Hi. In this lecture I want to continue our discussion of the prisoner's dilemma, and I want to focus on how we get cooperation in the prisoner's dilemma. And I'm gonna highlight seven ways that scholars have identified that cooperation can emerge in a prisoner's dilemma even though it's in no one's individual interest to cooperate necessarily. Now remember in the business dilemma it looks as follows: got two players, player one, player two and each one has two actions; they can cooperate or they can defect. It's in our collective interest, to have us both cooperate. We both get a payoff of four. But individually because six is bigger than four and two is bigger than zero, it's always in each player's interest to defect. But if we both defect, we both get payouts of two, which is worse than if we both cooperate. So individual interest don't line up with collective interest, So if the individual incentives point us towards defecting, collectively, we'd like to cooperate. So how do we get that cooperation? So to analyze that, I want to move to a somewhat simpler model. I'm going to assume that what happens is now is each person just has an action they can take. And if they take that action it has some costs and some benefits. So I assume if I cooperate that has a cost to me of C. But, it has a benefit to the person I'm, get, playing with of B. And I'm further going to assume that their benefit is larger than my cost. So, therefore, socially, we'd like me to cooperate because the other person's benefit is larger than my cost. Individually, I'd like to not cooperate because my cost is positive. So this captures, the [inaudible], the essence, really, of a prisoner's dilemma, right? Individually, I'd rather not cooperate. Socially, everyone would prefer if I do cooperate. So in this simpler setting, I wanna talk about the different ways in which we can get cooperation. I want to start by talking about some work by Martin Nowak. Now Martin has this wonderful book called Super Cooperators, where he goes in much more deta ils about these different mechanisms. So the language I'm gonna use comes from his book. The first way in which we can get cooperation is something like a prisoner's dilemma is through repetition. Now Nowak refers to this as direct reciprocity. What does he mean by that? What he means is, we're gonna play this game many times. So, if we're gonna play this game many times, I can recognize maybe it's in my interest to cooperate now, because if we meet next time, we'll cooperate in the future. So my colleague, Bob Axle rod, has a very simple strategy that can, that can induce cooperation and produce [inaudible] called tit for tat. We both start out cooperating, and as long as the other person keeps cooperating, we cooperate. If that person ever defects, then I defect. And this very simple strategy can keep us both cooperating provided we meet often enough. And that's the essence of direct reciprocity. Let's see why. So let's let p be the probability that we meet again. And let's let zero be our path if we deviate. And then, what's my payoff if I cooperate? Well, if I cooperate, it's gonna cost me C, however, if I'm gonna meet you later, there's some probability of meeting you later. And when we meet later, you can cooperate with me, then later I'll get a payoff of B. So, my payoff isn't just minus C whom defecting, it's minus C plus P times the benefit if we meet again. So, if that ends up being positive, what that means is that I should probably cooperate. >> And we can rewrite that as the property of meaning is bigger than C over B, than cooperation should emerge in the prisoner's dilemma. Let me give an example from my life that sort of explains how this works. I used to live in Los Angeles which is a huge city Pasadena, actually, And when we moved, my wife and I moved to Iowa City. And one of the first days I was in Iowa I was in the grocery store and I was just buying a couple of items. The woman in front of me in the grocery store had a cart full of food, and she said to me, why don't you go ahead? >> Now I w as shocked because no one in L.A. Ever let me jump ahead of them in line [laugh] at the grocery store no matter had much stuff they had in their carts. But the reason why she did this is not because people in Iowa are intrinsically nicer than the people in L. A., it's because of the fact that she knew she was likely to meet me again because Iowa City was a small town. So let's see how that works. It's just direct reciprocity. So let's suppose that the benefit to me of getting the jump in front of her was ten because she had this cart full food. Let's suppose the cost to her is only two because I only had a few items. So this ratio of cost to benefits is one over five. Well now the question is what's her likelihood of meeting me again? Well in a place like Los Angeles, if we're at the same grocery store, it could be maybe one in 1000. It's not very big. But in a town like Iowa City, there might be a 50 percent chance she's gonna see me again. [laugh] It's not a very big town. So, given that she's likely to see me again, in Iowa City, she's gonna cooperate. In L. A., she's not gonna cooperate. She has a greater likelihood of direct reciprocity, and direct reciprocity leads to cooperation. What's another method? Reputation. Now Nowak calls this indirect reciprocity. Cuz [inaudible] reputation is as follows. Instead of us directly meeting again, maybe I know the, and I get to know the woman who met, was in front of me in the grocery store, and I tell other people how nice she is. So she gets a reputation. So now we can do is we can say, let's let q, instead of it being the probability that we meet again, let it be the probability that her reputation gets out, And so what's going to happen is, now the cost of her to letting me go ahead is C. Her benefit is Q, the probability that her reputation gets known, times B, because if her reputation is known that she's a nice person, other people cooperate with her because they know she's going to cooperate with them or possibly someone else. So you're creating this sort of v irtuous cycle of people cooperating with one another. So again we get this same sort of inequality. As long as the value for reputation being known is bigger than that ratio C over B, we're going to get cooperation. So notice this subtle difference. In direct reciprocity I'm cooperating because I'm going to meet you again, and I think I'm going to get a payoff from you. In indirect reciprocity I'm hoping to get a reputation, a good reputation. I'm hoping that person will spread far and wide how cooperative I am. And then when somebody else meets me, they'll say, oh there's Scott. He's cooperative. I'll cooperate with him because he's such a nice person. And through indirect relationships we can induce cooperation. Here's a third, network reciprocity. So let's suppose that we got a set of cells in a network and we want to ask little cells cooperate with one another. Is it in their interest to cooperate with one another? What I'm gonna do is I'm gonna consider a very regular graph, number of certain networks with these different types of networks. A regular graph is [inaudible]. Everybody has the same number of neighbors. What we're gonna see is each person has K neighbors. If K is less than B over C, that's same ratio again, then what we're likely to get Is cooperation. Let's see why In this setting, I'm going to make a different assumption about behavior. What I'm going to assume that people are networks and they decide what behavior to follow based on how successful their neighbors are. So let's think of a simple network where the benefit of having someone cooperate with you is five, the cost of cooperating is two. And again, here, each person's connected to two people. So red is gonna denote defectors, and green is gonna denote cooperators. [inaudible] this long line of people. If you're a defector right here surrounded by two defectors, your path is gonna be zero. Cuz you're neither cooperating, nor is anybody who's playing against you cooperating, so your payoff is just a flat rate of zero. If you're over he re, and you're a cooperator, and you're playing with cooperators, your path is gonna be six. Why is that? You're gonna get plus five. >> From each of the two people playing with you, but you're playing two for corroborating each of those, so that gives you a payoff of six. So now we have to think about this person sitting here in the center. It's on the edge between defectors and cooperators. What are they gonna do? Well they're gonna look at the defector to their left, they're gonna say this person isn't cooperating with anybody, so they're not, it's not costing them anything, I'm cooperating with them, so they're gonna get a payoff of five. This person to my right is getting a payoff of six, as we talked about before. So when you look at this person in the center. They're deciding that if I defect, I might only get five, but if I cooperate, that person is getting six. So, their impressions are going to be that cooperating is better than defecting, so they're going to cooperate. Let's change the path. Let's suppose that the cost of cooperating. >> Is three. Well now, the payoffs to the defectors is going to be unchanged, Because they're not cooperating with anybody. But the payoffs to the cooperators are going to fall, by 2Y. Before they were six, but now they're going to be four. Let's see why that's true. Remember their benefiting, they're cooperating with two people, and two people are cooperating with them. So that means they're getting two benefits of five. But they're getting two costs of three, and we add that up to get four. Well now when this person looks to the right and their left they're going to see that well, the defectors look like they're doing better, and so this person is going to switch, and defect. Now, we can do more elaborate models. Let's suppose we now have K=4. So each person is playing against four people. So this person is playing against four people, as follows. Again, if this person is defecting, and all their friends are defecting, their path is gonna be zero, And if we look at som eone over here. >> Who's cooperating and let's suppose all four friends are cooperating, we can figure out their payoff as well, and they're going to get four times five, and they're going to get four times -two. So that's twenty. That's -eight. So the payoff is twelve. So if I look at this person here, their payoff is seven, but if they look at the cooperators they know, they say wow these people are getting twelve. If they look at the one defector they know they see this person's getting fifteen, and they're going to defect. So it's interesting here, is the benefits are five, the costs are two, and this person wants to defect. Whereas previously, we had the benefits of five cost her, too, but we were less connected. We only had two neighbors. You want it to cooperate. You see, as you get more connected, you have incentives to defect. Let's see why. We get this K less than B over C. And the way to think about that is you want to think about this one defector who's sort of sticking out. Who's in the midst of all these cooperators? So as in general you get K neighbors. And let's suppose you're surrounded by cooperators. What's your payoff going to be? It's going to be K times the quantity B minus C. Because everybody cooperates with you, so you're getting K times B. But you're cooperating with everybody else, so you're losing K times C. Let's suppose you're a boundary defector. [laugh]. So, somebody who's defecting, who's surrounded by cooperators, Well, then, your payoff is gonna be K minus one, all your other neighbors that are cooperating with you, times B. Now, let's look at when this is gonna be true. When is K times B minus C gonna be bigger than K minus one times B? Well, if we just do the math, we're gonna get B over C is gotta be bigger than K. And so, what we get is we get that, this is the inequality we get. That [inaudible] gotta be less than B over S. So again, very simple mathematics explains what's gonna be necessary for cooperation. Now, [inaudible], when you think about reputation, A really dense network, cause their reputation's more likely to spread. When we think about this network reciprocity story, we'd like to have a less dense network because there's less of an incentive to defect. So whether you want a rich network with lots of connections with high degree and high [inaudible] coefficient, or whether you'd like a sparse network, depends on the mechanism you're using to get cooperation. If you're relying on reputation, you want lots of clusters, lots of connections. If you're relying on network reciprocity, you'd prefer it to be starker. Next, Group selection, Group selection refers to the fact that it could be that the solution is on groups of people as opposed to individuals. And so groups of cooperators can win out. Here's the idea. Suppose you've got two groups of people. There's a red group here and a blue group here, And each group has to, within itself has some percentage of cooperators. So let's suppose the red group has 80 Percent Cooperators, And let's suppose the blue group has only 50 percent cooperators. Now let's suppose that the red group and the blue group go to war. Well who?s likely to win? The red group is likely to win because they've got more cooperators and they've got more cooperators and over time they've benefited more. They probably have more food. They probably have better technology. They probably have all sorts of stuff. So when you think of going to war, groups of cooperatives are likely to beat groups of defectors. So what's gonna happen is, even though it's the case that within the group. Defectors will do better when those groups go to war against each other, groups that have more cooperators are likely to win, And so what you get, is that by selection at the group level, if there's competition between these groups. As long as it's frequent enough. Then you could actually get a force towards cooperation. Last, you have kin selection. In kin selection, the idea is this. Is that different members of a species have different amounts of relatedness. And so if someone is my brother, or my offspring or second cousin, I may actually care about their benefit. So what we formally do is we have some measure of relatedness R. So for a child, that relatedness would be a half genetically. So if I could do something that benefits my child ten, and only costs me two, Maybe it's not the case but ten is bigger than two, Purely selfless, that'd be the case. But if I just take into congenetic relatedness, I, then I just think, is five bigger than two? This particular model has been used a lot in ecology, because you have some species, like ants and bees, where R is really, really high, and it's not surprising that within those species, you see lots of cooperation. Okay. Those are the [inaudible] five general ways; let's talk about two ways we can get cooperation in human societies. The first one is, laws and prohibitions, you can just make things illegal. So for example, it might be my interest to talk on the cell phone when I'm driving, but it's not in society's interest, because it increases the probability that somebody else is going to get injured. So we pass laws saying, it's not legal to talk on your cell phone when you're driving. Another thing we can do is create incentives. So, when I lived in Madison, Wisconsin, often times it would show a lot, and there was a law saying that 24 hours after the snow had stopped, you had to have your sidewalk shoveled. Now, the cost of shoveling my sidewalk was high for me, but other people are gonna benefit, And, the way that they, and they would benefit more. Then it cost me to shovel. In order to induce me to do it, the city basically said; if you don't shovel it you're gonna have to pay huge cost. You're gonna have to pay $100. And that fear of paying the $100 made me shovel my walk. So simple incentives, it wasn't illegal, I didn't have to shovel my walk I was just gonna have to pay a fine if I didn't. >> Okay. So we've seen a whole bunch of ways in which we can get cooperation [inaudible] dilemma. It can be repeated, direct reciprocity; it c ould be reputation, indirect reciprocity. >> Yeah. >> It can be a network effect. It can be group selection, where groups fight against each other, and so the groups that cooperate are likely to win. There can be kin selection, where what happens is that I cooperate with people who are like me. And then finally, we can have things like laws prohibit, just prohibiting things that aren't good, and we can have incentive structures, where we pay people to cooperate when they'd naturally be willing to defect. Okay, so that's the prisoner's dilemma, and how we can solve it. But that's a simple two by two interaction. Where we wanna go next is we wanna talk about larger prisoner's dilemmas, where there's lots of players. These are sometimes called collective action problems. Okay? Let's move on... Thank you.
