耶鲁大学博弈论公开课 - 第9集 进化稳定-合作，突变，与平衡

视频全文转录（中英双语）

以下为完整中英双语转录，已加标点。英文在前，中文在后，逐段对照。

由 Groq Whisper 转录 → M2.7 B 方案整文标点 + 分段 → M2.7 段号保留翻译 → 逐段对照。

[段 1]

So we’re going to do a new topic today. We’re going to talk about evolution. We’re going to talk about the connection between evolution and game theory. All right, now there’s going to be an extra, I already emailed you about this, there’s going to be an extra reading on this for the people who want it. I’m going to, there’s a reading packet that’s available to anybody who wants it. I emailed you all the websites where you can order that reading packet and find, and figure out where to pick it up. It’s not compulsory, you don’t have to look at that reading packet, it’s just it might help. In addition, as with last Wednesday, since some of the material here is new, I have written a handout that goes with this lecture, and that handout will appear magically this afternoon on the website All right So things today are fast Don worry There a handout that goes with it All right so why look at evolution in the context of game theory There are really two reasons. The first reason is because of the influence of game theory on biology. It turns out that in the last few decades, there’s been an enormous amount of work done in biology, in particular looking at animal behavior and using game theory to analyze that animal behavior. And just to give you a loose idea about how this works, the idea is to relate strategies with genes, or at least the phenotype of those genes, and to relate payoffs to genetic fitness.

[译文 1]

今天我们要讲一个新话题。我们要讨论进化，以及进化与博弈论之间的联系。好了，关于这个话题我已经发邮件告诉你们了，有一个额外的阅读材料，需要的人可以去看。我这里有一份阅读资料包，任何想要的人都可以获取。我已经给你们发了所有可以订购这份阅读资料包的网站，以及在哪里领取。它不是强制性的，你们不必去看，只是可能会有帮助。另外，和上周三一样，由于这里有些材料是新的，我为这次讲座准备了一份讲义，那份讲义今天下午会神奇地出现在网站上。好，今天的内容进度很快，别担心，有配套的讲义。那么，为什么要在博弈论的背景下讨论进化呢？实际上有两个原因。第一个原因是博弈论对生物学的影响。事实证明，在过去的几十年里，生物学领域做了大量的工作，特别是研究动物行为，并运用博弈论来分析这些动物行为。简单给你们一个大概的概念，其核心思想是将策略与基因联系起来，或者至少与这些基因的表现型联系起来，并将收益与遗传适应性联系起来。

[段 2]

All right, so the idea is that strategies are related to genes, and the payoffs in the games are related to genetic fitness. And the big idea of course is that strategies grow if they do well So strategies that do well in these games grow strategies that do less well die out All right so we going to be visiting that today One thing to bear in mind from the start is there’s an important difference between game theory as analyzed in animal behavior and biology and game theory that we’ve been doing so far in the class, and that is that we’re going to think of these behaviors, these strategies played by animals, not as chosen by reasoning individuals, but rather as being hardwired. So if strategies grow, it isn’t that some lion or ant has chosen that strategy, it’s simply that the ant or lion who has that gene that corresponds to that strategy grows and has many children. All right, so it’s important to… This is a new idea for us, that these strategies are hardwired. All right, so that’s interesting in and of itself, but there’s also an important influence the other way. So a second reason for studying this stuff is that there’s been an enormous influence from biology, or from evolutionary biology in particular, back on the social sciences This influence the other way uses evolution largely as a metaphor And you find this if you’re a political scientist, if you’re a historian, if you’re an anthropologist, if you’re a sociologist.

[译文 2]

好了，这个思想就是策略与基因相关联，博弈中的收益与遗传适应性相关联。当然，其核心思想是表现好的策略会增长。所以在博弈中表现好的策略会增长，表现不好的策略会消亡。好了，我们今天会讲到这些。首先需要记住的一件事是，动物行为和生物学中分析的博弈论与我们迄今为止在课堂上所做的博弈论之间有一个重要的区别，那就是我们将把动物的行为、动物使用的策略，不是看作由理性个体选择的，而是看作天生的。因此，如果策略增长，并不是某只狮子或蚂蚁选择了那个策略，而仅仅是拥有与该策略相对应基因的蚂蚁或狮子在增长并拥有许多后代。好了，重要的是……这对我们来说是一个新想法，这些策略是天生的。这个本身就很有趣，但另一方面也有重要的影响。研究这些内容的第二个原因是，生物学，或者说尤其是进化生物学，对社会科学产生了巨大的影响。这种反向的影响把进化主要当作一种隐喻来使用。如果你是政治学家、历史学家、人类学家或社会学家，你会发现这种情况。

[段 3]

Let me give you an example from economics since it’s perhaps closer to home. So here’s an example. You might imagine some firms in the marketplace. and you could think of these firms not necessarily reasoning out what is the most profitable strategy for them or what is the most cost reducing strategy for them however, they might just have rules of thumbs to select their strategies however, in a competitive marketplace survival of the fittest firms will lead to us ending up with a bunch of firms who have low costs and high profits alright, that make sense? so here, competition in the marketplace substitutes for competition in the jungle, as it were. All right? And the analog of a gene dying out is a firm going bankrupt. All right? So this is the kind of ideas we’re going to explore. These are the kind of ideas that are going to be in the background for today and Monday. Now before I go on, I want to put in a couple of kind of important riders. So there are two things that I am not. These ideas are going to be in the background for today and Monday. Now before I go on, I want to put in a couple of kind of important riders. So there are two things that I am not. The first thing I am not is I’m not a biologist.

[译文 3]

让我给你们举一个经济学的例子，因为它可能更贴近生活。是这样的。你可能会想象市场上的一些企业。你可以把这些企业不是看作必然在推理什么是它们最有利的策略或最降低成本的方法，而是可能只是有一些经验法则来选择它们的策略。然而，在竞争激烈的市场中，适者生存会使我们最终拥有一批低成本、高利润的企业。明白了吗？在这里，市场竞争代替了丛林中的竞争。基因消亡的类比就是企业破产。这就是我们要探讨的这类想法。这些想法将成为今天和周一讲座的背景。现在在我继续之前，我想加几个重要的补充说明。有两件事我不是。这些想法将成为今天和周一的背景。现在在我继续之前，我想加几个重要的补充说明。有两件事我不是。第一件事我不是生物学家。

[段 4]

I’m sure there are people in this room who are biologists, who know a lot more about the genetic side of this than I do. So I’m not going to attempt to teach you biology here. I’m going to focus on the game theory part. The second thing I’m not is I’m not an American citizen. And I worry slightly when we talk about evolution because I realize it’s very controversial here and some of you might be from places like, say, Kansas. And if you are from someplace like Kansas, then please don’t write to your senator about this. Or if they see the movie, just tell them it was some other English guy giving the lecture. All right. So what we’re going to actually do is we’re going to look at a highly simplified model of evolution, at least for today. We might get on to a more complicated model on Monday, but today it’s going to be very highly stylized. So this is going to be our extremely simplified model And this simplified model is going to do tremendous violence to the biology but it really just to fix some ideas okay So we going to focus on what we going to call within competition So the lions are competing against the lions and the ants are competing against the ants. And the way we’re going to think about this is we’re going to look at symmetric two-player games.

[译文 4]

我确信在座有些人可能学生物的，你们在遗传学方面知道的比我多得多。所以我不打算在这里教你们生物学。我会专注于博弈论部分。第二件事我不是美国公民。我有点担心当我们讨论进化论的时候，因为我知道这在美国非常有争议，你们中有些人可能来自像堪萨斯州这样的地方。如果你们来自像堪萨斯这样的地方，请不要给你们参议员写信反映这个。或者如果他们看了这个视频，就告诉他们这是另一个英国人在做讲座。好了，我们实际要做的，是我们将要审视一个高度简化的进化模型，至少今天是这样。我们周一可能会涉及更复杂的模型，但今天会是非常风格化的。所以这将是我们的极其简化的模型，这个简化模型会对生物学造成极大的简化，但它真的只是为了阐明一些想法。我们将专注于我们所说的种内竞争。狮子与狮子竞争，蚂蚁与蚂蚁竞争。我们思考这个问题的方式是，我们将审视对称的双人博弈。

[段 5]

So we look at very simple games. They’re only going to involve two players, and they’re going to be symmetric games, which means both players have the same strategies, both players have the same payoffs. All right? The way we’re going to think of this is we’re going to imagine that there’s a large population out there, each of whom is playing a particular strategy. And we’re going to assume that what happens is we randomly pick two people from that population, and pair them up. So everyone in this large population will be randomly paired with someone else. They’ll play the strategy that they are hardwired to play, and then we’ll see what happens. Alright so the idea here is there a large population of if you like animals that are hardwired to play particular strategies and we going to have random matching And the idea here is, when we do lots of these random matching, we’re going to keep track of the average payoffs. So what we’re going to focus on are the average payoffs of particular strategies when randomly matched in these games. All right? All right? And again, the underlying idea is that relatively successful strategies will grow, and relatively unsuccessful ones will decline. I’m not going to write that, that’s obviously the other part of that. So relatively successful strategies are growing, and relatively unsuccessful strategies are declining.

[译文 5]

我们来看非常简单的博弈。它们只涉及两个参与者，而且会是对称博弈，这意味着两个参与者有相同的策略，相同的收益。我们思考这个问题的方式是，我们想象有一大群人在外面，每个人都在执行一个特定的策略。我们假设发生的情况是，我们随机从这群人中挑选两个人，把他们配对。所以这个大群体中的每个人都会随机与另一个人配对。他们将执行他们天生的策略，然后我们看会发生什么。好了，这里的想法是，有一大群动物天生会执行特定的策略，我们将进行随机配对。这个想法是，当我们进行大量的随机配对时，我们将记录平均收益。所以我们要关注的是，在这些博弈中随机配对时特定策略的平均收益。好了好了，潜在的想法是，相对成功的策略将会增长，相对不成功的策略将会减少。我不打算再写一遍，那是显而易见的另一面。所以相对成功的策略在增长，相对不成功的策略在减少。

[段 6]

Now to keep things simple, I’m not going to do any dynamics here. That would take us beyond the math that a lot of you can do in the class. So we’re not going to worry about dynamics here, but this is the underlying dynamic. The idea is that if a strategy is successful, that strategy will grow in the population. If a strategy is unsuccessful, it will decline. So this isn horrible violence to biology yet It the next bit that the horrible violence So to keep things simple today we going to assume that there is no gene redistribution Alright? We’re going to assume that there’s no gene redistribution. So in principle, what we’re looking at is asexual reproduction. Which is clearly not a good model, but it will do for today. Alright, so you want to think about a practical example of asexual reproduction? Think about root vegetables or judging from our experiments with the dating game we better hope that’s true for economic majors as well. Alright, maybe the fact I’m focusing on asexual reproduction will get me off with the guys in Kansas. Maybe I won’t lose my green card after all. Alright, so that’s going to be our basic story and the basic idea we’re going to use is this. It’s an idea due to a guy called Maynard Smith in the 70s, although obviously the big idea goes back earlier.

[译文 6]

为了保持简单，我不打算在这里做任何动态分析。那会超出你们很多人在课堂上能做的数学范畴。所以我们这里不担心动态分析，但这就是潜在的动态。想法是，如果一个策略是成功的，那个策略将在种群中增长。如果一个策略不成功，它就会减少。这对生物学来说仍然有很大的简化，但下一部分才是更大的简化。为了保持简单，今天我们将假设没有基因重分配。好了，我们将假设没有基因重分配。所以原则上，我们看到的是无性繁殖。这显然不是一个好的模型，但今天先这样做。你们想要一个无性繁殖的实际例子吗？想想根茎类蔬菜，或者从我们约会游戏的实验来看，我们也最好希望经济学专业也是如此。好了，也许我专注于无性繁殖这件事能让我免于和堪萨斯州的人起冲突。也许我终究不会失去我的绿卡。好了，这就是我们的基本故事，我们要用的基本想法是这样的。这是一个由一个叫 Maynard Smith 的人在70年代提出的想法，虽然显然这个大想法可以追溯到更早。

[段 7]

So this is the big idea. Suppose that we imagine that there’s a particular game and there’s a large population, so think of yourselves as a large population, and suppose that the entire population, the entire population were all playing the same strategy, call it S. So they’re all hardwired to play the same strategy, S. And suppose now that there’s a mutation, right, so some small group start playing some other strategy, let’s call it S prime. what we want to ask is will that small mutation group the S prime strategy will it thrive or will it die out if it’s true that for all possible mutations, all possible little groups of mutation of people playing S prime, they’ll die out then we’ll say that the original strategy the strategy S is evolutionarily stable and we’re going to write that up more formally later on, but that’s just the basic idea and it was retained again. There’s a strategy out there that everyone’s playing, call it S. We’re going to look at a mutation, that’s S prime, so a small group of people are going to start playing S prime. They’re going to go on being randomly matched, everyone’s going to be randomly matched and we going to ask if the S prime group does well in which case they grow or does badly in which case they shrink and eventually die out If they die out we say that S was evolutionarily stable That true for all possible mutations.

[译文 7]

这就是核心思想。假设我们设想有一个特定的游戏和一个大群体，所以把你们自己想象成一个大群体，假设整个群体，整个群体都在使用同一个策略，称之为S。所以他们都被硬编码为使用同一个策略S。现在假设出现了一个突变，对吧，所以有一小群人开始使用另一个策略，我们称之为S’。我们想问的是，这个使用S’策略的小的突变群体，它会繁荣还是会消亡？如果对于所有可能的突变，所有可能的使用S’策略的突变小群体都会消亡，那么我们就会说原来的策略，即策略S，是进化稳定的，我们之后会更正式地写出这个定义，但这就是基本思想。有一个策略在那里，每个人都在使用，称之为S。我们要看一个突变，那就是S’，所以一小群人将开始使用S’策略。他们会继续被随机匹配，每个人都会被随机匹配，我们会问S’群体是否表现良好，如果表现良好他们就会增长，如果表现不好他们就会减少并最终消亡。如果他们消亡了，我们就说S是进化稳定的，这对于所有可能的突变都成立。

[段 8]

All right? And just notice when we’re randomly matching them, one thing to note is, since there’s only a small mutation to start with, most of the time when they’re randomly matched, they’re going to match against somebody who’s still playing S. Occasionally they’re going to meet one of the other mutants, but most of the time we’re going to have to worry about how the mutants do against the incumbent population. All right, so that was too abstract to really get one’s head around, so let’s try and do an example. All right, so let’s remove our motivation and get down to actually doing some work. All right, so we’re going to start with a very simple example that you’ll all recognize. recognize. Here’s a game, it’s a two by two game and the payoffs are as follows, 2, 2, 0, 3, 3, 0 and 1, 1. All right we call these strategies cooperate or defect C for cooperate D for defect C for cooperate D for defect All right and you all recognize immediately that this game is what This game is what This is Prisoner Dilemma right? So we’re going to start by imagining these animals playing Prisoner’s Dilemma. And to put it into context, imagine that these are a group of lions, all right? Again, leave aside the fact that it’s a sexual reproduction for a second.

[译文 8]

好吗？要注意的是，当我们随机匹配他们时，有一件事需要注意，那就是由于一开始只有一小部分突变，大部分时候当他们被随机匹配时，他们会匹配到仍在使用S的人。偶尔他们会遇到另一个突变体，但大部分时候我们必须担心突变体们如何对抗现有的群体。好，那刚才说的太抽象了，难以理解，所以我们试着来做一个例子。好，让我们抛开动机开始实际做一些工作。好，我们要从一个非常简单的例子开始，你们都会认出来的。这里有一个游戏，是一个2x2的游戏，收益如下，2,2，0,3，3,0和1,1。好，我们把这些策略称为合作或背叛，C表示合作，D表示背叛，C表示合作，D表示背叛。好，你们立刻就会认出这个游戏是什么，这个游戏是囚徒困境，对吧？所以我们要从想象这些动物在玩囚徒困境开始。为了把它放到具体情境中，想象这是一群狮子，好吧？同样，先忽略这是有性繁殖这个事实。

[段 9]

Imagine these are a group of lions, and cooperating means cooperating on the hunt means using a lot of energy going after, as you cooperate in a group while hunting, and defecting here would mean not working hard on the hunt, and then the other lions catch the antelope, whatever, and they’re just sharing in the spoils, so free-riding, basically. All right, or another example, think about ants with an ant nest, and imagine this ant nest has been attacked by, I don’t know what, some other creature, you could imagine that cooperating is joining in in defending the nest at the risk of being hurt, and defecting is running away. All right? All right? And we all know in this game roughly how to analyze it. I’m not going to go over that now. I want to ask a different question today. I want to ask in this model of asexual reproduction is cooperation evolutionarily stable Is cooperation evolutionarily stable? All right. Okay. So to try and illustrate that, let’s just think about this game being played for real out there. So what we’re going to do is we’re going to start off by imagining that you are all, I don’t know, ants, lions, what do you want to be? Ants, I think. Alright, so you’re, from up here you don’t like ants, right? You’re all ants and all of you have been hardwired to play the strategy C.

[译文 9]

想象这是一群狮子，合作意味着在狩猎中合作，意味着消耗大量能量去追捕，当你们一群人一起狩猎时，而这里的背叛意味着在狩猎中不努力工作，然后其他狮子捕获羚羊什么的，它们只是分享战利品，基本上就是搭便车。好，或者另一个例子，想想蚂蚁和蚁巢，假设这个蚁巢受到了某种东西的攻击，我不知道是什么，可能是其他生物，你可以想象合作就是参与保卫蚁巢，即使有受伤的风险，而背叛就是逃跑。好吗？好，我们在这种游戏中大致知道怎么分析。我现在不打算讲那个。我今天想问一个不同的问题。我想问在这个无性繁殖的模型中，合作是否是进化稳定的？合作是否是进化稳定的？好。那么为了说明这一点，让我们想想这个游戏在现实中是如何进行的。我们要做的第一步是想象你们所有人，我不知道，是蚂蚁、狮子，你们想成为什么？我认为蚂蚁。好，所以你们，从上面看你们不喜欢蚂蚁，对吧？你们都是蚂蚁，而且你们所有人都被硬编码为使用策略C。

[段 10]

Alright, you’ve been hardwired to play the strategy C. Alright, so life’s going on fine and let’s do some random matching here. So suppose that’s going to get me eventually. Suppose that we randomly match this ant, all right, so stand up a second, with, all right, so this ant gets randomly matched with this ant, all right, all right, so they now play this prisoner’s dilemma against one another, all right, but both of them have been hardwired. to cooperate, all right? Both have been hardwired to cooperate. So they both cooperate, is that right? All right, so this ant, whose name is? Lenore. Lenore? Lenore the ant cooperates, and? Wu. And Wu the ant cooperates. So they both cooperate. We can look at what their payoff is. their payoff is cooperate against cooperate, so they get two. So that’s pretty good. So they’re doing pretty well in terms of genetic fitness. And suddenly, there are two ants, right? So this ant produces another ant. Well, it’s only without a computer, okay? So this ant produces another ant, all right? And this ant produces another ant, sorry, all right? All right, so they’re doing fine, right? And the whole population is matching against other cooperative ants, and more ants are being produced, and everything’s fine and dandy. All right, everything looks good, all right? All right, sit down a second, sit down a second.

[译文 10]

好，你们被硬编码为使用策略C。好，所以生活正常进行，我们来做一些随机匹配。假设这最终会让我有所收获。假设我们随机匹配这只蚂蚁，好，站起来一下，好，所以这只蚂蚁被随机匹配到这只蚂蚁，好，好，所以他们现在互相玩这个囚徒困境，好，但他们两个都被硬编码了。合作，好？两个都被硬编码为合作。好，所以他们都合作，对吗？好，所以这只蚂蚁，叫什么名字？Lenore。Lenore？Lenore这只蚂蚁合作，还有？Wu。Wu这只蚂蚁合作。所以他们都合作。我们可以看看他们的收益是什么。他们的收益是合作对合作，所以他们得到2。所以这相当不错。就遗传适应性而言他们做得很好。突然间，有了这两只蚂蚁，对吧？所以这只蚂蚁产生了另一只蚂蚁。呃，没有电脑的话，好吧？所以这只蚂蚁产生了另一只蚂蚁，好？这只蚂蚁产生了另一只蚂蚁，抱歉，好？好，所以他们过得很好，对吧？整个群体都在和其他合作的蚂蚁匹配，更多的蚂蚁被产生，一切都很美好。好，一切都看起来很好，好？好，坐下吧，坐下吧。

[段 11]

Now imagine that there’s a mutation, a small mutation, and this small mutation involves a group of rather nasty uncooperative ants It a rather scary mutation So here our scary mutation Let use the right So the TAs are my scary mutation So all the TAs here are the scary mutation There another one over there All right, so there’s a bunch. Everyone else is still playing cooperate. There’s this vast number of cooperative ants, but now there’s this small number of ants with this mutation that says don’t cooperate, play D. All right, so let’s see what happens with random matching. So a lot of these cooperative ants are still matching with each other and doing fine, but that’s not the point. What we’re worried about is what’s going to happen to the mutants. All right, so let’s pick our representative nasty mutants, so that’s going to be Rahul, all right? All right, so Rahul is our scary mutant, all right? And let’s go back and randomly match him against one of our incumbent ants, our nice cooperative ants. So let’s pick this one, all right? So it just happens he’s randomly matched with this cooperative ant. And this cooperative ant, whose name is Nick, is hardwired, since he’s a regular ant, he’s hardwired to play cooperate. So he’s going to play cooperate. Say cooperate. All right, I should put the mic, never mind.

[译文 11]

现在想象出现了一个突变，一个小小的突变，这个小小的突变涉及一群相当不友好的不合作蚂蚁。这是一个相当可怕的突变。所以这里我们的可怕突变，让我用正确的说法。助教们就是我的可怕突变。所以所有的助教们就是那个可怕的突变。那里还有另一个。好，所以有一群人。其他人仍然在使用合作。有一大群合作的蚂蚁，但现在有一小群有这种突变的蚂蚁，突变的内容是不合作，使用D。好，让我们看看随机匹配会发生什么。很多这些合作的蚂蚁仍然互相匹配并且做得很好，但这不是重点。我们担心的是突变体们会发生什么。好，所以让我们选我们的代表性好斗的突变体，那就是Rahul，好？好，所以Rahul是我们的可怕突变体，好？让我们回去，把他随机匹配到我们的现有蚂蚁之一，我们友好的合作的蚂蚁。选这一只吧，好。碰巧他被随机匹配到这只合作的蚂蚁。这只合作的蚂蚁，叫什么名字？Nick，由于他是一只普通的蚂蚁，他被硬编码为合作。所以他要合作。说合作。好，我应该把麦克风，不用了。

[段 12]

And meanwhile, do we have a mic handy Thank you All right so he says cooperate but unfortunately he been matched against the nasty uncooperative ant It not that he nasty he just been hardwired to say Not cooperative Not cooperative to say defect Alright So what going to happen now Alright, so unfortunately, unfortunately for Nick, he, the cooperative ant, was matched against a not very nice ant. He was playing C, and Rahul was playing D, so Nick’s payoff is zero, so we went from one Nick to being no Nick. He got wiped out. alright, and Rahul meanwhile, he got a payoff of what what was Rahul’s payoff? Rahul’s payoff of 3, so suddenly there’s not just Rahul but there’s the mutation is growing, alright alright, and now we go on matching each period, some of you are going to be matched against other cooperative ants and you’re going to be doing fine, but every now and then, and actually increasingly often, you’re going to be matched against one of our mutants and those mutants are going to grow. Those mutants are going to go on growing. Now it’s true that every now and then a mutant’s going to meet another mutant. Is that right? But we don’t have to worry about that. All we have to worry about is are they going to die out early on?

[译文 12]

与此同时，我们有麦克风吗？谢谢。好，所以他说合作，但不幸的是他被匹配到了那个可恶的不合作蚂蚁。不是他可恶，他只是被硬编码为说不合作。不合作，也就是背叛。好，那么现在会发生什么呢？不幸的是，对Nick来说很不幸，他这只合作的蚂蚁，被匹配到了一只不太友好的蚂蚁。他出C，Rahul出D，所以Nick的收益是零，所以我们从有一个Nick变成没有Nick。他被淘汰了。好，而Rahul呢，他的收益是多少？Rahul的收益是3，所以突然之间不只是有Rahul，而且突变在增长，好，好，现在我们继续每个时期进行匹配，你们中的一些人会和其他合作的蚂蚁匹配，你们会做得很好，但时不时地，实际上越来越频繁，你们会被匹配到我们的突变体之一，而那些突变体会增长。那些突变体会继续增长。确实时不时地一个突变体会遇到另一个突变体。是这样吗？但我们不用担心那个。我们只需要担心他们一开始是否会消亡？

[段 13]

It’s going to be very rare for them to meet another mutant, at least early on. So the strategy cooperate is evolutionally stable if this small mutation there our small mutation disappears and doesn turn into a bigger mutation And of course in this case, as we just saw, the mutation didn’t die out, it actually grew. So to be formal about this, and we be careful, the strategy to cooperate to be evolutionally stable, it needs to be the case that the mutation actually dies out on average. And here, far from dying out, it grew. We now have lots of these nasty All right, people understand? All right, so let’s just, thanks. All right, so let’s just try and do that a little bit more formally on the board. All right, that was the basic idea, the basic graphics of the idea, but let’s look at it a little bit more formally. So we basically have cooperative ants and a few mutant defector ants out there, so mutant not cooperative ants, and what we want to keep track of are their average payoffs in random matches. All right? So the incumbent ants, the cooperative ants, they’re playing, we’re interested in their payoff, and they’re playing a population that’s mixed. Almost everybody in the population is cooperative. So let’s say 1 minus epsilon of the people they’re playing against, where epsilon is a small minus. that’s mixed.

[译文 13]

在早期，他们遇到另一个突变体的概率非常低。因此，合作策略是进化稳定的，如果这个小的突变——我们的小突变——消失且不会变成更大的突变的话。当然，在这种情况下，正如我们刚才看到的，突变没有灭绝，它实际上增长了。所以让我们更正式地说，我们要谨慎一点，合作策略要成为进化稳定的，必须满足突变平均来说会灭绝的条件。而在这里，它非但没有灭绝，反而增长了。现在我们有很多这样的讨厌鬼。大家理解吗？好的，那么让我们在黑板上更正式地做一下。好的，这是基本思想，基本图示，但让我们更正式地看一下。所以我们基本上有合作的蚂蚁和几个突变的不合作蚂蚁，所以我们想追踪的是它们在随机配对中的平均收益。好的？所以原住民蚂蚁，合作的蚂蚁，它们在玩，我们关心的是它们的收益，它们在和一个混合的种群对抗。几乎种群中的每个人都是合作的。所以假设 epsilon 是一个非常小的数字，他们对抗的人中有 1 减去 epsilon 也是合作的。

[段 14]

Almost everybody in the population is cooperative. So let’s say 1 minus epsilon of the people they’re playing against, where epsilon is a small number, a very small number, are also cooperative. But every now and then, like our poor friend Nick, every now and then they’re going to come across a nasty mutant. They’re going to come across Rahul. So what’s their payoff going to be on average, their average payoff? Okay, so 1 minus epsilon of the time, we can actually think of this mixed here, so it’s 1 minus epsilon and epsilon. 1 minus epsilon of the time, they’re going to meet another cooperative ant and get a payoff of 2, all right? But epsilon of the time, they’re going to meet Rahul and get a payoff of 0. Is that correct? So that’s their average payoff. Now how about the mutants’ payoffs? So the mutants, there aren’t many of them around, but they are also playing against the same mixed population. 1 minus epsilon of the time, they going to be matched against the cooperative ants and epsilon of the time the two mutants the two TAs are going to play off against each other Let have a look at what their payoff is So their payoff is 1 minus epsilon they get a payoff of 3 That’s what we saw when Rahul met Nick. And the other epsilon of the time, Rahul meets Jake or somebody, and has to make do with a payoff of 1.

[译文 14]

几乎种群中的每个人都是合作的。所以假设 epsilon 是一个非常小的数字，他们对抗的人中有 1 减去 epsilon 也是合作的。但时不时地，就像我们可怜的朋友 Nick，时不时地他们会遇到一个讨厌的突变体。他们会遇到 Rahul。那么他们的平均收益是多少？好的，所以 1 减去 epsilon 的时间，我们实际上可以这样想，这是混合的，所以是 1 减去 epsilon 和 epsilon。1 减去 epsilon 的时间，他们会遇到另一只合作的蚂蚁并获得 2 的收益，对吧？但 epsilon 的时间，他们会遇到 Rahul 并获得 0 的收益。这是正确的吗？那是他们的平均收益。那么突变体的收益呢？突变体周围没有很多，但它们也在和同一个混合种群对抗。1 减去 epsilon 的时间，它们会与合作的蚂蚁配对，epsilon 的时间两个突变体——两个 TA——会互相竞争。让我们看看它们的收益是多少。所以它们的收益是 1 减去 epsilon 获得 3 的收益。这就是我们看到的 Rahul 遇到 Nick 时的情况。另一个 epsilon 的时间，Rahul 遇到 Jake 或其他人，只能得到 1 的收益。

[段 15]

But if we keep track of this, what do we have? this equals 2 1 minus epsilon, and this equals 3 1 minus epsilon plus epsilon. Is that right? And clearly, I hope this is clear, the payoff to the mutant is bigger. Everyone happy with that? The payoff to the mutant is bigger? Which means the mutation is not going to die out. In fact, it’s going to grow. The mutation is not going to die out. So we can conclude that C is not evolutionarily stable. I’m going to start using ES for evolutionarily stable. All right. So in this particular game, evolution is not evolutionarily stable. So we might ask what is evolutionarily stable in this game Well it not going to be hard to figure it out but since there only one choice I claim that D is going to be evolutionarily stable But let’s just prove it. So is it going to be the case that these mutants will eventually take over the whole population, and everyone will end up looking like a TA and being uncooperative. So they’re busily asexually reproducing all the time, they get bigger and bigger, and is it in fact the case that once they’ve conquered everything that they’re evolutionarily stable? So is defect evolutionarily stable in this game? To figure that out, we have to do exactly the reverse experiment.

[译文 15]

但如果我们追踪这些，我们有什么？这个等于 2 乘以 1 减去 epsilon，这个等于 3 乘以 1 减去 epsilon 加 epsilon。对吗？显然，我希望这很清楚，突变体的收益更大。大家对这点满意吗？突变体的收益更大？这意味着突变体不会灭绝。事实上，它会增长。突变体不会灭绝。所以我们可以得出结论，C 不是进化稳定的。我开始用 ES 来表示进化稳定。好的。所以在这次游戏中，进化不是进化稳定的。那么我们可能会问，在这个游戏中什么是进化稳定的？好吧，这不会太难弄清楚，但由于只有一个选择，我认为 D 将是进化稳定的。但让我们证明一下。那么，这些突变体最终会占领整个种群，让每个人都变得像 TA 一样不合作，这种可能性存在吗？它们不断无性繁殖，变得越来越大，一旦它们征服了一切，它们在事实上是进化稳定的吗？所以在这个游戏中背叛是进化稳定的吗？为了弄清楚这一点，我们必须做完全相反的实验。

[段 16]

So the experiment now is, imagine everyone in the population, all of you guys, all the students in the room are nasty non-cooperative defecting ants. You’re going along and most of the time you’re meeting each other all the time for now and you’re getting a payoff of one and now there a mutation but this time the mutation isn the scary non mutation that we saw before with Rahul It going to be Myrto over there all right And Myrto is a nice mutation who cooperates all right So Myrto is our nice cooperative mutation, all right? Let me just get the camera to pan on poor little Myrto, all right? So there’s poor little Myrto, who’s the only little cooperative mutation in the room. Wave. All right, there we go. Okay, all right. All right, what’s going to happen to our cooperative mutation? All right, so most of the time, you non-cooperative people are matching up against each other, we can figure out what your payoff is. So the uncooperative incumbents, they’re playing a population that is 1 minus epsilon non-cooperative, and epsilon, epsilon of the time, they meet Merto. All right? So what’s their average payoff? So let’s be careful. Let’s switch things around up here, just to make sure we can see what happens. So switching things around on our chart up here, we’re now looking at the case where there’s one minus epsilon non-corporators and epsilon cooperators.

[译文 16]

所以现在的实验是，想象种群中的每个人，你们所有人，房间里所有的学生都是讨厌的不合作的背叛蚂蚁。你们继续前进，大多数时候你们一直互相见面，现在你们得到 1 的收益，现在有一个突变，但这次突变不是我们之前看到 Rahul 时那个可怕的突变。它将是 Myrto，在那边，好吗？Myrto 是一个好的合作突变，好吗？让我把镜头转向可怜的小 Myrto，好吗？所以这里有可怜的小 Myrto，她是房间里唯一的合作突变。挥挥手。好了，我们继续。好的，我们的合作突变会发生什么？好的，所以大多数时候，你们这些不合作的人互相配对，我们可以算出你们的收益是多少。所以不合作的原住民，它们对抗的种群是 1 减去 epsilon 的不合作者，epsilon 的时间，它们遇到 Merto。好的？那么它们的平均收益是多少？让我们仔细一点。让我们在这里换一下东西，只是为了确保我们能看到发生了什么。所以在黑板上的图表换一下，我们现在看的是 1 减去 epsilon 的不合作者和 epsilon 的合作者的情况。

[段 17]

All right, so these, you guys, you non-cooperative ants, are most of the time meeting the same All right. So these, you guys, you non-cooperative ants, are most of the time meeting each other, and when you meet each other, you’re getting a payoff of one. So one minus epsilon of the time, you’re getting a payoff of one. But epsilon of the time, you’re doing great, because you’re meeting Murtu, and unfortunately, you’re beating up on Murtu. All right. And how’s she doing? well she’s a cooperator and 1 minus epsilon of the time she’s meeting you guys and epsilon of the time she meets another nice cooperative ant like Jake so her payoff is 1 minus epsilon of the time she gets nothing and epsilon of the time she does pretty well and gets 2 but it’s only epsilon of the time is that right? so far so good So what’s going to happen to this mutation? Well, the incumbent population, their average payoff comes down to being 1 minus epsilon plus 3 epsilon. And Mertz payoff the mutant ends up being 2 epsilon All right So unfortunately unfortunately these nasty ants are thriving and Moto gets wiped out. So you can sit down again. All right? So what have we shown here? We’ve shown that defect, not cooperate, is evolutionally stable in this game. All right? Any mutation, there is only one possible mutation. any mutation gets wiped out.

[译文 17]

好的，你们这些不合作的蚂蚁，大多数时候遇到同样的——好的。所以你们，不合作的蚂蚁，大多数时候互相见面，当你们互相见面时，你们得到 1 的收益。所以 1 减去 epsilon 的时间，你们得到 1 的收益。但 epsilon 的时间，你们干得很好，因为你们遇到了 Murtu，不幸的是，你们打败了 Murtu。好的。她怎么样？嗯，她是一个合作者，1 减去 epsilon 的时间她遇到你们，epsilon 的时间她遇到另一个像 Jake 那样好的合作者，所以她的收益是 1 减去 epsilon 的时间她什么都得不到，epsilon 的时间她做得相当好并得到 2，但只有 epsilon 的时间，是这样吗？到目前为止还不错。那么这个突变会发生什么？嗯，原住民种群，它们的平均收益下降到 1 减去 epsilon 加 3 epsilon。而 Mertz 的收益——突变体最终是 2 epsilon。好的，所以不幸的是，不幸的是这些讨厌的蚂蚁正在蓬勃发展，Moto 被消灭了。所以你可以再坐下了。好的？那么我们在这里展示了什么？我们已经证明在这个游戏中，背叛——不合作——是进化稳定的。好的？任何突变，只有一个可能的突变。任何突变都会被消灭。

[段 18]

All right? Think about these two different models here. The first, well, not different models, but different metaphors. So we started out with this population of relatively nice people, and we had a nasty mutation. So think of the movie Alien. Actually, don’t, it’s a horrible movie, but all right. And if you remember that horrible movie, Alien or Species, think of the movie Species, an even worse movie. Think of the movie Species, that was Rahul, and he grew, albeit asexually, very rapidly. That’s a pretty scary movie, right? Conversely, when we had this population of nasty people and we tried to have a nice little invasion of a nice mutant, so think of the movie E.T., unfortunately E got squished All right So that the two extremes All right So what is the lesson that we can actually draw from this All right. So there’s a lesson here, and the lesson, I guess there’s going to be two lessons here, but let’s start with one. The first lesson here is that, well, let’s just put it on the board first of all. nature, the outcome of evolution, nature, can suck. So again, we could use a more formal term, but this is an important lesson.

[译文 18]

好的？思考一下这两个不同的模型。第一个，嗯，不是不同的模型，而是不同的比喻。所以我们开始时有一个相对好的人群，我们有一个讨厌的突变。所以想想电影《Alien》。实际上，别想，那是一部可怕的电影，但好吧。如果你记得那部可怕的电影，《Alien》或《Species》，想想电影《Species》，一部更糟糕的电影。想想电影《Species》，那就是 Rahul，他虽然是无性繁殖，但增长得非常快。那是一部相当可怕的电影，对吧？反过来，当我们有一个讨厌的人群，我们试图进行一次好的小的好的突变的入侵，想想电影《E.T.》，不幸的是 E 被压扁了。好的，这就是两个极端。好的，那么我们能从中得出什么教训？好的，这里有一个教训，我想这里会有两个教训，但让我们先从这一个开始。第一个教训是，嗯，让我们先把它写在黑板上。大自然，进化的结果，大自然可能会很糟糕。所以再说一次，我们可以用一个更正式的术语，但这是一个重要的教训。

[段 19]

There’s a tendency for some people to think that if something arises as a consequence of evolution, if something is natural, if something is in nature, it must therefore be good in some moral or other way, efficient or something. alright and what we’re seeing here is in this game the consequence of nature is a horrible consequence alright for those people who doubt that nature can be pretty unpleasant have a look at yesterday’s science page of the New York Times and read the piece about how baboons basically kill baby baboons the biggest cause of death among baby baboons is infanticide this is evolutionary stable it turns out and they explain why in the Times it isn pleasant alright so nature can suck Nature can be pretty inefficient Okay Now this raises a question in this particular game because look at the examples we started with. We started with lions who were thinking about, or not thinking about, they weren’t thinking about, they were hardwired to go after antelope and either cooperate or not. And we thought about ants who were hardwired to defend the nest or not. And we all know from watching endless nature shows as children on TV, that actually lions do cooperate when they go after antelope, and actually ants do defend the nest when it’s invaded by a spider or something. Is that right?

[译文 19]

有些人倾向于认为，如果某事物是进化的结果，是自然的，是存在于自然界中的，那么它在某种道德层面上或其他方面一定是有益的，或者说高效的。好吧，我们在这个博弈中看到的，自然的结果是可怕的后果——对于那些怀疑自然可以相当残酷的人，看看昨天《纽约时报》的科学版，读一读那篇关于狒狒如何基本上杀死幼狒狒的文章——幼狒狒死亡的最主要原因就是杀婴行为，而这在进化上是稳定的，事实证明如此，他们也在时报上解释了原因。这不令人愉快。好吧，所以自然可以很糟糕。自然可以相当低效。现在这提出了一个问题，在这个特定的博弈中，因为看看我们开始时的例子。我们从狮子开始，它们要么合作要么不合作地追逐羚羊——或者说它们并没有在思考，它们是被硬连线来追逐羚羊的。我们也思考了蚂蚁，它们被硬连线来保卫蚁巢或者不保卫。我们都从童年时在电视上看的无数自然节目中知道，实际上狮子在追逐羚羊时确实会合作，实际上蚂蚁在蚁巢被蜘蛛或其他东西入侵时确实会保卫蚁巢。是这样吗？

[段 20]

I’m looking for some nodding. Is that right? Yeah, okay, good. So what happened? What’s wrong with our model here? We’ve argued here that in this model, nature’s going to produce this non-cooperative behavior. We know there are examples of cooperation in nature. What’s going on? What are we missing? Yeah, can I get Can I get a mic in, it’s here. Let me do it, I’ll do it. Yeah, so what’s going on? Different communities might compete against each other, so one community that doesn’t have any of these mutations and stays cooperative might succeed, so communities in general don’t have as many mutations. Okay, so that’s an interesting idea. in the community that doesn’t have any of these mutations and stays cooperative might succeed, so communities in general don’t have as many mutations. Okay, so that’s an interesting idea. So part of this is because we’re focusing on within-species competition rather than across-species competition. That turns out to be a complicated idea and large literature on it, and it’s actually a little difficult. I was looking for a… But it’s a good suggestion. I was looking for something simpler, actually. Let me come back here. So, yeah? Do these species have ways to detect cheaters and punish them? Ah, okay, that might be a possibility, and there’s certainly some evidence of that, but not much. Again, we’re looking at pretty high primates by the time you get to that.

[译文 20]

我想看看有没有人在点头。是这样吗？是的，好吧，不错。那么发生了什么？我们的模型哪里有问题？我们在这里论证，在这个模型中，自然会产生这种非合作行为。我们知道自然界中存在合作的例子。到底是怎么回事？我们遗漏了什么？是的，我能拿个麦克风吗，在这呢。让我来，我来。好吧，所以怎么回事？不同的群体可能会相互竞争，所以一个没有任何这些突变的、保持合作的群体可能会成功，所以总体而言群体不会有那么多突变。好吧，这是一个有趣的想法。所以部分原因是因为我们关注的是物种内的竞争而不是物种间的竞争。事实证明这是一个复杂的想法，有很多相关文献，而且实际上有点困难。我本来在找……但这是个很好的建议。实际上我在找一个更简单的答案。让我回到这里。所以，嗯？这些物种有没有办法检测作弊者并惩罚他们？啊，好吧，这可能是一种可能性，确实有一些证据，但不是很多。同样，当我们到达那个阶段时，我们研究的是相当高级的灵长类动物。

[段 21]

Something simple, something you should all have on your mind all the time if you’re normal teenagers. The payoffs could change as the proportion of cheaters increases, so you’re not going to get the payoff of three if everyone’s cheating. All right, but it’s true that you’re going to get a different expected payoff, but that isn’t the payoff. I think it’s really simple that, let me repeat, I’m guessing is on all of your minds as teenagers, let’s see if that hint’s gonna get the right answer. Yeah. Answer not asexual Right right the point the assumption that really driving this here is the assumption of asexual reproduction right Sexual reproduction with gene exchange is going to make a difference Why I don want to delve too much into the biology here but the main why is what matters is survival of the gene not survival of the individual ant or individual lion. So if you have sexual reproduction, you get gene redistribution among children and cousins to a less extent, so provided the other ants in the nest are closely enough genetically related to you, and provided the lions are close enough related to you, then it may turn out that you will cooperate, and that in fact does change the payoff of the game. It doesn’t change the payoff for the individual lion or ant, it changes the payoff for the gene.

[译文 21]

一些简单的东西，如果你们是正常的青少年，应该一直记在心里的东西。收益可能会随着作弊者比例的增加而变化，所以如果每个人都在作弊，你就不会得到3的收益。好吧，但如果你作弊的话，你确实会得到不同的预期收益，但那不是原本的收益。我猜测这真的很简单，让我重复一下，我猜你们青少年都想到了，让我们看看这个提示能否得到正确答案。是的。答案不是无性繁殖。对，对，关键点，真正推动这里的是无性繁殖的假设。有性繁殖与基因交换会产生不同。为什么我不想在这里深入探讨生物学，但主要原因是——重要的是基因的生存，而不是个体蚂蚁或个体狮子的生存。所以如果你有性繁殖，你就会在孩子和表亲（程度较轻）之间进行基因重新分配，所以只要蚁巢中的其他蚂蚁在基因上与你足够接近，只要狮子与你足够接近，那么结果可能是你会合作，而实际上这确实改变了博弈的收益。它没有改变个体狮子或蚂蚁的收益，它改变的是基因的收益。

[段 22]

Now if we have time, we’ll come back and look at that on Monday, and even if we don’t have time, in the reading I’ve left for you, in the reading packet, it goes into that in some detail. It’s a much more complicated model, but that’s what we’re going to get. For the most part, that’s where we’re going to get cooperation from, rather than from the other points of view we made. So sex can make a difference here. But we’re going to stick with asexual reproduction because it’s easier to analyze, and stick with it for the minute. And what we’ve learned, the second lesson we’ve learned here, I’m going to generalize from this lesson, we learned that a strategy that was strictly dominated was not evolutionarily stable I only shown an example here but we extend from the example So if a strategy is strictly dominated, if a strategy is strictly dominated, and of course cooperates is strictly dominated here, then it is not evolutionarily stable at least in this simple game of just asexual reproduction. And even though I haven’t proved this here the idea is exactly the idea of this example. So let’s just try and talk it through. Suppose a strategy was strictly dominated. How do we know, how can we see that it will not be evolutionarily stable? Somebody? Where are my pre-med majors?

[译文 22]

现在如果有时间，我们周一会回来讨论这个问题，即使没有时间，在我留给你们阅读的材料中，在阅读资料包里有详细介绍。这是一种复杂得多的模型，但我们会得到它。在大多数情况下，我们将从那里获得合作，而不是从我们之前提出的其他观点。所以性可以在这里产生影响。但我们将坚持使用无性繁殖，因为它更容易分析，暂时坚持使用它。我们学到的第二个教训，我想从这里推广开来，我们学到的是，一个被严格占优的策略在进化上是不稳定的。我在这里只展示了一个例子，但我们可以从例子推广开来。所以如果一个策略被严格占优，当然合作在这里是被严格占优的，那么它在进化上是不稳定的，至少在这个简单的无性繁殖博弈中。虽然我没有在这里证明，但想法完全就是这个例子的想法。所以让我们试着来梳理一下。假设一个策略被严格占优。我们怎么知道，我们怎么能看出它不会是进化稳定的？有人吗？我的医学预科生在哪里？

[段 23]

I mean not pre-med measures, but my pre-med people they’re all hiding now I know you’re out there because I’ve seen your forms, but never mind, okay, so why is this dominated strategy not evolutionally stable I want to try this out yes the strategy that dominates it will be a successful mutation so the strategy that dominates the strategy that does the domination of this strictly dominated strategy would be a successful mutation if it enters it does well not just against this strategy but against any mix involving itself in this strategy All right? So the strictly dominator strategy will invade. All right? So we can’t have strictly dominated strategies surviving in evolution. All right. All right. Let’s do another example. And we’ll see if we can learn some more. So this is going to be a slightly more complicated example. Let’s have a 3x3 game. So here’s our 3x3 game. We’ll just label the strategies ABC and ABC. And once again, we’re going to focus on symmetric games, so this game will be symmetric. 2, 2, 0, 0, 0, 0. 0, 0, lots of 0s in this, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0. All right, this is a symmetric game. Okay, and if we look at this game a little bit, I don’t want to look at all of it. I want to ask the question, 0, 0.

[译文 23]

我是说不是医学预科的方法，但我那些医学预科的人都躲起来了，我知道你们在场，因为我看过你们的表格，但没关系。好吧，为什么这个被占优的策略在进化上不稳定？我想试着回答一下，是的，占优它的那个策略将是一个成功的突变。所以那个占优这个被严格占优策略的策略，如果它进入的话会表现良好，不仅对这个策略，而且对任何包含它自己和这个策略的混合策略都是如此。好吧？所以严格占优的策略将会入侵。好吧？所以我们不能让被严格占优的策略在进化中存活下来。好吧。好吧。让我们再看一个例子。我们看看能不能学到更多。所以这将是一个稍微复杂一点的例子。让我们来一个3x3的博弈。这是我们的3x3博弈。我们把策略标记为A、B和C。再说一次，我们关注的是对称博弈，所以这个博弈将是对称的。2, 2, 0, 0, 0, 0。0, 0，这里有很多0，0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0。好吧，这是一个对称博弈。好吧，如果我们稍微看看这个博弈，我不想全部看。我想问一个问题，0, 0。

[段 24]

This is a symmetric game. Okay. And if we look at this game a little bit, I don’t want to look at all of it. I want to ask the question, is strategy C evolutionarily stable? Strategy C evolutionarily stable. Let’s go back to our experiment. What does it involve? Imagine everyone in the room is hardwired to play C. Could that population be invaded by a mutation? So what do we think? What do we think? Is C evolutionary stable? Who thinks it is stable? Who thinks it’s not stable? So most of you think it’s not stable, so you must have some idea why. Why do we think it’s not stable? Let me just try and cold call a little bit. Anybody? Let me sample somebody. If it’s not stable, it must be it’s So who’s going to invade it? Can we try this gentleman? Have a guess who’s going to invade it? Strategy B. Strategy B. Okay, that’s correct. Okay, so your name is? Greg. So Greg is saying strategy B might invade here Let have a look Let see what happens All right So suppose there an invasion of B So again the TAs are playing B Everyone else is playing C. And let’s see how these incumbent genes do. So C is playing against 1 minus epsilon of the population who are playing C and epsilon who are playing B.

[译文 24]

这是一个对称博弈。好吧。如果我们稍微看看这个博弈，我不想全部看。我想问一个问题，策略C在进化上是稳定的吗？策略C在进化上是稳定的。让我们回到我们的实验。它涉及什么？假设房间里的每个人都被硬连线去玩C。这样的种群能被突变入侵吗？那我们怎么想？我们怎么想？C是进化稳定的吗？谁认为它是稳定的？谁认为它不稳定？所以你们大多数人认为它不稳定，所以你一定有某种想法为什么。为什么我们认为它不稳定？让我试着随便点几个人。有谁？让我抽样问一下。如果它不稳定，它一定是什么，所以谁会入侵它？我们能试试这位先生吗？猜猜谁会入侵？策略B。策略B。好吧，这是正确的。好吧，你叫什么名字？Greg。所以Greg说策略B可能会在这里入侵。让我们看看。让我们看看会发生什么。好吧，那么假设B入侵了。所以助教的角色在玩B，其他所有人都在玩C。让我们看看这些原有基因的表现。所以C在对抗人口中的1减去epsilon，这些人在玩C，还有epsilon的人在玩B。

[段 25]

And its average payoff, well, 1 minus epsilon of the time, it’ll be matched against essentially itself and get a payoff of 0. But epsilon of the time it will be matched against a B and get a payoff of 1. Everyone agree with that? So 1 minus epsilon of the time it meets itself and gets nothing. Epsilon of the time it gets lucky and it meets a TA and gets a payoff of 1. How about those the B invaders, the TAs in the class. So, one minus epsilon of the time, they’re going to meet Cs and epsilon of the time, they’re going to meet Bs. One minus epsilon of the time, therefore, the B is meeting a C and getting a payoff of one So one minus epsilon of the time the TA is meeting one of you And epsilon at the time the TA is meeting another TA playing B against B, and getting 0. So this works out as epsilon, and this works out as 1 minus epsilon. But notice that if epsilon is small, if there are just a small number of these mutations, 1 minus epsilon is bigger than epsilon. Is that right? And that means that this mutation will not die out. We can actually say a bit more. It’ll probably go on growing until it’s roughly half the population. But in particular, it won’t die out.

[译文 25]

它的平均收益是，在大多数时间（1 minus epsilon），它与自己匹配，获得0收益。但在 epsilon of the time，它与B匹配，获得1收益。大家都同意吗？所以，1 minus epsilon of the time，它遇到自己，什么也得不到。Epsilon of the time，它运气好，遇到TA，获得1收益。那么，那些B入侵者，也就是班里的TA，怎么样？所以，1 minus epsilon of the time，它们会遇到Cs，epsilon of the time，它们会遇到Bs。因此，1 minus epsilon of the time，B遇到C并获得1收益。所以，1 minus epsilon of the time，TA遇到你们中的一个。而在 epsilon of the time，TA遇到另一个TA，用B对抗B，获得0收益。所以，计算得出 epsilon，以及 1 minus epsilon。但注意，如果 epsilon 很小，即这些变异数量很少，那么 1 minus epsilon 大于 epsilon。对吗？这意味着这种变异不会灭绝。我们可以更进一步说。它可能会继续增长，直到大约 half the population。但尤其重要的是，它不会灭绝。

[段 26]

Since it won’t die out if the mutation doesn’t die out, we can conclude that C is not evolutionarily stable. Everyone okay with that? Everyone okay with that idea? Now this idea, this example is a little bit more complicated than the previous example because it turns out that the invading mutant population, the population B, is itself not evolutionarily stable. Everyone see that? C wasn evolutionarily stable because it got invaded by B who going to invade B Who going to invade B C is going to invade B. It turns out everything is exactly symmetric here. So even though we’re arguing that C is not evolutionarily stable because it’s invaded by B, it doesn’t have to be the case that the successful mutant is itself evolutionarily stable. So in this particular example, the invader the invader, namely B is itself not evolutionarily stable nevertheless it doesn’t die out when it invades a population of C. Alright. But there’s a second observation I want to draw off this one. I’ll give it a bit of chalk here. I’m not finding it. All right, there’s a second observation I want to draw up here. What do we observe? What else is true about both C and B? So what’s true about everyone playing C or everyone playing B? What else? Just going back to what we’ve learned in the course so far, what do we think about everyone playing C?

[译文 26]

由于突变不会消失，所以C不会灭绝，据此可以得出结论：C不是进化稳定的。大家都理解吗？大家都理解这个概念吗？现在这个概念，这个例子比前面的例子要复杂一些，因为事实证明，入侵的突变种群，即population B，是 itself not evolutionarily stable. 大家都看到了吗？C不是进化稳定的，因为它被B入侵了，而B又会入侵B，C又会入侵B。事实证明，这里的一切都是完全对称的。所以，虽然我们在论证C不是进化稳定的，因为它被B入侵了，但成功的突变体本身不一定是进化稳定的。所以在 这个特定例子中，入侵者，也就是B，itself not evolutionarily stable，然而当它入侵C种群时，它不会灭绝。好的。但这里我还想提出第二个观察。让我在黑板上写一下。我找不到（想说的了）。好的，这是我想在这里提出的第二个观察。我们观察到了什么？关于C和B，它们还有什么共同点？那么 everyone playing C或者everyone playing B时，什么是真的？还有什么？ just going back to what we’ve learned in the course so far，我们对 everyone playing C有什么看法？

[段 27]

B or everyone playing B? What else? Just going back to what we’ve learned in the course so far, what do we think about everyone playing C? So if everyone was playing C, we’d be looking at an object like this, C, C, right? And we might ask the question, is C, C what? Is it a Nash equilibrium? so is is CC a Nash equilibrium here we asked was it evolutionarily stable we found it wasn’t now let’s ask a different question and forget evolution for a second is CC a Nash equilibrium well that’s a question you all should be able to answer for the midterms so somebody tell me the answer it’s not a Nash equilibrium how do we know it’s not a Nash equilibrium What do we have to do to show it’s not an actual equilibrium? We have to show that there’s a profitable deviation, right? So shout it out. What’s the profitable deviation here? Profitable deviation is B. It’s not. No because B is a strictly profitable deviation B is a strictly profitable deviation. Notice that the thing that was a strictly profitable deviation was the same thing that would have invaded the population of C’s in the context of our ant’s nest, in the context of this classroom being an ant’s nest and looking at evolution. Is that right? so what have we just learned here? well this idea turns out to be general the idea is if a strategy S perhaps I should make this a lesson lesson if a strategy S is not Nash so in other words SS is not a Nash equilibrium it’s not a Nash equilibrium, then S is not evolutionarily stable.

[译文 27]

B还是每个人都玩B？还有别的吗？回到我们到目前为止在课程中学到的东西，我们对每个人都玩C有什么看法？所以如果每个人都玩C，我们看到的对象就像这样，C，C，对吧？我们可能会问这个问题，C，C是什么？是Nash均衡吗？所以CC是Nash均衡吗？我们之前问过它是否是evolutionarily stable，我们发现它不是。现在让我们问一个不同的问题，暂时忘记evolutionary，CC是Nash均衡吗？好吧，这是你们所有人都应该能够在midterms回答的问题，所以有人告诉我答案。它不是Nash均衡。我们怎么知道它不是Nash均衡？我们要做什么来证明它不是actual均衡？我们必须证明有一个profitable deviation，对吧？所以大声说出来。这里的profitable deviation是什么？Profitable deviation是B。不是。不，因为B是一个strictly profitable deviation B是一个strictly profitable deviation。请注意，strictly profitable deviation的东西与能够在C人群中入侵的东西是一样的，在我们的蚂蚁巢穴的背景下，在这个课堂是一个蚂蚁巢穴并从进化角度来看。是这样吗？那么我们刚刚学到了什么？好吧，这个想法原来是general的，这个想法是，如果一个策略S，也许我应该把它变成一个lesson lesson，如果一个策略S不是Nash，那么换句话说SS不是Nash均衡，它不是Nash均衡，那么S就不是evolutionarily stable。

[段 28]

If S is not Nash then S is not evolutionarily stable A little bit of logic and we flip that around and we say what that actually tells us is it equivalent to saying if S is evolutionarily stable, then S, S is a Nash equilibrium. That’s the equivalent way of saying that. All right, so what’s the idea here? Let’s look at the first line rather than the second line. It’s pretty easy to understand. The idea here is if a strategy is not Nash, what that means is, is there’s some other strategy that would be a strictly profitable deviation. And that’s enough to tell us that S cannot be evolutionarily stable. Why? Because take that same strategy that was a strictly profitable deviation, B in this case, and that strategy can be thought of as the mutation that’s going to invade the strategy we started from. Right, everyone see that? Say it once more. Suppose a strategy is, suppose a strategy S, S is not a Nash equilibrium. Right, that means there’s some other strategy, S prime say that is a strictly profitable deviation Now think of that strictly profitable deviation S prime as a mutant invasion so now Rahul is playing is hardwired to play S prime since it was strictly profitable as a deviation it going to be successful as an invader. So again, I haven’t formally proved it, but that’s exactly how the proof would run.

[译文 28]

如果S不是Nash均衡，那么S就不是进化稳定的。稍微用一点逻辑，我们把它反过来，这实际上告诉我们等价于说如果S是进化稳定的，那么S是Nash均衡。这是等价的表述方式。好的，这里的核心思想是什么？我们来看第一行而不是第二行。这很容易理解。这里的核心思想是，如果一个策略不是Nash均衡，这意味着存在另一个策略可以带来严格占优偏离。这就足以说明S不可能是进化稳定的。为什么？因为把这个严格占优偏离的策略——这里是B——可以把这个策略看作是将会入侵我们最初策略的突变。大家明白了吗？再说一遍。假设一个策略，假设策略S不是Nash均衡。这意味着存在另一个策略，我们姑且称之为S’，它是一个严格占优偏离。现在把这个严格占优偏离的S’看作一次突变入侵，所以现在Rahul被硬编码为玩S’，既然它作为偏离是严格占优的，它作为入侵者将会是成功的。所以再说一次，我还没有形式化地证明它，但这正是证明的思路。

[段 29]

So let’s pause for a second and just see where we are. what we’ve managed to show is there’s a connection here between dominance and evolutionary stable we’ve said if a strategy is not dominated it cannot be evolutionary stable and we’ve also begun to show a connection between Nash equilibrium and evolutionary stability so two of the ideas we’ve developed over the last six weeks are re-emerging in this completely other context of animal behaviour the idea of dominance and the idea of Nash. And so far, how far have we got? We’ve said that if something’s going to be evolutionarily stable, it better be the case that it’s also Nash. And that raises, I think, a natural question, which is, is the opposite true? It would be really pretty great if it were true the other way around. If I could show… that if a strategy was Nash, then it would necessarily also be evolutionarily stable. And that would be pretty cool, right? This idea we’ve spent a lot of time developing in the class turns out to be the key idea. Unfortunately, life isn’t quite so neat. Let’s see why. All right, so what I’m going to do is I’m going to show you an example, and what this example is going to illustrate is that we can find Nash strategies that are not evolutionarily stable, and the example is embarrassingly simple.

[译文 29]

所以让我们暂停一下，看看我们目前的位置。我们已经证明的是，在支配性（dominance）和进化稳定性（evolutionarily stable）之间存在联系。我们已经说过，如果一个策略不是被支配的，它就不可能是进化稳定的。我们也开始证明了在纳什均衡（Nash equilibrium）和进化稳定性之间存在联系。所以在过去六周中我们发展的两个概念——支配性的概念和纳什均衡（Nash）的概念——在这个完全不同的动物行为语境中再次出现。到目前为止，我们进展到哪里了？我们已经说过，如果一个策略要成为进化稳定的，它最好同时也满足纳什均衡。而这引发了一个自然的问题，反过来成立吗？如果我们能够证明——如果一个策略是纳什均衡，它必然也是进化稳定的——那就太好了，对吧？我们在这个课程中花了很多时间发展的这个概念成为了关键概念。不幸的是，事情并没有那么完美。让我们看看为什么。好吧，我要做的是给你们展示一个例子，这个例子要说明的是，我们可以找到是纳什均衡但不是进化稳定的策略，而且这个例子简单得令人尴尬。

[段 30]

Here’s a game. It’s a two-player, two-strategy game, and the payoffs are 1-1, 0-0, 0-0, 0-0. This is kind of an embarrassingly simple game. What are the Nash equilibria in this game? Let go through slowly If player Rho is playing A then column best response is to pick A And if row is playing B then column best response is either to play A or B All right, if column is playing A, then row’s best response is to choose A. If column is playing B, then row’s best response is either A or B. All right, this is kind of exercise that is almost second nature for you guys now. All right? So we can conclude by looking where these best responses coincide that the Nash equilibria here are AA and BB. Okay? Everyone happy with that? Should we be looking like this? Yeah, this is easy, right? All right. So, all right. But let’s look at this second Nash equilibria. the BB Nash Equilibrium, the one that’s down here. Is B evolutionarily stable? Well again let think about it let just think through the exercise Suppose the entire class were playing B All right Wake up the guy in the middle and tell him he playing B too There you go Him yeah Okay All right The whole class is playing B All right? And suppose there’s an invasion.

[译文 30]

这是一个博弈。这是一个双人双策略博弈，收益是1-1, 0-0, 0-0, 0-0。这是一个相当简单到令人尴尬的博弈。这个博弈中的纳什均衡是什么？让我们慢慢来分析。如果玩家Rho选择A，那么列的最佳应对就是选A。如果行选择B，那么列的最佳应对可以是选A或B。好的，如果列选择A，那么行的最佳应对就是选择A。如果列选择B，那么行的最佳应对可以是A或B。好的，这基本上是一个对你们来说已经近乎第二本能的练习了。对吧？所以我们可以得出结论，通过观察这些最佳应对的交点可以知道，这里的纳什均衡是AA和BB。明白吗？大家都清楚吗？我们应该这样分析吗？是的，这很简单，对吧？好的。那么，让我们来看看第二个纳什均衡。这个BB纳什均衡，就是下面这个。B是进化稳定的吗？好吧，再想想看，我们来走一遍这个练习。假设整个班级都在玩B。好的，叫醒中间那个人，告诉他他也在玩B。这样就行了他，好的。好的，整个班级都在玩B，对吧？如果出现一个入侵者。

[段 31]

What could the invasion be? The invasion better be an invasion of A’s. All right? What’s going to be the expected payoff or the average payoff of the incumbents of all of you in the class who are playing B? Well, without doing it too laboriously, 1 minus epsilon at a time, you’re going to meet another student, in which case you’ll be playing B against B, and your payoff will be 0. And epsilon at a time, you’re going to meet a TA, and these TAs are playing A, but again, you’ll get 0. So your average payoff will be what? 0. 0, all right? Your payoff will be 0. So if you’re incumbent against this mix, your payoff will be 0. and if you’re an invader, so how’s Rahul doing this time? Rahul’s playing A. 1 minus epsilon of the time, this A is playing against a B, right? So 1 minus epsilon of the time, he’s playing against a B right So 1 minus epsilon of the time Rahul meets a student and gets a payoff of 0 But epsilon of the time Rahul hits it lucky and Rahul meets another TA Rahul meets Jake And when he meets Jake, his payoff is 1. Is that correct? Is that correct? So his total average payoff is epsilon, which is bigger than 0. so indeed it turns out that Rahul’s gene is going to grow, the bees are going to shrink BB was Nash but it’s not evolutionarily stable it can be invaded everyone see that? everyone see how that invasion worked? so the key to that invasion was when Rahul met another incumbent student he did no better than the students did against students But on those rare occasions when Rahul met another TA, he made hay.

[译文 31]

入侵可能是什么？入侵最好是A的入侵，好吗？那么在你们班上所有玩B的人，作为既有者的预期收益或平均收益是多少？好吧，不用太费力地算，1减去epsilon的时间，你会遇到另一个学生，在这种情况下你是在用B对B，你的收益是0。而epsilon的时间，你会遇到助教，这些助教在玩A，但同样，你会得到0。所以你的平均收益是多少？0。0，好吗？你的收益是0。所以如果你作为既有者对抗这个混合，你的收益是0。而如果你是入侵者，那么Rahul这次表现得怎么样？Rahul在玩A。1减去epsilon的时间，这个A在对阵B，对吧？所以1减去epsilon的时间Rahul遇到一个学生得到0的收益。但epsilon的时间Rahul撞大运，Rahul遇到另一个助教。Rahul遇到Jake。当他遇到Jake时，他的收益是1。对吗？是这样吗？所以他的总平均收益是epsilon，这大于0。因此事实上Rahul的基因会增长，蜜蜂会缩小。BB是Nash但它不是进化稳定的，它可以被打入。大家都看到了吗？大家都看到这个入侵是如何运作的吗？所以这个入侵的关键是，当Rahul遇到另一个既有学生时，他并没有比学生对学生做得更好。但在那罕见的几次Rahul遇到另一个助教时，他占了便宜。

[段 32]

Or not hay, it’s asexual reproduction. He, he, he, you’re a payoff of one. All right? And that, those rare occasions were enough to make Rahul grow and thrive, whereas the bees, relatively speaking, shrink. All right, everyone happy with that? Yeah? Okay, so what we have here is an example of something that is Nash, but is not speaking, shrink. Alright, everyone happy with that? Yeah? Okay. So what we have here is an example of something that is Nash, but is not evolutionarily stable. Can anyone say what’s special about this example? How did I rig this example? There’s something really kind of knife-edge and rigged about this example. What’s rigged about this example? Anybody? I don’t want a cold call on this. No takers? Everyone’s kind of in that pre-midterm scared mode. Yeah? This guy is saving the class today. Someone else has to get in. Okay, what’s your name again? Stephen. So Stephen is saving the rest of you, but go ahead, Stephen. B is only a weak best response to anything. Good. So what’s true about this example is that although B is a best response against B, it only is so weakly. Is that right? It’s only so weakly. All right? If, in fact, we got rid of Nash equilibria that relied only on weak best responses, then we wouldn’t be able to produce this example.

[译文 32]

或者不是占便宜，是无性繁殖。他，他的收益是1。好吧？而那些罕见的情况足以让Rahul增长和繁衍，而蜜蜂，相对来说，在萎缩。好吧，大家都明白吗？嗯？好的，所以我们这里有一个Nash但不是进化稳定的例子。有谁能说出这个例子的特别之处吗？我是怎么设计这个例子的？这个例子有些非常临界和被设计的地方。这个例子被设计了什么？有人吗？我不想冷不丁点名。没人接招吗？大家都有点处于期中考试前的恐惧模式。是吗？这位在拯救全班。还得有人参与进来。好，你叫什么名字？Stephen。所以Stephen在拯救你们剩下的，但请说吧，Stephen。B只是对任何东西的弱最佳反应。很好。那么这个例子的真实情况是，虽然B是对B的最佳反应，但它只是弱最佳反应。对吗？它只是弱最佳反应。好吧？如果实际上我们去掉只依赖弱最佳反应的Nash均衡，那么我们就无法产生这个例子。

[段 33]

So in particular if we looked at Nash equilibria where the Nash strategy was strictly a best response it was strictly better than playing any other pure strategy those cases would be evolutionarily stable. Anyone remember what we call Nash equilibria, where the Nash strategy is a strict best response? We call them strict Nash. All right, so what is in fact true is that if S is a strict Nash equilibrium, by which I mean S is a strict best response to S, then S is a Nash equilibrium. That’s right, and S is evolutionary state. All right. Excuse me. Okay, so we’ve been pretty, we’ve been only semi-formal so far. I haven’t really put up a formal definition. And what I’ve been groping towards slowly is a connection between an idea growing out of biology, namely evolutionary stability, and an idea growing out of economics and mathematics, namely Nash equilibrium And what I want to do now is I want to show you formally how are these connected I not going to prove this part of the class because a proof is a bit lengthy but I will include a proof on the handout that’s going to be available this afternoon. I’m just going to argue it for now. There we go, that’s in short. All right? So what I want to do first is I want to write down a formal definition of evolutionary stability.

[译文 33]

特别是如果我们看那些Nash策略是严格最佳反应的Nash均衡，即严格优于玩任何其他纯策略的情况，那些情况将是进化稳定的。有人记得我们把Nash策略是严格最佳反应的Nash均衡叫什么吗？我们称之为严格Nash。好的，那么事实上正确的是，如果S是一个严格Nash均衡，意思是说S是对S的严格最佳反应，那么S是一个Nash均衡。是的，而且S是进化稳定的。抱歉。好的，所以我们目前一直相当，只是半形式化的。我还没有给出一个正式的定义。我一直在慢慢摸索的是连接来自生物学的一个概念——进化稳定性，和来自经济学和数学的一个概念——Nash均衡。我现在要做的是向你们形式地展示这些是如何连接的。我不打算在课堂上证明这部分，因为证明有点长，但我会在今天下午提供的讲义中包含一个证明。我现在只是论证一下。好的，就是这样。好的，那么我首先要做的是写下一个进化稳定性的正式定义。

[段 34]

And it’s a definition that comes out of biology, and it’s a little bit of a, you’re going to see, it’s a little bit of a mouthful, this definition. So for those people who are a little bit sleepy, you need to wake up a little bit, because this is going to be a little bit of a difficult definition. But this is the definition that came out of biology. This is a formal definition. it comes out of biology, and in particular, it’s due to a guy called Maynard Smith, who wrote this in 1972. So obviously the earlier idea is due to Darwin, but this formal idea is due to Maynard Smith. Alright, so here’s our formal definition. So in a symmetric two game so everything we been looking at so far the pure strategy, the pure strategy S, let’s call it S-hat, to give it a name, S-hat, is evolutionarily stable, and again I’ll just use ES for that, But let me be a little bit nerdy here. I’m going to say impure strategies. It’s evolutionarily stable in pure strategies. I’m putting this in because we’re going to come back on Monday and consider mixed strategies. So it’s evolutionarily stable in pure strategies if… Just leave a little bit of a space here. I’m going to need a little bit of space between these next two lines.

[译文 34]

这是一个来自生物学的定义，你们会看到，这个定义有点拗口。所以对于那些有点困的人，你们需要清醒一点，因为这将是一个有点难的定义。但这是来自生物学的定义。这是形式定义。它来自生物学，特别是，它归功于一个叫Maynard Smith的人，他在1972年写下了这个。所以更早的思想归功于达尔文，但这个形式思想归功于Maynard Smith。好的，那么这是我们的正式定义。在一个对称博弈中，我们到目前为止一直在看的一切，纯策略，纯策略S，让我们叫它S-hat，给它一个名字，S-hat，是进化稳定的，我再次用ES来表示。但让我在这里稍微啰嗦一点。我要说不纯策略。它在纯策略中是进化稳定的。我这样写是因为我们周一还要回来考虑混合策略。所以它在纯策略中是进化稳定的如果……在这里留一点空间。我需要在下面两行之间留一点空间。

[段 35]

Leave a little space here. So here’s our big mouthful. So for example, what we need is 1 minus epsilon of the payoff of playing S hat against S hat, plus epsilon of the payoff of playing S hat against S prime has to be strictly bigger than 1 minus epsilon of the payoff of playing S prime against S hat plus epsilon of the payoff of playing S prime against itself. this has to be true for all possible deviations S prime but we also need the extra line and for all mutation sizes epsilon less than some epsilon bar. Now, this is where I need to go back and just be a bit more careful. So I’m going to write in this extra line, there exists an epsilon bar. All right? That’s why I left that extra line there. So this looks like very nerdy things Let talk our way through it What it saying It saying that S hat is evolutionarily stable if against all possible mutations so all possible versions of Rahul, S prime, all right, the payoff of S hat against the subsequently mixed population is bigger than the payoff of S prime against the subsequently mixed population. Let’s just see why that’s so. So this on the left is the payoff of S hat against a population in which 1 minus epsilon of the population like it is playing S hat.

[译文 35]

在这里留一点空间。所以这是我们的拗口定义。那么例如，我们需要的是1减去epsilon乘以玩S-hat对S-hat的收益，加上epsilon乘以玩S-hat对S’的收益，必须严格大于1减去epsilon乘以玩S’对S-hat的收益，加上epsilon乘以玩S’对自己的收益。这必须对所有可能的偏离S’成立，但我们还需要额外的一行，而且对所有突变规模epsilon小于某个epsilon bar。现在，这是我需要回去更仔细对待的地方。所以我要写上这额外的一行，存在一个epsilon bar。好的？那就是为什么我留了那个额外的行。这看起来像是非常书呆子的东西。让我们来解释一下。它说，如果对抗所有可能的突变，即所有可能的Rahul版本，S’，好吧，S-hat的收益对抗随后混合的人群大于S’的收益对抗随后混合的人群。让我们看看为什么是这样。左边这个是S-hat对抗一个人群的收益，其中1减去epsilon的人像往常一样在玩S-hat。

[段 36]

And epsilon of the population is like Rahul. 1 minus epsilon of the time, it meets something like itself. And epsilon of the time, it meets a TA. Conversely, on the right-hand side, we have the payoff to the mutation. the payoff for the mutation is 1 minus epsilon of the time the TA meets a student and gets the payoff of S prime against S hat epsilon of the time the TA meets itself and gets a payoff of S prime against S prime and we have this inequality we had before it says the mutation, the invader has to do worse, so it dies out alright so the nasty bit of this definition the part that a little bit if it takes a while to get your head around is this qualifier about the size of mutations And all it saying and we ignore the math of it all it saying is this better be true for all small mutations. You can think of that as saying, this has to be true for all small mutations. So this definition from biology is exactly, it exactly mimics the argument we went through several times now, both using the class and using the board and figuring out average payoffs. This is an ugly definition. Everyone agree this is kind of ugly? So now what I’m going to do is I’m going to give you an entirely different definition.

[译文 36]

有epsilon的人是像Rahul这样的人。1减去epsilon的时间，它遇到像它自己的东西。epsilon的时间，它遇到一个助教。反过来，在右边，我们有突变的收益。突变的收益是1减去epsilon的时间助教遇到一个学生，得到S’对S-hat的收益。epsilon的时间助教遇到自己，得到S’对S’的收益。我们有这个不等式，就像之前一样，它说突变，入侵者必须做得更差，所以它会消亡。好的，所以这个定义的棘手之处，有点需要一点时间才能理解的部分，是关于突变规模的限定符。它所说的，以及我们忽略所有数学部分，就是这对所有小突变都必须成立。你可以这样理解，这必须对所有小突变成立。所以这个来自生物学的定义正是，它精确地模仿了我们多次进行的论证，无论是用课堂还是用黑板，计算平均收益。这是一个丑陋的定义。大家都同意这有点丑陋吗？那么现在我要做的是给 你们一个完全不同的定义。

[段 37]

It’s also going to be ugly, but a little bit less ugly. So think of that as definition one, and it came from biology. It came from this paper in 1972. I think it was in Nature by Maynard Smith and co-authors. And think of this other definition that’s coming out of economics. So definition two. A strategy S hat oh I should have the same qualifier in a symmetric two game okay same thing to start off with a strategy S hat is ES in pure strategies so same as we had before, if two things. thing number one, let’s call it A, if S hat S hat is a Nash equilibrium, is a symmetric, obviously, is a symmetric Nash equilibrium of the game. We’re not done yet. Let me just write down what it means to be a symmetric Nash equilibrium of the game. What does that mean? that means i.e. the payoff of S hat against S hat must be at least as big as the payoff of S prime against S hat for all S prime. That’s a standard thing we see many, many times. It says S hat is the best response against S hat. Not quite done yet. All right. That’s a standard thing we see many, many times. It says f hat is a best response against f hat. All right? Not quite done yet.

[译文 37]

这也将是丑陋的，但会稍微不那么丑陋。所以把这个当作第一个定义，它来自生物学。它来自1972年的这篇论文，我想是发表在《自然》杂志上的，由Maynard Smith及其合著者撰写。现在想想另一个来自经济学的定义。第二个定义。一个策略S hat，我应该在对称二人博弈中加上相同的限定符，好，和之前一样，两件事。第一件事，我们称之为A，如果S hat S hat是一个Nash均衡，显然，是该博弈的一个对称Nash均衡。我们还没完。让我写下成为对称Nash均衡意味着什么。这意味着什么？也就是说，S hat对抗S hat的收益必须至少与S prime对抗S hat的收益一样大，这是我们多次看到的标准表述。它表明S hat是对抗S hat的最佳反应。还没完全结束。好吧，这是我们多次看到的标准表述。它表明f hat是对抗f hat的最佳反应。好吗？还没完全结束。

[段 38]

And b, what else do we need? We need if this weakening equality I wrote above is actually an equality, if the payoff of S hat against itself is actually equal to the payoff of S prime against S hat, then the payoff of S hat against S prime must be bigger than the payoff of S prime against itself. This is still a bit of a mouthful, but I claim it’s going to end up being a little simpler to think about. So it says again, it says S hat is evolutionally stable if it’s a Nash equilibrium, right, that’s basically this, and if it only a weak Nash equilibrium if it only a weak Nash equilibrium if there a tie then it better beat up on the mutant It better beat up on the mutant when it beats the mutant. All right? So, I’m going to try and give you an intuition as to why this is true in a minute. But first, I want to tell you why you should care about this. So, without getting too religious, I want to tell you why I think this is an exciting result? Why am I dragging you through more algebra than usual? So I think there are two reasons why this is an important result. The first is, it’s going to turn out that when we analyze games, it’s very easy to check definition two.

[译文 38]

还有B，我们还需要什么？我们需要，如果上面写的这个弱化等式实际上是等式，如果S hat对抗自身的收益实际上等于S prime对抗S hat的收益，那么S hat对抗S prime的收益必须大于S prime对抗自身的收益。这仍然有点拗口，但我声称它最终会更容易思考。所以它再说一次，它表明S hat是进化稳定的，如果它是一个Nash均衡，对吧，这基本上就是这个，而如果它只是一个弱Nash均衡，如果有平局，那么它最好打败突变体。当它打败突变体时，它最好打败突变体。好吗？我要试着给你们一个关于为什么这是真的的直觉。但首先，我想告诉你们为什么你们应该关心这个。所以，不说得太宗教化，我想告诉你们为什么我认为这是一个令人兴奋的结果？为什么我要带你们经历比平时更多的代数？我认为这是重要结果有两个原因。第一个是，当我们分析博弈时，检查定义二将是非常容易的。

[段 39]

It’s going to turn out it’s very easy to check definition two. It’s really rather a pain to check definition one. Why? Why? Because you’ve got to keep track of these epsilons and so on. But the fact that definition two is equivalent to definition, did I say that? That’s the point. The fact that definition one and two are equivalent, so one is equivalent to two, the fact that these definitions are equivalent means we only have to check the second definition And that easy So when biologists are setting up experiments involving wasps or ants or lions or chimpanzees provided they can check this in the game they done And that turns out to be easy. And we’ll see that on Monday. That’s the sort of instrumental reason. Now I want to give you the religious reason why you should care about this. Let me just try and… I often find that this appeals to like a third of the students and the other two-thirds of the students think I’m completely bonkers at this point. So that’s fine. You can think I’m bonkers. I’m fine. I want to guess for the third of you who are nerdy like me, I want you to sort of see the appeal of this. Here we have two ideas. One idea arises out of biology. I say it arose from Mainland Smith, but it really comes from Darwin.

[译文 39]

检查定义二将是非常容易的。而检查定义一确实非常痛苦。为什么？为什么？因为你必须跟踪这些epsilon等等。但定义二等价于定义一这个事实，我说了吗？这就是重点。这些定义是等价的，所以一等价于二，这些定义是等价的这个事实意味着我们只需要检查第二个定义，而这很容易。所以当生物学家设置涉及黄蜂、蚂蚁、狮子或黑猩猩的实验时，只要他们能在博弈中检查这一点，而事实证明这很容易。我们会在周一看到这一点。这是工具性的理由。现在我想给你们宗教性的理由，为什么你们应该关心这个。让我试着……我经常发现这吸引大约三分之一的学生，而另外三分之二的学生此时认为我完全疯了。没关系。你们可以认为我疯了。我没关系。我想为你们中像我一样怪异的的三分之一的人猜测，我想让你们看看这其中的吸引力。这里有两个想法。一个想法来自生物学。我说它来自Maynard Smith，但实际上它来自Darwin。

[段 40]

So it’s a 19th century idea in biology. It’s probably the most important idea in biology in the 19th century. It may be the most important idea in biology for 200 years. Is that a fair judgment? the notion of evolutionary stability. The other idea comes out of economics. It’s looking at Nash Equilibria and strict Nash Equilibria and so on. Up there we’ve got biology, down here we have economics. This is an idea that emerged in economics in the 20th century in the 1950s So roughly 80 years after Darwin This is a big idea in economics This is a big idea in biology So what appeals to me, because I’m kind of a nerdy kind of guy, is I think it’s kind of wonderful that those two ideas are almost the same. Right? I think it’s kind of, it’s a beautiful thing that those two ideas are kind of the same thing. And now you are looking at me and thinking I’m bonkers, okay? So let me push it harder. All right? So think back to the, think about the great intellectual coincidences of earlier periods. Think about the 17th century. In the 17th century, people figured out that the mechanical laws that governed the rotation of the planets were at least approximately the same as those that governed a clock. The basic laws of Newtonian physics were the same.

[译文 40]

所以这是生物学中一个19世纪的思想。它可能是19世纪生物学中最重要的思想。它可能是200年来生物学中最重要的思想。这是一个公平的判断吗？进化稳定性的概念。另一个想法来自经济学。它关注的是Nash均衡、严格Nash均衡等等。在上面我们有生物学，在下面我们有经济学。这是一个在20世纪50年代出现在经济学中的思想。所以大约在Darwin之后80年。这是经济学中的一个伟大思想。这是生物学中的一个伟大思想。吸引我的是什么，因为我是个有点怪异的人，我认为这两个想法几乎是相同的，这很美妙，对吧？我认为，这两个想法是同一种东西，这是一件美好的事情。现在你们看着我，认为我疯了，好吗？让我更用力地推它。好吗？回想一下，思考一下早期历史上伟大的知识巧合。思考17世纪。在17世纪，人们发现控制行星旋转的力学定律至少大致与控制时钟的定律相同。牛顿物理学的基本定律是相同的。

[段 41]

And they thought this was a wonderful thing. They had these completely different areas of intellectual pursuit, and they turned out to be the same. The scientific revolution said they were the same. And for those people who chose to continue believing in God at that point, they thought of God now as being what? as being a heavenly clockmaker. Right? It was a wonderful moment. So here, in our own time, or perhaps just before it, we see a similar thing. Right? Here we see, you know, one of the most important ideas in science coinciding with one of the most important ideas in economics. Right? And where does this lead us? Well, now, I guess, I mean, the bad news is this big… important ideas in science coinciding with one of the most important ideas in economics. And where does this lead us? Well, now I guess the bad news is this may make a lot of people doubt the existence of God, I guess. That’s going to get me in trouble. I didn’t say that. But at least if you do go on believing in God, you’re going to have to believe God’s an economist. That’s a pretty good thing. All right? All right? So that’s my little religious piece on this. All right, now I want to spend the last ten minutes having told you this wonderful thing, trying to convince you that it’s true.

[译文 41]

他们认为这是一件美妙的事情。他们有这些完全不同的知识追求领域，结果发现它们是相同的。科学革命说它们是相同的。对于那些选择继续相信上帝的人来说，他们此时认为上帝是什么？是天堂的钟表匠，对吧？那是一个美妙的时刻。所以在这里，在我们自己的时代，或者也许就在它之前，我们看到了类似的事情，对吧？在这里我们看到，科学的最重要的思想之一与经济学中最重要的思想之一重合，对吧？这把我们引向何方？好吧，现在我想说的是，坏消息是这可能让很多人怀疑上帝的存在，我想。这会让我惹上麻烦。我没说过那个。但至少如果你确实继续相信上帝，你将不得不相信上帝是一个经济学家。这是一件相当好的事情。好吗？好吧，所以这是我对这件事的一点宗教性的小论述。好吧，现在我想用最后十分钟告诉你们这个美妙的事情，试着说服你们这是真的。

[段 42]

And again, there’s a proof, there’s a proof on the handout that you can read on the web. I just want to give you an idea of why it’s true. Okay, so here we go. I need some space. Well, okay. All right, so what I want to convince you of is that this definition, this kind of game theory definition, this econ definition, implies the one above. The one above we’ve kind of already argued several times today corresponds to the notion of evolutionary stability. We talked about it several times. we wrote down these kind of equations several times. So I want to convince you that this one implies that one. Alright? Okay. So what do I have to convince you? So let’s imagine let fix there no way of getting me to see that one without Never mind. That was a bad idea. Never mind. Okay. Let’s go back again. The embarrassment of playing with heavy boards here. All right, let’s try and shoot this one up. if you ever think of asking your dean about some new technology for Yale we might think about having some slightly lighter blackboards just for the evolutionary stability of the average weakling economist alright alright so what we’re going to do is let’s fix a strategy S hat and suppose because S hat S hat is in fact Nash. So that’s what I want to convince you of is that S hat is going to be evolutionally stable.

[译文 42]

而且，还有一个证明，在讲义上有一个证明，你们可以在网上阅读。我只想给你们一个关于为什么这是真的的想法。好，我们开始吧。我需要一些空间。嗯，好吧。好吧，我想让你们相信的是，这个定义，这种博弈论定义，这种经济学定义，蕴含上面的那个。上面的那个我们已经多次论证它对应于进化稳定性的概念。我们已经讨论过它好几次了。我们多次写下这些方程。所以我想让你们相信这个蕴含那个。好吗？好吧，我需要说服你们什么？让我们想象一下让我固定一个策略S hat并假设因为S hat S hat实际上是Nash。所以我想让你们相信的是S hat将是进化稳定的。

[段 43]

So I claim there two possibilities There are two cases The two cases are either it the case that S hat against S hat does strictly better than S hat against S prime for all S prime So let’s just be careful. since we know it’s Nash, we already know that the payoff of S hat against itself is at least weakly better than any possible deviation. That’s what it means to be Nash, right? So everyone agree with that? We know that S hat is the best response to S hat, so it must be weakly better than any possible deviation. All right, so let’s take the first case where actually it’s strictly better. And let’s go back to our classroom example and suppose that the, let’s go back to the first definition by going back to our metaphor of the class. So you guys are all ants, and suppose that every student in the class is playing S-hat. You’re all playing S-hat, and suppose it’s the case that in fact the payoff of S-hat against S-hat is bigger than the payoff of S-hat against S-prime, and suppose that a mutation arises, sorry Rahul, here we go again, suppose a mutation arises, here’s Rahul, our mutation and he going to be randomly matched against one of you guys So let compare Rahul payoff against this gentleman payoff Your name is Pat.

[译文 43]

所以我断言有两种可能性。有两种情况。两种情况中的一种是，S hat 对阵 S hat 的表现严格优于 S hat 对阵 S prime 的情况，对所有 S prime 都是如此。让我们仔细一点。既然我们知道这是纳什均衡，我们已经知道 S hat 对阵自身的收益至少弱优于任何可能的偏离。这就是成为纳什均衡的含义，对吧？大家都同意吧？我们知道 S hat 是对 S hat 的最佳反应，所以它一定弱优于任何可能的偏离。好吧，让我们看第一个情况，即实际上它是严格更优的。让我们回到课堂的例子，假设我们回到第一个定义，回到我们班级的比喻。所以你们都是蚂蚁，假设班级里每个学生都在玩 S-hat。你们都在玩 S-hat，假设实际上 S-hat 对阵 S-hat 的收益大于 S-hat 对阵 S-prime 的收益，假设出现了一个变异，对不起 Rahul，又来了，假设出现了一个变异，这是 Rahul，我们的变异者，他将随机匹配对阵你们中的一个人。那么让我们比较 Rahul 的收益与这位先生的收益。你叫 Pat。

[段 44]

So Pat is our typical S-hat incumbent ant. Doesn’t look like an ant, but never mind. Stretch your imagination a bit. And most of the time, what’s happening? Most of the time is Pat is being matched against one of the rest of you. And when he’s matched against one of the rest of you, he’s getting the payoff of S-hat against S-hat, which is U-S-hat, S-hat. All right? That’s what he’s getting most of the time. All right? And every now and then, every now and then, he’s meeting a mutant. Okay, fine. Okay, fine. So every now and then, he’s getting a slightly different payoff. But most of the time, he’s getting a payoff which is U-S-hat, S-hat. All right? What about Rahul? So Rahul, okay, every now and then, he’s going to be lucky and meet another TA, but most of the time, almost all the time, he’s being matched against one of you. For example, he’s matched against Pat. And when he’s matched against Pat, his payoff is US hat S prime, which is lower. So Pat’s payoff almost all the time is US hat S hat. And Rahul’s payoff almost all the time is US hat S hat. So Pat’s payoff almost all the time is U S hat S hat, and Rahul’s payoff almost all the time is U S hat S prime. But our assumption is U S hat S prime is lower, so Rahul’s going to die out.

[译文 44]

所以 Pat 是我们典型的 S-hat 在位者蚂蚁。看起来不像蚂蚁，但没关系。稍微发挥一下你的想象力。大部分时间会发生什么？大部分时间是 Pat 与你们中的另一个人匹配对阵。当他与你们中的另一个人匹配对阵时，他获得的收益是 S-hat 对阵 S-hat，即 U-S-hat, S-hat。对吧？这就是他大部分时间获得的收益。对吧？偶尔，偶尔，他会遇到变异者。好的，好吧。好的，好吧。所以偶尔，他会获得稍微不同的收益。但大部分时间，他获得的收益是 U-S-hat, S-hat。对吧？Rahul 呢？Rahul，好吧，偶尔他会幸运地遇到另一个 TA，但大部分时间，几乎所有时间，他都与你们中的一个人匹配对阵。例如，他与 Pat 匹配对阵。当他与 Pat 匹配对阵时，他的收益是 US hat S prime，这较低。所以 Pat 的收益几乎所有时间都是 US hat S hat。而 Rahul 的收益几乎所有时间都是 US hat S hat。所以 Pat 的收益几乎所有时间是 U S hat S hat，而 Rahul 的收益几乎所有时间是 U S hat S prime。但我们的假设是 U S hat S prime 较低，所以 Rahul 将会灭绝。

[段 45]

Okay? Okay, so in this case, Rahul dies out because most of the time he’s playing an incumbent and just doing horribly. Okay? Sorry, Rahul. Okay? Okay? Everyone convinced by that? So in this case, the mutant dies out. So the mutant dies out because she or he meets S-hat often. The mutant meets the incumbent often and just does horribly against the incumbent. That’s the first case. That’s case one. And case two is, it could be the case that in fact Rahul does pretty well against the incumbents but unfortunately for Rahul it could be the case but in that case according to definition B Rahul doesn do so well against other TAs All right, so this is case two. Case one was Rahul gets beaten up whenever he meets one of you. Sorry, Rahul, getting beaten up a lot here, but Rahul gets beaten up when he meets one of you. And here’s the case where Rahul is doing fine against you, but he does horribly against incumbents. So that’s the harder case. Let’s talk through that case. So back up again. So Rahul, sorry, again, you’re up again. All right, so once again, you guys are all playing S-hats. there’s a mutation of Rahul’s playing S prime, all right? And let’s get Pat up again, all right?

[译文 45]

好吗？好的，所以在这个情况下，Rahul 灭绝了，因为大部分时间他都在对付一个在位者，表现得很糟糕。好吗？对不起，Rahul。好吗？好吗？大家都相信这个结论吗？在这个情况下，变异者灭绝了。所以变异者灭绝了，因为他或她经常遇到 S-hat。变异者经常遇到在位者，而且对付在位者时表现得很糟糕。这是第一种情况。这是情况一。情况二是，实际上 Rahul 对付在位者可能表现不错，但不幸的是，根据定义 B，Rahul 对付其他 TA 时表现不太好。好吧，这就是情况二。情况一是 Rahul 每次遇到你们中的一个人时都被打败。对不起，Rahul，在这里被打得很惨，但是 Rahul 遇到你们中的一个人时被击败。而这里的情况是，Rahul 对付你们时表现不错，但他对付在位者时表现得很糟糕。所以这是更难的情况。让我们讨论一下这种情况。再回顾一下。Rahul，对不起，再次，你又上场了。好吧，再一次，你们都在玩 S-hats。有一个 Rahul 的变异在玩 S prime，好吧？让我们再请 Pat 上场，好吧？

[段 46]

So when Pat meets one of you guys, so let’s pick out one of you guys, so let me pick up, sorry your name is Christine So when Pat meets Christine which is most of the time he does okay and when Rahul meets Christine which happens most of the time he does exactly the same So if you just compared them in their random matches against Christine or the rest of you they the same But every now and then, thanks Christine, every now and then, every epsilon of the time, rarely, Rahul is going to meet Jake, I don’t know if Jake’s writing, he’s going to meet Kai, alright alright alright and on those occasions on those occasions let’s see what happens so when Pat when Pat meets Kai he does pretty pretty well and he gets some big bonanza he gets a much bigger payoff than when Rahul meets Kai they just do horribly against each other alright and the fact that they do horribly against each other causes them to die out alright let’s talk this through there were two ways in which Rahul could die out one is he dies out because he does horribly against you and the second way he can die out is if he does equally well against you as you do against yourself but he does horribly when he meets the other TAs in either case you guys are evolutionarily stable and Rahul in trouble alright was that convincing enough in our group there a formal proof thanks guys there a formal proof in the handout but this will be enough to get us started Alright?

[译文 46]

所以当 Pat 遇到你们中的一个家伙时，让我们选出你们中的一个，对不起，你叫 Christine 所以当 Pat 遇到 Christine 时，这是大部分时间，他表现还不错，当 Rahul 遇到 Christine 时，这也是大部分时间发生的事，他表现完全一样。所以如果你只是比较他们在随机匹配对阵 Christine 或你们其他人时，他们是一样的。但偶尔，thanks Christine，偶尔，偶尔，每 epsilon 的时间，很少，Rahul 会遇到 Jake，我不知道 Jake 是否在写东西，他会遇到 Kai，好的好的好的，在那些场合，在那些场合让我们看看会发生什么，所以当 Pat 当 Pat 遇到 Kai 时，他表现非常好，他得到了一些大收获，他获得的收益比 Rahul 遇到 Kai 时大得多，他们只是互相表现得很糟糕，好吧，他们互相表现得很糟糕这一事实导致他们灭绝，好吧让我们详细讨论一下，Rahul 可能灭绝有两种方式，一种是他因为对付你们时表现得很糟糕而灭绝，第二种方式是他可能灭绝的方式是，如果他对你们的表现和你们对自己一样好，但当他遇到其他 TA 时表现得很糟糕，在这两种情况下，你们都是进化稳定的，Rahul 遇到麻烦了，好吧，在座的各位，这足够令人信服了吗，有一个正式证明，谢谢各位，handout 中有一个正式证明，但这足够让我们开始了，好吧？

[段 47]

So in this case, in this case, the mutant does okay against S, sorry, against S hat, but gets clobbered does badly, let’s say, against S prime. So mutants that are going to die out are those that do badly against incumbents or do badly against themselves. In either case, S hat will be evolutionarily stable. Now next time, I want to take this further in two directions. I want to see what happens about evolutionary stability in more complicated games, for example in cooperation games, and we’re going to see that evolution doesn’t do great in cooperation games. And we’re also, if we have time, going to look at sexual reproduction, which is probably what you guys are all interested in anyway. Alright, I’ll see you all on Monday.

[译文 47]

所以在这个情况下，在这个情况下，变异者对付 S，对不起，对付 S hat 时表现不错，但被痛击，表现糟糕，假设，对付 S prime 时。所以将要灭绝的变异者是那些对付在位者表现糟糕或对付自己表现糟糕的。在任何一种情况下，S hat 都将是进化稳定的。现在下次，我想从两个方向进一步探讨。我想看看在更复杂的游戏中进化稳定性会发生什么，例如在合作游戏中，我们会看到进化在合作游戏中表现不佳。如果时间允许的话，我们也将看看有性繁殖，这可能是你们所有人都感兴趣的。好吧，周一见。