耶鲁大学博弈论公开课 - 第12集 进化稳定-社会公约，侵略，和周期

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]

All right, so this was the definition we saw at the end last time. I tried to write it a bit larger. I just repeat the second of those definitions. This is the definition that connects the notion of evolutionary stability, for now in pure strategies with Nash equilibrium. And basically it says this, to check whether a strategy is evolutionally stable in pure strategies, first check whether S hat, S hat is a symmetric Nash equilibrium. And if it is, if it’s a strict Nash equilibrium, we’re done. And if it’s not strict, that means there’s at least some other strategy that would tie with S hat against S hat. then compare how s-hat does against this mutation with how the mutation does against itself. And if s-hat does better than the mutation than the mutation does against itself, then we’re okay. And one virtue of this definition is it very easy to check so let try an example to see that And also to get us back into gear and reminding ourselves what we’re doing a bit. All right, so in this example, it’s a sort of trivial game, but still, the game looks like this. All right, and we’re asked, since we’re asked the question, what is evolutionarily stable in this game? So no prizes for finding the symmetric Nash equilibrium in this game. Shout it out.

[译文 1]

好的，这是我们在上次课结束时看到的定义。我试着把它写大一点。我只是重复第二个定义。这是将纯策略的进化稳定性概念与纳什均衡联系起来的定义。基本上，它说的是这样，要检查一个策略在纯策略中是否是进化稳定的，首先检查 S hat，S hat 是否是一个对称纳什均衡。如果它是，如果它是一个严格的纳什均衡，我们就完成了。如果它不是严格的，这意味着至少有一些其他策略会与 S hat 对阵时打成平手。然后比较 s-hat 对这个变异策略的表现与变异策略对自身表现。如果 s-hat 比变异策略表现得更好，而变异策略对自身表现更差，那我们就没问题了。这个定义的一个优点是它非常容易检查，所以让我试着举一个例子来说明这一点，也让我们重新进入状态，提醒自己我们在做什么。好的，在这个例子中，这是一个有点 trivial 的游戏，但还是，游戏是这样的。好吧，既然我们被问到这个问题，在这个游戏中什么是进化稳定的？找出这个游戏中的对称纳什均衡并不难，大声说出来吧。

[段 2]

What’s the symmetric Nash equilibrium in this game? AA. Okay, so AA is a symmetric Nash equilibrium. That’s easy to check. So really the only candidates for an evolutionally stable strategy here is A. All right So the second thing you would check is is AA a strict Nash equilibrium So what does strict Nash equilibrium mean It means that if you deviate you do strictly worse All right So is AA strict Nash Well is it strict Nash or not It not right If you deviate to B you notice very quickly that UAA is equal to UBA Is that right It just a tie Both of these get one. So it’s not a strict Nash equilibrium. So we have to check our third rule. What’s our third rule, so we need to check we need to check how does A do against the possible deviation which is here, which is B here how does that compare with B against itself alright, so UAB the payoff to A against B is 1 and the payoff to B against itself is 0 1 is indeed bigger than 0 so we’re okay and A is in fact evolutionarily stable. Alright, so just a very, very simple example to illustrate how quick it is to check this idea. Alright, I want to spend all of today going over more interesting examples of having the payoff for having invested in this last time And to start off with let get rid of this rather trivial example I want to think about evolution as it’s often applied in the social sciences.

[译文 2]

这个游戏中的对称纳什均衡是什么？AA。好的，所以 AA 是一个对称纳什均衡。这很容易检查。所以实际上这里进化稳定策略的唯一候选者是 A。好的，你要检查的第二件事是 AA 是否是一个严格的纳什均衡。严格的纳什均衡是什么意思？它的意思是，如果你偏离，你就会做得更差。好的，那么 AA 是严格的纳什均衡吗？它是严格纳什还是不是？它不是严格的。如果你偏离到 B，你会很快注意到 UAA 等于 UBA。对吗？打成平手。两个都得到 1。所以它不是一个严格的纳什均衡。所以我们必须检查我们的第三条规则。我们的第三条规则是什么，所以我们需要检查我们需要检查 A 对可能的偏离策略（在这里是 B）的表现如何，这与 B 对自身表现相比如何。好的，所以 UAB，A 对 B 的收益是 1，而 B 对自身的收益是 0，1 确实大于 0，所以没问题，A 实际上是进化稳定的。好的，这只是一个非常非常简单的例子来说明检查这个概念有多快。好的，我想今天剩下的时间用来复习更有趣的例子，关于上次投入获得收益的例子。首先让我们摆脱这个相当 trivial 的例子，我想思考一下进化，因为它在社会科学中经常被应用。

[段 3]

So one thing you might talk about in the social sciences is the evolution of a social convention. Alright, sometimes you’ll read a lot in sociology or political science about the evolution of institutions or social conventions and things like this, maybe also in anthropology, and I want to see a trivial example of this just to see how this might work and see if we can learn anything. And the trivial example I’m going to think about is driving on the left or the right, on the left or right side of the road. Alright, so this is a very simple social convention, I think we all can agree, this is a social social convention. And let’s have a look at the payoffs in this little game. So you could imagine people drive on the left or drive on the right. Okay? And if you drive on the left and the other people are driving on the right, you don’t do great and nor do they. And if you drive on the right and they’re driving on the left, you don’t do great, and nor do they. All right? And if you both drive on the right, you do fine. But if you both drive on the left, you do a little better because you look a little bit more sophisticated. All right? Okay, so this is our little game.

[译文 3]

所以在社会科学中你可能会谈论社会惯例的进化。好的，有时候你会在社会学或政治学中读到很多关于制度或社会惯例的进化以及类似的东西，也许还有在人类学中，我想看看这方面的一个 trivial 例子，只是为了看看这可能会如何运作，以及我们是否能学到什么。我要考虑的这个 trivial 例子是靠左还是靠右行驶，在道路的左侧还是右侧。好的，这是一个非常简单的社会惯例，我想我们都能同意，这是一个社会惯例。让我们看看这个小游戏中的收益。你可以想象人们靠左行驶或靠右行驶。好的？如果你靠左行驶，而其他人靠右行驶，你做得不太好，他们也是。如果你靠右行驶，而他们靠左行驶，你做得不太好，他们也是。好的？如果你们都靠右行驶，你们做得很好。但是如果你们都靠左行驶，你们会做得更好一点，因为你们看起来更有品味一点。好的？好的，这是我们的小游戏。

[段 4]

And we can see this could be an evolutionary game, right? So you could imagine this emerging, you could imagine a government coming in and imposing a law saying everyone has to drive on the left or everyone has to drive on the right, but you could also imagine at the beginning of roads in different societies in different parts of the world, people just started doing something and then they settled down to one convention or another. And you can see how evolutionary stability is going to play a role here. Well, perhaps we should just work it through and see what happens. All right, so what are the potential evolutionarily stable things here? What are the potentially evolutionarily stable things? Let’s get some mics up. What liable to be evolutionarily stable in this setting Anyone Yeah there a I in here Where my other mic on that side Left left and right right are both candidates Good so our obvious candidates here are left left, and right, right. These are the two candidates, all right, or more formally, left is a candidate and right is a candidate, but you’re right to say that left, left, and right, right are both Nash equilibria in this game. Is that right? And much more, not only are they Nash equilibria, but what kind of Nash equilibria are they? They’re strict Nash equilibria.

[译文 4]

我们可以看到这可能是一个进化游戏，对吧？所以你可以想象这会自然出现，你可以想象政府介入并颁布法律说每个人都必须靠左行驶或每个人都必须靠右行驶，但你也可以想象在道路的早期，在世界的不同地区，人们只是开始做某件事，然后他们安定下来形成一种惯例或另一种。你可以看到进化稳定性将如何发挥作用。好吧，也许我们应该只是把它推演一遍，看看会发生什么。好的，这里的潜在进化稳定的东西是什么？什么是潜在进化稳定的东西？让我们把麦克风传起来。这在这种设置中很可能是进化稳定的，任何人？是的，这里有一个？在那边我还有另一个麦克风。左左和右右都是候选者。好的，所以我们的明显候选者是左左，和右右。这两个候选者，好的，或者更正式地说，左是一个候选者，右是一个候选者，但你说得对，左左和右右都是这个游戏中的纳什均衡。对吗？而且更重要的是，它们不仅仅是纳什均衡，它们是什么样的纳什均衡？它们是严格的纳什均衡。

[段 5]

In fact, they’re strict. Both are strict. So indeed, left is evolutionarily stable and right is evolutionarily stable. All right, let’s just talk through the intuition for that and make it kind of clear. So suppose you’re in a society in which everybody drives on the left. So this is England. And suppose a mutation occurs, and the mutation you could think of as an American tourist. The American tourist is dropped into English society, hasn’t read the guidebook carefully, starts driving on the right, and what happens to the tourists? They die out pretty rapidly And conversely if you drop an unobservant Brit into America and they drive on the right they going to get squashed pretty quickly So it kind of clear why everyone driving on the left or why everyone driving on the right is, each of these are perfectly good evolutionary stable social conventions. All right, but despite that, despite that simplicity, there’s kind of a useful lesson here. So the first lesson here is you can have multiple evolutionarily stable settings. You can have multiple social conventions that are evolutionarily stable. We shouldn’t be surprised particularly if we think there’s some evolutionary type force, some sort of random dynamic going on that’s generating social conventions that are then stable. We should not be surprised to see different social conventions in different parts of the world.

[译文 5]

事实上，它们是严格的。两者都是严格的。所以确实，左是进化稳定的，右也是进化稳定的。好的，让我们只是谈谈这背后的直觉，让它变得清晰。所以假设你在一个每个人都靠左行驶的社会中。这是英国。假设发生了一个变异，你可以把它想象成一个美国游客。这个美国游客被投放到英国社会中，没有仔细阅读旅游指南，开始靠右行驶，游客会发生什么？它们会相当迅速地灭绝。相反，如果你把一个不细心的英国人放到美国，他们靠右行驶，他们很快就会被压扁。所以很清楚为什么每个人都靠左行驶或者为什么每个人都靠右行驶，这两者都是完全好的进化稳定的社会惯例。好的，但是尽管如此，尽管这很简单，这里有一个有用的教训。所以第一个教训是，你可以有多个进化稳定的设置。你可以有多个进化稳定的社会惯例。如果我们有某种进化类型的力量，某种随机动态正在产生然后稳定下来的社会惯例，我们不应该特别惊讶。在世界的不同地区看到不同的社会惯例，我们不应该感到惊讶。

[段 6]

And in fact, we do. We do see parts of the world, like England and Japan and Australia, where they drive on the left, and we see parts of the world, like France and America, where they drive on the right. All right so we can have multiple evolutionary stable conventions All right And there another lesson here which is you could imagine a society settling down to a social convention down here. You could imagine ending up at a social convention of right-right. What do we know about the social convention of everyone driving on the right? We know it’s worse than the social convention of everyone driving on the left. at least in my version so what do we regard, what we see here we can see that they’re not necessarily efficient these need not be equally good so it’s hard to resist saying that American society’s driving habits if we think about the alternatives to evolution are a good example of unintelligent design. All right? All right? All right? So when you’re talking about this in a more, a less formal way in your anthropology or political science or sociology classes, you want to have in the back of your mind what we mean by evolutionary stability and that it doesn’t necessarily mean that we’re going to arrive at efficiency. All right? or political science or sociology classes, you want to have in the back of your mind what we mean by evolutionary stability and that it doesn’t necessarily mean that we’re going to arrive at efficiency.

[译文 6]

事实上，我们确实看到了。我们确实看到世界上有些地方，比如英国、日本和澳大利亚，他们靠左行驶，我们也看到世界上有些地方，比如法国和美国，他们靠右行驶。好的，所以我们可以有多个进化稳定的惯例。好的，这里还有另一个教训，你可以想象一个社会安定下来，形成一种社会惯例在这里。你可以考虑最终到达右右的社会惯例。我们对这个所有人都靠右行驶的社会惯例了解什么？我们知道它比所有人都靠左行驶的社会惯例要差。至少在我的版本中是这样的，那么我们认为，我们在这里看到的是，它们不一定是有效率的，这些不一定是同样好的，所以我们很难抗拒说，美国社会的驾驶习惯，如果我们考虑进化的替代方案，是一个不佳设计的很好的例子。好的？好的？好的？当你用一种更不正式的方式在你的，人类学或政治学或社会学课堂上谈论这个时，你想要在心里记住我们所说的进化稳定性是什么意思，以及它不一定意味着我们会达到效率。好的？或者政治学或社会学课堂上，当你谈论这个时，你想要在心里记住我们所说的进化稳定性是什么意思，以及它不一定意味着我们会达到效率。

[段 7]

And in some sense, it is exactly analogous to what we discussed when we looked at games like this with rational players playing, if you think about coordination games. These are essentially coordination games. All right, that was a fairly straightforward example. To leave it up there, let me… there’s another board here somewhere, here we go. Let’s look at yet another example, slightly more interesting. Alright. So today I’m going to spend most of today just looking at examples and talking about them. Alright, so here’s another example of a game we might imagine. And once again, it’s a 2 by 2 game, nice and simple game. All right, and here the payoffs are as follows. So down the diagonal we have 0 0 0 0 And off the diagonal we have 2 1 and 1 2 All right So what is this game essentially It’s very, very similar to a game we’ve seen already in class. Anybody? This is essentially Battle of the Sexes, right? I’ve taken the Battle of the Sexes game, the dating game, I’ve just twiddled it around to make it symmetric. All right? So this is a symmetric version of Battle of the Sexes. or our dating game. All right? And you could think of this game also in the context of driving around. All right? So you could imagine that one version of this game, you sometimes hear this game referred to as chicken.

[译文 7]

从某种意义上说，这与我们之前讨论的理性玩家进行这类博弈的情况完全类似，如果你考虑一下协调博弈的话。这些本质上就是协调博弈。好的，这是一个相当简单的例子。让我把它留在那里，我……这里还有另一块白板，来，我们看另一个例子，稍微有趣一点。好，今天我大部分时间会用来举例子并讨论它们。好，这是我们可能想象的另一个博弈例子。再说一遍，这是一个2x2的博弈，非常简单。好的，这里的收益如下。对角线上是0 0 0 0，对角线外是2 1和1 2。这个博弈本质上是什么？这与我们课上已经见过的一个博弈非常相似。有人知道吗？这本质上是"性别之战"，对吧？我把"性别之战"博弈、约会博弈拿过来，只是调整了一下使其对称。明白吗？所以这是"性别之战"或我们约会博弈的对称版本。你也可以在驾驶场景中考虑这个博弈。你可能听说这个博弈有时被称为"胆小鬼博弈"。

[段 8]

All right? What’s the game chicken when cars are involved? Anybody? What’s the game chicken when cars are involved? Now, it’s probably a good thing if people don’t know this. Okay, no one knows this? Okay, somebody knows it. Can I get the mic way back there? Yeah, shout out. . Right, right. So you could imagine here we are on a road that perhaps isn big enough to have driving on the left and driving on the right These two cars face each other they drive towards each other both of them going in the middle of the road and the loser is the person who swerves first You could think of A as being the aggressive strategy of not swerving, and B as being the less aggressive, the more benevolent strategy, if you like, of swerving. So the best thing for you is for you to be aggressive and the other person to swerve, and then at least they remain alive, And conversely, if you’re benevolent and they swerve, you remain alive but they win. And unfortunately now, if you’re both aggressive, you get nothing. And the way we’ve written this game, if you’re both benevolent, you get nothing. You can imagine making some more negatives here. So this is a game that seems kind of important in nature, not just in teenage male behavior, but in animal behavior. since we’re starting to talk about aggression and non-aggression.

[译文 8]

好的？当涉及汽车时，“胆小鬼博弈"是什么？有人知道吗？当涉及汽车时，“胆小鬼博弈"是什么？现在，如果人们不知道这个可能是一件好事。好的，没人知道这个？好的，有人知道。我能让那边后面的人拿着麦克风吗？是的，大声说出来。是的，是的。你可以想象我们行驶在一条可能不够宽、无法同时容纳左侧和右侧行驶的道路上。这两辆车面对面，相向而行，两辆车都在路中间行驶，输的人是第一个转向的人。你可以把A理解为不转向的激进策略，B理解为不那么激进的、更仁慈的策略，如果你愿意的话，意思是转向。所以对你来说最好的情况是你激进而对方转向，这样至少他们还活着。相反，如果你仁慈而他们转向，你活着但他们赢了。不幸的是，如果你们都激进，你们什么都得不到。按照我们写这个博弈的方式，如果你们都仁慈，你们也什么都得不到。你可以想象在这里加一些更负面的东西。所以这是一个在自然界中似乎很重要的博弈，不仅仅是在青少年男性行为中，也在动物行为中，因为我们开始讨论攻击性和非攻击性了。

[段 9]

All right. Okay, so what’s evolutionarily stable in this game? Well, remember, our starting point is what? Our starting point is to look for symmetric Nash equilibria. Okay so are there any symmetric Nash equilibria in this game And if so what are they Anybody Everybody Wow There are some Nash equilibria in this game and if so what are they Anybody Well there are some Nash Equilibria in pure strategies there are some Nash Equilibria in this game for example AB is a Nash Equilibrium, and BA is a Nash Equilibrium, but unfortunately they’re not symmetric, right, they’re not symmetric. And so far, we’re focusing on games that are symmetric, right, there’s this random matching, there’s no asymmetry in the roles of the row and column player, although in the handout, not the handout, in the reading packet I made for you, they do also look at some asymmetric versions of games, but for now we’re just looking at symmetry. So neither AB nor BA will serve our purpose because they’re not symmetric Nash equilibria. In fact, in fact, there is no symmetric, there’s no symmetric So, it’s a very generic pure strategy Nash equilibrium in this game. So, it can’t be that if this was a species, if it was an animal that came in, So it can’t be, it can’t be that if this was a species, if this was an animal that came in, that had two possible strategies, aggression or passivity, it can’t be the case in this particular game that you end up with 100% aggression out there, 100% aggressive genes out there, or 100% unaggressive genes out there.

[译文 9]

好的。那么，在这个博弈中什么是进化稳定的？嗯，记住，我们的出发点是什么？我们的出发点是寻找对称的纳什均衡。好的，在这个博弈中有对称的纳什均衡吗？如果有，它们是什么？有人知道吗？哇，这个博弈中有一些纳什均衡，如果有的话，它们是什么？有人吗？嗯，在纯策略中有一些纳什均衡，例如AB是一个纳什均衡，BA也是一个纳什均衡，但不幸的是它们不是对称的，对吧，它们不是对称的。到目前为止，我们关注的是对称的博弈，对吧，这是随机匹配的，行玩家和列玩家的角色没有不对称，虽然在讲义中，不是讲义，是在给你们准备的阅读材料中，他们确实也看了一些博弈的不对称版本，但目前我们只看对称性。所以AB和BA都不能满足我们的目的，因为它们不是对称的纳什均衡。事实上，事实上，没有对称的，没有对称的。所以，这是这个博弈中一个非常通用的纯策略纳什均衡。所以，不可能是这样，如果这是一个物种，如果这是一种动物，有两种可能的策略，攻击性或被动性，在这种特定的博弈中，你不可能最终得到100%的攻击性，100%的攻击性基因，或者100%的非攻击性基因。

[段 10]

In either case, if you had 100% aggressive genes out there, then it would be doing very, very badly, and you’d get an invasion of passive genes. And if you had 100% passive genes out there, you’d get an invasion of aggressive genes. You can’t have a pure ESS, a pure evolutionally stable gene mix out there. All right. So what does that suggest? It suggests we should start looking at mixed strategies. there is a mixed strategy. There is a mixed, I’ll have the word symmetric first, there is a symmetric mixed strategy Nash equilibrium in the game And we could go through and work it out We all know now probably you been laboring through the homework assignments so you all know how to find a mixed strategy Nash equilibrium in this game. You have to set the mix to make the other player indifferent between her two strategies. But this is a game we’ve seen already. It’s essentially Battle of the Sexes. All right, so we probably remember from a week ago what that equilibrium mix is. Can anyone remember what the equilibrium mix is in battle of the sexes? It’s two-thirds, one-third. Turns out that two-thirds, one-third, two-thirds, one-third is a Nash equilibrium here. So if you go back to your notes a week ago, you’ll find something very much like that was a Nash equilibrium in the original version of Battle of the Sexes, a week ago it would have been two-thirds, one-third, one-third, two-thirds, because things weren’t symmetric.

[译文 10]

无论哪种情况，如果你有100%的攻击性基因，那么它会表现得很糟糕，你会遭到被动基因的入侵。如果你有100%的被动基因，你就会遭到攻击性基因的入侵。你不可能有一个纯的ESS，一个纯的进化稳定基因组合在那里。好的。那这意味着什么？意味着我们应该开始考虑混合策略。有一个混合策略。有一个混合的，我先说对称这个词，在这个博弈中有一个对称的混合策略纳什均衡。我们可以逐步计算出来。我们现在可能都知道，你们一直在努力完成作业，所以你们都知道如何在这个博弈中找到混合策略纳什均衡。你必须设置混合比例，使另一个玩家在她的两个策略之间保持无差异。但这已经是我们见过的博弈了。它本质上是"性别之战”。好的，所以我们可能记得一周前那个均衡混合是什么。有人记得"性别之战"中的均衡混合是什么吗？是三分之二，三分之一。事实证明，三分之二，三分之一，三分之二，三分之一在这里是一个纳什均衡。所以如果你回去看一周前的笔记，你会发现一个非常类似的东西，在"性别之战"的原始版本中是一个纳什均衡，一周前会是三分之二，三分之一，三分之一，三分之二，因为当时不对称。

[段 11]

Now I’ve made things symmetric. It’s just two-thirds, one-third for both players. You can check it at home. All right? So what’s this telling us? It’s telling us that there’s at least an equilibrium in this game in which two of the genes are aggressive and one of the genes are unaggressive All right But does that mean anything What could that mean In terms of biology what could that mean Well, so far we’ve been looking at evolutionary stable pure strategies. And evolutionary stable pure strategies correspond to evolutionary stable situations in nature which are what are called monomorphic. and monomorphic monomorphic means one shape or you only get one type out there but you can also have situations in nature where there are actually stable mixed types and they’re called polymorphic polymorphic It’s probably not hyphenated, it’s probably just one word. So you can have a monomorphic population. That’s what we’ve focused on so far. But you could also have a mixed population. All right? Now for this to be a mixed population we better change the definition accordingly All right We come back and talk about what it means in a second a bit more but first we better make sure we have an okay definition for it So what I going to do is I going to drag this definition down and do something you’re not meant to do usually in teaching.

[译文 11]

现在我已经把它对称化了。就是双方都是三分之二，三分之一。你可以在家验证。好的。这告诉我们什么？告诉我们至少在这个博弈中有一个均衡，其中三分之二的基因是攻击性的，三分之一的基因是非攻击性的。但这是什么意思？在生物学上这可能是什么意思？嗯，到目前为止我们一直在看进化稳定的纯策略。进化稳定的纯策略对应于自然界中进化稳定的情况，即所谓的单态性，单态性意味着一种形态，或者你只能得到一种类型在那里，但你也可以有自然界中实际存在稳定混合类型的情况，它们被称为多态性，多态性。可能没有连字符，应该就是一个词。所以你可以有一个单态性种群。这是我们到目前为止关注的。但你也可以有一个混合种群。现在，为了使其成为混合种群，我们最好相应地改变定义。我们回过头来再讨论一下这意味着什么，但现在我们最好确保有一个可以接受的定义。所以我要做的是把这个定义拖下来，做一些你通常在教学中不被允许做的事情。

[段 12]

I’m going to use the eraser to correct my definition. All right, so here I have my pure strategy definition, and let me just change it into a strategy, into a definition that will allow for these polymorphic populations. so I’m going to change this into a p hat I’m going to change this pure into mixed and everywhere you see an s hat I’m going to put a p hat and here too and everywhere you see an s prime I’m going to put a p prime and the reason I’m doing this this way is I want to emphasize that there’s nothing new here I’m just writing down the same definition as we had before except I’m now allowing for the idea allowing for the idea of populations being mixed and I’m also just note in passing I’m also allowing for the possibility that a mutation might be mixed I’m allowing for the idea of populations being mixed, and I’m also, just note in passing, I’m also allowing for the possibility that a mutation might be mixed. A mutation might be mixed. Did I catch them all? All right, I’ve just gone through the definition you have, and I’ve switched everything from pure to mixed. Let’s put it up there again. Okay. So in our example, does the mix 2 thirds, 1 third satisfy the definition above? Let’s go through carefully.

[译文 12]

我要用橡皮擦纠正我的定义。好的，这里我有我的纯策略定义，让我把它改成一个允许这些多态种群的定义。我要把它改成一个p hat。我要把这个"纯"改成"混合”，你看到的每个s hat我都改成p hat，这里也一样，你看到的每个s prime我都改成p prime。我这样做的原因是，我想强调这里没有什么新东西。我只是写下我们之前有过的同样的定义，只不过我现在允许种群是混合的想法。我也顺便提一下，我也允许突变可能是混合的可能性。突变可能是混合的。我都改过来了吗？好的，我只是把你们的定义过了一遍，把所有从纯改成混合。让我们再把它放上去。好的。那么在我们的例子中，三分之二，三分之一的混合是否满足上面的定义？让我们仔细检查一下。

[段 13]

So 2 thirds, 1 third is a Nash equilibrium. So we’ve satisfied part A. It’s a symmetric Nash equilibrium. So we’re okay there. Is this equilibrium a strict equilibrium? Is this mixed population, 2 thirds aggressive and 1 third unaggressive, or this mixed strategy, is it a strict equilibrium? How do deviations do against it? Anybody Equally well Yeah go ahead Equally well Equally well equally well thank you So right so it can be it can be a strict Nash equilibrium because if we deviated to A, we’d do as well as we were doing in the mix, or if we deviated to B, we’d do as well as we were doing in the mix. What we’re saying is an A mutation does exactly as well against this mix as the mix does against itself, and a B mutation does exactly as well. In fact, that’s how we constructed the equilibrium in the first place, right? We chose a P that made you indifferent between A and B. All right? So in a mixed dash equilibrium, it can’t be strict. Cannot be strict since it is mixed. In a mixed dash equilibrium, a genuinely mixed dash equilibrium, By definition, you’re indifferent between the strategies in the mix. All right? So to show that this is, in fact, evolutionarily stable, we’d have to show rule B. So we need to show, we need to check, give us some room here, we need to check how the payoff of this strategy let call it p how p does against all possible deviations and compare that with how those deviations do against themselves.

[译文 13]

所以三分之二、三分之一是一个纳什均衡。所以我们满足了A部分。这是一个对称纳什均衡。所以这方面没问题。这个均衡是严格均衡吗？这个混合群体，三分之二是攻击性的，三分之一是非攻击性的，或者这个混合策略，它是严格均衡吗？偏离它的行为表现如何？谁来说说？都一样好，嗯说吧，都一样好，都一样好，谢谢。所以对，它可以是一个严格纳什均衡，因为如果我们偏离到A，我们的表现和混合策略中的表现一样好，或者如果我们偏离到B，我们的表现也和混合策略中的表现一样好。我们要说的是，A突变体对这个混合策略的表现，正好和混合策略对自身的表现一样好，B突变体也是一样。事实上，这就是我们最初构造这个均衡的方式，对吧？我们选择了一个P值，使你在A和B之间无差异。对吧？所以在混合均衡中，它不可能是严格的。因为它是混合的，所以不可能严格。在一个真正的混合均衡中，根据定义，你在混合中的策略之间是无差异的。对吧？所以要证明这实际上是进化稳定的，我们必须证明规则B。所以我们需要证明，我们需要检查，给我们一些空间，我们需要检查这个策略——我们称之为p——对所有可能的偏离策略的表现如何，并与这些偏离策略对自身表现进行比较。

[段 14]

We’d have to make this comparison. How does this mix do against all other possible mixes versus how those mixes do against themselves? And we’d have to do this, unfortunately, we’d have to check this for all possible, and now we have to be careful, all possible mixed mutations, P prime. All right, so that would take us a while. it’s actually possible to do if you do enough math it isn’t so hard to do so rather than prove that to you let me give you a heuristic argument why this is the case I’m trying to convince you without actually proving it that this is indeed the case so here we are with this population exactly two thirds of the population is aggressive and one third of the population is passive and suppose suppose there is a mutation P prime that is more aggressive than P hat It’s a relatively aggressive mutation. For example, this mutation may be 100% aggressive or at least it may be very, very highly aggressive. It may be 90% aggressive or something. Now I want to argue that that aggressive mutation is going to die out. And I’m going to argue it by thinking about this rule. So I want to argue that the reason this very aggressive mutation dies out is because the aggressive mutation does very badly against itself. Is that right?

[译文 14]

我们必须进行这种比较。这个混合策略对所有其他可能的混合策略的表现如何，与那些混合策略对自身表现相比？不幸的是，我们必须这样做，我们必须检查所有可能的，现在我们必须小心，所有可能的混合突变，P’。好吧，这会花我们一些时间。如果你做足够的数学，实际上是可能做到的，并不难做到，所以与其向你证明，让我给你一个启发式论证为什么情况是这样的。我试图说服你，而不需要实际证明，情况确实如此。现在我们面对这个群体，正好三分之二是攻击性的，三分之一是被动的。假设，假设有一个突变P’比P hat更具攻击性。这是一个相对具有攻击性的突变。例如，这个突变可能是100%具有攻击性，或者至少可能是非常、非常具有攻击性的。可能是90%具有攻击性或类似。现在我想论证那个具有攻击性的突变将会灭绝。我要用这个规则来论证。所以我想论证这个非常具有攻击性的突变灭绝的原因是，具有攻击性的突变对自身表现非常糟糕。对吧？

[段 15]

If you have a very aggressive mutant, the very aggressive mutants do very, very badly against themselves. They get zero. And that’s going to cause them to die out. What about the other extreme? What about a deviation that’s very passive? so a very nice mutation of very passive types for example it could be 100% B or you know 99% B or 98% B how will that do? well it turns out in this game again it doesn’t do very well against itself and in addition the original mix the mixed p-hat that is more aggressive out in this game again, it doesn’t do very well against itself, right? And in addition, in addition, the original mix, the mixed p-hat that is more aggressive than this very passive mutation, does very well against the mutation, right? So the mix that’s in there, the mix in the population already is relatively aggressive compared to this very passive mutation, and so that, the incumbents, the relatively aggressive incumbents are doing very, very well on average against the mutation, and hence, once again, this equality holds. So just heuristically, without proving it, more aggressive mutations are going to lose out here because they do very badly against themselves, and more passive mutations are going to do badly because they make life easy for p-hat, which is more aggressive than that. So it wasn’t a proof, it’s a heuristic argument, and it turns out indeed to be the case.

[译文 15]

如果你有一个非常具有攻击性的突变体，这些非常具有攻击性的突变体对自身表现非常、非常糟糕。他们得到零。这将导致他们灭绝。另一个极端呢？如果偏离策略是非常被动的呢？一个非常被动的类型的好的突变，例如它可能是100%B，或者你知道99%B或98%B，它会表现如何？嗯，事实证明在这个博弈中，它对自身表现也不好，而且此外，原来的混合策略，即比这个非常被动的突变更具攻击性的混合p-hat，在这个博弈中再次对自身表现不好，对吧？而且此外，此外，原来的混合策略，即比这个非常被动的突变更具攻击性的混合p-hat，对这个突变表现非常好，对吧？所以在其中的混合策略，群体中已有的混合策略相对于这个非常被动的突变来说是相对具有攻击性的，所以，那些在位者，相对具有攻击性的在位者，平均而言对这个突变表现非常、非常好，因此，再一次，这个等式成立。所以仅仅是启发式地，不去证明它，更具攻击性的突变将在这里失利，因为它们对自身表现非常糟糕，而更被动的突变将表现糟糕，因为它们让p-hat的日子好过，而p-hat比那个更具有攻击性。所以这不是证明，是一个启发式论证，事实证明确实如此。

[段 16]

So in this particular game, a game you can imagine in nature, again involving aggression and passivity within this species, it turns out that in this example, the only equilibrium is a mixed equilibrium with two aggressive and one unaggressive And this raises the question what does it mean What does it mean to have a mix in nature So it could mean two different things. It could mean that the gene itself is randomizing. It could mean that the strategy played by the particular ant, squirrel, lion or spider is actually to randomize. That’s possible. but there’s another thing it could mean that’s probably a little bit more important what’s the other thing it could mean it could mean that in the stable mix the evolutionally stable population for this particular spider say it could be that there are actually two types surviving stably in these proportions if you go back to what we said about mixed strategies a week ago we said one of the possible interpretations of mixed strategies is not that people are necessarily randomizing, but that you see a mix of different strategies in society. And again, in nature, one of the impossible interpretations here, the polymorphic population interpretation, is that rather than just have all of the species look and act alike, it could be there a stable mix of behaviors and or appearances in this species So let me try and convince you that that not an uninteresting idea So again, with apologies, I’m not a biologist.

[译文 16]

所以在这个特定的博弈中，一个你可以在自然界中想象的博弈，再次涉及这个物种内的攻击性和被动性，事实证明在这个例子中，唯一的均衡是一个混合均衡，三分之二具有攻击性，三分之一不具攻击性。这引发了一个问题，这是什么意思？在自然界中有混合是什么意思？这可能意味着两件不同的事情。它可能意味着基因本身在随机化。它可能意味着特定的蚂蚁、松鼠、狮子或蜘蛛所采取的策略实际上是随机化。这是可能的。但还有另一件事它可能意味着，这可能更重要一点，另一件事它可能意味着在稳定的混合中，对于这个特定的蜘蛛来说，进化稳定的群体，可能实际上有两种类型以这些比例稳定存活。如果你回到我们一周前关于混合策略所说的，我们说混合策略的一种可能的解释不是人们一定在随机化，而是在社会中看到不同策略的混合。同样，在自然界中，这里一种不可能的解释是，多态群体解释，即不是让所有物种看起来和行动都相似，而是可能在这个物种中存在行为和/或外观的稳定混合。让我试着说服你这不是一个无聊的想法。同样，抱歉，我不是生物学家。

[段 17]

I spent a bit of time on the web this weekend trying to come up with good examples for you. And the example I really wanted to come up with I couldn’t find on the web, which makes me think maybe it’s apocryphal. But I’ll tell you the story anyway. It’s not entirely apocryphal. It may just be that my version of it’s apocryphal. So this particular example I have in mind is to do with elephant seals. And I think even if it isn’t true of elephant seals, it’s definitely true of certain types of fish. So the elephant seals make a better story. So imagine that these elephant seals, it turns out that there are two possible mating strategies for male elephant seals. By the way, do you all know what elephant seals are? They’re these big, you know, people looking blankly at me. You have some rough image in your mind of an elephant seal, you’ve all seen enough nature shows, right? Yeah? Yes, no, yes? Okay, so there are two male mating strategies for the, for the male elephant seal. One successful male mating strategy is to be the head the dominant or a dominant elephant dominant male elephant seal and have a as it were harem of many female elephant seals with which the male mates with. For the males in the room, don’t get too happy.

[译文 17]

这周末我在网上花了一些时间为你们寻找好的例子。而我真的想要找到的那个例子我在网上找不到，这让我觉得可能只是传言。但我还是告诉你们这个故事。它不全是传言。可能只是我那个版本是传言。我脑海中的这个特定例子与象海豹有关。而且我认为即使它对象海豹不成立，它对某些类型的鱼肯定成立。所以象海豹的故事更好。想象这些象海豹，事实证明雄性象海豹有两种可能的交配策略。顺便问一下，你们都知道象海豹是什么吗？它们是这些大的，你知道，人们茫然地看着我。你脑海中有一个象海豹的大致印象，你们都看过足够多的自然节目，对吧？是的？不，是的？好吧，所以雄性象海豹有两种交配策略。一种成功的雄性交配策略是成为一头占主导地位的雄性象海豹的头领，拥有一个所谓的后宫，里面有许多雌性象海豹，雄性与她们交配。对于在座的男性们，别太高兴了。

[段 18]

These are elephant seals, right? They’re not you guys, right? So one possible successful strategy is to be a successful bull elephant seal and have many, many, many, essentially, wives. So to be polygamous. And presumably to do that well, a thing that would go well with that strategy is to be huge. So you could imagine this successful male elephant seal being an enormous animal. It looks like a sort of linebacker in football and basically fights off all other big elephant seals that show up. But it turns out, I think I’m right in saying, if I did my research correctly, this is true among northern elephant seals but not true among southern elephant seals. We’re talking about the Arctic, not the Antarctic. But someone’s going to correct me. Once this is on the web, I’m going to get floods of emails saying I’ve got this wrong. Never mind. Okay, so it turns out that this is not quite evolutionarily stable. Why is this not evolutionarily stable? I’m going to get floods of emails saying I’ve got this wrong. Never mind. Okay, so it turns out that this is not quite evolutionarily stable. All right? Why is this not evolutionarily stable? So what’s the alternative male strategy that can successfully invade the large bull harem keeper elephant seal? Any guesses? Anyone looking for a successful career as an elephant seal?

[译文 18]

这些是象海豹，对吧？他们不是你们各位，对吧？所以一种可能成功的策略是成为一头成功的雄性象海豹，拥有许多、许多、许多，本质上是妻子。所以是一夫多妻的。而且 presumably 要做好这一点，与该策略配合良好的事情是体型巨大。所以你可以想象这头成功的雄性象海豹是一只巨大的动物。它看起来像橄榄球中的线卫，基本上击退所有出现的大象海豹。但事实证明，我认为我说的应该是对的，如果我的研究正确的话，这在北象海豹中成立，但在南象海豹中不成立。我们说的是北极，不是南极。但会有人来纠正我。一旦这个放到网上，我会收到大量电子邮件说我搞错了。没关系。好的，所以事实证明这并不完全是进化稳定的。为什么这不完全是进化稳定的？我会收到大量电子邮件说我搞错了。没关系。好的，所以事实证明这并不完全是进化稳定的。对吧？为什么这不完全是进化稳定的？那么，什么样的替代雄性策略可以成功入侵那头大型雄性象海豹的后宫守护者？有什么猜测吗？有谁在寻找一个成功的象海豹职业生涯吗？

[段 19]

As a male elephant seal? Say again? Female looking elephant seal. Good, good. Good, thank you, thank you. Did people catch that? So a good alternative strategy is to be a male elephant seal who looks remarkably like a female elephant seal. All right? So they’re looking like a linebacker. They look like a wide receiver. All right? All right? I’m probably offending somebody on the football team. But you get the idea. And what do they do? they sneak in among these large numbers of male elephant seals, and they just mate with a few of them. All right? So they look like a female elephant seal, they can hide among the female elephant seals in the harem and they mate with a few of them And provided this is successful enough it be evolutionary stable for the female elephant seal to want to mate with that too Alright Now I forget if actually this is exactly right but it certainly I did enough research over the weekend to know it’s right at least in some species and the nicest part of this story is at least some biologists have a nice technical name for this strategy that was well described by our friend at the back and the name for this strategy is SLF and since we’re on film I’m going to tell you what the S and the L are but you’re going to have to guess the rest so this is this is sneaky this is little and you can guess what that is So this turns out to be actually quite a common occurrence.

[译文 19]

作为一只雄性象海豹？再说一遍？看起来像雌性象海豹的。好的，好的。好的，谢谢，谢谢。大家听到了吗？所以一个很好的替代策略是成为一只看起来非常像雌性象海豹的雄性象海豹。好吗？它们看起来像橄榄球后卫。它们看起来像外接手。好吗？好吗？我可能冒犯了橄榄球队里的某个人。但你们懂我的意思了。它们做什么呢？它们悄悄混入这些大量雄性象海豹中，然后只与其中几只交配。好吗？它们看起来像雌性象海豹，它们可以藏身在雌性象海豹群中与几只交配。只要这足够成功，从进化角度来说对于雌性象海豹想要与这样的雄性交配就是稳定的。好的，我忘了这是否完全正确，但我肯定在周末做了足够的研究知道至少在某些物种中这是对的，这个故事最精彩的部分是至少有些生物学家对这个策略有一个很好的技术名称，这被我们后排的朋友很好地描述了，这个策略的名称是SLF，既然我们在录像，我要告诉你们S和L是什么，但你们得猜剩下的部分，所以这是sneaky这是little你们可以猜那是什么。所以这实际上是一个相当常见的现象。

[段 20]

It’s been observed in a number of different species, perhaps not with the full added colour I just gave to it. All right? Okay, so having convinced you that polymorphic populations can be interesting, let go back to a more subtle case of aggression and non because that seems to be one of the most important things we can think of in animal behavior So let go back and look at a harder example of this of where we started So as these examples get harder, they also get more interesting, so that’s why I want to get a little bit harder. So the chicken game, the battle of the sexes game, is not a particularly interesting version of aggression and non-aggression. Let’s look at a more general version of aggression versus non-aggression. And let’s look at a game that’s been studied a lot by biologists and a little bit by economists called Hawk-Dove. And again, just to stress, we’re talking about, we’re only talking about within species competition here, So I don’t mean hawks versus doves. I mean thinking of hawk as being an aggressive strategy and dove as being a dovish, a passive strategy. All right? So here’s the game. And now we’re going to look at more general payoffs than we did before. All right So this is the hawk strategy This is the dove strategy hawk and dove and the payoffs are as follows v plus c sorry start again v minus c over 2 and v minus c over 2, and here we get v over 2 and v over 2, and here we get v and 0 and here we get 0 and v.

[译文 20]

这已经在许多不同的物种中被观察到，也许不是我刚才描述的那个完整版本。好吧？好的，既然已经让你们相信多态种群可以很有趣，让我们回到一个更微妙的攻击与非攻击的案例，因为这似乎是我们思考动物行为时最重要的事情之一。所以让我们回到我们开始的那个更难的例子。随着这些例子变得越来越难，它们也越来越有趣，这就是为什么我想让它们再难一点。所以小鸡博弈、性别战争博弈并不是一个特别有趣的侵略与非侵略版本。让我们看一个更通用的攻击与非攻击版本。让我们看一个被生物学家研究很多、被经济学家研究一点的博弈，叫做鹰鸽博弈。再强调一次，我们说的只是同物种内的竞争，所以我不是指鹰对鸽。我是把鹰想成一种攻击性策略，把鸽想成一种鸽派的、被动的策略。好吗？游戏是这样的。现在我们要看比之前更通用的收益。好吧，这是鹰策略，这是鸽策略，鹰和鸽的收益如下：v加c，对不起重新来，v减c除以2和v减c除以2，这里我们得到v除以2和v除以2，这里我们得到v和0，这里我们得到0和v。

[段 21]

All right, so this is a generalization, a more interesting version of the game we saw already. Let’s just talk about it a little bit. So the idea here is there’s some potential battle that can occur among these two animals, and the prize in the battle, the prize is V. So V is the victor’s spoils, and we’re going to assume that V is positive. And unfortunately, if the animals fight, so if the hawk meets another hawk, and they fight one another, then there are costs of fighting. So the costs of fighting are C, and again, we’ll assume that they’re positive. So this is the cost of fighting. And this more general format is going to allow us to do two things. a positive, right? So this is the cost of fighting. All right? And this more general format is going to allow us to do two things. We’re going to look and ask what is going to be evolutionarily stable, all right, including mixtures now, and we’re also going to be allowed to ask, able to ask, what happens, what will happen to the evolutionarily stable mix as we change the prize or as we change the cost of fighting. All right? Seems a more interesting, richer game. Okay, so let’s start off by asking, could we have an evolutionary stable population of doves? All right?

[译文 21]

好的，这是一个概括，是我们之前看到的博弈的一个更有趣的版本。让我们来讨论一下。这里的概念是，这两只动物之间可能会发生一些潜在的战斗，战斗的奖品是V。所以V是胜利者的战利品，我们假设V是正的。不幸的是，如果动物打斗，如果鹰遇到另一只鹰，它们互相打斗，那么就有战斗成本。战斗成本是C，我们同样假设它们是正的。这是战斗成本。这个更通用的格式将允许我们做两件事。我们要问什么会是进化稳定的，好的，包括混合策略在内，我们也可以问，当我们改变奖品或改变战斗成本时，进化稳定的混合会发生什么。好吧？看起来是一个更有趣、更丰富的博弈。好的，让我们先问一个问题，我们能否有一个进化稳定的鸽群？好吗？

[段 22]

So is D an evolutionary stable strategy? I’ll start using the term ESS now. So ESS means an evolutionary stable strategy. Is D an ESS? So in this game, could it be the case that we end up with a population of doves? Seems a nice thing to imagine but is it going to occur in this game in nature What do people think How do we go about checking that How do we go about checking that What the first step First step is to ask, is Dove-Dove a Nash equilibrium? If it’s evolutionarily stable, in particular, Dove-Dove would have to be a Nash equilibrium. That’s going to make it pretty easy to check. So is Dove-Dove a Nash equilibrium in this game? It’s not. It’s not, right? But why not? Because if you had a mutation, I’m tempted to say deviation, but I won’t think of it as a mutation. If I had a mutation of hawks, the hawk mutation against the doves is getting V, whereas dove against dove is only getting V over 2. So it’s not Nash. alright so we can’t have an evolutionarily stable population of doves and the reason is there will be a hawk mutation, an aggressive type will get in there and grow much like we had last week when we dropped Rahul into the classroom in Prisoner’s Dilemma and he grew alright or his type grew okay so second question is is HAWK an evolutionary stable strategy Is HAWK an evolutionary stable strategy?

[译文 22]

所以D是进化稳定策略吗？我现在开始使用ESS这个术语。ESS意味着进化稳定策略。D是ESS吗？在这个博弈中，我们最终会得到一个鸽群吗？看起来是个很好的设想，但这会在自然界这个博弈中发生吗？人们怎么想？我们如何检查？我们如何检查？第一步是什么？第一步是问，Dove-Dove是纳什均衡吗？如果它是进化稳定的，特别是Dove-Dove必须是一个纳什均衡。这将使检查变得相当容易。所以Dove-Dove在这个博弈中是纳什均衡吗？不是。不是，对吧？但为什么不是？因为如果你有一个变异，我很想说偏差，但我不会把它想成变异。如果我有鹰的变异，鹰的变异对抗鸽获得V，而鸽对鸽只获得V除以2。所以它不是纳什。好吧，所以我们不能有一个进化稳定的鸽群，原因是会有鹰的变异，一种攻击性类型会进入并增长，就像上周我们把Rahul放入囚徒困境的教室一样，他增长了，或者说他那种类型增长了，好的第二个问题是鹰是进化稳定策略吗？鹰是进化稳定策略吗？

[段 23]

So how do we check this? Well, we have to look at it once again and ask the question. The first question to ask is, is HH a Nash equilibrium? All right. So is it a Nash equilibrium? well I claim it depends I claim it’s a Nash equilibrium provided V minus C over 2 is at least as large as 0 is that right is that right, it’s a Nash equilibrium it’s a symmetric Nash equilibrium provided Hawk against Hawk does at least as well as Dove against Hawk so the answer is yes if V minus C is at least as big as 0. All right? So now we have to think very carefully because there’s two cases. So case one is the easy case which is when v is strictly bigger than C If V is strictly bigger than C then V minus C over 2 is strictly positive Is that right In which case what kind of a Nash equilibrium is this? It’s strict, right? So if V is bigger than C, then hawk-hawk is a strict Nash equilibrium. It’s a strict Nash equilibrium. Alright? The second case is if v is equal to c, if v is equal to c, then v minus c over 2 is actually equal to 0, which is the same as saying that the payoff of Hawke against hawk is equal to the payoff of dove against hawk.

[译文 23]

所以我们如何检查这个？嗯，我们必须再看一遍并问这个问题。第一个要问的问题是，HH是纳什均衡吗？好的。它是纳什均衡吗？我声称这取决于，我声称如果V减C除以2至少和0一样大，它就是纳什均衡，对吗？对吗？它是一个纳什均衡，如果鹰对鹰至少和鸽对鹰一样好，那么它就是一个对称的纳什均衡，所以答案是如果V减C至少和0一样大。好的？现在我们必须非常仔细地思考，因为有两种情况。第一种情况是简单的情况，即当v严格大于C时。如果V严格大于C，那么V减C除以2严格为正。对吗？在这种情况下，这是一种什么类型的纳什均衡？是严格的吗？是的，所以如果V大于C，那么鹰-鹰就是一个严格的纳什均衡。这是一个严格的纳什均衡。好的？第二种情况是如果v等于c，如果v等于c，那么v减c除以2实际上等于0，这就等于说鹰对鹰的收益等于鸽对鹰的收益。

[段 24]

Is that correct? All right. So in that case, what do we have to check? Oh, I’ve deleted it now. You have to come from your notes. What do I have to check in the case in which there’s a tie like that? What do I have to check? check, oh I’ve deleted it now, it seems to come from your notes, what do I have to check in the case in which there’s a tie like that? What do I have to check? I have to check, in this case I need to check, I need to check how hawk does against dove, because dove will be the mutation, I need to compare that with the payoff of dove against dove. All right. Okay. So how does hawk do against dove? What’s the payoff of hawk against dove? Anybody? Payoff of hawk against dove, it shouldn’t be that hard, it’s on the board. Hawk against dove. Shout it out. V. V, thank you. So this is v. And how about the payoff of dove against Delve? V over 2. So which is bigger, V or V over 2? V is bigger because it’s positive, right? So it’s, okay, so it’s, so this is bigger, so we’re okay. So what have we shown? We’ve shown, we shown let just draw it over here we shown that if V is at least as big as C then H is an evolutionary stable strategy.

[译文 24]

对吗？好的，那么在这种情况下，我们要检查什么？哦，我已经删掉了。你得从你的笔记里找。在出现这种平局的情况下我要检查什么？我要检查什么？哦，我已经删掉了，似乎你得从你的笔记里找。在出现这种平局的情况下我要检查什么？我要检查什么？我得检查，在这种情况下我需要检查，我需要检查鹰对鸽会怎样，因为鸽会是那个变异，我需要把它与鸽对鸽的收益进行比较。好的。好的。鹰对鸽的表现如何？鹰对鸽的收益是多少？有人吗？鹰对鸽的收益，不应该那么难，就在黑板上。鹰对鸽。喊出来。V。谢谢。所以这是v。鸽对鸽的收益呢？V除以2。哪个更大，V还是V除以2？V更大，因为它是正的，对吧？所以它是，哦，好的，所以这是更大的，所以我们没问题。那么我们证明了什么？我们证明了，让我们就在这里画出来，我们证明了如果V至少和C一样大，那么H是一个进化稳定策略。

[段 25]

So in this game, in this setting in nature, if the size of the prize to winning the fight is bigger than the costs that would occur if there is a fight, then it can occur that all the animals in this species are going to fight in an evolutionally stable setting. Let me say it again. If it turns out in this setting in nature that the prize to winning the fight is bigger, or at least as big as the cost of fighting, then it will turn out that it will be evolutionally stable for all the animals to fight. The only surviving genes will be the aggressive genes. All right? And what does that mean? So what typically do we think of as the payoff to fight and the cost of fighting? Let’s put this in a biological context. The fight could be about what? It could be let go back to where we started from it could be males fighting for the right to mate with females That going to be pretty important for genetic fitness It could be females fighting over the right to mate with males It could also be fighting over, for example, food or shelter. If the prize is large and the cost of fighting is small, you’re going to see fights in nature. But we’re not done yet. Why are we not done?

[译文 25]

所以在这个博弈中，在这种自然设定下，如果赢得打斗的奖赏大于打斗可能产生的代价，那么这个物种的所有动物都可能会在进化稳定的环境中打斗。让我再说一遍。如果在这种自然设定中，赢得打斗的奖赏更大，或者至少与打斗的代价一样大，那么对于所有动物来说，打斗就会是进化稳定的。唯一能够存活下来的基因将是攻击性基因。明白了吗？这意味着什么？通常我们认为打斗的收益和打斗的代价是什么？让我们把它放在生物学背景下。打斗可能为了什么？可能是让我们回到起点，可能是雄性为了与雌性交配的权利而打斗。这对基因适应性非常重要。雌性也可能为了与雄性交配的权利而打斗。也可能为了食物或住所而打斗。如果奖赏很大而打斗的代价很小，你会在自然界看到打斗。但我们还没有结束。为什么还没有结束？

[段 26]

Because we’ve only considered the case when V is bigger than C. So we also need to consider the case when C is bigger than V. This is the case where the cost of fighting are high relative to the prize in the particular setting we’re looking at. So again, let’s go back to the example. Suppose the cost of fighting could be that the animal could lose a leg or even its life, and the prize is just today’s meal, there are perhaps other meals out there, then we’d expect something different to occur. However, we’ve already concluded that even in this setting, it cannot be the case only to have doves in the population. We shown that even in the case where the costs of fighting are high relative to the prizes it cannot be evolutionarily stable only to have dove genes around passive genes around So in this case, it must be the case that if anything is evolutionarily stable, it’s going to be what? It’s going to be a mix. It’s going to be a mix. So in this case, we know that H is not ESS, and we know that D is not ESS. So what about a mix? What about some mix P hat? All right. We can actually put the P hat in here. We can imagine looking for a mix P hat 1 minus P hat. that will be stable.

[译文 26]

因为我们只考虑了V大于C的情况。所以我们还需要考虑C大于V的情况。这就是打斗代价相对于奖赏较高的情形，在我们所观察的特定环境中。好的，让我们再回到例子。假设打斗的代价可能是动物会失去一条腿甚至丧命，而奖赏只是一顿今天的饭，可能还有其他的饭，那么我们会预期发生不同的情况。然而，我们已经得出结论，即使在这种情况下，群体中不可能只有鸽子存在。我们已经证明，即使在打斗代价相对于奖赏较高的情况下，也不可能进化稳定地只有鸽子这种被动基因存在。所以在这种情况下，如果有什么是进化稳定的，那一定是什么？那一定是一种混合。那一定是一种混合。所以在这种情况下，我们知道H不是ESS，我们知道D不是ESS。那么混合呢？P hat这样的混合呢？好的，我们实际上可以把P hat放在这里。我们可以想象寻找一个混合 P hat 1减P hat，那将是稳定的。

[段 27]

Now how do we go about finding a possible mixed population that has some chance, has some hope of being evolutionarily stable? So here we are, we’re biologists, we’re about to do an experiment, we’re about to either experiment or we’re about to go out and do some field work out there, and we want to set things up, and we’re asking the question, stable. So here we are, we’re biologists, we’re about to do an experiment, we’re about to go out and do some field work out there, and we want to set things up, and we’re asking the question, what’s the mix we expect to see? What’s the first exercise we should do here? Well, if it has any hope to be evolutionarily stable, what does it have to be? It has to be a symmetric Nash equilibrium. So the first step is step one find a symmetric mixed Nash equilibrium in which people will be playing p hat 1 minus p hat it’s symmetric so both sides will be playing this okay so this is a good review for the exam on Wednesday how do I go about finding a mixed equilibrium here? Shouldn’t be too many blank faces. This is likely to come up on the exam on Wednesday. Let’s get some cold calling going on here. How do I find a mixed strategy? Just find anybody.

[译文 27]

现在我们如何找到一个可能有希望成为进化稳定的混合群体？所以我们在这里，我们是生物学家，我们要做一个实验，我们要么做实验要么我们要出去做一些野外工作，我们想要设置条件，我们问这个问题，稳定的。所以我们在这里，我们是生物学家，我们要做一个实验，我们要出去做一些野外工作，我们想要设置条件，我们问这个问题，我们预期看到什么样的混合？我们应该做的第一个练习是什么？如果它有任何希望成为进化稳定的，它必须是什么？它必须是一个对称的纳什均衡。所以第一步是第一步找到一个对称的混合纳什均衡，人们将玩 P hat 1减P hat，这是对称的，所以双方都将玩这个，好，所以这是星期三考试的很好的复习内容。我如何在这里找到混合均衡？不应该有太多空白的表情。这很可能会出现在星期三的考试中。让我们来进行一些冷call。我如何找到混合策略？随便找个人就行。

[段 28]

How do I find a mixed strategy equilibrium Just use the other player payoff I use the other player payoff and what do I do with the other person payoffs Oh you set them equal Set them equal, okay? Okay, so here it’s a symmetric game. It’s really, you know, there’s only one population out there. So what I need, I need the payoff of Hawk against P hat, or P hat 1 minus P. I need this to be equal to the payoff of Dove against this P. alright so the payoff of hawk is going to be what it’s reading up from up there let’s use our pointer so hawk p hat of the time will meet another hawk and get this payoff alright so they’ll get so p of the time they’ll get a payoff of v minus c over 2 and 1 minus p hat of the time they’ll meet a dove and get a payoff of v. And dove against this same mix p hat 1 minus p p hat of the time they meet a hawk and get nothing And 1 minus p hat of the time, they’ll meet another dove and get v over 2. OK, everyone happy with the way I did that? That should be pretty familiar territory to everybody by now. Is that right? Okay, so I’m going to set these two things equal to each other, since they must be equal if this is in fact a mixed strategy equilibrium.

[译文 28]

我如何找到混合策略均衡？用对方玩家的收益，我用对方玩家的收益，然后我对对方玩家的收益做什么？哦，你让它们相等。让它们相等，好的？好的，这里是一个对称博弈。实际上，你知道，那里只有一个人口。所以我需要的是什么，我需要Hawk对P hat的收益，或者 P hat 1减P。我需要这个等于Dove对这个P的收益。好的，所以Hawk的收益将是什么，它从上面读上来的，让我们用我们的指示器，所以Hawk有P hat的时间会遇到另一个Hawk并得到这个收益，好的，所以他们会有P的时间得到V减C除以2的收益，有1减P hat的时间会遇到一只Dove并得到V的收益。而Dove对同样的混合P hat 1减P，P hat的时间他们会遇到一只Hawk并得到零，1减P hat的时间，他们会遇到另一只Dove并得到V除以2。好的，每个人对我这样做的方式满意吗？这对大家来说应该是相当熟悉的领域了。是对的吗？好的，所以我将把这两个东西设为相等，因为如果这确实是一个混合策略均衡的话，它们必须相等。

[段 29]

Alright, and then I’m going to play around with the algebra, but so as to save time, I did it at home. So trust me on this. Trust me, alright, this is implication with the word trust on top of it, right? Trust me that I got the algebra right, or check me at home. this is going to turn out to imply that P hat equals V on C. P hat equals V on C. All right? So it turns out that there is in fact a mixed Nash equilibrium There a mixed Nash equilibrium which is of the following form V on C and 1 minus V on C played by both players. Okay? V on C, 1 minus V on C played by both players. Alright? Is this a strict Nash equilibrium? I found the Nash equilibrium. Is it strict? Everyone should be shouting it out. Is it strict? It can’t be strict because it’s mixed, right? By definition it’s not. It can’t be strict because we know that deviating to H or for that matter deviating to D yields the same payoff. So it can’t be a strict Nash equilibrium. So we need to check something. So we need to check. It’s not strict. Not strict. So we need to check. whether u of p hat against p prime is bigger than u of p prime against itself and we need to check this for all possible mutations p prime and again that would take a little bit of time to do in class so just trust me on it and once again let me give you mutation is p prime.

[译文 29]

好的，然后我要做代数运算，但为了节省时间，我在家做了。所以相信我。相信我，好吧，这是包含信任这个词的推论，对吧？相信我代数是对的，或者回家检查我。这将会得出P hat等于V除以C。P hat等于V除以C。好的？所以实际上确实存在一个混合纳什均衡，存在一个以下形式的混合纳什均衡：V除以C和1减V除以C，由双方玩家执行。好？V除以C，1减V除以C，由双方玩家执行。好？这是严格纳什均衡吗？我找到了纳什均衡。它是严格的吗？每个人都应该喊出来。它是严格的吗？不可能是严格的，因为它是混合的，对吧？根据定义它不是。它不可能是严格的，因为我们知道偏离到H或者就此偏离到D会得到相同的收益。所以它不可能是严格纳什均衡。所以我们需要检查一些东西。所以我们需要检查。它不是严格的。不是严格的。所以我们需要检查。是否u of P hat对P prime大于u of P prime对自己，我们需要对所有可能的突变P prime检查这一点，再次这需要在课堂上花一点时间做，所以相信我就对了，再让我给你一个启发式论证我之前给你的。它本质上是相同的论证。所以我之前给你的启发式论证是，想象一个P prime，一个比我们候选均衡更具攻击性的突变。如果它更具攻击性，那么它对抗自己时会表现得很糟糕。因为在这种情况下C大于V，它实际上会得到负收益。既然它对抗自己得到负收益，结果这将导致它灭绝。相反，想象一个相对温和的突变，相对鸽派的突变，这个突变对现有者非常有利，因为现有者基本上会打败它或者对它得到很高的分数。所以再一次，更鸽派的突变将灭绝。好的，所以这又不是一个证明，但要相信这个论证，你知道我们需要展示这个，但它确实原来是这种情况，好的，所以我们在这里展示了我没有证明最后一部分，但我们论证的是，在打斗代价大于赢得打斗奖赏的情况下，我们最终不会得到100%的鸽子。

[段 30]

And again, that would take a little bit of time to do in class, so just trust me on it. And once again, let me give you the heuristic argument I gave to you before. It’s essentially the same argument. So the heuristic argument I gave to you before was, imagine a p prime, a mutation, that is more aggressive than our candidate equilibrium. If it’s more aggressive, then it’s going to do very, very badly against itself. because C is bigger than V in this case it’s actually going to get negative payoffs against itself. Since it gets negative payoffs against itself it turns out that will cause it to die out. Conversely, imagine a mutation that’s relatively soft that’s relatively dovish this mutation is very good for the incumbents because the incumbents essentially beat up on it or score very highly on it. So once again, the more dovish mutation will die out. all right so again that isn’t a proof but trust the argument just you know we need we need to show this but it does in fact turn out to be the case all right so what have we shown here I didn prove the last bit but what we argued is that in the case in which the costs of fighting in nature are bigger than the prizes of winning the fight it is not the case that we end up with 100% doves.

[译文 30]

再次，这需要在课堂上花一点时间做，所以相信我就对了。再让我给你一个启发式论证我之前给你的。它本质上是相同的论证。所以我之前给你的启发式论证是，想象一个P prime，一个比我们候选均衡更具攻击性的突变。如果它更具攻击性，那么它对抗自己时会表现得很糟糕。因为在这种情况下C大于V，它实际上会得到负收益。既然它对抗自己得到负收益，结果这将导致它灭绝。相反，想象一个相对温和的突变，相对鸽派的突变，这个突变对现有者非常有利，因为现有者基本上会打败它或者对它得到很高的分数。所以再一次，更鸽派的突变将灭绝。好的，所以这又不是一个证明，但要相信这个论证，你知道我们需要展示这个，但它确实原来是这种情况，好的，所以我们在这里展示了我没有证明最后一部分，但我们论证的是，在打斗代价大于赢得打斗奖赏的情况下，我们最终不会得到100%的鸽子。

[段 31]

So we don’t end up with no fights, for example. No fights is not what we would expect to observe in nature. And we don’t end up with 100% fights. 100% fights is not what we’d expect to see in nature. what we end up with is a mixture of hawks and doves such that V over C is the proportion of hawks. So the fights that occur are essentially V over C squared. We can actually observe those fights in nature. Okay, what lessons can we draw from this, sort of biology lessons? All right, so we used a lot of what we learned in the last day or so to figure out what the ESS was there. We kind of did the nerdy part. Now let’s try and draw some lessons from this. All right. So the first thing we know is I’ve hidden what we cared about here. So we know that if V is smaller than C, then the evolutionarily stable mixed population has V on C hawks. So let’s just see how much of this makes sense. So as V goes up, as the prizes go up, if you took the same species and put them into a setting in which the prizes tended to be larger what would we expect to see? Do we expect to see the proportion of hawks go up or down?

[译文 31]

所以我们最终不会没有争斗。比如，没有争斗不是我们在自然界中所期望观察到的。我们也不会得到100%的争斗。100%的争斗也不是我们在自然界中所期望看到的。我们最终得到的是鹰和鸽子的混合体，其中V除以C就是鹰的比例。所以发生的争斗本质上就是V除以C的平方。我们实际上可以在自然界中观察到这些争斗。好的，我们可以从中得出什么教训呢，生物学的教训？好的，我们用了过去一天左右学到的很多东西来找出那个ESS是什么。我们做了那个比较学术的部分。现在让我们试着从中得出一些教训。好的。我们首先知道的是我把我们关心的东西藏在这里了。所以我们知道，如果V小于C，那么进化稳定的混合种群中鹰的比例是V除以C。让我们看看这有多少道理。随着V增加，随着奖励增加，如果你把同一物种放在奖励倾向于更大的环境中，我们会期望看到什么？我们期望看到鹰的比例增加还是减少？

[段 32]

Up, right? As V goes up we see more hawks. What else do we see Not so surprisingly So ultimately as C goes up we look at settings where the cost of fighting is higher we tend to see more more doves alright and this is E and S so more hawks in the evolutionally stable mix and more doves in the evolutionally stable mix Now it’s possible, of course, that the species in question can recognize these two different situations and be coded differently, to behave differently in these two different situations, but that’s beyond the class by now. Perhaps a more interesting observation is about the payoffs. Let’s look at the actual genetic fitness of this species overall. So in this mix, what is the payoff? Well, how are we going to figure out what is the payoff So the payoff in this mix, we can actually construct by looking at the payoff to dove. It doesn’t really matter whether you get the payoff to dove or the payoff to hawk. So let’s look at the payoff to dove. So the payoff was what? It was, if you were dove, then 1 minus v on c of the time. All right, so the payoff, the payoff was what? It was, if you were a dove, then 1 minus v on c of the time, you met another dove, and in that instance, you got a payoff of v on 2.

[译文 32]

增加，对吧？随着V增加，我们看到更多的鹰。还有什么我们看到的？不太令人惊讶的是，随着C增加，我们在争斗成本更高的环境中倾向于看到更多更多的鸽子，好吧，这是E和S，所以在进化稳定混合中鹰更多，在进化稳定混合中鸽子更多。当然，这个相关物种有可能识别这两种不同情况并进行不同编码，在这两种不同情况下表现不同，但这已经超出了现在的课程范围。也许更有趣的观察是关于收益。让我们看看这个物种的整体实际遗传适应度。在这个混合体中，收益是多少？好吧，我们如何计算收益呢？在这个混合体中，我们可以通过观察鸽子的收益来构建收益。查看鸽子的收益还是鹰的收益实际上并不重要。所以让我们看看鸽子的收益。收益是多少？它是什么？如果你是一只鸽子，那么在1减去V除以C的时间里。好的，收益是什么？它是什么？如果你是一只鸽子，那么在1减去V除以C的时间里，你遇到另一只鸽子，在这种情况下，你得到了V除以2的收益。

[段 33]

And it must be the payoff to being a dove is the same since they’re mixing. All right, so this is the payoff. All right, so what’s happening to this payoff as we increase the cost of fighting? what happens as C rises? So just to note out what happens as C goes up. So you might think naively, you might think that if you’re in a setting, be it a social evolutionary setting or a biology and nature evolutionary setting, you might think that as the cost of fighting goes up for you guys in society or for the antelopes or lions we’re talking about, you might think that the payoffs in society go down. Cost of fighting go up more limbs get lost and so on Sounds like that going to be bad for the overall genetic fitness of the species But in fact we don find that What happens as C goes up The payoff goes up, right? As C goes up, the payoff goes up. If we take C bigger, this begets smaller, which means this is bigger. Everyone see that? Everyone see that? So just look at that term, 1 minus v over c times v over 2, it’s actually increasing, it’s increasing in c. So how does that work? As the cost of fighting go up, it’s true that if you do fight, you’re more likely to lose a finger or a limb or a claw or a, thank you, whatever those things are called, or a foot, whatever it is you’re likely to lose.

[译文 33]

成为鸽子的收益必须相同，因为它们在混合。所以这是收益。好的，当我们增加争斗成本时，这个收益会发生什么变化？当C上升时会发生什么？注意到当C上升时会发生什么。你可能天真地认为，你可能认为如果你处于一种环境中，无论是社会进化环境还是生物学和自然的进化环境，你可能认为随着争斗成本上升，对你们社会中的人或我们谈论的羚羊或狮子来说，收益会下降。争斗成本上升，更多的肢体失去等等。听起来这对你的物种的整体遗传适应度不利。但实际上我们发现，随着C上升，收益上升，对吧？随着C上升，收益上升。如果我们取更大的C，这就产生更小的，这意味着这个更大。大家都看到了吗？大家都看到了吗？只看那个项，1减去V除以C乘以V除以2，它实际上在增加，它在C中增加。所以这是怎么回事？随着争斗成本上升，确实，如果你真的争斗，你更可能失去一根手指或一条肢体或一只爪子或一只脚，等等，你可能失去的。无论它是什么，它是什么。

[段 34]

But the number of fights that actually occur in this evolutionarily stable mix goes down, and it goes down sufficiently much to compensate you for that. Kind of a remarkable thing. So these animals that actually are going to lose a lot through fighting are actually going to do rather well overall because of that mixed effect. If you like, it’s one of those strategic effects again. Now of course that raises a question which is what would happen if a particular part of the species evolved that had lower cost of fighting that could regrow a leg It sounds like that would do pretty well, and that would be bad news for the species as a whole. Third thing we can observe here is what’s sometimes called identification. Identification. So what does identification mean here? It means that by observing the data in nature, by going out and filming these animals behaving for hours and hours, or changing their setting in a lab and seeing how they interact, or changing their setting in a field and seeing how they interact, we can actually observe something, namely the proportion of fights. Perhaps we can do better now and actually look at their genetics directly, since science has evolved, and we can actually back out the V and the C. By looking at the proportion of hawk genes out there, or hawkish behavior out there, we can actually identify what must be the ratio of V to C.

[译文 34]

但是实际上在这个进化稳定混合中发生的争斗数量下降，而且下降得足够多，足以补偿你。有点了不起的事情。所以这些实际上会因争斗而失去很多的动物，实际上会做得相当好，因为那种混合效应。如果你愿意，这是又一次战略效应。现在，当然这引发了一个问题，那就是如果物种的某个特定部分进化出更低的争斗成本，能够再生一条腿，那会发生什么？听起来这会做得相当好，而且对整个物种来说是个坏消息。我们在这里可以观察到的第三件事是，有时被称为识别。识别。所以这里的识别是什么意思？它的意思是通过观察自然界中的数据，通过出去拍摄这些动物的行为数小时，或者在实验室中改变它们的环境并观察它们如何互动，或者在野外改变它们的环境并观察它们如何互动，我们实际上可以观察到一些东西，即争斗的比例。

[段 35]

We can tell what the ratio V over C is from looking at data We started off with a little matrix I just written in V and C I didn put any numbers in there We can tell what V is we can tell what C is but we can tell what the ratio is by looking at real-world data. So if you spend enough hours in front of the TV watching nature shows, you could back this out. Not literally. You need to actually do some serious work. So this is a useful input of theory into empirical science. You want theory to be able to set up the science so you can back out the unknowns in the theory. And that’s called identification, not just in biology, but in economics as well. All right. Now there’s one other thing you’d like theory to be, other than identifiable. You’d like theory to be testable. You’d like theory to make predictions that were kind of outside of the sample you started with. If I’m saying familiar to everybody in the room, this is a very familiar idea, I’m hoping, to everybody. slightly philosophy but very familiar idea if you have a new theory it’s one thing for that theory to explain the existing facts but you’d like it to predict new facts why? because it might be a little bit too easy to reverse engineer a model or theory to fit existing facts but if it has to deal with new facts that’s kind of exciting Why?

[译文 35]

也许我们现在可以做得更好，直接观察它们的遗传学，因为科学已经发展了，我们可以实际推导出V和C。通过观察那里的鹰基因或鹰类行为的比例，我们实际上可以识别V与C的比例必须是什么。我们可以通过查看数据来判断V除以C的比例。我们从一个小矩阵开始，我只是写了V和C，我没有在那里放任何数字。我们可以判断V是什么，我们可以判断C是什么，但我们可以通过查看真实世界的数据来判断这个比例。所以如果你花足够的时间在电视前观看自然节目，你可以推导出这个比例。字面上不是。你需要实际上做一些严肃的工作。所以这是理论对经验科学的有用输入。你希望理论能够建立科学，以便你能推导出理论中的未知数。这被称为识别，不仅在生物学中，也在经济学中。好的，你现在还希望理论是另一个东西，除了可识别之外。你希望理论是可测试的。你希望理论做出你开始时的样本之外的预测。

[段 36]

Because it’s a little easy to, it might be a little bit too easy to reverse engineer a model or theory to fit existing facts, but if it has to deal with new facts, that’s kind of exciting. Right? That’s a real test. That make sense? So you might ask about this theory, you might say, well it’s just a whole bunch of just-so stories. I don’t know, I mean, may I just-so story? It’s a just-so stories, children’s stories written by Kipling. And people sometimes accuse a lot of evolutionary theory as being just those stories, because you know what the fact is already, and you reverse the game, you come up with a story afterwards. That doesn’t sound like good science. So you’d like this theory, this theory that matches game theory with evolution, to predict something that we hadn’t seen before, had not seen before, and then we can go out and look for it, and see if it’s there. And that’s exactly what we now have. So our last example is a slightly more complicated game again.

[译文 36]

如果我对房间里的每个人都熟悉，这是一个非常熟悉的想法，我希望。对稍微哲学但非常熟悉的想法，如果你有一个新理论，这个理论解释现有事实是一回事，但你希望它预测新事实为什么？因为reverse engineer一个模型或理论来适应现有事实可能有点太容易了，但如果它必须处理新事实，那是有点令人兴奋的为什么？因为reverse engineer一个模型或理论来适应现有事实可能有点太容易了，但如果它必须处理新事实，那是有点令人兴奋的。对吧？那是一个真正的测试。有道理吗？所以你可能会问这个理论，你可能会说，呃，这只是一堆just-so故事。我不知道，我可以说just-so故事吗？这是一个just-so故事，是吉卜林写的儿童故事。人们有时指责很多进化论就像那些故事一样，因为你知道事实是什么，然后你反转游戏，之后提出一个故事。这听起来不像好科学。所以你希望这个理论，这个将博弈论与进化相匹配的理论，预测一些我们之前没见过的、没见过的东西，然后我们可以出去寻找，看看它是否在那里。这正是我们现在拥有的。所以我们的最后一个例子是一个稍微更复杂的游戏。

[段 37]

And the slightly more complicated game has three strategies and the strategies are called well I’ll tell you what the strategies are called in a second actually I’ll give you the payoffs first of all so once again this is a game about different forms of aggression and we’ll look at the we’ll look at other interpretations in a second and once again V is going to be the prize for winning 0 is going to be the prize for losing and 1 is if it’s a tie I’m just short sighted enough that I can’t read my own writing so I hope I got this right there we go so this is the game and we’re going to assume that the prize V is somewhere between 1 and 2 so V you can think of as winning 0 is losing and one is if it’s a tie. Does anyone recognize what this game essentially is? It’s essentially rock, paper, scissors. Now it turns out that when biologists play rock, paper, scissors, they give it a different name. They call it scratch, bite and trample Scratch bite and trample is essentially the tactics of the Australian football team All right so scratch bite and trample are the three strategies and it a little bit like rock, papers, and scissors. How do we change it? First, we added one to all the payoffs to make sure there’s no negatives in there, and second, we added a little bit more than one to winning, right?

[译文 37]

那么稍微复杂一点的博弈有三种策略，策略的名字叫什么我等一下再告诉你，我先给你们收益。所以这又是一个关于不同形式攻击的博弈，我们等会儿再看其他解释。V是获胜的奖励，0是失败的奖励，1是平局。我的眼光还不够长远，读不懂自己的字迹，所以我希望我写对了。好，这就是这个博弈，我们假设奖励V在1和2之间。V你可以认为是获胜，0是失败，1是平局。有人能认出这个博弈本质上是什么吗？它本质上是石头、布、剪刀。生物学家玩石头、布、剪刀时给它起了个不同的名字，他们称之为抓、咬、踩。抓、咬、踩基本上就是澳大利亚足球队的战术。抓、咬、踩是三种策略，有点像石头、布、剪刀。我们怎么改动的呢？第一，我们给所有收益加1以确保里面没有负数，第二，我们给获胜的奖励加的比1稍微多一点，对吧？

[段 38]

If we add one to everything, sorry, a little bit less than one to winning, right? So If we added 1 to everything, then V would have been 2, but we’ve kept V somewhere between 1 and 2. So this is certainly a game you could imagine in nature. There’s three possible strategies for this species, and the payoff matrix happens to look like this. So where’s my prediction going to come from? Well, since this is rock, paper, scissors, we know that there’s really only one hope for an evolutionarily stable strategy. right, since it’s essentially rock, paper, scissors, what would, if there is an evolutionary stable strategy or an evolutionary stable mix, what must it be? One third, one third, one third. Right so the only hope the only hope for an ESS is one third one All right let put that in here So one-third, one-third, one-third. And you can check at home that that indeed is a mixed strategy equilibrium. All right? And the question is, is this evolutionarily stable? Is this evolutionarily stable? All right? So we know it’s a Nash equilibrium, that I’ve given you, and we know it’s not a strict Nash equilibrium. Everyone okay with that? It can’t be a strict Nash equilibrium because it’s mixed. All right? So if this is an ESS, it must be the case we have to check that, let’s call this p-hat again like we’ve been doing, right?

[译文 38]

如果我们给所有东西加1，对不起，是给获胜的奖励加的比1稍微少一点，对吧？如果我们给所有东西加1，那么V就会是2，但我们让V保持在1和2之间。所以这确实是你可以在自然界中想象的一个博弈。这个物种有三种可能的策略，收益矩阵碰巧看起来是这样的。那么我的预测从哪里来呢？既然这是石头、布、剪刀，我们知道对于进化稳定策略来说实际上只有一种希望，对吧？既然本质上这是石头、布、剪刀，如果存在进化稳定策略或进化稳定的混合，那它必须是什么？三分之一，三分之一，三分之一。唯一的希望，唯一的ESS希望是三分之一，三分之一。让我把它写在这里。三分之一，三分之一，三分之一。你可以在家里检查这确实是一个混合策略均衡，对吧？问题是，这在进化上稳定吗？这在进化上稳定吗？我们知道这是一个纳什均衡，我已经给你们了，我们知道它不是一个严格的纳什均衡。大家对此没问题吧？因为它是混合的，所以不可能是严格的纳什均衡。所以如果这是一个ESS，我们必须检查，必须检查那个情况，我们把这个叫做p-hat，就像我们之前做的那样，对吧？

[段 39]

We have to check that the payoff from p-hat against any other p-prime would have to be bigger than the payoff from P’ against itself. We’d need that to be the case. All right, so let P’ be scratch. Let P’ be scratch. All right, so let’s compare these things. So U of P hat against scratch is what? Well, you’re playing against scratch. You are a third scratch, a third bite, a third trample. So a third of the time you get 1, a third of the time you get nothing, and a third of the time you get V. Is that right? So your payoff is 1 plus V over 3. How would we do if we’re scratch against scratch? The payoff of scratch against scratch is what? No prizes for this. What’s the payoff of scratch against scratch? 1. Which is bigger, 1 plus V on 3 or 1? Well, look, V is less than 2, right? v is less than 2, so 1 plus v on 3 is less than 1, so 1 is bigger. So in this game, the only hope for an evolutionarily stable mix was a third, a third, a third, and it isn’t stable. So here’s an example, example, in this example, there is no evolutionarily stable strategy. There’s no evolutionary stable mix. And then the obvious question is, what does that mean in nature?

[译文 39]

我们必须检查p-hat对任何其他p’的收益必须大于p’对自身的收益。我们需要那个条件成立。好的，所以让p’是抓。让p’是抓。好的，让我们比较这些东西。U of p-hat 对抓是什么？你在对付抓。你是三分之一抓、三分之一咬、三分之一踩。所以三分之一的时间你得到1，三分之一的时间你什么都得不到，三分之一的时间你得到V。对吗？所以你的收益是1加上V除以3。如果我们让抓对付抓会怎样？抓对抓的收益是什么？这没有奖励。抓对抓的收益是什么？1。哪个更大，1加上V除以3还是1？V小于2，对吧？v小于2，所以1加上v除以3小于1，所以1更大。所以在这个博弈中，进化稳定混合的唯一希望是三分之一、三分之一、三分之一，而它不稳定。所以这是一个例子，在这个例子中，没有进化稳定策略。没有进化稳定的混合。那么显而易见的问题就是，这在自然界中意味着什么？

[段 40]

Can we find a setting that looks like rock, paper, scissors in nature in which nothing is evolutionary stable If nothing is evolutionary stable what going to happen We going to see cycling around You going to see a lot of the scratch strategy followed by a lot of the trample strategy followed by a lot of the bite strategy and so on all right so it turns out that’s exactly what you see when you look at these example I’ve left you on the web there’s an article in nature in the mid-90s that looks at a certain type of lizards and these lizards come in three colors I forget what the colors are no I wrote it down one is orange one is yellow and one is blue and these lizards have three strategies one the orange lizard is like our big elephant bull it likes to keep a harem of many or a large territory with many female lizards in it to meet with but that can be invaded by our SLF strategy which turns out to be the yellow lizard. The yellow lizard can invade and just mate with a few of these female lizards. All right? But when there are too many of these sneaky yellow lizards, then it turns out that they can be invaded by a blue lizard, and the blue lizard has much smaller territories.

[译文 40]

我们能在自然界中找到看起来像石头、布、剪刀的情形吗？在这种情况下没有任何东西是进化稳定的。如果没有任何东西是进化稳定，会发生什么？我们会看到循环。你会看到很多抓策略，然后被很多踩策略取代，然后被很多咬策略取代，如此循环。所以事实证明这正是你在这些例子中看到的，我在网上给你们留了一篇文章，是90年代中期的《自然》杂志，讲述了一种特定类型的蜥蜴，这些蜥蜴有三种颜色我忘了是什么颜色，不，我写下来了一种是橙色一种是黄色还有一种是蓝色。这些蜥蜴有三种策略。橙色蜥蜴就像我们的大象公牛，它喜欢保持一个有很多雌性蜥蜴的后宫或一个有很多雌性蜥蜴的大型领地以供交配，但这可以被我们的SLF策略入侵，结果证明SLF策略就是黄色蜥蜴。黄色蜥蜴可以入侵，只需要与这些雌性蜥蜴中的几个交配。当这些偷偷摸摸的黄色蜥蜴太多的时候，结果发现它们可以被蓝色蜥蜴入侵，而蓝色蜥蜴的领地要小得多。

[段 41]

It’s almost monogamous. All right? So what happens in nature is you get a cycle, orange invaded by yellow, invaded by blue, harem keeper invaded by sneaky, invaded by monogamous, invaded by harem keeper again. And indeed, the population does cycle around exactly as predicted by the model. So here’s an example of evolutionary theory via game theory making a prediction that we can actually go off and test and find. This, for biologists, was like finding a black hole. It’s a really cool thing. All right, we’ll leave evolution here. Midterm on Wednesday, we’ll come back to do something totally different next week. See you on Wednesday.

[译文 41]

它几乎是一夫一妻制。所以自然界中发生的是你会得到一个循环，橙色被黄色入侵，黄色被蓝色入侵，后宫守护者被偷偷摸摸的入侵，被一夫一妻制的入侵，再被后宫守护者入侵。确实，人口正好按照模型预测的那样循环。所以这是进化论通过博弈论做出预测的一个例子，我们可以实际走出去测试和发现。这对生物学家来说，就像是发现了黑洞。真的是很酷的东西。好的，我们在这里结束进化论的话题。期中考试在周三，下周我们回来做完全不同的事情。周三见。