Okay, let's make a start. 好吧 开始上课
So I hope everyone had a good break. 希望大家都过了个愉快的假期
We're going to spend this week looking at repeated interaction. 这周我们要讲的内容是重复博弈
协调: 飞天宇 浅浅 时间轴: Money1026 校对: harvey66
翻译: Ee不舍 harvey66 Dante袁 Flower一休
We already saw last time, before the break, that once we repeat games... 放假前 我们已经讲过了 一旦进行重复博弈...
once games go on for a while... we can sustain behavior that's quite interesting. 一旦博弈持续一段时间... 我们就可能做出很有意思的行为
So, for example, before the break, we saw that we could sustain fighting 比如说 放假前 我们说过两个理性的博弈方
by players that were rational in a war of attrition. 可能持续恶性竞争 进行一场长久的消耗战
Another thing we learned before the break is 放假前说的另外一个知识点是
when we're analyzing these potentially very long games, 当我们在分析潜在的长期重复博弈时
it helps sometimes to break the analysis up into 把博弈分解到每个阶段 可能更有帮助
what we might call "stage games", each period of the game, 每个阶段的博弈叫做"阶段博弈"
and break the payoffs up into: the payoffs that are associated with that stage; 此外 收益也分为几个部分: 与该阶段相关的收益
payoffs that are associated with the past, 与过去相关的收益
they're sunk, so they doesn't really matter; 这部分沉没了 所以没有影响
and payoffs that are going to come in the future from future equilibrium play. 还有未来会得到的收益 来自未来的均衡博弈
So those are some ideas we're going to pick up today, 这些概念都是我们今天应该掌握的
but for the most part what we do today will be new. 但今天要讲的内容大多是新知识
Now whereas last time we focused on fighting, 尽管上次我们专注于竞争
for the whole of today I want to focus on the issue of cooperation. 但今天的主要内容是关于合作的
In fact, for the whole of this week, I want to focus on the issue of cooperation. 实际上 整个星期 我们都要讲合作
The question behind everything this week is going to be: 这周要讲的内容背后 有个核心问题 就是
can repeated interaction among players, 博弈方之间进行的重复博弈
both induce and sustain cooperative behavior, or if you like "good behavior"? 能不能激发 并促使合作行为持续下去? 你们也可以叫它"道德的行为"
Our canonical example is going to be Prisoners' Dilemma. 一个经典的例子是囚徒困境
Way back in the very first class, we talked about Prisoners' Dilemma, 在第一节课中 我们讨论了囚徒困境
and we mentioned that playing the game repeatedly might 我们也提到过 重复这个博弈
be able to get us out of the dilemma. 有可能使我们走出这个困境
It might be able to enable us to sustain cooperation. 可能让我们持续合作
And what's going to be good about that is not just sustained cooperation, 重复博弈的好处是 不仅能让我们长久地合作
but sustained cooperation without the use of outside payments 而且 这种合作不需要额外的付出
such as contracts or the Mafia or whatever. 比如合同 或者利用帮会 或者其它什么
So why does this matter? 这有什么关系呢?
Well one reason it matters is that most interactions in society, 因为 在这个社会里的绝大多数相互关系
either don't or perhaps even can't rely on contracts. 都不是 甚至是无法 建立在契约上的
Most relationships are not contractual. 大多数的关系没有合同约束
However, many relationships are repeated. 但是 很多关系都是长久的
So this is going to be of more importance perhaps in general life... 所以 在一般生活中...
though perhaps less so in business... 也许在生意场上要淡化一些...
more important in general life than thinking about contracts. 重复博弈的应用 都比总想着依靠合同更重要
So let's think about some obvious examples, think about your own friendships. 我们举些明显的例子 比如你们的友谊
I don't know if you have any friendships... I assume you do... 我不知道你们有没有友谊... 就假设你们有...
but for those of you who do your friendships are typically not contractual. 对有的那些人 你们的友谊都不是建立在合同上的
You don't have a contract that says if you're nice to me, I'll be nice to you. 没有订立合同的意思是 如果你对我好 我也会对你好
Similarly, think about interactions among nations. 国家之间的相互关系也是相似的
Interactions among nations typically cannot be contractual 即使可以订一些条约
because there's no court to enforce those would-be contracts, 国家之间的关系 通常也无法用合同约束
although you can have treaties I suppose. 因为没有法院使这些所谓的合同生效
But most interactions among nations... 但国家之间的关系...
cooperation among nations is sustained by the fact 国家之间的合作能够持久
that those relationships are going to go forever. 是因为 他们的关系是永远存在的
Even in business, even where we have contracts, 即使是在商业中 即使是在我们签订了合同的地方
and even in a very litigious society like the U. S. 即使是在美国
which is probably the most litigious society in the world, 这样一个可能是全世界最喜欢诉讼的社会
we can't really rely on contracts for everyday business relationships. 我们都不能在日常关系中总是依靠合同
So, in some sense, we need a way to model a way to sustain cooperation 所以 在某种意义上 我们要找个方法建立一个模型
and good behavior that forms, if you like, the social fabric of our society 使长期合作和道德行为成为可能 构建一个社会结构
and prevents always going to court about everything. 并且 防止什么事情都要上法庭
Now, why might repeated interaction work? 那么 为什么重复博弈会起作用呢？
Why do we think way back in day one of the class 我们回到第一节课
that repeated interaction might be able to enable us to behave well, 重复博弈可能会让我们做出合乎道德的行为
even in situations like Prisoner's Dilemma's or situations involving moral hazard 即使是在囚徒困境中 或 有道德风险的情况下
where bad behavior is going to occur in one shot games? 又或者 在那些背叛行为可能会出现的单次博弈中
So the lesson we're going to be sort of underlying things today 今天的课 以及这周的课
and all week is this one. 基本上是建立在这点上的
In ongoing relationships 在长期关系中
the promise of future rewards 未来收益的承诺
and the threat of future punishments may... 以及未来惩罚的威胁 可能...
let's be careful... may sometimes provide incentives for good behavior today. 我们别说得那么绝对... 可能会激发今天的道德行为
Just leave a gap here in your notes because we're going to come back to this. 这个地方 你们在笔记本上留几行 等会我们回过头来讲这个
So this is a very general idea. 这是个非常普遍的概念
And the idea is that future behavior in the relationship 这个概念是 在相互关系中
can generate the possibility of future rewards and/or future punishments, 未来行为可能导致 未来收益或者未来惩罚的产生
and those promises or threats may sometimes provide incentives for people 这些承诺或者威胁可能刺激人们
to behave well today. 在现在做出良好的行为
The reason I want to leave a gap here is I want... 我让你们在这里空几行是想要...
part of the purpose of this week's lectures is to try and get beyond this. 这周课程的部分任务是试着掌握比它复杂的东西
This is kind of, almost a platitude, right? 这有点老生常谈 是吧?
Most of you knew this already. 大部分人已经知道这个了
So I want to get beyond this. 所以我希望你们学到更多
I want to see when is this going to work. 什么时候它会奏效
When is it not going to work? 什么时候它不管用?
How is it going to work? 它是怎么起作用的?
So I don't want people to leave this week of classes or leave the course, thinking: 我不希望你们抱有这种想法: 因为我们之间的博弈是长期的 所以没关系
oh well, we're going to interact more than once so everything's fine. 因此你们这周就要逃课 或者放弃这门课程
That's not true. 这是不对的
I want to make sure we understand when things work, how they work, 我想确保 你们能明白它们什么时候有用 怎么起作用的
and more importantly, when they don't work and how they don't work. 更重要的是 什么时候不管用 为什么不管用了
So we're going to try and fill in the gap 随着今天的课程继续下去
that we just left on that board as we go on today. 我们要试着填满黑板上的空白处
Nevertheless, we do have this very strong intuition that 尽管这样 直觉还是告诉我们
repeated interaction will get us, 重复博弈会和原来一样
as it were, out of the Prisoner's Dilemma. 让我们走出囚徒困境的
So why don't we start with the Prisoner's Dilemma. 那我们就从囚徒困境开始吧
I'll put this up out of the way and we'll come back to it. 我先把这个放在一边 一会儿再回来讨论它
Let's just remind ourselves what the Prisoner's Dilemma is 我们先回忆一下什么是囚徒困境
because you guys are all full of turkey and cranberry sauce 你们这些家伙满脑子都是火鸡和酸果酱
and you've probably forgotten what Game Theory is entirely. 可能把整个博弈论都忘光了
Let's name these strategies, rather than alpha and beta, 这两个策略叫做合作和背叛
let's call them cooperation and defect. 不要再叫α和β了
And that will be our convention this week. 这周的课程里我们都这么叫
We'll call them cooperation and defect. 叫它们合作和背叛
This is Player A and this is Player B, 这是博弈方A和博弈方B
and the payoffs are something like this (2, 2), (-1, 3), (3, -1) and (0, 0). 收益是 (2, 2) (-1, 3) (3, -1) 和 (0, 0)
It doesn't have to be exactly this but this will do. 不一定非得是这几个数字 但用它们就行了
This is the game we're going to play. 这就是我们要进行的博弈
And, to try and see if we get cooperation out of it 现在 如果我们通过进行重复博弈
by having repeated interaction, 不止玩一次
we're going to play it more than once. 看看我们能否在这个博弈中合作
So let me go and find some players to play here. 那我就找几个人来玩玩吧
This should be a familiar game to everybody here. 这对大家来说都是很熟悉的游戏了
So why don't I pick some people who are kind of close to the front row? 我叫几个前排的人
So what's your name again? 我又忘了你叫什么
I'm using the green one over here. So say again. 我用的是绿色的话筒 再说一遍
Brooke. 我叫Brooke
Brooke. 哦 Brooke
Okay so Brooke is going to be Player B. 好的 那么Brooke是博弈方B
And I've forgotten your name, by this stage I should know it. 我也不记得你的名字了 虽然上了那么久的课
Patrick you're going to be Player A. Patrick 你是博弈方A
And the difference between playing this game now 现在你们要玩的这个游戏
and playing this game earlier on in the class is 和你们原来在课堂上玩的不同之处在于
we're going to play not once, but twice. 我们不止玩一次 而是要玩两次
We're going to play it twice. 我们要玩两次
So write down what you're going to do the first time and show it to your neighbor. 写下你第一次博弈的策略 给你旁边的人看
Don't show it to each other. 但你们俩不要互相看
And let's find out what they did the first time. 现在我们看看他们第一次是怎么选择的
So is it written down? 写好了吗?
Something's written down? 好了没?
So Brooke. 那么 Brooke
I cooperated. 我合作
You cooperated. 你合作
Patrick? Patrick呢?
I defected. 我背叛
Patrick defected, okay. Patrick背叛了 好的
Okay let's play it a second time. 好的 我们玩第二次
So write down what you're going to do the second time. 写下你们第二次的选择
Brooke? Brooke选什么?
This time I'm going to defect. 这次我背叛
Me too. 我也是
All right, so we had the play this time... let's just put it up here... 好的 我们玩过这个游戏了... 我们写在这儿...
so when we played it this time, we had A and B. 当我们进行这个博弈的时候 有博弈方A和B
And the first time we had (defect, cooperate) 第一次结果是 (合作, 背叛)
and the second time we had (defect, defect). 第二次是 (背叛, 背叛)
Let's try another pair. 我们再叫另外一对
We'll just play this a couple times and we'll talk about it. 我们先玩几次 再来讨论
So that's fair enough. 公平起见
Why don't we go to your neighbors? 我们叫坐在你们旁边的人来吧?
That's fair enough. 这样比较公平
It's easy. 很容易
So you are? 你叫?
Ben. 我叫Ben
That's a good name, very good okay. 很好的名字 很好 好的
You are? 你叫?
Edwina. 我叫Edwina
Edwina, Edwina and Ben okay. Edwina Edwin和Ben 好的
So we're going to make Ben Player B and Edwina Player A. 那么Ben就是博弈方B Edwina是博弈方A
And why don't you write down what you're going to do for the first time. 写下你们在第一次博弈中的选择
Again, we're going to play it twice. 一样 我们还是玩两次
Why don't we mix it up, we can play it three times. 增加点难度 我们玩三次吧
We'll play it three times this time okay. 这回我们玩三次
We'll play it three times. 这个博弈我们进行三次
Both people are happy with the decisions. 两个人都没意见
Okay so the first time Edwina what did you choose? 好的 那么第一次 Edwina你选择什么?
Defect. 背叛
Cooperate. 合作
All right so we had... let's put down this time... 好的 结果是... 我们先记下来...
so we've got Edwina will Play A and Ben B. Edwina是博弈方A Ben是B
And we had (cooperate, defect). 结果是 (合作, 背叛)
Second time please, Edwina? 第二次呢 Edwina?
Cooperate. 合作
Defect. 背叛
Okay, so we're going to and fro now. 我们现在要陷入循环了
So this was cooperate and defect and one more time: 所以这次的结果是合作和背叛 再一次:
write it down. 写下来
Both players written down? 两个人都写好了吗?
Edwina? Edwina选什么?
Cooperate. 合作
Defect. 背叛
Okay, so you we flipped round again okay. 好的 我们又绕回来了
Okay, so we're seeing some pretty odd behavior here. 那么 这次我们看到了很奇怪的选择
Who did what that time? 这次是谁选了什么?
Edwina what did you do? Edwina你怎么做的?
I cooperated. 我合作
So we had this. 所以结果是这样
Is that right? 对吧?
So keep the microphones for a minute, and we'll just talk about it a second. 先拿着麦克风 我们讨论一下
All right, so first of all let's start with Ben here. 那好 先从Ben开始吧
Ben you were cooperating on the first go. Ben 你在第一轮中选择合作
So why did you choose to cooperate the first turn? 为什么你要在第一轮选择合作?
Shout out so everbody can hear you. 大声点 让大家都听到
I felt that if I established a reputation for cooperating 我觉得 如果我树立了想合作的名声
we could end up in the (cooperate, cooperate). 我们就能得到 (合作, 合作) 的结果
All right, so you thought that by playing cooperate early 好的 你认为早点合作
you could establish some kind of reputation. 可以树立好名声
And what about later on when you played defect thereafter, 那么之后的背叛呢?
what were you thinking there? 你当时怎么想的?
I realized that she established a reputation for defecting a second time. 我意识到她会再次树立一个背叛的坏名声
All right, so you switched strategies mid-course. 所以你中途改变了策略
Edwina you started off by defecting. Edwina 你一开始就背叛了
Why did you start off by defecting? 为什么?
Shout it out so people can hear you. 说大声点 让大家都能听见
Because his friend defected so I thought he might defect. 因为他的朋友背叛了 所以我觉得他可能也会
Okay, his friend defected. 好的 他的朋友背叛了
Okay, so he's been tainted by his friend there. 那么 是他的朋友连累了他
There's a shortage of space in the class. 这个教室小了点
They could have just been sitting next to each other. 他们只能相互挨着坐
Thereafter you cooperated. 之后你合作了
Why was that? 为什么?
Because I thought he cooperated. Maybe he was going to keep cooperating. 因为我想他第一次合作了 或许他可能继续合作下去
All right, so in fact your reputation works in some sense. 好的 实际上你树立的名声起到了一点作用
By cooperating early you convinced Edwina you would cooperate. 第一次你的合作 使Edwina相信了你接下来会合作
And then you went on cooperating even after he defected, 而即使接下来他背叛了 你还是继续合作
so what were you doing in the third round? 那么在第三轮中 你是怎么想的?
Shout out. 大声说
I thought he might cooperate because I cooperated. 我想第二轮中我合作了 他可能也会合作
All right, he might come back. 好的 你指望他回心转意
Let's talk about it to your neighbors. 那么接下来 问一下你旁边的同学
So Brooke, shout out why you cooperated in the first round. Brooke 第一轮你为什么要合作 大声说出来
Because I was hopeful that he would cooperate. 因为我对他选择合作抱着很大希望
You were hoping he would cooperate. All right, and why did you defect thereafter? 你希望他会合作 好的 那为什么后来你又背叛了?
Because I thought he would continue to defect after he defected. 因为我觉得他背叛了一次以后 就会继续背叛下去
Because he defected, he would continue to defect. 因为他背叛了 就会背叛下去
Patrick, you're the person who just defected throughout here. Patrick 你是自始至终都选择背叛的人
Grab the mic that's next to you. 拿你边上的那个话筒
Why did you just defect? 为什么你选择背叛?
It's such a short game that it makes sense to defect in the last period 这个博弈很短 在最后一轮里 背叛才是明智的
so the second last period and the first period. 第二阶段和第一阶段是一样的
All right, that's an interesting idea. 嗯 这个观点有意思
So Patrick's saying, actually if we look at the last period of this game, Patrick的意思是 事实上 如果我们只看这个博弈的最后一个阶段
if we look at this last period of the game, 如果只看博弈的最后一阶段
what does the game look like in the last period? 在最后一阶段 这个博弈是什么?
It's a single period game. 就是单次博弈
In the last period, this actually is the game. 在最后一阶段 这实际上就是那个博弈
If I drew out the game with two periods, 想把这个两阶段博弈画出来
it would be kind of a hard thing to draw, 有点困难
it would be an annoying diagram to draw. 画出来会是个很复杂的图
But in the last period of the game, 但在博弈的最后一阶段
whatever happened in the first period is what? 第一阶段发生的结果都怎么了?
It's sunk, is that right? 都过去了 是吧?
Everything that happened in the first period is sunk. 第一阶段发生的都过去了
So in the last period of the game 所以在博弈的最后一阶段
these are the only relevant payoffs, is that right? 只考虑和这阶段有关的收益了 对吧?
Since these are the only relevant payoffs looking forward, 既然 在博弈的最后一阶段
in the last period of the game, 期望收益就是当前的有关收益
we know that there's actually a dominant strategy. 那我们就知道 一定存在一个优势策略
And what is that dominant strategy in the last period of the game? 而在博弈最后阶段的优势策略是什么
To do what? 选择什么?
In Prisoner's Dilemma, what's the dominant strategy? 在囚徒困境中 优势策略是什么?
Shout it out. 大声点
Defect, okay. 背叛 没错
So what we should see in this game... 所以我们在这个博弈中应该看到的结局...
we didn't actually because we had some kindness over here from Edwina 没有出现 因为Edwina太善良了
but what we should see in general... 但一般来说 我们应该看到的是...
is we know that in the last period of the game, 是我们知道在最后一轮博弈中
in period two, we're going to get both people defecting. 在第二轮 大家都会选择背叛
The reason we're going to get both people defecting is 大家都会选择背叛的原因是
because the last period of the game is just a one shot game. 最后一轮博弈其实就是一轮一次性博弈
There's nothing particularly exciting about it. 这没什么好奇怪的
There is no tomorrow, and so people are going to defect. 没有明天了 所以大家会背叛
But now let's go back and revisit some of the arguments that Edwina, Brooke, and… 不过我们现在要会过去看看他们的论证 Edwina Brooke 还有...
I've forgotten what your neighbor is called again, Ben, 我又忘了你边上那位的名字 Ben
I should remember that… and Ben said earlier. 我应该记得... Ben 他们之前说过
They gave quite elaborate reasons for cooperating: 他们详细阐述了合作的理由:
cooperating to establish reputation; 合作可以建立声誉
cooperating because the other person might cooperate; whatever. 你合作是因为别人也可能合作 不管是什么
But most of these behaviors were designed to either induce 但是大多数行为都是为了诱使
or promise cooperation in period two, is that right? 或承诺在第二轮中合作 对吗?
What we've just argued is that in period two everyone's going to defect. 我们刚才已经论证了在第二轮中 每个人都会背叛
Period two is just a trivial one stage Prisoner's Dilemma. 第二轮只是囚徒困境中的一个微小环节
We actually analyzed it in the very first week of the class. 我们在第一周的课程中就已经分析过了
And provided we believe these payoffs, we're done. 假如我们相信这些收益 问题就迎刃而解了
Period two people are going to defect. 在第二轮大家会背叛
Since they're going to defect in period two, 因为他们会在第二轮背叛
nothing I can do in period one is going to affect that behavior 我在第一轮的所作所为就无济于事
and therefore I should defect also in period one. 所以在第一轮我就会背叛
In order to belabor this point, we can actually draw up 为了着重说明一下这点 我们来画张表
what the matrix looks like in period one… so let's do that… 看看第一轮是什么样子的... 我们来画...
using the style we did last week… two weeks ago… before we went away. 用上周... 两周前... 我们放假前的方法
So here once again is the matrix we had before, 这就是我们以前的表格
and I want to analyze the first stage game. 我要分析一下第一阶段的博弈
In the first stage game, what I'm going to do is 在第一阶段中 我们要做的是
I'm going to add in the payoffs I'm going to get from tomorrow. 加入明天我会得到的收益
The payoffs I'm going to get from tomorrow are from tomorrow's equilibrium. 明天得到的收益来自明天的均衡
Well this isn't going to do very much for me as we'll see 我们会看到 这并不麻烦
because I'll get 2 + 0 tomorrow because we know I'm playing defect tomorrow, 因为我明天会得到 2 + 0 因为明天我会背叛
2 + 0 tomorrow, - 1 + 0 tomorrow, 3 + 0 tomorrow, 明天2 + 0 明天3 + 0
3 + 0 tomorrow, - 1 + 0 tomorrow, and 0 + 0 tomorrow, 0 + 0 tomorrow. 明天3 + 0 明天- 1 + 0 明天0 + 0 明天0 + 0
So just as we did with the war of attrition game two weeks ago, 就像两周前我们在消耗战中看到的那样
we can put in the payoffs from tomorrow, 我们可以加入明天的收益
we can roll back those equilibrium payoffs to today, 我们可以把那些均衡收益放到今天
it's just in this particular exercise it's rather a boring thing, 这里确实很无聊
because we're just adding 0 to everything. 因为我们只是在所有数后面加了个0
When I add 0 to everything, I then just cancel out the zeros 在都加0的情况下 我也可以把那些0减掉
and I'm back where I started, and of course I should defect. 那就回到起点了 当然我会背叛
So what I'm going to see is because I'm going to defect anyway tomorrow, 所以我看到的会是 因为明天我无论如何都会背叛
today is just like a one shot game as well. 今天也就像是一个单次博弈了
And I'm going to get defect again. 我还会背叛
Now here we played the game twice and got defect, defect, 我们进行了两轮博弈 结果是背叛 背叛
what about if we played the game three times? 如果进行3轮博弈会怎样呢?
It's the same thing. 一回事
We didn't play three times but we did play the game three times 我们没进行3轮博弈 我们确实进行了3轮博弈
between Edwina and Ben. 由Edwina和Ben进行的
There we know we're going to defect in the third round. 我们知道我们在第三轮会背叛
Therefore we may as well defect in the second to last round. 所以可能在前面一轮就背叛
Therefore we may as well defect in the first round. 所以可能在第一轮就背叛
If we played it five times, we know we're going to all defect in the fifth round. 如果我们进行5轮 我们知道在第五轮大家会背叛
Therefore we may as well defect in the fourth round. 所以我们可能会在第四轮背叛
Therefore we may as well defect in the third, and so on. 所以我们可能会在第三轮背叛 以此类推
If we played it 500 times, we wouldn't have time in the class, 如果我们进行500轮 课堂上没这么多时间
but if we played it 500 times, we know in that 500th period 不过如果我们进行500轮 我们知道第500轮
it's a one shot game and people are going to defect. 就是一个单次博弈 大家会背叛
And therefore, in the 499th period people are going to defect. 所以 在第499轮大家会背叛
And therefore in the 498th period people are going to defect, and so on. 所以 在第498轮大家会背叛 以此类推
So the problem here is that we get unraveling, 所以 这样我们就得以解脱了
something we've seen before in this class, we get unraveling from the back. 跟我们之前在这个课堂上看到的一样 我们通过回溯得以解脱
I have a worry that there might only be one L 我有点担心 在美式英语里
in unraveling in America, is that right? unraveling只有一个L 对吗?
How many L's do we put in unraveling in America? 美式英语里unraveling有几个L?
One, I've just come back from England 一个 我刚从英国回来
and my spelling is somewhere in the mid-Atlantic right now. 我现在的拼写还在大西洋中间
I'll leave it as one. 我就写一个吧
All right: unraveling from the back. 好的 通过回溯得以解脱
Essentially this is a backward induction argument, 基本上 这是一种逆向归纳法
only instead of using backward induction we're really using sub-game perfection. 只不过没用逆向归纳 而是采用逐层完善法
We're looking at the equilibria in the last games 我们来审视最后一轮的均衡
and as we roll back up the game, we get unraveling. 然后向前回溯 找到解决的办法
So here's bad news. 所以 这是个坏消息
The bad news is, we'd hoped that by having repeated interaction 这个坏消息就是 我们希望在囚徒困境中
in the Prisoners' Dilemma, we would be able to sustain cooperation. 通过重复博弈能维持合作
That's been our hope since day one of the class. 我们从这堂课的第一天就这样希望
In fact, we stated it kind of confidently in the first day of the class, 实际上 我们在第一天就非常自信这样表白了
and we kind of intuitively believe it. 而且 直觉上我们也这样相信
But what we're discovering is, 但是我们却发现
even if you played this game for 500 times and then stopped, 即便是重复500次后停止
you wouldn't be able to sustain cooperation in equilibrium 你仍然不能在均衡中维持合作
because we're going to get unraveling in the last stage and so on and so forth. 因为我们会在最后一轮得以解脱 等等
So it seems like our big hope that repeated interaction would 所以 似乎通过重复博弈在社会上
induce cooperation in society is going down the plug hole. 诱导合作的宏伟希望就要落空了
That's bad. 太糟糕了
So let's come back and modify our lesson a little bit. 所以 我们回来稍微修改一下课程
So what went wrong here was, in the last period of the game, 这里出了什么错 在最后一轮博弈中
there was no incentives generated by the future, 没有来自未来的激励
so there was no promise of future rewards or future punishments, 所以没有对未来的回报或惩罚的承诺
and therefore cooperation broke down and then we had unraveling. 所以合作破裂 我们得以解脱
So the lesson here is what? 那我们学到了什么?
The lesson is: but for this to work it helps to have a future. 学到的是: 这有助于拥有未来
This whole idea of repeated interaction was the future 重复博弈的全部所在就是
was going to create incentives for the present. 未来为现在提供了动机
But if the games come to an end, there's going to be some point 但是 如果这个博弈走到尽头 确实会有这一时刻
when there isn't a future anymore and then we get unraveling. 不再有未来 而我们得以解脱
Now this is not just a formal technical point to be 这不仅仅是耶鲁这座象牙塔里的
made in the ivory tower of Yale. 古板的技术性观点
This is a true idea. 它有现实意义
So for example, if we think about CEO's or presidents, 例如 我们可以想想CEO们 总统们
or managers of sports teams, there's a term we use, there's a word we use… 体育队的管理者 我们用一个词 一个词...
at least in the States… to describe such leaders 至少在美国是... 形容领导人的
when they're getting towards the end of their term and everyone knows it. 当他们任期将满并且尽人皆知
What's the expression we use? 我们怎么表达?
"Lame duck." 跛鸭
So we have this lame duck effect. 我们把它叫做跛鸭效应
The lame duck effect at the end of somebody's term 人们任期最后阶段的跛鸭效应
undermines their ability to cooperate, 损害了他们的合作能力
their ability to provide incentives for people to cooperate with them, 损害了他们为别人提供合作动机的能力
and causes a problem. 从而造成问题
So this lame duck effect affects presidents 所以跛鸭效应影响总统
but it also affects CEO's of companies. 也影响公司的CEO
But it's not just leaders who run into this problem. 但并不是只有领导人会遇到这个问题
So if you have an employee, if you're employing somebody, 如果你有一个雇员 如果你雇人
you may have a contract with the person you're employing, 你和你雇用的人可能有个合同
but basically you're sustaining cooperation with this person 基本上你会和他合作
because you interact with them often. 因为你们经常相互沟通
You know you're always going to interact with them. 你知道你会一直跟他们沟通
But then this employee approaches retirement. 但是 这个员工快退休时
Everyone knows that in April or something they're going to retire, 大家都知道在4月什么的 他们要退休了
then the future can't provide incentives anymore. 那么 未来不再能提供动机了
And you have to switch over from the implicit incentive of knowing 你就不得不转变 从含蓄地暗示
you're going to be interacting in the future, 你们未来仍会合作
to an explicit incentive of putting incentive clauses in the contract. 到清晰地把激励条款写进合同
So retirement can cause, if you like, a lame duck effect. 所以 退休可以导致跛鸭效应
This is even true in personal relationships. 甚至在个人关系中也是如此
In your personal relationships with your friends, 在你同你的朋友的关系中
if you think that those friendships are going to go on for a long time, 如果你认为你们的友谊会持续很长一段时间
be they with your significant other or just with the people you hang out with, 不管是对你重要的人还是你喜欢一起玩的人
you're likely to get a lot of cooperation. 你们很可能会合作得很好
But if, as with perhaps most economics majors, 但是如果 就像经济学专业人士
most of your significant others are only going to last for a day at most, 多数对你重要的人 最多只能持续几天
you're not going to get great cooperation. 你们不太会合作得很好
You're going to get cheating. 你们会相互欺骗
No one's rising to that one but I guess it's true. 没人表示反对 我猜大概是正确的
So what do we call these: "economics majors' relationships." 所以我们把它叫做: "经济学专业人士关系"
These are kind of "end effects." 这是一种"末端效应"
All of these things are caused by the fact 所有这些都起因于
that the relationship is coming to an end. 他妈的关系走到尽头了
And once the relationship is coming to an end, 一旦关系走到尽头
all those threats and promises of future behavior, 所有那些未来行为的威胁和承诺
implicit or otherwise, are going to basically disappear. 不管含蓄与否 基本都消失了
So at this point we might think the following. 因此 我们考虑以下问题
You might conclude the following. 你可以得出以下结论
You might conclude that if a relationship has a known end, 如果一中关系有个已知的终点
if everyone knows the relationship's going to end at a certain time, 如果每个人都知道 在某个时间这种关系会终止
then we're done and 那我们就结束了
we basically can't sustain cooperation through repeated interaction. 我们基本上就不能通过重复博弈来维持合作
That's kind of what the example we looked at seems to suggest. 我们研究的例子似乎说明了这点
However, that's not quite true. 但是 这并不完全正确
So let's look at another example where a relationship 我们再看一个例子
is going to have a known end 有个已知的终点
but nevertheless we are able to sustain some cooperation. 但仍能维持一定的合作
And we'll see how. 我们来看一下
So again, I'm being careful here, I've said it helps to have a future. 这里还要注意一下 我说的是帮助维持合作
I haven't said it's necessary to have a future. 我没说一定会有一个未来
So that's good news for the Economics majors again. 这对经济学专业人士来说是个好消息
So let's do this example to illustrate that, even a finite interaction, 我们以这个例子来说明一下 即使有限互动
even an interaction that's going to end and everyone knows it's going to end, 即使互动有个终点 而且大家都知道
might still have some hope for cooperation. 仍可能有一定的希望来合作
So look at this slightly more complicated game here. 看一下这个稍微复杂一点的博弈
And this game has three strategies 这个博弈有3中策略
and we'll call them A, B, and C for each player. 我们称之为A B和C
The payoffs are as follows (4, 4), (0, 5), (0, 0), 收益如下 (4, 4), (0, 5), (0, 0)
down here we'll do (0, 0), (0, 0) and (3, 3), 下面一行 (0, 0), (0, 0) and (3, 3)
and in the middle row (5, 0), (1, 1), (0, 0). 中间一行 (5, 0), (1, 1), (0, 0)
We're going to assume that this game, 我们假定这个博弈
just like we did with the first time we did Prisoner's Dilemma, 跟我们第一次应对囚徒困境一样
this game is going to be played twice. 博弈进行两次
It's going to be repeated, it's going to be played twice, repeated once. 会重复 进行两次 重复一次
So let's just make sure we understand what the point of this game is. 我们来确保都明白这个博弈的目的
In this game, in the one shot game I hope it's clear that (A, A) 在这个博弈中 这个单次博弈中 我希望弄明白(A, A)
is kind of the cooperative thing to do. 是一种合作
We'd like to sustain play of (A, A), 我们希望维持(A, A)
because then both players get 4 and that looks pretty good for everybody. 因为双方都得到4 对每个人都不错
However, in the one shot game (A, A) is not a Nash equilibrium. 但是 在单次博弈中 (A, A)不是一个纳什均衡
Why is (A, A) not a Nash equilibrium? 为什么(A, A)不是一个纳什均衡?
Let me grab those mikes again. 再用一下话筒
Why is (A, A) not a Nash equilibrium? 为什么(A, A)不是一个纳什均衡?
Anybody? 谁来?
I'm even getting to know the names at this stage of the term. 这时候我差不多要知道你们的名字了
This is Katie, right? 你是Katie 是吗?
So shout out. 大声说
The best response to the other guy playing A is playing B. 别人选A的话 你最好的选择是B
Good, so if I think the other person's going to play A, 很好 如果我们认为另外一个人会选A
I'm going to want to defect and play B and obtain a gain, a gain of 1. 我会背叛 选B来获得超额收益 多得1
So basically I'll get 5 rather than 4. 所以 一般我会得到5 而不是4
I'll defect to playing B and get 5 rather than 4 for a gain of 1, is that right? 为了1分额外的收益 我会背叛选B得5分而不是4分 对吗?
So (A, A) is not a Nash equilibrium in the one shot game… 所以(A, A)不是一个纳什均衡...
we're sometimes going to call it… it's fine… in the one shot game. 我们有时把它叫做... 好... 在单次博弈中
So now imagine we play this game twice. 现在 设想一下如果进行两次博弈
Instead of just playing once, we're going to play this game two times. 不是一次 我们要进行两次
I'll come back to that. 我等下回来说
Before I do that what are the pure strategy Nash equilibria in this game? 先说一下这个博弈中 纯粹的纳什均衡是什么?
Anybody? 谁说一下?
So the Nash equilibria in this one shot game are (B, B) and (C, C). 这个博弈里的纳什均衡是(B, B)和(C, C)
There's some mixed ones as well but this will do. 这里有点混淆 不过没问题
So (B, B) and (C, C) are the pure strategy Nash equilibria. (B, B)和(C, C)是纯粹的纳什均衡
Now consider playing this game twice. 现在考虑一下进行两轮博弈会是什么样
And last time we looked at a game played twice, it was Prisoner's Dilemma, 上一次进行两轮是在囚徒困境时
and we noticed that we couldn't sustain cooperation 我们注意到无法维持合作
because in the last stage people weren't going to cooperate 因为在最后一轮 大家不会合作
and hence in the first stage people weren't going to cooperate. 因而在第一轮 大家就不会合作
But let's look what happens here. 但是我们看看这里会发生什么
If this game is played twice 如果进行两轮博弈
is there any hope of sustaining cooperation, i.e. A, in both stages? 有没有什么维持合作的希望 就是说两轮都选A?
Could we have people play A in the first stage 我们能让大家在第一轮选A
and then play A again in the second stage? 而第二轮也选A吗?
So Patrick's shaking his head. Patrick在摇头
So that's right to shake his head. 摇头是对的
So let me grab the other mike. 再拿一个麦克风来
So why is that not going to work? 为什么不会奏效?
Why can't we get people to cooperate and play A in both periods? 为什么不能让大家合作 在两轮中都选A?
Shout out. 大声说
In the second period you're still going to defect and play B. 在第二轮中 你仍然会背叛而选B
Good, so in the second period, exactly the argument that Katie produced 很好 在第二轮中 就像刚才Katie
just now in the one shot game applies 在单次博弈中论证的那样
because the second period game is a one shot game. 因为第二轮博弈就是一个单次博弈
So we've got no hope of sustaining cooperation in both periods. 所以 我们没有希望能在两轮中维持合作
Let's call this cooperation. 我们称之为合作
We can't sustain (A, A) in period two, in the second period. 我们无法在第二阶段维持(A, A)
However, I claim that we may be able to get people to cooperate 然而 我说了我们有可能让人们
in the first period of the game. 在第一阶段合作
Now how are we going to do that? 我们该怎么做?
So to see that let's consider the following strategy. 让我们考虑一下下面的策略
But consider the strategy… the strategy's going to be… 考虑一下这种策略... 这种策略将会是...
play A and then play C if (A, A) was played; 先选A 如果是(A, A)就选C
and play B otherwise. 否则就选B
So this strategy is an instruction telling the player how to play. 所以 这种策略是一种指示 告诉选手怎样进行下去
Now, before we consider whether this is an equilibrium or not, 在我们考虑这是否是一个均衡之前
let's just check that this actually is a strategy. 我们先确定 这确实是一种策略
So what does a strategy have to do? 那一种策略必须做什么?
It has to tell me what I should do… 必须告诉我怎么做
it should give me an instruction… at each of my information sets. 应该给我一个指示... 根据我得到的信息
In this two period game, each of us, 在这个两阶段的博弈中 我们每个人
each of the players in the game has two information sets. 每个博弈方 有两套信息
They have an information set at the beginning of the game 在博弈开始时有一套信息
and they have another information set at the beginning of period two. 在第二阶段开始时 又有一套信息
Is that right? 对吗?
So it has to tell you what to do at the first information set 所以这个策略必须告诉我 根据不同阶段的信息我应该怎么做
and at the second information set and it does. 而它的确是这样
It says play A at the first one, and then at the beginning of period two… 它说在第一阶段 还有在第二阶段开始时...
I said there's only one information set there, 我刚才说一套信息
but actually there's nine possible information sets depending on 但实际上 根据第一阶段发生的情况
what happened in the first period. 有9套可能的信息
So each thing that happened in the first period is associated 所以 在第一阶段的每一个选择
with a different information set. 都有一套相应的信息
I always know what happened in the first period. 我一直都能知道第一阶段发生了什么
And at each of those nine information sets it tells me 这9套信息被选择的情况 告诉我
what to do at the beginning of period two. 该在第二轮的开头做些什么
In particular, it says if it turns out that (A, A) was played then play C now. 详细地说 它规定了 如果选了(A, A)游戏 接着就会选C
And otherwise, all the other eight possible information sets 否则 如果我发现身处其它8套信息中
I could find myself in, play B. 就选B
So this is a strategy. 所以这就是一种策略
Now of course the big question is, is this strategy an equilibrium, 当然重点问题就是 这种策略是一种均衡吗
and in particular is it a sub-game perfect equilibrium? 进而言之 这个子博弈它是不是完美的平衡呢?
Let me a bit more precise: if both players were playing this strategy 说得再确切些: 如果博弈方都运用这种策略的话
would that be a sub-game perfect equilibrium? 会不会使子博弈达到完美的平衡呢?
Well let's have a look. 就让我们来看一看吧
Of course I don't see it now so let's pull both these boards down. 当然现在还看不到 所以我得把这两块黑板拉下来
So to check whether this is a sub-game perfect equilibrium, 那么要看这个子博弈是否平衡
we're going to have to check what? 要看什么东西呢?
We're going to have to check that it induces Nash behavior in each sub-game. 我们就要看在每以个子博弈中 有没有引起纳什行为
I think the battery is going on that. 好像电池没电了
Shall I get rid of that. 我得处理一下
Okay, I'm going to shout. 好 我会大声点
Can people still hear me? 大家能听到吗?
People in the balcony can they hear me? 楼上的同学能听见吗?
Yup, okay. 好 行了
So we're going to have to see if we can sustain Nash behavior 那么我们就要看看在每一个子博弈中能否
in each sub-game. 维持纳什行为
So let's start with the sub-games associated with the second period. 让我们先从第二轮的子博弈开始看
Technically, there are nine such sub-games depending on what happened in the past, 准确说来 根据第一阶段的情况
depending on what happened in the first period. 应该有九个这样的子博弈
There's a sub-game following (A,A). (A,A)之后有一个子博弈
There's a sub-game following (A,B). (A,B)之后有一个子博弈
There's a sub-game following (A,C), and so on. (A,C)之后有一个子博弈 以此类推
So for each activity in the first period, 所以对于第一局中的每一个行为
for each profile in the first period there's a sub-game. 对于第一阶段中的每种情况 都有一个子博弈对应
However, it doesn't really matter to distinguish 但是 没有必要非常仔细地区分
all of these sub-games particularly carefully here, 所有这些子博弈
since the costs from the past, 由于过去的成本
what happened in the past is sunk, so we'll just look at them as a whole. 也就是过去所发生的已经沉没 所以我们就把它们当做一个整体来看
So in period two after (A,A)... 所以在第二局 (A,A)之后...
so you have one in particular of those nine sub-gam... 你就有了九种子博弈中的一种...
after (A,A), this strategy induces (C, C). 在(A,A) 之后 这种策略激发了(C, C)
If people play A, if both people play A in the first period, 如果都选A 如果所有人都在第一局选A的话
then in the sub-game following people are supposed to play (C, C). 那么之后大家都会在后面的子博弈中选(C, C)了
Is that a Nash equilibrium of the sub-game? 那么这种子博弈是一种纳什均衡吗?
Well was (C,C) a Nash equilibrium? 那么(C,C) 会是一种纳什均衡吗?
Yeah, it's one of our Nash equilibria. 没错 它就是我们纳什均衡其中的一种
Let's just look up there, we've got it listed. 大家来往上看 这已经列好了
There it is. 就在这儿
So we're playing this Nash equilibrium. 所以我们进行的是纳什均衡
So that is a Nash equilibrium, so we're okay. 那么有纳什均衡的话 我们就不用担心了
After the other choices in period one, 在第一轮的其他选择之后
this strategy induces (B, B). 会激发 (B, B)
That's good news too because (B, B), we already agreed, 这是个好消息 因为(B, B) 我们已经认同了
was a Nash equilibrium in the one shot game. 在单次博弈中它是一个纳什均衡了
So in all of those nine sub-games, the one after (A,A), 所以在这所有九种子博弈中 (A,A)之后的一种
and the eight after everything else, (A,A)之后的八种策略
we're playing Nash behavior, so that's good. 我们有了纳什均衡了 非常好
What about in the whole game? 那么从整体来看博弈呢?
In the whole game, starting from period one, 整体上 从第一轮开始
we have to ask do you do better to play the strategy as designated, 我们就得问 你是按指定的那样
in particular to choose A, or would you do better to defect? 去选择A呢 还是最好选背叛?
Well let's have a look. 就让我们来看一看
So if I choose A... remember the other person is playing this strategy... 那么如果我选择A... 别忘记另一个人正在执行这种策略...
so if I choose A then my payoff in this period comes from (A,A) and is 4. 那么如果我选择A 我在这轮的结果就是(A,A) 也会就是4
If I choose A then we're both playing A in this period and I get 4. 如果我选择A 我们都在这局执行A 我将得到4
Tomorrow, according to this strategy… 明天 按照这种策略
tomorrow since we both played A... both of us will now play C. 明天 因为我们都选了A... 我们两个都会选C
Since we're both playing C, I'll get an additional payoff of 3. 而我们都选C的话 我们又会得到3的收益
So tomorrow (C, C) will occur and I'll get 3 for a total of 7. 所以明天选(C, C) 我们将得到3 从而总共得到了7
What about if I defect? 如果我背叛会怎样呢?
We could consider lots of possible defections, 有多种背叛可能性供考虑
but let's just consider the obvious defection. 我们就考虑最明显的一种吧
You can check the other ones at home. 其他的大家可以课余时间考虑
So if I defect and choose B now, then, in this period, 如果我背叛 选择B 那么 在这一阶段
I will be playing B and my opponent or my pair will be playing A. 我选B 我的对手或伙伴 将会选A
So in this period I will get 5. 那么在这一局中我会得到5
And tomorrow, since (A,A) did not occur, 而明天 因为不会出现(A,A)
both of us will play B and get a continuation payoff of 1. 我们都会选B 然后得到后续收益1
So the continuation payoff this time will be following 那么这次选(B, B)的后续收益
from (B, B), and I'll get 1. 是我将得到1
Why don't I do what I've been doing before in this class 就让我像以前一样
and put boxes around the continuation payoffs, 把明天的后续收益给框出来
just to indicate that they are, in fact, continuation payoffs. 标明它们是后续收益
So if I play A, I get 4 now and a continuation payoff of 3, for a total of 7. 那么我选A 得到4 明天又得到3 总共是得到了7
If I play B now, I gain something now, I get 5 now, 如果我选B 会多得一些 得到了5
but tomorrow I'll only get 1 for a total of 6. 但是明天我只拿到了1 总共得到了6
So in fact, 7 is bigger than 6, 那么事实上 7是大于6的
so I'm okay and I won't want to do this defection. 所以我认可 不会选背叛
I just want to write this one other way because it's going to be useful for later. 我想要用另一种方式把这种策略写下来 因为后面会有用
So one other way to write this, 那么用另一种方式来写
I think we've convinced ourselves that this is an equilibrium, 我相信大家都已经确定这是一种均衡了
but one other way to write this is, 但是从另一种方式来写
and it's a more general way in repeated games, 其实这种方式在重复博弈中显得更具普遍性
is to write it explicitly comparing the temptations to cheat today 这种方式是 明确地比较今天背叛的诱惑
with the rewards and punishments from tomorrow. 和来自明天的奖励或惩罚
So what we want to do is, in general, we can just rewrite this as checking that 所以我们要做的就是 从整体来看 我们这样重写
the temptation to cheat or defect 审视今天欺骗或背叛的诱惑
today is smaller than 小于
the value of the reward minus the value of the punishment. 回报值减去惩罚值的数值
But the key words here are: 注意这里的关键词是:
defecting occurs today; rewards and punishments occur tomorrow. 今天背叛 明天的回报和惩罚
If we just rewrite it this way, 如果我们写成这种形式
we'll see exactly the same thing, just rearranging slightly. 我们所看到的还是一样的东西 只是稍作了点重新安排
The temptation to defect today is I get 5 rather than 4, 今天背叛 我得到5而不是4
or if you like a gain of 1. 也可以说成是多得了1
And the value of the reward tomorrow... 而明天的奖励值是...
the reward was to play (C, C) tomorrow and get 3. 奖励是在明天选(C, C)时得到的3
The value of the punishment tomorrow was to play (B, B) tomorrow 惩罚值则是在明天选(B, B) 时
and get 1, and that difference is 2. 得到的1 两者的差距是2
So here the fact that the temptation is outweighed by the difference 所以这里可以看到 这种诱惑是没有
between the value of the reward and the value of the punishment is 奖励值和惩罚值之间的差值大的
what enabled us to sustain cooperation. 所以可以让我们维持合作
I'm just writing that in a more general way 我只是把它用一种更概括的方式写出来
because this is the way that we can apply in games from here on. 因为这种方式更有利于我们以后应用
We're going to compare temptations to cheat with tomorrow's promises. 我们将用背叛的诱惑与明天的承诺来做比较
Patrick, let me get you a mike. Patrick 我给你找个麦克风吧
I don't understand why it's reasonable to think 我不明白为什么在第二阶段
you would play (B, B) in the second period though. 选(B, B) 会是合理的选择
In the second period you have a temptation to play (C, C) 在第二阶段中就算对手背叛你
even if the person defected on you. 你还是可以选(C, C) 啊
Good, that's a very good point. 很好 这是一个很好的观点
So what Patrick's saying is, Patrick的意思是
it's all very well to say we're sustaining cooperation in the first period here, 如果我们在第一局里保持合作状态固然很好
but the way in which we sustained cooperation was by going along with, 但是我们保持合作的方式
as it were, the punishment tomorrow. 伴随着明天的惩罚
It required me, tomorrow, to go along with the strategy of choosing B 它要求我 在明天 在我在第一轮选背叛的情况下
if I cheated in the first period. 同样地去选择B的策略
I want to answer this twice, 我想用两种角度回答这个问题
once disagreeing with him and once agreeing with him. 同意他的角度和不同意他的角度
So let me just disagree with him first. 那么先让我用不同意他的角度来说
So notice tomorrow, if the other person, 请注意 如果明天对手
the other player is going to play B then I'm going to want to play B. 另一博弈方选B那么我也选B
So the key idea here is, as always in Nash equilibrium, 所以重点就在于 在纳什平衡中
if I take the other person's play as given 如果我把对手的行为当做既定的
and just look at my own behavior... 再来看自己的行为...
if I think the other person is playing this strategy 如果我认为对手在选这种策略
and hence he's going to play B tomorrow after I've cheated... 那么所以在我作弊之后的第二天 他就会去选B...
then I want to be play B myself. 然后我自己就会选B
So that check is just our standard check, 所以那种检查只是我们最标准的检查
and actually that's the check that makes sure that 而事实上正是那种检查确定了
it really is a sub-game perfect equilibrium. 它确实是一种绝对平衡的子博弈
We're not putting some punishments down the tree 我们不会对不平衡的博弈
that are arising out of equilibrium. 做出任何惩罚
It has to be I want to do tomorrow what I'm told to do tomorrow. 就是我明天不得不做我应该做的事
So that idea seems right and I'm glad Patrick raised it 所以那种说法看起来没有错 我很高兴Patrick把它提了出来
because that was the next thing in my notes. 因为那正是我下一个想讲的
I want to go along with this punishment 我想要跟惩罚一起讲的
because if the other person's playing B I want to play B myself. 因为如果对手想选B那我也会选B
Nevertheless, I think Patrick's onto something 然而 我还是认为Patrick得想法是有依据的
and let me come back to it in a minute. 让我待会再说这个
What I want to do before I do that 在那之前我还是想要
is just draw out a general lesson from this game. 从这个博弈中引出一个普遍的结论
The general lesson is we can sustain cooperation 就是甚至在有限重复的博弈中
even in a finitely repeated game, 我们还是能够保持合作状态
but to do so we need there to be more than one Nash equilibrium in the stage game. 但是要这样的话 我们需要在阶段性博弈中有多个纳什均衡
What we need there to be is several Nash equilibria, 我们所需要的是有不止一个的纳什均衡
one at least of which we can use as a reward 其中至少有一个可以作为奖励
and another one which we can use as a punishment. 至少有一个可以作为惩罚
So even if a game is only played a finite number of times, 所以即使一个只进行有限几次的博弈
if there are several equilibria in the stage game, 如果在阶段性博弈中有一个以上的均衡
both (B,B) and (C,C), we can use one of them as a reward (B,B) 和(C,C) 其中一个可以被我们当做奖励
and the other one as a punishment, 而另一个则被我们当做惩罚
and use that difference to try and get people to resist temptations today. 并且利用这种差异 来使人们在当天不屈从于诱惑
So that's the general idea here, and let's just write that down. 那么总体思想就出来了 我来写下来吧
Patrick don't let me get away with not coming back to your point, Patrick 我不会忘记回到你的观点的
I want to come back to it in a second. 我一会儿就会回到你的观点上来
So the lesson here is, 所以这里的结论就是
if a stage game... 如果一个阶段性博弈...
a stage game is the game that's going to be repeated... 一个阶段性博弈就是重复博弈...
if a stage game has more than one Nash equilibrium in it, 如果阶段性博弈有不止一个的纳什均衡
then we may be able 那么我们就可以
to use the prospect of playing 通过在明天
different equilibria tomorrow 使用不同的均衡
to provide incentives... 来提供一些激励...
and we could think of these incentives as rewards and punishments… 我们可以把这些激励手段看成是 对今天合作的...
for cooperation today. 奖励和惩罚
In the game we just saw, there were exactly 在刚才的博弈中我们看到 在子博弈中
two pure strategy Nash equilibria in the sub-game. 存在两种绝对的纳什均衡
We used one of them as a reward and the other one as a punishment, 我们把其中之一当做奖励而另一种则当做惩罚
and we were able to sustain cooperation in a sub-game perfect equilibrium. 并且我们可以在绝对均衡的子博弈中保持合作
Now, a question arises here, and I think it's behind Patrick's question, 现在 问题出现了 我觉得这和Patrick的问题有关系
and that is how plausible is this? 问题就是这有多可信呢?
How plausible is this? 这有多可信呢?
Formally, if we write down the game and do the math, this comes out. 形式上 如果我们把博弈写下来并且算一算 就出现了这个问题
But how plausible is this as a model of what's going on in society? 但是在一个真实社会里 这种模型有多少可信度呢?
I think the worry... I'm guessing this is worry 我认为这个顾虑... 我猜想这也是Patrick的问题
that was behind Patrick's question... is this. 所关心的... 就是这个
Suppose I'm playing this game with Patrick 假设我在和Patrick进行这种博弈
and suppose Patrick cheats on me the first period, 假设Patrick在第一阶段背叛了我
so Patrick chooses B while I wanted him to choose A in the first period. 也就是Patrick在我要他在第一局选A的时候选了B
Now in the second period, according to the equilibrium instructions, 现在在第二局中 根据均衡理论
we're supposed to play (B,B) 我们应该选(B,B)
and get payoffs of 1 rather than (C,C) and get payoffs of 3. 并且得到了1 而不是选(C,C) 得到3
So let's make that visible again. 所以让我们再次把它写下来
But suppose Patrick comes to me in the meantime. 但是假设Patrick期间来找过我
So between period one and period two, 也就是说在第一局和第二局之间
Patrick shows up at my office hours and he says: Patrick到我的办公室并跟我说:
yeah, I know I cheated on you yesterday, 没错 我前一天是背叛你了
but why should we punish ourselves today? 但是为什么今天要去惩罚我们自己呢?
Why should we, both of us, lose today by playing the (B,B) equilibrium? 为什么我们 我们两个人 要选(B,B)均衡作为惩罚呢?
Why don't we both switch to the (C,C) equilibrium? 为什么我们不一起改为选(C,C) 呢?
After all, that's better for both of us. 毕竟这样对我们俩都有好处啊
Patrick's saying to me, it's true that I cheated you yesterday, Patrick说的意思就是 我确实在前一天背叛你了
but "let bygones be bygones," or "why cry over spilt milk," 但是 " 过去了就过去了" 或 " 凡事要向前看"
or he'll use some other saying plucked out of the book of platitudes, 或者他把其他在书上看到的名言警句都说一遍
and say to me: well why go along with the punishment. 然后跟我说: 没必要一起受罚嘛
Let's just play the good equilibrium now. 让我们一起选更好的平衡
And, if I look at things and I say well, 然后 我想了想 说好吧
actually, it's true I got nothing in the first period 事实上 我在第一局是没得到多少
because Patrick kind of cheated me in the first period... 因为Patrick背叛我了
so it's true I got nothing yesterday... 确实我在前一天没得到什么...
and it's true it was Patrick who caused me to get nothing yesterday, 确实也是Patrick让我变成这样的结果的
but nevertheless that's a sunk cost 但是木已成舟
and I'm comparing getting 1 now with getting 3 now. 我现在只考虑是得到1还是3的问题了
Why don't we just go along and get 3? 为什么我们干脆不一起得到3呢?
Moreover, I'm not in danger of being cheated again 再说 我也不用害怕被他背叛了
because if Patrick believes I'm going to play C, he's going to play C too. 因为如果Patrick相信我选C 那么他也会选C的
So that kind of argument involves what? 所以这里面包含一个什么呢?
It involves some kind of communication between stages, 包含了两个阶段之间的交流
but it sounds like that's going to be a problem. 但是这听起来又不太对
Why? 为什么?
Well, suppose it's the case that 假设在这场博弈中
we are going to get communication between periods 我们在局之间进行了交流
and suppose it's the case that someone with the gift of the gab, 并且假设是个伶牙俐齿的人
someone on his way to law school like Patrick, 是像Patrick一样要去考法学院的人
is going to be able to persuade me to go back 它肯定能够说服我
to the good equilibrium for everybody in period two, 在第二阶段里选择对大家都好的平衡
then we know we're going to play the good equilibrium in period two 那么我们就知道 我们在第二阶段会选更好的平衡
and now we've lost any incentive to cooperate in period one. 现在我们在第一局中 已经失去了合作的激励了
The only reason I was willing to cooperate in period one was 在第一阶段中唯一让我去选合作的原因是
because the temptation to defect was outweighed by the difference 背叛的诱惑小于
between the value of the reward and the value of the punishment. 奖励值和惩罚值的差值
If we're going to get the reward anyway, I'll go ahead and defect today. 如果我们无论如何都要那个回报的话 我就会在当天背叛
So the problem here is this notion of "renegotiation," 所以这里体现的问题就是" 再谈判" 的概念
this notion of communicating 这种两阶段之间的交流
between periods can undermine this kind of equilibrium. 会破坏掉这种平衡
There's a problem that arises if we have renegotiation. 如果我们进行再谈判就会导致这个问题
So there may be a problem of renegotiation. 所以再谈判也许存在某种问题
Now, this probably may not be such a big problem. 这也许不是什么大问题
For example, it may be, say, I'll be so angry at Patrick 比如说 也许 我因为Patrick在第一阶段背叛我
because he screwed me over in period one that 而对他非常非常地生气
I won't go along with the renegotiation. 我不想跟他再谈判
It may also be the case, 也会出现这种情况
and we'll see some examples of this on the homework assignment, 在家庭作业中我也安排了类似的例子
that the many equilibria in the second stage of the game are not such that 在博弈的第二阶段里很多平衡不那么...
a punishment for Patrick is also a punishment for me. 对Patrick 的惩罚 同样也是对我的惩罚
What really caused the problem here was, 这里面关键的原因就是
in trying to punish Patrick, I had to punish myself. 为了惩罚Patrick 我也同样要承受惩罚
But you could imagine games, 但是你们也可以在下次家庭作业中
or see some concrete examples on the next homework assignment, 想象到或者看到一些真实的例子
in which punishing Patrick is rather fun for me, 惩罚Patrick对我有利
and punishing me is rather fun for Patrick, 而惩罚我也对Patrick有利
and that's going to be much harder to renegotiate our way out of. 那么这就很难使我们的再谈判成功
There was a question, let me get a mike out to the question. 有人在提问 麻烦给他一个麦克风
Yeah? 对吧?
If we're ruling out renegotiation, 如果我们排除了再谈判
can't we devise a strategy for Prisoner's Dilemma as well 即使没有多重纳什均衡
even though it doesn't have multiple Nash equilibriums? 我们还能不能设计出一种策略 解决囚徒困境?
Yeah, okay good, so the issue there is, in Prisoner's Dilemma, 是的 很好 这里的问题是 在囚徒困境中
we established in the first week that 我们在第一周就规定了
if we're not allowed to make side payments, 如果不允许我们进行旁支付
we're not allowed to bring in outside contracts, 也不允许我们引进外部合同的话
then no amount of communication is going to help us. 那再多的交流也无济于事
So you're right if we can rely on the courts 你说得不错 如果我们能依赖法院
or the mafia to enforce the contracts 或者帮会来强制执行合同
that would be fine and then communication would have bite. 那就没问题了 交流也就起到作用了
But you remember way back in the first week 但大家都记得在第一周的时候
when we tried to talk our way out of bad behavior in the Prisoner's Dilemma 当我们在囚徒困境中 试图不采用坏行为而找到走出困境的办法
it didn't help precisely because it's a dominant strategy. 但没有起到作用 因为这是一个优势策略
Whereas, here, Patrick's conversation, 然而 这里 在Patrick的对话中
Patrick's verbal agreement to play the other equilibrium Patrick参与其它均衡时的口头协议
is an agreement to play a Nash equilibrium. 就是一种纳什均衡的协议
That's what is getting us into trouble. 它会让我们陷入困境之中
So what may help us here, what may avoid renegotiation is simply 所以最简单有效能避免再谈判的方法
I'm not going to go along with that renegotiation... 就是不再进行再谈判...
I'm too angry about having been cheated on... 我很讨厌被欺骗...
and it may be for other reasons it may actually be that I enjoy the punishment. 基于另外一些原因 我喜欢惩罚
Nevertheless, this is a real problem in society 然而 这一个存在于社会中的问题
and I don't think we should pretend that this problem isn't there. 我觉得我们不应该熟视无睹
So a good example is in bankruptcy, 破产就是个很好的例子
which is one of those words I can never spell. 这是一个我总拼不好的单词
It seems I have too many consonants in it, is that right? 好像辅音太多了 是吗?
It's approximately right anyway. 差不多是对了
So bankruptcy law in the U.S. 美国的破产法
for the last 200 odd years has gone through cycles. 在过去200多年中循环往复
One way to view these cycles is, they're cycles of relaxing the law 看待这些循环的一个方法是 当要让贷款者的生活好过些时
and making life "easier for borrowers" and then tightening up again. 法律就放松了 而紧接着又开始收紧了
This is not a recent phenomenon, this is not only a recent phenomenon, 这种现象不是现在才有的 这不是一种新的现象
this occurred throughout the nineteenth century. 它产生并贯穿于整个十九世纪
So what typically happened was there was either explicit renegotiation 所以要么是各党派之间进行再谈判
between parties or renegotiation through Act of Congress or sometimes 要么就是通过国会法抑或
through the Acts of the States, 联邦法进行再谈判
in which bankrupt debtors were basically let off or given easier terms. 宽恕债务人或给他们更宽松的条款
The argument was always the same. 争论从未改变过
These people are not going to pay back now. 这些人不会再还钱了
It's clear from the nineteenth century, 这从十九世纪开始就很明确了
often if you were bankrupt you were in jail, actually worse than that. 通常破产了就得去坐牢 甚至更惨
Sometimes in the nineteenth century in England not only 在十九世纪的英格兰 如果你破产的话
if you were bankrupt were you in jail 不仅要去坐牢
but your creditors were having to pay the fees to feed you in jail. 你的债权人还要支付你在监狱的开销
So there you were sitting in jail, 你在坐牢
you weren't paying that money back to your creditor, 你不用把这钱还给你的债权人
and you're actually costing money to your creditor by being in jail. 事实上你坐牢的时候在还花你债权人的钱
This seems like a situation that you want to renegotiate your way out of. 这种情况下你就会想进行再谈判了
You say, hey let's let these guys out of jail. 你会说 嘿 让这些人出狱吧
Let them be productive again, and then they'll pay back part of the loans. 让他们继续干活 那样他们就能还上一部分贷款了
So you had these waves of bankruptcy reform in which 你推动的这场破产法改革
the debtors' prisons were closed down, 去关闭债务人的监狱
people were let out, people were relieved of debt. 大家都出狱了 不用还贷了
What's the problem with doing that? 这样做的问题是什么?
That seems like a good idea right. 这似乎是个好主意
After all, you don't want all these people bankrupt, 毕竟你不想让所有这些人都破产
in debt, not paying money back to their creditors anyway. 欠债 还不起钱
That doesn't seem like a good situation in society. 对社会来说这不是一个好的局面
It seems like a renegotiation that's a win-win situation: 看起来再谈判是一种双赢的局面
it's better for everybody. 对大家都更有好处
What's the problem with it though? 但问题是什么?
Let's get a mike down here. 我来拿个麦克风
What's the problem with this? 这样做的问题是什么?
It incentivizes bankruptcy. 它鼓励了破产行为
Right, it creates an incentive for people not to repay in the first place. 没错 它一开始的时候就建立了一种鼓励人们不用还款的机制
It creates an incentive for people to take big risks now, 他鼓励人们去冒大的风险
and hence, it makes bankruptcy, if you like, 因此它会导致破产
or makes non-repayment of debt more likely. 或者说更倾向于不偿还债务
So this has been going on for a while, but you see it very much today 这已经持续了一段时间 但在如果你看了过去几周报纸上的
if you read the financial pages of the papers in the last few weeks. 金融板块后 你会发现越来越多这样的现象
There's a big worry in the U.S.right now 如今在美国 人们非常担忧一件事
about people failing to repay what kind of debt? 大家都担心还不起哪一种债务?
What kind of debt is the big worry about? 哪一种债务是大家最担心的?
Mortgage debt, right, so those people 抵押债券 对吧 所以那些
who are house owners failing to pay back mortgage debt and, 还不起抵押债券的业主
equally worrying, financial institutions that have a lent a lot of, 同样担心放出了大量贷款的金融机构
for example, sub-prime debt now find themselves in financial trouble. 举例来说 次贷发现自己已经陷入了金融困境
You're going to read a lot in the papers 你会在报纸上看到越来越多的报道
about not letting people out lightly out of those situations of being in debt, 是不让人们轻易摆脱欠债的困境
or not letting people out lightly out of bankruptcy. 或者是不让人们轻易走出破产困境
The term you're going to hear is "bail out." 你听到的术语会是"救济"
So bail out... the argument you're going to read is, 救济... 你看到的争论会是
you don't want the government 你不想让政府
or the central bank bailing out those financial institutions 或者央行来救济那些
who have apparently taken too large risks on sub-prime mortgage debts, 明显放了太多次贷的金融机构
even though we all agree it's better right now 即使我们都同意说现在
for those financial institutions not to go under. 不让那些金融机构破产是更好的选择
Why are we not going to... 我们为什么不...
Even though it's better for everybody for it not to go under, 即使它们不破产对大家是更好的选择
why are we not going to bail them out? 我们为什么不帮他们摆脱困境?
Because it undermines the incentives for them not to make bad loans to start with. 因为它一开始就鼓励了他们制造坏账
To a lesser extent you're going to hear that on the debtor side as well. 而在债务人这边 这样的言论就不多了
You're going to hear some people say 你会听到有些人说
we shouldn't be bailing out people who took on bad loans, 我们不应该救济那些借了次贷的人
took on bad mortgages to finance their houses, again for bail out reasons. 借了次贷买房子 还要救济
So this is an important trade off. 所以这是个重要的权衡
If you go on to law school, 如果你去法学院
you're going to see a lot about this kind of discussion, 你会听到很多这样的讨论
and this is the discussion of trading off ex-ante efficiency 讨论这种权衡的事前有效性
and ex-post efficiency. 和事后有效性
Sometimes, as Patrick has pointed out in the game just now, 有时 正如Patrick刚刚在这个博弈中指出的那样
the ex-post efficient thing to do is to go back to the good equilibrium, 事后有效性是回到道德的均衡
or if you like to bail out these firms who've made bad loans. 或者你也可以心甘情愿地救济放次贷的公司
However, from an ex-ante point of view, 从事前的观点来看
it creates bad incentives for people to make those loans in the first place; 它一开始就建立了一个鼓励人们制造坏账的机制
and, in the ex-ante point of view, 站在事前的角度
it created the incentive for people to defect in the first period of that game. 它在博弈的开始阶段就鼓励人们背叛
So this theme of ex-ante versus ex-post efficiency 这个事前有效和事后有效的话题
is not one we're going to go into anymore in this class, 并不是在我们这门课中要讨论的对象
but it should be there in the back of your minds 但当几年之后你们结束了在法学院的学业后
when you all end up in law school in a few years time. 这些应该深刻在你们的记忆之中
Okay, so, so far what have we done? 好了 到目前为止我们做了些什么?
We've been looking at repeated interaction 我们研究了重复互动
and seeing if it can sustain cooperation. 看它是否能够维系合作
The first thing we learned was that 我们学到的第一件事情是
if the repeated interaction is a finite interaction, 如果这个重复互动是一个有限的互动
if we know when it's going to end... 如果知道了它将会结束...
we know when the interaction's going to end... 我们知道当互动要结束的时候...
then sustaining cooperation is going to be hard 那维系合作将会很难
because in the last period there will be an incentive to defect. 因为在最后一轮的时候很有可能会背叛
We saw we could get around that to some extent 我们发现当博弈存在着多重均衡的时候
if games have multiple equilibria, 我们或多或少可以应付一下
but in a game like Prisoner's Dilemma, we're really in trouble. 但在像囚徒困境那样的博弈中 我们真的陷入了麻烦
Things will unravel from the back. 要通过回溯来解决
So now let's mix things up a little bit 现在让我们把事情弄得复杂一些
by looking at a more complicated variety of repeated interactions. 我们来研究一个更复杂的重复互动
Rather than just play the game once or twice, or three times, 我们不再进行一次 两次或者三次博弈了
let's play the game under the following rules. 我们按照如下的规则来进行博弈
We'll go back to our same players, how many mikes are still out here? 我们还是请相同的博弈者来参与 那儿有几个麦克风?
I took them both back, is that right? 我都拿回来了 是吧?
I'm taking both the green and the blue mike, 我拿了绿色和蓝色的麦克风
and I'm giving them back to our players. 我把他们交到我们的博弈者手中
So this is to Brooke and this is to Patrick. 这是Brooke 这是Patrick
And we're going to have Brooke and Patrick play Prisoner's Dilemma again. 我们要让Brooke和Patrick再一次参与囚徒困境的博弈
I'm hoping I haven't deleted it. 我希望我还没擦掉
Maybe I did. 可能擦掉了
It doesn't matter we know the payoffs. 没关系 大家都知道它的收益
We're going to have them play Prisoner's Dilemma again, but this time, 我们要让他们再进行一次囚徒困境的博弈 但这一次
in between every play of the game, I'm going to toss a coin. 在每一轮博弈之间 我要掷一次硬币
Actually I'll toss the coin twice and 事实上我要掷两次硬币
if that coin comes up heads both times then the game will end, 如果两次都正面 那就结束博弈
but otherwise they'll play again. 否则的话他们要再进行一次
So everyone understand what we're going to do? 大家都明白我们要做什么了吗?
We're going to play Prisoner's Dilemma. 我们要进行囚徒困境的博弈
At the end of every period I'll toss a coin twice. 在每一轮结束后我都要掷两次硬币
I might get Jake to toss it. 我可以让Jake来掷
Jake will toss a coin twice. Jake会掷两次硬币
If it comes up heads both times the game's over but otherwise the game continues. 如果两次都正面 就结束博弈 否则的话继续下去
So both Brooke and Patrick should get ready to play, Brooke和Patrick应该都准备好了吧
and the payoffs of this game are just what we had before. 这次博弈的收益和我们之前的相同
So let's just remind ourselves what the payoffs of that game are. 大家都回忆一下这个博弈中的收益
So we've got cooperate, defect, cooperate, defect, 我们会有合作 背叛 合作 背叛
(2, 2), (-1, 3), (3, -1) and (0, 0). 分别为(2, 2) (-1, 3) (3, -1)和(0, 0)
And we'll keep score here: so this Brooke and Patrick. 我们在这里记分 分别是Brooke和Patrick
So, putting pressure on these guys, 好 给他们点压力
let's write down what you're going to do the first time. 让我把你们第一次的选择写下来
Brooke? Brooke呢?
Defect. 背叛
Patrick? Patrick呢?
Cooperate. 合作
All right. 很好
I think we're getting some payback from earlier, right. 我觉得我们从之前的内容获得了些收益 对吧
Round two. 第二轮
Are you going to toss the coin? 你不掷硬币了吗?
Oh I have to toss the coin, you're absolutely right, thank you. 哦 我得掷硬币 没错 谢谢
Now I have to find a coin. 我先要找一枚硬币
Look at that, thank you Ale. 看那个 谢谢Ale
Twice: toss it twice. 两次 掷两次
Heads, heads again, so the game is over. 正面 还是正面 所以博弈结束了
That didn't last long. 太短了吧
Just for the sake of the class, let's pretend that it came up tails. 为了把课上下去 我们假装这一次是反面吧
Okay we'll cheat a little bit. 我们作弊一下
Okay, so we're playing a second time... just with a little bit of cheating. 我们再来一次... 稍微作弊一下
I need someone else, someone less honest to toss the coin. 我要再找个人 不要真么诚实地掷硬币的人
Brooke what do you choose? Brooke 你的选的什么?
Oh I'm defecting. 哦 我选背叛
Defecting again, Patrick? 又是背叛 Patrick呢?
Cooperate. 合作
Cooperate, Patrick seems very trusting here, 合作 Patrick看起来还是很可靠的
all right let's toss the coin a third time. 好的 我们再掷一次硬币
All right, Brooke? 好了 Brooke呢?
I'm going to defect again. 我继续选背叛
Defect. 背叛
All right, 好的
so this time we'll end it. 这次我们就结束了
So what happened this time, let's just talk about it a bit. 这次发什么了什么 我们来讨论一下
So Brooke and Patrick were playing, Patrick cooperated a bit in the beginning, Brooke和Patrick参与博弈 开始的时候Patrick选择合作
Brooke's defected throughout. Brooke每次都是背叛
Brooke why did you defect? Brooke 你为什么选择背叛?
Shout out so everyone can hear you. 大声带你说出来 让大家都听得到你
Why did you defect right from the start of the game? 为什么在这场博弈中你一直选的是背叛?
Because last time it didn't work so well cooperating. 因为上一次我选合作的时候结果不怎么好
Last time it didn't work so well, okay. 上一次结果不怎么好 好的
Fair enough but even after Patrick was sort of cooperating you went on defecting. 有道理 但当Patrick一直选合作的时候你还是选背叛
So why then? 这又是为什么?
Because I wanted to get the higher payoff, 因为我想获得更高的收益
I thought either he would continue cooperating and I could defect, and... 我想他 要么他一直选合作而我选背叛 要么...
All right, if he had gone on cooperating, which in fact he did. 好的 如果他继续选合作的话 事实上他就是这么做的
Patrick why were you cooperating early on here? Patrick 开始的时候你为什么选合作?
Shout out so people can hear you. 大点声让大家都听得到你
So with a two head rule, like you have a 75% chance at having another game. 由于有了硬币规则 你有75%的机会会进行下一轮博弈
So with those payoffs, even one period the payoff of cooperating twice 所以两次都选合作的收益
is the same as defecting once, 和选一次背叛的收益是相等的
so it's better if you can continue cooperating, 所以你最好继续合作
and the percentage is high enough that it would make sense to do so. 而且这个概率也足够高 所以这么做是有道理的
All right, if you figure there's a good enough chance of getting... 好的 如果你算出有足够多的几率会得到...
even after Brooke's defected the first period you went on cooperating, 即使当Brooke在第一轮的时候选了背叛 你还是会继续选合作
but then after the second period you gave up and started defecting. 但是第二轮过后 你放弃了 开始选背叛了
If it had gone on to the fourth period what would you have done? 如果到了第四轮你会选什么?
Defected. 背叛
You would have defected again, all right. 你还是会选背叛 对吧?
Fifth period? 第五轮呢?
Well if she kept defecting, I would keep defecting. 如果她一直选背叛的话 我也会一直选背叛的
All right, so what Patrick's saying is he started off cooperating 好的 Patrick说他开始的时候选择合作
but once he saw that Brooke was defecting, he was going to switch to defect. 但当他一旦发现Brooke选背叛的时候 他也会转向背叛
And basically as long as she went on defecting, 也就是说只要她继续选背叛
he was going to stick with defecting. 他也会坚持选背叛
Let's try a different pair. 让我们再来换一组
So why don't we switch it over to your partners there. 直接换你们的邻座好了
So Ben here and Edwina. Ben和Edwina
So why don't you stand up. 你们站起来一下
I want everybody to see these people. 我想让大家都看到你们
So stand up a second. 站起来一下
So these are our players, 这就是我们的博弈者
I want people at the back to essentially know who are playing, 我想让后面的人最起码知道是谁在进行博弈
this is Edwina and this is Ben. 这是Edwina 这是Ben
So Edwina... sit down so you can actually write things down. Edwina... 坐下吧 你们还要写东西
So Edwina and Ben, Edwina, have you both written down a strategy? Edwina和Ben Edwina 你们都写下一个策略了吗?
Ben, have you written down a strategy? Ben 你把策略写下来了吗?
Edwina what did you choose? Edwina 你选的什么?
Cooperate. 合作
So Edwina's cooperating, Ben? Edwina选的是合作 Ben你呢?
Cooperate. 合作
Okay, let's toss the coin. 好的 让我们来掷硬币
So we're okay, so we're still playing. 很好 我们要继续下去
Edwina? Edwina呢?
Cooperate. 合作
Ben? Ben呢?
I chose cooperate. 我选择合作
All right, so they're cooperating. 好的 他们合作了
Tails again, so you're still playing. 又是反面 你们还得继续下去
Cooperate. 合作
All right, so they're still cooperating. 好的 他们仍是合作
Some pain in the voice this time. 那个声音有点痛苦啊
Heads and then tails, write down what you're going to do. 正面 反面 把你们要做的写下来
Edwina? Edwina呢?
Defect. 背叛
Ben. Ben呢
Cooperate. 合作
Things were going so nicely there. 事情发展得很顺利
We had such a nice class going on there. 这节课上得很顺利
All right, so we're still playing. 好的 我们继续下去
Edwina? Edwina呢?
Defect. 背叛
Ben? Ben呢?
Defect. 背叛
All right, Jake? 好的 Jake呢?
Tails, tails, we're still going. 反面 反面 我们还得继续
Defect. 背叛
All right, let me stop it there, we'll pretend that we had two heads. 好的 到此结束吧 我们就当两个都是正面好了
So let's talk about this. 我们来讨论一下
We had some cooperation going on here, both people started cooperating. 还是有几次合作的 开始的时候大家都选合作
So Ben, why did you cooperate at the beginning? Ben 开始的时候你为什么选择合作呢?
Well, going along with Patrick's reasoning I felt that 就像Patrick所说的那样 我觉得
if we could have the cooperate, 如果我们可以合作的话
cooperate in the long term with the 75% chance of continuing playing, 我们有75%的可能性继续进行下去 长远来看我们可以合作
that it would be a worthwhile investment. 那合作就是一个值得选择的投资
All right. 没错
Until I realized that Edwina had started defecting. 直到我发现Edwina开始背叛了
Let's come back a second. 让我们回头看一下
Let's get you guys to stand up so people can hear you. 你们不妨站起来说 这样大家都能听到
When you stand up you shout more. 站起来说可以更大声
So stand up again. 再站一下吧
Edwina, so you also started cooperating, why did you start cooperating? Edwina 开始的时候你还是选择合作 这是为什么呢?
For the same reason. 基于同样的理由
Same reason, okay. 同样的理由 好的
So the key thing here is why did you start defecting? 这里的关键问题是为什么你后来选择了背叛?
You heard the big sigh in the class. 你听得到全班人都在叹气
Why did you start defecting at this stage? 这一次开始的时候你为什么选背叛?
Because we'd had so many, 因为我们经历了那么多次
I mean the coin toss had to come to heads, heads sometime, 我是说一定要掷出正面才行
- so I started thinking that maybe... - The reversion to the mean of the coin. - 所以我开始想可能... - 掷硬币的回归平均数
Yeah, I just thought that it. 没错 我就是那么想的
I thought, I mean, I don't know. 我想 我是说 我也不知道
So what did I say about the relationships of Economic majors that are in the class? 我是怎么说经济学专业的人际关系的呢?
Anyway, all right, so Edwina defected 好的 Edwina选择了背叛
and then Ben you switched after, why did you switch? 接着Ben也改变了 你为什么改变了呢?
Because once Edwina started defecting I felt that 因为当Edwina开始选背叛的时候我觉得
we'd revert back to the defect, defect equilibrium. 我们回到了那个背叛的均衡
All right, so thank you guys. 好的 谢谢你们
So there's another good strategy here. 所以还有一个好的策略
People started off cooperating and I claim that at least Ben... 人们开始的时候选择合作 我是说至少Ben...
Ben can contradict me in a second... 过会儿Ben可以来反驳我...
but I think Ben's strategy here was something like this. 但我觉得Ben的策略跟有点像这样
I'm going to cooperate and 我去选择合作
I'm going to go on cooperating as long as we're cooperating. 只要我们都合作 那我将继续选择合作
But if at some point Edwina defects... or for that matter I defect... 但如果某一轮Edwina背叛了... 或者说是我背叛了...
then this relationship's over and we're going to play defect forever. 那这种关系就瓦解了 我们会永远选择背叛
Is that right? 没错吧?
That's kind of a rough description of your strategy? 这可以算是对这种策略的大致描述吧?
Edwina was more or less playing the same thing. Edwina差不多也是这么做的
In fact it was her who defected, but once she defected she realized that 事实上是她背叛的 但是当她背叛的时候她就意识到了
it was over and she went on defecting. 这种关系的瓦解 并选择一直背叛下去
So this strategy has a name. 这个策略有一个名字
Let's just be clear what the strategy is. 让我们来弄明白这是怎样的一种策略
This strategy says play C which is cooperate, 这种策略中 选择C 即合作
and then play C if no one has played D; 如果没人选D的话就选择C
and play D otherwise. 否则的话就选择D
So start off by cooperating. 开始的时候选择合作
Keep cooperating as long as nobody's cheated. 只要没人欺骗的话 就一直合作下去
But if somebody cheats, this relationship's over: 但如果有人欺骗的话 这种关系就结束了
we're just going to defect forever. 我们会一直选择背叛
Now this strategy is a famous strategy. 这是一个著名的策略
It has a name. 它有一个名字
Anyone know what the name is? 有谁知道它的名字吗?
This is called the "Grim Trigger Strategy." 我们称其为"变节触发策略"
So this strategy again, it says we're going to cooperate, 这种策略说 我们会一直合作下去
but if that cooperation breaks down ever, 但如果这种合作的关系一旦瓦解
even if it's me who breaks it down, then I'm just going to defect forever. 即使是我破坏的这种关系 我也得一直背叛下去
Now, we're going to come back next time to see if this is an equilibrium, 下面我们要回过头看看这是否是一个均衡
but there's a few things to do first. 但是 首先我们要做的是
First let's just check that it actually is a strategy. 首先让我们验证一下这的确是一个策略
What does it mean to be a strategy again? 是一个策略呢意味着什么呢?
It has to tell us what to do at every information set I could find myself at. 它应该告诉我们 在每个处境 根据每套信息 我们应该怎么做
And this game is potentially infinite, 这种博弈可能会是无限进行下去的
so potentially there's an infinite number of information sets I could reach. 所以可能会有无限多套信息
So you might think that writing down a strategy that 你会认为一旦这个博弈无限进行下去的时候
gives me an instruction at every single information set 根据每一套信息
is going to be incredibly complicated once 给出一个指令
we go to games that are potentially infinite, 会是相当复杂的事
because there needs to be an infinite number of instructions. 因为这需要有无数多的指令
But it turns out, actually it's possible to write down 但事实证明 把这种简单策略
such strategies rather simply, 写下来是有可能的
at least if they're simple strategies. 至少在简单策略的情况下是这样
This example is one. 这就是一个例子
This tells me what to do at the first information set, it says play C. 它让我知道了面对第一套信息时该怎么做 应该选C
It then tells me for every information set I find myself at, 然后告诉我 根据每一套信息 我该怎么做
in which only cooperation has ever occurred in the history of the game, 当这个博弈中只出现C的时候
I'm going to go on cooperating: play C. 我会一直选合作 即C
And it says for all other histories, 它告诉我在其他情况下
for all other information sets I might find myself at, play D. 在面对其它的信息时我应该选D
So it really is a strategy. 这的确是一种策略
Now this is very different behavior... we played with the same players... 这儿有些很不同寻常的行为... 参与博弈的人相同...
this kind of behavior is very different, in both games actually, 这种行为很不同寻常 两次游戏中都是
is very different than the behavior we saw in the game that ended, 和刚才那种博弈中的行为不同
the game with two periods or three periods. 2或3个回合就结束的博弈大不相同
What is it essentially that made this different? 到底是什么造成了这种差异呢?
What's different about this way of playing Prisoner's Dilemma, 这和囚徒困境中的做法有什么差异呢
where we had Jake toss the coin versus the way we played before 我们让Jake掷硬币和我们之前进行的
and we just played for five periods and then stopped? 五次后结束的博弈有什么区别呢?
What's different about it? 两者之间有何不同?
Somebody? 谁知道呢?
Let's talk to our players, 让我们和博弈者来谈谈
Patrick why is this different? Patrick 为什么不一样呢?
We don't when the game is going to end or if it's going to end, 我们不知道这个博弈什么时候会结束 或者说是它是否会结束
so there's no last period. 不存在最后一轮
Good, so our analysis of the game before, 很好 我们对之前那个博弈的分析
the analysis of the Prisoner's Dilemma 对囚徒困境的分析
when we knew it was going to end after two periods, 当我们知道还有两轮就要结束的时候
after five periods, whatever it was, was we all knew it was going to end. 或者是还有五轮 随便啦 总之我们知道它肯定会结束
There was a clearly defined last period. 有一个明确的结束阶段
When people are going to retire, 当人们即将退休的时候
we know the month in which they're going to retire. 我们知道退休的确切月份
When President's are going to step down, 当总统任期快要结束的时候
we know they're going to step down that period. 大家都知道什么时候会结束
When CEO's are going to go, we know they're going to go... 当CEO要走的时候 我们知道他们肯定会走...
or acctually we don't always know they're going to go but let's just pretend we do. 抑或我们不总能得知他们会走 但让我们假装一下我们是知道的
So what's different about this game is, 这个博弈中不一样的地方在于
every time we play the game, there is a probability, 每次进行这个博弈的时候 都存在一种可能性
in this case a .75 probability that the game 在这个博弈中有75%的可能性
is going to continue to the next period. 会继续进行下一轮
Every time we play the game, 每一次进行博弈的时候
with probability of .75 there's going to be a future. 我们都有75%的可能性会进行下一轮
There's no obvious last period from which 没有明显的最后一轮
we can unravel the game in the way we did before. 可以让我们像以前那样得以解脱
Just to remind ourselves, the way in which cooperation... 提醒大家一下 合作以某种方式...
our analysis of cooperation... 我们对合作的分析...
broke down in the finitely repeated Prisoner's Dilemma, 在有限重复次数的囚徒困境中瓦解了
was when we looked at the last period, we know people are going to defect. 当我们面对最后一轮的时候 我们知道大家会选背叛
And once that thread is loose we can unravel it 一旦理出线索
all the way back to the beginning. 我们就可以通过回溯去解决
But here, since there is no last period that 但在这里 因为没有最后一轮
unraveling argument never gets to hold. 所以永远不能理出线索
Now instead we're able to see strategies emerge like the Grim Trigger Strategy, 而我们可以看到类似于变节触发策略的出现
and notice that the Grim Trigger Strategy 并注意到变节触发策略
has a pretty good chance of actually sustaining cooperation. 很有可能来维持合作
So in particular, as long as people play this strategy they are cooperating. 在这种情况下 只要人们一直采用这种策略 就可以合作了
It turns out that Edwina eventually gave up that strategy, 结果表明Edwina最终放弃了这种策略
but had she gone on playing it, they would have gone on cooperating forever. 但只要她一直进行下去 他们就能一直合作下去
But of course there's a question here, 当然这里也有问题
and the question is: is this in fact an equilibrium? 问题是 这真的是一种均衡吗?
We know that if people play this way, we get cooperation, but the question... 大家知道如果人们按照这种方式进行下去 我们可以进行合作 但问题是...
the thousand dollar question or whatever... is: is this an equilibrium? 或者说这里的关键问题是... 这是一种均衡吗?
So what do we have to do to check whether this is an equilibrium or not? 我们如何来验证这是一个均衡呢?
We have to mimic the argument we had before. 我们要效仿一下之前的论证
We have to compare the temptation to defect today 我们要比较一下今天背叛的诱惑
and compare that with the value of the reward (to cooperating) 拿它和明天的奖励(合作)
and the value of the punishment (from defecting) tomorrow. 以及惩罚(背叛)的价值做比较
So this basic idea is going to re-emerge. 所以这里的基本思路就会重现
Having said that, let me now delete it so I have some room. 说了这么多 我先擦掉腾出点空间
To show this is an equilibrium, we need to show that the temptation to defect... 要证明这是一种均衡 我们就要证明选择背叛的诱惑...
the temptation to cheat in the short term... 背叛的短期诱惑...
is outweighed by the difference between the value of the reward 不及
and the value of the punishment. 将来的奖励与惩罚的价值差
All right, so let's set that up. 好的 我们来算一下
Let's put the temptation here first. 我们先给出这里的诱惑
So the temptation in Prisoner's Dilemma, the temptation to cheat today is what? 所以在囚徒困境中的诱惑 今天选择欺骗的诱惑是什么?
I'll get 3 rather than 2, is that right? 我会得到3而不是2 对吧?
So if I defect... when Edwina defected: 所以如果我背叛的话... 当Edwina背叛的时候
here's Edwina defecting in this period... Edwina在这一轮背叛了...
she got a payoff of 3 rather than the payoff of 2 她获得的收益是3
she would have got from cooperating. 而不是选择合作后获得的2
So the temptation here is just 3-2 and let's be clear, 这里的诱惑就是3-2 大家要明白
this is a temptation today and we want to compare this 这就是今天的诱惑 我们要把它和
with the value of the reward minus the value of the punishment, 奖励和惩罚的价值之差做比较
but the key observation is that these occur tomorrow. 但关键问题是这些要明天才会发生
So since they occur tomorrow we have to weight them a little bit lower. 既然它们要明天才会发生 所以我们不得不降低它们的价值
So in general, the way in which 所以通常
we're going to weight them tomorrow is we're going to discount them 我们处理明天的收益的办法 就是打一个折扣
just like we did in our bargaining game. 就像我们在还价博弈中做的那样
We're going to weight tomorrow's payments by δ, 我们把明天的收益乘以δ
where δ< 1. 其中δ< 1
Now why is δ< 1? 为什么δ< 1呢?
Why are we weighing tomorrow less than payment today? 为什么明天的收益比重少于今天的呢?
Why are payments tomorrow worth less than payments today? 为什么明天的收益比今天少呢?
Because tomorrow might not happen. 因为明天可能不会发生
There are other reasons why, by the way. 还有其它一些原因
It might be that we are impatient to get the money today. 可能是我们没有耐心去获取今天的钱
Edwina just wanted the payoff in a hurry, Edwina只是想尽快获取收益
or it might be that she wanted to take the payment today 又或许是她今天就要这个收益
and put it in the bank and earn interest. 把它存到银行里去赚利息
There are other reasons why money today 还有一些其它的理由来说明为什么
might be more valuable than money tomorrow, 今天的钱比明天的更有价值
but, in games, the most important reason is: tomorrow may not happen. 但是在博弈中 最重要的原因是明天可能不会发生
By tomorrow you might be dead, or, if not dead, 到了明天你或许会死去 又或许你不会死
at least Jake's thrown two heads in the coins. 至少Jake扔的两个硬币都是正面的
So δ is less than 1 because the game may end. 所以δ小于1 因为博弈可能会结束
Now, what's the value of the reward? 回报的价值有多大?
The value of the reward is going to be the value of C "for ever," 回报的价值会是"永远"选C的价值
but you want to be careful about "for ever." 但你们要留心一下这里的"永远"
It's C for ever, but of course it isn't really for ever 是一直选C 但当然了 不太可能真的一直这样下去
because the game may end. 因为博弈可能会结束
So by "for ever" I mean until the game ends. 所以这里的"永远"我指的是直到博弈结束
Let me be a bit more careful actually, it's (C, C) isn't it? 我再来说得详细一点 是(C, C) 对吧?
The value of (C, C)... (cooperate, cooperate)... for ever. 一直是(C, C)... (合作, 合作)... 的价值
Here we're going to have the value of (D, D) for ever. 这里我们会一直获得(D, D)的价值
And once again, the for ever here means until the game ends. 再说一遍 这里的永远指的是直到博弈结束
So this is the calculation we're going to have to do. 这就是我们要进行的计算
We're going to have to compare the temptation, that was easy, 我们要来比较一下这种诱惑 很简单
that was just 1 with the discounted difference 就是比较一下
between the value of cooperation and the value of defecting. 折扣过的合作和背叛之间的差价和1的大小
Let's do the easy bits now, 我们来算一下简单的部分
and then we'll leave you in suspense until Wednesday. 之后我们就要到周三再来讨论了
So let's do all the easy bits. 我们来算一下简单的部分
So what's this δ in this case? 这个例子中的δ是什么?
In this particular game what was the probability that the game 在这个博弈中 有多大的可能性
was going to continue? 博弈会继续下去?
What was the probability that the game was going to end? 博弈结束的可能性又有多大?
The probability of it ending was .25, so δ here was .75, that's easy. 结束的可能性是25% 所以δ是75% 很简单
The second bit that's relatively easy 第二个简单的问题是
is what's the value of playing (D, D) until the game ends? 在博弈结束之前选择(D, D)的价值有多大?
Once people have cheated you're going to play D for ever... 一旦有人欺骗了 你就会一直选择D...
here we are: Edwina's cheating here. 这里的情况就是Edwina选择了欺骗
You're going to get (D, D) in this period, (D, D) in this period, 你们在这一轮选择(D, D)
and so on and so forth until the game ends. 会直到博弈结束
In each of those periods you're going to earn 0, so this is just 0. 其中的每一轮你得到的都是0 所以这里是0
Which leaves us with a messy bit: 有点混乱的是:
what's the value of cooperating forever? 一直合作下去的价值是是什么?
Let's try and do it. 我们来看一下
We've got one minute. 我们还剩一分钟
Let's do it. 算一下
So in every period in which we both cooperate what do we earn? 在双方都选择合作的那些轮次中 我们得到了什么?
Throughout the beginning of the game: 在这个博弈的开始阶段
we cooperated in the first period; 第一轮大家选择了合作
now in the second period, we cooperate again. 第二轮大家又选择了合作
What payoff do we get from cooperating again? 再一次合作的收益是什么?
We get 2 and then Jake tosses his coin and with probability δ 我们得到了2 接着Jake掷硬币 有δ的可能性
we continue and we're going to cooperate again. 我们会继续进行下去并再次合作
So with probability δ 有δ的可能性
we cooperate again and get what payoff the next period? 我们会再次合作 那下一轮会获得多少收益呢?
2 again, and then Jake tosses the coin again, 还是2 接着Jake又掷了一次硬币
so now he's tossed the coin twice, so with probability δ2 这时候他掷了两次硬币 即有δ平方的可能性
we're still playing and we get 2, 我们会继续进行下去并获得2
and then Jakes tosses the coin again 接着Jakes又掷了一次硬币
and it comes up other than heads, heads again, 仍然不是两个正面
that's with probability δ3 we get 2 and so on. 即我们有δ三次方的可能性得到2并继续下去
So your exercise between now and Wednesday is to figure out 你们周三之前要做的练习就是算出
what the value of cooperation forever is: 一直合作的价值是多少
figure out this equation and find whether in fact 算出这个方程 并找出人们进行合作
it was an equilibrium for people to cooperate. 究竟上是不是一个均衡
We'll pick it up on Wednesday. 我们周三继续讲下去
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