The economic concept of game theory can be readily applied to evolution and behavior. By analyzing encounters between organisms as a mathematical "game," important information such as fitness payoffs and the proportions of "strategies" played by each group within a population can be inferred. While oftentimes these games are too simplified to apply directly to actual examples in nature, they are still useful models that help convey important concepts.
Krebs, John R. and Nicholas B. Davies. An Introduction to Behavioral Ecology, chapter 7
April 17, 2009
Professor Stephen Stearns: Okay, this is the second lecture on behavior. And today what I want to do is I want to give you one of the major analytical tools for dealing with behavior, which is evolutionary game theory. And before I get into the body of the lecture, basically I want to tell you at the beginning of the lecture where it came from--it came out of Economics--and I want to tell you that I'm going to give you two examples of particular games; one is the hawk-dove game, the other is the prisoner's dilemma.
And it turns out that neither of these games is really directly testable with good biological behavioral examples. So instead of actually testing these ideas with biology, what I'm going to show you is how biology introduces interesting qualifications to the assumptions of the games. And so I'll give you the two games, and then I'll give you a series of biological examples, and then comment on how that really changes our thinking about the assumptions of the games.
But before I go into all of that, I want to signal--and I'll come back to this in the last slide--that evolutionary game theory is one of the parts of evolutionary biology and behavior that connects this field to economics and political science, and that the prisoner's dilemma model, which I will present in the middle of it, actually is a particular embodiment of the tragedy of the commons, which is, of course, affecting the way that we use all of our natural resources, and methods for solving the tragedy of the commons are actually central issues in both economics and in political science. So this is actually an area in which there are strong trans-disciplinary connections of ideas.
So the basic idea behind an evolutionary game is that what you do depends on what everybody else is doing, and that means it's going to be frequency dependent. In other words, if I decide to be aggressive, in a certain environment, the success of that will depend upon the frequency with which I encounter resistance. So game theory is fundamentally frequency dependent. And the central idea in evolutionary game theory is that of the evolutionary stable strategy. So I'm going to show you that that, in fact, is equivalent to a Nash equilibrium.
So when you're playing a game against another member of a population it's not like playing a game against the abiotic environment, because your opponents can evolve. It's not like you're playing a game against, you know, in a sense of staying in the game against winter temperatures, or something like that. You actually have an opponent that has a strategy, and the strategy can change.
So that makes the whole analysis of games the analysis of a move and a counter-move, and that counter-move can either be dyadic, where you're playing against one other player, or it could be that you could conceive of it as playing against the entire population. So there are some nuances there.
And in that sense evolutionary game theory is really very fundamentally co-evolutionary; it's always about how your strategy co-evolves with the other strategies that might pop up in the population. Okay? But it's strategies within a population. Co-evolutionary game theory is really not applied so much to one species evolving against another. It's usually, how will my behavior do against the other behaviors that are present in the population?
Okay, where did it come from? Well here are some of--here's a little gallery of heroes. Basically it comes out of von Neumann and Morgenstern's book on game theory, which was published I think in 1944, as then further developed by people like John Nash and Reinhardt Selten. So these guys more or less founded it.
And you can actually go into Maynard Smith's book on evolutionary game theory and pull out of the appendix of von Neumann and Morgenstern one of the payoff matrices that they use; I mean, you can see that these guys were actually studying that book, and then developing it in an evolutionary context.
John von Neumann was a Hungarian genius who managed to show how some of the basic problems in quantum mechanics could be connected and explained; he did that back in 1929, and then he went on more or less to invent the idea of an operating system for computers. So he contributed very greatly to the conceptual underpinnings of the information revolution; and he also invented game theory. So John von Neumann was a bright guy.
John Nash, of course, is famous from the book and the movie A Beautiful Mind, as the fellow who saw at Princeton that the stable solution to a game that is being played between two parties is that the stable solution is the one that you play when everybody else is playing their best possible strategy. Okay? That's the insight that he had in the bar at Princeton, as a graduate student, and then he succumbed to his schizophrenia and didn't really recover until he was in his sixties.
And Reinhardt Selten is a German professor who developed game theory in the context of economics, and generalized it into all sorts- using all sorts of alternative assumptions. And these two guys shared the Nobel Prize in Economics for it. So that's where it came from.
And these are some of the key events that I've just gone over. And then these ideas were developed by George Price, a really remarkable guy, and John Maynard Smith, and applied in biological behavior. And John's book came out in 1982. So it had a lot of impact. And if John had just been willing to recognize and acknowledge that in fact an ESS is a Nash equilibrium, he probably would've shared in that Nobel Prize. But he didn't.
This is John. He was a very good dishwasher. I spent a lot of time with him. So that's one of my photos. And really the key figure that stimulated John, just as he stimulated Bill Hamilton, was George Price. And George Price was developing these ideas in the context of the puzzle of altruism and cooperation; how did cooperation and altruism ever come to be in an evolutionary context?
And you'll see that when we come to the prisoner's dilemma and I talk about Axelrod's experiment with competing different strategies against each other on a computer, that Price really actually contributed twice to the solution, or to our thinking, about this problem of where did altruism and cooperation come from? One, in the context of game theory, and once in the context of kin selection and hierarchical selection.
Okay, here are the basics. The basic thing that you ask in game theory is can any conceivable alternative invade? And this turns out to be why--by invade I mean will a mutation come up that modifies behavior, and if it comes up, will it increase in the population? If it's going to increase in the population, it will do so because it has greater lifetime reproductive success.
So if that gene affects a behavioral strategy in such a way that over the course of the lifetime it increases the reproductive success relative to other strategies, that will be what we call invasion. So if alternatives cannot invade, then that means that the resident strategy, the one that's already there, is an evolutionary stable strategy. So the stability means stability against invasion; stability against alternatives.
Now you might ask yourself, how do we know what all the alternatives are? And the answer is in reality we don't. But in theory we can imagine, if we restrict our attention to a certain scope of possible behaviors, that the alternatives are all the possible combinations of behaviors within that restricted set. Okay?
So that is actually the thing that's going on. The theorist is sitting there and saying, "Should I be more aggressive or less aggressive?" Well all the possible behaviors consist of not being aggressive at all, or being very aggressive, and everything in between. So those would be the ones that you tested against. You'll see how that works when we go through a couple of examples.
So the ESS is then a strategy that resists invasion, and it turns out that it's exactly the same thing as a Nash equilibrium. So when John Nash solved this problem for game theory, back in Princeton in I think 1951, he in fact was also at the same time solving the problem that Maynard Smith and George Price posed in I think 1973; just in a different context.
So here's a simple game, and this is one of the first that Price and Maynard Smith cooked up, to try to illustrate how you would apply this thinking to animal behavior. And they called it the hawk-dove game. So two animals come together, and they're going to fight over a resource, and that resource has value V, and that means that the fitness of the winner will be increased by V. Okay? The loser doesn't have to have zero fitness, it's just what--it's the increment in fitness which is determined by this particular encounter that we're talking about.
So we say, "Well, they can have one of two strategies; they can be hawks or doves." And the idea is that the hawk strategy is that you escalate and you continue to fight either until you're injured, in which case you have to back off because you can't fight anymore, or until you win and the opponent retreats, in which case you get the whole thing. And the dove strategy is that you go up and you display, and if the opponent escalates, you back off immediately and run away, and if the opponent doesn't escalate you'll see that you'll share the resource. Okay?
If two hawks encounter each other, then one or both are going to be injured, and the injury will reduce fitness by a certain cost. So being a hawk has a benefit in that you can be aggressive and acquire resources, but it has a cost in that if you run into another hawk, you can get beaten up and injured.
So this is sort of the fundamental intellectual construct of game theory; it's a payoff matrix. And the idea is that it lays out, for the things on the left, what happens to them when they interact with the things on the right. Okay? So when a hawk interacts with a hawk, this is its payoff. When it interacts with a dove, this is its payoff. When a dove interacts with a hawk, this is its payoff, and when it interacts with a dove, that's its payoff. I'm going to take you through that.
So if a hawk encounters a hawk, it has a fifty percent chance of winning and a fifty percent chance of being injured. So its payoff is one-half of the benefit minus the cost. So you just see we're kind of averaging that payoff over many such possible encounters. So the assumption here is that hawks are total blockheads and they escalate blindly; they disregard differences in size and condition; they're really stupid, they just go in there and they fight for the resource, and they don't have any nuance to them at all.
The dove will give up the resource. If a hawk encounters a dove, it gets the resource, the dove gets zero. So it gives it up and the hawk gets it; and that's what these entries in the matrix mean. Okay? So the hawk is encountering the dove, the dove is encountering the hawk, the hawk gets V, the dove gets zero.
That doesn't mean it has zero fitness, it just means that its fitness doesn't change because of the encounter. It doesn't get anything in addition, but it also doesn't lose anything. So you can think of the dove as a risk-averse strategy. When a dove meets a dove, they share it. They sort of shake hands and say, "Hey, 50/50."
Now if a strategy is going to be stable, then it must be the case that if almost all members of the population adopt it, then the fitness of the typical member is greater than that of any possible mutant; otherwise a mutant could invade, and that would mean the strategy wasn't stable.
So in this case if we let W of H be the fitness of the hawk, and W of D be the fitness of the dove, and E of H,D be the payoff to an individual adopting a hawk against a dove--and we have two possible strategies, I and J; so these are going to be, in general, what we've instanced by hawk and dove here--I is going to be stable if the fitness of I is greater than the fitness of J. And if the mutant J is at very low- when we assume the mutant is at very low frequency.
So if I is going to be stable, at very low frequency, then when I encounters I, it has a higher fitness than when J encounters I. Or when I encounters I, it has the same fitness as when J encounters I. And when I encounters J, it has a greater fitness than when J encounters J. Okay? So this is just a way of being very careful and logical about laying out the different possible relationships of fitness on encounters.
Now what happens? Well dove is not an ESS. If a population is 100% doves and one hawk pops up, it's going to interact almost all with doves. It's not going to run into any hawks. It's just going to go around beating up doves and taking away the spoils. So it will invade. Hawk will be an ESS if the payoff of an encounter is greater than the cost of the encounter. Okay?
Now even if the population is 100% hawks, and every other individual it encountered is somebody that fights and beats you up, that will be stable if V is greater than C. But what happens if V is less than C? Well if the cost of injury is high, relative to the reward of victory, then we expect mixed strategies.
That means the following: if we--well I'll ask you to play this. Just think about this situation a little bit, and I would like you to take just a moment to explain what's going on to each other, and then I'll ask one of you to tell me what happens when you start with 100% doves and a hawk mutant pops up, and another of you to tell me what happens when you have 100% hawks and a dove mutant pops up, when this condition is the case--okay?--when it really hurts a hawk to encounter another hawk?
So take a minute to describe to your partner what that frequency dependence is like, and then I will ask one of you to replay each of those cases. Okay?
[Class members confer with one another]
[Professor Stearns speaks with someone off lecture topic]
Professor Stephen Stearns: Okay, let's go. Who would like to explain what happens when you have a population which is 100% hawks and a dove crops up as a mutation? What happens?
Student: Mutation by the hawk doesn't, because the doves are able to [inaudible].
Professor Stephen Stearns: Okay. Why did that happen?
Student: Because every time it repeats it, every time [inaudible], and there's never a situation where [inaudible].
Professor Stephen Stearns: Well actually it doesn't happen Manny; it's not quite like that. Remember what the payoff is for the dove. When the dove encounters a hawk, its payoff- its fitness doesn't alter. Okay? Another idea.
Student: It will increase in its cost because since there's a dove reaching out at the hawk as a dove, it gains I guess a lot more than it gains just because of the hawk. And you said the fitness of the dove wouldn't change. So that one dove will at least be present. So it will [inaudible].
Professor Stephen Stearns: Yes, what's the average fitness of the hawks in that population?
Student: One-half V minus C.
Professor Stephen Stearns: Yes, and V is less than C?
Professor Stephen Stearns: So it's a negative isn't it? Is 0 bigger than a negative number? What happens to the doves? They increase. They increase because they actually don't bear any cost at all when they run into a hawk; their fitness is not decreased. And so basically they are a neutral allele that's introduced into the population, and if they have perfect heredity, they start reproducing. Right?
And basically what's going on is that the hawks are mutilating each other. They're damaging each other so much that even though the dove's fitness is zero on this scale, it's still greater than the hawk's, which when the hawks are mostly encountering hawks is negative, on this scale. Okay, now let's turn it around. What happens when it's all doves and a hawk enters the population; what happens? We have a population that we've just made in our mind. It's 100% doves and a hawk comes in.
Student: It really depends if it's a hawk.
Professor Stephen Stearns: Go further.
Student: So if there's one hawk, [inaudible].
Professor Stephen Stearns: Yeah, it goes like gangbusters. It only meets doves; it never gets beaten up by another hawk.
Student: So if there's [inaudible].
Professor Stephen Stearns: Yes, and so it just keeps going. Right. So from either side, from either 100% doves or from 100% hawks, the vector is towards the middle somewhere; and where it's going to stabilize depends on the relationship of V and C. Okay?
That's why this is a mixed strategy. Neither strategy is an evolutionarily stable strategy. The only reason that they can persist, with this relationship of V to C, is that they come to some kind of intermediate frequency. If there's too many hawks the doves will win out, and if there are too many doves the hawks will win out. Okay now, so that's one- that's an example of a game that will result in a mixed strategy.
Now let's look at the prisoner's dilemma. Okay? So this is the payoff matrix for player one, this is the strategy of player one, and this is the strategy of player two. And C stands for cooperate and D stands for defect. So basically the reason this game is set up this way is that it's trying to show you that it would be better for both players to cooperate, but both players are actually motivated to defect, and so if you have short-term selfishness, which is determining the outcome, defection will win over cooperation. So that you will not, in this circumstance, just playing this game one shot, you will not get the evolution of cooperation and altruism out of the prisoner's dilemma. Instead you will get the tragedy of the commons.
So the entries here. This is the expected value of cooperator playing cooperator; cooperator playing defector; defector playing cooperator; and defector playing defector. And if we put in some particular numbers that actually represent an instance of the general conditions, these particular numbers are chosen in such a way that defection will in fact be selected. So cooperation will be an ESS if the expected value of C playing C is greater than the expected value of D playing C. Okay?
And that's not the case. D will be an ESS if the expected value of D playing D is greater than the expected value of C playing D; which is true. Now look at the payoffs: 3 is greater than 1. If the population were all cooperators, everybody would get 3. If the population is all defectors, everybody only gets 1.
But because of the way the payoff matrix is set up, with the interactions between the cooperators and the defectors, this is the evolutionary stable strategy, and this one, which is great for the group, is not stable against invasions by defectors, because the payoff to a defector, who is playing against a cooperator, is even greater. But when a defector plays against a defector, life gets pretty unpleasant. So in fact this is the tragedy of the commons.
So the general condition for this, okay, if we do the algebra rather than the arithmetic, is that the stable strategy is always to defect from the social contract, always not to cooperate, if T is greater than (>) R, R > P, P > S, and R > than the average of S and T. So that this all has been analyzed in detail, and this is sort of the paradigmatic social science game that is used in many contexts.
Now what if you play it again and again? This was the first idea about how even in this circumstance, even if you're playing a prisoner's dilemma game, with rewards set up like this, you could get the evolution of cooperation. Just do it again and again. Okay? So you're not just playing once, you're playing many times against the same person.
And a very simple strategy turned out to work. Bill Axelrod, at the University of Michigan--he's a political scientist--said, "I want to hold a computer tournament, and I want everybody around the world who's interested in this issue to send me their computer program to play against other computer programs, in an iterated prisoner's dilemma." And it turned out that a very simple one did extremely well, and that is Tit-for-Tat. So you cooperate on the first move--if you've run into a defector, you get beat up by him; if you run into a cooperator you win, both of you win. And then you do whatever the guy did last time.
So the essential features of Tit-for-Tat, that make it work, is that it retaliates but it's forgiving, it doesn't hold a grudge. Okay? The other guy defects on you, you're going to punish him. If he switches to cooperation, you say, "Oh fine, I don't hold a grudge, I'll cooperate with you on the next time."
So after a huge amount of research, it turns out that there are some extremely nuanced and complicated strategies that can do a little bit better than Tit-for-Tat. But the appeal of Tit-for-Tat is its simplicity. It doesn't take very much cognitive power to implement this behavioral strategy. It doesn't take very much memory to implement it. Okay? It seems to be something which is simple and robust and that wins.
Now as soon as you put in space, you can get a much more complex strategy. And the take-home from that is that if you have what's called population viscosity, which means that particular individuals tend to encounter each other more spatially than if they're just randomly mixed up in the population, that promotes cooperation.
So Martin Novak, up at Harvard--actually he was at Princeton when he did this--he came up with a whole lot of nice, two-dimensional representations of those games. And this is one possible outcome here. Okay? So here blue is a cooperator--and these guys are, by the way, playing prisoner's dilemma, and they're playing prisoner's dilemma against their neighbors; they're not playing randomly in space, they're actually playing against the neighbors that are physically sitting right there.
So blue represents somebody that was a cooperator on the previous round and is a cooperator now; whether they retain in the population or not depends on whether they're losing or winning in the encounters. Green is a cooperator that was a defector.
And here you can see that here are some cooperators that have won against some defectors, and they're forming a little ring right around that little blue island of cooperation. And red is a defector that was a defector, and yellow is a defector that was a cooperator. And what happens in this particular game is that the percent cooperation goes up, comes down, stabilizes right at 30%.
So in a situation in which--the prisoner's dilemma suggests, if you just consider an interaction in isolation, it's going to be 100% defectors. Just putting in space and giving individuals a chance to interact repeatedly with other individuals creates a situation where often cooperators are actually interacting with cooperators, and they're getting a win, and as soon as they build up a little spatial island of cooperation, they do great. So they hold their own in a sea of defectors, just due to the two-dimensional nature of the interaction.
Okay, so thus far in the lecture I have give you pretty abstract mathematical kinds of stuff. And what I now want to do is go into a series of biological examples. And the biological examples are not direct tests of evolutionary game theory. What they are is the application of game theoretical thinking to biological contexts, that then inform us about the assumptions that we're making in the games.
And one of the early applications was to the bowl-and-doily spider. So this is a female bowl-and-doily spider. It doesn't really show her bowl and her doily, but basically what she does is she spins a web that looks like a bowl, and then it has a layer down below it that looks like a doily, and she puts up trip lines that go up above it, so that insects that are flying along hit one of these trip lines and fall into the bowl, and she's sitting on the doily and she comes up and grabs it. Okay? And it's on the doily that the mating interactions take place.
So this figure should look pretty familiar to you from the last lecture. Insemination in spiders works pretty much like insemination in dung flies. The probability that a male will fertilize eggs increases from the start of copulation up to a certain point, where he's getting perhaps 90 or 95% of them. So he's getting diminishing marginal returns as he sits on the female.
And in this study the contrast was between a resident male who actually had gone in and had successfully displayed to the female, and knew whether or not copulation was actually now taking place, or was going to take place, and a new intruder who's coming in.
So if the resident is somebody who's going to have this experience, and the intruder coming in has no idea what's been going on--okay?--we will assume that the intruder knows nothing about the shape of this curve; it may know that the curve has started but it knows nothing about the shape. So it has to simply make an assumption about the average value of that female--the intruder is a male.
Whereas the resident knows what this curve is, then the payoff to the resident gets greater and greater, the longer the copulation goes on, and then it drops towards the end of the copulation. It's already gotten 90% of the eggs. Okay? And so actually this happens pretty quickly, which is kind of nice; I mean, if you're doing a behavioral study in the field, it's nice to have it over quickly so you can get your data quickly.
Only seven minutes after the beginning of insemination that female doesn't have very much more added value to that male and he would be better going off--as you know from the marginal value theorem, that line, at the tangent, is going to be crossing probably somewhere up in here--it's good for him to jump off, try to go find another female.
Whereas that intruder coming in, not having so much information about the system, at least on a simple assumption, just thinks, oh, the female has a certain kind of average value, and I'm not- I haven't been able to copulate with her yet, so this is what I expect. Okay?
Well if you look at the actual behavior of spiders. This is the observed, this is the predicted. So the predicted is that the percentage of fights that would be won by the resident would go up; he would fight really hard if he were interrupted, after a certain point in copulation, assuming he could start over again, and that after he had been copulating for seven minutes, he wouldn't care anymore. So the prediction is intensity of fighting would peak and then drop; and the observed values seem to follow that pretty well.
So here's the twist on game theory. The cost-benefit ratio in the payoff matrix is being altered by the behavior of copulation, and one of the participants knows and the other one doesn't. So the thing that this example introduces into evolutionary game theory is the whole issue of who has information on the potential payoffs of the game, and it shows that that makes a big difference. Okay? And that's not in the assumptions of hawk versus dove; it's not in the assumptions of the prisoner's dilemma. This is some important aspect of biology that alters that analysis.
So this just runs through what happens. You put both males in at the start, the bigger male will win. Okay? So if neither of them has any information, any more information than the other, the big one wins. If they're the same size, the fights are settled by what's the difference in reward? Okay? So the resident will fight longer and will be more likely to win at the end of the pre-insemination phase, but intruders are more likely to win after seven minutes of insemination.
If a resident is smaller than the intruder, they persist longer, when the reward was greater. So you will find weenie little runts fighting great big bullies if they know something about the reward they're going to get. And if the costs and benefits are nearly identical, they'll fight until one or both are seriously injured or in fact dead. So this is another way, of course, of underlining that the payoff in evolution is number of offspring, not personal survival. So they're willing to risk a lot, if there's a lot on the line.
So that's one biological example, and that's the bowl-and-doily spider. This next example has to do with Harris sparrows. And again it has to do with information, but now it also has something to do with honest signaling and perception. So there is a sense here in which what you're going to see is simple-minded sparrows getting really ticked off at being deceived. Okay?
So this is a study done by Seivert Rohwer, who's in the museum at the University of Washington in Seattle, and what he noticed was that if you just go out in Nature, you see a lot of variation in how dark the heads of the males are, and that these guys with the dark heads are dominant and they win most of the fights. And by the way, you see a lot of this in birds, that they have a signal that they can give that is a signal of their condition and of the likelihood that they might be able to win a fight if they got into it.
So here are some of the experiments; and so I put up Appearance and Reality. Okay? I don't know if Harris's sparrows analyze the problem philosophically in terms of appearance and reality, but they certainly react to appearance and reality.
So what Seivert did was he experimentally treated subordinates, either by painting them black, or by injecting testosterone, or by painting them black and injecting them with testosterone. So if you paint them black, they look dominant; they behave- they do not behave dominant because they don't know they've been painted black, and they don't have the testosterone in their system. Okay? Do they rise in status? No.
If you inject them with testosterone, they behave like they're dominant, but they don't have the signal that they're dominant and they get beaten up, they do not rise in status. Because basically what they're doing is they're behaving in a very--according to bird lore--they're behaving in a very deceptive fashion.
But if you do both things, you paint them black and you inject them with testosterone, then you do to them essentially what evolution and their development has already done to them, which is that the black is actually naturally expressed in male individuals that have higher testosterone levels: they look dominant, they behave dominant and they rise in status. So this is a very interesting observation right here. Okay?
Now that's one twist on evolutionary game theory. It says that your perception of your opponent, and your understanding of whether he's trying to deceive you or not, is an important thing. However, there is another issue, and that is that it always pays to assess before escalating.
When I was in grad school we had a Great Pyrenees, a great big dog, Aikane. His head stood about this high; he weighed about 130 pounds. And we were living in a suburb of Vancouver, British Columbia. And I was out for a walk one day with Aikane, and a great big aggressive male German Shepherd came around the corner, about fifty feet away, and each dog went on alert, ears went up, hair went up on the back.
They started barker ferociously at each other. They rushed, at high speed, at each other. I was thinking, "Oh my God, I'm going to have to pull a fight apart." They went by each other, like ships in the night, went about fifty feet down the road, and they both urinated on a post and trotted proudly away. They had managed to avoid serious damage.
Well that's what's going on with Red Deer. If Red Deer get into a fight where they are really about equally matched, they can end up locking antlers in such a way that they can't extricate themselves and they will actually starve to death. Also, if they are swinging those nice pointed antlers around in a fight, they can rip out the eye of an opponent, they can get a wound that will be infected, and they'll get bacterial sepsis and die from an infection. So fights are dangerous. But fights are the only way they can get babies.
So what they do is they first do a lot of assessing. They approach each other and they first roar. So if you're around moose in the fall, or deer in the fall, you will hear roaring, and that's what they're doing. Basically the ability of a male deer to make sound is pretty directly proportional to how good- what kind of shape he's in.
So if that sound is equally impressive, then they get into a thing where they do a parallel walk; they actually walk next to each other, kind of sizing each other up. And it turns out that if they're very closely matched in size, these parallel walks can go on for four or five hours. They'll just be wandering all over the landscape, trying to see who's going to give up first. Okay? So that's the parallel walk.
And in a certain number of cases one stag will say finally, "Well, looks like I'm going to lose this one, it's not worth fighting." And then finally after doing that parallel walk thing, if it's not resolved by then, they will fight, and one will win and one will withdraw.
So the point of this is that actual fights among animals are much more nuanced than the simple Hawk-Dove game would ever have you believe. And I think it's probably true, throughout all sorts of tradeoffs in evolutionary biology, that every time there is a significant cost, there will be some modification of behavior, or some way of tweaking that cost, that will arise, that will reduce the cost. So this is all cost reduction. Okay? They need to get their mating, but they're going to do it in a way that's not going to kill them, if at all possible. And that's not in the simple assumptions of any of the evolutionary games that I showed you.
So how solid are the assumptions of this whole way of looking at the world? Well it turns out that the assumption is being made is that you've got a big randomly mixed population. If you put in kin selection, so that the opponents can be related to each other, so that a brother might be fighting with a brother, the analysis gets complicated, but the result's simple: if you're related to the other player you're nicer. That's not surprising.
If you have repeated contests and there's an opportunity to learn, the results will change. Okay? If there's no learning, then having the series of contests really doesn't make any difference. So it is the ability to learn and to remember that turns the repeated prisoner's dilemma into a situation in which cooperation can evolve. So you have to have some cognitive capacity to do that.
If the population is very small, mutants might not be rare, and the basic model has to be altered.
It turns out that asexual reproduction doesn't matter too much. The sexual system--we usually get to the ESS if the genetic system will produce it; you know, will allow it--is you have more genes affecting a trait, it becomes more likely that the population will hit the ESS.
If you have asymmetry in the contest, that will--as we've seen with the bowl-and-doily spider, and with the size of the contest, size of body size, and with badge size in sparrows and in deer--that will change the outcome.
If you analyze pair-wise contest versus playing against the whole population, it turns out in general a mutant really is playing against the whole population. There you actually have to do it on a computer usually; you can't--it's hard to analyze analytically. But it doesn't make a huge difference. Okay?
So the take-home points that I want you to get from evolutionary game theory is that this is a tool, it's an abstract tool, and it is probably the tool of choice anytime you're looking at frequency dependent evolution of phenotypes. It is very often good for your mental health, as an evolutionary biologist or behaviorist, to test some property against the invasion of all possible mutants. That's a very useful criterion.
So, for example, if you are thinking about those red grouse in Scotland who are out in the fall in a big assembly, and somebody says, "Oh, the reason that they do that is so that next spring they won't reproduce so much." And you ask yourself, "What if a mutant crops up in that population that doesn't think like that and it's just going to reproduce like gangbusters, no matter how dense the population is?" That little thought process tells you the explanation that was being given doesn't work, because that selfish mutant will invade. Okay? So it's a very useful criterion.
And I'd like to recommend Ben Polak's course. Ben is a very good teacher. Ben's gotten teaching awards. He teaches an Econ course on game theory, and it will lead you through this stuff. And Ben is very good at actually having you do homework assignments in which you solve games; which is more than this course has time for. So if you want to get your head around this, I recommend Ben's course. And next time we're going to do mating systems and parental care.
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