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Okay, so in terms of our network formation models I want to talk a little

bit about now about repeated games and networks.

So, we've been talking about games on networks and now what we're going to do

is sort of wrap things back and in particular we'll be looking at a very per

special application in order to talk about this that a favor exchange.

But the idea here is going to be to understand the co-determination of

network structure and behavior. So the network structure affects what

behavior goes on, but the behavior actually affects what the link structure

looks like. And so here, when we talk about favor

exchange, the links are actually going to be defined by the fact that people are,

are trading favors. And so the behavior and the, the network

structure are, are going to be determined in one equilibrium notion together.

And what we'll do is, is look at a simple model of this and then try and understand

what the predictions are in terms of data and have a, have a quick look at some

data that goes along with that. So I'm going to talk a little bit about a

recent model, by myself and Tomas Rodriguez Barraquer and, and Shu Tan.

And, you know, the idea here is that, you know, when you talk about favor exchange

a lot of, of different kinds of relations, you know, interactions by

people are not contractable. If someone comes in to ask to borrow a

book, or CD from you, or, or something, you don't write down a contract,

generally you lend it to them and you anticipate that the reciprocation and,

and repeated interaction means that they'll, they'll return the favor at some

point in time. so, so things are going to be

self-enforcing. And in particular, you know, we'll be

looking at, at different kinds of favor exchange, borrowing and lending money,

kerosene, rice and so forth in the Indian villages that we've talked about before.

And, you know, the idea here is going to be how does successful favor exchange

depend on and influence the network structure?

So are how these two things intertwined? so again, we, you know, we have these

different borrowing. Who would you go to if you needed to

borrow 50 rupees for a day? So we have a network of, of different

borrowing relationships. who comes to you for kerosene?

who would you go to for, for medical help and so forth.

So we've got a set of networks we can look and, and what we're going to try and

do is make sense out of what's actually going on in these networks, which might

be hard to, to understand otherwise. And just in terms of background one, one

thing that has been looked at when you think about social enforcements.

So the social capital literature, Coleman, Bourdieu, Putnam, a whole

serious of authors, have talked about how enforcement depends on the structure of,

of interactions and the ideas, you know, the ones in strong positions then that

leads to better behaviors. And one interpretation of that has been

in terms of clustering. So the idea of it, and actually this was

best articulated by Coleman in a paper in 1988, and, and what he was talking about

is saying, you know, if, if we look at a given individual i And we sort of ask,

what are the incentives of i to behave well in a society?

Coleman's point was that if j and k, if, if two of i's friends are, know each

other and are friends with each other, then that helps them put pressure on i

and, and can help them enforce behavior and make sure that i behaves, so, okay?

And so one thing we want to do is, is just look at a, a simple, you know, model

to see whether that prediction comes out of, when we look at this and in

particular, you know this was suggestive of the different.

When we looked at clustering coefficients earlier in the course, you know we found

that these clustering coefficients tended to be quite high relative to a random

network, and so the idea was that there was some structure going on in terms of

clustering that was really being represented in the data.

And so what we're going to do here is actually explicitly model favor exchange,

and then look at those networks and try and understand what will come out, what,

what's the predicted structure, according to the model?

And then does that relate to clustering, does it relate to something else, what,

what kind of sense can we make of things? We're going to work with a really simple

model. you can enrich this substantially and,

and we actually do in the paper we wrote, so there's a much richer version of the

model there. I'm going to give you a simple vanilla

version of the model. So in, in this simple version basically

favors are worth some value v they cost some value c.

it's socially valuable to do a favor so the value of getting a favor is more than

the cost of doing it. So it makes sense for society to be doing

favors for each other, the person who needs to borrow something, really needs

it more than the person who's got it and so there is a net positive value for

lending something from one person to another or doing a favor for a given

individual. we have a discount factor.

So, people will look at values over time and today, a favor today is worth more

than a favor tomorrow, which is worth more than a favor and so forth, so each

day, the value goes down by some delta, so a favor a week from now is worth delta

to the seventh, and so forth. Okay?

so the prob, we'll also have these favors and the needs for favor arise randomly.

So there will be p, a probability p that some individual i needs a favor from j in

a given period. And generally we're going to treat this

as, instead of having multiple favors all needed at once.

We'll treat this P as being very small so that the probability that more than

favor's needed in a given period is basically negligible and so, we'll look

at the favor arrival process. And so, you can think of this as a, a

[UNKNOWN] arrival process with very small time windows.

Okay, so, favor needs arise at random to at most one of, of two people at a time.

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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.

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so here, you know, it's a small network so I'm not sure you could take 100%, but

here's a network where you actually get every single, business dealing at, a

family in common. And so, you know, when you look at, at

the support level of, just by businesses its actually 80%.

But then, when you add in the marriages, it goes up to 100%, so the marriages are

actually filling in to make sure that some of the business dealings are, have

friends in common. So for instance, you know, here's a

business relationship that doesn't have a friend in common in terms of business

dealings, but then once you go back and fill in the marriage dealings, the

marriages then you end up having a friend in common for these two, and so, you

know, a node in common, Okay, conclusions this robust enforcement

gives us what we call social quilts, we have these links that are supported on a

local level. It gives us some theory for friends in

common, which differs significantly from clustering.

So, it's a different measure of a, of a network, and the support at least is high

in the favor exchange data. This isn't a test of this because we're

not quite sure what the costs and benefits of the favors were, this could

be arising for other reasons but at least what we see is that there's a prediction

that comes out of the theory, and we see that that prediction is, is reasonably

sustained in the the, the data. I think more generally the kind of thing

that, that we should take away from this is that you know, we can begin to build

models more explicitly of what we think is going on in the network, and not only

take into account how the network's going to effect that behavior, but also think

what that means in terms of the behavior impacting the structure of the network.

What kinds of network structures do we need in order for certain behaviors to be

present? So here, this is done in terms of favor

exchange, but you could also think of, you know, people maintaining

relationships in a business setting, to make sure that they're going to have

information flows, or be able to call on other individuals to do trades, and so

forth. What kinds of networks do we need in

those settings, how does that impact both the behavior and the network structure?

So there's a rich set of settings in which this kind of technique can be used.

this is just one example of that..