Why is it weak? Well, it's only looking at pairs of

individuals. It's only looking at one link at a time.

And it just makes sure that there's no single link that would be better deleted.

And no link that's not present that would be better to add.

Okay. But often this already, is fairly

powerful. So sort of the minimal set of

requirements that we might think of in terms of stability.

It'll often began to narrow things down. Now, there's all kinds of other

variations. So this is this concept came out of the

paper with Ashford Lowenski. so the Ashford Lowenski 96 paper, people

have looked at a lot of other variations on these kinds of things, we'll talk

about some of them but, you know this, this will give us some basics to work

with. Okay.

So, so now when we go back to that example we had before both are Nash

equilibria, but this is the only pairwise stable[UNKNOWN], right.

So both people would gain by adding this pairwise ability just says that this is

the only stable network. Okay.

So let's take a look at this in action. So let's look at a slightly richer

example and we'll walkabout where these numbers come from a little bit later.

But let's suppose that we have a situation where everybody's symmetric, if

if nobody's connected, they all get payoffs of zero.

So, we'll just normalize payoffs with no connections to zero.

Let's suppose that if you form a relationship with one other person, you

get a value of three each. So, if two people form a dyad, they get a

value of three. So, if both sets of people formed

relationships and we have three to four people and we have two relationships, and

everybody gets a three. if, if we add a link to this network

where these two individuals now decide to form a link together, then their payoffs

go up so now they have two relationships each.

They get a marginal benefit, a bit little more, they get 3.25.

But lets suppose these people are jealous, they don't like their friends to

have new friends. So this is different then the connections

model this is a situation where now I'm, I'm losing time with my friend because

now their spending more time with somebody else so they get a value of two.

now these people if they connect to each other, they get more value.

But then these people are losing value because now their friends are spending

time with other individuals. So we can think of this, this will come

out of a collaboration network where if people I'm collaborating with are

collaborating with other people, then that means we spend less time together, I

get less value... So this is one where we've got negative

impact of other people forming new relationships.

And so you can go through and, and have different paths here, and here, you know,

when these people now form a relationship, their value goes from 2.5

up to 2.78. But these people are, go down from 2.5 to

2, so they, they're losing more time[INAUDIBLE] and so forth.

And then these people form a relationship.

They go up from the 2 to 2.3. And so forth.

Okay, so this is a very simple setting. And what we see in this setting.

In terms of the, value. the arrows represent.

Moves from one network to another network.

Which would be improving, or it would sort of means that this, this network is

not stable because the individuals here would gain by adding a link, and then

this one's not stable, and this one's not stable, right?

So each one of these is pointing to a new one, and we end up with the only pairwise

stable network for this set of payoffs. You, given all the permutations of these

things you're going to end up with everyone connected and everyone getting

2.33, okay. So, that's the pairwise stable network in

this set. Okay.

Now the interesting thing is well, they, they're getting, they're worse off than

they would have been had they stopped here.

The difficulty is, this is not stable in the sense of individual incentives.

People who have incentives can move on from there.

So let's talk about that a little bit in more detail.

So let's talk about he efficency, and contrast that with the individual

incentives. Ok so pair wise stability handles

individual incentives. Now lets talk about evaluating overall

welfare. So one notion that comes out of economics

due to[UNKNOWN] and the late 19th century is known as Pareto efficiency.

And what does Pareto efficiency mean? It says that a network is Pareto

efficient if there is not some other network for which everybody is at least

as well off, and somebody, some of the individuals are strictly better off.

Okay? So there's not something that one can do

which is unambiguously better for everybody.

Nobody suffers and some people are made better off.

So if something is not Pareto efficient then society really has better options.

Just unambiguously better options. If something is Pareto efficient then it

means that if somebody gains by move, by some change, somebody else loses.

Okay? So Pareto efficiency is a weak notion of

efficiency. There can be lots of pareto efficient

outcomes but it it does rule some things out.

So it's going to rule out things which are just unambiguously bad and you can do

better by. Okay?

Now when we look at a stronger notion. Instead of just keeping track of well

some people are just better off or is everybody better off or not.

Sometimes we have choices to make, that some people are going to be better off,

and some people are going to be worse off.

we could talk about just a, a stronger notion of efficiency, which we'll refer

to as efficiency, or we could refer to as strong efficiency.

If G is a maximizer of the overall sum of, of payoff.

Okay. So you know, this would be Pareto if, if

you allow it for people just to move utility back and forth.

You can always make, you know, if you make everybody if you make the some

better off then, then you could make everybody better off by, by making

appropriate transfers. But more generally this is just going to

be a notion which is known as Utilitarianism.