Game Theory Lecture 2 | Game Theory

Chapter 1.

第一章。

Recap of Previous Lecture: Prisoners' Dilemma and Payoffs [00:00:00] Professor Ben Polak: Okay, so last time we looked at and played this game.

上一讲回顾：囚徒困境和回报 [00:00:00] Ben Polak 教授：好的，上次我们看了并玩了这个游戏。

You had to choose grades, so you had to choose Alpha and Beta, and this table told us what outcome would arise.

你必须选择等级，所以你必须选择 Alpha 和 Beta，这张表告诉我们会出现什么结果。

In particular, what grade you would get and what grade your pair would get.

特别是，你会得到什么等级，以及你的搭档会得到什么等级。

So, for example, if you had chosen Beta and your pair had chosen Alpha, then you would get a C and your pair would get an A.

因此，举例来说，如果您选择了 Beta，而您的搭档选择了 Alpha，那么您将获得 C，而您的搭档将获得 A。

One of the first things we pointed out, is that this is not quite a game yet.

我们首先指出的一件事是，这还不是一款游戏。

It's missing something.

它缺少一些东西。

This has outcomes in it, it's an outcome matrix, but it isn't a game, because for a game we need to know payoffs.

这里面有结果，它是一个结果矩阵，但它不是游戏，因为对于游戏我们需要知道收益。

Then we looked at some possible payoffs, and now it is a game.

然后我们研究了一些可能的回报，现在这是一个游戏。

So this is a game, just to give you some more jargon, this is a normal-form game.

所以这是一个游戏，只是给你一些更多的术语，这是一个正常形式的游戏。

And here we've assumed the payoffs are those that arise if players only care about their own grades, which I think was true for a lot of you.

在这里，我们假设玩家只关心自己的成绩，就会产生回报，我认为这对你们很多人来说都是如此。

It wasn't true for the gentleman who's sitting there now, but it was true for a lot of people.

对于现在坐在那里的那位先生来说不是这样，但对于很多人来说却是这样。

We pointed out, that in this game, Alpha strictly dominates Beta.

我们指出，在这场比赛中，Alpha 严格压制 Beta。

What do we mean by that?

我们这么说是什么意思？

We mean that if these are your payoffs, no matter what your pair does, you attain a higher payoff from choosing Alpha, than you do from choosing Beta.

我们的意思是，如果这些是你的回报，那么无论你的搭档做什么，选择 Alpha 都会比选择 Beta 获得更高的回报。

Let's focus on a couple of lessons of the class before I come back to this.

在我回到这个话题之前，让我们先关注一下本堂课的几节课。

One lesson was, do not play a strictly dominated strategy.

一个教训是，不要采用严格占优的策略。

Everybody remember that lesson?

大家还记得那节课吗？

Then much later on, when we looked at some more complicated payoffs and a more complicated game, we looked at a different lesson which was this: put yourself in others' shoes to try and figure out what they're going to do.

很久以后，当我们研究一些更复杂的收益和更复杂的游戏时，我们看到了一个不同的教训：设身​​处地为别人着想，尝试弄清楚他们会做什么。

So in fact, what we learned from that is, it doesn't just matter what your payoffs are -- that's obviously important -- it's also important what other people's payoffs are, because you want to try and figure out what they're going to do and then respond appropriately.

所以事实上，我们从中学到的是，你的回报是什么并不重要——这显然很重要——其他人的回报是什么也很重要，因为你想尝试弄清楚他们将要做什么，然后做出适当的反应。

So we're going to return to both of these lessons today.

所以我们今天将回到这两课。

Both of these lessons will reoccur today.

这两个教训今天都会再次发生。

Now, a lot of today is going to be fairly abstract, so I just want to remind you that Game Theory has some real world relevance.

现在，今天的很多内容都相当抽象，所以我只是想提醒您，博弈论具有一些现实世界的相关性。

Again, still in the interest of recapping, this particular game is called the Prisoners' Dilemma.

再次，为了重述一下，这个特殊的游戏被称为“囚徒困境”。

It's written there, the Prisoners' Dilemma.

那里写着，囚徒困境。

Notice, it's Prisoners, plural.

注意，它是复数的囚犯。

And we mentioned some examples last time.

我们上次提到了一些例子。

Let me just reiterate and mention some more examples which are actually written here, so they'll find their way into your notes.

让我重申并提及一些实际写在这里的示例，以便它们可以进入您的笔记。

So, for example, if you have a joint project that you're working on, perhaps it's a homework assignment, or perhaps it's a video project like these guys, that can turn into a Prisoners' Dilemma.

因此，举例来说，如果您正在开展一个联合项目，也许这是一项家庭作业，或者也许是像这些人一样的视频项目，这可能会变成囚徒困境。

Why?

为什么？

Because each individual might have an incentive to shirk.

因为每个人都可能有逃避的动机。

Price competition -- two firms competing with one another in prices -- can have a Prisoners' Dilemma aspect about it.

价格竞争——两家公司在价格上相互竞争——可能会陷入囚徒困境。

Why?

为什么？

Because no matter how the other firm, your competitor, prices you might have an incentive to undercut them.

因为无论其他公司（即您的竞争对手）如何定价，您都可能有动机低于他们。

If both firms behave that way, prices will get driven down towards marginal cost and industry profits will suffer.

如果两家公司都这样做，价格将被压向边际成本，行业利润将受到影响。

In the first case, if everyone shirks you end up with a bad product.

在第一种情况下，如果每个人都推卸责任，你最终会得到一个糟糕的产品。

In the second case, if both firms undercut each other, you end up with low prices, that's actually good for consumers but bad for firms.

在第二种情况下，如果两家公司互相压价，最终会得到低价，这实际上对消费者有利，但对公司不利。

Let me mention a third example.

我再提第三个例子。

Suppose there's a common resource out there, maybe it's a fish stock or maybe it's the atmosphere.

假设那里有一种共同资源，可能是鱼类资源，也可能是大气。

There's a Prisoners' Dilemma aspect to this too.

这也有囚徒困境的一面。

You might have an incentive to over fish.

您可能有过度捕鱼的动机。

Why?

为什么？

Because if the other countries with this fish stock--let's say the fish stock is the Atlantic--if the other countries are going to fish as normal, you may as well fish as normal too.

因为如果其他国家有这种鱼群——假设鱼群是大西洋——如果其他国家照常捕鱼，你也可以照常捕鱼。

And if the other countries aren't going to cut down on their fishing, then you want to catch the fish now, because there aren't going to be any there tomorrow.

如果其他国家不打算减少捕捞，那么你现在就想抓鱼，因为明天就没有鱼了。

Another example of this would be global warming and carbon emissions.

另一个例子是全球变暖和碳排放。

Again, leaving aside the science, about which I'm sure some of you know more than me here, the issue of carbon emissions is a Prisoners' Dilemma.

再说一遍，抛开科学不谈，我相信你们中的一些人比我更了解科学，碳排放问题是一个囚徒困境。

Each of us individually has an incentive to emit carbons as usual.

我们每个人都有像往常一样排放碳的动机。

If everyone else is cutting down I don't have too, and if everyone else does cut down I don't have to, I end up using hot water and driving a big car and so on.

如果其他人都减少开支，我也没有，如果其他人都减少开支，我就不必这样做，我最终会使用热水并驾驶一辆大汽车等等。

In each of these cases we end up with a bad outcome, so this is socially important.

在每一种情况下，我们都会得到不好的结果，所以这对于社会来说很重要。

This is not just some abstract thing going on in a class in Yale.

这不仅仅是耶鲁大学课堂上发生的一些抽象的事情。

We need to think about solutions to this, right from the start of the class, and we already talked about something.

我们需要从课程一开始就考虑这个问题的解决方案，我们已经讨论过一些事情。

We pointed out, that this is not just a failure of communication.

我们指出，这不仅仅是沟通失败。

Communication per se will not get you out of a Prisoners' Dilemma.

沟通本身并不能让你摆脱囚徒困境。

You can talk about it as much as you like, but as long as you're going to go home and still drive your Hummer and have sixteen hot showers a day, we're still going to have high carbon emissions.

你可以随心所欲地谈论它，但只要你回家后仍然开着你的悍马，每天洗十六次热水澡，我们的碳排放量仍然很高。

You can talk about working hard on your joint problem sets, but as long as you go home and you don't work hard, it doesn't help.

你可以谈论努力做联合题，但只要你回家不努力，那就没有帮助。

In fact, if the other person is working hard, or is cutting back on their carbon emissions, you have every bit more incentive to not work hard or to keep high carbon emissions yourself.

事实上，如果对方正在努力工作，或者正在减少碳排放，那么你就有更多的动机不努力工作或自己保持高碳排放。

So we need something more and the kind of things we can see more: we can think about contracts; we can think about treaties between countries; we can think about regulation.

所以我们需要更多的东西，以及我们可以看到更多的东西：我们可以考虑合同；我们可以考虑国家之间的条约；我们可以考虑监管。

All of these things work by changing the payoffs.

所有这些事情都是通过改变收益来发挥作用的。

Not just talking about it, but actually changing the outcomes actually and changing the payoffs, changing the incentives.

不仅仅是谈论它，而是实际上改变结果、改变回报、改变激励。

Another thing we can do, a very important thing, is we can think about changing the game into a game of repeated interaction and seeing how much that helps, and we'll come back and revisit that later in the class.

我们可以做的另一件事，一件非常重要的事情，是我们可以考虑将游戏变成重复互动的游戏，看看这有多大帮助，我们稍后会在课程中回来重新讨论。

One last thing we can think of doing but we have to be a bit careful here, is we can think about changing the payoffs by education.

我们可以想到做的最后一件事是，我们可以考虑通过教育来改变回报，但在这里我们必须要小心一点。

I think of that as the "Maoist" strategy.

我认为这是“毛主义”战略。

Lock people up in classrooms and tell them they should be better people.

把人们关在教室里，告诉他们应该成为更好的人。

That may or may not work -- I'm not optimistic -- but at least it's the same idea.

这可能有效，也可能无效——我并不乐观——但至少是同样的想法。

We're changing payoffs.

我们正在改变回报。

So that's enough for recap and I want to move on now.

回顾一下就足够了，我现在想继续。

And in particular, we left you hanging at the end last time.

特别是，我们上次让您悬而未决。

We played a game at the very end last time, where each of you chose a number -- all of you chose a number -- and we said the winner was going to be the person who gets closest to two-thirds of the average in the class.

上次我们在最后玩了一个游戏，你们每个人都选择了一个数字——你们所有人都选择了一个数字——我们说获胜者将是最接近全班平均分三分之二的人。

Now we've figured that out, we figured out who the winner is, and I know that all of you have been trying to see if you won, is that right?

现在我们已经弄清楚了，我们已经弄清楚了谁是获胜者，我知道你们所有人都一直想看看自己是否获胜，是吗？

I'm going to leave you in suspense.

我会让你保持悬念。

I am going to tell you today who won.

今天我要告诉你谁赢了。

We did figure it out, and we'll get there, but I want to do a little bit of work first.

我们确实解决了这个问题，我们会实现这一目标，但我想先做一些工作。

So we're just going to leave it in suspense.

所以我们就留个悬念吧。

That'll stop you walking out early if you want to win the prize.

如果你想赢得奖品，这会阻止你早早走出去。

So there's going to be lots of times in this class when we get to play games, we get to have classroom discussions and so on, but there's going to be some times when we have to slow down and do some work, and the next twenty minutes are going to be that.

因此，在这堂课上，我们会有很多时间玩游戏、进行课堂讨论等，但有时我们必须放慢速度做一些工作，接下来的二十分钟就是这样。

So with apologies for being a bit more boring for twenty minutes, let's do something we'll call formal stuff.

因此，抱歉二十分钟的时间有点无聊，让我们做一些我们称之为正式的事情。

In particular, I want to develop and make sure we all understand, what are the ingredients of a game?

特别是，我想开发并确保我们都了解游戏​​的要素是什么？

So in particular, we need to figure out what formally makes something into a game.

因此，我们特别需要弄清楚是什么让某些东西正式成为游戏。

The formal parts of a game are this.

游戏的正式部分是这样的。

We need players -- and while we're here let's develop some notation.

我们需要玩家——趁我们在这里，让我们开发一些符号。

So the standard notation for players, I'm going to use things like little i and little j.

所以对于玩家来说，标准符号，我将使用像小 i 和小 j 这样的东西。

So in that numbers game, the game when all of you wrote down a number and handed it in at the end of last time, the players were who?

那么在那个数字游戏中，上次大家写下一个数字并交出的游戏中，玩家是谁？

The players were you.

球员就是你。

You'all were the players.

你们都是玩家。

Useful text and expression meaning you plural.

有用的文本和表达方式意味着您的复数。

In the numbers game, you'all, were the players.

在数字游戏中，你们都是玩家。

Second ingredient of the game are strategies.

游戏的第二个要素是策略。

(There's a good clue here.

（这里有一个很好的线索。

If I'm writing you should be writing.) Notation: so I'm going to use little "si" to be a particular strategy of Player i.

如果我在写，你也应该在写。） 符号：所以我将使用小“si”作为玩家 i 的特定策略。

So an example in that game might have been choosing the number 13.

因此，该游戏中的一个例子可能是选择数字 13。

Everyone understand that?

大家明白了吗？

Now I need to distinguish this from the set of possible strategies of Player I, so I'm going to use capital "Si" to be what?

现在我需要将其与玩家 I 的可能策略集区分开来，因此我将使用大写“Si”表示什么？

To be the set of alternatives.

成为一组替代方案。

The set of possible strategies of Player i.

玩家 i 的可能策略集。

So in that game we played at the end last time, what were the set of strategies?

那么我们上次打到最后的那个游戏，是怎样的一套策略呢？

They were the sets 1, 2, 3, all the way up to 100.

分别是第 1、2、3 组，一直到 100 组。

When distinguishing a particular strategy from the set of possible strategies.

将特定策略与一组可能策略区分开来时。

While we're here, our third notation for strategy, I'm going to use little "s" without an "i," (no subscripts): little "s" without an "i," to mean a particular play of the game.

当我们在这里时，我们的第三个策略符号，我将使用不带“i”的小“s”（无下标）：不带“i”的小“s”，表示游戏的特定玩法。

So what do I mean by that?

那么我这是什么意思呢？

All of you, at the end last time, wrote down this number and handed them in so we had one number, one strategy choice for each person in the class.

上次最后，你们所有人都写下了这个数字并交给了他们，这样我们就为班上的每个人提供了一个数字，一个策略选择。

So here they are, here's my collected in, sort of strategy choices.

所以它们就在这里，这是我收集的一些策略选择。

Here's the bundle of bits of paper you handed in last time.

这是你上次递过来的一叠纸片。

This is a particular play of the game.

这是一个特殊的游戏玩法。

I've got each person's name and I've got a number from each person: a strategy from each person.

我有每个人的名字和每个人的号码：每个人的策略。

We actually have it on a spreadsheet as well: so here it is written out on a spreadsheet.

实际上，我们也将其记录在电子表格中：所以这里将其写在电子表格上。

Each of your names is on this spreadsheet and the number you chose.

您的每个名字和您选择的号码都在此电子表格中。

So that's a particular play of the game and that has a different name.

这是游戏的一种特殊玩法，并且有不同的名称。

We sometimes call this "a strategy profile." So in the textbook, you'll sometimes see the term a strategy profile or a strategy vector, or a strategy list.

我们有时称之为“战略概况”。因此，在教科书中，您有时会看到术语“策略配置文件”、“策略向量”或“策略列表”。

It doesn't really matter.

这并不重要。

What it's saying is one strategy for each player in the game.

它所说的是游戏中每个玩家的一种策略。

So in the numbers game this is the spreadsheet -- or an example of this is the spreadsheet.

在数字游戏中，这就是电子表格——或者电子表格就是一个例子。

(I need to make it so you can still see that, so I'm going to pull down these boards.

（我需要做到这一点，这样你仍然可以看到它，所以我要拆下这些板。

And let me clean something.) So you might think we're done right?

让我清理一些东西。）所以你可能认为我们已经完成了？

We've got players.

我们有球员。

We've got the choices they could make: that's their strategy sets.

我们已经掌握了他们可以做出的选择：这就是他们的策略集。

We've got those individual strategies.

我们有这些单独的策略。

And we've got the choices they actually did make: that's the strategy profile.

我们已经得到了他们实际做出的选择：这就是战略概况。

Seems like we've got everything you could possibly want to describe in a game.

似乎我们已经拥有您想要在游戏中描述的一切。

What are we missing here?

我们在这里缺少什么？

Shout it out.

大声喊出来。

"Payoffs." We're missing payoffs.

“回报。”我们缺少回报。

So, to complete the game, we need payoffs.

因此，为了完成游戏，我们需要回报。

Again, I need notation for payoffs.

同样，我需要收益符号。

So in this course, I'll try and use "U" for utile, to be Player i's payoff.

因此，在本课程中，我将尝试使用“U”表示效用，作为玩家 i 的回报。

So "Ui" will depend on Player 1's choice …

所以“Ui”将取决于玩家 1 的选择……

all the way to Player i's own choice …

一直到玩家我自己的选择……

all the way up to Player N's choices.

一直到玩家N的选择。

So Player i's payoff "Ui," depends on all the choices in the class, in this case, including her own choice.

因此，玩家 i 的收益“Ui”取决于班级中的所有选择，在本例中，包括她自己的选择。

Of course, a shorter way of writing that would be "Ui(s)," it depends on the profile.

当然，更短的写作方式是“Ui(s)”，这取决于配置文件。

So in the numbers game what is this?

那么在数字游戏中这是什么？

In the numbers game "Ui(s)" can be two things.

在数字游戏中，“Ui(s)”可以是两件事。

It can be 5 dollars minus your error in pennies, if you won.

如果你赢了，可能是 5 美元减去你的错误（以美分计）。

I guess it could be something if there was a tie, I won't bother writing that now.

我想如果有平局的话可能会有所不同，我现在不会费心写这个。

And it's going to be 0 otherwise.

否则将为 0。

So we've now got all of the ingredients of the game: players, strategies, payoffs.

现在我们已经获得了游戏的所有要素：玩家、策略、收益。

Now we're going to make an assumption today and for the next ten weeks or so; so for almost all the class.

现在我们将在今天和接下来的十周左右做出一个假设；所以对于几乎​​所有的班级来说。

We're going to assume that these are known.

我们假设这些都是已知的。

We're going to assume that everybody knows the possible strategies everyone else could choose and everyone knows everyone else's payoffs.

我们假设每个人都知道其他人可以选择的可能策略，并且每个人都知道其他人的回报。

Now that's not a very realistic assumption and we are going to come back and challenge it at the end of semester, but this will be complicated enough to give us a lot of material in the next ten weeks.

现在这不是一个非常现实的假设，我们将在学期结束时回来挑战它，但这将足够复杂，足以在接下来的十周内为我们提供大量材料。

I need one more piece of notation and then we can get back to having some fun.

我还需要一个符号，然后我们就可以继续享受乐趣了。

So one more piece of notation, I'm going to write "s-i" to mean what?

那么还有一个符号，我要写“s-i”来表示什么？

It's going to mean a strategy choice for everybody except person "i." It's going to be useful to have that notation around.

这将意味着除了“我”之外的每个人的策略选择。拥有这个符号将会很有用。

So this is a choice for all except person "i" or Player i.

所以这是除了人“i”或玩家 i 之外的所有人的选择。

So, in particular, if you're person 1 and then "s-i" would be "s2, s3, s4" up to "sn" but it wouldn't include "s1." It's useful why?

因此，特别是，如果您是第 1 个人，那么“s-i”将是“s2、s3、s4”直到“sn”，但它不会包括“s1”。有用为什么？

Because sometimes it's useful to think about the payoffs, as coming from "i's" own choice and everyone else's choices.

因为有时考虑来自“我”自己的选择和其他人的选择的回报是有用的。

It's just a useful way of thinking about things.

这只是思考事物的一种有用的方式。

Now this is when I want to stop for a second and I know that some of you, from past experience, are somewhat math phobic.

现在我想停下来，根据过去的经验，我知道你们中的一些人有些数学恐惧症。

You do not have to wave your hands in the air if you're math phobic, but since some of you are, let me just get you all to take a deep breath.

如果你有数学恐惧症，你不必在空中挥舞双手，但既然你们中的一些人有数学恐惧症，那么让我让你们所有人深呼吸。

This goes for people who are math phobic at home too.

这也适用于在家有数学恐惧症的人。

So everyone's in a slight panic now.

所以现在大家都有点恐慌。

You came here today.

你今天来了。

You thought everything was going to fine.

你以为一切都会好起来的。

And now I'm putting math on the board.

现在我把数学放在黑板上。

Take a deep breath.

深吸一口气。

It's not that hard, and in particular, notice that all I'm doing here is writing down notation.

这并不难，特别是请注意，我在这里所做的只是写下符号。

There's actually no math going on here at all.

实际上这里根本没有数学。

I'm just developing notation.

我只是在开发符号。

I don't want anybody to quit this class because they're worried about math or math notation.

我不希望任何人因为担心数学或数学符号而退出这门课。

So if you are in that category of somebody who might quit it because of that, come and talk to me, come and talk to the TAs.

所以，如果你属于那种可能因此而放弃的人，请来跟我谈谈，来和助教谈谈。

We will get you through it.

我们会帮助您度过难关。

It's fine to be math phobic.

有数学恐惧症没关系。

I'm phobic of all sorts of things.

我对各种事情都感到恐惧。

Not necessarily math, but all sorts of things.

不一定是数学，而是各种各样的事情。

So a serious thing, a lot of people get put off by notation, it looks scarier than it is, there's nothing going on here except for notation at this point.

这是一件严肃的事情，很多人都被符号吓倒了，它看起来比实际更可怕，此时除了符号之外什么也没有发生。

So let's have an example to help us fix some ideas.

让我们举一个例子来帮助我们修正一些想法。

(And again, I'll have to clean the board, so give me a second.) I think an example might help those people who are disturbed by the notation.

（再说一遍，我必须清理电路板，所以请稍等一下。）我认为一个例子可能会帮助那些被符号困扰的人。

So here's a game which we're going to discuss briefly.

这是我们要简要讨论的一个游戏。

It involves two players and we'll call the Players I and II and Player I has two choices, top and bottom, and Player II has three choices left, center, and right.

它涉及两个玩家，我们称之为玩家 I 和 II，玩家 I 有两个选择，上和下，玩家 II 有左、中、右三个选择。

It's just a very simple abstract example for now.

目前这只是一个非常简单的抽象示例。

And let's suppose the payoffs are like this.

让我们假设回报是这样的。

They're not particularly interesting.

它们并不是特别有趣。

We're just going to do it for the purpose of illustration.

我们这样做只是为了说明。

So here are the payoffs: (5, -1), (11, 3), (0, 0), (6, 4), (0, 2), (2, 0).

所以这里是收益：(5, -1), (11, 3), (0, 0), (6, 4), (0, 2), (2, 0)。

Let's just map the notation we just developed into this game.

让我们将刚刚开发的符号映射到这个游戏中。

So first of all, who are the players here?

那么首先，这里的玩家是谁？

Well there's no secret there, the players are -- let's just write it down why don't we.

好吧，这并没有什么秘密，球员们——让我们把它写下来，为什么不呢。

The players here in this game are Player I and Player II.

这个游戏中的玩家是玩家 I 和玩家 II。

What about the strategy sets or the strategy alternatives?

策略集或策略选择怎么样？

So here Player I's strategy set, she has two choices top or bottom, represented by the rows, which are hopefully the top row and the bottom row.

所以这里玩家 I 的策略集，她有两个选择顶部或底部，由行表示，希望是顶行和底行。

Player II has three choices, this game is not symmetric, so they have different number of choices, that's fine.

玩家二有三个选择，这个游戏不是对称的，所以他们有不同数量的选择，这很好。

Player II has three choices left, center, and right, represented by the left, center, and right column in the matrix.

玩家II有左、中、右三个选择，用矩阵中的左、中、右列来表示。

Just to point out in passing, up to now, we've been looking mostly at symmetric games.

顺便指出一下，到目前为止，我们主要关注的是对称游戏。

Notice this game is not symmetric in the payoffs or in the strategies.

请注意，这个游戏的收益或策略并不对称。

There's no particular reason why games have to be symmetric.

游戏必须对称并没有什么特别的原因。

Payoffs: again, this is not rocket science, but let's do it anyway.

回报：再说一次，这不是火箭科学，但无论如何我们还是要做吧。

So just an example of payoffs.

这只是回报的一个例子。

So Player I's payoff, if she chooses top and Player II chooses center, we read by looking at the top row and the center column, and Player I's payoff is the first of these payoffs, so it's 11.

所以玩家 I 的收益，如果她选择顶部而玩家 II 选择中心，我们通过查看顶行和中间列来读取，玩家 I 的收益是这些收益中的第一个，所以它是 11。

Player II's payoff, from the same choices, top for Player I, center for Player II, again we go along the top row and the center column, but this time we choose Player II's payoff, which is the second payoff, so it's 3.

玩家 II 的收益，来自相同的选择，玩家 I 位于顶部，玩家 II 位于中间，我们再次沿着顶行和中间列进行操作，但这次我们选择玩家 II 的收益，这是第二个收益，所以它是 3。

So again, I'm hoping this is calming down the math phobics in the room.

再说一次，我希望这能平息房间里的数学恐惧症。

Now how do we think this game is going to be played?

现在我们认为这个游戏将如何进行？

It's not a particularly interesting game, but while we're here, why don't we just discuss it for a second.

这不是一个特别有趣的游戏，但既然我们在这里，我们为什么不讨论一下呢？

If our mike guys get a little bit ready here.

如果我们的麦克风人员在这里做好准备的话。

So how do we think this game should be played?

那么我们认为这个游戏应该怎么玩呢？

Well let's ask somebody at random perhaps.

好吧，也许我们随机询问某人。

Ale, do you want to ask this guy in the blue shirt here, does Player I have a dominated strategy?

Ale，你想问这个穿蓝色衬衫的家伙，玩家 I 有占优策略吗？

Student: No, Player I doesn't have a dominated strategy.

学生：不，玩家 I 没有占优策略。

For instance, if Player II picks left then Player I wants to pick bottom, but if Player II picks center, Player I wants to pick center.

例如，如果玩家 II 选择左侧，则玩家 I 想选择底部，但如果玩家 II 选择中心，则玩家 I 希望选择中心。

Professor Ben Polak: Good.

本·波拉克教授： 很好。

Excellent.

出色的。

Very good.

非常好。

I should have had you stand up.

我应该让你站起来。

I forgot that.

我忘记了。

Never mind.

没关系。

But that was very clear, thank you.

但这非常清楚，谢谢。

Was that loud enough so people could hear it?

声音足够大让人们都能听到吗？

Did people hear that?

人们听到了吗？

People in the back, did you hear it?

后面的人，你们听到了吗？

So even that wasn't loud enough okay, so we we really need to get people--That was very clear, very nice, but we need people to stand up and shout, or these people at the back can't hear.

所以即使声音还不够大，好吧，所以我们真的需要有人——这非常清楚，非常好，但我们需要人们站起来喊，否则后面的人听不到。

So your name is?

那么你的名字是？

Professor Ben Polak: What Patrick said was: no, Player I does not have a dominated strategy.

Ben Polak 教授：Patrick 说的是：不，玩家 I 没有占优策略。

Top is better than bottom against left -- sorry, bottom is better than top against left because 6 is bigger than 5, but top is better than bottom against center because 11 is bigger than 0.

顶部对左边比底部好——抱歉，底部比顶部对左好，因为 6 比 5 大，但顶部比底部对中心好，因为 11 比 0 大。

Everyone see that?

大家看到了吗？

So it's not the case that top always beats--it's not the case that top always does better than bottom, or that bottom always does better than top.

所以，上层并不总是胜过上层，也不是上层总是比下层做得更好，或者下层总是比上层做得更好。

What about, raise hands this time, what about Player II?

那么，这次举手，玩家二号呢？

Does Player II have a dominated strategy?

玩家 II 有占优策略吗？

Everyone's keeping their hands firmly down so as not to get spotted here.

每个人都把手压得很低，以免被发现。

Ale, can we try this guy in white?

艾尔，我们可以试试这个穿白衣服的家伙吗？

Do you want to stand up and wait until Ale gets there, and really yell it out now.

你想站起来等艾尔到达那里，现在就大声喊出来吗？

Student: I believe right is a dominated strategy because if Player I chooses top, then Player II will choose center, and if-- I'm getting confused now, it looks better on my paper.

学生：我相信右是占优策略，因为如果玩家 I 选择顶部，那么玩家 II 将选择中心，如果 - 我现在很困惑，它在我的纸上看起来更好。

But yeah, right is never the best choice.

但是，是的，正确的选择永远不是最好的选择。

Professor Ben Polak: Okay, good.

本·波拉克教授：好的，很好。

Let's be a little bit careful here.

让我们在这里小心一点。

So your name is?

那么你的名字是？

Professor Ben Polak: So Thomas said something which was true, but it doesn't quite match with the definition of a dominated strategy.

本·波拉克教授：托马斯说的是真的，但它与占优策略的定义不太相符。

What Thomas said was, right is never a best choice, that's true.

托马斯说的是，正确永远不是最好的选择，这是事实。

But to be a dominated strategy we need something else.

但要成为主导策略，我们还需要其他东西。

We need that there's another strategy of Player II that always does better.

我们需要玩家 II 有另一种总是做得更好的策略。

That turns out also to be true in this case, but let's just see.

在本例中事实证明也是如此，但让我们看看。

So in this particular game, I claim that center dominates right.

所以在这场特殊的比赛中，我认为中锋在右翼占主导地位。

So let's just see that.

让我们看看吧。

If Player I chose top, center yields 3, right yields 0: 3 is bigger than 0.

如果玩家 I 选择顶部，则中心产生 3，右侧产生 0：3 大于 0。

And if Player I chooses bottom, then center yields 2, right yields 0: 2 is bigger than 0 again.

如果玩家 I 选择底部，则中心产生 2，右侧产生 0：2 又大于 0。

So in this game, center strictly dominates right.

所以在这场比赛中，中锋严格控制右翼。

What you said was true, but I wanted something specifically about domination here.

你说的是真的，但我想要一些关于这里的统治的具体内容。

So what we know here, we know that Player II should not choose right.

所以我们在这里知道，我们知道玩家 II 不应该选择正确。

Now, in fact, that's as far as we can get with dominance arguments in this particular game, but nevertheless, let's just stick with it a second.

现在，事实上，这就是我们在这个特定游戏中所能得到的关于主导地位的论点，但尽管如此，让我们先坚持一下。

I gave you the definition of strict dominance last time and it's also in the handout.

上次我给了你们严格支配的定义，讲义里也有。

By the way, the handout on the web.

顺便说一句，网上的讲义。

But let me write that definition again, using or making use of the notation from the class.

但让我再次编写该定义，使用或利用类中的符号。

So definition so: Player i's strategy "s'i" is strictly dominated by Player i's strategy "si," and now we can use our notation, if "UI" from choosing "si," when other people choose "s-i," is strictly bigger than UI(s'i) when other people choose "s-i," and the key part of the definition is, for all "s-i."

所以定义如下：玩家 i 的策略“s'i”严格受玩家 i 策略“si”支配，现在我们可以使用我们的符号，如果其他人选择“s-i”时选择“si”的“UI”严格大于其他人选择“s-i”时的 UI(s'i)，并且定义的关键部分是，对于所有“s-i”。

So to say it in words, Player i's strategy "s'i" is strictly dominated by her strategy "si," if "si" always does strictly better -- always yields a higher payoff for Player i -- no matter what the other people do.

所以用语言来说，玩家 i 的策略“s'i”严格受她的策略“si”支配，如果“si”总是做得更好——总是为玩家 i 带来更高的回报——无论其他人做什么。

So this is the same definition we saw last time, just being a little bit more nerdy and putting in some notation.

这与我们上次看到的定义相同，只是更书呆子一点，并添加了一些符号。

People panicking about that, people look like deer in the headlamps yet?

人们对此感到恐慌，人们看起来像头灯里的鹿吗？

No, you look all right: all rightish.

不，你看上去还不错：还不错。

Let's have a look at another example.

让我们看另一个例子。

People okay, I can move this?

大家好，我可以移动这个吗？

All right, so it's a slightly more exciting example now.

好吧，现在这是一个更令人兴奋的例子。

So imagine the following example, an invader is thinking about invading a country, and there are two ways -- there are two passes if you like -- through which he can lead his army.

想象一下下面的例子，一个入侵者正在考虑入侵一个国家，他可以通过两种方式领导他的军队——如果你愿意的话，可以有两个通道。

You are the defender of this country and you have to decide which of these passes or which of these routes into the country, you're going to choose to defend.

你是这个国家的捍卫者，你必须决定要选择保卫哪些通行证或进入该国的路线。

And the catch is, you can only defend one of these two routes.

问题是，你只能防守这两条路线之一。

If you want a real world example of this, think about the third Century B.C., someone can correct me afterwards.

如果你想要一个现实世界的例子，想想公元前三世纪，事后有人可以纠正我。

I think it's the third Century B.C.

我认为这是公元前三世纪

when Hannibal is thinking of crossing the Alps.

当汉尼拔考虑穿越阿尔卑斯山时。

Not Hannibal Lecter: Hannibal the general in the third Century B.C..

不是汉尼拔·莱克特：汉尼拔是公元前三世纪的将军。

The one with the elephants.

那个有大象的。

Okay, so the key here is going to be that there are two passes.

好的，所以这里的关键是有两次通过。

One of these passes is a hard pass.

这些传球之一是硬传球。

It goes over the Alps.

它翻越阿尔卑斯山。

And the other one is an easy pass.

另一种是轻松传球。

It goes along the coast.

它沿着海岸行驶。

If the invader chooses the hard pass he will lose one battalion of his army simply in getting over the mountains, simply in going through the hard pass.

如果入侵者选择了艰难的关口，那么他仅仅翻过山脉，仅仅通过艰难的关口就会损失一个营的军队。

If he meets your army, whichever pass he chooses, if he meets your army defending a pass, then he'll lose another battalion.

如果他遇到你的军队，无论他选择哪个关口，如果他遇到你的军队守卫一个关口，那么他就会损失一个营。

I haven't given you--I've given you roughly the choices, the choice they're going to be for the attacker which pass to choose, and for the defender which pass to defend.

我没有给你——我已经给了你粗略的选择，进攻者将选择传球，防守者将选择传球。

But let's put down some payoffs so we can start talking about this.

但让我们先记下一些回报，这样我们就可以开始讨论这个问题了。

So in this game, the payoffs for this game are going to be as follows.

所以在这个游戏中，这个游戏的收益如下。

It's a simple two by two game.

这是一个简单的二对二游戏。

This is going to be the attacker, this is Hannibal, and this is going to be the defender, (and I've forgotten which general was defending and someone's about to tell me that).

这将是攻击者，这是汉尼拔，这将是防御者，（我忘记了哪位将军在防御，有人将要告诉我这一点）。

And there are two passes you could defend: the easy pass or the hard pass.

您可以防守两种传球：简单传球或困难传球。

And there's two you could use to attack through, easy or hard.

您可以使用两种方式进行攻击，无论是简单的还是困难的。

(Again, easy pass here just means no mountains, we're not talking about something on the New Jersey Turnpike.)

（再次强调，这里的轻松通过仅意味着没有山脉，我们不是在谈论新泽西收费公路上的事情。）

So the payoffs here are as follows, and I'll explain them in a second.

所以这里的回报如下，我稍后会解释它们。

So his payoff, the attacker's payoff, is how many battalions does he get to bring into your country?

所以他的回报，攻击者的回报，是他能带多少个营进入你的国家？

He only has two to start with and for you, it's how many battalions of his get destroyed?

他一开始只有两个营，对你来说，他有多少个营被摧毁？

So just to give an example, if he goes through the hard pass and you defend the hard pass, he loses one of those battalions going over the mountains and the other one because he meets you.

举个例子，如果他通过了硬山口，而你防守了硬山口，他就会失去一个翻山营和另一个营，因为他遇到了你。

So he has none left and you've managed to destroy two of them.

所以他已经没有了，而你已经成功摧毁了其中两个。

Conversely, if he goes on the hard pass and you defend the easy pass, he's going to lose one of those battalions.

相反，如果他进行硬传球，而你防守轻松传球，他就会失去其中一个营。

He'll have one left.

他还剩一张。

He lost it in the mountains.

他在山里把它弄丢了。

But that's the only one he's going to lose because you were defending the wrong pass.

但这是他唯一会输掉的一球，因为你防守了错误的传球。

Everyone understand the payoffs of this game?

大家都明白这个游戏的收益了吗？

So now imagine yourself as a Roman general.

现在想象自己是一名罗马将军。

This is going to be a little bit of a stretch for imagination, but imagination yourself as a Roman general, and let's figure out what you're going to do.

这可能有点超出你的想象力，但想象一下你自己是一位罗马将军，让我们看看你要做什么。

You're the defender.

你是后卫。

What are you going to do?

你会怎样做？

So let's have a show of hands.

让我们举手吧。

How many of you think you should defend the easy pass?

你们中有多少人认为您应该防守轻松传球？

Raise your hands, let's raise your hands so Jude can see them.

举起你们的手，让我们举起你们的手，让裘德能够看到它们。

Keep them up.

让他们保持下去。

Wave them in the air with a bit of motion.

轻轻地在空中挥动它们。

Wave them in the air.

在空中挥舞它们。

We should get you flags okay, because these are the Romans defending the easy pass.

我们应该给你旗子，因为这些是罗马人在防守简单的传球。

And how many of you think you're going to defend the hard pass?

你们中有多少人认为自己会捍卫硬通行证？

We have a huge number of people who don't want to be Roman generals here.

我们这里有很多人不想成为罗马将军。

Let's try it again, no abstentions, right?

我们再试一次，没有人弃权吧？

I'm not going to penalize you for giving the wrong answer.

我不会因为你给出错误答案而惩罚你。

So how many of you think you're going to defend the easy pass?

那么你们中有多少人认为自己会捍卫轻松传球呢？

Raise your hands again.

再次举起双手。

And how many think you're going to defend the hard pass?

有多少人认为你会捍卫硬通行证？

So we have a majority choosing easy pass -- had a large majority.

所以我们有大多数人选择轻松通行证——有绝大多数。

So what's going on here?

那么这是怎么回事呢？

Is it the case that defending the easy pass dominates defending the hard pass?

是不是防守易传优于防守硬传？

Is that the case?

是这样吗？

Is it the case that defending the easy pass dominates defending the hard pass?

是不是防守易传优于防守硬传？

You can shout out.

你可以大声喊出来。

No, it's not.

不，不是。

In fact, we could check that if the attacker attacks through the easy pass, not surprisingly, you do better if you defend the easy pass than the hard pass: 1 versus 0.

事实上，我们可以检查一下，如果攻击者通过轻松传球进攻，毫不奇怪，如果你防守轻松传球，你会比防守硬传球做得更好：1 比 0。

But if the attacker was to attack through the hard pass, again not surprisingly, you do better if you defend the hard pass than the easy pass.

但如果进攻者要通过硬传球进攻，同样毫不奇怪，如果你防守硬传球，你会比防守简单传球做得更好。

So that's not an unintuitive finding.

所以这并不是一个不直观的发现。

It isn't the case that defending easy dominates defending hard.

并非易守胜过难守。

You just want to match with the attacker.

你只想与攻击者匹配。

Nevertheless, almost all of you chose easy.

尽管如此，几乎所有人都选择了简单。

What's going on?

这是怎么回事？

Can someone tell me what's going on?

有人能告诉我发生了什么事吗？

Let's get the mikes going a second.

让我们把麦克风调一下。

So can we catch the guy with the--can we catch this guy with the beard?

那么我们能抓住那个留着胡子的人吗？我们能抓住这个留胡子的人吗？

Just wait for the mike to get there.

只需等待麦克风到达即可。

If you could stand up: stand up and shout.

如果你能站起来：站起来喊。

There you go.

就这样吧。

Student: Because you want to minimize the amount of enemy soldiers that reach Rome or whatever location it is.

学生：因为你想尽量减少到达罗马或任何地点的敌方士兵的数量。

Professor Ben Polak: You want to minimize the number of soldiers that reach Rome, that's true.

本·波拉克教授：你想尽量减少到达罗马的士兵数量，这是真的。

On the other hand, we've just argued that you don't have a dominant strategy here; it's not the case that easy dominates hard.

另一方面，我们刚刚指出，这里没有占优策略；并非易主难。

What else could be going on?

还有什么可能发生的事情？

While we've got you up, why don't we get the other guy who's got his hand up there in the middle.

当我们把你扶起来的时候，为什么我们不让另一个把手放在中间的人呢？

Again, stand up and shout in that mike.

再次站起来，用麦克风大声喊叫。

Point your face towards the mike.

将脸朝向麦克风。

Good.

好的。

Student: It seems as though while you don't have a dominating strategy, it seems like Hannibal is better off attacking through--It seems like he would attack through the easy pass.

学生：看起来好像虽然你没有主导策略，但汉尼拔似乎最好通过进攻——看起来他会通过轻松的传球进攻。

Professor Ben Polak: Good, why does it seem like that?

Ben Polak 教授：很好，为什么看起来是这样？

That's right, we're on the right lines now.

没错，我们现在走的是正确的路线。

Why does it seem like he's going to attack through the easy pass?

为什么他看起来像是要通过轻松传球进攻？

Student: Well if you're not defending the easy pass, he doesn't lose anyone, and if he attacks through the hard pass he's going to lose at least one battalion.

学生：好吧，如果你不防守简单的通道，他不会失去任何人，如果他通过困难的通道进攻，他将失去至少一个营。

Professor Ben Polak: So let's look at it from--Let's do the exercise--Let's do the second lesson I emphasized at the beginning.

本·波拉克教授：所以让我们从——让我们做练习——让我们做我一开始强调的第二课。

Let's put ourselves in Hannibal's shoes, they're probably boots or something.

让我们设身处地为汉尼拔着想，它们可能是靴子之类的。

Whatever you do when you're riding an elephant, whatever you wear.

无论你骑大象时做什么，无论你穿什么。

Let's put ourselves in Hannibal's shoes and try and figure out what Hannibal's going to do here.

让我们设身处地为汉尼拔着想，试着弄清楚汉尼拔在这里要做什么。

So it could be--From Hannibal's point of view he doesn't know which pass you're going to defend, but let's have a look at his payoffs.

所以可能是——从汉尼拔的角度来看，他不知道你要防守哪一个关口，但让我们看看他的回报。

If you were to defend the easy pass and he goes through the easy pass, he will get into your country with one battalion and that's the same as he would have got if he went through the hard pass.

如果你要防守简单的通道，而他通过了简单的通道，他将带着一个营进入你的国家，这与他通过困难的通道时得到的结果是一样的。

So if you defend the easy pass, from his point of view, it doesn't matter whether he chooses the easy pass and gets one in there or the hard pass, he gets one in there.

因此，如果你防守轻松传球，从他的角度来看，无论他选择轻松传球并进一球还是选择硬传球，他都会进一球。

But if you were to defend the hard pass, if you were to defend the mountains, then if he chooses the easy pass, he gets both battalions in and if he chooses the hard pass, he gets no battalions in.

但如果你要防守硬山口，如果你要防守山地，那么如果他选择易山口，他就会让两个营进入，如果他选择硬山口，他就不会进入任何营。

So in this case, easy is better.

所以在这种情况下，简单更好。

We have to be a little bit careful.

我们必须要小心一点。

It's not the case that for Hannibal, choosing the easy pass to attack through, strictly dominates choosing the hard pass, but it is the case that there's a weak notion of domination here.

对于汉尼拔来说，选择易传进攻并不是严格支配选择硬传，而是这里存在弱支配观念。

It is the case -- to introduce some jargon -- it is the case that the easy pass for the attacker, weakly dominates the hard pass for the attacker.

情况确实如此——引入一些术语——情况就是攻击者的简单传球弱于攻击者的硬传球。

What do I mean by weakly dominate?

我所说的弱支配是什么意思？

It means by choosing the easy pass, he does at least as well, and sometimes better, than he would have done had he chosen the hard pass.

这意味着通过选择轻松传球，他至少比选择困难传球做得更好，有时甚至更好。

So here we have a second definition, a new definition for today, and again we can use our jargon.

所以这里我们有第二个定义，今天的新定义，我们可以再次使用我们的行话。

Definition- Player i's strategy, "s'i" is weakly dominated by her strategy "si" if--now we're going to take advantage of our notation--if Player i's payoff from choosing "si" against "s-i" is always as big as or equal, to her payoff from choosing "s'i" against "s-i" and this has to be true for all things that anyone else could do.

定义 - 玩家 i 的策略“s'i”弱于她的策略“si”，如果——现在我们要利用我们的符号——如果玩家 i 选择“si”对抗“s-i”的收益始终与她选择“s'i”对抗“s-i”的收益一样大或相等，并且对于其他任何人都可以做的所有事情都必须如此。

And in addition, Player i's payoff from choosing "si" against "s-i" is strictly better than her payoff from choosing "s'i" against "s-i," for at least one thing that everyone else could do.

此外，对于至少一件其他人都可以做的事情，玩家 i 选择“si”而不是“s-i”的回报严格优于她选择“s'i”而不是“s-i”的回报。

Just check, that exactly corresponds to the easy and hard thing we just had before.

检查一下，这与我们之前的简单和困难完全对应。

I'll say it again, Player i's strategy "s'i" is weakly dominated by her strategy "si" if she always does at least as well by choosing "si" than choosing "s'i" regardless of what everyone else does, and sometimes she does strictly better.

我再说一遍，玩家 i 的策略“s'i”弱于她的策略“si”，如果她总是选择“si”至少比选择“s'i”好，无论其他人做什么，有时她会做得更好。

It seems a pretty powerful lesson.

这似乎是一个非常有力的教训。

Just as we said you should never choose a strictly dominated strategy, you're probably never going to choose a weakly dominated strategy either, but it's a little more subtle.

正如我们所说，您永远不应该选择严格占优策略，您可能也永远不会选择弱占优策略，但它有点微妙。

Now that definition, if you're worried about what I've written down here and you want to see it in words, on the handout I've already put on the web that has the summary of the first class, I included this definition in words as well.

现在这个定义，如果你担心我在这里写下的内容，并且想以文字形式看到它，在我已经放在网上的包含第一堂课摘要的讲义上，我也以文字形式包含了这个定义。

So compare the definition of words with what's written here in the nerdy notation on the board.

因此，将单词的定义与黑板上书呆子符号中写的内容进行比较。

Now since we think that Hannibal, the attacker, is not going to play a weakly dominated strategy, we think Hannibal is not going to choose the hard pass.

现在既然我们认为攻击者汉尼拔不会采取弱占优策略，我们认为汉尼拔不会选择硬传。

He's going to attack on the easy pass.

他将通过轻松传球发起进攻。

And given that, what should we defend?

鉴于此，我们应该捍卫什么？

We should defend easy which is what most of you chose.

我们应该捍卫简单，这是你们大多数人选择的。

So be honest now: was that why most of you chose easy?

所以现在说实话：这就是你们大多数人选择简单的原因吗？

Yeah, probably was.

是的，可能是。

We're able to read this.

我们能够读到这个。

So, by putting ourselves in Hannibal's shoes, we could figure out that his hard attack strategy was weakly dominated.

因此，通过设身处地为汉尼拔着想，我们可以发现他的强攻策略是弱占优的。

He's going to choose easy, so we should defend easy.

他会选择easy，所以我们应该防守easy。

Having said that of course, Hannibal went through the mountains which kind of screws up the lesson, but too late now.

当然，话虽如此，汉尼拔穿过山脉，这有点搞砸了课程，但现在已经太晚了。

Now then, I promised you we'd get back to the game from last time.

那么现在，我向你保证我们会回到上次的比赛。

So where have we got to so far in this class.

那么到目前为止，我们在这堂课上已经到了什么程度了？

We know from last time that you should not choose a dominated strategy, and we also know we probably aren't going to choose a weakly dominated strategy, and we also know that you should put yourself in other people's shoes and figure out that they're not going to play strongly or strictly or weakly dominated strategies.

从上次我们就知道你不应该选择占优策略，我们也知道我们可能不会选择弱占优策略，我们也知道你应该设身处地为别人着想，弄清楚他们不会玩强或严格或弱占优策略。

That seems a pretty good way to predict how other people are going to play.

这似乎是预测其他人将如何玩游戏的好方法。

So let's take those ideas and go back to the numbers game from last time.

因此，让我们带着这些想法回到上次的数字游戏。

Now before I do that, I don't need the people at home to see this, but how many of you were here last time?

在我这样做之前，我不需要家里的人看到这个，但是上次你们中有多少人在这里？

How many of you were not.

你们中有多少人不是。

I asked the wrong question.

我问错了问题。

How many of you were not here last time?

上次你们有多少人没来？

So we handed out again that game.

所以我们再次分发了那个游戏。

We handed out again the game with the numbers, but just in case, let me just read out the game you played.

我们再次分发了带有数字的游戏，但为了以防万一，让我读出你玩的游戏。

This was the game you played.

这就是你玩的游戏。

"Without showing your neighbor what you are doing, put it in the box below a whole number between 1 and a 100.

“在不向邻居展示你在做什么的情况下，将其放入 1 到 100 之间的整数下方的方框中。

We will (and in fact have) calculated the average number chosen in the class and the winner of this game is the person who gets closest to two-thirds times the average number.

我们将（事实上已经）计算出班级中选择的平均数字，而这场游戏的获胜者是最接近平均数字三分之二的人。

They will win five dollars minus the difference in pennies."

他们将赢得五美元减去便士差额。”

So everybody filled that in last time and I have their choices here.

所以每个人都在上次填写了这个，我在这里有他们的选择。

So before we reveal who won, let's discuss this a little bit.

因此，在我们揭晓谁获胜之前，让我们先讨论一下这个问题。

Let me come down hazardously off this stage, and figure out--Let's get the mics up a bit for a second, we can get some mics ready.

让我冒险离开这个舞台，然后想办法——让我们把麦克风调高一点，我们可以准备一些麦克风。

So let me find out from people here and see what people did a second.

因此，让我从这里的人们那里了解一下，看看人们做了什么。

You can be honest here since I've got everything in front of me.

你可以在这里说实话，因为我已经把一切都摆在我面前了。

So how many of you chose some number like 32, 33, 34?

那么你们中有多少人选择了像 32、33、34 这样的数字？

One hand.

一只手。

Actually I can tell you, nine of you did.

事实上我可以告诉你，你们中有九个人做到了。

So should I read out the names?

那么我应该读出这些名字吗？

Should I embarrass people?

我应该让别人难堪吗？

We've got Lynette, Lukucin, we've Kristin, Bargeon; there's nine of you here.

我们有 Lynette、Lukucin、Kristin、Bargeon；这里有你们九个人。

Let's try it again.How many of you chose numbers between 32 and 34?

让我们再试一次。你们中有多少人选择了 32 到 34 之间的数字？

Okay, a good number of you.

好吧，你们中有很多人。

Now we're seeing some hands up.

现在我们看到一些人举手了。

So keep your hands up a second, those people.

所以，那些人，请举起手来。

So let me ask people why?

那么让我问问人们为什么？

Can you get your hand into the guy?

你能把手伸进那个家伙吗？

What's your name?

你叫什么名字？

If we can get him to stand up.

如果我们能让他站起来的话

Stand up a second and shout out to the class.

站起来并向全班同学大声喊叫。

What's your name?

你叫什么名字？

Professor Ben Polak: Chris, you're on this list somewhere.

本·波拉克教授：克里斯，你在这个名单上的某个地方。

Maybe you're not on this list somewhere.

也许您不在这个列表中的某个地方。

Never mind, what did you choose?

没关系，你选择了什么？

Professor Ben Polak: Okay 30, so that's pretty close.

Ben Polak 教授：好的，30，非常接近了。

So why did you choose 30?

那你为什么选择30呢？

Student: Because I thought everyone was going to be around like the 45 range because 66 is two-thirds, or right around of 100, and they were going to go two-thirds less than that and I did one less than that one.

学生：因为我认为每个人都会在 45 范围内，因为 66 是三分之二，或者说是 100 左右，他们会比这个少三分之二，而我比那个少了一个。

Professor Ben Polak: Okay, thank you.

本·波拉克教授：好的，谢谢。

Let's get one of the others.

我们去拿其他的一个吧。

There was another one in here.

这里还有另外一个。

Can you just raise your hands again, the people who were around 33, 34.

那些33岁、34岁左右的人，你们能再举手吗？

There's somebody in here.

这里有人。

Can we get you to stand up (and you're between mikes).

我们可以让你站起来吗（你在麦克风之间）。

So that would be--Yep, go ahead.

那么那就是——是的，继续吧。

Shout it out.

大声喊出来。

What's your name first of all?

首先你叫什么名字？

Professor Ben Polak: Ryan, I must have you here as well, never mind.

Ben Polak 教授：Ryan，我也必须让你在这里，没关系。

What did you choose?

你选择了什么？

Professor Ben Polak: You are Ryan Lowe, okay.

Ben Polak 教授：你是 Ryan Lowe，好吧。

Good, go ahead.

好，继续吧。

Student: I thought similar to Chris actually and I also thought that if we got two-thirds and everyone was choosing numbers in between 1 and 100 ends up with 33, would be around the number (indiscernible).

学生：我的想法实际上与 Chris 类似，我还认为，如果我们得到三分之二，并且每个人都选择 1 到 100 之间的数字，最终得到 33，那么就会在这个数字附近（难以辨认）。

Professor Ben Polak: So just to repeat the argument that we just heard.

本·波拉克教授：重复一下我们刚才听到的论点。

Again, you have to shout it out more because I'm guessing people didn't hear that in room.

再说一次，你必须大声喊出来，因为我猜人们在房间里没有听到。

So I'll just repeat it to make sure everyone hears it.

所以我会重复一遍，以确保每个人都听到。

A reason for choosing a number like 33 might go as follows.

选择 33 这样的数字的原因可能如下。

If people in the room choose randomly between 1 and 100, then the average is going to be around 50 say and two-thirds of 50 is around 33, 33 1/3 actually.

如果房间里的人在 1 到 100 之间随机选择，那么平均值将约为 50，而 50 的三分之二实际上约为 33，即 33 1/3。

So that's a pretty good piece of reasoning.

所以这是一个很好的推理。

What's wrong with that reasoning?

这种推理有什么问题吗？

What's wrong with that?

这有什么问题吗？

Can we get the guy, the woman in the striped shirt here, sorry.

我们能找到那个穿条纹衬衫的男人吗？抱歉。

We haven't had a woman for a while, so let's have a woman.

我们已经有一段时间没有女人了，所以我们要有一个女人吧。

Thank you.

谢谢。

Student: That even if everyone else had the same reasoning as you, it's still going to be way too high.

学生：即使其他人都和你有同样的推理，它仍然太高了。

Professor Ben Polak: So in particular, if everyone else had the same reasoning as you, it's going to be way too high.

本·波拉克教授：所以特别是，如果其他人都和你有同样的推理，那就太高了。

So if everyone else reasons that way then everyone in the room would choose a number like 33 or 34, and in that case, the average would be what?

因此，如果其他人都这样推理，那么房间里的每个人都会选择 33 或 34 这样的数字，在这种情况下，平均值是多少？

Sorry, that two-thirds of the average would be what?

抱歉，平均值的三分之二是什么？

Something like 22.

大约22。

So the flaw in the argument that Chris and Ryan had -- it isn't a bad argument, it's a good starting point -- but the flaw in the argument, the mistake in the argument was the first sentence in the argument.

所以克里斯和瑞安论证中的缺陷——这不是一个糟糕的论证，这是一个很好的起点——但是论证中的缺陷、论证中的错误是论证中的第一句话。

The first sentence in the argument was, if the people in the room choose random, then they will choose around 50.

争论的第一句话是，如果房间里的人随机选择，那么他们会选择 50 左右。

That's true.

确实如此。

The problem is that people in the room aren't going to choose at random.

问题是房间里的人不会随机选择。

Look around the room a second.

环视一下房间。

Look around yourselves.

看看你自己的周围。

Do any of you look like a random number generator?

你们中有人看起来像随机数生成器吗？

Actually, from here I can see some of the people, but I'm not going to put.

事实上，从这里我可以看到一些人，但我不会放。

Actually looking at some of your answers maybe some of you are.

实际上看看你们的一些答案也许你们中的一些人是。

On the whole, Yale students are not random number generators.

总体而言，耶鲁大学的学生并不是随机数生成器。

They're trying to win the game.

他们正在努力赢得比赛。

So they're unlikely to choose numbers at random.

所以他们不太可能随机选择数字。

As a further argument, if in fact everyone thought that way, and if you figured out everyone was going to think that way, then you would expect everyone to choose a number like 33 and in that case you should choose a number like 22.

作为进一步的论证，如果事实上每个人都这么想，并且如果你发现每个人都会这么想，那么你会期望每个人都选择像 33 这样的数字，在这种情况下你应该选择像 22 这样的数字。

How many of you, raise your hands a second.

你们中有多少人请举手。

How many of you chose numbers in the range 21 through 23?

你们中有多少人选择了 21 到 23 范围内的数字？

There's way more of you than that.

你的数量远不止这些。

I'll start reading you out as well.

我也会开始给你读。

Actually about twelve of you, raise your hands.

实际上大约有十二个人，请举手。

There should be twelve hands going up somewhere.

某处应该有十二只手举起来。

There's two, three hands going up, four, five hands going up.

有两、三只手举起，四、五只手举起。

There's actually 12 people who chose exactly 22, so considerably more if include 23 and 21.

实际上有 12 个人选择了 22 个，所以如果包括 23 和 21 的话，数量会多得多。

So those people, I'm guessing, were thinking this way, is that right?

所以我猜那些人是这样想的，对吗？

Let me get one of my 22's up again.

让我再次拿起我的 22 号球衣。

Here's a 22.

这是22。

You want to get this guy?

你想得到这个人吗？

What's your name sir?

先生，你叫什么名字？

Stand up and shout.

站起来大声喊。

Student: I chose 22 because I thought that most people would play the game dividing by two-thirds a couple of times, and give numbers averaging around the low 30's.

学生：我选择 22 是因为我认为大多数人都会玩几次除以三分之二的游戏，并给出平均在 30 左右的数字。

Professor Ben Polak: So if you think people are going to play a particular way, in particular if you think people are going to choose the strategy of Ryan and Chris, and choose around 33, then 22 seems a great answer.

Ben Polak 教授：所以，如果你认为人们会以某种特定的方式进行游戏，特别是如果你认为人们会选择 Ryan 和 Chris 的策略，并选择 33 左右，那么 22 似乎是一个很好的答案。

But you underestimate your Yale colleagues.

但你低估了你的耶鲁同事。

In fact, 22 was way too high.

事实上，22 太高了。

Now, again, let's just iterate the point here.

现在，让我们再次重申一下这一点。

Let me just repeat the point here.

让我在这里重复一下这一点。

The point here is when you're playing a game, you want to think about what other people are trying to do, to try and predict what they're trying to do, and it's not necessarily a great starting point to assume that the people around you are random number generators.

这里的要点是，当您玩游戏时，您想要考虑其他人正在尝试做什么，尝试并预测他们正在尝试做什么，并且假设您周围的人是随机数生成器并不一定是一个很好的起点。

They have aims- trying to win, and they have strategies too.

他们有目标——努力获胜，他们也有策略。

Let me take this back to the board a second.

让我再把这个问题带回到董事会。

So, in particular, are there any strategies here we can really rule out?

那么，具体来说，是否有我们可以真正排除的策略？

We said already people are not random.

我们已经说过，人不是随机的。

Are there any choices we can just rule out?

有什么选择是我们可以排除的吗？

We know people are not going to choose those choices.

我们知道人们不会做出这些选择。

Let's have someone here.

我们这里有人吧。

Can we have the guy in green?

我们可以请那个穿绿衣服的人吗？

Wait for Ale, there we go.

等一下阿雷，我们就出发了。

Good.

好的。

Stand up.

起来。

Give me your name.

告诉我你的名字。

Professor Ben Polak: No one is going to choose a number over 50.

Ben Polak 教授：没有人会选择超过 50 的数字。

Okay, I was going--okay that's fair enough.

好吧，我要去——好吧，这很公平。

Some people did.

有些人做到了。

That's fair enough.

这很公平。

I was thinking of something a little bit less, that's fine.

我想少一点，没关系。

I was thinking of something a little bit less ambitious.

我正在考虑一些不那么雄心勃勃的事情。

Somebody said 66.

有人说66。

So let's start analyzing this.

那么我们就开始分析这个吧。

So, in particular, there's something about these strategy choices that are greater than 67 at any rate.

因此，特别是，这些策略选择无论如何都大于 67。

Certainly, I mean 66 let's go up a little bit, so these numbers bigger than 67.

当然，我的意思是 66，让我们增加一点，所以这些数字比 67 更大。

What's wrong with numbers bigger than 67?

大于 67 的数字有什么问题？

What's wrong with--Raise your hands if you have answer.

有什么问题吗——有答案的请举手。

What's wrong?

怎么了？

Can we get the guy in red who's right close to the mike?

我们能找到那个靠近麦克风的红衣人吗？

Stand up, give me your name.

站起来，告诉我你的名字。

Stand up.

起来。

Shout it out to the crowd.

向众人大声喊出来。

Professor Ben Polak: Good, so even if everyone in the number--everyone in the room didn't choose randomly but they all chose a 100, a very unlikely circumstance, but even if everyone had chosen 100, the highest, the average, sorry, the highest two-thirds of the average could possibly be is 66 2/3, hence 67 would be a pretty good choice in that case.

Ben Polak 教授：很好，所以即使房间里的每个人都不是随机选择的，而是都选择了 100，这是一种非常不可能的情况，但即使每个人都选择了 100，最高的平均值，抱歉，平均值的最高三分之二可能是 66 2/3，因此在这种情况下 67 将是一个相当不错的选择。

So numbers bigger than 67 seem pretty crazy choices, but crazy isn't the word I'm looking for here.

因此，大于 67 的数字似乎是相当疯狂的选择，但“疯狂”并不是我在这里要找的词。

What can we say about those choices, those strategies 67 and above, bigger than 67, 68 and above?

对于那些选择、那些 67 及以上、大于 67、68 及以上的策略，我们能说些什么呢？

What can we say about those choices?

对于这些选择我们能说些什么呢？

Somebody right behind you, the woman right behind you, shout it out.

你身后的某个人，你身后的那个女人，大声喊出来。

Professor Ben Polak: They have no payoffs.

本·波拉克教授：他们没有任何回报。

What's the jargon here?

这里的行话是什么？

Let's use our jargon.

让我们使用我们的行话。

Somebody shout it out, what's the jargon about that?

有人喊出来，这是什么行话？

They're dominated.

他们被统治了。

So these strategies are dominated.

所以这些策略占主导地位。

Actually, they're only weakly dominated but that's okay, they're certainly dominated.

事实上，他们只是弱占主导地位，但没关系，他们确实占主导地位。

In particular, a strategy like 80 is dominated by choosing 67.

特别是像80这样的策略，选择67占主导地位。

You will always get a higher payoff from choosing 67, at least as high and sometimes higher, than the payoff you would have got, had you chosen 80, no matter what else happened in the room.

选择 67 时你总会获得更高的回报，至少与选择 80 时的回报一样高，有时甚至更高，无论房间里发生了什么。

So these strategies are dominated.

所以这些策略占主导地位。

We know, from the very first lesson of the class last time, that no one should choose these strategies.

我们从上次课程的第一课就知道，任何人都不应该选择这些策略。

They're dominated strategies.

它们是占优策略。

So did anyone choose strategies bigger than 67?

那么有人选择大于 67 的策略吗？

Okay, I'm not going to read out names here, but, turns out four of you did.

好吧，我不会在这里读出名字，但是，事实证明你们四个都读出了。

I'm not going to make you wave your--okay.

我不会让你挥手——好吧。

So okay, for the four of you who did, never mind, but …

好吧，对于你们四个人来说，没关系，但是……

well mind actually, yeah.

其实介意吧，是的。

So once we've eliminated the possibility that anyone in the room is going to choose a strategy bigger than 67, it's as if those numbers 68 through 100 are irrelevant.

因此，一旦我们消除了房间里的任何人都会选择大于 67 的策略的可能性，就好像 68 到 100 之间的数字是无关紧要的。

It's really as if the game is being played where the only choices available on the table are 1 through 67.

就好像在玩游戏时，桌面上唯一可用的选择是 1 到 67。

Is that right?

是这样吗？

We know no one's going to choose 68 and above, so we can just forget them.

我们知道没有人会选择 68 及以上，所以我们可以忘记他们。

We can delete those strategies and once we delete those strategies, all that's left are choices 1 through 67.

我们可以删除这些策略，一旦删除这些策略，剩下的就是选项 1 到 67。

So can somebody help me out now?

那么现在有人可以帮我吗？

What can I conclude, now I've concluded that the strategies 68 through 100 essentially don't exist or have been deleted.

我能得出什么结论，现在我得出的结论是，策略 68 到 100 基本上不存在或已被删除。

What can I conclude?

我能得出什么结论？

Let me see if I can get a mike in here.

让我看看这里是否可以拿个麦克风。

Stand up and wait for the mike.

站起来等待麦克风。

And here comes the mike.

麦克风来了。

Good.

好的。

Shout out.

喊出来。

Student: That all strategies 45 and above are hence also ruled out.

学生：因此，45 及以上的所有策略也被排除。

Professor Ben Polak: So Henry is saying once we've figured out that no one should choose a strategy bigger than 67, then we can go another step and say, if those strategies never existed, then the same argument rules out -- or a similar argument rules out -- strategies bigger than 45.

Ben Polak 教授：亨利的意思是，一旦我们发现没有人应该选择大于 67 的策略，那么我们就可以进一步说，如果这些策略从未存在过，那么同样的论点就排除了——或者类似的论点排除了——大于 45 的策略。

Let's be careful here.

让我们小心点。

The strategies that are less than 67 but bigger than 45, I think these strategies are not, they're not dominated strategies in the original game.

小于67但大于45的策略，我认为这些策略不是，它们不是原游戏中的占优策略。

In particular, we just argued that if everyone in the room chose a 100, then 67 would be a winning strategy.

特别是，我们刚才认为，如果房间里的每个人都选择 100，那么 67 将是一个获胜策略。

So it's not the case that the strategies between 45 and 67 are dominated strategies.

所以45到67之间的策略并不是占优策略。

But it is the case that they're dominated once we delete the dominated strategies: once we delete 67 and above.

但情况是，一旦我们删除了占主导地位的策略，它们就会占主导地位：一旦我们删除 67 及以上。

So these strategies -- let's be careful here with the word weakly here -- these strategies are not weakly dominated in the original game.

所以这些策略——让我们小心这里的“弱”这个词——这些策略在原始游戏中并不是弱占主导地位的。

But they are dominated -- they're weakly dominated -- once we delete 68 through 100.

但一旦我们删除 68 到 100，它们就处于主导地位——它们处于弱主导地位。

So all of the strategies 45 through 67, are gone now.

因此，所有策略 45 到 67 现在都消失了。

So okay, let's have a look.

那么好吧，让我们来看看。

Did anyone choose -- raise your hands, Be brave here.

有没有人选择——举起手来，勇敢一点。

Did anyone choose a strategy between 45 and 67?

有人选择45到67之间的策略吗？

Or between 46 and 67?

或者46到67之间？

No one's raising their hand, but I know some of you did because I got it in front of me, at least four of you did and I won't read out those names yet, but I might read them out next time.

没有人举手，但我知道你们中的一些人举手了，因为我把它放在我面前，至少有四个人举手了，我还不会读出这些名字，但下次我可能会读出它们。

So four more people chose those strategies.

于是又有四个人选择了这些策略。

Now notice, there's a different part of this, this argument.

现在请注意，这个论证有不同的部分。

The argument that eliminates strategies 67 and above, or 68 upwards, that strategy just involves the first lesson of last time: do not choose a dominated strategy, admittedly weak here, but still.

消除策略67及以上，或68以上的论点，该策略只是涉及上次的第一课：不要选择占优策略，诚然这里很弱，但仍然如此。

But the second slice, strategies 45 through 67, getting rid of those strategies involves a little bit more.

但第二部分，即策略 45 到 67，摆脱这些策略涉及的内容更多。

You've got to put yourself in the shoes of your fellow classmen and figure out, that they're not going to choose 67 and above.

你必须设身处地为同学着想，并弄清楚他们不会选择 67 及以上。

So the first argument, that's a straight forward argument, the second argument says, I put myself in other peoples shoes, I realize they're not going to play a dominated strategy, and therefore, having realized they're not going to play a dominated strategy, I shouldn't play a strategy between 45 and 67.

所以第一个论点，这是一个直接的论点，第二个论点说，我把自己放在别人的立场上，我意识到他们不会采取主导策略，因此，在意识到他们不会采取主导策略后，我不应该采取 45 到 67 之间的策略。

So this argument is an 'in shoes' argument.

所以这个论点是一个“穿鞋”的论点。

Now what?

现在怎么办？

Where can we go now?

我们现在可以去哪里？

Yeah, so let's have the guy in the beard, but let the mike get to him.

是的，所以让我们让那个留胡子的家伙，但让麦克风传到他身上。

Yell out your name.

大声喊出你的名字。

Student: You just repeat the same reasoning again and again, and you eventually get down to 1.

学生：你只要一次又一次地重复同样的推理，最终就会得到1。

Professor Ben Polak: We'll do that but let's go one step at a time.

Ben Polak 教授：我们会这么做，但我们要一步一步来。

So now we've ruled out the possibility that anyone's going to choose a strategy 68 and above because they're weakly dominated, and we've ruled out the possibility that anyone's going to choose a strategy between 46 and 67, because those strategies are dominated, once we've ruled out the dominated strategies.

所以现在我们排除了任何人选择 68 及以上策略的可能性，因为它们是弱支配的，并且我们排除了任何人选择 46 到 67 之间策略的可能性，因为一旦我们排除了支配策略，这些策略就被支配了。

So we know no one's choosing any strategies above 45., It's as if the numbers 46 and above don't exist.

所以我们知道没有人选择45以上的任何策略。就好像46及以上的数字不存在一样。

So we know that the highest anyone could ever choose is 45, and two-thirds of 45 is roughly …

所以我们知道，任何人可以选择的最高值是 45，而 45 的三分之二大约是……

someone help me out here …

有人帮我这里…

30 right: roughly 30.

30 右：大约 30。

So we know that all the numbers between 45 and 30, these strategies were not dominated.

所以我们知道，45到30之间的所有数字，这些策略都不占优势。

And they weren't dominated even after deleting the dominated strategies.

即使删除了占优策略后，它们也不再占优。

But they are dominated once we deleted not just the dominated strategies, but also the strategies that were dominated once we deleted the dominated strategies.

但是，一旦我们不仅删除了被支配策略，而且删除了被支配策略后，它们也被支配了。

I'm not going to try and write that, but you should try and write it in your notes.

我不会尝试把它写下来，但你应该尝试把它写在你的笔记中。

So without writing that argument down in detail, notice that we can rule out the strategies 30 through 45, not by just examining our own payoffs; not just by putting ourselves in other people's shoes and realizing they're not going to choose a dominated strategy; but by putting our self in other people's shoes while they're putting themselves in someone else's shoes and figuring out what they're going to do.

因此，无需详细写下该论点，请注意，我们可以排除策略 30 到 45，而不仅仅是检查我们自己的收益；不仅仅是设身处地为别人着想，意识到他们不会选择占主导地位的策略；而是当他们设身处地为别人着想并弄清楚他们要做什么时，我们要设身处地为别人着想。

So this is an 'in shoes', be careful where we are here, this is an 'in shoes in shoes' argument, at which point you might want to invent the sock.

所以这是一个“鞋中鞋”，小心我们在这里，这是一个“鞋中鞋”的论点，此时你可能想发明袜子。

Now, where's this going?

现在，这是要去哪里？

We were told where it's going.

我们被告知它要去哪里。

We're able to rule out 68 and above.

我们可以排除 68 及以上的情况。

Then we were able to rule out 46 and above.

然后我们就可以排除 46 及以上的情况。

Now we're able to rule out 31 and above.

现在我们可以排除 31 及以上的情况了。

By the next slice down we'll be able to eliminate -- what is it -- about 20 and above, so 30 down to above 20, and this will be an 'in shoes, in shoes, in shoes'.

到下一个切片时，我们将能够消除——这是什么——大约 20 及以上，所以 30 下降到 20 以上，这将是“在鞋子里，在鞋子里，在鞋子里”。

These strategies aren't dominated, nor are they dominated once you delete the dominated strategies, nor once we dominated the strategies dominated once we've deleted the dominated strategies, but they are dominated once we delete the strategies that have been dominated in the--you get what I'm doing here.

这些策略不是占主导地位的，一旦你删除了占主导地位的策略，它们也不是占主导地位的，一旦我们删除了占主导地位的策略后，它们就不再占主导地位了，但是一旦我们删除了已经占主导地位的策略，它们就被占主导地位了——你明白我在这里所做的。

So where is this argument going to go?

那么这个争论将会走向何方呢？

Where's this argument going to go?

这场争论将走向何方？

It's going to go all the way down to 1: all the way down to 1.

它将一路下降到 1：一路下降到 1。

We could repeat this argument all the way down to 1.

我们可以重复这个论证直到 1。

Notice that once we've deleted the dominated strategies, you know I had said before about four people chose this strategy, and in here, about four people chose this strategy, but in this range 30 through 45, I had lots of people.

请注意，一旦我们删除了主导策略，你知道我之前说过大约有四个人选择了这个策略，在这里，大约有四个人选择了这个策略，但在 30 到 45 的范围内，我有很多人。

How many of you chose a number between 30 and 45?

你们中有多少人选择了 30 到 45 之间的数字？

Well more than that.

还不止这些。

I can guarantee you more than that chose a number between 30 and 45.

我可以向你保证，你会选择 30 到 45 之间的数字。

In fact, the people who chose where we started off 33 chose in that range.

事实上，选择我们起点的人 33 都选择了这个范围。

A lot more of you chose numbers between 20 and 30, so we're really getting into the meat of the distribution.

很多人选择了 20 到 30 之间的数字，所以我们真正进入了分布的核心。

But we're seeing that these are choices, that perhaps, are ruled out by this kind of reasoning.

但我们看到这些选择也许被这种推理所排除。

Now, I'm still not going to quite reveal yet who won.

现在，我仍然不打算透露谁赢了。

I want to take this just one step more abstract.

我想把这一点更抽象一些。

So I want to just discuss this a little bit more.

所以我想多讨论一下这个问题。

I want to discuss the consequence of rationality in playing games, slightly philosophical for a few minutes.

我想讨论玩游戏时理性的后果，稍微有点哲学性几分钟。

So I claim that if you are a rational player, by which I mean somebody who is trying to maximize their payoffs by their play of the game, that simply being rational, just being a rational player, rules out playing these dominated strategies.

所以我认为，如果你是一个理性的玩家，我的意思是那些试图通过玩游戏来最大化他们的收益的人，那么只要是理性的，只是作为一个理性的玩家，就排除了玩这些占主导地位的策略。

So the four of you who chose numbers bigger than 67, whose names I'm not going to read out, maybe they were making a mistake.

所以你们四个选择了大于 67 的数字，我不会读出他们的名字，也许他们犯了一个错误。

However, the next slice down requires more than just rationality.

然而，下一个切片需要的不仅仅是理性。

What else does it require?

它还需要什么？

Yes, can I get this guy again, sorry?

是的，我可以再去找这个人吗，抱歉？

Shout out your name again, I've forgotten it.

再喊一遍你的名字，我忘了。

Student: The assumption that your opponents are being rational as well.

学生：假设你的对手也是理性的。

Professor Ben Polak: Good.

本·波拉克教授： 很好。

To rule out the second slice, I need to be rational myself, and I need to know that others are rational.

为了排除第二部分，我自己需要保持理性，并且我需要知道其他人是理性的。

That's illegible, but what it says is rational and knowledge that other people are rational.

这是难以辨认的，但它所说的是理性的，并且知道其他人是理性的。

Now how about the next slice after that?

现在接下来的切片怎么样？

Well now I need to be rational, I need to know that other people are rational, and I need to know that other people know that other people are rational.

现在我需要保持理性，我需要知道其他人是理性的，我需要知道其他人知道其他人是理性的。

So to get this slice, this next slice here, I need rationality; as some of you know that's widely criticized in the social sciences these days.

因此，为了得到这片、下一片，我需要理性；正如你们中的一些人所知，这在当今社会科学界受到了广泛的批评。

Are we right to assume that people are rational?

我们假设人是理性的对吗？

To get this slice I need rationality, I need knowledge of rationality, let's call that KR and I need knowledge of knowledge of rationality.

为了得到这个切片，我需要理性，我需要理性知识，我们称之为 KR，我需要理性知识知识。

As I go down further, I'm going to need rationality, I need to know people are rational; I need to know that people know that people are rational, and I need to know that people know that people know that people are rational.

当我进一步深入时，我将需要理性，我需要知道人们是理性的；我需要知道人们知道人们是理性的，我需要知道人们知道人们知道人们是理性的。

Now let's just make this more concrete for you.

现在让我们为您讲得更具体一些。

These people, the four people who chose this, they made a mistake.

这些人，选择这个的四个人，他们犯了一个错误。

What about the four people who chose numbers between 45 and 67?

那么选择 45 到 67 之间数字的四个人呢？

What can we conclude about those people?

关于这些人我们能得出什么结论？

The people who chose between 45 and 67?

那些选择 45 和 67 之间的人？

Should I read out their names?

我应该读出他们的名字吗？

No, I won't, perhaps I better not.

不，我不会，也许我最好不要。

What can we conclude about these people?

关于这些人我们可以得出什么结论？

Yeah.

是的。

We're never going to get the mike to this -- try and get the mike in there.

我们永远不会把麦克风放到这里——尝试把麦克风放在那里。

Come forward as far as you can and then really shout, yep.

尽可能向前走，然后大声喊，是的。

Professor Ben Polak: Right, right.

本·波拉克教授：对，对。

It's not necessarily that the four people who chose between 46 and 67 are themselves "thick," it's that they think the rest of you are "thick." Down here, this doesn't require people to be thick, or to think the rest of you are thick, they're just people who think that you think, sorry, they're just people who think that you think that they're thick and so on.

选择46和67之间的四个人不一定是他们自己“厚”，而是他们认为你们其他人都是“厚”。在这里，这并不要求人们很厚，或者认为你们其他人很厚，他们只是认为你认为，对不起，他们只是认为你认为他们很厚等等的人。

But again, all the way to 1 we're going to need very, very many rounds of knowledge, of knowledge, of knowledge …

但同样，一直到 1，我们将需要非常非常多轮的知识、知识、知识……

of rationality.

的理性。

Does anyone know what we call it if we assume an infinite sequence of "I know that you know that I know that you know that I know that you know that I know that you know" something?

如果我们假设一个无限序列“我知道你知道我知道你知道我知道你知道我知道你知道”某事，有谁知道我们称之为什么？

What's the expression for that?

这句话的表达方式是什么？

Believe it or not, technical expression.

不管你信不信，技术表达。

The technical expression of that in philosophy is common knowledge, which I can never spell, so I'm going to wing it.

哲学中这一点的技术表达是常识，我永远无法拼写它，所以我将即兴发挥。

Common knowledge is: "I know something, you know it, you know that I know it, I know that you know it, I know that you know that I know it, etc., etc. etc.: an infinite sequence.

常识是：“我知道某事，你知道它，你知道我知道它，我知道你知道它，我知道你知道我知道它，等等，等等：一个无限的序列。

But if we had common knowledge of rationality in this class, then the optimal choice would have been 1.

但如果我们对此类的合理性有常识，那么最优选择就是 1。

How many of you chose 1?

有多少人选1？

Look around the room.

环视房间。

Let's just pan the room.

让我们平移房间。

Keep your hands up a second.

请举起双手。

How many of you chose 1?

有多少人选1？

So actually a lot of you chose 1.

所以实际上很多人都选择了1。

1 was the modal answer in this class.

1 是这堂课的模态答案。

A lot of you chose 1.

很多人都选择了1。

So those people did pretty well.

所以那些人做得很好。

They must have done--they must be thinking they're about to win…

他们一定已经这么做了——他们一定认为自己即将获胜……

but they didn't win.

但他们没有赢。

So it turns out that the average in this class, the average choice was about 13 1/3, which means two-thirds of the average was 9.

所以事实证明，这个班的平均选择大约是 13 1/3，这意味着平均数的三分之二是 9。

Two-thirds of the average was 9 and some of you chose 9, so if you are here, stand up.

三分之二的平均值是 9，你们中的一些人选择了 9，所以如果您在这里，请站起来。

The following people chose 9, that's not right, where are the people who chose 9?

下面的人选了9，不对，选9的人在哪里？

I've got them here somewhere?

我把它们放在某个地方了？

I'm sorry there's so many pages of people.

抱歉，人太多页了。

Here we go.

我们开始吧。

The following people chose 9.

下面的人选择了9。

So stand up if you're here and if you're that person's roommate if they're not here.

因此，如果您在这里，请站起来；如果您是该人的室友，如果他们不在这里，请站起来。

So Leesing Chang: is Leesing Chang here?

那么Leesing Chang：Leesing Chang 在吗？

Stand up if you're here.

如果你在这里，请站起来。

A G.

A.G.

Christopher Berrera: you can stand up, if you're here.

Christopher Berrera：如果你在的话，你可以站起来。

And William Fischel: are you here?

威廉·菲舍尔：你在吗？

I don't know if he is here.

我不知道他是否在这里。

Jed Glickstein: are you here?

杰德·格利克斯坦：你在吗？

Jed Glickstein: stand up if you're here.

Jed Glickstein：如果你在这儿，请站起来。

And Jeffrey Green: stand up if you're here.

杰弗里·格林：如果你在这儿，请站起来。

And Allison Hoyt: stand up if you're here.

艾莉森·霍伊特：如果你在这儿，请站起来。

No Allison Hoyt, okay.

没有艾莉森·霍伊特，好吧。

There's John Robinson.

那是约翰·罗宾逊。

All right so these people, stay up a second so the camera can see you.

好吧，这些人，请稍等一下，以便摄像机可以看到您。

There you go, all the way around.

就这样，一路走来。

Wave.

波。

Wave to mom at home.

在家向妈妈挥手。

Can we get a round of applause for our winners?

我们能为获奖者赢得热烈的掌声吗？

So Jude has trustworthily brought back the five dollars.

所以裘德可靠地带回了五美元。

I've got to focus for a second just to get it.

我必须集中注意力才能得到它。

Here is the five dollars, we're going to tear this into nine pieces, except I'd get arrested and deported if I did that, so we're going to find a way to break this into change later.

这是五美元，我们要把它撕成九块，除非我这样做会被捕并被驱逐出境，所以我们稍后会找到一种方法将其分解成零钱。

Come and claim it afterwards, but you're all entitled to whatever a ninth, whatever that fraction of five dollars is.

过后再来领取吧，但是你们都有权得到九分之一，无论五美元的零头是多少。

Okay, so why was it after all that work -- why was it that 1 wasn't the winning answer?

好吧，那么为什么在做了这么多工作之后——为什么 1 不是获胜的答案呢？

Why wasn't 1 the winning answer?

为什么 1 不是获胜答案？

Let's have someone we haven't had before.

让我们来找一个我们以前没有过的人吧。

Can we get the mike in way in the back there?

我们可以把麦克风放在后面吗？

Can we get the mike in there on the row you're on?

我们可以把麦克风放在你所在的那排吗？

See if you can point.

看看能不能指点一下。

Actually good.

其实很好。

Stand up.

站起来。

Shout.

喊叫。

Shout away.

喊走。

Student: 1 would have been the winning answer had everyone assumed that the average would have been constantly compounded down to 1, but since a couple of people chose the, I mean not incorrect answers, but the higher averages, then it was pushed up to 13.

学生：如果每个人都假设平均值会不断地复合到 1，那么 1 本来就是获胜答案，但由于有几个人选择了，我的意思不是错误的答案，而是更高的平均值，然后它被推高到 13。

Professor Ben Polak: Right, so to get all the way, -- good -- so to get all the way -- thank you -- So to get all the way to 1, we need a lot.

Ben Polak 教授： 是的，所以要一直达到 1，我们需要很多。

We need not just that you're all rational players, not just that you know each other's rational, but you know everyone else's rational.

我们不仅需要你们都是理性的参与者，不仅需要你们知道彼此的理性，而且还需要你们知道其他人的理性。

I mean I know you all know each other because you've met at Yale, but you also know each other well enough to know that not everyone in the room is rational, and you're pretty sure that not everyone knows that you're rational and so on and so forth.

我的意思是，我知道你们都认识彼此，因为你们在耶鲁大学见过面，但你们也足够了解彼此，知道并不是房间里的每个人都是理性的，而且你很确定不是每个人都知道你是理性的，等等。

It's asking a lot to get to 1 here, and in fact, we didn't get to 1.

在这里达到 1 要求很高，但事实上，我们没有达到 1。

In previous years we were even higher, so this was low this year.

往年我们甚至更高，所以今年很低。

In 2003, the average was eighteen and a half.

2003年，平均年龄为十八岁半。

And in 2004, it was twenty-one and a half.

2004年，这个数字是二十一岁半。

And in 2005, we had a class that didn't trust each other at all I guess, because the average was twenty-three.

我想，2005 年，我们班上的学生彼此之间根本不信任，因为平均人数是 23 人。

And this year, it was thirteen and a third.

今年，已经是十三又三分之一了。

We're getting better there I think.

我认为我们正在变得更好。

One nice thing, by the way -- this is just chance I think -- the median answer in the class was nine, which is spot on, so the median hit this bang on.

顺便说一句，一件好事——我认为这只是偶然——班上答案的中位数是九，这是正确的，所以中位数正好符合这个要求。

Now what I wanted you to do, is I want you all to play again.

现在我希望你们做的是，我希望你们都能再次参加比赛。

We haven't got time to do this properly, even though I've given you the sheets.

尽管我已经给了你床单，但我们没有时间正确地做这件事。

So write down -- don't tell this to your neighbors -- write down a number.

所以写下来——不要告诉你的邻居——写下一个数字。

Don't talk among yourselves that's cheating.

不要互相谈论欺骗的事情。

Write down a number.

写下一个数字。

If you haven't got a sheet in front of you, just write it on your notepad.

如果您面前没有纸，只需将其写在记事本上即可。

Write down a number.

写下一个数字。

Has everyone written down a number?

大家都写下数字了吗？

I'm going to do a show of hands now.

我现在要举手。

How many -- we'll get the camera on you -- how many of you chose a number higher than 67?

有多少人——我们会把摄像机对准你们——你们中有多少人选择了高于 67 的数字？

Oh there's some spoil makers in the class.

哦，班上有一些破坏者。

How many of you chose a number higher than 20?

你们中有多少人选择了大于 20 的数字？

How many of you chose a number higher than 10?

你们中有多少人选择了大于 10 的数字？

How many chose a number between 5 and 10?

有多少人选择了 5 到 10 之间的数字？

How many chose a number between 0 -- I'm sorry -- between 1 and 5?

有多少人选择了 0 之间的数字——抱歉——1 到 5 之间的数字？

How many of you, excluding the people who chose 1 last time, how many of you chose a number that was lower than the number you chose last time?

你们中有多少人，除了上次选择1的人之外，有多少人选择了比上次选择的数字更低的数字？

Now keep your hands up a second.

现在请举起手来。

So almost all of you came down.

所以你们几乎都下来了。

Why?

为什么？

Why are seeing this massive contraction?

为什么会出现如此大规模的收缩？

I'm guessing the average number in the class now is probably about 3 or 4, maybe even lower.

我猜现在班上的平均人数可能是三四个人左右，甚至可能更低。

Why are we seeing this massive contraction in the numbers being chosen?

为什么我们会看到被选择的人数大幅减少？

The woman in green, I've forgotten your name, I'm sorry?

绿衣女子，我忘记你的名字了，抱歉？

Student: Because we've just sat in lecture and you've told us we're not being rational if we pick a high number.

学生：因为我们刚刚听完讲座，你告诉我们，如果我们选择一个很高的数字，我们就不是理性的。

Professor Ben Polak: So part of it is, you yourselves have figured out, some of you, that you shouldn't choose a high number.

本·波拉克教授：所以部分原因是，你们中的一些人自己已经发现，你们不应该选择一个很高的数字。

What else though?

还有什么？

What else is going on here?

这里还发生了什么？

Let's get somebody.

我们找个人吧。

There's a guy waving an arm out there.

有一个人在那里挥舞着手臂。

Do you want to stand up behind the hat?

你想站在帽子后面吗？

You.

你。

Professor Ben Polak: It's true we've repeated it.

Ben Polak 教授：我们确实重复了这一点。

It's true we repeated it but what is it about repeating it?

确实我们重复了，但是重复又是什么呢？

What is it about talking about this game that makes a difference?

谈论这款游戏有何不同之处？

Let me hazard a guess here.

让我在这里大胆猜测一下。

I think what makes a difference is not only do you, yourselves, know better how to play this game now, but you also know that everybody around you knows better how to play the game.

我认为重要的不仅是你自己现在更好地知道如何玩这个游戏，而且你还知道你周围的每个人都更好地知道如何玩这个游戏。

Discussing this game raised not just each person's sophistication, but it raised what you know about other people's sophistication, and you know that other people now know that you understand how to play the game.

讨论这个游戏不仅提高了每个人的成熟度，还提高了你对其他人成熟度的了解，并且你知道其他人现在知道你了解如何玩这个游戏。

So the main lesson I want you to get from this is that not only did it matter that you need to put yourself in other people's shoes and think about what their payoffs are.

因此，我希望您从中得到的主要教训是，您不仅需要设身处地为他人着想，思考他们的回报是什么，这一点很重要。

You also need to put yourself into other people's shoes and think about how sophisticated are they at playing games.

你还需要设身处地为别人着想，想想他们玩游戏的水平有多高。

And you need to think about how sophisticated do they think you are at playing games.

你需要考虑一下他们认为你玩游戏有多熟练。

And you need to think about how sophisticated do they think that you think that they are at playing games and so on.

你需要考虑一下他们认为你认为他们在玩游戏等方面有多复杂。

This level of knowledge, these layers of knowledge, lead to very different play in the game.

这种知识水平，这些知识层次，导致了游戏中截然不同的玩法。

And to make this more concrete, if a firm is competing against a competitor it can be pretty sure, that competitor is a pretty sophisticated game player and knows that the firm is itself.

更具体地说，如果一家公司正在与竞争对手竞争，那么可以肯定的是，该竞争对手是一个非常老练的游戏玩家，并且知道该公司就是它自己。

If a firm is competing against a customer -- let's say for a non-prime loan -- perhaps that assumption is not quite so safe.

如果一家公司正在与客户竞争——比如说非优质贷款——也许这种假设就不那么安全了。

It matters in how we take games through to the real world, and we're going to see more of this as the term progresses.

重要的是我们如何将游戏带入现实世界，随着学期的进展，我们将会看到更多这样的情况。

Now I've got five minutes, do I have five minutes left?

现在我还有五分钟，还剩五分钟吗？

So I've got five minutes to take a little small aside here.

所以我有五分钟的时间在这里稍微讨论一下。

We've been talking about knowledge and about common knowledge.

我们一直在谈论知识和常识。

I just want to do a very quick experiment, so everyone stay in their seat.

我只想做一个非常快速的实验，所以每个人都坐在座位上。

I'm going to get two T.A.'s up here, why don't I get Ale and Kaj up here.

我要带两个助教到这里，为什么我不带艾尔和卡吉到这里呢。

And I wanted to show that common knowledge is not such an obvious a concept, as I've made it seem on the board.

我想表明，常识并不是一个像我在黑板上表现的那样明显的概念。

Come up on the stage a second.

一会儿上台吧。

You can leave the mike its okay.

你可以把麦克风留下来就可以了。

Here we have two of our T.A.'s, actually these are the two head T.A.'s, and I want you to face forward so you don't see what I'm doing.

这里有我们的两个助教，实际上这是两个助教的头，我希望你面朝前方，这样你就看不到我在做什么。

I'm about to put on their heads a hat.

我正要把帽子戴在他们头上。

Here's a hat on Ale's head, and here's a hat on Kaj's head.

这是 Ale 头上的帽子，这是 Kaj 头上的帽子。

Let's move them this way so they're in focus.

让我们这样移动它们，以便它们成为焦点。

Now you can all see these hats, and if they turn around to each other, they can see each other's hat.

现在你们都可以看到这些帽子，如果他们转向对方，他们就可以看到彼此的帽子。

Now I want to ask you the question here.

现在我想在这里问你这个问题。

Here is a fact, so is it common knowledge that -- is it common knowledge that at least one of these people has a pink hat on their head?

这是事实，那么，这些人中至少有一个人头上戴着一顶粉红色的帽子，这是常识吗？

Is it common knowledge?

这是常识吗？

So I claim it's not common knowledge.

所以我声称这不是常识。

What is known here?

这里知道什么？

Well I'll reveal the facts now: that in fact Ale knows that Kaj has a pink hat on his head.

好吧，我现在就揭露事实：事实上，Ale 知道 Kaj 头上戴着一顶粉色帽子。

So it's true that Ale knows that at least one person in the room has a pink hat on their head.

所以艾尔确实知道房间里至少有一个人头上戴着一顶粉红色的帽子。

And it's true that Kaj knows that Ale has a pink hat on his head.

Kaj 确实知道 Ale 头上戴着一顶粉色帽子。

They both look absurd, but never mind.

他们看起来都很荒谬，但没关系。

But notice that Ale doesn't know the color of the hat on his own head.

但请注意，艾尔不知道自己头上帽子的颜色。

So even though both people know, even though it is mutual knowledge that there's at least one pink hat in the room, Ale doesn't know what Kaj is seeing.

因此，尽管两个人都知道，尽管双方都知道房间里至少有一顶粉色帽子，但 Ale 并不知道 Kaj 看到了什么。

So Ale does not know that Kaj knows that there's a pink hat in the room.

所以 Ale 不知道 Kaj 知道房间里有一顶粉色帽子。

In fact, from Ale's point of view, this could be a blue hat.

事实上，从艾尔的角度来看，这可能是一顶蓝帽子。

So again, they both know that someone in the room has a pink hat on their head: it is mutual knowledge that there's a pink hat in the room.

再次，他们都知道房间里有人头上戴着一顶粉红色的帽子：房间里有一顶粉红色的帽子是共同的知识。

But Ale does not know that Kaj knows that he is wearing a blue, a pink hat, and Kaj does not know that Ale knows that Kaj is wearing a pink hat.

但Ale不知道Kaj知道他戴着蓝色、粉色的帽子，Kaj也不知道Ale知道Kaj戴着粉色的帽子。

Each of their hats -- each of their own hats -- might be blue.

他们的每一顶帽子——他们自己的每一顶帽子——可能都是蓝色的。

So notice that common knowledge -- thanks guys -- common knowledge is a rather subtle thing, thank you.

所以请注意常识——谢谢大家——常识是一个相当微妙的东西，谢谢。

Common knowledge is a subtle thing.

常识是一件微妙的事情。

Mutual knowledge doesn't imply common knowledge.

共同知识并不意味着共同知识。

Common knowledge is a statement about not just what I know.

常识不仅是关于我所知道的事情的陈述。

It's about what do I know the other person knows that I know that the other person …and so on and so forth.

这是关于我知道对方知道什么，我知道对方……等等。

Even in this simple example, while you might think it's obviously common knowledge, it wasn't common knowledge that there was a pink hat in the room.

即使在这个简单的例子中，虽然您可能认为这显然是常识，但房间里有一顶粉红色的帽子并不是常识。

Does anybody have smaller siblings or children of their own.

有人有较小的兄弟姐妹或自己的孩子吗？

They can have a pink hat at the end of the class?

他们可以在课程结束时戴上粉红色的帽子吗？

We'll see you on Wednesday.

我们周三见。