Hi. Welcome back. Remember that we're talking about Lyapunov functions, and they're this intuitively very simple thing. So a Lyapunov function would be a case where you get a maximum value and if the process stops, that's fine. But if it doesn't stop, it goes up by some fixed amount. Therefore eventually it's going to hit this thing and stop. Which means that if a system has a Lyapunov function associated with it, it goes to equilibrium. Now if it doesn't have a Lyapunov function, or we can't think of one, then maybe it goes to equilibrium, maybe it doesn't. The point is, if we can construct a Lyapunov function, then it definitely goes to equilibrium. That's a useful thing to know. What we want to do in this lecture is give an example of another process, in this case it's gonna be an exchange market, that has a Lyapunov function, so therefore goes to equilibrium. Then I'm gonna problematize a little bit and show that, well, why might the system not go to equilibrium. So, create another sort of market that doesn't go to equilibrium, and we'll see why is that the case. Why, what has to be in the way from us to create a Lyapunov function. So, what prevents us from constructing a really simple Lyapunov function and showing, yes this system goes to equilibrium. Interestingly enough, we can relate this back to something we studied earlier in the course: Chris Langton's lambda parameter. And that was in the context, remember those very abstract one-dimensional cellular automata models? We're actually going to bring that logic back into play here, to understand something about Lyapunov functions. So let's get started. What's an exchange market? An exchange market consists of a situation where people just bring stuff. So this is in Bologna, Italy. People bring fish, other people bring money, people bring baskets, and you trade things. So when he brings their stuff in a cart, and they trade with other people who brought their stuff in a cart, and at the end of the day people go home. We want to ask, is that system gonna go to an equilibrium, or are people just going to keep trading things constantly all throughout the day and is it going to be some muddled mess of behavior? Well, let's think through it. What are our assumption for the model? First is, each person just brings a wagon full of stuff. Second assumption: We're gonna assume that you only trade with someone if you're happier with what you have now, than what you had before. Otherwise, why would you trade? And the third assumption, hidden in here a little bit, is: We're gonna assume that, you have the increase your happiness by some fixed amount. X. So there's some amount, and I probably should have called this k, 'cause I was using k before. So let's cross out that X and call it k. There's some fixed amount of k you have to be happier by in order to make the trade. Now why would I assume that? Cause I'm assuming that there's some transaction costs. So if I don't, if I'm trading a basket for some fish, it's got to be the case that I want the fish by at least, you know, the cost of going through that whole trading thing in order to get the fish for the basket. So this k is just the cost of trade. And that's going to be important, because we need that in order for our Lyapunov function to work, because we need happiness to go up by at least some amount K. Let's recall what is a Lyapunov function? It's some function F that has two assumptions: Assumption one, it's got a maximum or minimum value. Here's we're gonna assume a maximum. Assumption two, that either the process is at an equilibrium, or, if it moves, the Lyapunov function, the value, the function F, goes up in value by at least some amount k. Those are the two assumptions. So let's think about what a Lyapunov function here might be. Well, here in this exchange market, we can let it be the total happiness of the people. That's it, very straightforward. Total happiness of the people is a Lyapunov function. Let's think about it. Does it satisfy these two assumptions? Does it have a minimum, a maximum value? Sure, because people brought, just, wagon loads worth of stuff in. There's only so much happiness that can go around, right? People can only get so happy with a fixed amount of stuff. So if you give everybody exactly what they wanted and added that up, that'd be the maximum happiness you could possibly get. So there's some maximum, you don't know how to achieve it, but there's definitely a maximum, 'cause you've just got some wagons full of stuff. Assumption two, there's some k such that, if the process doesn't stop, happiness goes up by at least k. Well, that also holds, remember, because if we assume that you only trade if you're improving by some amount k, because there's some cost of doing a transaction. So you're only going to trade with someone if you're happier, so therefore every trade makes people happier and there's a maximum, so therefore we're going to get an equilibrium, so this function's going to work. Let me add a little more detail. Two pieces of detail. First, I'm going to say that NW here stands for nuclear weapons, and O stands for oil. NK here stands for North Korea and I stands for Iraq. So, North Korea is gonna say to Iraq, "I'll give you some nuclear weapons for oil", and Iraq says to North Korea, "That's great, here's some oil. Give us some nuclear weapons." And U, who's not involved, I'm going to assume they're the United States. So this is no longer people getting fish for baskets. This is North Korea getting some oil, and Iraq getting some nuclear weapons. Let's think about our happiness function. North Korea is happier. They got rid of some nuclear weapons. They got a bunch of oil to help their economy. Iraq's happier. They got rid of oil and now they can better defend themselves to get nuclear weapons. The United States who is actor U here, even though they're not materially engaged in the transaction, they're affected. There's an externality. They're less happy. And because they're less happy, that means that total happiness didn't necessarily go up. Because Germany, France, England, Brazil, Venezuela, all sorts of people could be less happy. And if all sorts of people are less happy, this trade may mean total happiness went down. And if total happiness went down, that may mean that other people then have to make other trades as they try and make total happiness go up. So what can happen is, we don't know for sure whether the system is ever going to stop if we continue to churn, because we can't put a Lyapunov function on the process. Let's think of some other situations: Political coalitions. When party A merges with Party B, party C may be upset. Total happiness isn't going up. What about mergers within firms? You think of firms, you know, you think of, their Lyapunov function could be profitability, it could be firm happiness, it could be firm security. But when two firms merge, that can make other firms less profitable, less secure, reduce their market share, whatever. And so what you could get is that that process may still churn. Same is true with political alliances. If you think of one country forming an alliance with another country, that could make other countries less secure. And that could mean that there is no Lyapunov function. The process is gonna continue to churn. Finally, this is when I talk about my other graduates quite a bit, what about dating? You think when two people decide to date, they're both happier. Or when two people break up, presumably they're both happier. But that could affect other people who are friends of those people, who maybe wanted to date one of those people, and it's not clear. Maybe dating has a Lyapunov function, maybe happiness is a Lyapunov function for dating, maybe it's not. It depends on the size of the externalities. What have we got? We looked at exchange markets and we said, if we had a total happiness function, as a Lyapunov function, works great, happiness just goes up, process has to stop, we get an equilibrium. Then we said, not all exchange markets. If there's externalities, like, with North Korea trading oil for nuclear weapons for oil with Iraq, other people could be affected. And that could mean that happiness doesn't necessarily go up. Total happiness could go down, and the process could keep churning. And we related that to Langton's lambda parameter from that simple cellular automata model. This is very interesting, right? That cellular automata model, which was very abstract, told us that systems that, where behavior isn't influenced by others, tend to go to equilibrium; systems where my behavior and actions are influenced by others, tend to be more likely to be complex or random. And that same logic applies here. We can apply a Lyapunov function to things like, you know, changing which location, when I go shopping for fish, and when I go to the bookstore, or trading fish for baskets. Because my actions don't materially affect other people, or if they do, they make them happier. Right? So, in each case: When I move from a crowded place to a less crowded place, I make everybody happier. And when I trade fish for a basket, I make the person I trade with happier. So, I'm not changing the, I'm not lowering the happiness of anyone else, there's no externality going on. But, in the cases where there are these externalities, now when I take an action I materially affect other people in the opposite direction, with these negative externalities. And that means that they may then want to change what they're doing, which could mean that the system keeps moving. So, what do we have? We've got that, without externalities, or with only positive externalities in the case of finding a maximum, what you're gonna get is that, it's easy to construct a Lyapunov function, and boom, you get there. The system's gonna stop. But if there's these negative externalities, I'm making myself happier but I'm gonna make other people less happy, then the system could continue to churn, and we may not be able to say whether or not the system's gonna go to equilibrium or whether it's gonna be complex. But we do have some intuitions. And those intuitions suggest that markets, simple markets trading goods should go to equilibrium, should constantly improve. People choosing routes should constantly improve. But things like international alliances or coalitions within political parties, or possibly even dating, that these things may be more complex, and certainly that's how it appears out there in the real world. Okay, thanks.
