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

Â bit about now about repeated games and networks.

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

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

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

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

Â network structure and behavior. So the network structure affects what

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â depend on and influence the network structure?

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

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

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

Â borrowing relationships. who comes to you for kerosene?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â particular, you know this was suggestive of the different.

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

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

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

Â clustering that was really being represented in the data.

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

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

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

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

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

Â model. you can enrich this substantially and,

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

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

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

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

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

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

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

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

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

Â individual. we have a discount factor.

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

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

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

Â to the seventh, and so forth. Okay?

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

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

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

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

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

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

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

Â [UNKNOWN] arrival process with very small time windows.

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

Â 6:54

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

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

Â delta, delta squared, delta cubed and so forth.

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

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

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

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

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

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

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

Â Okay. So when can you sustain favor exchange?

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

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

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

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

Â perpetuity, so the value from tomorrow onwards.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â going to be able to sustain favor exchange.

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

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

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

Â collapses, and so this whole triangle disintegrates.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â relationships, whatever. so links are maintained if people

Â mutually agree. And so after this process, depending on

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

Â repeats itself. Okay?

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

Â networks if we look at this process, over time.

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

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

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

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

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

Â which are both going to support favor exchange.

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

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

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

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

Â different ways. Now if they fail to perform that favor

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â who are actually quite far away in the network.

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

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

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

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

Â you know, define robustness against the contagion.

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

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

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

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

Â originally who are going to end up losing relationships.

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

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

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

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

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

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

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

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

Â Okay? So support means that two individuals

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

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

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

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

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

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

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

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

Â friend in common. [INAUDIBLE].

Â Okay? So, so this says that,

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

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

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

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

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

Â common. And that's different.

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

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

Â how many of their friends are friends of each other?

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

Â in common. Okay?

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

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

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

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

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

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

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

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

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

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

Â is going to be lower. Okay?

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

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

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

Â 21:47

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

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

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

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

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

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

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

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

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

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

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

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

Â local level. It gives us some theory for friends in

Â common, which differs significantly from clustering.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â forth. What kinds of networks do we need in

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

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

Â this is just one example of that..

Â