Hi. In this lecture, we're gonna extend the core idea of the prisoner's dilemma, where it's in my individual benefit. To [inaudible], to defect. But collectively, we're better off if we cooperate. What I mean by extend is this. In the prisoner's dilemma, it's a two player game. But more generally, we can think of N person prisoner's dilemma, where, when I cooperate, lots of people benefit, not just one. And what we wanna see is how those differ from the standard prisoner's dilemma, and see how those also sort of apply in some real world cases. So the first thing we're gonna talk about is what is called a collective problem. In a collective action problem, I make some choice. Do I contribute? Now, there's a cost for me to contribute. But there's a benefit to everyone else. So, how do we write this? We write this as follows. We say, let's let X J be the action. >> Of person j. And what we'll assume is that XJ is some amount of effort. It could be anywhere between zero and one. It's how much you are contributing to the public good. Now there's a cost. So the more I contribute XJ, the more it costs me. So my payoff is negative in my action. So therefore you think I should choose XJ to be zero, and that's, sometimes like defecting in the prisoner's dilemma. However we also assume, that there's a benefit, Which is the sum. This, this sign here means the sum. It's the sum, of all of the Xi so of all the people in society, so collective we'd be better off if everybody put it on big XI and XI be one but individually we don't want to. >> Now what we assume here is we see this term beta is between zero and one. If beta were bigger than one, that would mean to be that my interest to cooperate. Because it would it, my cost would be minus XJ, but then I'd be getting beta times XJ, and beta's bigger than one, so I'd be better off. So we see that beta's in zero one. So my collective benefit from doing this thing is actually less than my individual cost. Let's look at an example. Let's suppose that there's ten people, and we're going to assume that beta is equal to.6. Now let's with everybody else is cooperating. So, let's suppose I'm X one, and that X two equals X three equals X ten, equals one. So, everybody else is doing one. So my payoff if I choose X one equals zero is just gonna equal zero plus.6 times nine, which is 5.4 If everybody else is cooperating. So there's nine people cooperating so I'm gonna get.6 times that which is 5.4. Which is great, If I were to cooperate. It's gonna cost me one, and what I'm gonna get though, is I'm gonna get. Point six times ten which is six minus one which is five, 5.4 is bigger than five so it's in my interest to put XY equals zero. But collectively everybody else would be better off if I would choose XY equal to one. So again this a lot like the [inaudible] dilemma but instead of me playing against one person I am playing against a whole bunch of other people. Where does this apply, well think of things like carbon emissions. It's in our collective interest if we reduce carbon emissions, so if I sort of, you know, put forth effort to, you know, put less carbon in the air. But it's costly for me to do that. And so, what happens is over time, we've had lots more carbon emissions, probably more than we'd ideally like to have, because it's a collective action problem. Now, sometimes these are called free rider problems. And they're called free rider, because it's in their interest to free ride off the good will of everyone else. I'd like everyone else to cooperate, but I don't want to. Now, there's another type Of sort of N-person prisoner's dilemma, which is called a common pool resource problem. And this differs a little bit from the collective action problem. Let me explain how. In a common pool resource problem, we've got something. Let's say, like, cod in the ocean. And if we fish those too much the population gets smaller and the population can't reproduce itself. So what we're trying to do is manage some resource that has the possibility of reproducing itself. Let's see how this would work. Let's suppose now that I've got to decide how much cod to eat, And so we'll let XJ equal the amount of cod that I eat, how much I fish out. And we'll let X be the total consumed. That's the sum of what everybody does. And what we'll assume that the amount of cod available next period is just the cod that was available this period minus what was eaten squared. So we'll assume the can mate and, you know, reproduce and make more cod. And so the more we eat, the fewer there are to reproduce. So, let's do an example. Let's suppose that the cod population is 25, and let's suppose that we eat twenty. So if we eat twenty, and how much gonna be available next print, well just 25 minus twenty squared which is 25 so we've got a nice stable equilibrium here. We're gonna have 25 cod this period, 25 cod next period, 25 cod the period after that, everything's gonna be perfectly great. But what if I say, you know, I'm gonna have a party, let's have a big cod fest, and lets you know, I'm just gonna consume more cod than I'm supposed to. And that's gonna drive. X one up to 21. So now collectively we're getting 21 units of cod. Well how much are we going to have next time? Well next time we're going to have 25 minus 21 squared, which is four squared, which is sixteen. Well if we've got sixteen, and we've got a problem because in the past we've been eating twenty units of cod. 20's bigger than sixteen. We're going to run out. And the cod's all going to go away. And you might think, okay, that's, doesn't make any sense, but if we think about it, there's lots of cases of that happening. There's a famous book written by Jared Diamond called Collapse where he talks exactly about this problem. That you've got a society of people whether it's the people in Finland, whether it's the people in, you know, [inaudible], or these people on Easter Island. And what you get. Is over fishing, over consumption and eventually the resource goes away. So in the case of Easter Island what you see is the population was getting bigger, bigger and bigger , and then there was a collapse and the collapse happened when the forests were completely destroyed. >> They kept, sort of harvesting too much wood, and the forests couldn't reproduce themselves, and what happened is, they had a collapse of the society. If you look at present day Philippines, this is a graph showing the total forest covering and you see that it was very high, and then suddenly, it's fallen away really quickly, again, over harvesting the forest. Now the Philippines likely a part of the global economy, they can trade for things that they used to get from the forest, but if they were isolated like the people on Easter Island, they too would have collapsed. Previous lecture we talked about, look we've got all these ways to overcome these things. Repeated games, reputation, networks, group selection, kin selection, incentives, prohibitions, all sorts of stuff that we can do in order to get cooperation. Well those prefer sort of two person problems in most cases, and they ignore a lot of the particulars. So we think about con, that's a very different thing than thinking about forest, and this is gonna be a very different thing than say, cattle grazing on the commons. So when we think about these collective action problems or these common police horse problems we realize that each one in and of itself is particular. So the way we're gonna have to solve it, we're gonna have to do cooperation is gonna depend on the circumstances of the problem itself. The particulars are gonna matter. And that's where we're gonna go in the next lecture. We're gonna talk about how you solve collective action problems and chemical resource problems in the real world. All right. Thank you.