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Hello there. So I'm back again.

Â And we are here now to talk about strategic network formation, and

Â strategic meaning people are actually making choices.

Â And the strategic aspect means that they have to worry about what other people are

Â doing. And they care actually about things

Â beyond just the, possibly the direct relationships they are involved in with.

Â So we end up with a rich context in which to analyze network formation.

Â So in terms of our outline of where we are in, in the course, we've gone through

Â and we talked quite a bit about random network models.

Â After we did our background and fundamentals.

Â 0:38

Now we're moving into the course where we're, we're dealing with, again, network

Â formation. But choices of individuals and then, once

Â we've got this behind us, we'll start looking at how networks influence

Â behavior. So in order to understand it, network

Â Economic game theoretic models of network formation, this idea that people are

Â making choices. The, the basic techniques are, we're

Â going to think of, of nodes now as actors that are actively making choices.

Â And there's going to be costs and benefits.

Â I'll, I'll often refer to these as players, so they could be nodes, they

Â could be players, agents. And, it'll be fairly vague, because

Â sometimes they might be countries, sometimes they might be people, choosing

Â friends. Sometimes they might be researchers

Â choosing collab or a, collaborators. Sometimes it might be firms choosing with

Â whom to have a research and development agreement.

Â you know, it could be a whole series of different entities.

Â It might be an employee thinking about a company to work for.

Â so the, the ideas here are that this could be fairly rich, but the main item

Â is that there's actually choice. And what we'll be doing is contrasting

Â the incentives of the individuals to form relationships with what might be best for

Â society, what we'll call social efficiency.

Â What's the overall optimum in terms of networks compared to what people will do

Â when left to their own devices to form networks.

Â So that's the, the basic underlying theme.

Â Now when we, when we, once we go in this direction, we've got all kinds of

Â modeling choices to make. So how do we model the incentives to form

Â in several links. first of all there is consensus needed,

Â is this directed or undirected, network. So, you know, does somebody have to say

Â yes to need to be my friend? Or can I just be friends with somebody

Â other wise. Often in things like citation networks, I

Â can sight somebody else without having their permission.

Â but forming a friendship, an alliance, a calibration agreement, we're going to

Â have consensus. can people coordinate changes in the

Â network, so can we, we at one time say, okay look, I'll form an alliance with

Â you. But only if we can also ally with someone

Â else at the same time. is the process dynamic or static?

Â Is it sort of are we going to think of a lot of people coming together, and

Â forming friendships at the same time or is this on going.

Â How sophisticated are agents? Are these people who are calculating you

Â know, in the case of international agreements, you know.

Â There people calculating what's the value of these, how should they structured or

Â are these. You know, people you know, kids in high

Â school bumping into somebody and, and you know, forming a, a friendship.

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How much are they forward looking about this?

Â what do they know when they're making a decision?

Â Do they know a lot about the structure? Do they make errors?

Â What's going on? What's generating value?

Â I mean, where are people getting value from this?

Â Where are the benefits coming from? What are the costs?

Â Can people compensate each other? So you know, if, if, somebody's a very

Â valuable friend can I do favors for them to make it more worthwhile?

Â are there different intensities of links? Are we going to think of zero one?

Â There's all kinds of modeling choices which come up and which are very

Â interesting. A lot of these different things have been

Â looked at in the literature. I'm not going to go through all of these

Â different things in detail in the, in the short time that we have together

Â But, instead I'm going to give you some basic feeling for how these kinds of

Â things are modeled, and what the issues are.

Â And then you can, you know, dig into the literature as you see fit.

Â in, in seeing, you know, how people have dealt with a lot of these things.

Â So, we'll do some examples of different variations of these things.

Â But we're not going to go through all of the different issues that you can think

Â of associated with this. Okay, now questions to keep in mind as we

Â are going through this. Which networks are going to form, are

Â some more stable than others, are ones that form going to be the right ones from

Â society's perspective. If they're not, are they way off, or are

Â they pretty close? if somebody wanted to come in and improve

Â the networks, can they do so? So, for instance, if the government

Â decides that there's not enough collaborations between firms in terms of

Â research. And development, can they subsidize

Â things to make more of these research and development alliances come true.

Â can such models provide insight into observed characteristics?

Â So these will be fairly stylized models so we can actually say something about

Â them. Can we take these to data?

Â Is there, are there ways of enriching these to the point that they begin to be,

Â data friendly. Okay, so what I'm going to start with is

Â just a basic approach in terms of, of how we would represent this stuff.

Â And, and it comes out of a work I did with Asher Wolinsky in, in the mid 1990s,

Â or early 90s but 96 by the time it was published.

Â so what, what we're looking at is we're actually going to have a network now

Â generate some payoff to given individuals.

Â So we'll keep track of a utility that a person I or noda I gets from graph G.

Â So an agent I is going to get a payoff, if the network turns out to be g.

Â And for the simplest version of this, we'll think of undirected, but you, you

Â can, you know can do directed, weighted networks and so forth.

Â The, the, the, but, for illustration purposes we'll take this to be undirected

Â network formation. Okay.

Â So let's start with a simple example that came out of the work with Asher in the

Â 90s. So this is just one of the examples that

Â we use to sort of illustrate some of the ideas.

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So this is called the connections model. It's a very simple model that gives the

Â idea that I get benefits from having friends.

Â I also get benefits from having friends of friends.

Â So the simplest possible network thing is there's values to having friends of

Â friends and friends of friends of friends and so forth.

Â And so we'll think of Sum Delta as a parameter which is going to capture how

Â that decays. So if you remember decay centrality there

Â was some decay, decay centrality is actually based on this model.

Â So the benefit parameters for I for a connection between I and J is going to be

Â some value of Delta. And there can be costs so the value of a,

Â of a, link between I and J, is, could also have some costs associated with it.

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Now the general version of this model you could have the delta depend on I and j as

Â well. But um,let's, let's, for now let's just

Â take delta to be some parameter. costs between I and J.

Â And when we're thinking about different individuals I could be directly connected

Â to somebody. I could be indirectly connected to

Â somebody. And what's going to happen is the

Â benefits you get from being connected in a network are going to be proportional to

Â the the deltas raised to the power of how far away you are.

Â So for instance if delta is .5 then I get .5 from the direct friendship .25 from an

Â indirect friendship and so forth. .125 from from a friendship of .3 whereas

Â if its .9 then it falls off more gradually right?

Â An indirect friendship is almost worth as much whereas here is worth half as much

Â and so forth. Right, so, so we're, we're sort of moving

Â at, at different rates depending on how big delta is.

Â And then the costs are just going to be costs for maintaining direct friendships.

Â 8:24

So, if we take a syre-, symmetric version of this model, everybody has the same

Â delta. and so the benefit's of a friend of a

Â friend is dealt a square, and so forth cost's and seagrade of is zero.

Â So if there's just one link in the world, then both of the individuals get a

Â benefit of delta and a cost of c. So their net utility, the worth of this

Â network to them, is delta minus c, in this case everybody else is just getting

Â zero. They're not connected, they're not

Â getting any cost's, they're not getting any benefit's.

Â So this would be you know just one link just delta minus c.

Â Okay so what happens is we add a link. So now we add a link between now one

Â connects to two and one connects to three.

Â one is now involved in two relationships. So they're getting two benefits two

Â deltas paying two costs two c's. Let's look the more interesting utility

Â let's say person two, who's now getting a delta from their direct connection

Â between one and two. And they're also getting an indirect

Â connection. So they're getting a delta squared from

Â this indirect connection to person three.They're still only paying to

Â maintain one relationship. But now, the fact that 1 has this other

Â friend, is beneficial to 2. So, I might, 2 might be getting extra

Â information from 3 through 1. Or getting favors from 3 through 1.

Â Now, delta squared, given that delta is less than 1, is worth less than delta.

Â So, 2 doesn't get as much value from the, the indirect value.

Â Of the relationship to 3, but they still get some value.

Â So the idea here is, you know, for any number configuration, we can assign

Â values. Right?

Â So now if we add another link here, then we can go through.

Â Right now one has two direct connections, one indirect connection of length two

Â Four has a connection to two. A connection to one, a connection to

Â three, delta, delta squared, delta cubed minus c.

Â And so for every network configuration, we can talk about how much value each

Â individual in the network gets. And so, once we've got that then we can

Â talk about, well, which is the most valuable network for society?

Â Which is the one that each individual would like to be in?

Â which is the ones they'll choose, which links will they choose to form if they

Â can make these decisions? so that's the basic idea.

Â All right? And you can just keep adding links, keep

Â doing calculations. As you go through, for each different

Â configuration you can do different calculations.

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Now here you knwo for instance three when counting the distance to four the way

Â that we work this model is that three counts four as a delta squared.

Â And doesnt account for the fact that they could also have other paths that are

Â longer to get to four. So they have a delta cubed, to 5, and a

Â delta squared to 4, and then the two deltas to the two people they're directly

Â connected with. But they're not counting these extra

Â paths that they have. Now you could enrich this model to have

Â multiple path variations and so forth. The basic ideas here are already going to

Â be captured in the fact that, you know, direct relationships are valuable.

Â Indirect relationships are, are slightly less valuable.

Â you could allow for multiple paths to matter.

Â that something that would enrich this. Okay, so major questions we have.

Â Which networks are best for society, and which ones are going to be formed by the

Â people involved? So we'll, we'll take a look at both of

Â those next. And the idea is going to be that we're

Â going to have to first of all model how the network formation process works.

Â And then also have some ways of evaluating the overall societal benefits

Â from, from the network.

Â