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Hey folks? So, we're back again.

We're going to be talking a little bit about refinement of pairwise stability now,

and an alternative way of modeling network formation.

This is known as Pairwise Nash Stability.

The idea here is, we're thinking just about

other possible methods of modeling network formation. There are many different ways.

So, I wanted to just give you some feeling for the fact that there's not

going to be any single notion which is perfect for analyzing network formation,

but there might be various ones,

and different ones are going to have different strengths and weaknesses.

So, in terms of going beyond pairwise stability,

part of the idea here is that we want to allow individuals,

for instance, to make multiple changes to links,

not just one at a time.

Already in pairwise stability,

we allowed for two people to act,

you could allow for many more people.

There's a whole series of other questions that we can ask.

We'll talk a little bit about some of these in coming videos, existence, dynamics,

stochastic stability, forward looking, directed,

there's a lot of topics here,

and a fairly rich literature.

I'm going to give you a peek at some of those.

So, let's think about this alternative way that we

thought of early on of forming networks,

which was just to think of each person

saying who they're willing to form relationships with,

and then having relationships form if and only if both people named each other.

So, this is a game, actually,

Roger Myerson talked about this game in the early 1990s,

an announcement game of forming relationships.

So, we can use that here in this network formation setting.

Players can simultaneously announce which their preferred set of neighbors are.

So, each person, i,

makes an announcement, let's call this set Si.

This is the set of other nodes that i wants to be linked to.

So, for instance, person seven could say that they want to be linked with persons one,

two, five, 11, and so forth.

So, they're saying these are my preferred neighbors.

Then, the network that forms as a function of the profile,

the full vector of all the announcements made by

different individuals are the links such that j was named by i,

and i was named by j.

So, this is consensual network formation,

you form a relationship if and only if both people named each other.

So, what's nash stability then?

Well, just take a nash equilibrium of that announcement game,

and we'll look at pure strategies.

So, this is a situation where the utility that a given individual

gets from the network that forms under the announcements that are there,

is at least as good as any thing that they could get by changing their announcements.

So, they might want to announce, for instance,

they can't add some new links announcements that they didn't make,

and do better, and they can't

delete some of the announcements they did make, and do better.

So, we say that a network is nash stable if and only if

no player wants to delete some set of his or her links.

So, that's going to be equivalent to having this be a nash equilibrium of this game.

So, nash stability basically looks at a given network and says,

''Does anybody want to take some subset of

network links that are there and delete them?''

So, the set of pure strategy nash equilibria of this game are going to be equivalent to

the networks where no player wants to

deviate from the links that they have and delete some of them,

but it doesn't ask about adding mutually.

So, if we look at a very simple example.

So, look at this example here. What do we have?

All individuals separately get zero,

a pair of individuals gets one.

If you end up forming a full triad,

then you end up getting payoffs of one each.

In this situation, if you end up in a two-link setting,

then you get minus one.

So, this is a setting where,

when we look at the nash stable networks, what do we end up with?

We end up with three of them.

So, it's not terribly predictive,

we end up with three possible networks that could be nash stable.

Now, if you look at the comparison between these in pairwise here,

let's just go through why these are nash stable.

Why is this nash stable?

This is sort of a coordination failure,

nobody manages to name anybody else,

and nobody thinks anybody's going to name anybody else.

So, everybody, each Si,

is equal to the empty set.

Nobody names anybody, and now,

if nobody named me,

I can't form a link anyway.

So, I might as well have named the empty set.

This is a nash equilibrium.

These two players are getting one,

they don't want to deviate and add the third player,

it doesn't make any sense, they're happy.

This is a nash equilibrium,

everybody's getting one, there's no better payoff they could get.

This one is not, and why isn't it?

Well, this person must be announcing.

So, if we call this player one,

player one must be announcing player two,

they could deviate and not announce player two,

and they would be better off because their payoff will go from minus one to one.

So, this one is the only one that's not nash stable.

Okay. So, what do we end up with?

We end up with three nash stable networks.

If we look at the pairwise stable networks here,

well, this one's not pairwise stable.

All right.

So, this is not pairwise stable.

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For the same reason, it wasn't nash stable,

this person can delete a link.

This one's not pairwise stable

because these two individuals would both strictly gain from adding a link.

So, pairwise stability rules this one out,

whereas nash stability did not.

That was part of the reason that we went to pairwise stability,

because it eliminated this problem that we have with coordination failures leading

to networks that really don't make a whole lot of

sense in terms of if anybody really could communicate,

they'd rather form the links that are giving them positive payoffs.

This one's clearly going to be pairwise

stable because it gives everybody a maximum payoff,

there's no better thing they could do.

What about this one? Is it pairwise stable?

Well, if we think about deleting link,

nobody wants to delete a link.

Do any two players want to add a link?

Well, if they added a link, so,

if these two players added the link for instance, what would happen?

They would go to payoff of minus one.

So, indeed this one is pairwise stable.

They wouldn't want to do that.

This one is pairwise stable.

So, what we end up with is a situation where we have three nash stable networks,

and then two Pairwise Stable Networks.

So, the pairwise stable Networks are picking

a subset of what the nash stable networks are.

So, we could ask, which ones are both pairwise stable and nash stable?

It will be these two, and those we could call pairwise nash stable networks.

Well, here pairwise stability,

already was just picking a subset,

so there wasn't really any reason to look at

nash stability in addition to pairwise stability,

because it wasn't narrowing things at all.

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But more generally, if we look at pairwise nash stability, so,

we ask for something to be both pairwise stable and nash stable,

we can end up, in some cases,

with more of a refinement.

So, let's look at a slightly different example.

So, here's a situation where,

if everyone is separate,

they get zero, as before,

one link leads to payoff of one.

Here, this situation is one where two links together lead to minus two,

and three links together,

lead everyone to a payoff of minus one.

So, in this setting,

when we look at the nash stable networks,

this is slightly different than our previous example.

The only ones that are nash stable are this one and this one.

So, we've got this one,

we still got the problem of coordination failure,

we're getting the nash stable here.

This one is also nash stable.

This is not nash stable anymore, because now,

somebody could just sever both of their links,

and get a zero instead of a minus one.

So, they would be better off.

This one's clearly not nash stable,

because by severing both links and getting zero,

they'd be better off as well.

So, in this situation,

what we end up with is,

the only nash stable networks are the two at the bottom.

When we look at pairwise stability,

we end up with this one being pairwise stable,

nobody wants to add or delete a link from here.

That's clear, that's the maximum possible payoffs that people could get.

But this one ends up being pairwise stable as well. Why is that?

Well, if anybody deleted one of their links,

just severed one link,

they would end up getting a minus two.

So, they would end up being at the end of a triangle,

they would go to a minus two pay off.

So, they don't want to sever

a single link even though they could benefit from severing multiple links.

So, pairwise stability was only looking one link at a time,

and it didn't allow people to say, ''Look,

I'd be better off severing two links.''

So, nash stability allows for multiple link changes,

but then has this coordination failure problem.

Pairwise stability only looks at one link at a time,

and so might miss some deviations where you

could delete multiple links and be better off.

But putting them together in this case,

we end up with a simple conclusion that seems

to be the right network in this setting to expect to form,

which is the one that's both pairwise stable.

So, nobody wants to add a link,

and nobody wants to delete multiple links.

That combination of pairwise nash stability

ends up being more selective than either of the two,

and in some senses, is picking out more sensible prediction in this particular example.

So, that's useful to have some refinements of pairwise stability,

alternative methods of modeling network formation,

and captures as multiple link changes.

You could do all kinds of other variations.

You could allow additions of links plus deletions of some links,

you could allow for larger coalitions.

So, there's a whole series of different ways

in which you can embellish these definitions,

and of course, game theorists love to

work with different definitions and see what they give.

So, there's a non-trivial literature which looks at

enriching a set of ways in which we model network formation.