In the last lecture, we explained the basic framework of evolutionary game theory. So the idea of game theory can be applied to biology where plants, animals, and insects have absolutely no capacity for strategic thought. But even in those instances, The concept of Nash equilibrium is useful. The outcome of evolution, biological evolution, is very much likely to be a Nash equilibrium. So in the last lecture of this week, I'm going to give you an example where insect behavior is well predicted by a Nash equilibrium. So I'm going to explain that fig wasps play a particular Nash equilibrium. Okay, so the basic logic of evolutionary game theory is that the outcome of biological evolution is a Nash equilibrium of a game played by genes. So let's apply this idea to the case of fig wasps. All right, so this is the picture of fig wasp- wasps. And female is, is here. And she has a long tail. And the male has two types. One type of male has no wings. The other type has wings, okay? So, male fig wasps have two possible strategies determined by genes. Either to have wings or to have no wings. Two strategies for males. Okay. And those two, two strategies let's say. Staying strategies, because males without wings stay in a fig. And the other strategy for male is flying strategies. Those two strategies are very much likely be determined by genes. Okay. So let's cons-, let's first examine the behavior of female fig wasps. A certain fraction of females stay in fig for a long time. They linger, certain fraction of females linger in fig for a long time. And a certain fraction of female fly out very quickly, okay? And the relative frequency of those females flying out quickly. And the relative frequency of those females staying in the fig. That depends on local conditions. So you have many possibilities. Lots of female flying out or lots of female staying. So there are lots of different situations. And let's suppose for the sake of argument that this behavior of female is fixed. And what kind of evolutionary forces operate on the selection of better strategy for the male. Okay? So this is what I'm going to show. So now let's suppose that- so we are going to consider a games played by two genes. Staying gene produces staying males, and flying genes, you know provides males with wings. So, let's suppose that flying males are rare in the original situation. So, the gene pool for, you know, staying gene is large. And the population of flying, gene is very small. Let's suppose this is true in the initial situation. So, what's going to happen? Okay. A certain fraction of female go out of fig quickly, and certain fraction of female are staying. And if there are lots of staying males, there is a fierce competition to win females attention here, okay? But there, okay. So staying males have a very small chance of mating with a female. A fig is overcrowded if there are lots of staying males. On the other hand, for our flying male, well, this guy is surrounded with, surrounded by lots of beautiful ladies. And he has much higher chance of mating. Okay. So when fraction of flying male is small, then flying males have better chance of getting lots of offsprings. So let me summarize the situation by means of a simple diagram. So on horizontal axis, I'm going to measure the fraction of flying males. So it's in between zero and one and I just explained what happened if the fraction of a flying male is small. Okay? So vertical axis measures the fitness of each strategy, flying and staying, and it’s the number of offsprings. So as I explained, when the fraction of flying males is small, then flying males have a better chance of getting many offsprings. Okay? Where flying males are rare, a flying male have a higher fitness than staying males. And therefore, the fraction of flying male is going to increase. Okay, so this arrow indicates the nature of natural selection. When the fraction of flying male is small. Okay, so what happens if there are lots flying males. This situation is opposite. Okay? Again, certain fractions of females are quickly going out of fig, and certain fraction linger in- in the fig. So, if there are lots of flying males there's a fierce competition here and those flying males have, have a very low chance of getting females attention. And this guy here it's like heaven you know? And this staying male have a very much higher chance of, of, getting offsprings. So the situation is opposite. When fraction of flying male is large flying males have less offspring. So that's the situation here. When the fraction is high fitness of flying male is small and fitness of staying male is larger. Therefore, the fraction of flying male is going to increase- is going to decrease. Okay? So evolutionary force go, you know, by means of evolutionary selection, fraction of flying male decreases here. So eventually. Evolutionary force leads to a point where both males, flying males and staying males have an equal chance of getting offsprings. So at this stationary point, no male can increase its payoff by switching to the other strategy. And it's a kind of Nash equilibrium, and it should be the outcome of biological evolution for the fig- fig wasp males. Okay, so this story explains why you have two males. Flying males and, staying males. But there could be other possible explanations why they are two different kind of males. So the question now is, is there any testable implication of Nash equilibrium that we can test by means of data? Okay. So, well, if you examine this situation carefully, Nash equilibrium provides you with the precise prediction about the fraction of flying male. So let me, let me explain what it, it is. Let me lay it out, confirm. By data this specific prediction of Nash equilibrium. Okay so we first considered the situation when fraction of flying male is small. Okay. So in this situation flying male is surrounded by lots of females. And the flying males have more offspring. So this is not the stationary point of evolution- Evolutionary forces. Okay? So now flying males are going to have lots of offsprings and the fraction of flying males and staying males are going to change. So this is not the end point of evolutionary process. So what would the end point of evolutionary process look like? Well this is the answer. So certain fraction of females are quickly going out from the fig. So let's say three out of five. Those females are quickly going out of the fig. And fraction of staying female is two out of five. And this is the Nash equilibrium situation, so five out of Three males are flying. And two, I'm sorry- Three out of five males are flying. And that two out of five males are staying. Okay, in this situation, they have equal chance of mating, and it's a Nash equilibrium situation. Okay, so Nash equilibrium should have the following property. A certain fraction of females are flying out, say three out of five. And also a fraction of flying male, three out of five, they are equal. Okay, only in that case you reach a Nash equilibrium. Okay, so this is specific prediction given by biological model. Fraction of female quickly and flying out of fig should be equal to the fraction of flying male. This prediction can be tested by collecting data, okay? Let's say, let's see what actually happened. Okay so this graph with- Adopted from the original paper by biologist Hamilton. Here we have fraction of flying male, and here we measure the fraction of female quickly flying out from fig. Well, this is a result. So, you have lots of data points. And if Nash equilibrium is true, all data points should be on this 45 degree line. Well, it's not a perfect fit, but data points are closer, very close to the 45 degree line. And therefore we can conclude that the fig wasps playing the Nash equilibrium of a game played by genes. This is an example of an application of game theory to biology.