Okay so let's have a quick look at Pairwise Stability in the connections model. So, here we are now looking at the same range where [COUGH] for low costs we had complete networks being efficient, medium cost stars, high costs empty networks. And when we start thinking about pairwise stable. Now we're asking which ones are ones that people want to form when they have the choice of adding links or deleting links. And what we'll find is that indeed low cost, the complete network is going to be pairwise stable. So we end up with the, the right network in that situation, and in fact the complete network is, is going to be uniquely pairwise stable. when we end up with a very high cost, the empty network is pairwise stable, and that's going to be the only network that's pairwise stable. In the middle cost range things now are going to break into two different pieces. one is sort of a medium low cost, so c is less than delta, so it's still valuable to have a connection. but bigger than delta minus delta squared. So we're in a situation where it's valuable to have connections but it's not worth it to shorten indirect connections necessarily to direct ones. The star network turns out to be pairwise stable. There can also be other pairwise stable networks, so it's not the only pairwise stable. So you can have some inefficient networks for some parameter values also being pairwise stable. The interesting case breaks in to this part where now we dealing with a medium, high cost. Here c is bigger than delta but still less than delta plus n minus 2 delta squared over 2. So this is a situation where the star is efficient. So, we, society would like to start a form but the star network is not pairwise stable. And in particular, nonempty pairwise stable networks are going to be over-connected and may include too few agents. Okay, so what does all of that mean? Basically, what that means is now c is bigger than delta, so the cost the only reason you want to have a relationship is if it's bringing also some indirect benefits with it. So having a, a relationship with somebody that only brings that one person, it's not going to be worth it because c is bigger than delta, okay. So the only reason you're going to form relationships is if you're also getting indirect benefits in this region. And in that case a star is still efficient but a star is not going to be pairwise stable. What does no loose ends mean? It means there's no individual that's going to want to connect to some, some other individual that doesn't bring them any indirect benefits. They'd rather sever that link because they're only getting a delta here and it's not worth it. So that certainly means that a star is not going to be worthwhile, right. The center is not going to be willing to take these other individuals who don't bring them anything besides themselves. So for instance if we look at you know, a star in this case. Here what do we get? We get the payoff to the center, 3 deltas minus 3cs. It's not going to be pairwise stable if delta is less than c. They're paying too much cost, they're not getting any indirect benefits, the center is, says it's not worth it. Yet the overall payoff has these extra delta squares. And so the peripheral players are actually getting indirect benefits and the center does not get those. So the center is willing to sever the links even though the outside players, the peripheral players, would rather have the center maintain the star. So, here we get a simple inefficiency. and now we see why there might be a difference between what's socially efficient. It, it, the value is highest from this and yet the center isn't seeing enough of the benefits and so says forget it, it's not worth it. the peripheral players are seeing indirect benefits, I'm not seeing them and, and severs the links. Okay? So that's the, the difference between the, uh,efficiency and the, pairwise stable networks. Here's an example, you can play with this example if you like. Let delta be bigger than c and, in this case smaller than delta plus delta squared plus delta cubed, times 1 minus delta squared. six individuals, and there's a unique non-empty pairwise stable network architecture, which looks like this. So it doesn't look like a star, it, it actually looks like a ring or a circle. And the idea here is that each individual is willing to have these other relationships even though c is bigger than delta. So c is higher than delta, but that's because they get indirect benefits from having these. So by having these they, they get indirect benefits as well and that makes it worthwhile and it all hooks together. Nobody wants to add any extra links across because the value of, of adding that link is not worth the cost. It doesn't change the indirect relationships to anybody else but adds an extra cost. And it's not worth shortening that path from a delta cubed to a single one. So this, in this setting you can check is the unique pairwise stable nonempty network and we end up with something that is not a star. Right, so not a star and yet pairwise stable. Okay, so what we've seen is we've seen a conflict between what's efficient from society's perspective and what's pairwise stable from individual's perspective in terms of how they'd like to form or delete links. And this is going to be a theme that runs through a lot of the strategic network formation literature and a lot of the models. We're generally going to see some distance, or differences between what's pairway stable and what's efficient. And on occasion you'll find these two coinciding, but often we won't given the externalities that are present in the society. So the next thing we'll look at is, is looking at other kinds of models, some different kinds of externalities. And seeing how inefficiencies might arise in those other settings.