed a link.

Â So they all link to the center. Okay.

Â 7:12

Well what this says is this one is the only one of the stars that's going to be

Â Nash stable. And why is that?

Â because if you think about it, here the center is bearing a cost by connecting to

Â somebody, and they're only seeing one delta benefit.

Â So that's less than the cost, they would rather sever that link.

Â So they're in a situation where they don't want to be maintaining

Â relationships to other peripheral agents. If the peripheral people don't bring them

Â indirect benefits. This one is stable.

Â Why? Because by accessing the center these

Â people are getting all kinds of indirect benefits of paths of length too right.

Â So they've got a bunch of indirect they get a delta plus n minus 2 delta squared

Â by forming the one link minus the cost. So that peripheral agents are willing to

Â keep those things because they're they're getting this extra cost.

Â when we look at a situation where the, when the cost becomes high enough, , then

Â we end up with a situation where if things can be, have complete networks, be

Â efficient, but not equalibria. So,Nash stable networks are going to

Â depend on the exact cost structure. What, what's a little bit different here

Â then wha, was before is that the, the relative costs are going to select out

Â who might be forming the links. And so you can have some prediction about

Â links going in one direction or the other, even though flow might go in both

Â directions. Okay?

Â So let's talk briefly about an, another version of a directed model, and this is

Â a one way flow model. So now we have to keep track of directed

Â flow. And this is when I form a link that I can

Â access this person, I can listen to them and I can listen, I hear things

Â indirectly that they listen to or here. So if I connect to somebody and they're

Â connected to somebody else and so forth. Then the benefits flow back to me from

Â these connections. But the fact that I connect it to them

Â doesn't give them this person any benefits unless they've also connected

Â back to me or have some indirect connection back to me.

Â Okay, so thing flow in the direction and so the person who bares the cost also

Â gets the benefits. The other person does not get the

Â benefits. Okay?

Â So one example of this is a directed connections model with no decay, where

Â the delta is basically 1. And in that situation then, the utility

Â becomes just the number of people that you can access via directed paths, minus

Â however many links I formed out, outward. So what's my outward degree in the

Â network times the cost, right. So I'm bearing a number.

Â So if I have three different links then I'm going to have an out degree of 3 and

Â how many people I reach is going to depend on how many people they're

Â connected to. Right?

Â So in this case, it might be that I can reach, 1, 2, 3 directly, plus 4, 5, 6, 7,

Â 8 in total. And, so my reach would be eight.

Â Okay, so in that situation, then you've got some number of people that you access

Â for your, your links. Minus your degree times the cost for

Â degree is now measured in outward degree. Okay, so, very simple payoff structure.

Â And efficient networks in this world, as long as c is less than n minus 1, are

Â going to be wheels. In particular, wheels that have a

Â particular direction to them. You should be careful because those

Â wheels also mean something else in, in graph theory.

Â But Bala and Goyal used to work wheels so, basically this is a directed graph

Â where each node points to a subsequent node which points back.

Â And so, each one here can access all of the others.

Â So if you want to think what's the best way to do this in so that each node can

Â access every other node. So we got maximum reach, everybody's

Â reaching n minus 1. We're doing this with only n links, so

Â everybody's sponsoring one link and we're getting the maximum reach possible in

Â this world. Right?

Â So, and each, each link is responsible then for connecting a given individual to

Â n minus 1 other individuals. Right?

Â So we, we've got a situation where we have the most efficient architecture.

Â possible being a wheel. Or if the cost goes above this, then it

Â doesn't make sense to have any links. And, and then you're better off with the

Â empty network.

Â [BLANK_AUDIO]

Â no stable networks when cost is, is low enough.

Â then the wheels turn out to be the only Nash stable networks that are strict Nash

Â equilibria. So that everybody has a strict incentive

Â to keep the the nodes that they links that they have.

Â 12:35

if, if the cost gets to a higher range then, wheels and empty networks are the

Â only strict, Nash stable networks. So then you get stuck at the empty

Â network, because you, nobody wants to form a ring to somebody else unless, they

Â bring in direct benefits. so, so basically what we've got here is a

Â situation where again we have some conflict between stability and

Â efficiency. You can go through and convince yourself

Â of, of why these things are true. the strictness is important here in

Â getting these things. There's lots of, of networks which are

Â Nash equilibria that are not strict Nash equilibria.

Â So for instance here we've got a situation where this is Nash stable.

Â You can check that nobody wants to delete any link that they have, so for instance

Â 2 accesses 1 and also accesses 3 and 4 via this link, so they're happy.

Â 3 can reach everybody via the links, but they have to have 2 links because they

Â have to reach 4 as well as, as 2 and 1, and, and so forth.

Â You can go through and check this. But, 1 is actually indifferent between

Â where they place this last link, right? They're indifferent between having it

Â here. They could also put it to 4 instead.

Â And they could access everybody through either connecting to 3 or connecting to 4

Â since there's direct links in both directions between 3 and 4.

Â Okay? So this is Nash stable, but not strictly

Â Nash stable. So the if you check on the wheel, if we

Â go back to the wheel then that's a a situation where nobody.

Â So if we did this one in terms of a wheel, now anybody who changes the the

Â links of severed this one and put it somewhere else.

Â You would actually lose access to somebody.

Â So they would no longer be able to be given the direction of the network now,

Â the one way flow part they would no longer be able to access that node.

Â So strictness is important in, in making those results come out.

Â Okay, so that's just a glimpse at directed network formation.

Â Theres going to be different applications, and I think that's one

Â thing that's important to emphasize here is that which.

Â Model that we should be using has nothing to do with whether we like unilateral or,

Â or mutual consent formation as a, as a model.

Â What's important is, what does the application actually demand?

Â So, if we're dealing with situations with alliances, or friendships, or social

Â relationships. A whole series of things you really need

Â to have mutual consent and, and two way formation processes.

Â If you're doing things like citing an article, forming a link on a web page.

Â then you can deal with unilateral network formation, and so it's not a question of

Â which is a better model. there are different models and they fit

Â different applications. And some network settings are ones that

Â are naturally directed and unilateral. others are ones where mutual consent is

Â really needed, and a lot, a lot of social settings are going to fit into that

Â category. And so which model you use really depends

Â on the application. It's not an issue of which one's you

Â know, sort of a, a nicer model; they're, they're a different models.

Â Okay. so that sort of takes us through a lot of

Â modeling. Of, of, of strategic network formation

Â and some basic looks at different issues. And we'll have a couple of looks at

Â models for fitting these things in in dealing with data.

Â And then we're going to turn to diffusion as the next major topic.

Â So we'll start working. This has been, so far, we've been looking

Â at network formation, and the next major subject is now, given the network, what's

Â actually happening on that network? How do we understand behavior on that

Â network? And what the consequences of different

Â network formations for the actual behaviors that result.

Â