If we do this in perpetuity, so we expect these things to keep rising over time

then you know just summing the series of these things times delta times one minus

delta, delta squared, delta cubed and so forth.

So the value of this in perpetuity is just going to be 1 minus 1 over 1 minus

delta times this. So this, this multiplying this thing by 1

over 1 minus delta is just capturing the fact that we've got this you know, in the

first period. So this just equal to 1 plus delta plus

delta squared plus delta cubed. So we're getting this today, tomorrow,

the next day, and so forth in perpetuity. Okay?

So this is the value of perpetual relationship is exactly this.

Okay. So when can you sustain favor exchange?

Well, now, if somebody's called on to do a favor.

They can look at what's the value, now the worst I can do by not providing the

favor is lose this relationship, so what's the value of the relationship in

the future? Well, it's delta times the value in

perpetuity, so the value from tomorrow onwards.

What's the cost? Well, I have to pay this today.

So as long as the cost is less than the value of the future relationship, then

you would want to provide the favor. But, if the cost is bigger than the value

of the future relationship, you couldn't sustain it.

And, indeed, you can check that, that you know, if you write this down as a

repeated game, and people can provide favors for each other, you can enforce

favor exchange if and only if the cost is, is does not exceed the value of the

future relationship. Okay, so that's with just two people

exchanging favors, fairly simple idea. as long as the value of the future

relationship's sufficiently high, people provide favors with the threat of losing

the relationship otherwise. So we could have a situation where we'll

keep providing favors as long as, as everyone does it in the past.

If somebody stops the relationship dissolves, we're no longer friends, and

so you're going to lose the value of that in the future.

Okay now, what's the value of putting this in a network?

But the value of putting this in a network is sometimes, these favors can be

quite costly. So it could be that somebody has a crop

failure, and I have to run them a lot of money, or you know, do, give them a lot

of help. in that situation, to do, what's, what's

the incentive to provide the favor? Well, if we're in a social network, then

it can be that the value of providing a favor is increased by the fact that

instead of losing just one friendship, if I don't behave well and, and follow the,

you know, keep providing favors when I'm asked, I could lose multiple friendships.

So let's look at a situation where we have three people in a triad here and

what we do is ostracize anybody who does not perform a favor.

So if anybody's called on to do a favor and they say no, then both friendships

are, are severed there. So in this case Now, the cost only has to

be less than two times this value of a virtual friend, friendship.

So it's easier to satisfy this relationship, or this, incentive

constraint now because the value is, is, increased in terms of how many

friendships I might lose. In the future, I could be ostracized by

both of the other agents. So in this case if one is called on to do

a favor for two if one doesn't do the favor for two, one loses two friendships

and therefore this incentive constraint is that the cost only has to be less than

two times the value of the friendships. Now, of course in this situation, once

that really, these two relationships are, are gone then it could be also that two

and three are no longer going to be able to do favors for each other, because if

they couldn't satisfy, if c was greater than 1 times this, then they're no longer

going to be able to sustain favor exchange.

So, in this case, the whole triangle would collapse because once one is

ostracized, and two and three no longer have enough of an incentive on a

bilateral basis to keep the friendship going and, and so the whole thing

collapses, and so this whole triangle disintegrates.

Okay, so that, when you can just define this as a game, and so let's look at a

simple game where, basically, at any given period at most, one person is

called on to perform a favor for somebody else in their neighborhood.

So we'll think of p as being small. So somebody in their neighborhood of the

network at a given time. And then, the idea is that i is either

going to keep the relationship going, meaning provide the favor and keep it

going, or can just say sorry it's over, I'm going to sever this relationship and

not provide the favor. So here we'll have, you either keep

relationships, meaning, you keep providing the favor or you sever the

relationship meaning, no favors between those two individuals in the future,

okay? So to make the game simple, we'll just

have that be the choice. Either maintain a relationship, do the

favors or stop the relationship, don't do the favors.

Then others can respond. So they can announce, this can react.

So suppose I don't do a favor, others can see that and, and sever their links to

me, or it might be that I do the favor and they can decide to keep their

relationships, whatever. so links are maintained if people

mutually agree. And so after this process, depending on

what everybody does we end up with a new network of gt plus 1 and then the process

repeats itself. Okay?

And so then we can ask which neigh-, which networks will be equilibrium

networks if we look at this process, over time.

So let, let's take a quick peek, at, two different networks that could be

sustained in equilibrium in situations where two -- losing two friendships is,

is bigger than the cost of doing a favor. But the cost of doing a favor is bigger

than the value of one relationship. So you need two to, the, two friendships

lost to, to give incentives. So we've got two different networks here,

which are both going to support favor exchange.

And the idea is basically going to be that if somebody doesn't perform a favor

they're going to lose two friendships instead of just one.

And so imagine, for instance, that this person one, here, is called on to do a

favor for person, say, two, right? So we could do it in either of these two

different ways. Now if they fail to perform that favor

they are going to actually be losing two different relationships not just one and

that's what is keeping them making sure they provide the favor but then we can

look at this, what's the subsequent implications of the fact that we've lost

this now, okay? So if we look at this, now these two

individuals, this person for instance can't be trusted anymore because they

only have 1 friendship left. So the next time they're called on to do

a favor they're not going to do a favor, so effectively, this relationship is

going to have to disappear because it's not enough, to, it can't maintain itself

any longer. Similarly this one's going to have to

disappear, right? This person can't be trusted.

And this one, and so forth, right? So what we end up with, is effectively,

those disappearing, and then over here, we can see that that's going to have

further implications, right? We're going to have other relationships

which are no longer sustainable given that we've lost part of our network.

And so in this setting we have a widespread contagion, so that the fact

that this person did not perform a favor, ended up having consequences for people

who are actually quite far away in the network.

And it reached somebody that was at, you know, the distance 3 away from them in

terms of the network. In this situation it stopped right here.

Effectively this part was lost, but then the rest of the network.

is maintained, right? So this is a situation where we could,

you know, define robustness against the contagion.

To say that a network will be robust if the favor, the failure to perform a favor

only impacts the direct neighbors of the original players who, who did not perform

or, or lost links. So we don't have a widespread contagion.

It's only people who are actually directly connected to that individual

originally who are going to end up losing relationships.

So the impact of some sort of deletion or perturbation is local.

So that would say that, that when we go back to these networks, this one is

going to be robust. This one is not in this, in this

particular manner, right? So, one of them fell apart, the other one

had only local things. Okay, so now a quick definition and then

the results on this. So we'll say that a link in a network is

supported, i j have is, is a link that's supported, if there's some k, such that

both i k is in g, and j k is in g, so having a friend in common.

Okay? So support means that two individuals

have a friend in common. so the implications of this game are,

that if we look at a situation where there's no players, no pair of players

could sustain favor exchange in isolation, so you need at least two, a

threat of two or more, link deletions in order to keep somebody honest.

Then, the networks that are robust have to have all of their links supported,

okay? So every link is going to have to have at

least one friend in common, and sometimes, if, we're looking at

situations where we've got more, you might have to even have more than one

friend in common. [INAUDIBLE].

Okay? So, so this says that,

Support is actually a, a property that's going to come out of this particular

favor exchange game. So if you look at it, this favorite

exchange game, the networks that are stable are going to have a very

particular structure to them, and in particular, when we're looking at two

individuals, a and b. They're going to have to have a friend in

common. And that's different.

So let's just emphasize the difference between this and the usual measure, which

involves, clustering, where we are looking at a given individual and saying,

how many of their friends are friends of each other?

Here, we're just saying, if two people are friends, they have to have a friend

in common. Okay?

So even though both of these involve triangles, they're having different

implications. One's saying, if you have a link, you

have to other, two other links present. And this one says that you could have,

if, if you look at friends, they have to be connected to each other.

And so, for instance, you know, here's a network where every single link is

supported. So, every time you look at a link,

there's a friend in common, but the clustering in this network is only less

than 50%, because there's a number of individuals for instance, this

individual, who has friends who don't know each other, right?

So there's one, two, and three two and three are not connected so the clustering

is going to be lower. Okay?

And so basically what this says is we ought to see high support.

That's what the theory says. But it it provided that favors are costly

enough then you should see high support. So a joint hypothesis.