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

Â a little bit about the refinement of Pairwise Stability now,

Â an alternative way of modeling network formation.

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

Â we're thinking just about

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

Â And so I wanted to just give you some feeling for

Â the fact that there's not going to be any single notion which is

Â perfect for analyzing work formation but there might be

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

Â And so in terms of going beyond pairwise stability,

Â part of the idea here is that we want to

Â allow individuals for instance to make multiple changes to lengths,

Â not just one at a time.

Â Maybe you know it already in pairwise stability we allowed for two people to act.

Â You could allow for many more people.

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

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

Â you know: existence, dynamics,

Â stochastic stability, forward looking, directed.

Â There's a lot of topics here and a fairly rich literature.

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

Â So let's think about this alternative way that we thought of

Â early on of forming networks which was just to think

Â of each person saying who they're willing to form relationships with and then

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

Â So this is a game.

Â Actually Roger Myerson talked about this game in

Â the early 1990s of an announcement game of forming relationships.

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

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

Â So each person, i,

Â makes an announcement, it's called a set Si.

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

Â So for instance, you know,

Â set person seven could say that they want to be linked with persons one, two,

Â five, 11 and so forth so they're saying these are my preferred neighbors.

Â And then the network that forms as a function of the profile,

Â the full vector of all the announcements made by different individuals,

Â are the links such that j was named by i and i was named by j.

Â So this is consensual network formation,

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

Â Okay, so what's Nash stability then?

Â Well just take a Nash equilibrium of

Â that announcement game and we'll look at pure strategies.

Â So this is a situation where the utility that

Â a given individual gets from the network that forms under

Â the announcements that are there is at least as good

Â as any thing that they could get by changing their announcements.

Â So they might want to announce for instance that they can't add some new links,

Â announcements that they didn't make and do better and

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

Â So we say that a network is Nash stable if and only if

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

Â So that's going to be equivalent to having this be a Nash equilibrium of this game.

Â So Nash stability basically looks at a given network and says "Does anybody

Â want to take some subset of links that are there and delete them?"

Â So the set of all Nash equilibria

Â of pure strategy Nash equilibria could be of this game are going to be equivalent to

Â the networks where no player wants to

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

Â But it doesn't ask about adding mutually.

Â So if we look at a very simple example,

Â so look at this example here.

Â What do we have? All individuals separately get zero.

Â A pair of individuals gets one.

Â If you end up forming a full triad then you end up getting payoffs of one each.

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

Â So this is a setting where when we look at

Â the Nash stable networks where do we end up with?

Â We end up with three of them.

Â So its not terribly predictive.

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

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

Â let's just go through why these are Nash stable.

Â Why is this Nash stable?

Â This is sort of a coordination failure.

Â Nobody manages to name anybody else and

Â nobody thinks anybody is going to name it anybody else.

Â So everybody, each Si is equal to the empty set.

Â Nobody names anybody and now if nobody named me

Â I can't form a link anyway so I might as well name the empty set.

Â This is a Nash equilibrium.

Â These two players are getting one.

Â They don't want to deviate and the third player doesn't make any sense, they're happy.

Â This is a Nash equilibrium,

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

Â This one is not and why isn't it?

Â Well this person must be announcing.

Â So if you call this player one,

Â player one must be announcing player two.

Â They could deviate and not announce player two and

Â they would be better off because their pay off would go from minus one to one.

Â So this one is the only one that's not finished. So what do we end up with?

Â We end up with three Nash stable networks.

Â If we look at the pairwise stable networks here.

Â Well this one's not pairwise stable, right?

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

Â This person can delete a link.

Â This one is not pairwise stable because these two individuals would

Â both strictly gain from adding a link so

Â pairwise stability rules this one out whereas Nash stability did not.

Â And that was part of the reason that we wanted

Â pairwise stability because it eliminated this problem that we have

Â with coordination failures leading to networks that really

Â don't make a whole lot of sense in terms of if anybody really can communicate.

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

Â This one's clearly going to be pairwise

Â stable because it gives everybody a maximum payoff.

Â There's no better thing they could do.

Â What about this one? Is it pairwise stable?

Â Well, if we think about deleting a link,

Â nobody wants to delete a link,

Â do any two players want to add a link?

Â Well if they added a link,

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

Â They would go to payoff of minus one so indeed this one is pairwise stable.

Â They wouldn't want to do that.

Â And this one is pairwise stable.

Â So what we end up with is a situation where we have three Nash stable networks and then

Â two pairwise stable networks so

Â the pairwise stable networks are picking a subset of what the Nash stable networks are.

Â And so we could ask which ones are both pairwise stable and Nash stable.

Â It will be these two and those we could call pairwise Nash stable networks.

Â Well, here pairwise stability

Â already was just picking a subset so there wasn't really any reason

Â to look at Nash stability in addition to pairwise stability

Â because it wasn't narrowing things at all.

Â But more generally, if we look at pairwise Nash stability,

Â so we ask for something to be both pairwise stable and Nash stable,

Â we can end up in some cases with more of a refinement.

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

Â So here's a situation where if everyone is separate they get zero as before.

Â One link leads to payoff of one.

Â Here this situation is one where two links together lead to

Â minus two and three links together lead everyone to a payoff of minus one.

Â So in this setting,

Â when we look at the Nash stable networks,

Â this is slightly different than our previous example.

Â The only ones that are Nash stable are this one and this one.

Â So we've got this one,

Â we still get the problem of a coordination failure,

Â we're getting the Nash stable here.

Â This one is also Nash stable.

Â This is not Nash stable anymore because now somebody could just sever both of

Â their links and get a zero instead of a minus one so they would be better off.

Â And this one's clearly not Nash stable because by

Â severing both links and getting zero they'd be better off as well.

Â So in this situation what we end up with is

Â the only Nash stable networks are the two at the bottom.

Â And when we look at pairwise stability we end up with this one being pairwise stable.

Â Nobody wants to add or delete a link from here, that's clear.

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

Â But this one ends up being pairwise stable as well.

Â And why is that?

Â Well if anybody deleted one of their links,

Â just severed one link,

Â they would end up getting a minus two.

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

Â they would go to a minus two payoff.

Â And so they don't want to sever

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

Â So pairwise stability was only looking one link at

Â a time and it didn't allow people to say "Oh look I'd be better

Â off severing two links so Nash stability allows for

Â multiple link changes but then has this coordination failure problem.

Â Pairwise stability one looks one link at a time and so might

Â miss some deviations where you could delete multiple links and be better off.

Â But putting them together in this case we end up

Â with a simple conclusion that seems to sort of be

Â the right network in this setting to expect

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

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

Â And that combination of

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

Â two and in some senses is

Â picking out a more sensible prediction in this particular example.

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

Â alternative methods of modeling network formation and,

Â you know, it captures these multiple linked changes.

Â You could do all kinds of other variations,

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

Â you can allow larger coalitions.

Â So there's a whole series of different ways in which you can embellish these definitions.

Â And of course game theorists love to

Â work with different definitions and see what they give.

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

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

Â