耶鲁大学博弈论公开课 - 第4集 作弊，惩罚和外包

视频全文转录（中英双语）

以下为完整中英双语转录，已加标点。英文在前，中文在后，逐段对照。

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

[段 1]

So last time, we were focusing on repeated interaction, and that’s what we’re going to continue with today. There’s lots of things we could study under repeated interaction, but the emphasis of this week is, can we attain and achieve cooperation in business or personal relationships without contracts by use of the fact that these relationships go on over time? And our central intuition, where we started from last time, was perhaps the future of a relationship can provide incentives for good behavior today. Can provide incentives for people not to cheat. So specifically, let’s think of an example, and we’ll go back to where we were last time. Specifically, suppose I have a business relationship, an ongoing business relationship with Jake. And each period I’m supposed to supply Jake with some inputs for his business, let’s say some fruit, and each period he’s supposed to provide me with some input for my business, namely vegetables. And clearly there are opportunities here in each period for us to cheat. We could cheat both on the quality of the fruit that I provide or the quantity of the fruit that I provide to Jake and he can cheat on the quantity or quality of the vegetables that he provides to me Our central intuition is perhaps what can work is, what can give us good incentives is, the idea that if Jake cooperates today, then I might cooperate tomorrow.

[译文 1]

上次我们专注于重复互动，今天我们将继续探讨这个主题。在重复互动的主题下我们可以研究很多东西，但本周的重点是：在商业或人际关系中，我们能否通过关系会持续下去这一事实，在没有合同的情况下获得并实现合作？我们的核心直觉——也就是上次我们出发的地方——是：也许一段关系的未来可以为今天的好行为提供激励。可以为人们不作弊提供激励。具体来说，让我们思考一个例子，我们回到上次讨论的地方。具体来说，假设我与Jake有一层商业关系，一段持续的商业关系。每个时期我都应该向他提供一些业务投入，比如一些水果，每个时期他也应该向我提供一些业务投入，也就是蔬菜。显然，在每个时期我们都有作弊的机会。我可能在提供给他的水果的质量或数量上作弊，他也可能在提供给我的蔬菜的质量或数量上作弊。我们的核心直觉是，也许有效的方法是：如果Jake今天合作，那么我明天可能会合作。

[段 2]

I might not cheat tomorrow. And conversely, if he cheats and provides me with lousy vegetables today, I’m going to provide him with lousy fruit tomorrow. And similarly for me, if I provide Jake with lousy fruit today, he can provide me with lousy vegetables tomorrow. So what do we need? We need the difference in the value of the promise of good behavior tomorrow and the threat of bad behavior tomorrow to outweigh the temptation to cheat today. I’m going to gain by providing him with the bad fruit or fewer fruit today, bad fruit because those I’d otherwise have to throw away. So that temptation to cheat has to be outweighed by the promise of getting good vegetables in the future from Jake, and vice versa So here that idea on the board What we need is the gain if I cheat today to be outweighed by the difference between the value of my relationship with Jake after cooperating and the value of my relationship with Jake after cheating tomorrow. All right. Now, what we discovered last time, this was an idea I think we kind of knew. We might have known it since the first week. But we discovered last time, somewhat surprisingly, that life is not quite so simple. And in particular, what we discovered was we need these to be credible. So there’s a problem here of credibility.

[译文 2]

我明天可能不会作弊。反之，如果他今天作弊，给我提供了劣质蔬菜，我明天就给他提供劣质蔬菜。同样地，如果我今天给Jake提供了劣质水果，他明天可以给我提供劣质蔬菜。那么我们需要什么？我们需要承诺良好行为明天与威胁恶劣行为明天之间的价值差异来超过今天作弊的诱惑。我可以通过给他提供坏水果或更少的水果来获得好处，坏水果是因为那些我本来要扔掉的。所以这种作弊的诱惑必须被从Jake那里获得好蔬菜的承诺所抵消，反之亦然。这里黑板上的想法是：我们需要的是，如果我今天作弊获得的收益，被我与Jake合作后的关系价值与我今天作弊后明天的关系价值之间的差异所抵消。好吧。现在，我们上次发现了这个想法，我认为我们某种程度上知道它。我们可能从第一周就知道了。但我们上次发现，出乎意料的是，生活并不那么简单。特别是，我们发现我们需要这些是可信的。所以这里有一个可信度的问题。

[段 3]

So in particular, if we think of the value of the relationship after cooperating tomorrow as being a promise, and the value of the relationship after cheating as being a threat, We need these promises and threats to be credible. We need to actually believe that they’re going to happen. And one very simple area where we saw that ran immediately into problems was if this repeated relationship, although repeated had a known end And why did known ends cause problems for us Because in the last period in the last period of the game we know that whatever we promised to do or whatever we threatened to do, in the last period, once we reach that last period, in that sub-game, we’re going to play a Nash equilibrium. What we do has to be consistent with our incentives in the last period. So in particular, if there’s only one Nash equilibrium in that last period, then we know in that last period that’s what we’re going to do. So if we look at the second to last period, we might hope that we could promise to cooperate if you cooperate today, tomorrow, or you could promise to punish tomorrow if you cheat today. But those threats won’t be credible because we know that tomorrow you’re just going to play whatever that Nash equilibrium is. That lack of credibility means there’s no scope to provide incentives today for us to cooperate, and we saw things unravel backwards.

[译文 3]

特别是，如果我们把明天合作后的关系价值看作一个承诺，把作弊后的关系价值看作一个威胁，我们需要这些承诺和威胁是可信的。我们需要真正相信它们会发生。有一个非常简单的领域，我们看到它立即遇到了问题，那就是如果这种重复关系虽然是重复的，但有一个已知的结束日期。已知结束日期为什么会给我们带来问题？因为在最后一个时期，在游戏的最后一个时期，我们知道无论我们承诺做什么或威胁做什么，在最后一个时期，一旦我们到达那个最后时期，在那个子游戏中，我们要进行纳什均衡。我们做什么必须与我们在最后一个时期的激励相一致。所以特别是，如果最后一个时期只有一个纳什均衡，那么我们知道在最后一个时期那就是我们要做的。所以如果我们看倒数第二个时期，我们可能希望如果我们今天合作，你明天可以合作，或者你可以承诺如果你今天作弊，明天惩罚你。但这些威胁不会是可信的，因为我们知道明天你只会玩那个纳什均衡。这种缺乏可信度意味着没有空间为今天提供激励让我们合作，我们看到事情从后向前崩溃。

[段 4]

So the way in which we ensure that we’re really focusing on credible promises and credible threats here is by focusing on sub-game perfect equilibrium, the idea that we introduced just before the Thanksgiving break. And we know that sub-game perfect equilibria have the property that they have Nash behavior in every sub-game, so in particular in the last period of the game and so on. So what we want to be able to do here is try to find scope for cooperation in relationships without contracts, without side payments by focusing on sub-game perfect equilibria of these repeated games. And right at the end last time, we said, okay, let’s move away from the setting where we know a game is going to end, and let’s look at a game which continues, or at least might continue. So in particular, we looked at the problem of the prisoner’s dilemma, which was repeated with the probability that we called delta each period. The probability delta of continuing So every period we going to play Prisoner Dilemma However, with probability 1 minus delta, the game might just end every period. All right. And we already noticed last time some things about this. The first thing we noticed was that we can immediately get away from this unraveling argument. Because there’s no known end to the game, we don’t have to worry about that thread coming loose and unraveling all the way back.

[译文 4]

因此，为了确保我们真正关注可信的承诺和可信的威胁，我们通过关注子博弈完美均衡来实现这一点，这是我们在感恩节假期前引入的思想。我们知道子博弈完美均衡具有这样的特性：它们在每个子博弈中都具有纳什行为，特别是在游戏的最后一个时期等等。所以我们在这里想要做的是，通过关注这些重复游戏的子博弈完美均衡，在没有合同、没有附带支付的关系中寻找合作的空间。就在上次结束时，我们说，好吧，让我们从我们知道游戏将要结束的环境中走出来，让我们看看一个可能继续的游戏。所以特别是，我们研究了囚徒困境问题，这是一个以我们称为δ的概率每个时期重复的囚徒困境。继续的概率δ。所以每个时期我们要玩囚徒困境游戏。然而，以1减去δ的概率，游戏可能每个时期都会结束。好吧。上次我们已经注意到一些关于这一点的事情。我们注意到的第一件事是我们可以立即摆脱这种崩溃的论点。因为游戏没有已知的结束日期，我们不必担心那个线程松动并一直崩溃回去。

[段 5]

So at least there’s some hope here to be able to establish credible promises and credible threats later on in the game that will induce good behavior earlier on in the game. All right, so that’s where we were last time. And here is the prisoner’s dilemma. We saw this last time. All right, and we actually focused on a particular strategy, but before I come back to the strategy that we focused on last time, let’s just see some things that won’t work, just to reinforce the idea. So here’s a possible strategy in the prisoner’s dilemma. A possible strategy in the prisoner’s dilemma would be cooperate now and go on cooperating regardless of what anyone does. So just cooperate forever regardless of the history of the game Now if two players if Jake and I are involved in this business relationship which has the structure of a prisoner dilemma and both of us play this strategy of cooperate now and cooperate forever no matter what clearly that will induce cooperation. That’s the good news. The problem is, that isn’t an equilibrium. It’s not even a Nash equilibrium, let alone a sub-game perfect equilibrium. Why is it not a sub-game perfect equilibrium? Because in particular, if Jake is smart, and he is, Jake will look at this equilibrium and say, Ben is going to cooperate no matter what I do, so I may as well cheat.

[译文 5]

所以至少在这里有一些希望，能够在游戏后期建立可信的承诺和可信的威胁，这将促使游戏早期的好行为。好吧，这就是上次我们的位置。这是囚徒困境。我们上次看到过这个。好吧，在我们回到上次关注的策略之前，让我们先看看一些不起作用的东西，以加强这个想法。所以这是在囚徒困境中的一种可能策略。在囚徒困境中的一种可能策略是现在就合作，然后继续合作，无论任何人做什么。所以无论游戏的历史如何，永远合作。现在如果两个玩家如果Jake和我参与这个具有囚徒困境结构的商业关系，并且我们两个都玩这种策略：现在就合作，无论如何永远合作，显然这将促使合作。这是好消息。问题是，这不是均衡。它甚至不是纳什均衡，更不用说子博弈完美均衡了。为什么它不是子博弈完美均衡？因为特别是，如果Jake很聪明，而且他确实很聪明，Jake会看这个均衡并说，Ben会无论如何都合作，所以我不如作弊。

[段 6]

And in fact, I may as well go on cheating. So Jake has a very good deviation there, which is simply to cheat forever. So the strategy, cooperate now and go on cooperating no matter what, doesn’t contain incentives to support itself as an equilibrium. And we need to focus on strategies that contain subtle behavior that generates promises of rewards and threats of punishment that induce people to actually stick to that equilibrium behavior. All right, so everyone clear that cooperating no matter what, you know, it sounds good, but it isn’t going to work. People aren’t going to stick with that. So instead what we focused on last time and actually we had some players who seemed to actually they moved now but they seem to actually be playing this strategy we focused on what we called the grim trigger strategy And the grim trigger strategy is what It says in the first period, cooperate, and then go on playing cooperate as long as nobody has ever defected, nobody has ever cheated. But if anybody ever plays D, anybody ever plays the defect strategy, then we just play D forever. So this is a strategy. It tells us what to do at every possible information set. It also, if two players are playing the strategy, has the property that they will cooperate forever. That’s good news. And what we left ourselves last time was checking that this actually is an equilibrium.

[译文 6]

事实上，我不如一直作弊。所以Jake有一个很好的偏离，那就是简单地永远作弊。所以这个策略，现在就合作然后无论如何继续合作的策略，并不包含激励来支持自己作为一个均衡。我们需要关注那些包含微妙行为的策略，这些行为产生奖励的承诺和惩罚的威胁，促使人们真正坚持那个均衡行为。好吧，大家清楚了吗？无论如何都合作，你知道，这听起来很好，但不会起作用。人们不会坚持这一点。所以相反，我们上次关注的是，实际上我们有一些似乎真的在移动的玩家，但他们似乎真的在玩这个策略，我们关注的是我们所说的冷酷触发策略。冷酷触发策略是什么？它说的是在第一个时期合作，然后继续玩合作，只要没有人曾经背叛，没有人曾经作弊。但如果任何人曾经玩D，任何人曾经玩背叛策略，我们就永远玩D。所以这是一个策略。它告诉我们每个可能的信息集该做什么。如果两个玩家都在玩这个策略，它也有这样的特性：他们会永远合作。这是好消息。我们上次留给自己的是检查这实际上是否是一个均衡。

[段 7]

Or more generally, under what conditions is this actually an equilibrium? We were halfway through that calculation last time. So what we need to do is we need to make sure that the temptation of cheating today is less than the value of the promise minus the value of the threat tomorrow. We did parts of this already. Let’s just do the easy parts. So the temptation today is if I cheat today, I get three. Professor Ben Polak The temptation today is if I cheat today I get 3, whereas if I went on cooperating today I get 2. So the temptation is just 1. Watch the threat. The threat is playing D forever. So this is actually the value d, dd forever. Now, I want to be careful about forever. When I say forever, I mean until the game ends, because eventually the game’s going to end. But let’s use the code forever to mean until the game ends. And what’s the promise? The promise is the value of continuing incorporation, so the value of CC forever. That’s what this bracket is. That’s what this bracket is and it’s still tomorrow. So let’s go on working on this. So the value of cooperating forever is actually, let’s make it a bit more detailed this is the value of getting 2 in every period So it the value of two forever And this is the value of zero forever All right.

[译文 7]

或者更一般地，在什么条件下这实际上是一个均衡？我们上次已经完成了一半的计算。所以我们需要做的是确保今天作弊的诱惑小于承诺的价值减去明天的威胁的价值。我们已经做了其中一部分。让我们只做简单的部分。所以今天的诱惑是，如果我今天作弊，我得到3。Ben Polak教授今天的诱惑是，如果我今天作弊我得到3，而如果我继续合作我得到2。所以诱惑就是1。注意威胁。威胁是永远玩D。所以这实际上是价值d，dd永远。现在，我要小心这个"永远"。当我说永远时，我的意思是直到游戏结束，因为游戏最终会结束。但让我们用"永远"这个代号来表示直到游戏结束。承诺是什么？承诺是继续合作的价值，所以是CC永远的价值。这就是这个括号里的内容。这就是这个括号里的内容，它还是明天。让我们继续做这个。所以永远合作的价值实际上是，让我们更详细地做一下，这是每期得到2的价值所以是2永远的价值而这是0永远的价值好的。

[段 8]

Okay, so the value of zero forever, that’s pretty easy to work out. I get zero tomorrow. I get zero the day after tomorrow. I get zero the day after the day after tomorrow. or more accurately, I’d get 0 tomorrow. I’d get 0 the day after tomorrow if we’re still playing. I’d get 0 the day after the day after tomorrow if we’re still playing, and so on. But that isn’t a very hard calculation. This thing is going to equal 0. So this object here, let’s use my other colored chalk, this object here is just 0. Is that right? This object here is 3 minus 2. I can do that one in my head. That’s 1. All right? So what I’m left with is the value of getting 2 forever. And that requires a little bit more thought, but let’s do that one bit of algebra because it’s going to be useful throughout today. So this thing here, the value of 2 forever, is what? Well, I’d get 2. That tomorrow And then assuming I still playing the day after tomorrow so I need to discount it with probability delta I still playing the day after tomorrow and I get 2 again And the day after the day after tomorrow I’m still playing with the probability that the game didn’t end tomorrow or didn’t end the next day.

[译文 8]

好的，0永远的价值，这很容易算出来。明天我得到0。后天我得到0。大后天我得到0。或者更准确地说，如果游戏还在进行的话，明天我得到0。如果游戏还在进行的话，后天我得到0。如果游戏还在进行的话，大后天我得到0，等等。但这不是一个很难的计算。这个东西等于0。所以这个东西，让我用另一种颜色的粉笔，这个东西就是0。对吗？这个东西是3减2。这个我可以在脑子里算。是1。好的？所以我剩下的是永远得到2的价值。这需要多想一下，但让我们做一下这部分代数，因为它今天会很有用。这个东西，永远得到2的价值，是什么？我会得到2。明天然后假设我后天还在玩所以我需要用概率delta折现我后天还在玩我再次得到2大后天我还在玩的概率是游戏明天没有结束或者后天没有结束。

[段 9]

So that’s with probability delta squared and again I get 2. And then the day after, what is it? This is tomorrow, the day after tomorrow, the day after the day after tomorrow, this is the day after the day after the day after tomorrow, which is delta cubed 2 and so on. All right, everyone happy with that? All right, so starting from tomorrow, starting from tomorrow, if we play CC forever, I’ll get 2 tomorrow, 2 the day after tomorrow, 2 the day after the day after tomorrow, and so on, and I just need to take into account the fact that the game may end between tomorrow and the next day, the game may end between the day after tomorrow and the day after the day after tomorrow and so on. All right, everyone happy with that? All right, so what is the value, what is this thing? Let’s call this x for a second. We’ve done this once before in the class, but let’s do it again anyway. This is the value, this is the geometric sum. Some of you may even remember from high school how to do a geometric sum but let do it slowly So to work out what x is what I going to do is I going to multiply x by delta I going to multiply x by delta So what delta x So this 2 here will become a 2 delta, and this delta 2 here will become a delta squared 2, and this delta squared 2 will become a delta cubed 2, and this delta cubed 2 will become a delta to the 4 2 and so on.

[译文 9]

所以那是概率delta平方，再次我得到2。然后第二天，这是明天，后天，大后天，这是大大后天，是delta立方2，以此类推。好的，大家都明白吗？好的，从明天开始，从明天开始，如果我们永远玩CC，我会明天得到2，后天得到2，大后天得到2，以此类推，我只是需要考虑游戏可能在明天和后天之间结束的可能性，游戏可能在大后天和大大后天之间结束的可能性，等等。大家都明白吗？好的，这个价值是多少，这个东西是什么？让我们暂时叫它x。我们之前在课上做过一次，但还是让我们再做一遍。这是价值，这是一个几何级数。你们有些人可能还记得高中时怎么做几何级数，但让我们慢慢来。要算出x是什么，我要做的是把x乘以delta。我要把x乘以delta所以这是delta x所以这个2会变成2 delta，这个delta 2会变成delta平方2，这个delta平方2会变成delta立方2，这个delta立方2会变成delta的4次方2，以此类推。

[段 10]

And now what I’m going to do is I’m going to subtract the second of those lines from the first of those lines. So what I’m going to do is I’m going to subtract x minus delta x. So I’m going to subtract the second line from the first line, and when I do that I’m going to notice, I hope, that this 2 delta is going to cancel with this 2 delta, and this delta squared 2 is going to cancel with this delta squared 2, and this delta cubed 2 is going to cancel with this delta cubed 2, and so on. What I’m going to get left with This delta squared 2 and this delta cubed 2 is going to cancel with this delta cubed 2 and so on. All right, so what I’m going to get left with is what? Everything’s going to cancel except for what? Except for that first 2 there. All right, so this is just equal to 2. All right, now this is a calculation I can do. So I’ve got x is equal to 2 divided by 1 minus delta. All right, so just to summarize the algebra, getting 2 forever, that means 2 plus delta 2 plus delta squared 2 plus delta cubed 2, etc. The value of that object is 2 over 1 minus delta. 2 over 1 minus delta.

[译文 10]

现在我要做的是从第一行减去第二行。所以我要做的是减去x减去delta x。我要从第一行减去第二行，当我这样做的时候，我希望注意到这个2 delta会和这个2 delta抵消，这个delta平方2会和这个delta平方2抵消，这个delta立方2会和这个delta立方2抵消，以此类推。我会得到什么剩下的呢？所有这些都会抵消除了什么？除了第一个2。好的，所以这等于2。好的，这是一个我可以做的计算。所以我得到x等于2除以1减delta。好的，总结一下代数，永远得到2，这意味着2加delta 2加delta平方2加delta立方2，等等。那个东西的价值是2除以1减delta。2除以1减delta。

[段 11]

All right, so we can put that in here as well. So this object here, 2 over 1 minus delta, is the value of 2 forever. Now before I go on to a new board, I want to do one other thing. On the left-hand side, I’ve got my temptation. That was 1. I got the value of cooperating forever starting from tomorrow which is 2 over 1 minus delta and I got the value of defecting forever starting from tomorrow which is 0 However, all of these objects on the right-hand side, they start tomorrow, whereas the temptation today is today. today. Temptation today happens today. These differences in value start tomorrow. Since they start tomorrow, I need to discount them because we don’t know that tomorrow is going to happen. The world may end, or more importantly, the relationship may end between today and tomorrow. So how much do I have to weight them by? By delta. I need to multiply all of these lines by delta and so on. All right, now this is now a mess, so let’s go to a new board. All right, and let’s summarize what we now have. Now what we doing here is asking is it the case that if people play the grim trigger strategy that that is in fact an equilibrium that is a way of sustaining cooperation And the answer is, we need 1, that’s our temptation, to be less than, big bracket, it.

[译文 11]

好的，我们也可以把这个放进去。所以这个东西，2除以1减delta，是永远得到2的价值。现在在我转到新白板之前，我想做另一件事。在左边，我有我的诱惑。那是1。我得到了从明天开始永远合作的价值，那是2除以1减delta，我得到了从明天开始永远背叛的价值，那是0。然而，右边所有这些东西，它们从明天开始，而诱惑是今天。今天。今天的诱惑发生在今天。这些价值的差异从明天开始。既然它们从明天开始，我需要折现它们，因为我们不知道明天会发生。明天和今天之间，世界可能会结束，或者更重要的是，关系可能会结束。所以我需要给它们多大的权重？用delta。我需要把所有的这些行乘以delta，以此类推。好的，现在这变成了一团乱，所以让我们转到新白板。好的，让我们总结一下我们现在有的。我们在这里做的是问，如果人们玩冷酷策略，这实际上是一个均衡吗？这是一种维持合作的方式吗？答案是，我们需要1，那是我们的诱惑，要小于大括号里的它。

[段 12]

2 over 1 minus delta, that’s the value of cooperating forever starting from tomorrow. Minus 0, that’s the value of defecting forever starting tomorrow. And this whole thing is multiplied by delta because tomorrow may not happen. Okay, everyone happy with that so far. I’m just collecting up the terms that we did slowly just now. Now what I want to do is, put a question mark here because I don’t know whether it is, I’m going to solve this for delta. When I solve this for delta, I’ll probably get it wrong, but let’s be careful. This is equivalent to saying, I have to worry about the zeroes, So this is a common for saying 1 minus delta is less than 2 delta And it also equivalent to saying therefore delta is greater than or equal to a third Delta is greater than or equal to a third. Everyone happy with that? Let me just turn my own page. Alright. So what have we shown so far? We’ve shown that if we’re playing the grim trigger strategy and we want to deter people from doing what? From defecting from this strategy in the very first period, then we’re okay, provided delta’s bigger than a third. But at this point, some of you could say, yeah, but that’s just one of the possible ways I could defect from this strategy.

[译文 12]

2除以1减delta，那是明天开始永远合作的价值。减0，那是明天开始永远背叛的价值。而整个东西要乘以delta，因为明天可能不会发生。好的，大家都明白到目前为止吗？我只是在收集我们刚才慢慢做的项。现在我要做的是在这里放一个问号，因为我不知道它是不是，我要解这个delta。当我解这个delta时，我可能会算错，但让我们仔细点。这等价于说，我必须担心零，所以这通常是说1减delta小于2 delta这也等价于说因此delta大于等于三分之一。delta大于等于三分之一。大家都明白吗？让我翻一下我自己的讲义。好的。那么我们到目前为止证明了什么？我们证明了如果我们正在玩冷酷策略，并且我们想要阻止人们做什么？从这个策略的第一期就开始背叛，那么只要delta大于三分之一，我们就没问题。但此时，你们中的一些人可能会说，是的，但这只是我从这个策略背叛的一种可能方式。

[段 13]

After all, the defection we just considered, the move away from equilibrium we just considered, was what? thought we considered my cheating today, but thereafter I reverted back to doing what I was supposed to do. I went along with playing D thereafter. So the particular defection we looked at just now was in period 1 I’m going to defect, but thereafter I’m actually thereafter. So the particular defection we looked at just now was in period one, I’m going to defect, but thereafter, I’m actually going to do what the equilibrium strategy tells me to do. I’m going to go along with the punishment. I’m going to go along with the punishment and play my part of DD forever. So you might want to ask, why would I do that? Why would I go along? I cheated the first time, but now I’m doing what the strategy tells me to do. It tells me to play D. Why am I going along with that? You could consider going away from the equilibrium by defecting, for example, in period 1, and then in period 2 do something completely different, like cooperating. So we might want to worry. play, how about playing D now and then C in the next period, and then D forever. That’s just some other way of defecting So far we said I going to defect by playing D and then playing D forever But now I saying let play D now then play a period of C and then D forever Is that going to be a profitable deviation Well, let’s see what I’d get if I do that particular deviation.

[译文 13]

毕竟，我们刚才考虑的背叛，我们刚才考虑的偏离均衡的举动，是什么？我们考虑了今天我作弊，但之后我恢复做我应该做的事。之后我就按部就班地选择D。所以我们刚才看的这个具体背叛发生在第1期，我要背叛，但之后我实际上要按照均衡策略告诉我的去做。我要接受惩罚。我要接受惩罚，永远配合DD。所以你可能会问，我为什么要这样做？我为什么要接受？我第一次作弊了，但现在我在做策略告诉我要做的事。它告诉我要选D。我为什么要接受这个？你可以考虑通过在第1期背叛来偏离均衡，然后在第2期做一些完全不同的事情，比如合作。所以我们可能想担心……玩法，如果现在选D然后下一期选C，然后永远选D呢？那只是另一种背叛方式。到目前为止我们说我将通过选D然后永远选D来背叛。但现在我说让我现在选D然后选一期C然后永远选D。这会是一个有利可图的偏离吗？好，让我们看看如果我做出这个特定的偏离会得到什么。

[段 14]

What play is that going to induce? Remember, the other player is playing equilibrium. So that play is going to induce, in the first period, I’m playing D and Jake’s playing C. In the second period, Jake’s going to start punishing me. So he’s going to play D. And according to this deviation, I’m going to play C. So in the second period, I’ll play C and Jake will play D. And in the third period and thereafter, we’ll just play D, D, D, D, D, D, D. So this is just some other deviation other than the one we looked at. So what payoff do I get from this? What payoff will I get from this? I get 3 in the first period, just as I did for my original defection. That’s good news. But now in the second period, in the second period, discounted I actually get minus 1 I actually doing even worse in the second period because I cooperating while Jake defecting And then the third period I get 0 and the fourth period, I get 0, and so on. So the total payoff to this defection is 3 minus delta. Now that’s even worse than the defection we considered to start with. The defection we considered to start with, I got 3 in the first period, and thereafter I got 0. Now I got 3 in the first period, minus 1 in the second period, and then 0 thereafter.

[译文 14]

这个玩法会诱导出什么？记住，另一个玩家在玩均衡策略。所以这个玩法会诱导出，在第一期，我选D而Jake选C。在第二期，Jake会开始惩罚我。所以他要选D。根据这个偏离，我要选C。所以在第二期，我选C而Jake选D。在第三期及之后，我们就一直选D、D、D、D、D、D、D。所以这只是我们之前看的那个偏离之外的另一种偏离。所以我从这个偏离中得到什么收益？我从这个偏离中得到什么收益？第一期我得到3，就像我最初背叛时一样。这是好消息。但现在在第二期，折现后我实际上得到负1。我在第二期实际上做得更差，因为我在合作而Jake在背叛。然后第三期我得到0，第四期我得到0，以此类推。所以这个背叛的总收益是3减delta。现在这比我们最初考虑的背叛还要糟糕。我们最初考虑的背叛，我在第一期得到3，之后得到0。现在我在第一期得到3，第二期得到负1，然后之后都是0。

[段 15]

So this defection in which I defect, this move away from equilibrium in which I cheat in the first period and then don’t go along with the punishment, I don’t, in fact, play D forever, is even worse. Is that right? It’s even worse. So what’s the lesson here? The lesson here is the reason that I’m prepared to go along with my own punishment and play D forever after a defection is what? If Jake is going to play D forever I may as well play D forever Is that right Now the way of saying this is the only way which I could possibly hope to have a profitable deviation given that Jake going to revert to playing D forever is for me to defect on Jake once, and then go along with playing D forever. There’s no point, once he’s playing D, there’s no point in me doing anything else. So this is worse. This is even worse. This defection is even worse. And more generally, the reason this is even worse is because the punishment we looked at before, which was DD forever, the punishment DD forever is itself an equilibrium. It’s credible because it’s itself an equilibrium. So unlike in the finite repeated games we looked at last time, unlike in the two period or the five period repeated games, here the punishment really is a credible punishment because what I’m doing in the punishment phase is playing an equilibrium.

[译文 15]

所以这个背叛——我在第一期作弊然后不接受惩罚、不永远选D的偏离均衡——甚至更糟。对吗？甚至更糟。所以这节课的教训是什么？教训是，我愿意接受自己的惩罚并在背叛后永远选D的原因是什么？如果Jake要永远选D，我不如也永远选D。对吗？这样说来，鉴于Jake要恢复永远选D，我唯一可能希望有利可图的偏离方式就是在Jake身上背叛一次，然后接受永远选D。一旦他在选D，我做其他任何事都没有意义。所以这个更糟。这个甚至更糟。这个背叛甚至更糟。更一般地，这甚至更糟的原因是，我们之前看的惩罚，即永远DD，本身就是一个均衡。它是可信的，因为它本身就是一个均衡。所以不像我们上次看的有限重复博弈，不像两期或五期的重复博弈，这里的惩罚确实是一个可信的惩罚，因为我在惩罚阶段做的事情是在玩一个均衡。

[段 16]

There’s no point considering any other deviation. other than playing D once, and then just going on playing D. So that’s one other possible deviation, but there are others you might want to consider. So far, all we’ve considered is what? we considered the deviation where in the very first period I cheat on Jake and then I just play D forever. But what about the second period? Another thing I could do is how about cheating not in the first period of the game, but in the second. So according to this strategy what I’m going to do, the first period of the game I’ll go along with Jake and cooperate, but in the second period I’ll cheat on him. Now how am I going to check whether that a possible a good deviation or not How do I know that not going to be a good deviation Well, we already know that I’m not going to want to cheat in the first period of the game. I want to argue that exactly the same analysis tells me I’m not going to want to cheat in the second period of the game. Why? Because once we reach the second period of the game, it is the first period of the game. Once we reach the second period of the game, looking from period 2 onwards is exactly the same as it was when we looked from period 1 initially.

[译文 16]

考虑任何其他偏离都没有意义。除了选一次D然后继续永远选D。所以那是另一个可能的偏离，但还有其他的你可能想考虑。到目前为止，我们考虑的是什么？我们考虑的是在第一期欺骗Jake然后永远选D的偏离。但第二期呢？另一件我能做的事是不要在游戏的第一期作弊，而是在第二期。根据这个策略，我要做什么，游戏的第一期我要配合Jake合作，但在第二期我要欺骗他。现在我要如何检查这是否是一个可能的好偏离？我怎么知道这不会是一个好偏离？好，我们已经知道我不会想在游戏的第一期作弊。我想论证，同样的分析告诉我，我也不会想在游戏的第二期作弊。为什么？因为一旦我们到达游戏的第二期，它就是游戏的第一期。一旦我们到达游戏的第二期，从第2期开始看和最初从第1期看完全一样。

[段 17]

So say again, what we argued before was, on the board I’ve now covered up, what we argued before was, no, it’s here, what we argued before was I’m not going to want to cheat in the very first period of the game provided delta’s greater than a third. And I want to claim that that same argument tells me I’m not going to want to cheat in the second period of the game, provided delta is bigger than the third, and I’m not going to want to cheat in the fifth period of the game, provided delta is bigger than the third. Because this game from the fifth period on or the 500th period on or the 1 period on looks exactly the same as it does from the beginning So what neat about this argument is the same analysis says this is not profitable if delta is bigger than a third. All right. So what have we learned here? I want to show some nerdy lessons and then some actual sort of real-world lessons. So let’s start with the nerdy lessons. The nerdy lesson is this Grimm strategy works because both, let’s put it up again so we can actually see it, this grim strategy, here it is, it works because both the play that it suggests if we both cooperate and the play that it suggests if we both defect are themselves equilibrium These are credible threats and credible promises because what you end up doing in the both in the promise and in the threat, is itself equilibrium behavior.

[译文 17]

再说一遍，我们之前论证的是，在我刚才遮住的板子上，我们之前论证的是，不，是这里，我们之前论证的是，只要delta大于三分之一，我就不想在游戏的第一期作弊。我想声称同样的论证告诉我，只要delta大于三分之一，我就不想在游戏的第二期作弊，而且只要delta大于三分之一，我就不想在游戏的第五期作弊。因为这个游戏从第五期开始或从第500期开始或从第一期开始，看起来和从开始看完全一样。这个论证的精妙之处在于，同样的分析表明，如果delta大于三分之一，这是不有利可图的。好，那么我们学到了什么？我想展示一些技术性的教训，然后是一些实际的世界教训。好，让我们从技术性教训开始。技术性教训是，这个grim策略有效，因为两者——让我再把它放出来这样我们能看到——这个grim策略在这里，有效是因为它建议的两种玩法——如果我们双方都合作时它建议的玩法，以及如果我们双方都背叛时它建议的玩法——本身都是均衡。这些是可信的威胁和可信的承诺，因为你在承诺和威胁中最终所做的本身就是均衡行为。

[段 18]

All right, that’s good. All right, the second thing we’ve learned, however, is for this to work, we need delta to be bigger than a third. We need the probability of continuation to be bigger than a third. So leaving aside the nerdy stuff for a second, you’ll have more practice on the nerdy stuff on the homework assignment, the lesson is we can get cooperation in the prisoner’s dilemma using the Grimm trigger. Remember the Grimm trigger strategy is cooperate until someone defects and then defect forever. So we get cooperation in the prisoner’s dilemma using the Grimm trigger as a sub-game perfect equilibrium. So this is an equilibrium strategy. That’s good news. Provided that Professor Ben Polak Professor Ben Polak Let’s try and generalize that lesson away from the prisoner’s dilemma. So last time, our lesson was about what in general could we hope for in ongoing relationships. So let’s put down a more general lesson that refines what we learned last time. So the more general lesson is, let me do it here, in an ongoing relationship, in an ongoing relationship, let me mimic exactly the words I used last time. So for an ongoing relationship, for an ongoing relationship, to provide incentives for good behavior today it helps right what we wrote last time was it helps for that relationship to have a future That’s what we wrote last time, but now we can refine this.

[译文 18]

好，那很好。好，然而我们学到的第二件事是，要使这有效，我们需要delta大于三分之一。我们需要继续的概率大于三分之一。好，暂时把技术性的东西放在一边，你们在作业中会有更多技术性东西的练习，教训是我们可以用grim触发器在囚徒困境中获得合作。记住grim触发器策略是合作直到有人背叛，然后永远背叛。所以我们用grim触发器作为子博弈完美均衡在囚徒困境中获得合作。所以这是一个均衡策略。这是好消息。假设Ben Polak教授Ben Polak教授……让我们把这个教训从囚徒困境中推广出去。所以上次，我们的教训是关于在持续关系中我们通常能希望什么。好，让我们写下更一般的教训来完善我们上次学到的东西。更一般的教训是，让我在这里写，在持续关系中，在持续关系中，让我完全模仿我上次用的措辞。所以对于持续关系，对于持续关系，为了给今天的好行为提供激励，它有助于……我们上次写的是它有助于那个关系有一个未来。那是我们上次写的，但现在我们可以完善这一点。

[段 19]

It helps for there to be a high probability high probability that the relationship will continue. All right, so the specific lesson for Prisoner’s Dilemma and the Grimm-Pregor strategy is we need delta, the probability of continuation to be bigger than a third, but the more general intuition is if we want my ongoing business relationship with me and Jake to generate good behavior, so I’m going to provide him with good fruit and he’s going to provide him with good vegetables, we need the probability that that relationship will continue to be reasonably high And I claim this is a very natural intuition Why Because the probability that the relationship will continue is the weight that you put on the future The probability that the relationship will continue this thing. This is the weight you put on the future. The more weight I put on the future, the more likely, the easier it is for the future to give me incentives to behave well today. The easier it is for those to overcome the temptations to cheat today. That seems like a much more general lesson than just the prisoner’s dilemma example. Let’s try to push this to some examples and see if it rings true. The lesson we’ve got here is to get cooperation in these relationships, we need there to be a high probability, a reasonably high probability that they’re going to continue.

[译文 19]

这段话对于建立合作关系非常有帮助，因为关系继续的概率必须足够高。具体来说，囚徒困境和格林-Pregor策略的核心教训是，我们需要δ（继续的概率）大于三分之一。但更普遍的直觉是：如果我希望与Jake之间的持续业务关系能够产生良好行为——我向他提供优质水果，他向我提供优质蔬菜——那么这种关系继续的概率必须足够高。我认为这是一个非常自然的直觉。为什么？因为关系继续的概率就是你放在未来的权重。这个概率就是你放在未来的权重。你把越多的权重放在未来，未来就越容易给你动力让你今天表现良好，就越容易让你克服今天作弊的诱惑。这似乎是一个比囚徒困境例子更普遍的教训。让我们把这个应用到一些例子中，看看它是否成立。我们在这里得到的教训是，要在这些关系中获得合作，我们需要有很高的概率，即相当高的概率，他们将会继续。

[段 20]

We know exactly what that is for prisoner’s dilemma, but the lesson seems more general. So here’s two examples. How many of you seniors? Quite a few of you are seniors. Keep your hands up a second Of those of you who are seniors we can pan these guys Let have a look at them all right actually we get mister almost in to stand up need to work a bit here all right now that now the tricky question the tricky personal question how many of you who are seniors are currently involved in personal and you have a have a significant other stay standing up if you’re still if you’re if you have a significant other look at this This is pathetic. What have I been saying about economic majors? All right. So let’s just think about it. So stay standing a second. Let’s get these guys to think about it in a second. All right. So seniors who are involved in ongoing relationships with significant others, what do we have to worry about those seniors? Well, these seniors are about to depart from the beautiful confines of New Haven. And they’re going to take jobs in different parts of the world.

[译文 20]

我们确切知道这对于囚徒困境意味着什么，但这个教训似乎更具普遍性。好，这里有两个例子。你们当中有多少是大四学生？相当多的人是大四学生。手举高一点，稍等一下在你们这些大四学生中，我们可以扫视一下这些人让我们看看他们，好吧，实际上我们请那位先生站起来，需要稍微配合一下，好，现在那个棘手的问题，那个涉及个人生活的棘手问题——你们这些大四学生中，有多少目前正处于恋爱关系中，有另一半的，请继续站着——如果你还有——如果你有另一半的话，看看这个这太可悲了。我一直在跟经济学专业的学生说什么来着？好吧，让我们想想这个。所以继续站一会儿。让我们让这些人想一想。好了，那么那些正处于恋爱关系中的大四学生，我们有什么需要担心的呢？嗯，这些大四学生即将离开纽黑文这个美丽的地方。他们将去世界各地工作。

[段 21]

And the problem is, some of them are going to take jobs in New York while their significant other takes a job in San Francisco or Baghdad whatever it’s not hope that better take off like that but London to be safe all right now if it’s the case if it’s the case that you are going to take a job in New York next year and your significant other is going to take a job in Baghdad or London anyway far away what does that do to the are going to take a job in New York next year, and your significant other is going to take a job in Baghdad or London, or anyway, far away, what does that do to the, in reality, being cynical a little bit, what does that do to the probability that your relationship’s going to last? It makes it go down, right? It makes it go down. It lowers the probability that your relationship’s going to continue, right? So what is the prediction? Let’s be mean here. How many if you have to, these are the people with significant others who are seniors, how many of you are going to be separated by a long distance from your significant others next period? Well, one of them, one of them at the back, okay, one guy, no, two, there we go, sort of honesty here, three, four of them, right?

[译文 21]

问题在于，他们中的一些人要去纽约工作，而他们的伴侣则要去旧金山或巴格达——不管怎样——希望不要发展成那样，但要伦敦才安全。好吧，如果情况是这样的话，如果你明年要去纽约工作，而你的伴侣要去巴格达或伦敦工作，或者不管怎样，离得很远，这对你们关系的持续概率会有什么影响呢？稍微有点愤世嫉俗地说，这会让概率下降，对吧？这会让概率下降。它会降低你们关系继续的可能性，对吧？那么预测是什么呢？让我们无情一点。如果你不得不问的话，这些是伴侣是高年级学生的群体，你们中有多少人在下一个阶段会与伴侣分隔两地？嗯，其中一个人，后排的一个，好吧，一个人，不对，两个，这样比较诚实，三个，四个，对吧？

[段 22]

So what’s our prediction here? What does this model predict as a social science experiment? What does it predict? It predicts that for those of you who just raised your hands, those seniors who just raised their hands who are about to be separated by large distances, those relationships, each player in that relationship, is going to have a lower value on the future. So during the rest of your senior year, during the spring of your senior year, what the prediction of this model They going to cheat They going to right So we can actually do a controlled experiment What we should do here is we should keep track of the people here, the seniors who are about to have long-term relation, who are going to be separated. You can sit down now. I’m sorry to embarrass you all. We could keep track of those seniors who are about to be separated and go into long-distance relationship and those that are not, the people who are not are our control group. And we should see if during the spring semester, the people who are going to be separated cheat more often than the others. All right? So it’s a very clear prediction of the model that’s relevant to some of your lives. All right? Let me give you another example that’s less exciting, perhaps, but the same kind of thing.

[译文 22]

那么我们的预测是什么？这个模型作为一个社会科学实验，它预测了什么？它预测，对于刚才举手的那些人，那些即将因为远距离而分开的 seniors 之间的关系，关系中的每个人对未来的评价都会降低。所以在整个大四剩余的时间里，在大四的春季，这个模型的预测是什么？他们会去欺骗 他们会去 right 所以我们实际上可以做一个对照实验 我们应该做的是追踪这里的人，那些即将进入长期关系、即将分开的人。你可以坐下了。很抱歉让你们尴尬了。我们可以追踪那些即将分开并进入 long-distance relationship 的 seniors 和那些不会分开的，不分开的人就是我们的 control group。然后我们应该看看在春季学期，那些即将分开的人是否比其他人更频繁地作弊。明白吗？这是一个非常明确的模型预测，与你们一些人的生活相关。明白了吗？让我给你们另一个例子，也许没那么激动人心，但属于同样的情况。

[段 23]

Consider the relationship that I have with my garage mechanic. I should stress this is not a significant other relationship. All right? So I have a garage mechanic in New Haven. And that garage mechanic fixes my car. And we have an ongoing business relationship. He knows that whenever my car needs fixing, even if it’s just a small thing like an oil change, I’m going to go to him and have him fix it, even though it might be cheaper for me to go to Jiffy Lube or something. All right? So I going to take my car to him to be fixed and he going to make some money off me on even the easy things What do I want in return for that I want him to be honest And if all I need is an oil change I want him to tell me that And if what I actually need is a new engine, he tells me I need a new engine. So my cooperating with him is always going to him, even if it’s something simple. And his cooperating with me is his not cheating on fixing the car. He knows more about the car than I do. But now what happens if he knows either that I’m about to leave town, which is the example we just did, or more realistically, he kind of knows that my car’s a lemon and I’m about to get rid of it anyway.

[译文 23]

考虑我和我的修车师傅之间的关系。我需要强调这不是恋爱关系。好吗？所以我在New Haven有一个修车师傅。那个修车师傅帮我修车。我们有一种持续的商业关系。他知道每次我的车需要修理时，即使只是像换机油这样的小事，我也会去找他修理，即使去Jiffy Lube可能会更便宜。好吗？所以我要把车交给他修理，即使是很简单的事情他也能从我这里赚到钱。作为回报我想要什么？我希望他诚实。如果我只是需要换机油，我希望他告诉我就是这样。如果我实际需要的是新发动机，他会告诉我需要新发动机。所以我的合作方式是总是找他，即使是很简单的事。而他的合作方式就是修理汽车时不欺骗我。他比我更懂车。但现在如果他知道我要离开 town 了——就是我们刚才举的例子——或者更现实一点，他知道我的车是个次品而我反正要处理掉它，会发生什么？

[段 24]

Once I get a new car, I’m not going to go to him anymore because I have to go to the dealer to keep the warranty intact. So if he knows that my car is about to break down anyway, and he knows that I know the car’s about to break down anyway, and my lemon of a car is about to be passed on probably to all my graduate students, then what’s going to happen? What’s going to happen? So I’m going to have an incentive to cheat because I’m going to start taking my useless car to Jiffy Lube for the oil changes, and he’s going to have an incentive to cheat. He’s going to start telling me, you know, you really need a new engine or a new clutch It a manual so I have a clutch It a real car So I going to need a new clutch rather than just tightening up a bolt So once again the probability of the continuation of the relationship as it changes, it leads to incentives to cheat. It leaves that relationship breaking down. That’s the content, that’s the real world content of the math we just did. Let’s try and push this a little further. Now, what we’ve shown is that the Grimm trigger works provided delta is bigger than 1 third. And delta being bigger than 1 third doesn’t seem like a very large continuation of probability.

[译文 24]

一旦我买了新车，我就不会再去找他了，因为我必须去经销商那里做保养才能保持保修有效。所以如果他知道我的车反正快坏了，他也知道我知道我的车快坏了，而我那辆柠檬车（次品车）很可能会传给我所有的研究生，那会发生什么呢？会发生什么呢？所以我就会有作弊的动机，因为我开始会把我的破车送到Jiffy Lube换机油，而他就会有作弊的动机。他会开始跟我说，你知道吗，你真的需要一个新发动机或者一个新的离合器——那是手动挡的，所以我需要一个新离合器，而不是仅仅紧一个螺栓。所以再一次，随着这段关系继续下去的可能性发生变化，它会导致作弊的动机。它最终会让这段关系破裂。这就是内容，这就是我们刚才数学运算的真实世界内容。让我们再深入一点。我们已经证明，Grimm trigger（格林姆触发策略）在delta大于1/3时有效。而delta大于1/3看起来并不是一个很大的继续概率。

[段 25]

So just having a probability of 1 third that the relationship continues allows the Grimm trigger to work. That seems good news for the Grimm trigger. However, in reality, in the real world, The grim trigger might have some disadvantages. So let’s just think about what the grim trigger is telling us in the real world. It’s telling us that if even one of us cheats just a little bit, I just provide one item of rotten fruit to Jake, or he gives me one too few branches of asparagus in his provisions to me, then we never do business with each other again ever. shake, or he gives me one too few branches of asparagus in his provisions to me, then we never do business with each other again ever. It’s completely the end. We just never cooperate again. And that seems a little bit drastic. It’s a little bit draconian, if you like. So in particular, in the real world, there’s a complication here. In the real world, every now and then, one of us is going to cheat by accident. That day, I didn’t have my glasses on and I put in a rotten apple in the apples I supplied to Jake. And in the fruit, he was counting out the asparagus and he lost count at 1,405 and he gave me one too few. And so we might want to worry about the fact that the grim trigger, it’s triggered by any amount of cheating and it’s very drastic.

[译文 25]

所以，只要关系继续的概率达到三分之一，Grimm触发器就能发挥作用。这对Grimm触发器来说似乎是好消息。然而，在现实中，在现实世界里，Grimm触发器可能有其缺点。让我们想想Grimm触发器在现实世界中告诉我们的含义。它告诉我们，如果我们中的任何一个人只是稍微作弊一点点，比如我给Jake提供了一件腐烂的水果，或者他在给我的配给中少给了几根芦笋，那么我们就永远不会再次做生意了。摇手，或者他在给我的配给中少给了几根芦笋，那么我们就永远不会再次合作了。一切都彻底结束了。我们再也不会合作了。这似乎有点极端。如果可以这么说的话，这有点过于严苛。因此，特别是在现实世界中，这里有一个复杂的问题。在现实世界中，我们时不时会因为意外而作弊。那天，我没戴眼镜，把一个烂苹果放进了我提供给Jake的苹果里。在水果方面，他在数芦笋时，在1405这个数字上数错了，给了我少了一根。所以我们可能需要担心的是，Grimm触发器会受到任何程度的作弊触发，而且它非常极端。

[段 26]

It says we never do business again. The grim trigger is the analogue of the death penalty. It’s the business analogue of the death penalty. It’s not I’m going to kill Jake if he gives me one, two, three branches of asparagus, but I’m going to kill the relationship. For you seniors or otherwise who are involved in personal relationships, it’s the equivalent of saying if you even see your partner looking at someone else and alone sitting next to them in class the relationship over It seems drastic So we might be interested because mistakes happen because misperceptions happen we might be interested in using punishments that are less draconian than the Grimm-Trigger, less draconian than the death penalty. Is that right? So what I want to do is, I want to consider a different strategy, a strategy other than the Grimm-Trigger strategy, and see if that could work. All right. So, where shall I start? Let’s start here. All right. Okay, so again, what I’m going to revert to is the math and the nerdiness of our analysis of the prisoner’s dilemma, but I want you to have in mind business relationships, your own personal relationships, your friendships, and so on. Everything, more or less everything you do in life involves repeat interactions. So have that in the back of your mind, but let’s be nerdy now. So what I want to consider is a one period punishment.

[译文 26]

它说我们永远不再做生意。Grimm触发器是死刑的模拟版本。它是商业领域的死刑模拟。不是我要因为Jake给我一到三根芦笋就杀他，而是我要扼杀这段关系。对于你们这些涉及人际关系的学长或其他人来说，这相当于说如果你看到你的伴侣独自和另一个人坐在教室里看着对方，关系就结束了。这似乎很极端。所以我们可能感兴趣的是，因为错误会发生，因为误解会发生，我们可能对使用比Grimm触发器更不那么严苛的惩罚措施感兴趣，比死刑更不那么严苛的惩罚。这是正确的吗？所以我想做的是，我想考虑一种不同的策略，一种除了Grimm触发器策略之外的策略，看看它是否能起作用。好的。那么，我从哪里开始呢？让我们从这里开始。好的，好的，所以，我要再次回到我们分析囚徒困境的数学和宅男宅女的一面，但我希望你们心里记住的是商业关系、你自己的个人关系、你的友谊等等。生活中你做的几乎所有事情都涉及重复互动。所以把这个记在心里，但现在让我们变得宅一点。所以我想考虑的是一种单期惩罚。

[段 27]

A one period punishment. So how are we going to write down a strategy that has cooperation but a one period punishment So here the strategy It says it kind of a weird thing but it works, play C to start and then play C if, here’s the tricky thing, it’s going to seem weird but trust me for a second, play C if either CC or DD were played last. So if in the previous period either both people cooperated or both people defected, then we’ll play cooperation this period. and play D otherwise. Play D if either CD or DC were played last. All right, let’s just think about this strategy for a second. What does this strategy mean? So provided people start off cooperating and they go on cooperating if both people play both Jake and I play this strategy in fact we cooperate forever Is that right So I claim this is a one period punishment strategy. Let’s just see how that works. So suppose Jake and I are playing this strategy, we’re supposed to play C every period, and suppose, deliberately or otherwise, I D. So now in that period in which I play D, the strategies played were D by me and C by Jake. So next period, what does this strategy tell us both to play? So it was D by me and C by Jake, so this strategy tells us to play D.

[译文 27]

一种单期惩罚。那么我们要如何写出一种策略，让它能够促进合作，但只有单期惩罚呢？所以这里的策略——它说起来有点奇怪，但它有效——开始时选择合作，然后如果出现以下情况就选择合作：这里有个棘手的部分——它看起来会很奇怪，但请暂时相信我——如果上一轮是CC或DD，就选择合作。所以如果在前一个时期，两个人都合作了，或者两个人都背叛了，那么我们这个时期就选择合作。否则就选择背叛。如果上一轮是CD或DC，就选择背叛。好的，让我们想想这个策略一会儿。这个策略意味着什么？所以只要人们开始合作并继续合作——如果两个人，Jake和我，都玩这个策略——我们实际上会永远合作。这是正确的吗？我声称这是一个单期惩罚策略。让我们看看它是如何运作的。假设Jake和我正在玩这个策略，我们应该每个时期都选择合作，假设我故意或以其他方式选择了背叛。所以现在在我选择背叛的那个时期，策略执行结果是：我选择了背叛，Jake选择了合作。那么下一个时期，这个策略告诉我们两个都选择什么？所以是我选择了背叛，Jake选择了合作，所以这个策略告诉我们下一个时期选择背叛。

[段 28]

So next period, both of us will play D. Both of us will be uncooperative for precisely that period, that next period. Now what about about the period after that? The period after that, Jake will have played D, I will have played D, so this is what will have happened. We both played D, and now it tells us to cooperate again. Okay, everyone happy with that? All right, so this strategy I’ve written down, it seems kind of cumbersome, but what it actually induces is exactly a one-period punishment. If Jake cheats Happy with that? So this strategy I’ve written down, it seems kind of cumbersome, but what it actually induces is exactly a one-period punishment. If Jake is the only cheat, then we both defect for one period and go back to cooperation. If I’m the only person who cheats, then we both defect for one period and go back to cooperation. It’s a one-period punishment strategy. And of course the question is, the question you should be asking is, is this going to work? Is this an equilibrium? So let’s just check. Is this an SP? Is it an equilibrium? So what do we need to check? We need to check, as usual, that the temptation is than or equal to the value of the promise of continuing incorporation the value of the promise minus the value of the threat And once again, we have to be careful because the temptation occurs today, and this difference between values occurs tomorrow.

[译文 28]

所以下一个时期，我们两个都会选择背叛。我们两个在这个下一个时期都会不合作，正好是那个时期。现在关于那个时期之后呢？在那个时期之后，Jake会选择背叛，我也会选择背叛，所以这就是将会发生的情况。我们两个都选择了背叛，现在它告诉我们再次合作。好的，每个人对此都满意吗？好的，所以这个我写下的策略，它看起来有点麻烦，但实际上它所产生的结果正是单期惩罚。如果Jake是唯一的作弊者，那么我们两个都会背叛一个时期，然后回到合作。如果我是唯一一个作弊的人，那么我们两个都会背叛一个时期，然后回到合作。这是一种单期惩罚策略。当然问题是，你应该问的问题是，这能行得通吗？这是一个均衡吗？让我们检查一下。这是一个SP吗？它是一个均衡吗？那么我们需要检查什么？我们需要像往常一样检查，诱惑是否大于或等于继续合作承诺的价值——承诺的价值减去威胁的价值。同样，我们必须小心，因为诱惑发生在今天，而这种价值差异发生在明天。

[段 29]

Is that right? So this isn’t nothing new. This is what we’ve always written down, so we have to check. So the temptation for me to cheat today, that’s the same as it was before. It’s 3 minus 2. The fact that tomorrow is going to give me a delta here. Here’s our square bracket. So what’s the value of the promise? So provided we both go on cooperating, we’re going to go on cooperating forever, in which case we’re going to get 2 forever. Is that right? So this is going to be the value of 2 forever starting tomorrow, and again forever means until the game ends. The value of the threat is what Be a bit careful now It the value of so what going to happen If I cheat then tomorrow we both going to cheat Tomorrow we’re both going to cheat. So tomorrow, what am I going to get tomorrow? Zero. So it’s the value of zero tomorrow and then 2, we’re both going to cheat, we’re both going to play D, and then the next period what’s going to happen? We’re going to play C again, and from there on we’re going to go on playing C. So it’s going to be the value of 0 tomorrow and then 2 forever starting the next day. That’s what we have to evaluate. So 3 minus 2, I can do that one again, that’s 1.

[译文 29]

这是正确的吗？所以这没有什么新东西。这和我们一直写下的东西一样，所以我们必须检查。所以我今天作弊的诱惑，和以前一样，是3减2。事实是明天会给我一个delta在这里。这是我们的方括号。那么承诺的价值是多少？所以只要我们两个继续合作，我们会永远继续合作，在这种情况下我们会永远得到2。这是正确的吗？所以这将是从明天开始的永远2的价值，同样，永远意味着直到游戏结束。威胁的价值是多少？现在要小心一点——威胁的价值是，所以如果我作弊会发生什么？如果我作弊，那么明天我们两个都会作弊。明天我们两个都会作弊。所以明天我明天会得到什么？零。所以这是明天零的价值，然后2，我们两个都会作弊，我们两个都会选择背叛，然后下一个时期会发生什么？我们会再次选择合作，从那里开始我们会继续选择合作。所以这将是明天零的价值，然后从第二天开始的永远2的价值。这是我们必须计算的。3减2，我可以再算一遍，那是1。

[段 30]

So what’s the value of 2 forever? We did that already today, what was it? It’s in your notes, actually it’s on the board, it’s that x up there, what is it? Here it is 2 forever we figured out the value of it before and it was 2 over 1 minus delta So the value of 2 forever is going to be 2 over 1 minus delta. How about the value of 0, so starting from tomorrow, I’m going to get 0 and then with one period delay I’m going forever. Well, two forever, we know what the value of that is. It’s 2 over 1 minus delta. But now I get it with one period delay. So what do I have to multiply it by? By delta. Good. So the value of zero tomorrow and then two forever starting the next day is delta times 2 over 1 minus delta. And here’s the delta coming from here, which just takes into account of all this analysis is starting tomorrow. All right? Just to summarize, this is my temptation today. This is what I’ll get starting tomorrow if I’m a good boy and cooperate. And this is the value of what I’ll get if I cheat today. Starting tomorrow, I’ll get nothing and then I’ll revert back to cooperation. And since all of these values in this square bracket start tomorrow, I’ve discounted them by delta.

[译文 30]

那么永远2的价值是多少？我们已经算过了今天，它是多少？它在你的笔记里，实际上它在黑板上，就是那里的x，它是多少？就是这个——永远2我们之前算出了它的价值，是2除以1减delta。所以永远2的价值将是2除以1减delta。零的价值怎么样，所以从明天开始，我会得到零，然后延迟一个时期我会永远得到2。嗯，永远2，我们知道它的价值是多少。是2除以1减delta。但现在我要延迟一个时期才能得到它。那么我需要乘以什么？乘以delta。好的。所以明天零的价值，然后从第二天开始的永远2的价值是delta乘以2除以1减delta。而这里的delta来自这里，它只是考虑了这整篇分析是从明天开始的。好的吗？总结一下，这是今天我的诱惑。如果我是个好孩子并合作的话，这是我明天开始会得到的。如果我今天作弊，这是我所得到的价值。明天开始，我会什么都得不到，然后我会恢复到合作。由于方括号中的所有这些价值都是从明天开始的，我已经用delta将它们折现了。

[段 31]

All right, now this requires some math. So… bear with me while I probably get some algebra wrong, and please can I get the TAs to stare at me a second, because I’ll probably get this wrong. Okay, so what I’m going to do is, I’m going to look at my notes, I’m going to cheat, that’s what I’m going to do. Okay, so what I’m going to do is I’m going to have 1 1 is less than or equal to, I’m going to get a common factor of 2 over 1 minus delta and delta. So I’m going to have 2 delta over 1 minus delta and that’s going to leave inside the square brackets, this is a 1 and this is a delta. delta here was that delta there, and I took out a common factor of 2 from this bracket. Are we okay with the algebra? Just algebra, nothing fancy going on there. So that’s good because now the delta the 1 minus delta cancels right This cancels with this so this tells us we okay provided 1 provided a half is less than or equal to delta And we’re done. All right? Okay, so don’t worry too much about the algebra. Trust me on the algebra a second. All right, let’s just worry about the conclusion. What’s the conclusion? conclusion. The conclusion is that this one period punishment is an SPE, it will be enough, one period of punishment will be enough to sustain cooperation in my prisoner’s dilemma repeated business relationship with Jake, or in the senior’s relationships with their significant others, provided Delta is bigger than a half.

[译文 31]

好的，这需要一些数学计算。所以……请忍受我可能犯的一些代数错误，助教们能看我一眼吗，因为我可能会出错。好的，我来看看我的笔记，我这是要作弊，就是这样。好的，我要做的是，我有1，1是小于等于的，我要提取一个公因子2除以1减delta和delta。所以我会有2 delta除以1减delta，然后在方括号里会剩下这个1和这个delta。这里的delta就是那个delta，我从括号里提取了一个公因子2。我们对代数没问题吧？只是代数，没什么复杂的。这样很好，因为现在delta和1减delta消掉了，对吧？这个和这个消掉了，所以这告诉我们，只要1，只要二分之一小于等于delta就可以了，我们就完成了。好吗？好的，所以不要太担心代数。代数方面相信我一下。好的，我们只关心结论。结论是什么？结论。我们得出的结论是，这个一期惩罚是一个序贯均衡，它就足够了，一期惩罚就足以维持与Jake的重复囚徒困境商业关系中的合作，或者学长学姐与恋人之间的关系中的合作，只要delta大于二分之一。

[段 32]

What did Delta need to be for the Grimm strategy? A third. So what did we learn here? What did we learn? We learned so, notably, what we learned was that for the Grimm strategy, we needed delta bigger than a third. For the one period punishment, we needed delta to be bigger than a half. But what the more general lesson The more general lesson is if you use a softer punishment if you use a softer punishment a less draconian punishment for that to work we going to need a higher delta Is that right? Is that right? So what we’re learning here is there’s a trade-off. There’s a trade-off in incentives. And the trade-off is, if you use a shorter punishment, a less draconian punishment. Instead of cutting people’s hands off or killing them or never dealing with them again, you just don’t deal with them for one period. That’s okay, provided there’s a slightly higher probability of the relationship continuing. So shorter punishments are okay, but they need, well, the implication sign isn’t really necessary there, They need more value, more weight, more weight delta on the future. I claim that very intuitive What it saying is we always trading things off in the incentives We trading off the ability to cheat and get some cookies today versus waiting and we hope getting cookies tomorrow So if in fact the reward or the difference between the reward and the punishment isn’t such a big deal, isn’t too big, and the punishment is just I’m going to give you one fewer cookies tomorrow, then you better be pretty patient not to go for the cookies today.

[译文 32]

格林策略需要Delta是多少？三分之一。所以我们学到了什么？我们学到了什么？我们学到了，值得一提的是，对于格林策略，我们需要delta大于三分之一。对于一期惩罚，我们需要delta大于二分之一。但更普遍的教训是什么呢？更普遍的教训是，如果你使用更温和的惩罚，如果你使用更温和的惩罚，一个不那么严酷的惩罚，要使其生效，我们需要更高的delta。对吗？是这样吗？所以我们在这里学到的是，有一个权衡。在激励方面有一个权衡。这个权衡是，如果你使用更短的惩罚，一个不那么严酷的惩罚。不是砍掉人的手或杀死他们或再也不和他们做生意，而只是一期不和他们打交道。只要关系继续的可能性稍微高一点，那也可以。所以更短的惩罚是可以的，但它们需要，呃，那里其实不需要蕴含符号，它们需要更多的价值，更多的权重，未来的权重delta更高。我认为这非常直观。它说的是，我们总是在激励方面进行权衡。我们在权衡作弊的能力和今天获得一些饼干 versus 等待并希望明天获得饼干。所以如果实际上奖励或奖励和惩罚之间的差异不是那么重要，不太大，而且惩罚只是我明天少给你一块饼干，那你最好非常有耐心才不会去拿今天的饼干。

[段 33]

I was about to say those of you who have children, and I’m probably the only person in the room with children, that cookie example will resonate. For the rest of you, wait until you get there, you’ll discover that in fact cookies are the right example. So shorter punishments, less draconian punishments, less reduction in your kids’ cookie ration tomorrow is only going to work and you’re going to sustain good behavior provided those kids put a high weight on tomorrow. In that case, it isn’t that the kids are worried about the relationship breaking down. I mean, you’re stuck with your kids. It’s just that they’re impatient. OK, so we’ve been doing a lot of formal stuff here, and I want to go on doing formal stuff. But what I want to do now is spend the rest of today looking at an application. An application that’s, I hope, going I’m doing a lot of formal stuff here, and I want to go on doing formal stuff, but what I want to do now is spend the rest of today looking at an application. An application that’s, I hope, going to convince you that repeated interaction really matters. This is assuming that the one about the seniors and their boyfriends and girlfriends wasn’t enough. Okay, so the application is going to take us back a little bit because it’s going to, what I want to talk about is repeated moral hazard.

[译文 33]

我刚要说，你们当中有孩子的人，我是房间里可能唯一一个孩子的人，那个饼干例子会引起共鸣。对你们其余的人来说，等到你们有了孩子再说，你们会发现实际上饼干是正确的例子。所以更短的惩罚，更温和的惩罚，明天减少你孩子的饼干配给，只有在那些孩子高度重视明天的情况下才会有效，你们才能维持好行为。在这种情况下，不是孩子们担心关系破裂。我的意思是，你们和孩子们绑在一起了。只是他们没有耐心。好的，我们这里做了很多形式化的工作，我还想继续做形式化的工作。但我现在想做的是花剩下的时间看一个应用。一个应用，我希望这个应用能说服你们，重复互动真的很重要。这是假设那个关于学长学姐和他们的男朋友女朋友的例子还不够。好的，这个应用会让我们稍微回顾一下，因为它要讨论的是重复道德风险。

[段 34]

And moral hazard is something we discussed the first class after the midterm. So what I want to imagine is that you are running a business in the U.S. and you are considering making an investment in an emerging market and again so as not to offend anybody who watches this on the video let’s just call that emerging market Fredonia right rather than give it a name like Kazakhstan oh no no it’s like a name like something other than Fredonia all right all right all right so Fredonia for those who don’t know is a is a republic in a in a Marx Brothers film right so you thinking of outsourcing some production of what part of what your business is to this to Fredonia And the reason you thinking of doing this outsourcing what makes it attractive is that wages are low in Fredonia So you get this outsourced in Fredonia, you think you’re going to get it done cheaply. The downside is, because Fredonia is an emerging market, the court system doesn’t operate very well, and in particular, it’s going to be pretty hard to enforce contracts and to jail people and so on. in Fredonia. So you’re considering outsourcing. The plus is, from your point of view, the plus is wages are cheap where you’re going to get this production done. The downside is it’s going to be hard to enforce contracts, so this is an emerging market.

[译文 34]

道德风险是我们在期中考后的第一节课讨论的内容。所以我想让你们想象一下，你们正在美国经营一家企业，你们正在考虑在一个新兴市场进行投资，同样地，为了不冒犯任何观看视频的人，我们就把那个新兴市场叫做弗雷多尼亚好了，而不是给它起一个像哈萨克斯坦这样的名字，哦不不不，就像一个除了弗雷多尼亚之外的名字，好的，好的，好的，所以弗雷多尼亚，对于不知道的人来说，是一部马克思兄弟电影中的一个共和国。所以你们考虑把这个业务的一部分外包给弗雷多尼亚。你们考虑这样做的原因是弗雷多尼亚的工资很低。所以如果你在弗雷多尼亚外包，你认为你可以廉价完成。坏处是，因为弗雷多尼亚是一个新兴市场，法庭系统运作得不太好，特别是，执行合同和监禁人们等等会相当困难。所以你们正在考虑外包。好的一面是，从你们的角度来看，好的一面是工资便宜，你们可以在那里完成生产。坏处是执行合同会很难，所以这是一个新兴市场。

[段 35]

So what you’re considering doing is employing an agent, and you’re going to pay that agent W. So W is the wage if you employ them. And I’ll put this up in a tree in a second. Let’s assume that the going wage in Fredonia is one. All right, let’s normalize it. All right, so the going wage in Fredonia is one. And let’s assume that to get this outsourcing to work, you’re going to have to send some resources to your agent, your employee in Fredonia, and let assume that the amount you going to have to send over there is equivalent to another one All right so the going wage in Fredonia is one and the amount you going to have to send over there is equivalent to another one So the going wage is one and the amount you going to have to invest in giving this agent materials or machinery is another one And let’s assume that this project is a pretty profitable project. So if the project succeeds, if the project goes ahead and succeeds, it’s going to generate a gross revenue of four. Of course, you have to invest one so that’s a net revenue of three for you. But nevertheless, there’s a big potential return here. The bad news is that your agent in Fredonia can cheat on you. In particular, what he can do is he can simply take the one that you’ve sent to him, sell those materials on the market, and then go away and just work on his normal job anyway.

[译文 35]

所以你们考虑做的是雇佣一个代理人，你们要付给那个代理人W。所以W是工资，如果你们雇佣他们的话。我一会儿会把这个画成树形图。让我们假设弗雷多尼亚的现行工资是1。好的，让我们把它标准化。好的，弗雷多尼亚的现行工资是1。让我们假设为了让这个外包工作进行下去，你们必须向你们在弗雷多尼亚的代理人和员工发送一些资源，让我们假设你们必须发送过去的数量相当于另一个1。好的，弗雷多尼亚的现行工资是1，你们必须发送过去的数量相当于另一个1。所以现行工资是1，你们投资给这个代理人的材料或机械是另一个1。让我们假设这个项目是一个相当有利可图的项目。所以如果项目成功了，如果项目进行下去并成功了，它将产生4的总收入。当然，你们必须投资1，所以那是你们3的净收入。但尽管如此，这里有一个很大的潜在回报。坏消息是你们在弗雷多尼亚的代理可以欺骗你们。特别是，他可以做的是，他可以简单地拿走你们发送给他的那个东西，在市场上卖掉那些材料，然后走掉，继续做他的正常工作。

[段 36]

So he can get his normal wage of 1, just doing his normal job, whatever that was, and he can steal the resources from you. So let’s put this up as a kind of tree. This is a slight cheat, this tree, but we’ll see why in a second. So your decision is to invest and set W So if you invest in Fredonia you invest and set W set the wage you going to pay him The going wage is 1 but you could set a different wage, or you could just not invest. If you don’t invest, you get nothing and your agent in Fredonia just gets the going wage of 1. If you do invest in Fredonia and set a wage of W, your agent has a choice. Either he can be honest or he can cheat. If he cheats, what’s going to happen to you? You had to invest 1 in sending it over there. You’re going to get nothing back, so you’ll get minus 1. And he will go away and work his normal job and get 1 and in addition he’ll sell your materials. So he’ll get a total of 1 plus 1 is 2. Thank you. He’ll get a total of 2. On the other hand, if he’s honest, then you’re going to get a return of 4 minus the 1 you had to invest minus whatever wage you paid to him.

[译文 36]

所以他可以得到他正常的工资1，只做他的正常工作，不管那是什么，然后他可以偷走你们的资源。让我们把这个画成一个树形图。这个树形图是一个小小的作弊，但我们一会儿会明白为什么。你们的决定是投资并设定W，所以如果你们在弗雷多尼亚投资，你们投资并设定W，设定你们要付给他的工资。现行工资是1，但你们可以设定不同的工资，或者你们可以不投资。如果你们不投资，你们什么都得不到，你们在弗雷多尼亚的代理只得到现行工资1。如果你们在弗雷多尼亚投资并设定工资W，你们的代理有一个选择。他要么可以诚实，要么可以作弊。如果他作弊，你们会怎么样？你们必须投资1发送过去。你们什么都拿不回来，所以你们会得到负1。他会走掉做他的正常工作得到1，另外他会卖掉你们的材料。所以他总共会得到1加1等于2。谢谢。他总共会得到2。另一方面，如果他诚实，那么你们会得到4的回报减去你们必须投资的1减去你们付给他的任何工资。

[段 37]

So your return will be 3 minus the wage you pay him. You’re only going to pay him once the job’s done. 3 minus W. And he’s going to get W. He’s done his job. He hasn’t exercised his outside option. He hasn’t sold any materials. So he’ll just get W. Now, I’m slightly cheating here, because this isn’t really the way the tree looks, because I can choose different levels of W. So this upper branch where I invest and set W, there’s actually a continuum of such branches, one for each possible W I could set. But for the purpose of today, this is enough. This gives us what we need to see. So let’s imagine that this is a one-shot investment. What I want to learn is, in this one-shot investment, I invest in Fredonia, I hire my agent once. What I want to learn is, how much do I have to pay that agent to actually get the job done. And remember the starting position. The starting position is, it looks very attractive. It looks very attractive because the returns on this project are four or four minus one give me the investment so that the surplus available on this project is three minus the wage and the going wage was just one So it looks like there lots of profit around to make this outsourcing profitable I mumbled that, so I tried again.

[译文 37]

所以你的回报将是3减去你支付给他的工资。你只在工作完成后才支付给他。3减W。他将得到W。他已经完成了他的工作。他没有行使他的外部选择权。他没有出售任何材料。所以他只会得到W。现在，我在这里稍微作弊了一下，因为这棵树实际上不是这个样子的，因为我可以选择不同的W水平。在我投资并设定W的这个上部分支，实际上有这样分支的连续统，每个可能的W设定都有一个。但对于今天的目的，这已经足够了。这给了我们看到我们需要的东西。让我们假设这是一次性投资。我想知道的是，在这次一次性投资中，我在Fredonia投资，我雇佣我的代理人一次。我想知道的是，我必须支付多少才能让他真正完成工作。记住最初的立场。最初的情况看起来非常有吸引力。看起来非常吸引人，因为这个项目的回报是4，减去1等于投资，所以这个项目可用的剩余是3减工资，而现行工资只有1。所以看起来有很多利润可以使这种外包有利可图。我说含糊了，所以我又试了一次。

[段 38]

So the reason this looks attractive is, the going wage is just one, so if I just pay him one, and he does the project, then I’ll get a gross return of 4 minus the one I invested minus the one that I had to pay him for a net return of 2. It seems like a very, it’s a 100% profitable project. It looks very attractive. What’s the problem? The problem is if I only set, this is going to be a backward induction, if I set the wage equal to the going wage, so if I set W equal to 1, what will my agent do? He’s going to cheat. The problem is if I set W equal to 1, which is the going wage, the going wage in Fredonia, the agent will cheat And if he cheats I just lose my investment So how much do I have to set the W to Let look at this We have to set W What I need is I need his wage to be big enough so that being honest and going on with my project outweighs his incentive to cheat. I need W to be bigger than 2. Is that right? I need W to be at least as big as 2. So in setting the wage in equilibrium, in equilibrium, what are we going to do? I’m going to set a wage, let’s call it W star, equal to 2 plus a penny.

[译文 38]

所以这看起来有吸引力的原因是，现行工资只有1，所以如果我只支付他1，他完成这个项目，那么我将获得总回报4减去我投资的1再减去我支付给他的1，净回报是2。这看起来是一个非常百分之百有利可图的项目。看起来非常有吸引力。问题是什么？问题是，如果我只设定，这是逆向归纳，如果我把工资设定为现行工资，如果我把W设定为1，我的代理人会做什么？他会作弊。问题是如果我把W设定为1，也就是Fredonia的现行工资，代理人会作弊。如果他作弊，我就损失了我的投资。所以我必须把W设定为多少？让我们看看。我们必须设定W。我需要的是，他的工资要足够高，以至于诚实并继续我的项目要胜过他作弊的动机。我需要W大于2。对吗？我需要W至少等于2。所以在设定均衡工资时，在均衡中，我们要做什么？我将设定一个工资，让我们称之为W star，等于2加一分钱。

[段 39]

Is that right? So this is an exercise which we visited the first day, the first day after the midterm. This is about incentive design. It’s about incentive design. In this one-shot game, which we can easily solve by backward induction, I’m going to need to set a wage equal to 2, and then he’ll work. All right So in a minute we going to look at the repeated version of this But before we do let just sum up where we are so far What is this telling us It telling us that when you invest in an emerging market where the courts don work so they’re not going to be able to enforce this guy to work well, in particular he can run off with your investment, even though wages are low, so it seems very attractive to do outsourcing, if you worry about getting incentives right, you’re going to have to pay an enormous wage premium to get the guy to work. So the going wage in Fredonia was one, but you had to set a wage equal to two, a 100% wage premium to get the guy to work. So the wage premium in this emerging market is 100%. You’re paying two even though the going wage is one. And by the way, this is not an unreasonable prediction. If you look at the wages paid by European and American companies in some of these emerging markets, which have very, very low going wages.

[译文 39]

对吗？所以这是一个我们在第一天，即期中考试后的第一天做过的练习。这是关于激励设计的。这是关于激励设计的。在这个一次性游戏中，我们可以很容易地用逆向归纳法求解，我需要把工资设定为2，然后他就会工作。好吧，一会儿我们要看这个的重复版本。但在此之前，让我们总结一下我们目前在哪里。这告诉我们什么？它告诉我们，当你在一个法院不工作的新兴市场投资时，所以他们将无法强制这个人好好工作，特别是他可以带着你的投资跑掉，即使工资很低，所以做外包看起来非常有吸引力，如果你担心激励机制是否正确，你将不得不支付巨额工资溢价来让这个人工作。所以Fredonia的现行工资是1，但你必须把工资设定为2，一个百分之百的工资溢价来让这个人工作。所以在这个新兴市场，工资溢价是百分之百。你支付2，即使现行工资是1。顺便说一句，这不是一个不合理的预测。如果你看看欧洲和美国公司在一些这些新兴市场支付的工资，这些市场的现行工资非常非常低。

[段 40]

And look at the wages that are actually being paid. by the companies that are doing outsourcing. You see enormous wage premium. There’s enormous premium over and above the going wage. All right. Now what I want to do, I want to put that up there, and I want to revisit exactly the same situation. But now we’re going to introduce the wrinkle of the day. What’s the wrinkle of the day? The wrinkle of the day is you’re not only going to invest in Fredonia today, but if things go well, you’ll invest tomorrow. And if things go well again, you’ll invest the day after, at least with some significant probability. So the wage premium we just calculated was the one-shot wage premium. It was getting this job, this single one-shot job, outsourced to Fredonia. And now I want to consider how much you’re going to have to pay, what a wage is going to be in Fredonia, in the foreign investment sector, if instead of just having a one one investment you investing for the long term You going to be in Fredonia for a while So consider repeated interaction with probability delta of continuing So we don’t know that you’re going to go on in Fredonia. Things might break down in Fredonia because there’s a coup. It might break down in Fredonia because the American administration says you’re not allowed to do outsourcing anymore.

[译文 40]

然后看看那些实际支付的工资。那些正在进行外包的公司支付的工资。你看到巨额工资溢价。远远高于现行工资的溢价。好吧。现在我想做的是，我想把这个放在那里，我想重新审视完全相同的情况。但现在我们要引入今天的转折。今天的转折是什么？今天的转折是，你不仅今天要在Fredonia投资，如果事情进展顺利，你明天还会投资。如果事情再次进展顺利，你后天还会投资，至少有相当大的概率。所以我们刚才计算的是一次性工资溢价。这是把这工作，这个单一的一次性工作，外包给Fredonia。现在我想考虑你将必须支付多少，Fredonia的外国投资部门的工资会是多少，如果你不是只有一次性投资而是长期投资，你将在Fredonia待一段时间。所以考虑以概率delta继续的重复互动。我们不知道你是否会继续在Fredonia。事情可能在Fredonia破裂，因为那里发生了政变。也可能在Fredonia破裂，因为美国政府说你不再被允许做外包了。

[段 41]

All sorts of things might happen. But with some probability delta, the relationship’s going to continue. So repeated interaction with probability delta. And let’s redo the exercise we did before to see what wage you’ll have to charge. So our question is, what wage, let’s call it W double star now, we called it previously W double star, let’s call it W double star, what wage will you pay? And the way we going to solve this is exactly using the methods we learned in this class So what we going to compare is the temptation to cheat today and we better make sure that that’s less than delta times the value of continuing the relationship minus the value of the relationship. Let’s call this tomorrow. What’s happening now is once again I’m employing my agent in Fredonia and provided he does a good job, I’ll employ him again tomorrow, at least with probability delta. But if he doesn’t do a good job, if he runs off with my investment and doesn’t do my job, what What am I going to do Well what would you do You fire him right You fire him Is that right So the punishment it clear what the punishment is going to be here The punishment is if he doesn do a good job, you fire him. The value of ending the relationship, this is firing.

[译文 41]

各种各样的事情都可能发生。但以某种概率delta，关系将继续。所以以概率delta的重复互动。让我们重做我们之前做的练习，看看你会必须收取什么工资。所以我们的问题是，什么工资，我们现在称之为W双星，我们之前称之为W双星，让我们称之为W双星，你会支付什么？我们将如何解决这个问题正是使用我们在这门课中学到的方法，所以我们将比较的是今天作弊的诱惑，我们最好确保这小于delta乘以继续关系的价值减去关系的价值。让我们称之为明天。现在发生的事情是，我再次在Fredonia雇佣我的代理人，只要他做好工作，我明天将再次雇佣他，至少以概率delta。但如果他没有做好工作，如果他带着我的投资跑掉而不做我的工作，我该做什么？嗯，你会做什么？你解雇他，对吧？你解雇他。对吗？所以惩罚是明确的，这里的惩罚是什么。如果他没有做好工作，你就解雇他。结束关系的价值，这是解雇。

[段 42]

And this is continuing. So let’s just work out what these things are. So his temptation to cheat today, if he cheats today, he doesn’t get my wage, but he does run off my cash and he does go and do his job at the going wage. So if he cheats today, he gets two, he’s stolen my cash and he’s going off and working at the going wage, but he doesn’t get what I would have paid him, W double star, if the job was well done. We need this to be less than the value of continuing the relationship. Well let’s do the easy bit first. What does he get if we end the relationship? What does he get if we end the relationship? He’s been fired, so he’ll just work at the going wage forever. So this is the value of the He’s been fired, so he’ll just work at the going wage forever. So this is the value of 1 forever, or at least until the end of the world. And this is the value of 1. As long as he stayed employed by me, what’s he going to get paid every period? What do you get paid? W double star. It’s the value of W double star forever. Let me cheat a little bit and assume that the probability of some coup happening that ends our relationship exogenously is the same probability of the coup happening and ending his own going wage endogenous space, we can use the same delta.

[译文 42]

而这是继续。让我们算出这些东西是什么。所以他今天作弊的诱惑，如果今天作弊，他得不到我的工资，但他确实卷走了我的现金，他确实去做他的工作获得现行工资。所以如果今天作弊，他得到2，他偷了我的现金，他去以现行工资工作，但他得不到如果工作做得好我会支付给他的W双星。我们需要这个小于继续关系的价值。让我们先做简单的部分。如果结束关系，他得到什么？如果结束关系，他得到什么？他被解雇了，所以他将以现行工资永远工作。所以这是价值1永远，或者至少直到世界末日。这是价值1。只要他继续受雇于我，每个时期他会得到什么报酬？你得到什么报酬？W双星。是W双星永远的价值。让我稍微作弊一下，假设某种政变发生的概率会外生地结束我们的关系，这与政变发生并内生地结束他自己的现行工资的概率相同，我们可以使用相同的delta。

[段 43]

Let’s just do some math here. What’s the value of W double star forever? So remember the value of 2 forever was what? 2 over 1 minus delta. So what the value of W double star forever W double star over okay so this is going to be W double star over 1 minus delta And what’s the value of 1 forever? 1 over 1 minus delta. The whole thing is multiplied by delta and this is 2 minus W double star. Now I need to use some algebra to solve for W double star. I can’t even say it. So let’s try and do that. So I claim that this is the same as 1 minus delta 2 minus 1 delta W double star is less than W double star delta minus delta times 1. Everyone okay with that? One more line let me just sort out some terms here So taking this on the other side I have 1 minus Professor Ben Polak Professor Ben Polak Professor Ben Polak So 1 minus delta 2 plus delta 1 has to be less than or equal to W double star delta plus 1 minus delta W double star, which is equal to W double star. So I’m going to just check my algebra at home, but I think that’s right. So the last two sets were just algebra, nothing fancy.

[译文 43]

让我们在这里做点数学。W double star 的长期价值是多少？记住，2 的长期价值是多少？2 over 1 minus delta。那么 W double star 的长期价值是多少？它等于 W double star over 1 minus delta。1 的长期价值又是多少？1 over 1 minus delta。整个式子乘以 delta，得到的是 2 minus W double star。现在我需要用代数解出 W double star。我甚至说不出它的名字。让我来试试。于是我声称这等价于 1 minus delta 2 minus 1 delta W double star 小于 W double star delta 减 delta 乘以 1。大家明白吗？再来一行，让我把一些项整理一下。于是把它移到另一边，我有 1 minus Professor Ben Polak Professor Ben Polak Professor Ben Polak 因此 1 minus delta 2 plus delta 1 必须小于等于 W double star delta 加 1 minus delta W double star，等于 W double star。所以我回家后会检查一下代数，但我觉得是对的。最后两组只是代数，没什么特别的。

[段 44]

What have we learned? We’ve learnt that the wage I have to pay this guy, the wage I have to pay him, lies somewhere between 2 and 1. It lies somewhere between 2 and 1, but we can do a bit better than that. Let’s just delete everything here. So in particular if delta is equal to 0 what W double star If delta is equal to 0, W double star is equal to what? Somebody? Equals to 2, right? Equals to 2. And that’s what we had before in the one-shot game, where there was no possibility of continuing the relationship tomorrow, I had to pay him a wage of 2 or if you like a wage premium of 100%. If there’s no chance of continuing this relationship, if delta is equal to 0, we find again that I’m paying a 100% wage premium. Let’s take the other extreme. If delta is equal to 1, so I just know this relationship’s going to continue. If delta is equal to 1, so there’s no probability of the world ending or there being a coup, then what’s double double star? It’s equal to 1. What’s that? What’s 1? It’s the going wage, right? It’s the going wage. So this is the going wage. If I know for sure we’re going to continue forever, I can get away with playing the going wage, at least on the limit.

[译文 44]

我们学到了什么？我们学到，我必须支付给这个人的工资，介于 2 和 1 之间。介于 2 和 1 之间，但我们还能做得更好一点。让我们把所有东西删掉。特别是，如果 delta 等于 0，W double star 是多少？如果 delta 等于 0，W double star 等于多少？有人回答吗？等于 2，对吧？等于 2。这正是我们在一次性博弈中的结果，当时不存在明天继续这段关系的可能性，我必须支付他 2 的工资，或者说 100% 的工资溢价。如果没有继续这段关系的可能，即 delta 等于 0，我们再次发现我在支付 100% 的工资溢价。让我们看另一个极端。如果 delta 等于 1，意味着我知道这段关系会继续。如果 delta 等于 1，即没有世界末日或政变的概率，那么 W double star 是多少？它等于 1。这是什么？它是现行工资，对吧？它是现行工资。于是这就是现行工资。如果我确定我们会永远继续下去，我至少在极限情况下可以用现行工资来应付。

[段 45]

If we know we’re not going to continue, then I have to play the one-shot wage. But let’s look at a more interesting intermediate case. Suppose delta is equal to a half. There’s just a half probability. It’s pretty low. There’s a half probability that your company, American Widgets, is going to stay in Fredonia. With probably a half, it’s going to be done next period. Probably a half, it’s going to stay. What does that do to the wage? What happens to the wage in this case in which there’s probably about half of American widgets staying in Fredonia? It’s a half between 2 and 1, which is therefore 1 and a half. Or another way of saying that is the wage premium is now only 50%. What have we learned from this example? Just an example of using repeated gains. Well the first thing we learned is it kind of easy to get used to it It easy to use this technology of comparing temptations to cheat with values of continuing in a cooperative relationship versus the value of the punishment which in this case was just firing the guy But more specifically in this example, we’ve learned that even a relatively small probability of this relationship continuing, continuing, so this is good news for those of you who are seniors and about to move to San Francisco while your significant other is going to London, even a small probability of the relationship continuing drastically reduces the wage premium.

[译文 45]

如果我们知道不会再继续，那我只能采用一次性工资。但让我们看一个更有趣的中间情形。假设 delta 等于 0.5。有一半的概率。相当低。有一半的概率，你们的公司 American Widgets 将继续留在 Fredonia。大概有一半的可能，下个时期会被结束。也大概有一半的可能，会留下来。这对工资有什么影响？在这种情况下，大约有一半的 American Widgets 留在 Fredonia，工资会是多少？它是 2 和 1 之间的中点，即 1.5。换句话说，工资溢价现在只有 50%。从这个例子中我们学到了什么？这只是利用重复收益的一个例子。首先，我们学到的是，这种比较诱惑与继续合作关系的价值以及惩罚价值（这里惩罚就是直接解雇这个人）的方法很容易上手。但更具体地说，在这个例子中我们学到，即使这段关系继续的概率相对较小——这对你们当中即将搬到旧金山而你们的另一半要去伦敦的大四学生来说是好消息——即使很小的继续概率也会大幅降低工资溢价。

[段 46]

The amount you have to pay your significant other not to cheat on you as they go off to London or San Francisco is drastically lower if there’s some probability, in this case just a half, of continuing. Before you leave us with one more thought, how does this all work? To summarize, to get good behavior in these continuing relationships, there has to be some reward tomorrow. That reward needs to be higher if the weight you put on tomorrow, the probability of continuing tomorrow is lower. The less likely tomorrow is to occur, the bigger the reward has to be tomorrow. We’re going to have to charge wage premium to employ people in Fredonia, but those premium will come down once we realize that we’re in established relationships in Fredonia, once American firms are established and not fly-by-night operations in Fredonia. Whether that’s good news or bad news for Fredonia, we’ll leave there.

[译文 46]

如果你要去伦敦或旧金山，而你的另一半有一定的（这里只是一半的）继续概率，那么你要支付给他们的工资——防止他们出轨的金额——会大幅降低。在你们离开之前，再留给大家一个想法，这一切是如何运作的？总结来说，要在这些持续关系中获得良好行为，必须有明天的奖励。如果你们对明天的重视程度（即明天继续的概率）越低，所需的奖励就必须越大。明天越不可能出现，奖励就必须越大。我们将不得不在 Fredonia 雇佣人员时收取工资溢价，但一旦我们认识到我们在 Fredonia 已建立了稳固的关系，一旦美国公司在 Fredonia 扎根而非临时经营，这些溢价就会下降。至于这对于 Fredonia 来说是好消息还是坏消息，我们暂且不论。