These three are not stable. Okay, and let's just go through and try

to understand why is this stable? What's the logic, here?

So, let's suppose that we perturb this a little bit.

So for instance we take a little bit of epsilon away, so we take this person down

to 1 minus epsilon. And we bump this person up to epsilon.

And let's, let's just keep this person constant for now.

if we do that, and now we look at the best responses to this.

Well, this person has already a neighbor providing at least one.

They're still at 2 minus epsilon in their neighborhood.

They're way bigger than the x star of one.

They're getting too much action. They can get rid of their epsilon and go

back to zero. So, they're going to tend to go right

back to zero. Once they go back to zero, this person's

going to go back to one, and we end up going back to the, to the situation that

we have. So here the fact that I'm already getting

too much means even if you perturb that I'm going to want to go right back to

zero. So the people that are in the

neighborhood of these specialists are going to stay there if they have at least

two specialists in their neighborhood they're going to go back to zeroes.

And that means that specialists are going to be forced back to one because

all their neighbors are providing zeroes. Whereas this one is unstable, and it's

unstable for the reason the dyad was, right?

We could make this a 1 minus epsilon and you know then, move this guy to epsilon,

this guy to epsilon. Once that happens, well then these people

are happy but this person's going to want to decrease more and, and it would

actually move eventually towards this equillibrium.

you know we could change this by moving this to minus epsilon, add an epsilon

over here. that would still be an equilibrium and so

forth. So, so the only ones that turn out to be

stable, turn out to be the ones that every, that, that every specialist.

Has specialized equilibria and such that every non-specialist has at least two

specialists in their neighborhood. Okay, so you can go through a sketch of

the proof of this basically the stability of such equilibria for, for perturbations

of this for the non-specialist, the best response they go right back to zero, and

you converge right back. For any other equilibrium, there is

somebody providing goods as a non-specialist and then you perturb that

agent up and the other neighbors down just the way we did, and basically you

can, you knoq work through that kind of logic, so the idea of the proof is much

similar to what we just worked through in terms of the examples, and you can deduce

this proposition in that manner. Okay, so what says is that is that these

specialized equilibria are somehow special in this model.

So you get, they're stable in the sense that if you do perturbations then you're

going to get pushing people towards that. Let me just sort of make one comment on

this, and sort of an interesting contrast.

another way we could think about, you know, so here we've got a game with

multiple equilibria. There's lots of different equilibria in

this game. And, you know, part of the reason that,

that one's interested in applying the stability concepts is just to find out

are some of these more naturally equilibria than others?

And so if we perturb them and we get back to them, we say, well, that seems to be a

more natural one than another one. another way we can think about stability

is instead to think, think about pairwise stability, right?

So suppose your links are costly, and if I'm specializing, and I've got a bunch of

friends who aren't specializing, I'm providing all the good.

They're not doing anything. I would start dropping links to the

non-specialists. Right?

So in some sense in that world, what you would have is the only reason people

would be willing to, to maintain links would be if their friends were also

providing some of the local public goods. So I normally want to keep these

friendships if they're actually providing some value.

And so there, what you're going to get is the non specialized equilibria are

going to be the only, stable ones. So, what this says is that, you know?

It depends very much on what kind of stability notion you put in, as to what

you might select out. And whether you think about perturbing

the, the network, or perturbing the actions.

You could get very different kinds of conclusions.

And so, you know, ultimately I think what this says is there's still a lot we need

to understand about these kinds of games. they are tractable games.

They have some interesting features. Some of them tie back to some of the

things like the best-shot public goods games and so forth and they, they can be

you know, interesting to analyze further. Okay.

So, so that's one example of these kinds of games.

if we begin to introduce heterogeneity to these kinds of things that's going to

change the nature of equilibria. We could begin to enrich things that way.

There's a lot of enrichments we can make in these kinds of games.

And what we are going to do next is look at another class of games.

And this other class of games is going to be one where we allow for strategic

complements and there the games again are going to be nicely behaved in the sense

it will be much easier to get a full characterization of what the equilibria

look like and, and it'll be nice we'll have some nice relationships to the

network structure and that'll allow us then to.