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Hi. In previous lectures we've talked about culture as being many coordination

Â games. What I want to do in this lecture is introduce yet another model of culture

Â that's based on those coordination games, that brings in some other features. Now

Â this model was developed by Jenna Bednar, who's a professor of political

Â science at the University of Michigan, along with me and two of our graduate

Â students, Aaron Bramson and Andrea Jones-Rooy. What this model tries to do is

Â include two things that were missing from the previous models. What are those two

Â things? Well, the first one is this notion of coherence and consistency; you remember

Â Trilling's definition of what culture is. Remember when we talked about making a

Â coherent life? Well that coherence can be seen, remember if we look at the data from

Â Ron Inglehart's World Value Survey, we see there's a lot of consistencies across

Â countries. So we see Protestant Europe looks the same. Catholic Europe looks the

Â same. The Islamic countries look the same. And what that means is, there's a

Â consistency to behavior. So if you're in Sweden let's say and you see how people

Â behave in one context, you can predict pretty well how their gonna behave in

Â another context. It's that consistency we wanna try and understand. There's

Â something else we'd like to try and understand, and that is the tremendous

Â heterogeneity within cultures. So remember, when we looked at Sweden, it wasn't

Â everybody was at one spot, there was a giant scattershot of behaviors. That

Â was like the "There is no Great Blue Heron" idea. And the same was true when we looked at

Â Zimbabwe. Zimbabwe wasn't just one point in the lower left hand corner. It was a

Â whole spread. So we'd like, if we like a model that gives us the consistency. So

Â why is Sweden way up here and Zimbabwe way down here? And we'd also like a model that

Â gives us some sort of heterogeneity within culture. Well here's the funny thing about

Â models. That's sort of an easy thing to do, because all you have to do is make

Â assumptions so that you get those results. So for example, all we have to do to get

Â consistency is to say, let's let the values that people coordinate on, the

Â actions, the behaviors, [inaudible] they are, have some meaning, and let's assume

Â that people desire some consistency. So then what we'll probably get is that

Â people have preferences for consistency, we should get some consistency. Let's also

Â throw in just a tiny little bit of innovation. And what we can do is we can ask okay, how

Â much innovation or errors do we have to have when people are trying to copy other

Â people or coordinate with other people, and trying to be consistent, to get the sort

Â of variation you see in those pictures? Right, so look at these pictures. We see

Â massive variations. So that would suggest, what we probably have to have is

Â substantial rates of innovation or errors in order to generate that. So we want the

Â model for us to explain, how does this work? Do we have to have a lot of

Â innovation/errors or just a little bit? And how much consistency do we

Â need in order to get consistency of the model? So here's what we gonna do. We're

Â just going to change the model by assuming one extra thing. We're going to assume, in

Â addition to people trying to coordinate, we're going to assume that they're also trying to be

Â consistent, and I'll explain what that means in a second. And then we'll also

Â just throw in these tiny errors. So remember our coordination rule, how does

Â that work? There's two people, they meet, they each have this vector of actions or

Â beliefs or attitudes, whatever you want to call them. And when the leader and

Â follower meet, they look at the second dimension, let's say the follower says,

Â well you're a three, I'm a one, I'll switch that to a three. That's what

Â coordination is, you switch your action. You put the ketchup where your friends put

Â the ketchup. What would consistency be? Well, consistency would just be this: you

Â look at yourself, now these values, 5, 3, 1, 4, have meaning. Five is close to

Â five, four is close to four. And you look and you think, I'm five on the first, I'm

Â one on the second. And that doesn't make any sense, so you switch and become five

Â on both. What would that be? Let me give you an example. So suppose that. >>

Â You are from a family that's pretty reserved and so you don't even hug your

Â parents very often. And then you go to college and all your friends are hugging

Â each other. So, now you got hugging behavior five with your friends. Well,

Â then you go home and you realize you're hugging someone you just met a month ago

Â to say good-bye and you're not even hugging your own mother. And so, you've

Â taken this action one, which is not hugging. So, what happens is that you've

Â switched. You say you know what I'm hugging my friends I might as well hug my

Â mother. And you become five, five and you change your behavior. Become more

Â consistent in terms of how you behave. So, those are the two rules. We're going to

Â assume that people try and coordinate with other people, and they want to try and be consistent. So

Â what should happen? Well, you'd expect to get, you'd expect to get consistent

Â coordinated behavior. You'd expect the system to just, boom, run right in. Well

Â here's what's funny. So when you write in this model, Jenna and I and Aaron and

Â Andrea [inaudible] as well, that it happened, this happened, but it took a

Â long, long time. The process took an incredibly long time to converge. And the

Â other thing is, when we threw in tiny errors. When we threw in just little, tiny

Â errors, not big errors, then we got those big clouds that you saw in the picture. So

Â when you see those clouds, they have all this spread in Sweden, all this spread in

Â Zimbabwe, you might think, well a lot of people have to be innovating. They're

Â trying new things, they're making errors. Instead, it turns out only a small number

Â of people have to, to get that level of heterogeneity within the community. So, that

Â was a surprising result. By constructing the model, we saw that tiny, tiny amounts

Â of error innovation leads to these big spreads. Now the question is why, what's

Â causing them? What's giving us this giant spread like this with just tiny amounts of

Â errors? Well here's where we have to think through the model. So let's suppose our

Â model converge to everybody being five on everything. Now let's suppose somebody

Â decides to innovate a little bit and so they change to six on this thing. So they just

Â try something new. We might think that should go away because they're trying

Â to be consistent with themselves, then maybe they'll turn that into a five, or they

Â meet somebody else who'll turn it into a five. There's another possibility,

Â and that is if they meet somebody else to copy the six, and there's a third

Â possibility, which is, they themselves can do something else to a six, if the six

Â works. So what you get is that six can spread this way, can spread sort of, you

Â think of it, vertically across society, it can also spread horizontally, so it can

Â also spread within the person. So you get these two directions you can go. You can

Â go horizontally within a person, and it can go vertically across people. What this

Â means is that these innovations, or these mistakes, can spread in two directions.

Â Which means the process isn't gonna converge very fast. And it's gonna mean

Â that you're gonna get a lot of spread, because errors are going to propagates in

Â many directions. See, if someone make a mistake. They'll innovate here and it

Â spreads both within the person and across people, and if another error happens here,

Â that could spread within the person and across people, and then you've got all this

Â heterogeneity spreading out. Now at the same time, though, the system's gonna

Â maintain substantial levels of consistency. Most of the things are gonna

Â be 5's, but there's gonna be 6's and 7's everywhere. So this is a, just a

Â picture of it. What you'd like to do is understand, can we understand this at a

Â deeper level. Can we mathematically understand why small amounts of behav-, of

Â error innovation are causing this big spread? We've got a picture of it, but

Â it'd be nice to nail that down. Well, here's how we nail it down. We construct

Â an even simpler model. So sometimes, when you wanna understand a process, what you

Â do is construct an incredibly simple model. So here's what were gonna do. We're

Â gonna construct a model with two agents, two games, and two actions. So people are

Â deciding whether to hug or bow with their family, or hug or bow with their friends,

Â that's it. So now we can write down all possible states of the world. So let's

Â think of it this way, here's game one and here's game two. So we'll call them the,

Â the column, the column one game, column two game. Here's person one. And here's

Â person two. So, what you have is, it could be that both of them are doing the same

Â thing on both games. So, the system would be coordinated and consistent. It could

Â also be the case that one of them, the person two, is taking the red action on

Â both games, but person one is taking the green action on game two, so we could call

Â this "off by one", so there's one person changes that one thing, then everything

Â would be coordinated and consistent. Now another thing we could have, is we could

Â have that people could be consistent but not coordinated. So person one is green on

Â both things and person two is red on both. Finally, obviously next they could be

Â coordinated but not consistent, so they could both play the red action on game one

Â and the green action on game two, but they'd have a lot of cognitive dissonance,

Â because neither one is consistent. And then finally it could be a total mess,

Â neither person could be consistent and the two couldn't be coordinated. Let's

Â first look at this system without an error, with no noise terms. What do we get?

Â Well, here's what the dynamics would look like. And let's just look at the

Â consistency dynamic first. If we're in this state where people are off by one,

Â one thing that could happen is, this person, person two, could look at themself

Â and say, "am I consistent?" And they would say, "yes, I am consistent", and

Â nothing would change. Alternatively, person one could look at herself and she

Â could say, "am I consistent?" She could say, "oh my goodness, I'm not" and she could switch. Now

Â there's two possibilities, one thing she could do is she could switch and have them

Â both become red. And then what would happen is both people would be consistent,

Â but neither would be coordinated, or alternatively, she could look at herself

Â here and say, "oh, maybe I'll make them both green" and then the whole situation will be

Â coordinated and consistent. So each of these would happen with a probability of

Â half. That's the consistency dynamic. We could also write down the coordination

Â dynamic, and we can do it for all of these different states. Here's what this is

Â gonna look like. It's gonna look like a map where we write down all the different

Â states. And we can write down the probability of moving from those states to

Â other states. So let's just do a couple cases. Suppose we're here. This is the

Â case that's a total mess. Well, if someone looks at themselves, and decides to become

Â consistent, then, like, let's suppose it's this person on the bottom, person two. If

Â they become consistent, then we'll move into the "off by one" state. Or

Â alternatively, suppose that two people may decide to coordinate. Well then again,

Â we'll move into the "off by one" state. So no matter what happens, with probability

Â one where gonna move from this state, the "total mess" state into the "off by one"

Â state. >> Now, once we're in the "off by one" state, a whole bunch of stuff could happen.

Â One thing that could happen is we could stay in the "off by one" state. How

Â could that happen? Well, that could happen if this person looked at himself and

Â decided to be consistent. Or if these two people, if the people in game one decided

Â to coordinate because nothing would happen if you stayed in the state. Alternative

Â it would be possible to move over here to this side to the right, where both

Â people are consistent, but they're no longer coordinated. Alternately, it would

Â be possible to move down here where they're each coordinated, but they're not

Â consistent. And you could even move over to this state which is the coordinated

Â consistent state and this is the only one that's stable. Because if you're here,

Â you're not going to move out. Now when you look at this, this is a complicated map,

Â but you look at this and you think, wow this reminds me of something. It should

Â remind you of a Markov process. However, it differs from a Markov process, in that a

Â Markov process one of our assumptions was you can get from any state to any other

Â state. But that's not true of this state, this equilibrium state right here, once

Â you're there, you're stuck there. But remember in our model, though. This here,

Â I'm assuming there's no innovation. Remember in our model, there's a slight

Â chance of innovation or making an error, some epsilon. And if you make an

Â innovation error, you're going to move to the off by one state. So, that would put

Â an arrow back in this other direction. And now we have a Markov process. So the way

Â that John and Andrea and Aaron and I captured this process, when you try to model

Â it, was to say: Here's all these states. And now we've got a Markov

Â process. And once you've got a Markov process, guess what? We just write a big

Â matrix. We just write down all of our states. Here's our states at time t. And

Â here's our states at time t+1. And we say, what are the odds that you move

Â from one state to the other? And you just write a giant matrix with all those

Â numbers in it. And then you can analyze the matrix and figure out how much of the

Â spread you get. And what you get when you run that, is you get that small innovation

Â rates, really small innovation rates, lead to substantial heterogeneity. So our

Â model gave us a big surprise. So how do we have [inaudible]? When we think about cultures,

Â what we see is, we see differences between them. So people from

Â France behave differently than people from Mexico. We also see similarities within,

Â that's what we got in this last model. We see that, you know, that people within a

Â culturally, within an interacting group, become similar. You notice how we said

Â "similarities within", not "identical behavior within". There's a big, you know, spread, a

Â lot of within-group heterogeneity, within Sweden, within Zimbabwe, within Greece, within

Â Spain. And finally, some of that behavior is "interesting", in the sense that it

Â doesn't appear to be optimal from outside. Now if we go back to our model, if we go

Â back to this model of our Markov process and our consistent behavior, one way we

Â can get the interesting behavior is that it could be that maybe if I'm doing five, on

Â each of these things, that this five isn't optimal, it's not the right thing to

Â do. However, we're choosing to do five because of the fact it's consistent with

Â the other things that we do. What we've learned here is that we can explain

Â something like culture, not all of culture. But we can at least get some

Â insights into culture by thinking of it as coordination. By thinking of cultural

Â behavior as coordination on a whole range of activities. And if we're trying to

Â coordinate with other people, that can explain why people from different cultures

Â are different. Just 'cause we've manage to coordinate in different ways. There are

Â three ways we can coordinate on a wrong action. First, we could just

Â idiosyncratically coordinate on the wrong thing. Second, remember on the shaker/

Â bower example we could have had it be the case that payoffs changed over time, and

Â shaking was better originally, but bowing was better later. And then third, as we

Â just saw, it could be, in order to maintain consistency, we could choose a

Â behavior that's suboptimal in one domain because it makes us consistent with our

Â behavior in other domains. So what we've thought of here is, we've thought of

Â culture as multiple coordination games, where what we're trying to do is be

Â consistent. And that makes a lot of sense. 'Cause what we're really trying to do in

Â order to make a coherent life is to have somewhat consistent behavior across a

Â variety of domains. And also to get good outcomes, it's important to put the ketchup

Â in the same place, greet people the same way. We have to coordinate with people

Â around us. So these very simple models have explained differences within culture,

Â similarities across, and why we see "interesting" stuff. And the surprise, the

Â one surprise we got. In this last model, is that, you know, when we actually look

Â at the data we see, even though people are trying to coordinate and be consistent, we

Â see a big blob. We see a lot of within-group heterogeneity. We could explain that with a

Â model by saying, you know what? That happens even if there is just a small

Â amount of error experimentation, because of the fact that errors propagate in two

Â directions. And that was the intuition we had, and we could put that in a Markov

Â model, and by analyzing the Markov model we can see in fact that was the case. That

Â small errors due in fact propagate, and the equilibrium in that Markov model for

Â even small epsilon, has large levels of heterogeneity. So we can explain those

Â large levels of heterogeneity, or at least, you know, why a process like this

Â might produce them from the dynamics, by using the Markov model. And that's sort

Â of the last point here, is that when we look at these coordination games, it's easy to

Â write down the model if there's a bunch of people and they coordinate. But then what

Â do we do? What we could do is, which is great as first-off, we could use our Lyapunov model to show

Â that pure coordinating behavior creates a Lyapunov function. So the process is gonna

Â very quickly converge. It's people trying to coordinate, should very quickly

Â converge. Then when we threw in the error terms, and we threw in consistency as

Â well, we now no longer had a Lyapunov function, but we could use our Markov

Â model and explain why this system ended up going to an equilibrium with a lot of

Â heterogeneity. So one of the reasons why we wanna construct a bunch of models and

Â learn a bunch of models, is, we can use them sometimes to analyze other models. And

Â that's what we did here. So we used our Lyapunov and our Markov model to analyze

Â our culture models. If we didn't have those skills, if we didn't have the

Â Lyapunov model and the Markov model, we couldn't have done much with the culture

Â model. We just could of written it down and say, and said, well this sort of

Â explains some things, but we don't know what happens. But because we have those

Â other two models, we can analyze in full what does happen. So this concludes our

Â discussion of culture. It's a constrained discussion. There's a lot more to culture

Â than this, and that's why there's hundreds and hundreds of definitions, and we need

Â hundreds of definitions, because culture is a complex thing. But what these simple

Â models have allowed us to do, is understand some basic properties of cultures. Which

Â are, there's a lot of difference between cultures, and those difference may arise

Â because the fact that people need to coordinate within groups with which they interact.

Â There's also consistency within cultures, and that happens because it gets

Â cognitively easier to do the same behavior in lots of different domains. And then

Â third, we see a lot of heterogeneity within cultures. And that happens, as we

Â saw by using a Markov analysis on our model, because if people just make small

Â mistakes or occasionally try an innovation, those differences are going to

Â profligate through the population in two ways; within an individual and across

Â individuals. And that's going to give us a lot of within-culture heterogeneity. So

Â cultures differ between themselves. Cultures differ within themselves. But

Â they still have this consistency. They have what you might call a cultural

Â signature. These very simple models combined with our tools of Lyapunov

Â functions and Markov processes have helped us understand why that happens. All

Â right. Thank you.

Â