0:01

Okay so let's have a quick look at Pairwise Stability in the connections

Â model. So, here we are now looking at the same

Â range where [COUGH] for low costs we had complete networks being efficient, medium

Â cost stars, high costs empty networks. And when we start thinking about pairwise

Â stable. Now we're asking which ones are ones that

Â people want to form when they have the choice of adding links or deleting links.

Â And what we'll find is that indeed low cost, the complete network is going to be

Â pairwise stable. So we end up with the, the right network

Â in that situation, and in fact the complete network is, is going to be

Â uniquely pairwise stable. when we end up with a very high cost, the

Â empty network is pairwise stable, and that's going to be the only network

Â that's pairwise stable. In the middle cost range things now are

Â going to break into two different pieces. one is sort of a medium low cost, so c is

Â less than delta, so it's still valuable to have a connection.

Â 1:15

So we're in a situation where it's valuable to have connections but it's not

Â worth it to shorten indirect connections necessarily to direct ones.

Â The star network turns out to be pairwise stable.

Â There can also be other pairwise stable networks, so it's not the only pairwise

Â stable. So you can have some inefficient networks

Â for some parameter values also being pairwise stable.

Â The interesting case breaks in to this part where now we dealing with a medium,

Â high cost. Here c is bigger than delta but still

Â less than delta plus n minus 2 delta squared over 2.

Â So this is a situation where the star is efficient.

Â 2:03

So, we, society would like to start a form but the star network is not pairwise

Â stable. And in particular, nonempty pairwise

Â stable networks are going to be over-connected and may include too few

Â agents. Okay, so what does all of that mean?

Â Basically, what that means is now c is bigger than delta, so the cost the only

Â reason you want to have a relationship is if it's bringing also some indirect

Â benefits with it. So having a, a relationship with somebody

Â that only brings that one person, it's not going to be worth it because c is

Â bigger than delta, okay. So the only reason you're going to form

Â relationships is if you're also getting indirect benefits in this region.

Â And in that case a star is still efficient but a star is not going to be

Â pairwise stable. What does no loose ends mean?

Â It means there's no individual that's going to want to connect to some, some

Â other individual that doesn't bring them any indirect benefits.

Â They'd rather sever that link because they're only getting a delta here and

Â it's not worth it. So that certainly means that a star is

Â not going to be worthwhile, right. The center is not going to be willing to

Â take these other individuals who don't bring them anything besides themselves.

Â 3:14

So for instance if we look at you know, a star in this case.

Â Here what do we get? We get the payoff to the center, 3 deltas

Â minus 3cs. It's not going to be pairwise stable if

Â delta is less than c. They're paying too much cost, they're not

Â getting any indirect benefits, the center is, says it's not worth it.

Â Yet the overall payoff has these extra delta squares.

Â And so the peripheral players are actually getting indirect benefits and

Â the center does not get those. So the center is willing to sever the

Â links even though the outside players, the peripheral players, would rather have

Â the center maintain the star. So, here we get a simple inefficiency.

Â and now we see why there might be a difference between what's socially

Â efficient. It, it, the value is highest from this

Â and yet the center isn't seeing enough of the benefits and so says forget it, it's

Â not worth it. the peripheral players are seeing

Â indirect benefits, I'm not seeing them and, and severs the links.

Â Okay? So that's the, the difference between

Â the, uh,efficiency and the, pairwise stable networks.

Â Here's an example, you can play with this example if you like.

Â Let delta be bigger than c and, in this case smaller than delta plus delta

Â squared plus delta cubed, times 1 minus delta squared.

Â six individuals, and there's a unique non-empty pairwise stable network

Â architecture, which looks like this. So it doesn't look like a star, it, it

Â actually looks like a ring or a circle. And the idea here is that each individual

Â is willing to have these other relationships even though c is bigger

Â than delta. So c is higher than delta, but that's

Â because they get indirect benefits from having these.

Â So by having these they, they get indirect benefits as well and that makes

Â it worthwhile and it all hooks together. Nobody wants to add any extra links

Â across because the value of, of adding that link is not worth the cost.

Â It doesn't change the indirect relationships to anybody else but adds an

Â extra cost. And it's not worth shortening that path

Â from a delta cubed to a single one. So this, in this setting you can check is

Â the unique pairwise stable nonempty network and we end up with something that

Â is not a star. Right, so not a star and yet pairwise

Â stable. Okay, so what we've seen is we've seen a

Â conflict between what's efficient from society's perspective and what's pairwise

Â stable from individual's perspective in terms of how they'd like to form or

Â delete links. And this is going to be a theme that runs

Â through a lot of the strategic network formation literature and a lot of the

Â models. We're generally going to see some

Â distance, or differences between what's pairway stable and what's efficient.

Â And on occasion you'll find these two coinciding, but often we won't given the

Â externalities that are present in the society.

Â So the next thing we'll look at is, is looking at other kinds of models, some

Â different kinds of externalities. And seeing how inefficiencies might arise

Â in those other settings.

Â