耶鲁大学博弈论公开课 - 第7集 合作与结局

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]

Okay, let’s make a start. So I hope everyone had a good break. We’re going to spend this week looking at repeated interaction. Now we already saw last time before the break that once we repeat games, once games go on for a while we can sustain behavior that’s quite interesting So for example before the break we saw that we could sustain fighting by players that were rational in a war of attrition alright and another thing we learned before the break was when we’re analyzing these potentially very long games it helps sometimes to break it break the analysis up into what we might call stage games each each period of the game and break the payoffs up into the payoffs that are associated with that stage payoffs that are associated with the past but they’re sunk they don’t really matter and payoffs that are going to come in the future from future equilibrium play so those are some ideas we going to pick up today but for the most part what we do today will be new Now whereas last time we focused on fighting for the whole of today I want to focus on the issue of cooperation In fact, for the whole of this week I want to focus on the issue of cooperation. And the question kind of behind everything this week is going to be, can repeated interaction among players both induce and sustain cooperative behavior, or if you like, good behavior.

[译文 1]

好的，我们开始吧。希望大家休息得好。我们这周将探讨重复互动。我们之前在休息前已经看到，一旦游戏重复进行，一旦游戏持续一段时间，我们就能够维持一些非常有趣的行为。例如，在休息前我们看到，我们可以在消耗战博弈中维持理性玩家的战斗行为。在休息前我们学到的另一件事是，当我们在分析这些可能非常长的游戏时，有时把分析分解成我们可能称为阶段博弈的东西——游戏的每个时期——会有所帮助，而且把收益分解成与该阶段相关的收益、与过去相关的收益（但那是沉没成本，它们实际上并不重要）以及来自未来均衡策略的未来收益。这些是我们今天要继续的一些想法，但今天的大部分内容将是新的。上一节课我们专注于战斗，而今天我想专注于合作问题。实际上，这整个星期我想专注于合作问题。这周所有问题背后的核心问题是：玩家之间的重复互动是否能够诱导和维持合作行为，或者如果你愿意这么说的话，好的行为。

[段 2]

And our canonical example is going to be Prisoner’s Dilemma. Way back in the very first class, we talked about Prisoner’s Dilemma, and we mentioned that playing the game repeatedly might be able to get us out of the dilemma. It might be able to enable us to sustain cooperation. And what’s going to be good about that is not just to sustain cooperation, but to sustain cooperation without the use of outside payments, such as contracts or the mafia or whatever. So why does this matter? Well, one reason it matters is that most interactions in society either don’t or perhaps even can’t rely on contracts. Most relationships are not contractual However many relationships are repeated So this is going to be of more importance perhaps in general life perhaps less so in business more important in general life than thinking about contracts So just think about some obvious examples. Think about your own friendships. I don’t know if you have any friendships, I assume you do, but for those of you who do, your friendships are typically not contractual. You don’t have a contract that says, if you’re nice to me, I’ll be nice to you. Similarly, think about interactions among nations. Interactions among nations typically cannot be contractual because there’s no court to enforce those would-be contracts, although you can have treaties, I suppose. But most interaction among nations, cooperation among nations, is sustained by the fact that those relationships are going to go on forever.

[译文 2]

我们的典型例子将是囚徒困境。早在第一堂课上，我们就讨论过囚徒困境，并且我们提到过，重复进行这个游戏可能能够帮助我们摆脱困境。它可能使我们能够维持合作。这样做的好的地方不仅在于维持合作，而且在于维持合作时不使用外部支付手段，比如合同、黑手党或其他什么。那么为什么这很重要呢？嗯，它重要的一个原因是，社会中的大多数互动要么不能，要么甚至可能无法依赖合同。大多数关系都不是契约性的。然而许多关系是重复的。所以这在一般生活中可能更重要，也许在商业中不那么重要，在一般生活中比思考合同更重要。所以想想一些明显的例子。想想你自己的友谊。我不知道你有没有友谊，我假设你有，但对于你们当中那些有友谊的人来说，你们的友谊通常不是契约性的。你没有一份合同说，如果你对我好，我就会对你好。类似地，想想国家之间的互动。国家之间的互动通常不可能是契约性的，因为没有法院来执行那些所谓的合同，尽管我想你可以有条约。但大多数国家间的互动、国家间的合作，是由这些关系将永远持续下去这一事实来维系的。

[段 3]

And even in business, even where we have contracts, and even in a very litigious society like the US, which is probably the most litigious society in the world, we can’t really rely on contracts for everyday business relationships. So in some sense, we need a way to model a way to sustain cooperation and good behavior that forms, if you like, the social fabric of our society, absence always going to court about everything. Now why might repeated interaction work Why do we think way back in day one of the class that repeated interaction might be able to enable us to behave well even in situations like prisoner’s dilemmas or situations involving moral hazard, where bad behavior is going to occur in one-shot games? All right? So the lesson we’re going to be sort of underlying things today and all week is this one. In ongoing, in ongoing relationships, in ongoing relationships, the promise of future rewards and the threat of future punishment may, let’s be careful, may sometimes may sometimes provide incentives may sometimes provide incentives for good behavior today. And just leave a gap here in your notes, because we’re going to come back to this. So this is a very general idea. The idea is that future behavior in the relationship can generate the possibility of future rewards and or future punishments, and those promises or threats may sometimes provide incentives for people to behave well today.

[译文 3]

即便在商业领域，即便我们有合同，即便在一个诉讼频繁的社会，比如美国——这可能是世界上诉讼最多的社会——我们也不能真正依靠合同来维持日常的商业关系。所以在某种意义上，我们需要一种方式来建立和维持合作与良好行为，这种方式构成了我们社会的社会纽带，而不必凡事都对簿公堂。为什么重复互动可能有效？为什么我们从第一天上课就认为，重复互动也许能够让我们在囚徒困境或涉及道德风险的情况下表现良好——在这些一次性博弈中，不良行为是会发生的？明白了吗？所以我们今天和整个星期要讲的核心内容就是这个。在持续的、在持续的关系中，未来回报的承诺和未来惩罚的威胁也许——请注意——也许有时候也许有时候能够提供激励，也许有时候能够为今天的好行为提供激励。在这里留个空白，因为我们会回过头来讲这个。这是一个非常普遍的观点。观点是，关系中的未来行为可以产生未来回报和/或未来惩罚的可能性，而这些承诺或威胁也许有时候能够为人们今天的好行为提供激励。

[段 4]

And the reason I want to leave a gap here is I want part of the purpose of this week’s lectures to be trying to get beyond this. This is kind of almost a platitude. I think most of you knew this already. So I want to get beyond this. I want to see when is this going to work? When is it not going to work? How is it going to work? So I don’t want people to leave this week of classes or leave the course thinking, oh, well, this relationship, we’re going to interact more than once, so everything’s fine. That’s not true. We want to make sure that we understand when things work how they work and more importantly when they don work and how they don work Alright so we going to try and fill in the gap that we just left on that board as we go on today Alright, nevertheless, we do have this very strong intuition that repeated interaction will get us, as it were, out of the prisoner’s dilemma. So why don’t we start with the prisoner’s dilemma. I’ll put this up, out of the way, and we’ll come back to it. and let’s just remind ourselves what the prisoner’s dilemma is because you guys are all full of turkey and cranberry sauce and you’ve probably forgotten what game theory is entirely and let’s name these strategies rather than alpha and beta let’s call them cooperation and effect and that will be our convention this week we’ll call them cooperation and effect this is player A and this is player B And the payoff is something like this, 2, 2, minus 1, 3, 3, minus 1, and 0, 0.

[译文 4]

这里我留一个空白的原因在于，本周讲座的部分目的是试图超越这一点。这几乎是一个老生常谈的话题了。我想你们大多数人都已经知道了。所以我想超越这一点。我想知道什么时候这会起作用？什么时候不会起作用？它会如何起作用？所以我不想让人们在上完这周的课或离开这门课程时认为，哦，这种关系，我们要多次互动，所以一切都好。这不是真的。我们想确保自己理解事情在起作用时是如何起作用的，更重要的是，当它们不起作用时以及它们如何不起作用。好吧，所以我们今天会一边继续一边尝试填补我们刚才在黑板上留下的那个空白。好吧，尽管如此，我们确实有一个非常强烈的直觉，即重复互动将把我们从囚徒困境中解救出来——可以这么说。那么我们为什么不从囚徒困境开始呢？我把这个放上去，先放到一边，我们之后再回来。让我们提醒自己囚徒困境是什么，因为你们都吃满了火鸡和蔓越莓酱，你们可能已经完全忘了什么是博弈论。让我们给这些策略命名，不要叫alpha和beta，我们叫它们cooperation和defect，这将成为我们本周的惯例，我们叫它们cooperation和defect。这是player A，这是player B。收益是这样的：2, 2, -1, 3, 3, -1, 和0, 0。

[段 5]

It doesn’t have to be exactly this, but this will be. This is a game we’re going to play, and to try and see if we get cooperation out of it by having repeat interaction, we’re going to play it more than once. So let me go and find some players to play here This should be a familiar game to everybody here All right, so why don’t I pick some people kind of close to the front row. So what’s your name again? I’m going to use the green one over here. Say again? Brooks. Brooks. Okay, so Brooks is going to be player, I guess, B. You can make him player B. And I’ve forgotten your name. By this stage I shouldn’t know it. Patrick, you’re going to be player A. All right. And the difference between playing this game now and playing this game earlier on in the class is we’re going to play not once, but twice. We’re going to play it twice. So write down what you’re going to do the first time. Write down what you’re going to do the first time and show it to your neighbor. Don’t show it to each other. All right. And let’s find out what they did the first time. So is it written down? Something written down? So, Brooks, when? I cooperated. You cooperated. Patrick? I defected.

[译文 5]

这不一定要完全一样，但这样也可以。这是一个我们要玩的游戏，为了尝试通过重复互动来获得合作，我们会玩不止一次。所以让我去找一些玩家来玩这个游戏。这对大家来说应该是一个熟悉的游戏。好的，那我在前排附近选几个人吧。你叫什么名字来着？我用这边的绿色筹码。你再说一遍？Brooks。Brooks。好的，那Brooks来做玩家，我猜是B。你可以让他做玩家B。我忘了你叫什么名字了。到这个阶段我不应该知道的。Patrick，你要做玩家A。好的。那么现在玩这个游戏和之前在课堂上玩这个游戏的区别是，我们要玩的不是一次，而是两次。我们会玩两次。所以写下你第一次要怎么做。写下你第一次要怎么做，然后展示给你的邻居看。不要互相展示。好的。让我们看看他们第一次做了什么。所以，写下来了吗？有写下来的东西吗？Brooks，你呢？我选择了合作。你合作了。Patrick呢？我选择了背叛。

[段 6]

Patrick defected. Okay, okay, well, okay, let’s play the second time, all right? Second time, so write down what we’re going to do the second time. Brooks. This time I’m going to defect. Me too. All right so we had the play this time Let just put it up here So the play when we played it this time we had A and B and the first time we had defect cooperate and the second time we had defect defect All right Let’s try another pair. We’ll just play this a couple of times and we’ll talk about it. So yeah, that’s fair enough. Why don’t we go to your neighbors? That seems fair enough. It’s easy. So you are? Ben. Shout it out to people higher. Ben. That’s a good name. It’s good. Very good. Okay. And you are? Edwina. Edwina. Edwina and Ben. Okay, so we’re going to make Ben play a B and Edwina play a A. And why don’t you write down what you’re going to do for the first time. Again, we’re going to play it twice. All right. Why don’t we mix it up? We can play it three times. We’ll play it three times this time, okay? We’ll play it three times. All right, both people are happy with their decisions. Okay, so the first time, Edwina, what did you choose? Defect. Cooperate.

[译文 6]

Patrick选择背叛。好的，好的，嗯，好的，我们再来第二轮，好吗？第二轮，所以写下我们第二轮要做什么。Brooks。这次我要背叛。我也是。好的，那我们这次的结果是，我们把它放在这里。所以我们这次的结果是A和B，第一次我们有背叛、合作，第二次我们有两个背叛。好的，让我们换一组。我们会玩几次然后讨论一下。所以，是的，这样很公平。我们去你的邻居那边怎么样？这看起来很公平。很简单。那你是？Ben。大声告诉上面的人。Ben。这是个好名字。很好。非常好。好的，你呢？Edwina。Edwina。Edwina和Ben。好的，所以我们让Ben玩B，Edwina玩A。你为什么不写下你第一次要做什么。再一次，我们要玩两次。好的，我们为什么不换一下？我们可以玩三次。这次我们玩三次，好吗？我们玩三次。好的，两个人都满意他们的决定。好的，那么第一次，Edwina，你选了什么？背叛。合作。

[段 7]

All right, so we had that split down this time. So we’ve got Edwina A and Ben B, and we had cooperate, defect. All right. Second time, please. Edwina? Cooperate. Defect. Oh, okay. So we’re getting rid of going to and fro now. So this was cooperate and defect. And one more time. One more time, right down. Both players written down. Cooperate. Defect. Oh, okay, so we flipped around again. Okay, okay. So we’re seeing some pretty on behavior here. Who did what that time? Edwina, what did you do? So we had this, is that right? We had this. All right. So keep the microphone for a minute and we’ll just talk about it in a second. All right. So first of all, let’s start with Ben here. Ben, you were cooperating in the first go. All right. So why did you choose to cooperate the first turn? Shout out to people who can hear you. I felt that if I established a reputation for cooperating, we could end up in the cooperate, cooperate. Alright, so you thought that by playing cooperate early, you could establish some kind of reputation. And what about later on when you played defect thereafter what were you thinking there I realized that she established a reputation for defecting so I could Alright alright so you switched strategies mid Alright Edwina you started off by defecting.

[译文 7]

好的，这次我们有了那个结果。Edwina A 和 Ben B，合作、背叛。好，第二次，请。Edwina？合作。背叛。哦，好。所以我们不再来回切换了。所以这是合作和背叛。再来一次。再来一次，填下去。双方玩家都写下了。合作。背叛。哦，好，所以我们又反过来了。好，好。我们看到了一些相当的行为。谁在那轮做了什么？Edwina，你做了什么？我们有了这个，对吗？我们有了这个。好，把麦克风留一分钟，我们稍后会讨论。好，首先，让我们从 Ben 开始。Ben，你在第一轮合作了。好，那你为什么在第一轮选择合作？让能听到的人喊出来。我觉得如果我建立了一个合作的名声，我们就能达到合作、合作。好，所以你认为通过早期采取合作策略，你可以建立某种名声。那后来当你采取背叛时你在想什么？我意识到她建立了一个背叛的名声，所以我可以……好，好，所以你中途切换了策略。好，Edwina，你一开始采取了背叛。

[段 8]

Why did you start off by defecting? Shout out to people that hear you. Because his friend defected so I thought he might defect. Oh, because his friend defected. Okay, okay, that’s a bit tainted by his friend there. Okay, alright. It’s short of space in the classroom. They could have just been sitting next to each other. And thereafter you cooperated why was that? So in fact your reputation works in some sense by cooperating early you convinced that we knew you would cooperate and then you went on cooperating even after he defected so what were you doing in the third round? Shout out? I thought I thought he might cooperate because I cooperated. All right you might come back let’s talk to your neighbors so Brooks Brooks why shout out why you cooperated in the first round? Because I was hopeful that he would cooperate. You were hoping he would cooperate. All right. All right. And why didn’t you defect thereafter? Because I thought he would continue to defect after he defected. Because he defected and he continued to defect. And Patrick, you’re the person who just defected throughout here. Grab the mic. Who’s next to you? Why didn’t you just defect? This is a short game that makes sense to defect in the last period So the second to last period is the first Ah alright that an interesting idea So Patrick saying actually if we look at the last period of this game if we look at this last period of the game, what does the game look like in the last period?

[译文 8]

为什么你一开始采取背叛？让听到的人喊出来。因为他的朋友背叛了，所以我觉得他可能也会背叛。哦，因为他的朋友背叛了。好，好，这有点被他朋友影响了。好，好。教室里空间很挤。他们可能只是坐在一起。之后你合作了，为什么？你的名声在某种意义上起作用了，通过早期合作，你说服了……我们知道你会在第一轮合作，然后你继续合作，即使他背叛了。所以你在第三轮做什么？喊出来？我想我觉得他可能会合作，因为我合作了。好，你可能回来。让我们和你的邻居聊聊。Brooks, Brooks，为什么喊出来，为什么你在第一轮合作？因为我希望他会合作。你希望他会合作。好，好。之后你为什么不背叛？因为我觉得他背叛后会继续背叛。因为他背叛了，而且他继续背叛。Patrick，你是那个一直背叛的人。拿起麦克风。谁坐在你旁边？你为什么不直接背叛？这是一个短期游戏，在最后一期背叛是有道理的。所以倒数第二期是第一期……好，那是一个有趣的想法。Patrick 说，实际上如果我们看这个游戏的最后一期，如果我们看这游戏的最后一期，在最后一期游戏看起来是什么样的？

[段 9]

A single period game. In the last period, this actually is the game, alright? If I drew out the game with two periods, it’d be kind of a hard thing to draw, it’d be kind of an annoying diagram to draw. but in the last period of the game, whatever happened in the first period is what? It’s sunk. Is that right? Everything happened in the first period is sunk. So in the last period of the game, these are the only relevant payoffs. Is that right? And since these are the only relevant payoffs looking forward, in the last period of the game, we know that there’s actually a dominant strategy. And what is that dominant strategy in the last period of the game? To do what? Imprisonment of the land. What’s the dominant strategy? Shout it out. Defect. Defect, okay. So what we should see in this game, we didn’t actually, we didn’t actually because we had some kindness over here from Edwina, but okay, but what we should see in general is we know that in the last period of the game, in period two we going to get both people defecting The reason we going to get both people defecting is because the last period of the game is just a one game All right There nothing particularly exciting about it There is no tomorrow, and so people are going to defect.

[译文 9]

一个单期博弈。在最后一期，这实际上是博弈本身，好吗？如果我画出两期的博弈，那会有点难画，会是一个有点烦人的图表。但在最后一期，无论第一期发生了什么是什么？那是沉没成本。对吗？第一期发生的一切都是沉没成本。所以在最后一期，这些是唯一相关的收益。对吗？既然向前看这些是唯一相关的收益，在最后一期，我们知道实际上有一个占优策略。那个在最后一期的占优策略是什么？要做什么？监禁土地。什么是占优策略？喊出来。背叛。背叛，好。所以我们在这个博弈中应该看到什么，我们实际上没有，因为我们有 Edwina 这里的一些善意，但好，但我们应该看到的一般情况是，我们知道在最后一期，在第二期，我们会让双方都背叛。我们会让双方都背叛的原因是最后一期只是一个一次性博弈。好，没有什么特别令人兴奋的。没有明天，所以人们会背叛。

[段 10]

But now, let’s go back and revisit some of the arguments that Edwina and Brooks and… I’ve forgotten what your neighbors called again. Ben. And Ben, I should remember that. And Ben said earlier, right? They gave quite elaborate reasons for cooperating. cooperating to establish reputation cooperating because the other person might cooperate whatever all right but most of these but most of these behaviors were designed to in to either induce or promise cooperation in period two is that right but we’ve just argued is that in period two everyone’s going to defect right period two is just a trivial one-stage prisoner’s dilemma we actually analyzed it the very first week of the class and provided we believe these payoffs we’re done period two people can defect since they’re going to effect in period two, nothing I can do in period one is going to affect that behavior, and therefore I should defect also in period one. If you want to belabor this point, we can actually draw up what the matrix looks like in period one. So let’s do that using the style we did last week. all right, before, two weeks ago, before we went away. All right, so here once again is the matrix we had before, and I want to analyze the first stage game. In the first stage game, what I’m going to do is I’m going to add in the payoffs I’m going to get from tomorrow.

[译文 10]

但现在，让我们回去重新审视 Edwina 和 Brooks 以及……我忘记了你的邻居叫什么名字。Ben。Ben，我应该记住的。Ben 之前说过，对吧？他们给出了相当详尽的合作理由。合作以建立名声合作，因为对方可能合作等等。好，但大多数这些行为，但大多数这些行为都是旨在在第二期诱导或承诺合作，对吗？但我们刚刚论证的是在第二期每个人都会背叛，对吗？第二期只是一个trivial的单期囚徒困境，我们实际上在第一周课就分析过了，而且如果我们相信这些收益我们在第二期完成后人们可以背叛，因为他们在第二期会这么做，第一期我做什么都不会影响那个行为，因此我也应该在第一期背叛。如果你想更详细地论证这一点，我们实际上可以画出第一期的收益矩阵是什么样子。让我们用上周的方式来做。好，两周前，在我们离开之前。好，这里又是我们之前的矩阵，我想分析第一阶段博弈。在第一阶段博弈中，我要做的是加入我明天将获得的收益。

[段 11]

The payoffs I’m going to get from tomorrow are from tomorrow’s equilibrium. Well, this isn’t going to do very much for me, as we’ll see, because I’ll get 2 plus 0 tomorrow, because we know I’m playing Defect tomorrow. 2 plus 0 tomorrow, minus 1 plus 0 tomorrow, 3 plus 0 tomorrow, 3 plus 0 tomorrow, minus 1 plus 0 tomorrow, and 0 plus 0 tomorrow, 0 plus 0 tomorrow. So just as we did with the war of attrition game two weeks ago, we can put in the payoffs from tomorrow. We can roll back those equilibrium payoffs to today. It’s just in this particular exercise, it’s rather a boring thing because I’m just adding 0 to everything. When I add 0 to everything and I then just cancel out the 0 I back where I started And of course I should defect So what I going to see is because I going to defect anyway tomorrow today is just like a one-shot game as well, and I’m going to get defect again. Now here we played the game twice and got defect, defect. What about if we played the game three times? It’s the same thing, right? We played the game three times, but we did play the game three times between Edwina and Ben. and there we know we’re going to defect in the third round, therefore we may as well defect in the second or last round, therefore we may as well defect in the first round.

[译文 11]

我明天将获得的收益来自明天的均衡。好，这不会对我有太大帮助，我们会看到，因为我明天会得到 2 加 0，因为我们知道明天我会采取背叛。明天 2 加 0，明天负 1 加 0，明天 3 加 0，明天 3 加 0，明天负 1 加 0，以及明天 0 加 0，明天 0 加 0。就像我们两周前在消耗战博弈中做的那样，我们可以放入明天的收益。我们可以把那些均衡收益回滚到今天。只是在这个特定的练习中，这是一件相当无聊的事情，因为我只是在给一切加 0。当我给一切加 0，然后我取消那个 0，我就回到了起点。我当然应该背叛。所以我看到的是，因为我明天反正要背叛，今天就像一个一次性博弈一样，我又要背叛了。现在我们玩了两次这个博弈，得到了背叛、背叛。如果我们玩三次呢？是一样的，对吧？我们玩了三次博弈，但 Edwina 和 Ben 之间我们确实玩了三次。我们知道我们会在第三轮背叛，因此我们不如在第二轮或最后一轮背叛，因此我们不如在第一轮背叛。

[段 12]

And if we get 5 times, we know we’re going to all defect in the fifth round, therefore we may as well all defect in the fourth round, therefore we may as well all defect in the third, and so on. If we play it 500 times, we wouldn’t have time in the class, but if we played it 500 times, we know in that 500th period it’s a one-shot game in people going to defect. And therefore in the 499th period, people are going to defect. and therefore in the 498th period people are going to defect and so on. So the problem here is that we get unravelling, what we’ve seen before in this class, we get unravelling from the back. I have a worry that there might only be one L in unravelling in America. Is that right? How many L’s have you put in unravelling in America? One I just come back from England and my spelling is somewhere in the middle antic right now I leave it as one All right unraveling from the back Essentially this is a backward induction argument Oh instead of using backward induction we’re really using sub-game perfection. We’re looking at the equilibria in the last games and as we roll back up the game we get unraveling. So here’s bad news. The bad news is we’d hoped that by having repeated interaction in the prisoner’s dilemma, we would be able to sustain cooperation.

[译文 12]

如果我们玩 5 次，我们知道我们会在第五轮全部背叛，因此我们不如在第四轮全部背叛，因此我们不如在第三轮全部背叛，以此类推。如果我们玩 500 次，我们在课堂上没有时间，但如果我们玩 500 次，我们知道在第 500 期这是一个一次性博弈，人们会背叛。因此在第 499 期，人们会背叛。因此在第 498 期人们会背叛，以此类推。所以这里的问题是，我们得到解构，我们之前在这个课堂上看到过的，我们从后面解构。我有一个担心在美国 unraveling 可能只有一个 L。对吗？你在美国的 unraveling 中放了多少个 L？一个。我刚从英国回来，我的拼写现在有点介于两者之间，我留作一个。好，从后面解构。从根本上说这是一个逆向归纳论证，哦，我们实际上使用的是子博弈完美。我们看最后博弈中的均衡，当我们回滚博弈时我们得到解构。所以这是坏消息。坏消息是我们曾希望通过在囚徒困境中的重复互动，我们能够维持合作。

[段 13]

That’s been our hope since day one of the class. In fact, we stated it kind of confidently in the first day of the class, and we kind of intuitively believe it, but what we’re discovering is even if you played this game for 500 times and then stopped, you wouldn’t be able to sustain cooperation in equilibrium because we’re going to get unraveling in the last stage and so on and so forth. All right? So it seems like our big hope that repeat interaction would induce corporation society is going down the plug hole. That’s bad. Alright? So let’s come back and modify our lesson a little bit. So what went wrong here was in the last period of the game there was no incentives generated by the future. Right So there was no promise of future rewards or future punishments And therefore cooperation broke down and then we had unraveling So the lesson here is what The lesson is, but for this to work, it helps to have a future. It helps to have a future. This whole idea of repeat interaction was the future was going to create incentives for the present, but if the game’s come to an end, there’s going to be some point when there isn’t a future anymore, and then we get unraveling. Now this is not just a formal technical point to be made in the ivory tower of Yale.

[译文 13]

这自开课第一天起就是我们的希望。事实上，在开课第一天我们就相当自信地陈述了这一点，而且我们凭直觉相信它，但我们的发现是，即使你玩这个游戏500次然后停下来，你也无法在均衡中维持合作，因为我们在最后阶段会出现崩溃等等。明白吗？所以看起来我们对重复互动会诱导合作社会的巨大希望正在付诸东流。这很糟糕。好吧？让我们回过头来稍微修改一下我们的课程。那么这里出了问题的是什么，在游戏的最后一个时期，没有来自未来的激励。明白吗？所以没有未来奖励或未来惩罚的承诺，因此合作崩溃了，然后我们出现了崩溃。所以这里的教训是什么，教训是，但为了让这个机制起作用，有一个未来会有所帮助。有一个未来会有所帮助。整个重复互动的想法是，未来会为现在创造激励，但如果游戏走到尽头，将会有某个时刻不再有未来，然后我们就会崩溃。这不仅仅是一个在耶鲁象牙塔里提出的形式技术问题。

[段 14]

This is a true idea. So, for example, if we think about CEOs or presidents or managers of sports teams, there’s a term we use for, there’s a word we use, at least in the States, to describe such leaders when they’re getting towards the end of their term and everyone knows it. What’s the expression we use? Lame duck. All right? So we have this lame duck effect. Okay. towards the end of their term and everyone knows it. What’s the expression we use? Lame duck. All right? So we have this lame duck effect. The lame duck effect at the end of somebody’s term undermines their ability to cooperate, their ability to provide incentives for people to cooperate with them, and causes a problem. So this lame duck effect affects presidents, but it also affects CEOs of companies. All right? But it’s not just leaders who run into this problem. So if you have an employee, if you’re employing somebody, and you may have a contract with the person you’re employing, but basically you’re sustaining cooperation with this person because you interact with them often, you know you’re always going to interact with them, but then this employee approaches retirement, everyone knows that in April or something they’re going to retire, then the future can’t provide incentives anymore, and you have to switch over from the implicit incentive of knowing you’re going to be interacting in the future to an explicit incentive of putting incentive clauses in the contract.

[译文 14]

这是一个真实的观念。比如，如果我们想到CEO或总统或运动队经理，有一个词我们用来描述这样的领导者，当他们接近任期结束而且每个人都知道的时候。我们用什么表达？跛脚鸭。好的？所以我们有这个跛脚鸭效应。好的。在他们任期接近尾声而且每个人都知道的时候。我们用什么表达？跛脚鸭。好的？所以我们有这个跛脚鸭效应。在某人任期结束时的跛脚鸭效应削弱了他们合作的能力，为人们提供合作激励的能力，并造成问题。所以这个跛脚鸭效应影响总统，但也会影响公司的CEO。好的？但不仅仅是领导者会遇到这个问题。如果你雇用某个人，你可能与这个人有合同，但你基本上是因为经常与他们互动而维持与这个人的合作，你知道你总是会与他们互动，但然后这个员工接近退休，每个人都知道他们将在四月或什么时候退休，那么未来无法再提供激励，你必须从知道未来会互动的隐性激励切换到在合同中加入激励条款的显性激励。

[段 15]

So retirement can cause, if you like, a lame duck effect. Retirements. And this is even true in personal relationships With your personal relationships with your friends if you think that those friendships are going to go on for a long time be they with your significant other or just with the people you hang out with you’re likely to get a lot of cooperation. But if, as with perhaps most economic majors, most of your significant others are only going to last for a day at most, you’re not going to get great cooperation, right? You’re going to get cheating. All right? All right? No one’s rising to that one, but I guess it’s true. So what do we call these? Economics majors relationships. These are kind of end effects. All of these things are caused by the fact that the relationship is coming to an end. And once the relationship is coming to an end, all those threats and promises of future behavior, implicit or otherwise, are going to basically disappear. So at this point, we might think the following. You might conclude the following. You might conclude that if a relationship has a known end if everyone knows the relationship is going to end at a certain time then we done And we basically can sustain cooperation through repeat interaction All right And that kind of what the example we looked at seems to suggest However, that’s not quite true.

[译文 15]

所以退休可以引起，如果你愿意的话，跛脚鸭效应。退休。这在个人关系中甚至也是真实的。与朋友的个人关系，如果你认为那些友谊会持续很长时间，无论是与你的另一半还是只是与你一起出去玩的人，你可能会获得很多合作。但如果你，像大多数经济学专业学生那样，大多数你的另一半最多只持续一天，你就不会得到很好的合作，对吧？你会得到欺骗。好的。好的。没人回应这个，但我觉得这是真的。那么我们称这些为什么？经济学专业学生的关系。这些是某种末期效应。所有这些事情都是由关系即将结束这一事实引起的。一旦关系即将结束，所有那些关于未来行为的威胁和承诺，隐性的或其他，基本上都会消失。所以此时，我们可能会想以下几点。你可能会得出以下结论。你可能会得出结论，如果一段关系有已知的结局，如果每个人都知道这段关系将在某个时候结束，那么我们就完成了，我们基本上可以通过重复互动来维持合作。好的，这正是我们看的例子所表明的。然而，这并不完全正确。

[段 16]

That’s not quite true. So let’s look at another example where a relationship is going to have a known end, but nevertheless we are able to sustain some cooperation, and we’ll see how. Okay? So again, I’ve been careful here. I’ve said it helps to have a future. I haven’t said it’s necessary to have a future. All right? So that’s good news for the economics majors again. All right? So let’s do this example to illustrate that even a finite interaction, even an interaction that’s going to end, and everyone knows it’s going to end, might still have some hope for cooperation. We look at this slightly more complicated game here, and this game has three strategies, we’ll call them A, B, and C, for each player. And the payoffs are as follows, 4 4 0 5 0 0 down here we do 0 0 0 33 and the middle row 50 11 00 alright and we’re going to assume that this game just like we did with the first with the first time we did Prisoner’s Dilemma this game is going to be played twice alright, it’s going to be repeated it’s going to be played twice, repeated once so let’s just make sure we understand what the point of this game is in this game, in the one shot game I hope it’s clear that AA is kind of the cooperative thing to do we’d like to sustain we’d like to sustain play of AA because then both players get 4 and that looks pretty good for everybody However, in the one-shot game, in the one-shot game, AA is not a Nash equilibrium.

[译文 16]

这并不完全正确。所以让我们看另一个例子，其中一段关系将有一个已知的结局，但我们仍然能够维持某种合作，我们会看到如何做到。好的？我在这里一直很谨慎。我说过有一个未来会有所帮助。我没有说过有一个未来是必要的。好的？对于经济学专业的学生来说，这是个好消息。好的？让我们做这个例子来说明，即使是一次有限的互动，即使是一次即将结束而且每个人都知道即将结束的互动，仍然可能有合作的一些希望。我们看看这个稍微复杂一点的博弈，这里，这个博弈有三个策略，我们称它们为A、B、C，每个玩家都有。收益如下，4 4 0 5 0 0 在下面我们做 0 0 0 3 3 以及中间行 5 0 1 1 0 0 好的，我们假设这个博弈就像我们第一次做的那样，就像我们第一次做囚徒困境时一样，这个博弈将进行两次，好的，它将重复一次，所以让我们确保我们理解这个博弈的要点，在这个博弈中，在一次性博弈中我希望AA是合作的事情是清楚的，我们想维持我们想维持AA的游戏，因为那样两个玩家都得到4，这对每个人来说看起来都很好。然而，在一次性博弈中，在一次性博弈中，AA不是一个纳什均衡。

[段 17]

Why is AA not a Nash equilibrium? Let me grab those mics again. Why is AA not a Nash equilibrium? I’ll just do it with one. Anybody? Yep. I’m even getting the names of the statements and terms. This is Katie, right? So shout out. The best response to the other guy playing A Now she can deliver him. I’ll just do it with one. Anybody? Yep. I’m even getting the names of the terms. This is Katie, right? So shout out. The best response to the other guy playing A is playing B. Good, good. So if I think the other person’s going to play A, I’m going to want to defect and play B and obtain a gain of 1. So I’ll get 5 rather than 4. I’ll defect to playing B and get 5 rather than 4 for a gain of 1. Is that right? All right. So AA is not a Nash equilibrium in the one-shot game. We’re sometimes going to call that, well, that’s fine, in the one-shot game. So now imagine we play this game twice. So just playing once, we’re going to play this game two times. All right. So now, I’ll come back to that. Before I do that, what are the pure strategy Nash equilibria in this game? Anybody? So BB is the Nash Equilibria. The Nash Equilibria in this one game are BB and CC BB there some mixed ones as well but this will do So BB and CC Alright BB and there some mixed ones as well but this will do Alright So BB and CC are the pure strategy in Nash Equilibrium Alright now consider playing this game twice.

[译文 17]

为什么AA不是一个纳什均衡？让我再拿一下麦克风。为什么AA不是一个纳什均衡？我只和一个说。任何人？是的。我甚至在获取声明和术语的名称。这是Katie，对吧？所以请大声说出来。对方玩A时的最佳应对。现在她可以给他。我只和一个说。任何人？是的。我甚至在获取术语的名称。这是Katie，对吧？所以请大声说出来。对方玩A时的最佳应对是玩B。好的，好的。所以如果我认为对方将玩A，我将要背叛并玩B，获得1的收益。所以我将得到5而不是4。我将背叛去玩B，得到5而不是4，获得1的收益。对吗？好的。所以在一次性博弈中AA不是一个纳什均衡。我们有时会称那个，呃，那没关系，在一次性博弈中是这样的。所以现在想象我们玩这个博弈两次。所以只玩一次，我们将玩这个博弈两次。好的。所以现在，我将回到那个。在我做那个之前，这个博弈中的纯策略纳什均衡是什么？任何人？所以BB是纳什均衡。这个博弈中的纳什均衡是BB和CC。BB还有一些混合的，但这样就可以了。所以BB和CC。好的，BB还有一些混合的，但这样就可以了。好的，所以BB和CC是纯策略纳什均衡。好的，现在考虑玩这个博弈两次。

[段 18]

Alright? Now last time we looked at a game played twice, well it was Prisoner’s Dilemma, and we noticed that we couldn’t sustain cooperation because in the last stage people weren’t going to cooperate and and hence in the first stage people weren’t going to cooperate. But let’s look what happens here. If this game is played twice, is there any hope of sustaining cooperation, i.e. A, in both stages? Could we have people play A in the first stage and then play A again in the second stage? So Patrick’s shaking his head, so that’s right. Shake his head if he grabbed the other leg. So why is that not going to work? Why can’t we get people to cooperate and play A in both periods? In the second period, you’re still in effect in play B. Good. Good. So in the second period, exactly the argument that Katie produced just now in the one-shot game applies because the second period game is a one-shot game. So we’ve got no hope of sustaining cooperation in both periods. Let’s call this cooperation. We can sustain We can sustain AA in period 2 in the second period However, I claim that we may be able to get people to cooperate in the first period of the game. Now how are we going to do that? So to see that, let’s consider the following strategy.

[译文 18]

好吗？上一次我们看了一个进行两次的博弈，呃，那是囚徒困境，我们注意到我们无法维持合作，因为在最后阶段人们不会合作，因此在第一阶段人们也不会合作。但让我们看看这里发生了什么。如果这个博弈进行两次，是否有任何维持合作，即A，在两个阶段的希望？我们能让人们在第一阶段玩A，然后在第二阶段再次玩A吗？Patrick在摇头，所以这是对的。如果他抓住了另一条腿就摇头。为什么这不会起作用？为什么我们不能让人们合作并在两个时期都玩A？在第二时期，你仍然在玩B。好的。好的。所以在第二时期，正如Katie刚才在一次性博弈中提出的论点完全适用，因为第二时期的博弈是一次性博弈。所以我们在两个时期都没有维持合作的希望。让我们称之为合作。我们可以维持我们可以在第二时期维持AA在第二时期。然而，我声称我们可能能够使人们在游戏的第一时期合作。现在我们要怎么做？所以为了看到这一点，让我们考虑以下策略。

[段 19]

So, but, consider the strategy. The strategy is going to be play A and then play C if A, A was played. and play B otherwise. Alright, so this strategy is an instruction telling the player how to play. Now before we consider whether this is an equilibrium or not we need to have some inverted commas here as well before we consider whether this is an equilibrium or not let just check that this actually is a strategy So what does a strategy have to do? It has to tell me what I should do, it should give me an instruction, at each of my information sets. In this two-period game, each of us, each of the players in the game, have two information sets. They have an information set at the beginning of the game, and they have another information set at the beginning of period 2. Is that right? So it has to tell you what to do at the first information set and at the second information set, and it does. It says play A at the first one. And at the beginning of period 2, well, now I said there’s only one information set there, but actually there’s nine possible information sets depending on what happened in the first period. So each thing that happened in the first period is associated with a different information set.

[译文 19]

那么，让我们来考虑这个策略。这个策略是：先出A，如果出了A A，则接下来出C；否则就出B。好，这个策略是一个指令，告诉玩家该怎么玩。现在，在我们考虑这是否是一个均衡之前，我们也需要在这里加上一些引号。在我们考虑这是否是一个均衡之前，让我们先确认这确实是一个策略。那么策略需要满足什么条件呢？它需要告诉我应该做什么，它应该给出一个指令，在我的每个信息集上。在这个两期博弈中，每个参与者、博弈中的每个玩家都有两个信息集。他们在博弈开始时有一个信息集，在第二期开始时还有另一个信息集。对吧？所以策略必须告诉你在第一个信息集上该怎么做，以及在第二个信息集上该怎么做，而它确实做到了。它说在第一个信息集上出A。而在第二期开始时，嗯，我刚才说那里只有一个信息集，但实际上有九种可能的信息集，取决于第一期发生了什么。所以第一期发生的每种情况都对应一个不同的信息集。

[段 20]

I always know what happened in the first period. And at each of those nine information sets, it tells me what to do at the beginning of period two. In particular, it says, if it turns out that A, A was played, then play C now. And otherwise, all the other eight possible information sets I could find myself in, play B. So this is a strategy. Now, of course, the big question is, is this strategy an equilibrium? And in particular, is it a sub-game perfect equilibrium? Is this a sub-game perfect equilibrium? Let me be a bit more precise. If both players were playing this strategy, would that be a sub-game perfect equilibrium? All right, well, let’s have a look. And of course, I can’t see it now, so let’s pull both these boys down. So to check whether this is a sub-game perfect equilibrium, we’re going to have to check what? We’re going to have to check that it induces Nash behavior in each sub-game. I think the battery’s going on, Matt. Shall I get rid of that? Sorry Okay I going to shout Can people still hear me People in the balcony can they hear me Yep okay So we going to have to see if we can sustain Nash behavior Yeah, thanks. We’re going to see if we can sustain Nash behavior in every sub-game.

[译文 20]

我总是知道第一期发生了什么。在这九个信息集中的每一个上，它都告诉我第二期开始时该怎么做。具体来说，它说，如果结果是A A被出了，那么现在就出C。否则，在我可能身处的那其他八个信息集中，出B。所以这是一个策略。当然，最大的问题是，这个策略是一个均衡吗？特别是，它是一个子博弈完美均衡吗？这是一个子博弈完美均衡吗？让我说得更精确一些。如果两个玩家都采用这个策略，那会是一个子博弈完美均衡吗？好，让我们来看一下。当然，我现在看不到，所以让我们把这两个都拉下来。为了检查这是否是一个子博弈完美均衡，我们需要检查什么？我们需要检查它在每个子博弈中是否诱导出纳什行为。我想电池快没电了，Matt。我把它关掉好吗？抱歉。好，我要喊了。大家还能听到我吗？包厢里的人能听到我吗？是的，好的。所以我们要看看是否能维持每个子博弈中的纳什行为。

[段 21]

So let’s start with the sub-games associated with the second period. Technically, there are nine such sub-games, depending what happened in the past, depending what happened in the first period. There’s a sub-game following AA, there’s a sub-game following AB, there’s a sub-game following AC, and so on. Should we be okay? Put it on a second. So for each activity in the first period, for each profile in the first period, there’s a sub-game. However, it doesn’t really matter to distinguish all of these sub-games particularly carefully here. since the cost from the past, what happened in the past, is sunk. So we’ll just look at them as a whole. So in period 2 after AA so one in particular of those nine sub after AA this strategy induces CC. If both people play A in the first period, then in the sub-game following, people are supposed to play CC. Is that a Nash equilibrium of the sub-game? Was CC a Nash equilibrium? Yeah, it’s one of our Nash equilibriums. Let’s look up there. We’ve got it listed. Here it is. So we’re playing this Nash equilibrium. So that is a Nash equilibrium, so we’re okay. All right? All right? after the other choices in period one, then this strategy induces BB. B That good news too because BB we already agreed was a Nash equilibrium in the one game All right so in all of those nine sub the one after AA and the eight after everything else we’re playing Nash behavior.

[译文 21]

让我们从第二期相关的子博弈开始。从技术上讲，有九个这样的子博弈，取决于过去发生了什么，取决于第一期发生了什么。有一个跟在A A之后的子博弈，有一个跟在A B之后的子博弈，有一个跟在A C之后的子博弈，以此类推。我们没问题吧？再放一下。所以对于第一期的每种行动，对于第一期的每个组合，都有一个子博弈。然而，在这里仔细区分所有这些子博弈并没有那么重要，因为来自过去的成本，即过去发生的事情，已经是沉没成本了。所以我们就把它们作为一个整体来看。那么在第二期，在A A之后，这九个中的一个，在这个特定的子博弈中，这个策略诱导出C C。如果两个人都在第一期出A，那么在这个子博弈中，人们应该出C C。这是一个纳什均衡吗？C C是一个纳什均衡吗？是的，这是我们的纳什均衡之一。让我们看上面。我们有列出。在这儿。所以我们正在玩这个纳什均衡。所以这是一个纳什均衡，我们没问题。好吧？好的。在第一期其他选择之后，这个策略诱导出B B。这也是好消息，因为B B我们已经同意是一个纳什均衡。所以在这九个中的每一个子博弈中，无论是跟在A A之后的那个，还是其他八个之后的那个，我们都在玩纳什行为。

[段 22]

So that’s good. What about in the whole game? In the whole game. And the whole game is starting from period one. We have to ask, do you do better to play the strategy as designated, in particular to choose A, or would you do better to defect? Well, let’s have a look. So if I choose A, then my… Remember, the other person is playing this strategy. So if I choose A, then my payoff in this period comes from AA a and is 4. If I choose a, then we’re both playing a in this period, and I get 4. And tomorrow, according to this strategy, tomorrow, since we both played a, both of us will now play C. Since we’re both playing C, I’ll get an additional payoff of 3. So tomorrow, CC will occur, and I’ll get 3 for a total of 7.

[译文 22]

所以这很好。整个博弈呢？整个博弈。整体博弈是从第一期开始的。我们必须问，按照指定的策略来做，特别是选择A，是否更好，还是背叛会更好？让我们来看一下。如果我选择A，那么我的……记住，对方在玩这个策略。所以如果我选择A，那么我在这期的收益来自A A，是4。如果我选择A，那么我们俩在这期都出A，我得到4。明天，根据这个策略，因为我们俩都出了A，我们俩现在都要出C。因为我们都出C，我额外得到3的收益。所以明天会出现C C，我得到3，总共是7。

[段 23]

What about if I defect? well we could consider lots of possible defections but let’s just consider the obvious defection you can check the other ones at home so if I defect if I defect and choose B now then in this period in this period I will be playing B and my opponent or my pair will be playing A so in this period I will get 5 and tomorrow since AA did not occur both of us will play B and get a continuation payoff of 1 so the continuation payoff this time will be following from BB and I get 1 Why don’t I do what I’ve been doing before in this class and put boxes around the continuation payoff, just to indicate that they are in fact continuation payoffs. alright so if I play A I get 4 now and a continuation payoff of 3 for a total of 7 if I play B now yeah I gain something now I get 5 now but tomorrow I’ll only get 1 for a total of 6 alright so in fact 7 is bigger than 6 so I’m ok and I won’t want to do this defection alright I just want to write this one other way because it’s going to be useful for later so one other way to write this I think we’ve convinced ourselves that this is an equilibrium but one other way to write this and it’s a more general way in repeated games is to write it explicitly comparing the temptations to cheat today with the rewards and punishments from tomorrow so what we want to do is in general we can just rewrite this as checking that the temptation to cheat or defect today is smaller than the value of the reward minus the value of the punishment.

[译文 23]

如果我背叛呢？嗯，我们可以考虑很多可能的背叛，但让我们只考虑明显的背叛，你可以在家里检查其他的情况。如果我背叛，如果我背叛现在选择B，那么在这期，在这期，我将出B，而我的对手或我的搭档将出A，所以在这期我将得到5，而明天因为没有出现A A，我们俩都将出B，得到1的延续收益。所以延续收益这次将来自B B，我得到1。为什么我不像之前在这门课上做的那样，在延续收益周围画方框呢，只是为了表明它们实际上是延续收益。好的，所以如果我出A，我现在得到4，延续收益是3，总共是7。如果我现在出B，嗯，我现在有所得，我得到5，但明天我只得到1，总共是6。好的，所以实际上7大于6，所以我没问题，我不想进行这个背叛。好的，我只想用另一种方式来写这个，因为它对后面会有用。所以另一种方式来写这个，我认为我们已经说服自己这是一个均衡，但另一种方式，而且是重复博弈中更通用的方式，是明确地比较今天的背叛诱惑与明天的奖励和惩罚。所以我们要做的是，一般来说，我们可以把它改写为检查背叛或今天作弊的诱惑是否小于奖励的价值减去惩罚的价值。

[段 24]

But the key words here are defecting occurs today, rewards and punishment occur tomorrow. Alright? If we just rewrite it this way, we’ll see exactly the same thing. Just rearranging slightly. The temptation to defect today is I get 5 rather than 4, or if you like a gain of 1. And the value of the reward tomorrow was the value of the reward, the reward was to play CC tomorrow and get three The value of the punishment tomorrow was to play BB tomorrow and get one right And that is two So here the fact that the temptation is outweighed by the difference between the value of the reward and the value of the punishment is what enabled us to sustain cooperation. I’m just writing that in a more general way because this is a way that we can apply in games from here on. We’re going to compare temptations to cheat with tomorrow’s promises. Patrick, yeah. Let me get you a mic. I don’t understand why it’s reasonable to think you would play BB in the second period, though. Instead of, in the second period, you have a temptation to play CC, even if the person defected on you. Good, good. That’s a very good point. So what Patrick’s saying is, how come, it’s all very well to say we’re sustaining cooperation in the first period here, but the way in which we sustained cooperation was by going along with, as it were, the punishment tomorrow.

[译文 24]

但这里的关键词是：背叛发生在今天，奖励和惩罚发生在明天。对吧？如果我们用这种方式改写，我们会看到完全相同的东西。只是稍微重新排列一下。今天背叛的诱惑是我得到5而不是4，或者说是一个加1的收益。明天的奖励的价值是奖励的价值，奖励是在明天出C C并得到3。明天的惩罚的价值是在明天出B B并得到1。好的，那就是2。所以在这里，诱惑被奖励的价值和惩罚的价值之间的差异所抵消，这正是使我们能够维持合作的原因。我只是用更一般的方式来写，因为这是我们从这里开始在博弈中应用的方式。我们将把作弊的诱惑与明天的承诺进行比较。Patrick，好的。让我给你一个麦克风。我不明白为什么你会认为在第二期会出B B。虽然，在第二期，你有一个出C C的诱惑，即使对方背叛了你。好的，好的。这是一个非常好的观点。所以Patrick说的是，这一切都很好说我们在第一期维持合作，但维持合作的方式是，可以说是，顺从明天的惩罚。

[段 25]

It required me tomorrow to go along with the strategy of choosing B if I cheated in the first period. And I want to answer this twice, once disagreeing with him, and once agreeing with him. So let me just disagree with him first. So notice tomorrow, if the other person, the other player, is going to play B, then I’m going to want to play B. So the key idea here is, as always in Nash equilibrium, if I take the other person’s player as given and just look at my own behavior, if I think the other person is playing the strategy, and hence he’s going to play B tomorrow after I’ve cheated, then I want to play B myself. So that check is just our standard check, and actually that’s a check that makes sure that it really is a sub-game perfect equilibrium. We’re not putting some punishments down the tree that are arising out of equilibrium. It has to be that I want to do tomorrow what I’m told to do tomorrow. So that idea seems right, and I’m glad Patrick raised it because it was the next thing in my notes. I want to go along with this punishment because if the other person’s playing B, I want to play B myself. All right? So that’s that. Nevertheless, I think Patrick’s onto something, and let me come back to it in a minute.

[译文 25]

这要求我在明天如果我在第一阶段作弊的话，要遵从选择B的策略。我想两次回答这个问题，一次不同意他的观点，一次同意他的观点。让我先不同意他。所以注意，明天如果另一个人，另一个参与者，要玩B，那我就会想玩B。所以关键想法是，正如纳什均衡一贯的情况，如果我把对方参与者视为既定前提，只看我的自己的行为，如果我认为对方在执行那个策略，因此他在我在第一阶段作弊之后明天会玩B，那么我自己就想玩B。所以那个检查就是我们的标准检查，而且实际上这是一个确保它真的是子博弈完美均衡的检查。我们没有在博弈树中放置一些不处于均衡状态的惩罚。必须是我明天想做我被告知明天要做的事情。所以这个想法看起来是对的，我很高兴Patrick提出来，因为这正好是我笔记中的下一个内容。我想要遵从这种惩罚，因为如果对方在玩B，我自己就想玩B。好吧？就这样。不过，我认为Patrick触及到了什么，让我稍后回来讨论它。

[段 26]

I’ll come back to it in a minute. All right? But what I want to do before I do that is just draw out a general lesson from this game And the general lesson is we can sustain cooperation even in a finitely repeated game but to do so, we need there to be more than one Nash equilibrium in the stage game. What we need there to be is several Nash equilibria, one at least of which we can use as a reward, and another one which we can use as a punishment. So even if a game is only played a finite number of times, if there are several equilibria in the stage game, let me just bring this down and show again, several equilibria in the stage game, both BB and CC, we can use one of them as a reward and the other one as a punishment and use that difference to try and get people to resist temptations today. All right? So that’s the general idea here. Okay. And let’s just write that down. But Patrick, don’t let me get away with not coming back to your point. I want to come back to it in a second. All right So the lesson here is if a stage game the stage game is the game that going to be repeated If a stage game has more than one Nash equilibrium in it, then we may be able to use the prospect of playing different equilibria different equilibria tomorrow to provide incentives and we can think of these incentives as rewards and punishments as rewards and punishments for cooperation today All right, in the game we just saw, there were exactly two pure strategy and natural equilibrium in the sub-game.

[译文 26]

我稍后会回来讨论它。好吧？但是在此之前我想做的是从这个博弈中得出一个一般性教训。一般性教训是我们即使在有限重复博弈中也能维持合作，但要做到这一点，我们需要阶段博弈中有不止一个纳什均衡。我们需要的是多个纳什均衡，其中至少有一个我们可以用作奖励，另一个可以用作惩罚。所以即使一个博弈只被玩有限次数，如果阶段博弈中有多个均衡——让我把它写下来再展示一下，阶段博弈中的多个均衡，B-B和C-C——我们可以用其中一个作为奖励，另一个作为惩罚，并利用这种差异来让人们抵制今天的诱惑。好吧，这就是这里的一般想法。好吧，让我们把它写下来。但是Patrick，别让我逃避不回到你的观点。我想稍后再回到它。好吧，这里的教训是，如果一个阶段博弈——阶段博弈就是将要重复的博弈——如果一个阶段博弈中有多个纳什均衡，那么我们也许能够利用明天玩不同均衡的前景来提供激励，我们可以把这些激励视为对今天合作的奖励和惩罚。好吧，在我们刚才看到的博弈中，子博弈中恰好有两个纯策略自然均衡。

[段 27]

We used one of them as a reward and the other one as a punishment, and we were able to sustain cooperation in a sub-game perfect equilibrium. liberal. All right, now a question arises here, and I think it’s behind Patrick’s question, and that is how plausible is this? How plausible is this? We really, it’s okay, formally, if we write down the game and do the math, this comes out, but how plausible is this as a model of what’s going on in society? All right, and I think the worry, I’m guessing this is the worry that was behind Patrick’s question, is this. Suppose I’m playing this game with Patrick, and suppose Patrick cheats on me the first period. So Patrick chooses B when I wanted him to choose A in the first period. And now in the second period, according to the equilibrium instructions, we’re supposed to play B-B and get payoffs of 1 rather than C-C and get payoffs of 3. Let’s make that visible again. But suppose Patrick comes to me in the meantime. So between period 1 and period 2, Patrick shows up at my office hours and he says, yeah, I know I cheated on you yesterday, but why should we punish ourselves today? Why should we both of us lose today by playing the BB equilibrium? Why don’t we both switch to the CC equilibrium?

[译文 27]

我们用一个作为奖励，另一个作为惩罚，我们能够在子博弈完美均衡中维持合作。自由的。好吧，现在这里出现了一个问题，我认为这在Patrick的问题背后，而这问题就是这有多合理？这有多合理？我们真的，从形式上讲，如果我们写下博弈并做数学计算，这确实会得出结果，但作为社会现象的模型，这有多合理？好吧，我认为这个担忧，我猜这就是Patrick问题背后的担忧，就是这个。假设我和Patrick在玩这个博弈，假设Patrick在第一阶段欺骗了我。所以Patrick在我要他第一阶段选择A时选择了B。现在在第二阶段，根据均衡指令，我们应该玩B-B并得到1的收益而不是C-C并得到3的收益。让我们再把它显示出来。但假设Patrick在此期间来找我。所以在第一阶段和第二阶段之间，Patrick出现在我的办公时间并说，是的，我知道我昨天欺骗了你，但为什么我们今天要惩罚自己？为什么我们两个人都要通过玩BB均衡而在今天亏损？为什么我们不都转到CC均衡？

[段 28]

After all, that’s better for both of us. It’s true that Patrick’s saying to me, it’s true that I cheated you yesterday, but let bygones be bygones, or why cry over spilt milk? He’ll use some other saying plucked out of the Book of Plattitudes and say to me, well, why go along with the punishment? Let’s just play the good equilibrium now. And if I look at things, I say, well, actually, it’s true I got nothing in the first period because Patrick kind of cheated me in the first period. So it’s true I got nothing yesterday. And it true it was Patrick who caused me to get nothing yesterday But nevertheless that a sunk cost I comparing getting 1 now with getting 3 now why don I just go along and get 3 And in fact moreover I not in danger of being cheated again because if Patrick believes I’m going to play C, he’s going to play C2. So that kind of argument involves some kind of communication between stages, but it sounds like that’s going to be a problem. Why? Well, suppose it’s the case that we are going to get communication between periods, and suppose it’s the case that someone with the gift of the gab, someone on his way to law school like Patrick, is going to be able to persuade me to go back to the good equilibrium for everybody in period two, then we know we’re going to play the good equilibrium in period two, and now we’ve lost any incentive to cooperate in period one.

[译文 28]

毕竟，这对我们两个都更好。Patrick对我说的是，真的，我昨天欺骗了你，但过去的事就让它过去吧，或者为什么要为打翻的牛奶哭泣？他会用一些从《陈词滥调之书》里摘出的其他谚语对我说，好吧，为什么要遵从惩罚？让我们现在就玩那个好的均衡。如果我看一下情况，我说，好吧，实际上，我在第一阶段确实什么都没得到，因为Patrick在第一阶段欺骗了我。所以确实我昨天什么都没得到。确实，是Patrick导致我昨天什么都没得到。但尽管如此，那是沉没成本，我比较现在得到1和现在得到3，为什么我不就遵从并得到3？而且事实上我不会再面临被骗的危险了，因为如果Patrick相信我要玩C，他会玩C2。所以这种论点涉及某种阶段之间的沟通，但这听起来会成为一个问题。为什么？好吧，假设我们确实会有阶段之间的沟通，假设有一个能说会道的人，一个像Patrick那样正在去法学院路上的人，将能够说服我在第二阶段回到对每个人都有利的好的均衡，那么我们知道我们将在第二阶段玩好的均衡，而现在我们失去了在第一阶段合作的任何动机。

[段 29]

The only reason I was willing to cooperate in period one was because the temptation to defect was outweighed by the difference between the value of the reward and the value of the punishment. If we’re going to get the reward anyway, I’ll go ahead and defect today. All right? So the problem here is this notion of renegotiation, this notion of communicating between periods can undermine this kind of equilibrium There a problem that arises if we have renegotiation All right So there may be a problem of renegotiation. alright this problem may not be such a big problem for example it may be I’ll say I’ll be so angry at Patrick because he screwed me over in period one that I won’t go along with the renegotiation and it may also be the case and we’ll see some examples of this on the homework assignments that the many equilibria in the second stage of the game are not such that a punishment for Patrick is also a punishment for me. What really caused the problem here was in trying to punish Patrick, I had to punish myself. But you could imagine games, and we’ll see some concrete examples in the next homework assignment, in which punishing Patrick is rather fun for me, and punishing me is rather fun for Patrick. And that’s going to be much harder to renegotiate our way out of.

[译文 29]

我愿意在第一阶段合作的唯一原因是因为作弊的诱惑被奖励的价值和惩罚的价值之间的差异所抵消。如果我们要得到奖励的话，我就在今天作弊了。好吧？所以这里的问题是重新谈判的概念，阶段之间沟通的概念可能会破坏这种均衡。如果我们有重新谈判，就会出现一个问题。好吧，所以可能存在重新谈判的问题。好吧，这个问题可能不是一个很大的问题，例如可能是这样，我将对Patrick非常生气，因为他在第一阶段坑了我，所以我不会遵从重新谈判，而且可能还有这样的情况，我们将在作业中看到一些例子，第二阶段博弈中的许多均衡并不是对Patrick的惩罚同时也是对我的惩罚。真正导致这里问题的是，在试图惩罚Patrick时，我不得不惩罚我自己。但你可以想象一些博弈，我们将在下次作业中看到一些具体例子，在那些博弈中惩罚Patrick对我来说是相当愉快的，而惩罚我对Patrick来说是相当愉快的。而那将更难通过重新谈判来摆脱。

[段 30]

There was a question Let me get a mic out to the question Ali back Yeah Go ahead Point to the microphone and shout If we’re ruling out renegotiation, can’t we devise a strategy for prisoners’ dilemma as well, even though it doesn’t have multiple Nash equilibriums? Okay, good. So the issue there is in prisoners’ dilemma, we established in the first week that if we’re not allowed to make side payments, we’re not allowed to bring in outside contracts, then no amount of communication is going to help us. So you’re right if we can rely on the courts or the mafia to enforce the contract, that would be fine, and then communication would have bite. But if you remember way back in the first week when we tried to talk our way out of bad behavior in the prisoner’s dilemma, it didn’t help precisely because it’s a dominant strategy. Whereas here, Patrick’s conversation, Patrick’s verbal agreement to play the other equilibrium is an agreement to play a Nash equilibrium. That’s what’s getting us into trouble. All right? All right? So what may help us here, what may avoid renegotiation, is simply I’m not going to go along with that renegotiation. I just am too angry about having been cheated on. And it may be, for other reasons, it may actually be that I enjoy the punishment. Nevertheless, this is a real problem in society, and it should retain.

[译文 30]

有一个问题。让我拿一个麦克风给后面的Ali。是的，请吧。指向麦克风并大声说。如果我们排除重新谈判，难道我们不能为囚徒困境设计一个策略吗，即使它没有多个纳什均衡？好吧，好的。那里的问题是，在囚徒困境中，我们第一周已经确定，如果我们不被允许进行旁支付，我们不被允许引入外部合同，那么再多的沟通也帮不了我们。所以如果我们能够依靠法院或黑手党来执行合同，那就没问题，那么沟通就会有咬劲。但是如果你记得早在第一周，当我们试图通过谈话摆脱囚徒困境中的不良行为时，那没有帮助正是因为它是一个优势策略。而在这里，Patrick的谈话，Patrick的口头同意玩另一个均衡是一个同意玩纳什均衡的协议。这就是让我们陷入麻烦的原因。好吧？好吧？所以可能帮助我们在这里、可能避免重新谈判的就是这样，我不去遵从那个重新谈判。我就是对于被骗感到太愤怒了。而且可能，出于其他原因，可能实际上我享受这个惩罚。尽管如此，这是社会中的一个现实问题，它应该保留。

[段 31]

I just am too angry about having been cheated on, and it may be, for other reasons, it may actually be that I enjoy the punishment. Nevertheless, this is a real problem in society, and we should pretend that this problem isn’t there. So a good example is in bankruptcy. Which is one of those words I can never spell. It seems to have too many consonants in it. Is that right? It’s approximately right, anyway. All right? So bankruptcy law in the U.S. for the last 200-odd years has gone through cycles. And one way to view these cycles is they’re cycles of relaxing the law and making life, quote, easier for borrowers, and then tightening up again. This is not only a recent phenomenon. This occurred throughout the 19th century. And so what typically happened was there was either explicit renegotiation between parties or renegotiation through acts of Congress or sometimes through the acts of the states in which bankrupt debtors were basically let off or given easier terms. And the argument was always the same. These people are not going to pay back now It clear from the 19th century often if you were bankrupt you were in jail actually worse than that but sometimes in the 19th century in England not only if you were bankrupt were you in jail, but your creditors were having to pay the fees to feed you in jail, so there you were sitting in jail, you weren’t paying that money back to your creditor, and you were actually costing money to your creditor by being in jail.

[译文 31]

我只是对被骗这件事太过愤怒了，而且可能出于其他原因，实际上可能是因为我享受惩罚。不过，这确实是社会中的一个真实问题，我们应该假装这个问题不存在。一个很好的例子就是破产。这个词我从来拼不对，似乎辅音字母太多了。是这样吗？大概差不多吧。好的吧。那么美国过去两百多年的破产法经历了周期。看待这些周期的一种方式是，它们是放松法律、让借款人的生活“更容易”的周期，然后又收紧。这不仅仅是最近的现象，在整个19世纪都发生过。通常发生的是，各方之间要么进行明确的重新谈判，要么通过国会法案重新谈判，有时通过州的法律重新谈判，破产债务人基本上被免除债务或获得更宽松的条款。理由总是一样的。这些人现在不会还钱了。19世纪的情况很清楚，如果你破产了，你会被关进监狱，实际上比这更糟，但在19世纪的英格兰，不仅破产会被关进监狱，债权人还得支付你在监狱里的伙食费，所以你就坐在监狱里，你不把钱还给债权人，而你的存在实际上是在让债权人赔钱。

[段 32]

This seems like a situation that you want to renegotiate your way out of. You say, hey, let’s let these guys out of jail, let them be productive again, and they’ll pay back part of the loans. So you have these waves of bankruptcy reform in which the debtors’ prisons were closed down, people were let out, people were relieved of debt. What’s the problem with doing that? That seems like a good idea, right? After all, you don’t want all these people bankrupt, in debt, not paying money back to their creditors anyway. That doesn’t seem like a good situation in society. It seems like a good renegotiation that’s a win-win situation. It’s better for everybody. What’s the problem with it, though? Yeah, let’s get a mic down here. What’s the problem with this? Yeah? It incentivizes bankruptcy. Right, it creates an incentive for people not to repay in the first place. It creates an incentive for people to take big risks now And hence it makes bankruptcy if you like or makes non of debts more likely So this has been going on for a while but you see it very much today if you read the financial pages of the papers in the last few weeks. There’s a big worry in the U.S. right now about people failing to repay. What kind of debt? What kind of debt is the big worry about?

[译文 32]

这看起来是一个你想要通过重新谈判来摆脱的局面。你说，嘿，让我们把这些家伙放出来，让他们重新变得有生产力，他们就会偿还部分贷款。所以你会看到这些破产改革的浪潮，债务人监狱被关闭，人们被放出来，债务被免除。这样做的问题是什么？这看起来是个好主意，不是吗？毕竟，你不会想让所有这些人都破产、负债、不还钱给债权人。那在社会中似乎不是一个好局面。这看起来像是一个双赢的好重新谈判，对每个人都有好处。但问题是什么呢？是的，让我们在这里放个麦克风。问题是什么？是的？它激励了破产。没错，它为人们首先不偿还创造了激励。它为人们现在冒大风险创造了激励，因此它使破产——如果你愿意这么说的话——或者使债务违约更可能发生。这已经持续了一段时间，但如果你在过去几周阅读报纸的金融版，你会非常明显地看到这一点。美国现在对人们无法偿还有很大的担忧。什么样的债务？什么样的债务是人们担忧的重点？

[段 33]

Mortgage debt, right? So both people who are house owners failing to pay back mortgage debt and equally worrying financial institutions that have lent a lot of, for example, subprime debt, now finding themselves in financial trouble. And you’re going to read a lot in the papers about not letting people out lightly out of those situations of being in debt, or not letting people out lightly out of bankruptcy. The term you’re going to hear is bailout. So bailout, the argument you’re going to read is, is you don’t want the government or the central bank bailing out those financial institutions who have apparently taken too large risks on subprime mortgage debt, even though we all agree it’s better right now for those financial institutions not to go under. And why are we not going to even though it better for everybody not to go under why are we not going to bail them out Because it undermines the incentives for them not to make bad loans to start with And to a lesser extent you going to hear that on the debtor side as well You going to hear some people say we shouldn be bailing out people who took on bad loans who took on bad mortgages to finance their houses, again, for bailout reasons. All right? So this is an important trade-off. If you go on to law school, you’re going to see a lot about this kind of discussion.

[译文 33]

抵押贷款债务，对吧？所以房主无法偿还抵押贷款债务的人和同样令人担忧的金融机构——他们借出了大量例如次级债——现在发现自己陷入财务困境。你会在报纸上读到很多关于不要轻易让人们摆脱债务困境，或者不要轻易让人们摆脱破产的说法。你会听到的词是救助。所以救助，你会读到的论点是，你不想让政府或央行救助那些显然在次级抵押贷款债务上承担了太大风险的金融机构，尽管我们都同意现在让这些金融机构不倒闭更好。为什么我们不会——虽然让每个人都不倒闭更好——为什么我们不会救助他们？因为这破坏了让他们一开始就不发放不良贷款的激励。在较小程度上，你也会在债务方听到这种说法。你会听到一些人说，我们不应该救助那些申请了不良贷款、申请了不良抵押贷款来资助买房的人，同样是出于救助的原因。好的吧？所以这是一个重要的权衡。如果你去上法学院，你会看到很多关于这类讨论的内容。

[段 34]

And this is the discussion of trading off ex-ante efficiency and ex-post efficiency. Sometimes, as Patrick’s pointed out in the game just now, the ex-post efficient thing to do is to go back to the good equilibrium, or if you like, to bail out these firms who’ve made bad loans. however from an ex-ante point of view it creates bad incentives for people to make those loans in the first place and the ex-ante point of view it created the incentive for people to defect in the first period of that game all right so this theme of ex-ante versus ex-post efficiency is not one we’re going to go into anymore in this class but it should be there in the back of your mind when you all end up in law school in a few years time okay so so far what have on this class, but it should be there in the back of your minds when you all end up in law school in a few years’ time. Okay. So, so far, what have we done? We’ve been looking at repeated interaction and seeing if it can sustain cooperation. And the first thing we learned was that if the repeated interaction is a finite interaction, if we know when it’s going to end, if we know when the interaction’s going to end, then sustaining cooperation’s going to be hard, because in the last period there’ll be an incentive to defect.

[译文 34]

这就是关于权衡事前效率和事后效率的讨论。有时候，正如帕特里克刚才在游戏中指出的，事后有效率的做法是回到好的均衡，或者如果你愿意的话，救助这些发放了不良贷款的firm。但从事前角度来看，这为人们首先发放那些贷款创造了坏激励，而从事前角度来看，它为人们在游戏的第一时期就背叛创造了激励。所以这个事前与事后效率的主题不是我们在这堂课里要深入讨论的，但当你们几年后都进入法学院时，它应该留在你们的脑海中。好的。那么到目前为止，在这堂课上我们做了什么？我们一直在研究重复互动，看它是否能维持合作。我们首先学到的是，如果重复互动是有限期的互动，如果我们知道它何时结束，如果我们知道互动何时结束，那么维持合作将会很困难，因为在最后一个时期会有背叛的激励。

[段 35]

We saw we could get around that to some extent if games have multiple equilibria, but in a game like Prisoner’s Dilemma, we’re really in trouble. Things will unravel from the back. All right? So now let’s mix things up a little bit by looking at a more complicated variety of repeated interactions. Rather than just play the game once or twice or three times, let’s play the game under the following rules. We’ll go back to our same players. How many mics are still out here? I took them both back, is that right? I’m taking both the green and the blue mic. I’m giving them back to our players. So this is to Brooks and this is to Patrick. And we’re going to have Brooks and Patrick play Prisoner Dilemma again I hoping I haven deleted it Maybe I did Alright it doesn matter We know the payoffs We’re going to have them play Prisoner’s Dilemma again, but this time, in between every play of the game, I’m going to toss a coin. Actually, I’ll toss the coin twice. And if that coin comes up heads both times, then the game will end. but otherwise they’ll play again so everyone’s how we’re going to do, we’re going to play prisoner’s dilemma at the end of every period I’ll toss a coin twice I might get Jake to toss it I’ll toss a coin twice, if it comes up heads both times, the game’s over but otherwise the game continues alright so both Brooks and Patrick should get ready to play and the payoffs of this game are just what we had before so let’s just remind ourselves what the payoffs of that game are so we’ve got cooperate, defect, cooperate, defect 2, 2, minus 1, 3, 3 minus 1, and 0, 0.

[译文 35]

我们看到如果游戏有多个均衡，我们可以在某种程度上解决这个问题，但在像囚徒困境这样的游戏中，我们真的陷入了麻烦。事情会从后面开始瓦解。好的吧。那么现在让我们通过研究一种更复杂的重复互动形式来混合一下。不只是玩一次、两次或三次游戏，让我们在以下规则下玩这个游戏。我们回到同样的players。有多少麦克风还在外面？我把两个都拿回来了，对吗？我把绿色和蓝色的麦克风都拿回来了。我把它们还给我们players。所以这是给布鲁克斯的，这是给帕特里克的。我们要让布鲁克斯和帕特里克再玩囚徒困境。我希望我没有删除它。也许我删了。没关系。我们知道收益。我们要让他们再玩囚徒困境，但这次，在每次游戏之间，我要抛一枚硬币。实际上，我会抛两次硬币。如果那枚硬币两次都出现正面，那么游戏就结束了。但否则他们会再玩。所以每个人——我们要怎么做，我们在每个时期结束时玩囚徒困境，我会抛两次硬币，我可能让杰克来抛，我会抛两次硬币，如果它两次都出现正面，游戏就结束了，但否则游戏继续。所以布鲁克斯和帕特里克都应该准备好玩，这个游戏的收益就像我们之前的一样，所以让我们只是提醒自己这个游戏的收益是什么。我们有合作、背叛、合作、背叛，2、2，-1、3，3、-1，和0、0。

[段 36]

And we’ll keep score here, so this is Brooks and Patrick. All right so putting pressure on these guys let write down what you going to do the first time Alright Brooks Defect. Defect, better? Cooperate. Alright, I think we’re getting some payback from earlier, right? Alright, round two. Are you going to toss a coin? Oh, I have to toss a coin, you’re absolutely right, thank you. I’m going to have to find a coin without… Ah, look at that, thank you. early. Twice, toss it twice. Heads. Heads again, so the game’s over, that didn’t last long. Just for the sake of the class, let’s pretend that it came up tails, okay? Okay, we’ll cheat a little bit, we’ll cheat a little bit, okay? So we’re playing a second time, just with a little cheating. I need someone less honest to toss the coin. Brooks, what do you choose? Oh, I’m defecting. Defecting again. All right. Patrick? Cooperate. Cooperate. Patrick seems very trusting here. All right. Let’s toss the coin a third time. I already did. It’s tails. Ah okay Good Thank you All right So Brooks I going to defect again Defect All right Okay, so this time we’ll end it. So what happened this time is, let’s talk about debates. So Brooks and Patrick were playing. Patrick cooperated a bit at the beginning. Brooks defected throughout.

[译文 36]

我们会在这里记分，所以这是布鲁克斯和帕特里克。好的，给这些家伙施压，让我写下你们第一次要做什么。好的，布鲁克斯。背叛。背叛，更好？合作。好的，我想我们从之前得到了一些回报，对吧？好的，第二轮。你要抛硬币吗？哦，我得抛硬币，你完全正确，谢谢。我得找一枚没有……啊，看那个，谢谢。早一点。两次，抛两次。正面。又一次正面，所以游戏结束了，没持续多久。只是为了这堂课的目的，让我们假装它出现的是反面，好吗？好的，我们会作弊一点点，我们会作弊一点点，好吗？所以我们在玩第二次，只是稍微作弊。我需要找个不太诚实的人来抛硬币。布鲁克斯，你选什么？哦，我在背叛。又一次背叛。好的。帕特里克？合作。合作。帕特里克在这里似乎非常信任。好的。让我们第三次抛硬币。我已经抛了。是反面。啊，好的。好的。谢谢。好的，布鲁克斯，我又要背叛了。背叛。好的。好的，这次我们会结束它。所以这次发生了什么，让我们谈谈辩论。布鲁克斯和帕特里克在玩。帕特里克在开始时合作了一点。布鲁克斯一直背叛。

[段 37]

Brooks, why did you defect? Shout out to everyone who can hear you. Why did you defect right from the start of the game? Because last time it didn’t work so well. Last time it didn’t work so well. Okay, okay, it’s fair enough. But even after Patrick started cooperating, you went on defecting. So why then? Because… Shout out. Because I wanted to get the higher payoff from… I thought either he would continue cooperating and I could defect. and All right, all right, so we’ve gone cooperating which in fact he did that he did Patrick why were you cooperating early on here shout out to people people here? So with a two head rules like you have a 75% chance of having another game so with those payoffs Even one period the payoff of cooperating twice is the same as defecting once. So, like, additional periods, it’s better if you can continue cooperating, and the percentage is high enough that… Alright, so you figure there’s a good enough chance of getting… Even after Brooks defected the first period, you went on cooperating, but then after the second period you gave up and started defecting, Had he gone in a fourth period, what would you have done? Defected. Defected again, all right. And fifth period? Well, if she kept defecting, I would keep defecting. All right, so what Patrick’s saying is he started off cooperating, but once he saw that Brooks was defecting, he was going to switch to defect.

[译文 37]

Brooks，你为什么背叛？从一开始你为什么要背叛？大声点让大家都听到。因为上次效果不太好。上次效果不太好。好吧，好吧，这很合理。但即使在Patrick开始合作之后，你还是继续背叛。那为什么呢？因为……大声点。因为我想获得更高的收益……我想要么他继续合作，我可以背叛。对对对，好好好，那我们开始合作，事实上他确实这么做了，他确实这么做了。Patrick，为什么你一开始选择合作？大声点让大家听到？因为按照两人都是正面的规则，你有75%的概率会有下一局游戏。按照这些收益，即使只有一期，合作两次的收益也和背叛一次一样。所以，加上额外的期数，如果你能继续合作就会更好，而且百分比足够高……好，所以你认为有足够好的机会获得……即使Brooks第一期背叛了，你还是继续合作，但第二期之后你就放弃了，开始背叛。如果进入第四期，你会怎么做？背叛。再次背叛，好。第五期呢？如果她继续背叛，我就继续背叛。好，所以Patrick说的是，他一开始合作，但一旦看到Brooks背叛，他就会转向背叛。

[段 38]

And basically, as long as she went on defecting, he was going to stick with defecting. All right, let’s try a different pair. So why don’t we switch it over to your partners there. So to Ben here and, I’m sorry. Edwina. Edwina, all right. All right. We’ll flip it twice. All right. All right So people if you want to just stand up I want to see these people So stand up a second So these are players I want people at the back to make sure we know who are playing This is Edwina and this is Ben All right good All right Okay. So Edwina, sit down so you can actually write things down. Okay. So, all right. So Edwina and Ben. Edwina, have you both written down the strategy? Ben, have you written down the strategy? Yep. Edwina, what did you choose? Cooperate. So Edwina’s cooperating. Ben? Cooperate. Cooperate. Uh-huh. Okay. Let’s toss a coin. So we’re okay. We’re okay. So we’re still playing. Okay. Edwina? Cooperate. Ben? I just cooperate. All right. So they’re cooperating. Tails again. So you’re still playing. Cooperate. Cooperate. All right. So they’re still cooperating. Some pain in the voice this time. Heads and then tails. Heads and then tails. So write down what you’re going to do. Edwina? Defect. Ben? Cooperate. All right, all right. So things were going so nicely there.

[译文 38]

基本上，只要她继续背叛，他就会坚持背叛。好，让我们换一组试试。我们把它换到你们那边的人。Ben在这里，还有，我很抱歉。Edwina。Edwina，好的。我们抛两次硬币。好的。大家如果想站起来就站起来，我想看看这些人。站起来一下。这些是玩家，我让后面的人确保我们知道谁在玩。这是Edwina和这是Ben。好，很好。Edwina，坐下来这样你可以写东西。好。那么，Edwina和Ben。Edwina，你们两个都写下策略了吗？Ben，你写下策略了吗？写好了。Edwina，你选择什么？合作。Edwina选择合作。Ben？合作。合作。嗯哼。好，我们抛硬币。好的。我们还在继续。好。Edwina？合作。Ben？我就是合作。好的，所以他们选择合作。再来一次反面。所以你们还在继续。合作。合作。好的，所以他们还在合作。这次声音里有点痛苦。正面然后反面。正面然后反面。所以写下你要做什么。Edwina？背叛。Ben？合作。好的，好的。事情本来进行得很顺利。

[段 39]

We had such a nice class going on there and everything All right All right so we still playing Edwina Defect Ben Defect All right all right Jake? Tails and tails. Tails and tails are still going. Defect. Defect. All right, let me stop it there. We’ll pretend that we had two heads. So let’s talk about this. We had some cooperation going on here. both people started cooperating so a band what why did you cooperate at the beginning shout out to people here why do you cooperate well going along with Patrick’s reasoning I felt that if we could have the cooperate cooperate in the long term with the 75% chance of continuing playing that it would be a worthwhile investment all right all right until I realized that we know I started defecting let’s come back to them second let’s get you guys to stand out it’s a beautiful people gonna hear you when you stand up you shout more so Stand up again. Yes, Edwina, why did you, so you also started cooperating. Why did you start cooperating? For the same reason. Same reason, okay, okay. So the key thing here is why did you start defecting? You heard the big sigh in the class. Why did you start defecting at this stage? Because we had so many I mean the coin toss had to come to head head sometime so I started thinking that This is reversion to the mean of the coin Yeah I just thought that I thought maybe I mean I don know All right, all right, all right.

[译文 39]

我们本来进行得很好，一切都很顺利。好的，好的，我们还在继续。Edwina背叛，Ben背叛。好的，好的。Jake？反面和反面。反面和反面还在继续。背叛。背叛。好，我就在这里停下来。我们假装是正面正面。让我们讨论一下这里。我们这里有一些合作行为。两个人都开始合作了，乐队，为什么你一开始选择合作？大声点让大家听到，你为什么选择合作？嗯，跟着Patrick的推理，我觉得如果我们能在这个75%概率继续游戏的长期合作中合作，那会是一个值得的投资。好的，好的，直到我意识到……我们知道，我开始背叛了。让我们回到他们那里。第二，让我们让你们站起来。当你们站起来时，你们喊得更大声，大家都听得到。所以再站起来。好的，Edwina，你也是一开始选择合作。你为什么开始合作？同样的原因。同样的原因，好的，好的。那么关键问题是，你为什么开始背叛？你听到教室里很大的叹气声。为什么你在这个阶段开始背叛？因为我们有太多次……我的意思是，抛硬币迟早会出现正面正面，所以我开始认为这是均值回归。对，我只是想……我的意思是，我不知道。好的，好的，好的。

[段 40]

So what did I say about the relationship of economic majors that are in the class? Anyway, all right. So Edwina defected, and then Ben, you switched after that. Why did you switch? because once Edwina started defecting I thought that we’d revert back to the defective equilibrium alright, alright, so thank you guys so let’s look at this strategy here people started off cooperating and I claim that at least Ben can contradict me a second but I think Ben’s strategy here was something like this I’m going to cooperate and I’m going to go on cooperating as long as we’re cooperating but if at some point if at some point Edwina defects or for that matter someone I defect then you know the game this relationship’s over and we’re going to play defect forever is that right that’s kind of a rough description of your strategy all right all right and Edwina was more or less playing the same thing in fact it was her who defected but once she defected she realized that it was over and he went on defecting all right so this And Edwina was more or less playing the same thing. In fact, it was her who defected, but once she defected, she realized that it was over and she went on defecting. All right? So this strategy has a name. Let’s just be clear what the strategy is.

[译文 40]

那么我对课上经济学专业学生之间的关系是怎么说的？不管了。好的。Edwina背叛了，然后Ben，你在之后也转换了。你为什么转换？因为一旦Edwina开始背叛，我认为我们会回到背叛均衡。好的，好的，非常感谢你们。让我们看看这个策略。人们一开始合作，我说至少Ben可以反驳我一下，但我认为Ben这里的策略是这样的：我会合作，只要我们继续合作我就会继续合作，但如果有某个时刻，如果某个时刻Edwina背叛，或者就此而言某人背叛，那么这场游戏就结束了，我们会永远背叛。这是对你策略的大致描述，对吗？好的，好的。而Edwina或多或少也在玩同样的东西。事实上是她先背叛的，但一旦她背叛了，她就意识到游戏结束了，然后她继续背叛。好的？那么这个策略有一个名字。让我们明确一下这个策略是什么。

[段 41]

This strategy says, play C, which is cooperate, and then play C if no one has played D, and play D otherwise. So start off by cooperating. Keep cooperating as long as nobody’s cheated. But if somebody cheats, this relationship’s over. We’re just going to defect forever. Now this strategy’s a famous strategy. It has a name. Anyone know what the name is? This is called the Grim Trigger Strategy. The Grim Trigger Strategy. the grim trade strategy So this strategy again it says we going to cooperate we’re going to cooperate, but if that cooperation breaks down ever, even if it’s me who breaks it down, then I’m just going to defect forever. All right? All right? Now, we’re going to come back next time and see if this is an equilibrium, but there’s a few things to do first. First, let’s just check that it actually is a strategy. What does it mean to be a strategy again? It has to tell us what to do at every information set I could find myself at. And this game is potentially infinite. So potentially there’s an infinite number of information sets I could reach. So you might think that writing down a strategy that gives me an instruction at every single information set is going to be incredibly complicated once we go to games that are potentially infinite. Because there needs to be an infinite number of instructions.

[译文 41]

这个策略说，玩C，也就是合作，如果没有一个人玩D，就继续玩C，否则就玩D。一开始选择合作。只要没有人作弊就继续合作。但如果有人作弊，这段关系就结束了。我们就永远背叛。这个策略是一个著名的策略。它有一个名字。有人知道它叫什么吗？这叫做冷酷触发策略。冷酷触发策略。那么这个策略再说一遍，我们选择合作，我们选择合作，但如果这种合作破裂了，即使是我打破的，我就永远背叛。好吗？好的。我们下次回来看看这是否是一个均衡，但首先有几件事要做。首先，让我们检查一下它确实是一个策略。成为一个策略意味着什么？它必须告诉我，在我可能发现的每个信息集上该怎么做。这个游戏可能是无限的。所以可能有无限数量的信息集我可能达到。所以你可能会认为，写下一个在每个信息集上都给我指令的策略，一旦我们进入可能是无限的游戏，将会非常复杂。因为需要无限数量的指令。

[段 42]

But it turns out, actually, it’s possible to write down such strategies rather simply, at least if they’re simple strategies. And this example is one. This tells me what to do at the first information set It says place C It then tells me for every information set I find myself at in which only cooperation has ever occurred in the history of the game I’m going to go on cooperating, play C. And it says for all other histories, for all other information sets I might find myself at, play D. So it really is a strategy. Now, this is very different behavior. we played with the same players this kind of behaviour is very different in both games actually is very different than the behaviour we saw in the game that ended the game with two periods or three periods what is it essentially that made this different? what’s different about this way of playing Prisoner’s Dilemma where we had Jake toss the coin versus the way we played before where we just played for five periods and then stopped what’s different about it? Somebody? Let’s talk to our players. OK, so is the mic still there? Patrick, do you have your mic still? Patrick, why is this different? We don’t know when the game is going to end or if it’s going to end. So there’s no last period.

[译文 42]

但事实证明，实际上，写下这样的策略是相当简单的，至少如果是简单策略的话。这个例子就是一个。它告诉我第一个信息集上该做什么。它说玩C。然后它告诉我，在我发现的每个信息集中，如果历史上只有合作发生，我就继续合作，玩C。它说对于所有其他历史，所有其他我可能发现自己的信息集，玩D。所以它确实是一个策略。现在，这种行为非常不同。我们用同样的玩家玩，这种行为在两个游戏中实际上非常不同。实际上，与我们在两期或三期就结束的游戏中看到的行为非常不同。是什么本质上让这个不同的？我们在玩囚徒困境时用Jake抛硬币的方式，与我们之前只玩五期然后停止的方式，有什么不同？有人知道吗？让我们和我们的玩家谈谈。好，麦克风还在吗？Patrick，你的麦克风还在吗？Patrick，为什么这个不同？我们不知道游戏什么时候会结束或者是否会结束。所以没有最后一期。

[段 43]

Good, good. So our analysis of the game before the analysis of the Prisoner of Salama when we knew it was going to end after two periods five periods whatever it was was we all knew it was going to end There was a clearly defined last period When people are going to retire, we know the month in which they’re going to retire. When presidents are going to step down, we know they’re going to step down that period. When CEOs are going to go, we know they’re going to… Well, we don’t always know they’re going to go, but let’s pretend we do. So what’s different about this game is every time we play the game, there is a probability, in this case a three-quarters probability, that the game is going to continue to the next period. Every time we play the game, with probability 0.75, there’s going to be a future. There’s no obvious last period from which we can unravel the game in the way we did before. Just to remind ourselves, the way in which our analysis of cooperation broke down in the finally repeated prisoner’s dilemma was that when we looked at the last period, We know people are going to defect and once that thread is loose we can unravel it all the way back to the beginning But here since there is no last period that unraveling argument never gets hold All right now instead We’re able to see strategies develops like the grim trigger strategy and notice that the grim trigger strategy never gets hold.

[译文 43]

好的，好的。所以我们对游戏的分析——在萨拉玛囚徒困境的分析之前——当时我们知道它会在两个时期、五个时期之后结束，无论怎样，我们都知道它会结束。有一个明确界定的最后时期。当人们要退休时，我们知道他们将在哪个月退休。当总统要下台时，我们知道他们将在那个时期下台。当CEO要离开时，我们知道他们要……好吧，我们并不总是知道他们要离开，但让我们假装我们知道。所以这个游戏的不同之处在于，每次我们玩这个游戏时，都存在一种概率，在这种情况下是四分之三的概率，游戏将继续到下一个时期。每次我们玩这个游戏时，有0.75的概率，会有一个未来。没有明显的最后时期让我们像之前那样解构游戏。让我们提醒自己，我们的合作分析在有限重复囚徒困境中是如何崩溃的：当我们看最后时期时，我们知道人们会背叛，一旦那根线松了，我们就可以一直解构到开头。但在这里，由于没有最后时期，那种解构论证从未站住脚。好吧，现在我们能够看到策略发展，比如冷酷触发策略，并注意到冷酷触发策略从未站住脚。

[段 44]

All right? Now instead, we’re able to see strategies developed like the grim trigger strategy, and notice that the grim trigger strategy has a pretty good chance of actually sustaining cooperation. So in particular, as long as people play this strategy, they are cooperating. As long as people play this strategy, they are cooperating. It turns out that Edwina eventually gave up that strategy, but had she got on playing it, they’d have gone on cooperating forever. all right but of course there’s a question here and the question is is this in fact an equilibrium right we know that people play this way we get cooperation but the question that the thousand dollar question whatever is is this an equilibrium so what do we have to do to check whether this is an equilibrium or not we have to mimic the argument we had before. We have to compare the temptation to defect today and compare that with the value of the reward to cooperating and the value of the punishment from defecting tomorrow All right so this basic idea is going to re Having said that let me now delete it so I have some room We’re going to have to, to show this is an equilibrium, we need to show that the temptation to defect, the temptation to cheat, in the short term, is outweighed by the difference between the value of the reward and the value of the punishment.

[译文 44]

好的？那么现在，我们能够看到像冷酷触发策略这样的策略被开发出来，并且注意到冷酷触发策略实际上有很大的机会维持合作。特别是，只要人们采用这个策略，他们就是在合作。只要人们采用这个策略，他们就是在合作。事实证明Edwina最终放弃了这个策略，但如果她一直采用这个策略，他们本可以永远继续合作下去。好的，但当然这里有一个问题，问题在于这实际上是一个均衡，对吧，我们知道人们这样做会得到合作，但问题是这是不是一个均衡，这是一个关键问题，所以我们必须做什么来检查这是否是一个均衡，我们必须模仿我们之前的论证。我们必须比较今天背叛的诱惑，比较合作的奖励价值和明天背叛的惩罚的价值。好的，所以这个基本思路会重新出现，话说回来让我把它删掉，这样我有一些空间，为了证明这是一个均衡，我们需要证明背叛的诱惑，欺骗的诱惑，在短期内，被奖励的价值和惩罚的价值之间的差异所抵消。

[段 45]

All right. So let’s set that up. Let’s put the temptation here first. So the temptation in prisoner’s dilemma, the temptation to cheat today is what? I’ll get three rather than two. Is that right? So if I defect, when Edwina defected, here’s Edwina defecting in this period, she got a payoff of 3 rather than the payoff of 2 she would have got from cooperating. So the temptation here is just 3 minus 2. And let’s be clear, this is a temptation today And we want to compare this with the value of the reward minus the value of the punishment But the key point observation is that these occur tomorrow. These occur tomorrow. All right. So since they occur tomorrow, we have to weight them a little bit lower. All right. So in general, the way in which we’re going to weight them tomorrow is we’re going to discount them. Just like we did in our bargaining game. We’re going to weight tomorrow’s payments by delta. Where delta is less than 1. All right. Now why is delta less than one? Why are we weighing tomorrow less than payments today? Why are payments tomorrow worth less than payments today? Because tomorrow might not happen. There are other reasons why, by the way.

[译文 45]

好的，让我们这样设置。我们先把这个诱惑放在这里。所以囚徒困境中的诱惑，今天背叛的诱惑是什么？我会得到3而不是2。对吗？所以如果我背叛，当Edwina背叛时，在这个时期Edwina背叛了，她得到了3的收益，而不是如果她合作会得到的2的收益。所以这里的诱惑就是3减2。让我们明确，这是今天的诱惑。我们想把这个和奖励的价值减去惩罚的价值进行比较。但关键的观察是这些发生在明天。这些发生在明天。好的，既然它们发生在明天，我们就必须把它们稍微降低权重。好的，一般来说，我们对明天权重的处理方式是，我们会折现它们。就像我们在议价博弈中做的那样。我们会用delta来对明天的收益进行加权，其中delta小于1。好的，为什么delta小于1？为什么我们对明天的支付比对今天的支付权重更低？为什么明天的支付不如今天的支付值钱？因为明天可能不会发生。顺便说一下，还有其他原因。

[段 46]

It might be that we impatient to get the money today and we just wanted to pay off in a hurry Or it might be that she wanted to take the payment today and put in the bank and earn interest there are other reasons why money today might be more valuable than money tomorrow but in games the most important reason is tomorrow may not happen right tomorrow by tomorrow you might be dead or if not dead at least Jake’s thrown out thrown two heads in the coins all right so Delta is less and 1 because the game may end. Because the game may end. Now, what’s the value of the reward? The value of the reward is going to be the value of C forever. But you want to be careful about forever. It’s C forever, but of course, it isn’t really forever because the game may end. so by forever I mean until the game ends let me be a bit more careful actually the CC isn’t it the value of CC cooperate, cooperate, forever and here we’re going to have the value of DD forever and once again since the forever here means until the game ends alright, so this is the calculate forever. Once again, since the forever here means until the game ends. All right. So this is the calculation we’re going to have to do.

[译文 46]

可能是我们急于拿到今天的钱，想赶紧付清。也可能是她想今天就把钱收进来存进银行赚取利息。还有其他原因解释为什么今天的钱比明天的钱更值钱。但在博弈中最重要的原因是明天可能不会来，对，到明天你可能已经死了。或者即使没死，至少Jake扔硬币扔出了两个正面（游戏结束）。好的，所以Delta小于1，因为游戏可能会结束。因为游戏可能会结束。那么，奖励的价值是多少？奖励的价值将是C的永久价值。但你要对forever一词谨慎。它是C的永久价值，但当然它并非真的是永久的，因为游戏可能会结束。所以我说的永久是指直到游戏结束。让我再谨慎一点。实际上，CC是CC合作、合作、永久价值，这里我们会有DD的永久价值。同样，由于这里所说的永久是指直到游戏结束，所以这就是计算永久值的方法。同样，由于这里所说的永久是指直到游戏结束。好的，这就是我们必须做的计算。

[段 47]

We’re going to have to compare the temptation, that was easy, that was just one, with the discounted difference between the value of cooperation and the value of defecting. All right. Let’s do the easy bits now, and then we’ll leave you in suspense until Wednesday. Let’s do all the easy bits. All right? So watch this delta in this case, in this particular game. What was the probability that the game was going to continue? What was the probability that the game was going to continue? What was the probability that the game was going to end? I said the probability of ending was 0.25, so delta here was 0.75. That’s easy. All right? the second bit that’s relatively easy is what’s the value of playing DD until the game ends once people have cheated you’re going to play D forever here we are, Ed Wiener’s cheating here you’re going to get DD in this period, DD in this period and so on and so forth until the game ends in each of those periods you’re going to earn 0 so this is just 0 which leaves us with the messy bit What the value of cooperating forever Let’s try and do it. We’ve got one minute, let’s do it. So in every period in which we both cooperate, what do we earn? So in the next period, we’re at the beginning of the game.

[译文 47]

我们必须将那个诱惑——那个很简单，就是一个数字——与合作价值减去背叛价值的贴现差值进行比较。好的。现在让我们先把简单的部分处理完，然后让你们悬着心等到周三。让我们把所有简单的部分都做掉。好吗？所以在这个案例中，在这个特定的博弈中，观察这个delta。这个博弈继续下去的概率是多少？这个博弈继续下去的概率是多少？博弈结束的概率是多少？我说过结束的概率是0.25，所以这里的delta是0.75。这个很简单。好吗？第二个相对简单的部分是：当人们已经背叛之后，DD（双方都背叛）的价值是多少，直到博弈结束。在这个例子中，Ed Wiener背叛了，你在这个时期会得到DD，在下一个时期也是DD，如此下去直到博弈结束，在每个这样的时期你将获得0，所以这直接就是0，这就只剩下复杂的部分了。合作永远的价值是多少？让我们试着算出来。我们还有一分钟，来做这个。所以在每一个我们都合作的时期，我们各获得多少收益？那么在下一个时期，我们处于博弈的开始。

[段 48]

We cooperated in the first period. Now in the second period we cooperate again. What payoff do we get from cooperating again? We get 2, right? We get 2. And then Jake tosses his coin, and with probability delta we continue. All right? And we’re going to cooperate again. So with probability delta we cooperate again and get what payoff the next period? 2 again. And then Jake tosses the coin again. All right? So now he’s tossed the coin twice. so with probability delta squared we’re still playing and we get 2 and then Jake tosses the coin again so it comes up other than head head again that’s with probability delta cubed we get 2 and so on alright so your exercise between now and Wednesday is figure out what the value of cooperation forever is figure out this equation and find what value whether in fact it was an equilibrium for people to cooperate Oh, pick a good one.

[译文 48]

我们在第一阶段合作了。现在第二阶段我们再次合作。再次合作我们得到什么收益？我们得到2，对吧？我们得到2。然后Jake掷硬币，以delta的概率我们继续。我们将继续合作。所以以delta的概率我们再次合作，下一个阶段得到什么收益？又是2。然后Jake再次掷硬币。好吗？所以现在他已经掷了两次硬币，所以以delta平方的概率我们仍在玩，我们得到2，然后Jake再次掷硬币，所以它出现正面而不是反面，以delta立方的概率我们得到2，以此类推。好了，所以你们现在到Wednesday之间的练习是找出永远合作的价值，写出这个等式，并找出这个值是否实际上是一个人们合作的均衡。哦，选一个好的。