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.