Hi. In previous lectures we've talked about culture as being many coordination games. What I want to do in this lecture is introduce yet another model of culture that's based on those coordination games, that brings in some other features. Now this model was developed by Jenna Bednar, who's a professor of political science at the University of Michigan, along with me and two of our graduate students, Aaron Bramson and Andrea Jones-Rooy. What this model tries to do is include two things that were missing from the previous models. What are those two things? Well, the first one is this notion of coherence and consistency; you remember Trilling's definition of what culture is. Remember when we talked about making a coherent life? Well that coherence can be seen, remember if we look at the data from Ron Inglehart's World Value Survey, we see there's a lot of consistencies across countries. So we see Protestant Europe looks the same. Catholic Europe looks the same. The Islamic countries look the same. And what that means is, there's a consistency to behavior. So if you're in Sweden let's say and you see how people behave in one context, you can predict pretty well how their gonna behave in another context. It's that consistency we wanna try and understand. There's something else we'd like to try and understand, and that is the tremendous heterogeneity within cultures. So remember, when we looked at Sweden, it wasn't everybody was at one spot, there was a giant scattershot of behaviors. That was like the "There is no Great Blue Heron" idea. And the same was true when we looked at Zimbabwe. Zimbabwe wasn't just one point in the lower left hand corner. It was a whole spread. So we'd like, if we like a model that gives us the consistency. So why is Sweden way up here and Zimbabwe way down here? And we'd also like a model that gives us some sort of heterogeneity within culture. Well here's the funny thing about models. That's sort of an easy thing to do, because all you have to do is make assumptions so that you get those results. So for example, all we have to do to get consistency is to say, let's let the values that people coordinate on, the actions, the behaviors, [inaudible] they are, have some meaning, and let's assume that people desire some consistency. So then what we'll probably get is that people have preferences for consistency, we should get some consistency. Let's also throw in just a tiny little bit of innovation. And what we can do is we can ask okay, how much innovation or errors do we have to have when people are trying to copy other people or coordinate with other people, and trying to be consistent, to get the sort of variation you see in those pictures? Right, so look at these pictures. We see massive variations. So that would suggest, what we probably have to have is substantial rates of innovation or errors in order to generate that. So we want the model for us to explain, how does this work? Do we have to have a lot of innovation/errors or just a little bit? And how much consistency do we need in order to get consistency of the model? So here's what we gonna do. We're just going to change the model by assuming one extra thing. We're going to assume, in addition to people trying to coordinate, we're going to assume that they're also trying to be consistent, and I'll explain what that means in a second. And then we'll also just throw in these tiny errors. So remember our coordination rule, how does that work? There's two people, they meet, they each have this vector of actions or beliefs or attitudes, whatever you want to call them. And when the leader and follower meet, they look at the second dimension, let's say the follower says, well you're a three, I'm a one, I'll switch that to a three. That's what coordination is, you switch your action. You put the ketchup where your friends put the ketchup. What would consistency be? Well, consistency would just be this: you look at yourself, now these values, 5, 3, 1, 4, have meaning. Five is close to five, four is close to four. And you look and you think, I'm five on the first, I'm one on the second. And that doesn't make any sense, so you switch and become five on both. What would that be? Let me give you an example. So suppose that. >> You are from a family that's pretty reserved and so you don't even hug your parents very often. And then you go to college and all your friends are hugging each other. So, now you got hugging behavior five with your friends. Well, then you go home and you realize you're hugging someone you just met a month ago to say good-bye and you're not even hugging your own mother. And so, you've taken this action one, which is not hugging. So, what happens is that you've switched. You say you know what I'm hugging my friends I might as well hug my mother. And you become five, five and you change your behavior. Become more consistent in terms of how you behave. So, those are the two rules. We're going to assume that people try and coordinate with other people, and they want to try and be consistent. So what should happen? Well, you'd expect to get, you'd expect to get consistent coordinated behavior. You'd expect the system to just, boom, run right in. Well here's what's funny. So when you write in this model, Jenna and I and Aaron and Andrea [inaudible] as well, that it happened, this happened, but it took a long, long time. The process took an incredibly long time to converge. And the other thing is, when we threw in tiny errors. When we threw in just little, tiny errors, not big errors, then we got those big clouds that you saw in the picture. So when you see those clouds, they have all this spread in Sweden, all this spread in Zimbabwe, you might think, well a lot of people have to be innovating. They're trying new things, they're making errors. Instead, it turns out only a small number of people have to, to get that level of heterogeneity within the community. So, that was a surprising result. By constructing the model, we saw that tiny, tiny amounts of error innovation leads to these big spreads. Now the question is why, what's causing them? What's giving us this giant spread like this with just tiny amounts of errors? Well here's where we have to think through the model. So let's suppose our model converge to everybody being five on everything. Now let's suppose somebody decides to innovate a little bit and so they change to six on this thing. So they just try something new. We might think that should go away because they're trying to be consistent with themselves, then maybe they'll turn that into a five, or they meet somebody else who'll turn it into a five. There's another possibility, and that is if they meet somebody else to copy the six, and there's a third possibility, which is, they themselves can do something else to a six, if the six works. So what you get is that six can spread this way, can spread sort of, you think of it, vertically across society, it can also spread horizontally, so it can also spread within the person. So you get these two directions you can go. You can go horizontally within a person, and it can go vertically across people. What this means is that these innovations, or these mistakes, can spread in two directions. Which means the process isn't gonna converge very fast. And it's gonna mean that you're gonna get a lot of spread, because errors are going to propagates in many directions. See, if someone make a mistake. They'll innovate here and it spreads both within the person and across people, and if another error happens here, that could spread within the person and across people, and then you've got all this heterogeneity spreading out. Now at the same time, though, the system's gonna maintain substantial levels of consistency. Most of the things are gonna be 5's, but there's gonna be 6's and 7's everywhere. So this is a, just a picture of it. What you'd like to do is understand, can we understand this at a deeper level. Can we mathematically understand why small amounts of behav-, of error innovation are causing this big spread? We've got a picture of it, but it'd be nice to nail that down. Well, here's how we nail it down. We construct an even simpler model. So sometimes, when you wanna understand a process, what you do is construct an incredibly simple model. So here's what were gonna do. We're gonna construct a model with two agents, two games, and two actions. So people are deciding whether to hug or bow with their family, or hug or bow with their friends, that's it. So now we can write down all possible states of the world. So let's think of it this way, here's game one and here's game two. So we'll call them the, the column, the column one game, column two game. Here's person one. And here's person two. So, what you have is, it could be that both of them are doing the same thing on both games. So, the system would be coordinated and consistent. It could also be the case that one of them, the person two, is taking the red action on both games, but person one is taking the green action on game two, so we could call this "off by one", so there's one person changes that one thing, then everything would be coordinated and consistent. Now another thing we could have, is we could have that people could be consistent but not coordinated. So person one is green on both things and person two is red on both. Finally, obviously next they could be coordinated but not consistent, so they could both play the red action on game one and the green action on game two, but they'd have a lot of cognitive dissonance, because neither one is consistent. And then finally it could be a total mess, neither person could be consistent and the two couldn't be coordinated. Let's first look at this system without an error, with no noise terms. What do we get? Well, here's what the dynamics would look like. And let's just look at the consistency dynamic first. If we're in this state where people are off by one, one thing that could happen is, this person, person two, could look at themself and say, "am I consistent?" And they would say, "yes, I am consistent", and nothing would change. Alternatively, person one could look at herself and she could say, "am I consistent?" She could say, "oh my goodness, I'm not" and she could switch. Now there's two possibilities, one thing she could do is she could switch and have them both become red. And then what would happen is both people would be consistent, but neither would be coordinated, or alternatively, she could look at herself here and say, "oh, maybe I'll make them both green" and then the whole situation will be coordinated and consistent. So each of these would happen with a probability of half. That's the consistency dynamic. We could also write down the coordination dynamic, and we can do it for all of these different states. Here's what this is gonna look like. It's gonna look like a map where we write down all the different states. And we can write down the probability of moving from those states to other states. So let's just do a couple cases. Suppose we're here. This is the case that's a total mess. Well, if someone looks at themselves, and decides to become consistent, then, like, let's suppose it's this person on the bottom, person two. If they become consistent, then we'll move into the "off by one" state. Or alternatively, suppose that two people may decide to coordinate. Well then again, we'll move into the "off by one" state. So no matter what happens, with probability one where gonna move from this state, the "total mess" state into the "off by one" state. >> Now, once we're in the "off by one" state, a whole bunch of stuff could happen. One thing that could happen is we could stay in the "off by one" state. How could that happen? Well, that could happen if this person looked at himself and decided to be consistent. Or if these two people, if the people in game one decided to coordinate because nothing would happen if you stayed in the state. Alternative it would be possible to move over here to this side to the right, where both people are consistent, but they're no longer coordinated. Alternately, it would be possible to move down here where they're each coordinated, but they're not consistent. And you could even move over to this state which is the coordinated consistent state and this is the only one that's stable. Because if you're here, you're not going to move out. Now when you look at this, this is a complicated map, but you look at this and you think, wow this reminds me of something. It should remind you of a Markov process. However, it differs from a Markov process, in that a Markov process one of our assumptions was you can get from any state to any other state. But that's not true of this state, this equilibrium state right here, once you're there, you're stuck there. But remember in our model, though. This here, I'm assuming there's no innovation. Remember in our model, there's a slight chance of innovation or making an error, some epsilon. And if you make an innovation error, you're going to move to the off by one state. So, that would put an arrow back in this other direction. And now we have a Markov process. So the way that John and Andrea and Aaron and I captured this process, when you try to model it, was to say: Here's all these states. And now we've got a Markov process. And once you've got a Markov process, guess what? We just write a big matrix. We just write down all of our states. Here's our states at time t. And here's our states at time t+1. And we say, what are the odds that you move from one state to the other? And you just write a giant matrix with all those numbers in it. And then you can analyze the matrix and figure out how much of the spread you get. And what you get when you run that, is you get that small innovation rates, really small innovation rates, lead to substantial heterogeneity. So our model gave us a big surprise. So how do we have [inaudible]? When we think about cultures, what we see is, we see differences between them. So people from France behave differently than people from Mexico. We also see similarities within, that's what we got in this last model. We see that, you know, that people within a culturally, within an interacting group, become similar. You notice how we said "similarities within", not "identical behavior within". There's a big, you know, spread, a lot of within-group heterogeneity, within Sweden, within Zimbabwe, within Greece, within Spain. And finally, some of that behavior is "interesting", in the sense that it doesn't appear to be optimal from outside. Now if we go back to our model, if we go back to this model of our Markov process and our consistent behavior, one way we can get the interesting behavior is that it could be that maybe if I'm doing five, on each of these things, that this five isn't optimal, it's not the right thing to do. However, we're choosing to do five because of the fact it's consistent with the other things that we do. What we've learned here is that we can explain something like culture, not all of culture. But we can at least get some insights into culture by thinking of it as coordination. By thinking of cultural behavior as coordination on a whole range of activities. And if we're trying to coordinate with other people, that can explain why people from different cultures are different. Just 'cause we've manage to coordinate in different ways. There are three ways we can coordinate on a wrong action. First, we could just idiosyncratically coordinate on the wrong thing. Second, remember on the shaker/ bower example we could have had it be the case that payoffs changed over time, and shaking was better originally, but bowing was better later. And then third, as we just saw, it could be, in order to maintain consistency, we could choose a behavior that's suboptimal in one domain because it makes us consistent with our behavior in other domains. So what we've thought of here is, we've thought of culture as multiple coordination games, where what we're trying to do is be consistent. And that makes a lot of sense. 'Cause what we're really trying to do in order to make a coherent life is to have somewhat consistent behavior across a variety of domains. And also to get good outcomes, it's important to put the ketchup in the same place, greet people the same way. We have to coordinate with people around us. So these very simple models have explained differences within culture, similarities across, and why we see "interesting" stuff. And the surprise, the one surprise we got. In this last model, is that, you know, when we actually look at the data we see, even though people are trying to coordinate and be consistent, we see a big blob. We see a lot of within-group heterogeneity. We could explain that with a model by saying, you know what? That happens even if there is just a small amount of error experimentation, because of the fact that errors propagate in two directions. And that was the intuition we had, and we could put that in a Markov model, and by analyzing the Markov model we can see in fact that was the case. That small errors due in fact propagate, and the equilibrium in that Markov model for even small epsilon, has large levels of heterogeneity. So we can explain those large levels of heterogeneity, or at least, you know, why a process like this might produce them from the dynamics, by using the Markov model. And that's sort of the last point here, is that when we look at these coordination games, it's easy to write down the model if there's a bunch of people and they coordinate. But then what do we do? What we could do is, which is great as first-off, we could use our Lyapunov model to show that pure coordinating behavior creates a Lyapunov function. So the process is gonna very quickly converge. It's people trying to coordinate, should very quickly converge. Then when we threw in the error terms, and we threw in consistency as well, we now no longer had a Lyapunov function, but we could use our Markov model and explain why this system ended up going to an equilibrium with a lot of heterogeneity. So one of the reasons why we wanna construct a bunch of models and learn a bunch of models, is, we can use them sometimes to analyze other models. And that's what we did here. So we used our Lyapunov and our Markov model to analyze our culture models. If we didn't have those skills, if we didn't have the Lyapunov model and the Markov model, we couldn't have done much with the culture model. We just could of written it down and say, and said, well this sort of explains some things, but we don't know what happens. But because we have those other two models, we can analyze in full what does happen. So this concludes our discussion of culture. It's a constrained discussion. There's a lot more to culture than this, and that's why there's hundreds and hundreds of definitions, and we need hundreds of definitions, because culture is a complex thing. But what these simple models have allowed us to do, is understand some basic properties of cultures. Which are, there's a lot of difference between cultures, and those difference may arise because the fact that people need to coordinate within groups with which they interact. There's also consistency within cultures, and that happens because it gets cognitively easier to do the same behavior in lots of different domains. And then third, we see a lot of heterogeneity within cultures. And that happens, as we saw by using a Markov analysis on our model, because if people just make small mistakes or occasionally try an innovation, those differences are going to profligate through the population in two ways; within an individual and across individuals. And that's going to give us a lot of within-culture heterogeneity. So cultures differ between themselves. Cultures differ within themselves. But they still have this consistency. They have what you might call a cultural signature. These very simple models combined with our tools of Lyapunov functions and Markov processes have helped us understand why that happens. All right. Thank you.
