Hi. In the previous lecture we talked about kind of a fun model, where students could be alert or bored. Remember, it was a Markov model. And in these Markov models, there's a set of states, in that case alert or bored. And then there's these transition probabilities, that give you the probability of moving from alert to bored. I want to move in this lecture to a slightly more complicated model. And it's going to be a model that involves countries. And these countries can either be free, partly free or not at all free and being run by dictators. Now I'm gonna look at the dynamics of that situation and it'll help us, sort of maybe learn a little bit more about how these Markov processes work, see how we can extend them to more dimensions and also cope with a somewhat counter-intuitive finding. Okay, so, let's start simply with even a, just a two-state democracy model. So we can imagine there's two types of countries: there's democracies and there's non-democracies, and assume that five percent of democracies switch into dictatorships every decade, and that twenty percent of dictatorships become democracies. So that'll be my assumption. And then, let's just walk through the logic. So how do we do this? Well, we're gonna write down a Markov transition matrix. So let's, let's start off by assuming we have 30 percent of countries that are in democracies, and 70 percent are dictatorships. So, of this 30 percent that are democracies, we know that 95 percent will stay democracies. Of the 70 percent that are dictatorships, we know that twenty, only twenty percent will become democracies. So to figure out how many new democracy's next time, we just take this row and multiply it by this column, so we get .95 times .3 plus .2. Right? Times .7. Put, make sure, parentheses around here. So this is going to be .285. And this is going to be .14. And this gives us .425. So we're going to get 43 percent of countries, in the next decade, are going to be democracies. And we could do this one more time, and say, well if we have 43 percent or 42.5 percent democracies, we multiply this row by the column, we're going to get that 52 percent are democracies next time. Now, if you looked at this trend, you say we start out with 30 percent democracies, and we go to 42%. Then we go from 42 percent to 52%, well, you might just sort of extrapolate, like, that looks like a linear trend. And eventually, we're gonna end up with everybody being a democracy. Yet we know that that's not true, right? We know that we can solve for the equilibrium, and that it's probably gonna involve some churn. So how do we solve for the equilibrium? Remember from the last statement, we just wanna take this row, times this column. We're now, instead of putting down a specific problem, we put p and 1-p, we want, after we multiply that through that we get the p, the same p back. So that means that's gonna be.95 times p, plus.2, times one minus p, should equal p. Alright, we've got a bunch of stuff and let's just multiply this by a 100 to get rid of everything. So we get 95 p plus 20 minus 20 p equals 100 p. Right? I just multiplied both sides by 100. Now if I bring everything over there I'm gonna get: 20 equals 25 p. So that means p equals four fifths. So here's sort of the surprising thing, we only end up with 80 percent democracies, even though, right, 95 percent of democracies stay democracies and 20 percent of dictatorships become democracies in each decade, we still end up with only 80 percent democracies. That's what's counter-intuitive, and that's why having this Markov model can be really useful, because it helps us, you know really figure out what's going to happen as opposed to just maybe extrapolating and thinking, boy, there's this big trend toward democratization. If things continue as they are, everything going to be a democracy. Well, now let's move to a more sophisticated model. Now, let's suppose that we classify countries not as just democracies or dictatorships, we have three categories. Free, partly free, and not free. And this is some data, actually, from Freedom House, that I, you know, just plugged in to Excel, and plotted out. And what we see is an increase in free countries, and a decrease in not free countries, and a slight decrease in partly free countries. And we could ask, what's likely to happen? Well, what you can do, if I plug in, sort of, in five year increments, the transition probabilities, and do some crude estimates, you sort of get the following. I get that each decade, five percent of free, and about fifteen percent of not free become partly free. So, those are those transition probabilities, And five percent of not free, and ten percent of partly free become free. And ten percent of partly free become not free. So, all sorts of transition probabilities are kinda complicated. The matrix is more useful. So, I can put free, partly free, and not free. Right here and then I can put free, partially free and not free here, and now I've just got three states, just like I had before except for instead of a two by two matrix, I've got a three by three matrix, same thing goes. Now it used to be, before we had computers, when you went to three by three and four by four matrices you'd just go, "oh no, it's going to be a lot of math, lot of algebra". And it was, but now that you've got, you know computers, it's very easy to just, you know make a huge matrix and solve for the equilibrium, there's nothing complicated about it. So what does that equilibrium look like? Well, all we do is take each one of these rows, and multiply by the columns. But now, the column has a p, a q, and a one minus p minus q, instead of just a p and a one minus p, and I want it to be the case so when I multiply this row times this column I get p. And when I multiply this row, by the column I get q. And when I multiply this row by this column, I get one minus p minus q. So a lot of algebra here. You can do it. And if you do it you get the following answer. You get that 62.5 percent of countries will be free, 25 percent will be partly free, and 12.5 percent will be not free. Now if you looked at that initial graph, you might have thought, oh look. There's this trend towards freedom. We're all gonna be free, But in fact if transition probabilities stay fixed, well fewer than two thirds of countries will be free. Again that's sort of surprising, 'cause if you look at this trend, you might guess, you come from 25% up to 45%. It looks like very soon we'll all be free. But in fact, assuming those transition probabilities stay the same, you end up with about 62 and a half percent being free. Here's what our model shows. Our model's not quite as good. Model sort of shows these general trends, like this. So if I plugged in that same initial condition and ran my model, I get this sort of picture. It doesn't look exactly the same as that picture, but it doesn't look bad. Another way to look at it is to say, if I feed those probabilities in and start at the initial case, how close does it get? And what you can see is the model comes up at the end of the 40 year period with values that are really close to what we saw in the real world. Now, the reason they're so close is because I estimated by transition probabilities from the actual data. So it's likely it would be really close like this. But what's more interesting is that the patterns look fairly similar, as well. Now does that mean we can buy into the 62.5 percent for sure? Probably not. You know, it doesn't mean it's going to be exactly right. But it does mean, unless those transition probabilities change in a very serious way, we're not going to get to 100 percent free countries. We're really more likely to see something like two thirds. So, what have we done? All we've done in this lecture is shown: we started out with that simple alert/bored model. That was a toy model. We often do that when constructing a model. We take something very simple, it's kind of fun, we can all understand it, and see how the process works. Once we understand the model, then we can take it to real problems, with more dimensions, and even tie it into real data, and get a deeper understanding of how the model works, and, in some cases, get fairly surprising solutions. Right, so here, the surprising result was, that even though there's a trend towards more free countries, if transition probably stay in the range we're in, we shouldn't expect to see everyone be free. We should only expect to see two-thirds of countries be free. Okay. Let's move on now, though, and talk about why that converges at 62 and a half percent. What's causing these Markov models to go to equilibrium? And we're gonna learn something called the Markov Convergence Theorem. Okay, thanks.