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Okay, so we've got some basic ideas behind strategic formation.

Â And now, we're going to talk about modeling how we measure efficiency and

Â also how we can model people's decisions form and delete links.

Â And so, in terms of modelling incentives and you know, just sort of keeping track

Â of these things, let me just first make a very simple point.

Â so let's think of a world where we need consensus to form links.

Â So, two individuals have to agree to be friends, and I can't form a friendship

Â with somebody else if they don't agree to be friends with me.

Â Now one way people might think of doing, of modeling this would be as just

Â modeling it as a game. And the simplest possible game you can

Â imagine is everybody just announces who they want to be friends with.

Â And then if both people, if two people both announce each other, then we form a

Â friendship between them, and if they don't both announce each other, then we

Â don't form a friendship. Okay.

Â Well, Nash equilibrium would be a situation where nobody wants to change

Â their list of announcements given the set of announcements of other individuals.

Â And so, l-, let's just talk through why that is doesn't really work so well as a

Â m-, model of, of network formation in these kinds of settings.

Â And imagine that we're in a setting where they're just two individuals.

Â And if they're separate, they get a value of 0, and if they're connected, they get

Â a value of 1. Okay, and now we have a game where they

Â beat, they simultaneously announce whether they're willing to form their

Â relationship. well both of these are Nash equilibria.

Â 1:44

it in this case it's a situation where if, if I don't think the other person's

Â going to, going to announce me, that doesn't matter what I do, I can't get the

Â friendship anyway, so I might as well not announce it.

Â So, there's an equilibrium where neither person announces the other, because they

Â can't you know, add or really change things.

Â There is also an equilibrium where both of them announce the friendship and then

Â it firms. And so there is two nash equilibria and

Â in this case that's somewhat problematic as a model here.

Â Because this is the simplest possible model and it predicts anything it could

Â happen in terms of linking to forming or not forming.

Â And yet, any reasonable, communication among these individuals should lead to

Â this link for me. Now there are ways of dealing with this

Â in terms of the Game Theory. We can put in stronger solutions concepts

Â and trying to do things that way, so instead of looking at Nash equilibrium we

Â can allow for slight errors by players and see what happens.

Â Or, but there are other examples that aren't quite as simple as this one which

Â give other solution concepts trouble. So what we're going to do is instead of

Â using off the shelf non cooperative gain theory, we're going to model incentives

Â using a very simple concept which we'll call pairwise stability...

Â 'Kay? So the idea here is, is simple.

Â What we'll do is, is we'll look at a network and we'll say that it is stable.

Â 3:15

In particular we'll say that it's Pairwise Stable.

Â If nobody who's involved in some particular relationship would gain by

Â deleting that relationship. Okay.

Â So in a situation where it takes mutual consent to form a link, either person can

Â get rid of the link. So if one person decides they don't want

Â to go and be friends with somebody else they sever that relationship.

Â So one person can delete a relationship and it takes two to form one.

Â So what we, what we do here is we have a situation where no single agent is

Â going to gain from deleting the link, and no two agents both gain from adding a

Â link with at least one strictly gaining. So we can worry about indifferences, but

Â the idea is that if two people both gain weekly, and somebody gains strictly, then

Â a relationship should form. Beneficial relationships should be

Â pursued when available, and ones that aren't beneficial should not be pursued

Â and should be deleted. Very simple concept in that already is

Â what we'll begin to, to put a lot of structure on networks.

Â So in terms of notation, pairwise stability is, is defined as follows.

Â So we'll say that the network, we'll say that g is pairwise stable, so this is

Â pairwise stability of a network g. It's stable if the utility of i for any

Â link that they're involved in is greater than or equal to the, what they would get

Â from deleting the link. Okay, if it was less they should have

Â deleted it. So for g to be stable it should be that

Â they get at least as much as they would get from deleting any of the links

Â they're in. Nobody gains from severing the link.

Â And if somebody really wants a new link, if somebody wants a new link it should be

Â that the other person didn't want it. Otherwise it should be added, ok?

Â So for this to be stable and and not to be subject to further changes, it should

Â that if somebody wants to add it, their partner doesn't want to add it.

Â So if some like i j is not in g. This, if i g is not in g then it should

Â be that if one person wants it the other person doesn't want it.

Â So it could be that neither person wants it, but it can't be that they're both

Â happy to have this link and it not to be in there.

Â Okay. So this is a very weak concept.

Â 5:36

Why is it weak? Well, it's only looking at pairs of

Â individuals. It's only looking at one link at a time.

Â And it just makes sure that there's no single link that would be better deleted.

Â And no link that's not present that would be better to add.

Â Okay. But often this already, is fairly

Â powerful. So sort of the minimal set of

Â requirements that we might think of in terms of stability.

Â It'll often began to narrow things down. Now, there's all kinds of other

Â variations. So this is this concept came out of the

Â paper with Ashford Lowenski. so the Ashford Lowenski 96 paper, people

Â have looked at a lot of other variations on these kinds of things, we'll talk

Â about some of them but, you know this, this will give us some basics to work

Â with. Okay.

Â So, so now when we go back to that example we had before both are Nash

Â equilibria, but this is the only pairwise stable[UNKNOWN], right.

Â So both people would gain by adding this pairwise ability just says that this is

Â the only stable network. Okay.

Â So let's take a look at this in action. So let's look at a slightly richer

Â example and we'll walkabout where these numbers come from a little bit later.

Â But let's suppose that we have a situation where everybody's symmetric, if

Â if nobody's connected, they all get payoffs of zero.

Â So, we'll just normalize payoffs with no connections to zero.

Â Let's suppose that if you form a relationship with one other person, you

Â get a value of three each. So, if two people form a dyad, they get a

Â value of three. So, if both sets of people formed

Â relationships and we have three to four people and we have two relationships, and

Â everybody gets a three. if, if we add a link to this network

Â where these two individuals now decide to form a link together, then their payoffs

Â go up so now they have two relationships each.

Â They get a marginal benefit, a bit little more, they get 3.25.

Â But lets suppose these people are jealous, they don't like their friends to

Â have new friends. So this is different then the connections

Â model this is a situation where now I'm, I'm losing time with my friend because

Â now their spending more time with somebody else so they get a value of two.

Â now these people if they connect to each other, they get more value.

Â But then these people are losing value because now their friends are spending

Â time with other individuals. So we can think of this, this will come

Â out of a collaboration network where if people I'm collaborating with are

Â collaborating with other people, then that means we spend less time together, I

Â get less value... So this is one where we've got negative

Â impact of other people forming new relationships.

Â And so you can go through and, and have different paths here, and here, you know,

Â when these people now form a relationship, their value goes from 2.5

Â up to 2.78. But these people are, go down from 2.5 to

Â 2, so they, they're losing more time[INAUDIBLE] and so forth.

Â And then these people form a relationship.

Â They go up from the 2 to 2.3. And so forth.

Â Okay, so this is a very simple setting. And what we see in this setting.

Â In terms of the, value. the arrows represent.

Â Moves from one network to another network.

Â Which would be improving, or it would sort of means that this, this network is

Â not stable because the individuals here would gain by adding a link, and then

Â this one's not stable, and this one's not stable, right?

Â So each one of these is pointing to a new one, and we end up with the only pairwise

Â stable network for this set of payoffs. You, given all the permutations of these

Â things you're going to end up with everyone connected and everyone getting

Â 2.33, okay. So, that's the pairwise stable network in

Â this set. Okay.

Â Now the interesting thing is well, they, they're getting, they're worse off than

Â they would have been had they stopped here.

Â The difficulty is, this is not stable in the sense of individual incentives.

Â People who have incentives can move on from there.

Â So let's talk about that a little bit in more detail.

Â So let's talk about he efficency, and contrast that with the individual

Â incentives. Ok so pair wise stability handles

Â individual incentives. Now lets talk about evaluating overall

Â welfare. So one notion that comes out of economics

Â due to[UNKNOWN] and the late 19th century is known as Pareto efficiency.

Â And what does Pareto efficiency mean? It says that a network is Pareto

Â efficient if there is not some other network for which everybody is at least

Â as well off, and somebody, some of the individuals are strictly better off.

Â Okay? So there's not something that one can do

Â which is unambiguously better for everybody.

Â Nobody suffers and some people are made better off.

Â So if something is not Pareto efficient then society really has better options.

Â Just unambiguously better options. If something is Pareto efficient then it

Â means that if somebody gains by move, by some change, somebody else loses.

Â Okay? So Pareto efficiency is a weak notion of

Â efficiency. There can be lots of pareto efficient

Â outcomes but it it does rule some things out.

Â So it's going to rule out things which are just unambiguously bad and you can do

Â better by. Okay?

Â Now when we look at a stronger notion. Instead of just keeping track of well

Â some people are just better off or is everybody better off or not.

Â Sometimes we have choices to make, that some people are going to be better off,

Â and some people are going to be worse off.

Â we could talk about just a, a stronger notion of efficiency, which we'll refer

Â to as efficiency, or we could refer to as strong efficiency.

Â If G is a maximizer of the overall sum of, of payoff.

Â Okay. So you know, this would be Pareto if, if

Â you allow it for people just to move utility back and forth.

Â You can always make, you know, if you make everybody if you make the some

Â better off then, then you could make everybody better off by, by making

Â appropriate transfers. But more generally this is just going to

Â be a notion which is known as Utilitarianism.

Â 11:58

[SOUND] Which means that you care just about the total utility or some weighted,

Â in this case the equal weights on all the individuals, of the utility in this

Â society. Okay so this is utilitarianism with equal

Â weights on everybody, I just care about total utility and I, I, I actually don't

Â care you know, some people might gain or lose but if overall it's better then I

Â want to go with that. Okay.

Â So this is a stronger notion that will narrow things down a little bit more.

Â So if we look back at the picture we looked at before, pairwise stability was

Â moving us to this complete network... If we look at the overall maximizer, the

Â overall maximizer would be here. so this is an efficient network and it's

Â also pareto efficient. this one is pareto efficient.

Â There's no other network which makes everybody better off.

Â This one is a better in terms of the overall sum.

Â Right this one gives us a higher sum than this one does.

Â But these people, some people go down and other people go up.

Â Okay, so this is the overall efficiency and it's Pareto efficient.

Â This one is Pareto Efficient, but not overall efficient in terms of, of

Â maximizing total sum of utilities. you know, this one is not Pareto

Â Efficient or Efficient. this is better for everybody and, and you

Â know, similar here, the, these are. So, here already in this example, we see

Â that what society would like to do in terms of picking something which

Â maximizes over your all utility Or even something which is[UNKNOWN] efficient.

Â They can end up things, which are worse, in the sense that everybody is worse off

Â than what would happen if the society could oppose the network.

Â And part of it is due to the fact that individuals aren't accounting for the

Â harm that they can afflict on others when they make their decision.

Â Right, so they're, they're are selfishly, maximizing their own payoffs and not

Â accounting for what that does to other individuals in the society.

Â And that's not unusual in, in a lot of studies and once we start looking at

Â individual incentives. they're going to be misaligned.

Â And especially in network contexts, where what individuals do.

Â and what benefits they get depends on the full structure of the network and what

Â other people were doing. It's not going to be unusual that we're

Â going to find some conflict between what is individuals are going to do and what

Â society would like them to do. What's going to be interesting is, is

Â trying to figure out when this happens, and to what extent it happens, and why it

Â happens, and. Whether interventions can help or not.

Â so there's going to be a whole series of questions but one of the basic themes

Â once we start looking at strategic network formation is there's payoffs

Â involved. And individuals are going to be forming

Â the relationships that they find beneficial, but what's good for them is

Â not necessarily good for the overall society because their actions have...

Â Implications for other people that they are not necessarily taking into account

Â when choosing those actions. Okay, we'll look into this a little more

Â detail now. We'll come back and look.

Â say the connection model, and some other models.

Â To try and analyze what are the efficient networks, what are the Pareto stable

Â networks. Do we see conflict and so forth.

Â And that'll be our next topic.

Â