0:00

Hi folks. So we're back again and we're going to be talking

a little bit about the refinement of Pairwise Stability now,

an alternative way of modeling network formation.

And this is known as Pairwise Nash Stability and the idea here is

we're thinking just about

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

And 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 work formation but there might be

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

And 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 lengths,

not just one at a time.

Maybe you know it 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,

you know: 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 of 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, it's called a set Si.

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

So for instance, you know,

set 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.

And 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.

Okay, 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 that they can't add some new links,

announcements that they didn't make and do better and

they can 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 links that are there and delete them?"

So the set of all Nash equilibria

of pure strategy Nash equilibria could be 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.

And in this situation if you end up in a tooling setting then you get minus one.

So this is a setting where when we look at

the Nash stable networks where do we end up with?

We end up with three of them.

So its not terribly predictive.

We end up with three possible networks that could be Nash stable.

Now if you look at the comparison between these and pairwise stable,

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 is going to name it 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 name the empty set.

This is a Nash equilibrium.

These two players are getting one.

They don't want to deviate and the third player 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 you 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 pay off would go from minus one to one.

So this one is the only one that's not finished. 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, right?

So this is not pairwise stable for the same reason it wasn't Nash stable.

This person can delete a link.

This one is 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.

And that was part of the reason that we wanted

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 can 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 a 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.

And 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.

And 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.

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 get the problem of a 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.

And 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.

And 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.

And 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 payoff.

And 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 "Oh 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 one looks 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 sort of be

the right network in this setting to expect

a form which is the one that's both pairwise stable.

So nobody wants to add a link and nobody wants to delete multiple links.

And that combination of

pairwise Nash stability ends up being more selective than either of the

two and in some senses is

picking out a 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,

you know, it captures these multiple linked changes.

You could do all kinds of other variations,

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

you can allow 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.