In the first part of this lecture, we were talking about the intersection between game theory and evolution. We focused on examples drawn from the notion of the selfish gene as it was described by Richard Dawkins. What I'd like to do now is continue on, so clear your mind with an entirely different example. So this does not have anything to do with the selfish gene. But the example that I'm going to talk about here is one called the evolutionarily stable strategy. That is to say we're going to be looking at the notion that we're interested in what is called an evolutionarily stable strategy, which is a strategy that an animal can play such that it cannot be successfully invaded by mutations to other behaviors. Think of it this way, there are certain situations in which animals of a particular species might play a game of some sort with each other. On those occasions, the animals have choices of strategies. Just as we've seen in game theory. If a strategy is an evolutionarily stable strategy, then an entire population of animals can be playing that strategy and they will not be successfully invaded by the other alternative strategies. In other words, a newborn animal, a mutation that's playing one of the other strategies, cannot perform better, cannot be more fit than the animals that are already in the population. Let me give you an example to show you how this works. This is a totally abstract example. It's due to a biologist named John Maynard Smith who really pioneered the study of game theory and evolution particularly with notions like this. Here's the idea. We have species of animal that occasionally will compete over a resource. They come up to a resource, two animals will come up to this resource and they will have to decide who gets the resource and who doesn't. So this is just a model. So the numbers here are chosen to make the model work, and the animal in question, you can just imagine a variety of situations. The resource in question, could be a mate, it could be food, it could be living space, could be a variety of things. Two animals come up to this resource and they now have to decide who gets the resource and who doesn't. So here is the matrix that we're going to draw out for this. We'll stipulate, and these are numbers that Maynard Smith chose, again, to make the example work, to make it interesting. We'll say that the resource is worth 50. Fifty what? Whatever, 50 points, 50 value units. If two animals fight and one gets injured, then the loser loses a 100 points or a 100 value units. We'll talk about that third thing in a moment. Here the idea is that a red individual, I mean, just think of the two individuals as labeled red and blue. They're two animals from the same species but we'll just call one red and call the other blue. They meet up over this resource. Now they have two strategies that they can employ the hawk strategy fights over the resource. It just starts to fight over the resource and then whoever wins gets the resource, and whoever loses gets injured and leaves the field and does not get the resource. Dove is a strategy that it's not going to fight over the resource. When it runs into a hawk, that is to say when it runs into an animal playing the hawk strategy, the dove will simply give up the resource. When an animal playing the dove strategy meets up with another animal playing dove, then they do a signal to each other that they're both doves, and then they essentially flip a coin over who gets the resource. So in this case, if two doves meet up, they'll do a display. That can be somewhat costly, and here we say it costs them 10 points. Then they flip a coin, and whoever wins gets the resource. Now, what happens, remember by the way that hawk and dove do not refer to different animals, they refer to different strategies that the animal of these species can play. These species can play either the hawk strategy or dove strategy. What happens when two hawks meet up with one another? Well, they fight it out over the resource. The one that wins gets the resource which is worth 50, the one that loses, gets injured and loses a 100. Assuming, as is reasonable, that when two hawks meet and fight, each one has a 50 percent chance of winning. Then the average payoff for a hawk hawk meeting, is negative 25. Why? Because think of it from the red hawk player's point of view. Half the time I get 50, half the time I lose a 100. Over many hawk hawk meetings, the average payoff is negative 25. What about when to doves meet up? Well, they both do this signal to each other, that costs them 10, and then they flip a coin one gets the resource worth 50, and one does not get the resource worth nothing. So in the bottom right corner here, the average to a dove dove meeting, well if I'm the red player playing dove if I happened to win the resource, then the payoff is 50 for the resource minus 10 for the display. If I don't win the coin flip, then I just lose 10 for the display. Since that's going to happen, each one is going to happen on average. Half the time, the average payoff to a dove dove meeting is 15. When a dove meets up with a hawk, the dove gets zero. When a hawk meets up with the dove, the hawk gets 50. Now, we've gone through this, we've set up this matrix, why go through this trouble? Here's the idea. Suppose we have a population of animals in this species, and everybody in the population is playing the hawk strategy. Would a newborn dove be successful in this world? A new born mutant dove, would they be successful in this world? Well, look at it this way. When hawks meet up with hawks on average they lose 25 points per encounter. The dove is only going to meet up with hawks. The world is filled with hawks. So when the dove meets up with hawks, they get zero. What that means is that the hawks are bashing each other and injuring each other. The doves are staying out of the fray. The dove is getting on average from each encounter zero while the hawks are losing 25. What that means is that hawk is not in an evolutionarily stable strategy. It can be successfully invaded by a newborn mutant dove. If a mutant dove arrives in this world, they'll do better than the hawks. They'll stay uninjured. What about a world of doves? Suppose you have a world of doves all playing and flipping coins and so forth. How would a newborn hawk do in that world? Well, the answer is pretty well. The hawk is meeting up only with doves, and it's getting 50 points per encounter. The doves are meeting up with each other, and getting 15 points on average per encounter. So the newborn hawk will be more successful. it'll be bullying around the peaceful doves and getting the resource every time. Therefore, dove is not an evolutionarily stable strategy. A world of doves can be successfully invaded by a newborn mutant hawk. So neither of these is an evolutionarily stable strategy. Are there ways around this? Well, there are a couple of suggestions we could make. We know that neither pure hawk nor pure dove is an ESS, as it's often abbreviated. Since each can be invaded by the other. One possibility is to say, suppose the population has a probability P of finding a hawk, and a probability one minus P of finding a dove. In other words, it's a mixed population with percentage P, we'll express it as a probability, percentage P of hawks and percentage one minus P of doves. Could that world be evolutionarily stable? In that case, it could be evolutionarily stable if we can find a value for p such that neither a newborn hawk nor a newborn dove would have an advantage. The way we do that is then to say, okay, suppose we have this world with probability P hawks, probability 1 minus P of doves, how would a newborn hawk do in this world? Well, with probability P, it runs into another hawk, and it loses 25 points on average per that kind of encounter. With probability 1 minus P, it runs into a dove and gets 50 points. What about doves? A newborn dove with probability P runs into a hawk and get zero, with probability 1 minus P runs into another dove and gets 15. If those two payoffs are equal, then neither the hawk nor the dove, a newborn hawk or dove has advantage in this world. You can solve for P in this case, that is, find the value of P that makes these two expressions equal. Turns out that P is 35-60th. So if the world consists of 35-60ths of hawks and 25-60ths of doves, then you have an evolutionarily stable community. You wouldn't necessarily call them evolutionarily stable strategy. But the mix of populations is evolutionarily stable. Another possibility would be rather than have percentage of pure hawks and a percentage of pure doves, is to have all the animals playing what's called a mixed strategy. We didn't really talk about this before in game theory, but the idea of a mixed strategy is that you as an individual when you come up to a resource, you will play hawk with a probability of P and dove with a probability of 1 minus P. In that case, we could do the exact same calculation. What we could find is that yes, there is a value of P such that if every animal is playing the same strategy but it's a mixed strategy, then that could be an evolutionarily stable strategy since neither a pure hawk, someone playing entirely hawk, or pure dove could gain an advantage here. So one way around this is to say, well, if we allow for mixed strategies, we could have an evolutionarily stable strategy. One that plays hawk some of the time and one that play doves some of time. Is there a pure strategy that might be evolutionarily stable? Well, Maynard Smith suggested a third strategy which he calls bourgeois. So what is the bourgeois strategy? In this case, the animal will play hawk if it knows that it is the first of the two individuals to be at this resource. In other words, if an animal thinks I'm first, and assume that's not in question. So an animal comes up to a resource, if I'm first, I'll play hawk, if I'm not first, I'll play dove. How does the bourgeois animal fare against these other strategies? Well, let's take a look. The earlier numbers here are just as they were before. That is the four numbers at the upper left of this matrix are unchanged. But let's see what happens when a bourgeois runs into a hawk. Well, half the time on average, the bourgeois says, I was at this resource first, I'm going to play hawk. It loses 25 points on average in those half of the encounters. What about when it thinks I'm second? In that case, it'll play dove and it'll get zero on average when it meets up with the hawk. In those half situations, the hawk will get the resource and the bourgeois will get nothing. So one more time, bourgeois meets with hawk, half the time, bourgeois fights thinking that it was there at the resource first. In those half of occasions, it loses 25 points. In the other half of occasions, it gets zero points. Therefore, the average payoff from a bourgeois meeting a hawk to the bourgeois player is negative 12.5. What about a bourgeois meeting up with a dove? In that case, half the time, the bourgeois will fight play like a hawk and get 50 points on those half occasions. On the other half occasions, it'll play dove with the dove and on average, it'll get 15 points. So the average result of a bourgeois meeting up with the dove is 25 plus 7.5 or 32.5. Finally, what happens when a bourgeois meets a bourgeois? On that occasion, if the animal we've called red, if the bourgeois meets up with another bourgeois, on half the occasions, it'll play hawk and the other bourgeois will play the dove because it'll know that the animal that we're calling red was there first. That's not in doubt. So half the time, bourgeois will play bourgeois and get 50 points by playing hawk, half the time the red bourgeois will play dove and get zero. So the average payoff of a bourgeois bourgeois meeting is 25. Now, why have we gone to all this trouble? We've invented these numbers, we've invented this animal. For now, the point is bourgeois an evolutionarily stable strategy? Well, suppose the whole population is playing bourgeois, then on average in each meeting they're getting 25 points. What would happen with a newborn hawk? A newborn hawk in this world would get, and I'll leave you to figure out the last two numbers in the right column here, but a newborn hawk meeting up with a world of bourgeois on average would get 12.5 points per encounter. A newborn dove meeting up with bourgeois on average would get 7.5 points per encounter. Neither of those is better than 25. So a population of bourgeois animals cannot be successfully invaded by either hawk or dove. Does this have any bearing on the actual world of biology? Well, it turns out that there's an animal that does in fact play a bourgeois strategy. This is a butterfly, particular species of butterfly that lives in Africa and in this case, they live in forests which have dense tree covering and they live near the ground. So the sun comes through the leaves of these trees and forms little patterns of sunspots, mostly it's shadow, but there are a little sunspots on the ground. These butterflies apparently like to hang out in the sunspots. That's the resource that we're talking about. Possibly it's warmer, it's more pleasant there, but in any event, these butterflies like to hang out in the sunspots. For our purposes, that'll be the resource. Turns out that when two butterflies meet over one of these sunspots, the one that was their first gets the resource. The two butterflies in this case do a little fluttering dance, and one of them remains. Through what I can only assume is infinite patience, the biologists who have studied these butterflies have determined that the butterfly who gets the resource is the one who was there first. In fact, biologists are more than just patient, they can also be a little mischievous. So when an experiment is set up where the two butterflies both think that they were there first, then this little fluttering dance takes a good deal longer than it does in the average case. It's almost as though the butterflies are having to make a difficult determination of who was there first. My suspicion is that they're registering the temperature of each other's wings. Because presumably, the one that was there at the sunspot first has slightly warmer wings, but I'm not sure. In any event, this is an animal that plays the bourgeois strategy. When it's at a resource, it will. Another animal comes up wanting that resource, the one that was their first gets it, and the one that was there second, goes away. So this an example of an evolutionarily stable strategy. This is not the only example of an interesting application of game theory to evolution, there are many, many more. Let me give you one more example, and having to do with what's called self-handicapping. So classic example is the peacock's tail. Peacock is at risk from a variety of predators, and for a peacock having this huge unwieldy tail is a handicap, it's a problem. It makes it more difficult to maneuver, to get away from predators, it makes you more noticeable. So in general, having a tail like this is not a good idea for the peacock as far as its encounters with predators are concerned. In that case, you might argue evolutionarily speaking, why does the peacock have this tail in the first place? If it reduces the bird's fitness, why would the peacock have this tail? The consensus among biologists is that the answer has to do not with natural selection but with sexual selection. The Peacock has this tail because the peahen likes it. Why would a peahen like to see a tail like this on a male bird? Well, in this case, you could anthropomorphize a little bit and say, what is the tail signaling to the peahen? What is the peacock's tail signaling to the peahen? It's a signal that essentially says, "I, the peacock I'm so healthy. I am so fit that I can carry this unwieldy thing around with me and still survive." It's a self-handicapping mechanism. It's showing that even with this unwieldy tail, and highly noticeable, and garish, even with this unwieldy tail, I can survive and I'm in good health. So this is a signal to the peahen of good health, and that's a plausible strategy to gain theoretic strategy. Self-handicapping works in a number of situations. We'll actually talk about another one a little bit later in the course. What about the gazelle at the bottom here? This gazelle is just starting to do something called stotting, and you can look this up. There are certain kinds of gazelles or antelopes that do this maneuver called stotting, S-T-O-T-T-I-N-G. When a herd of gazelles notices that there is a predator nearby like a leopard or a lion or something, some of the gazelles will start doing this stotting maneuver, and what is stotting? It's springing way up in the air, doing a huge jump way up in the air. It looks like a pointless maneuver, it looks like an energy wasting maneuver, why would the gazelle handicap itself by springing way up in the air when it sees a predator, wouldn't it be better off saving that energy to run away from the predator? Again, the answer is that this is a self-handicapping maneuver. Here, the signal is not for sexual selection, it's not a signal to a female or male gazelle, it's a signal to the predator. Basically, what it's saying again to put a personality behind this is, "I am so healthy and so fit that I can do this silly maneuver of springing way up in the air. You see, I am not afraid of being caught by you. You might as well not chase me because I am quite healthy and fit." A failed case to a predator is a highly costly event, they're running for a long time after a gazelle, and they don't get it, they don't catch it, so that's a problem for the predator. The predator is interested in this message, in other words, the predator does not want to chase a particularly fit or healthy gazelle. As it turns out, predators are far less likely to chase the gazelles that are stotting than other members of the herd. Stotting is a successful self-handicapping signal. It's a way of saying, "Don't bother, I'm a really fit animal, you won't be able to catch me." What about people? Do people ever do self-handicapping? Well, Jared Diamond in his book, The Third Chimpanzee suggests that for young people, historically, would be mostly males but it could also include females as well. But let's just say for the sake of argument, for young males smoking in mixed company, smoking in company of both males and females is a self-handicapping maneuver. It's a way of signaling to the young women who are there, "Look how fit I am, I am so fit that I can do this manifestly unhealthy thing. Yet, as you see, I appear to be healthy and in good shape." In other words, cigarette smoking is a self-handicapping maneuver that way. In this case, it's analogous to the peacock's tail. It's saying, "I'm so fit that I can do this really stupid thing but it doesn't seem to affect me." This is a plausible explanation, to really nail it down, you'd want to know certain things, like for example, thinking of cigarette smoking among young men as a self-handicapping maneuver. One thing that you would want to say is that, well, in that case, it's much more likely that young men would smoke in public than in private. What's the point of smoking alone in your room? You're not signaling to anyone in that case. So if you go with Jared Diamond's explanation, then that would seem to suggest that young men will smoke when they're in public much more than when they're in private, it wouldn't occur to them too much to smoke in private. But in public, it's a way of signaling to females primarily but also to males, "I'm an exceptionally fit individual." Again, it's a self-handicapping mechanism. This is not by the way an argument that smoking is a good idea. I think it's a terrible idea, one of the worst. But it's an evolutionary explanation, a game theoretic explanation of what might induce a young person to handicap themselves in this way.