Game Theory Lecture 8 | Game Theory

Chapter 1.

第一章。

Candidate - Voter Model: Standard Model [00:00:00] Professor Ben Polak: So last time we left things in the middle of a model, which was the candidate-voter model.

候选人 - 选民模型：标准模型 [00:00:00] Ben Polak 教授：所以上次我们把东西留在模型中间，即候选人选民模型。

What was--I don't want to go over the whole model again, but just to reiterate a little bit--what was different about that model from what we saw before--the main thing that was different was that the candidates cannot choose their positions.

我不想再次回顾整个模型，只是重申一下，该模型与我们之前看到的有什么不同，主要的不同之处在于候选人无法选择自己的职位。

If you like, every voter is a potential candidate but you know the positions of the voters.

如果你愿意，每个选民都是潜在的候选人，但你知道选民的立场。

Let me just bring out two lessons that we left hanging last time.

让我来回顾一下上次我们悬而未决的两个教训。

I want to just put them on the board to make sure they're in your notes.

我想把它们放在黑板上，以确保它们在你的笔记中。

So the first lesson is--one thing we saw already last time--is there can be lots of different Nash Equilibrium this model.

所以第一课是——我们上次已经看到的一件事——这个模型可以有很多不同的纳什均衡。

There are multiple possible Nash Equilibrium in this model and more to the point, not all of those equilibria have the candidates crowded at the center.

该模型中有多种可能的纳什均衡，更重要的是，并非所有这些均衡都让候选人挤在中心。

We saw early on in the classic Downs or median-voter model that that model predicted crowding the center.

我们很早就在经典的唐斯或中间选民模型中看到，该模型预测中间会拥挤。

This one doesn't, and we'll come back to that in a second.

这个没有，我们稍后会回到这个问题。

And a second thing we saw last time was that entry can--if you enter on the left one affect of entering on the left can be to make the candidate on the right more likely to win.

我们上次看到的第二件事是，如果你从左边进入，从左边进入的一个影响可能是让右边的候选人更有可能获胜。

Conversely, if you enter on the right--you're a right-wing voter candidate and you enter, that can make it more likely that the left wings are--can lead to the winner being more--being further from your ideal position.

相反，如果你从右翼进入——你是右翼选民候选人并且你进入，那么左翼更有可能——可能导致获胜者离你的理想位置更远。

Just to revisit this a little bit, let me go back and just illustrate those two points again.

为了稍微回顾一下这一点，让我回去再次说明这两点。

So I'll take a row further back this time and get a nice full row that we can see the whole of.

所以这次我会再往前走一排，得到一个漂亮的完整行，我们可以看到整个行。

This way, I'm going to take a row further back so that this time we have no confusion about what left wing and right wing is, at least almost no confusion.

这样，我要往前排一排，这样这次我们就不会混淆什么是左翼和右翼，至少几乎不会混淆。

So let me choose this row, this row good okay.

那么让我选择这一行，这一行好好吧。

I'm sorry, the people in the balcony are going to have to imagine this.

抱歉，阳台上的人必须想象这一点。

It's a penalty being in the balcony.

呆在阳台上是一种惩罚。

So this row.

所以这一排。

And here's my left wing of this row (for everyone who's in front of you which is almost everybody), and here's my right wing.

这是我这一排的左翼（对于你们前面的每个人，几乎是所有人），这是我的右翼。

Let's try and illustrate some equilibrium we saw last time.

让我们尝试说明一下我们上次看到的一些平衡。

So, in particular, if I can get the guy in the blue Yale shirt to stand up a second and the guy with his computer to stand up a second.

所以，特别是，如果我能让那个穿蓝色耶鲁衬衫的人站起来，让那个拿着电脑的人站起来。

Sorry.

对不起。

Let's assume all the seats are filled for now so--just to help me out a little bit since I'm doing this on the fly.

让我们假设现在所有座位都已满——只是为了帮助我一点，因为我正在匆忙中做这件事。

This, I'm going to claim is an equilibrium.

我认为这是一种平衡。

Notice that there are two candidates standing and notice that they are not particularly close to the center.

请注意，有两名候选人站着，并注意他们并不是特别靠近中心。

We have our sort of middle-of-the-democratic-party left candidates.

我们有民主党中间的左翼候选人。

So I'm tempted to give you a name but perhaps I will not.

所以我很想给你起个名字，但也许我不会。

And here's a middle-of-the-republican-party candidate.

这是一位中间派共和党候选人。

And this is the election.

这就是选举。

They're going to split the votes equally if I've actually chosen correctly.

如果我真的选择正确的话，他们会平分选票。

So, just to observe some things here.

所以，只是为了观察一些事情。

First, for this to be an equilibrium, it better be the case that they're symmetric on left and right.

首先，为了达到平衡，最好它们左右对称。

If they're not symmetric, then it isn't an equilibrium because one of those candidates is going to lose for sure and the way we set up the model that means that one of them will drop out: they will deviate to drop out.

如果它们不对称，那么就不是均衡，因为其中一位候选人肯定会失败，而我们建立模型的方式意味着其中一位候选人将退出：他们将偏离退出。

Is that correct?

这是正确的吗？

Let's also illustrate this.

我们也来说明一下这一点。

So, we've already illustrated that they're not particularly close to the center.

所以，我们已经说明它们并不是特别靠近中心。

Let's illustrate what I meant by somebody on the left causing the right wing to win.

让我们来说明一下我所说的左翼人士导致右翼获胜的意思。

So, if this was the election, here we are getting closer and closer to the election, and suddenly one of our left-wing guys, so Dennis Kucinich or something decides to enter--so let's suppose this is Dennis Kucinich and he enters.

所以，如果这是选举，我们离选举越来越近了，突然我们的左翼人士之一，丹尼斯·库西尼奇或其他人决定参加——所以让我们假设这是丹尼斯·库西尼奇，他参加了。

If Dennis Kucinich, our left-wing guy enters, there might be some sort of moral victory in entering, but the result of this will be that our right-wing candidate wins; everyone sees that now?

如果我们的左翼候选人丹尼斯·库西尼奇（Dennis Kucinich）加入，进入可能会带来某种道德上的胜利，但其结果将是我们的右翼候选人获胜；现在大家都看到了吗？

If these three guys are standing, Dennis Kucinich is going to steal some votes from our center-left candidate and cause our center-right candidate to win.

如果这三个人站起来，丹尼斯·库西尼奇将从我们的中左翼候选人那里窃取一些选票，并导致我们的中右翼候选人获胜。

Now, that may or may not be Mr. Kucinich's intention, but we should at least be aware of it.

现在，这可能是库西尼奇先生的意图，也可能不是，但我们至少应该意识到这一点。

So, this is a real effect.

所以，这是一个真实的效果。

Go back to the 2000 election and think about what happened when Nadar entered and "do the math" as it were, which people didn't do at the time apparently.

回到2000年的选举，想一想纳达尔上台后发生了什么，并“算算”，而当时人们显然没有这样做。

If you were listening to the newspapers or if you read The New York Times this morning or listened to the radio this morning, you'll find exactly the same debate is going on now in the republican party.

如果你今天早上正在听报纸，或者读《纽约时报》，或者今天早上听广播，你会发现共和党现在正在进行完全相同的辩论。

Some member--actually it's the other way around, it's the right wing this time; I suppose we should switch it around.

有的会员,其实是反过来的,这次是右翼,我想我们应该换个方式。

So the right--some people on the right of the republican party are saying that if it turns out that Giuliani, who currently is leading in the polls, wins the republican nomination they will run a third-party candidate.

因此，一些共和党右翼人士表示，如果目前民意调查中领先的朱利安尼赢得共和党提名，他们将竞选第三方候选人。

Of course the debate is actually in these terms.

当然，争论实际上是在这些方面。

If they run a right-wing third-party candidate--so that would be our guy over here, this might be--this might have payoff in terms of other things.

如果他们选举一位右翼第三方候选人——这可能就是我们这里的人——这可能会在其他方面带来回报。

But in terms of the election, it's going to result in Hilary winning.

但就选举而言，希拉里将获胜。

So we've seen these two effects.

我们已经看到了这两种效果。

They're very real effects.

它们是非常真实的效果。

We're not necessarily getting people crowding the center and we have to worry about, when we enter, causing the other wing to win.

我们不一定会让人们挤在中心，我们必须担心，当我们进入时，会导致另一侧获胜。

This isn't-- I'm not making a left wing/right wing politics argument here--this is true for both wings symmetrically.

这不是——我不是在这里进行左翼/右翼政治争论——这对于对称的两翼来说都是如此。

So, let's try and bring one more idea in here, which is where we ended up last time, which is just how far away from the center can we be?

那么，让我们尝试在这里再提出一个想法，这就是我们上次结束的地方，即我们离中心有多远？

So these guys sit down a second and let's stand up, Mr. Kucinich , again; I know that isn't your real name but never mind.

所以这些人坐下来，让我们再次站起来，库西尼奇先生；我知道那不是你的真名，但没关系。

Mr. Kucinich out here on the left and Mr. crazy-right-wing guy whose name I've-- who's the most crazy right wing guy of these candidates?

库西尼奇先生在左边，还有疯狂的右翼先生，我的名字是——谁是这些候选人中最疯狂的右翼人士？

I'm not--I'll get in trouble whoever I name--so whoever the most crazy right-wing candidate you can think of.

我不是——无论我点名谁，我都会惹上麻烦——所以无论你能想到的最疯狂的右翼候选人是谁。

and now we have the full spectrum represented with just the extremes standing.

现在我们拥有了完整的范围，仅代表了极端情况。

Here we have--they're symmetric around the center, but I claim this is not an equilibrium.

在这里，它们围绕中心对称，但我声称这不是平衡。

So the people in the balcony, I've got the extreme right and the extreme left standing here.

所以阳台上的人，我这里站着极右派和极左派。

Why is this not an equilibrium?

为什么这不是一个均衡？

Yeah I should--it's my fault I should have brought the mike.

是的，我应该——这是我的错，我应该带上麦克风。

Can I have the mike?

我可以拿麦克风吗？

I'm sorry my fault.

我很抱歉我的错。

Thanks.

谢谢。

This one.

这个。

So Katie, why is that not an equilibrium?

那么凯蒂，为什么这不是一个平衡呢？

Student: Because the person in the center could stand up and win the majority.

学生：因为中间的人可以站出来赢得多数。

Professor Ben Polak: Exactly, because the person in the center could stand up and win.

本·波拉克教授：没错，因为中间的人可以站起来并获胜。

Actually, it doesn't only have to be the person exactly in the center.

事实上，它不仅必须是正好位于中心的人。

A wide array of center candidates could deviate and win at this point.

众多中锋候选人可能会在这一点上偏离并获胜。

So, if this was the two candidates standing, and you imagine a third candidate standing, who for example, is this gentleman, if he was to stand, we could--he has to be a little bit closer to the center, let's say this guy, the guy in gray--fairly clearly he's going to end up winning.

所以，如果这是两个候选人站着，你想象第三个候选人站着，例如，这位先生，如果他要站起来，我们可以——他必须离中间更近一点，比方说这个人，那个穿灰色衣服的人——很明显他最终会获胜。

So this is the third lesson.

这是第三课。

If the candidates are too far apart we're going to see some center entry, which is going to win.

如果候选人之间的距离太远，我们就会看到一些中间的候选人，这将获胜。

Thank you guys.

谢谢你们。

So even though there isn't this full Downsian effect of pushing candidates towards the center, even though we don't have the median-voter theorem here, we still have part of the intuition surviving.

因此，即使不存在将候选人推向中间的完整唐斯效应，即使我们这里没有中间选民定理，我们仍然有部分直觉幸存。

The part of the intuition that's survives is, if the candidates are too far apart, then the center will enter and win.

直觉中幸存的部分是，如果候选人相距太远，那么中间派就会进入并获胜。

So there is something pulling people to the center still.

所以仍然有某种东西把人们拉向中心。

So a reasonable question here is just how far apart in equilibrium can the candidates be?

因此，这里一个合理的问题是，候选人之间的平衡能有多远？

We've established that we can have two candidates and they needn't both be at the center.

我们已经确定可以有两名候选人，但他们不必都在中间。

We've established that they can't be at the extremes.

我们已经确定他们不能走极端。

How far apart can they be?

他们可以相距多远？

Well, this is really just a--it's kind of a nerdy question to get it precisely but let's get it precisely nevertheless.

好吧，这实际上只是一个——精确地得到它是一个书呆子问题，但我们还是精确地得到它。

So let's have a look, let's use the other board.

那么让我们看看，让我们使用另一块板。

So, here's the full extent of our political spectrum from 0 to 1, and let me just try and illustrate how far apart these can be.

因此，这是我们从 0 到 1 的政治范围的完整范围，让我尝试说明一下这些范围之间的差距有多大。

What I'm going to do is I'm going to divide this into sixths - 1/6, 2/6, 1/2, 4/6, 5/6.

我要做的就是将其分成六等份 - 1/6、2/6、1/2、4/6、5/6。

So I claim--and I'll show afterwards--I'll claim that provided the two candidates aren't outside of 1/6 and 5/6 that that will be an equilibrium.

所以我声称——稍后我将证明——我将声称，只要两位候选人不在 1/6 和 5/6 之外，那将是一个均衡。

So, in particular, if the candidates are just inside 1/6 and just inside 5/6, or just more than 1/6 and just less than 5/6--so here's one of these candidates who's standing and here's the other one--then we'll be okay.

因此，特别是，如果候选人恰好在 1/6 以内且刚好在 5/6 以内，或者刚好超过 1/6 并且刚好小于 5/6，那么这里是其中一位站立的候选人，这是另一位，那么我们就可以了。

Now why?

现在为什么？

Why is that the right answer?

为什么这是正确答案？

Anyone want to try a guess?

有人想尝试一下猜测吗？

This was the question I sent you over the weekend.

这是我周末向您发送的问题。

I'm sure you were doing other things over the weekend but nevertheless, why is this answer?

我确信您周末在做其他事情，但是，为什么会是这个答案呢？

Well, let's see.

好吧，让我们看看。

What are they vulnerable too?

他们又有何脆弱之处？

They're vulnerable to deviation by somebody entering at the center.

他们很容易因为有人进入中心而发生偏差。

And if somebody enters at the center--what would make somebody enter at the center?

如果有人进入中心——什么会让某人进入中心？

They're going to enter at the center if they can win basically.

如果他们能基本获胜的话，他们就会进入中锋。

So if they enter at the center in this case, let's see how many votes everyone gets.

因此，如果他们在这种情况下进入中心，让我们看看每个人得到多少票。

So if we enter at the center here--here's our new candidate--who's thinking about entering at the center.

因此，如果我们进入这里的中心——这是我们的新候选人——他正在考虑进入中心。

So he's sort of thinking about it.

所以他正在考虑这个问题。

So what's his or her calculation going to be?

那么他或她的计算会是什么呢？

Well, let's look at what would have happened.

好吧，让我们看看会发生什么。

So all of these voters are going to vote for our left-wing candidate.

所以所有这些选民都会投票给我们的左翼候选人。

So these are going to vote for the left-wing candidate.

所以这些人将投票给左翼候选人。

All of these voters are going to vote for the right-wing candidate.

所有这些选民都会投票给右翼候选人。

They're all closest to the right-wing candidate.

他们都是最接近右翼候选人的人。

And the middle third--that's a third of the voters, another third--and the middle third (these ones here) are going to vote for the center candidate.

中间三分之一——即三分之一的选民，另外三分之一——中间三分之一（这里的这些人）将投票给中间候选人。

Is that right?

是这样吗？

So I've basically divided--the reason I divided it into sixths is I want to put everybody at the middle of a third.

所以我基本上已经划分了——我把它划分为六等分的原因是我想把每个人都放在三分之一的中间。

So here the left-wing candidate is at the middle of the left third; the right wing candidate is at the middle of the right third, and the center candidate is at the middle of the center third not surprisingly.

所以这里左翼候选人在左边三分之一的中间；右翼候选人位于右三分之一的中间，中间候选人位于中间三分之一的中间，这并不奇怪。

So if the center candidate enters, if they were exactly at 1/6 and 5/6, they would split the vote and the center candidate would win with probably 1/3.

因此，如果中间候选人进入，如果他们的得票率恰好是 1/6 和 5/6，那么他们将平分选票，中间候选人可能会以 1/3 获胜。

Is that right?

是这样吗？

But without worrying about that exact case suppose that this--as I claimed that the left-wing candidate is just slightly to the right of 1/6 and the right wing candidate is just slightly to the left of 5/6.

但不用担心具体情况，假设这一点——正如我所说，左翼候选人稍微偏向 1/6 的右侧，右翼候选人稍微偏向 5/6 的左侧。

Now this left-wing candidate gets a few extra votes here and the right wing candidate gets a few extra votes here (let's put them a bit harder), and you can see now that the center candidate isn't going to win.

现在，这位左翼候选人在这里获得了一些额外的选票，而右翼候选人在这里获得了一些额外的选票（让我们更难一点），你现在可以看到中间派候选人不会获胜。

Is that right?

是这样吗？

Because the left-wing candidate is getting slightly more than 1/3 of the votes; the right wing candidate's going to get slightly more than 1/3 of the votes; so the center candidate is going to get squeezed out.

因为左翼候选人获得了略多于1/3的选票；右翼候选人将获得略多于1/3的选票；所以中间的候选人将会被排挤。

So just a little bit of thinking about it, we can tell in this very simple model the furthest apart the left and right wing candidates can be is at 1/6 and 5/6.

因此，稍微思考一下，我们就可以在这个非常简单的模型中看出，左翼和右翼候选人之间的最远距离是 1/6 和 5/6。

They can't go to the extremes, but they're not pulled all the way to the center.

他们不能走极端，但也不会一直被拉到中间。

So that's just a little bit of nerdy math to confirm it, or nerdy thinking about it.

所以这只是一点点书呆子数学来证实它，或者书呆子思考它。

But coming back to our lesson, which is what I guess we care about-- coming back to our lessons, the third lesson here is if the candidates--if the two candidates are too extreme--where too extreme in this model meant beyond…

但是回到我们的课程，我想这就是我们关心的——回到我们的课程，这里的第三课是如果候选人——如果两位候选人太极端——在这个模型中太极端意味着超出……

meant less than 1/6 and more than 5/6--but are too extreme, someone in the center will enter.

意思是小于1/6和大于5/6——但是太极端了，中间会有人进入。

Again, if you look back in both American History and other countries' history, you'll see that when candidates are perceived to be too far apart there's been tremendous temptation for center parties with third parties to establish themselves in the center.

再说一次，如果你回顾美国历史和其他国家的历史，你会发现，当候选人被认为相差太远时，中间党派与第三方之间就会产生巨大的诱惑，让自己处于中间位置。

So again, with some biased towards England, this is what happened in English History during the Thatcher Period for example.

再说一遍，有些人对英国有偏见，这就是撒切尔时期英国历史上发生的事情。

The Thatcher government was perceived to be quite far on the right.

撒切尔政府被认为是相当右倾的。

The Labor Party at times could be quite far on the left.

工党有时可能相当左倾。

And we saw a center party set up in between them.

我们看到他们之间设立了一个中心派对。

Okay, so these seem to be the three main politics lessons of this model.

好的，这些似乎是该模型的三个主要政治教训。

Is everyone happy with that?

大家对此都满意吗？

I'm rushing it slightly because we said that already last time and we're happy with how it works?

我有点着急，因为我们上次已经说过了，而且我们对它的运作方式感到满意？

There's a also a Game Theory lesson that I want to just keep in mind here without--well, there's a Game Theory lesson here.

还有一门博弈论课程，我想在这里记住，但没有——嗯，这里有一门博弈论课程。

And the Game Theory lesson is that our method of finding equilibrium in this model, which was what--it's guess and check--is actually pretty effective.

博弈论的教训是，我们在这个模型中找到平衡的方法，也就是猜测和检查，实际上非常有效。

You might think that guess and check, since it doesn't sound like advanced mathematics, wouldn't be such a great way of going about solving games and thinking about them, and thinking about the real world.

你可能会认为猜测和检查，因为它听起来不像高等数学，所以不是解决游戏、思考游戏以及思考现实世界的好方法。

But actually, guess and check did pretty well here.

但实际上，猜测和检查在这里做得很好。

We were able to make sensible guesses pretty quickly.

我们能够很快做出合理的猜测。

We were pretty quickly able to figure out what was going on.

我们很快就弄清楚发生了什么事。

And the key here is what?

而这里的关键是什么？

Without writing it necessarily, the key here is: be systematic when you're guessing.

不一定要写出来，这里的关键是：猜测时要有系统性。

Make sure you've looked everywhere.

确保你已经到处寻找。

And second, be systematic, be careful when you're checking.

其次，要系统化，检查时要小心。

The big error is to ignore certain types of deviation.

最大的错误是忽略某些类型的偏差。

In this particular model, people very quickly realized that one possible deviation is for someone else to enter.

在这个特定的模型中，人们很快意识到一个可能的偏差是其他人进入。

But perhaps they're a little bit less good at spotting that another possible type of deviation is for somebody to drop out.

但也许他们不太擅长发现另一种可能的偏差类型是某人退出。

You want to look at all possible types of deviation.

您想要查看所有可能类型的偏差。

But if you're careful, this is a very effective method.

但如果你小心的话，这是一个非常有效的方法。

So, that's all I want to say about this politics model, and I want to move onto a model that perhaps has more to do with sociology.

所以，关于这个政治模型，我想说的就是这些，我想转向一个可能与社会学更多相关的模型。

We'll do a little tour of the Social Sciences here in showing how Game Theory can apply to each in turn.

我们将在这里对社会科学进行一些介绍，以展示博弈论如何依次应用于每个社会科学。

So to do that I'm going to--I'll put this up so you can still read it and work on this board.

因此，为了做到这一点，我将把它放出来，以便您仍然可以阅读它并在这个板上工作。

So, we're going to play another game this morning, and it's going to be completely different from the games we've played so far but it's still going to have lessons in it.

所以，今天早上我们要玩另一场比赛，这将与我们迄今为止玩过的比赛完全不同，但仍然会从中吸取教训。

It's going to be another location model, so that's a connection to what we've done before.

这将是另一个位置模型，因此这与我们之前所做的事情有联系。

But the idea of this game is as follows.

但这个游戏的想法如下。

We're going to imagine that there are two towns, two possible locations, and we'll call them East Town and West Town.

我们将想象有两个城镇，两个可能的位置，我们将它们称为东镇和西镇。

And we're going to assume that there are two types of people in the world.

我们假设世界上有两种类型的人。

So, there's two types of people, and these types of people are tall and short.

所以，有两种类型的人，这两种类型的人是高的和矮的。

Deliberately, this is arbitrary, right?

故意的，这是任意的，对吗？

East and West seems kind of meaningless, almost meaningless, in the nomenclature of the towns.

在城镇的命名中，东方和西方似乎毫无意义，几乎毫无意义。

I guess there's something about where they are.

我想他们的位置有一定的关系。

And tall and short is for pretty meaningless nomenclature for the people, except it says something about how tall they are.

高和矮对于人们来说是毫无意义的命名法，除了说明他们有多高之外。

The idea here is that people are going to choose where they live.

这里的想法是人们将选择他们居住的地方。

Let's assume that there's lots of people.

我们假设有很多人。

there's 100,000 of each type of person.

每种类型的人都有 100,000 个。

And let's assume that each town holds 100,000 people.

我们假设每个城镇有 100,000 人。

These are fairly big towns.

这些都是相当大的城镇。

So, the players in this game are going to be the people, the 200,000 people--200,000 tall people and 200,000 short people--but in a minute I'm going to tell you whether you're tall or short.

所以，这个游戏的参与者将是人，200,000 个人——200,000 个高个子和 200,000 个矮个子——但一会儿我会告诉你你是高还是矮。

So actually, you're going to be the players in this game.

所以实际上，你将成为这个游戏的玩家。

The strategies are going to be a choice of whether you choose East or West.

策略将取决于您选择东方还是西方。

So these are the players and these are going to end up being the strategies.

所以这些是参与者，这些最终将成为策略。

Each of you is going to choose do I want to live in the East Town or do I want to live in West Town?

你们每个人都要选择我想住在东镇还是想住在西镇？

So, as usual, what's missing is the payoffs.

所以，像往常一样，缺少的是回报。

To model the payoffs, let me first of all draw a picture and then come back and explain it.

为了模拟收益，让我首先画一张图，然后再回来解释它。

So, the picture is going to look like this.

所以，图片看起来像这样。

Here's the payoff picture, and it's a little complicated so just bear with me for a second.

这是回报图，有点复杂，请耐心等待。

So, on the horizontal axis I'm going to put the number of people of your type in the town you end up in.

因此，我将在横轴上填写您最终所在城镇中属于您类型的人数。

So on the horizontal axis is the number of your type in your town.

因此，横轴上是您所在城镇的类型编号。

So the most this could possibly be is 100,000 because that would say everybody is the same type as you in the town and the lowest it could possibly be is 0 because that would say that everyone in the town except for you is of the other type.

因此，最多可能是 100,000，因为这表示镇上的每个人都与您属于同一类型，而最低可能是 0，因为这表示镇上除了您之外的每个人都是另一种类型。

Is that right?

是这样吗？

I'm going to draw this payoff function.

我将画出这个收益函数。

This is going to be your payoff.

这将是你的回报。

This is going to be utility--the payoff of you.

这将是效用——你的回报。

Let's assume this is Type X, it doesn't really matter.

我们假设这是 X 型，这并不重要。

Let's assume this is the tall type payoff which is going to be symmetric for the other type as well.

我们假设这是高类型的回报，对于其他类型也将是对称的。

So the payoffs going to look like this.

所以回报看起来像这样。

Be careful.

当心。

Let me draw it and then explain it.

让我画出来然后解释一下。

That goes like this and then like this.

事情是这样的，然后又是这样的。

So the idea here--the idea is more important than the picture--the idea is, if you are a minority of 1 in your town, so everyone else is of the other type, you get a payoff of 0.

所以这里的想法——这个想法比图片更重要——这个想法是，如果你是你所在城镇中 1 的少数，那么其他人都是另一种类型，你得到的回报是 0。

If you are in the majority, and in fact, everyone in your town is of the same type then you get a payoff of a 1/2.

如果你占多数，而且事实上，你镇上的每个人都是同一类型，那么你会得到 1/2 的回报。

And if it's the case that everyone in your town--sorry, if it's the case that your town is exactly mixed, so half of your town is tall and half of your town are short, then you get a payoff of 1.

如果你的城镇中的每个人都是这种情况——抱歉，如果你的城镇完全是混合的，所以你的城镇一半高，一半矮，那么你得到的回报为 1。

I put a half here.

我在这里放了一半。

That's really the wrong thing to put here, it should be a 1/2 of 100,000 so I guess it's 50,000.

放在这里确实是错误的，它应该是 100,000 的 1/2，所以我猜是 50,000。

So if it's the case that 50,000 of the people in the town are your type and 50,000 are the other type, then you get the highest possible payoff which is 1.

因此，如果镇上有 50,000 人是你的类型，50,000 人是另一种类型，那么你将获得最高的可能回报，即 1。

So this is--we're going to assume that each and every one of you has this payoff.

所以这就是——我们假设你们每个人都有这样的回报。

Does that make sense?

这有道理吗？

So these are people--let's just try and get the idea across in words--these are people who would like to live in mixed towns, but if they're going to live in a town that's not mixed, they'd rather live in a town in which they're the majority.

所以这些人——让我们试着用语言来表达这个想法——这些人愿意生活在混合城镇中，但如果他们要生活在一个非混合城镇中，他们宁愿生活在一个他们占多数的城镇中。

Does that make sense?

这有道理吗？

They'd like to live in mixed towns, but if the town is not--if they had a choice between two towns and the towns are not evenly mixed, then theyd prefer to be in the majority town-- the town in which they're the majority.

他们喜欢生活在混合城镇中，但如果这个城镇不是——如果他们可以在两个城镇之间进行选择，并且这些城镇的混合程度不均匀，那么他们更愿意住在占多数的城镇——他们占多数的城镇。

Everyone understand this?

这个大家都明白了吧？

It's important; since we're about to play this game, it's important you're all on board here.

这很重要；因为我们即将玩这个游戏，所以大家都参与进来很重要。

So to play this game, I need to put down a few more rules.

所以要玩这个游戏，我需要再制定一些规则。

So the first rule is going to be that the choice is simultaneous and that's a little unrealistic because in practice, of course, people choose their towns sequentially, wherever they happen to be moving, but for the moment let's leave it as that.

因此，第一条规则是选择是同时进行的，这有点不切实际，因为在实践中，人们当然会依次选择他们的城镇，无论他们碰巧搬到哪里，但目前我们就这样吧。

Second, I need to just say what happens if too many people choose the same town.

其次，我只想说如果太多人选择同一个城镇会发生什么。

So if there's no room in a town, for example, if too many people chose East Town then we allocate the surplus randomly.

因此，如果一个城镇没有空间，例如，如果太多人选择东镇，那么我们会随机分配剩余的空间。

We ration the places randomly.

我们随机配给这些地方。

So then--we then randomize to ration; people understand that?

那么——然后我们随机分配；人们明白吗？

So, for example, there's 100,000 places in East Town, so if 150,000 people chose East Town, you're going to have a 2/3 chance of getting into East Town and the rest of you is going to be allocated to West Town.

比如说，东镇有10万个名额，如果有15万人选择了东镇，那么你们就有2/3的机会进入东镇，剩下的人就会被分配到西镇。

Does that make sense?

这有道理吗？

I just need something to make things add up.

我只需要一些东西来让事情加起来。

Okay, so to do this, I first of all need to decide who in the class is tall, and who in the class is short (and I'm going to grab that mike again).

好吧，为此，我首先需要决定班上谁高，谁矮（我要再次抓住那个麦克风）。

So let me just count backwards.

那么让我倒数一下。

I guess we'll ignore this row.

我想我们会忽略这一行。

So, 1, 2, 3, 4, 5, 6, 7, possibly 8.

所以，1、2、3、4、5、6、7，可能是 8。

Well, maybe this will work, so up to here this row and forwards you are short people.

好吧，也许这会起作用，所以到目前为止，这一排和前锋你们都是矮个子。

You understand that?

你明白吗？

Everyone forward of where I am now, this row included, is short.

我现在所在位置前面的每个人（包括这一排）都很矮。

The rest of you guys and the guys in the balcony too, who can't see me anymore, the guys in the balcony, you are tall.

你们其他人和阳台上的人，谁都看不到我了，阳台上的人，你们很高。

Short.

短的。

Tall.

高的。

Okay, so how many rows of these do you have?

好的，那么你有多少行呢？

One, two, three, four, five.

一、二、三、四、五。

Seven rows here, all right.

这里有七排，好吧。

One, two, three, four, five, six.

一、二、三、四、五、六。

We have seven rows of short people and the rest of you are tall and I'm going to hope I've split that evenly.

我们有七排矮个子，其余的人都很高，我希望我能平均分配。

I'm going to cheat a little bit--in a minute, you're going to have to choose which town you're going to live in and we'll do it by show of hands.

我要稍微作弊一下——一会儿，你必须选择你要居住的城镇，我们将通过举手的方式来完成。

But before we do that I'm going to cheat a little bit by giving each of you an initial position --;this is irrelevant but I just want to sort of set things up.

但在我们这样做之前，我会稍微作弊一下，给你们每个人一个初始位置——这无关紧要，但我只是想把事情做好。

So what we'll do is we'll put this row--so this row, one, two, three, four--these rows, your initial position is in East Town.

所以我们要做的就是放置这一行——所以这一行，一、二、三、四——这些行，你的初始位置是在东镇。

It doesn't matter, you can move, but your initial position is in East Town.

没关系，你可以移动，但你的初始位置是在东镇。

So East, East, East, East--everyone understand that?

所以东方，东方，东方，东方——大家都明白吗？

These three rows, this row, this row, and this row, you're in West Town.

这三排，这一排，这一排，这一排，你就在西镇了。

And these next four rows--so you guys, you guys, you guys, and you guys--let's do it again, this row, this row, this row and this row you're at East Town and--oh, I didn't notice this row--you're in East Town as well, up to here is East Town.

接下来的四排——所以你们，你们，你们，你们——让我们再做一次，这一排，这一排，这一排，这一排你们在东镇，哦，我没注意到这一排——你们也在东镇，到这里就是东镇。

And the rest of you is in West Town.

你们其余的人都在西镇。

Okay, everyone understand where they are now?

好了，大家明白现在在哪了吗？

Let's do a test with the camera watching.

我们用摄像头观看来进行测试。

So if you are currently in East Town raise your hands.

因此，如果您目前在东镇，请举手。

And if you're currently in West Town raise your hands.

如果您目前在西镇，请举手。

So I've set you up with a stripy pattern and notice that I've set things up so that slightly more of the short people were in East Town and slightly more of the tall people--if I did that right--were in West Town.

所以我给你设置了一个条纹图案，并注意到我已经设置了一些东西，以便稍微多一点的矮个子在东镇，而稍微多一点的高个子——如果我做得对的话——在西镇。

If I did that correctly--I'm not sure I did actually--but if I did that correctly, slightly more of my short people started life in East Town, and slightly more of my tall people started life in West Town.

如果我的做法正确的话——我不确定我是否真的这么做了——但如果我的做法正确的话，我的矮个子中的人会稍微多一些在东镇开始生活，而我的高个子中的人会稍微多一些在西镇开始生活。

Okay, so now think about it a second.

好吧，现在想一想。

Remember, these are your payoffs.

请记住，这些是你的回报。

That's a suspension of disbelief.

这就是怀疑的暂停。

These are your payoffs.

这些都是你的回报。

And in a minute I want all the people who are choosing East Town to raise their hands.

稍后我希望所有选择东镇的人都举手。

I'm going to do it at a count of three and don't cheat.

我数到三就做，不要作弊。

Don't look around you and see what's going on.

不要环顾四周，看看发生了什么。

You all close your eyes.

你们都闭上眼睛。

Everyone in the room close their eyes.

房间里的每个人都闭上了眼睛。

I can do anything now; I could be stripping naked.

我现在可以做任何事；我可能会脱光衣服。

So, you've all got your eyes closed.

所以，你们都闭着眼睛。

On the count of three, anyone who's choosing East Town should raise their hands 1, 2, 3--so this is my East Town.

数到三，选择东镇的人都举手1、2、3——这就是我的东镇。

People can open their eyes now and look at the East Town distribution.

人们现在可以睁开眼睛，看看东镇的分布了。

So I've got this big block of East Town here.

所以我这里有东镇的一大片街区。

I've got a little bit going on in the wings, a little bit of spillage here.

我的翅膀上发生了一些事情，这里有一些溢出。

Let's just make--let's just check it so those of you who didn't raise your hands just now are West Town.

让我们——让我们检查一下，以便那些刚才没有举手的人都是西镇。

So, West Town people raise their hands, so we've got a little bit of spillage here but basically I've got West Town over here.

所以，西镇的人们举手了，所以我们这里有一点溢出，但基本上我已经在这里得到了西镇。

Everyone understood just what happened?

大家都明白到底发生了什么事吗？

Everyone saw what happened?

大家都看到发生什么事了吗？

Let's do it again.

让我们再做一次。

That's where you are now; let's see what happens next time.

那就是你现在所在的地方；让我们看看下次会发生什么。

So once again close your eyes, don't communicate.

所以再次闭上眼睛，不要交流。

Everybody who's going to choose East Town raise their hands now.

现在所有要选择东镇的人都举手。

So I've got pretty much all of these rows now are in East Town and a little bit of spillage here, but basically I've got East Town in the front; I can't see the balcony.

所以我现在几乎所有这些行都在东镇，这里有一点溢出，但基本上我已经把东镇放在前面了；我看不到阳台。

What do I got in the balcony up there?

我在上面的阳台上有什么？

No hands up in the balcony; that's typical.

阳台上不许举手；这是典型的。

They live in the balcony, right?

他们住在阳台上，对吗？

So I've more or less got East Town here.

所以我或多或少已经了解了这里的东镇。

Where's my West Town?

我的西城在哪里？

Raise your hands if you're in West Town now.

如果你现在在西镇，请举手。

West Town: a little bit of spillage here but basically I've got West Town in the back.

西镇：这里有点溢出，但基本上我已经把西镇放在后面了。

Let's do it one more time, everyone understands where things were just now?

我们再来一遍，大家都明白刚才的情况了吧？

Am I leaving the hands up long enough?

我举手的时间够长吗？

Are they up long enough?

他们起床的时间够长吗？

Okay, so let's try it again.

好吧，让我们再试一次。

So East Town on the count of three 1, 2, 3.

于是东镇就数到了三个1、2、3。

So here's my East Town.

这就是我的东城。

I'm pretty much in the--among the East Town dwellers right now.

我现在几乎是东镇居民中的一员。

West Town raise your hands 1, 2, 3.

西镇举手1、2、3。

And I'm pretty much in the West Town dwellers right now and it's not at all stripy.

我现在几乎是西镇居民，那里一点也不条纹。

We started up with a stripy pattern and we ended up--we ended up how?

我们从条纹图案开始，最后——我们最后是怎样的？

We ended up with pretty much the whole front of the room choosing East Town, and the whole back of the room, maybe including the balcony, choosing West Town.

我们最终选择了东镇，几乎整个房间的前面，而房间的整个后面，可能包括阳台，选择了西镇。

The balcony were just flowing over.

阳台刚刚流过。

So everyone saw what happens.

所以每个人都看到了发生的事情。

Now, can anyone say what's happened there?

现在，有人能说出那里发生了什么吗？

Why did we end up like that?

为什么我们会变成这样？

How did we end up like that?

我们怎么就变成这样了？

We started off with a stripy pattern.

我们从条纹图案开始。

It was a little bit off from being kind of perfect, because all of you, by these preferences would prefer to be in a town that was exactly 50/50.

这离完美有点偏离，因为按照这些偏好，你们所有人都更喜欢住在一个完全是 50/50 的城镇。

It wasn't quite at 50/50 but it wasn't far off actually and we pretty quickly ended up with all these short people, you guys are short all living in East Town and all of you all tall people living in West Town.

虽然不是完全是 50/50，但实际上相差并不远，我们很快就遇到了所有这些矮个子，你们都住在东镇，都是矮个子，而你们所有的高个子都住在西镇。

So all of you pretty much ended up with a payoff of 1/2.

所以你们所有人最终几乎都获得了 1/2 的回报。

Some of you didn't, there are a few deviants.

你们中有些人没有，还有一些异常者。

Who are my deviants?

谁是我的异类？

Who are my guys--who are my tall guys who end up East Town?

谁是我的家伙——谁是我最终来到东镇的高个子家伙？

So these guys.

所以这些人。

It's fine, but they end up with a payoff close to 0.

这很好，但他们最终的回报接近于 0。

And who are my short deviants who ended up in West Town?

谁是我最后来到西镇的矮个子？

There's a few.

有几个。

Not many of them at all actually, but a few of them, but they end up with a payoff of 0 as well.

实际上，他们的数量并不多，但也有一些，但他们最终的回报也都是 0。

But pretty much everyone else was ending up with a payoff close to a 1/2, and splitting down the middle of the room.

但几乎其他人最终都获得了接近 1/2 的回报，并在房间中间分裂。

Now, why?

现在，为什么？

What do we call that?

我们称之为什么？

What do we call that process?

我们称这个过程为啥？

What's the outcome here?

这里的结果是什么？

Professor Ben Polak: So this is segregation.

本·波拉克教授：这就是种族隔离。

Right, we ended up getting the class to segregate on tall and short.

是的，我们最终让班级按照身高和矮子进行了隔离。

It wasn't that people wanted to segregate.

并不是人们想要隔离。

I gave you the preferences.

我给了你偏好。

I told you that you have to have these preferences, which may have actually been the preferences that favored being in a mixed town, but we very quickly settled down to a segregated distribution of the class.

我告诉过你，你必须有这些偏好，这实际上可能是有利于混合城镇的偏好，但我们很快就确定了班级的隔离分布。

Why?

为什么？

What led to that?

是什么导致了这一点？

Let's just talk it through a bit.

让我们稍微讨论一下。

Who has an opinion of what it was that kind of got us there.

谁对是什么让我们到达那里有意见？

Yes, so Patrick shout out for the classroom.

是的，所以帕特里克为教室大声喊叫。

This doesn't speak to the class.

这不是在课堂上讲的。

Student: I mean, I think what happened is basically because it was the first choice is simultaneous.

学生：我的意思是，我认为发生的事情基本上是因为它是第一选择是同时发生的。

Everyone in the slightly bad situation all changed to what was perceived to be a better situation.

每个人在稍微不好的情况下都变成了自己认为更好的情况。

So we sort of shot past the equilibrium where we were 50/50.

所以我们有点超出了 50/50 的平衡状态。

Professor Ben Polak: Right, so we--I cheated a little bit by starting you off away from the 50/50 point and you kind of went further very, very quickly.

Ben Polak 教授：是的，所以我们——我做了一点作弊，让你从 50/50 点开始，然后你就非常非常快地走得更远。

Now, I want to say that I think with a longer time period to play with I could have started you off just random, just choose whatever, and if we played enough times my guess is we would have ended up segregated anyway.

现在，我想说的是，我认为如果玩的时间更长，我可以让你随机开始，选择任何东西，如果我们玩了足够多的时间，我猜我们最终会被隔离。

It might not have been segregated with East here and West there, might have been the other way around, but I'm pretty sure we'd have ended up segregated anyway.

它可能没有被东部和西部那里隔离，也可能相反，但我很确定我们最终还是会被隔离。

I can't prove that because I can't--we wouldn't have enough time to do it.

我无法证明这一点，因为我不能——我们没有足够的时间来做到这一点。

I cheated by pushing us that way.

我通过那样的方式推动我们而作弊。

So what happened was--is this right?

那么发生的事情是——这是对的吗？

People who started off in the minority but not badly in the minority in--who were short guys in West Town--they moved to East Town.

那些一开始属于少数族裔但在少数族裔中还不错的人——他们在西镇都是矮个子——他们搬到了东镇。

That was largely the rows here; you guys moved to West Town.

这主要是这里的争吵。你们搬到西城了。

And guys who ended up--who I started off in the minority--I'm not even sure it was actually a minority--who were tall guys in East Town you guys drifted over to West Town.

最后的那些人——我一开始是少数派——我什至不确定这实际上是少数派——他们是东镇的高个子，你们这些家伙漂到了西镇。

Is that right?

是这样吗？

Okay, we'll talk about this more in a second, but let's just get some ideas down on the board.

好的，我们稍后会详细讨论这个问题，但让我们在黑板上写下一些想法。

Well, no--I don't know.

嗯，不——我不知道。

Actually, while I'm down here let's do a little bit more work.

事实上，当我在这里的时候，让我们做更多的工作。

So before we leave this, I mean before we leave talking about it, let's figure out what are the equilibria here.

所以在我们离开这个之前，我的意思是在我们离开谈论它之前，让我们弄清楚这里的平衡是什么。

So what's an equilibrium of this game?

那么这个博弈的均衡是怎样的呢？

It's a pretty simple game.

这是一个非常简单的游戏。

It has lots of people, two choices per person.

人很多，每个人有两种选择。

We're going to be doing guess and check here.

我们将在这里进行猜测和检查。

So what do people think the equilibrium is?

那么人们认为均衡是什么？

Let me get someone who hasn't spoken yet.

让我去找一个还没说话的人。

I surely shouldn't have to cold call here.

我当然不应该给这里打电话。

This is an easy one.

这是一件容易的事。

What's an equilibrium here?

这里什么是平衡？

Somebody?

有人吗？

What's an equilibrium here?

这里什么是平衡？

How about the gentleman here.

这里的绅士怎么样？

You want to have a guess at what the equilibrium is?

您想猜测一下平衡点是什么吗？

Professor Ben Polak: Everyone being segregated.

本·波拉克教授：每个人都被隔离。

Let's be a bit more precise.

让我们更精确一点。

So I claim there's two ways they can be segregated.

所以我认为有两种方法可以将它们隔离。

So spell out the two ways.

所以说一下这两种方式。

Student: If all of the tall people are in West Town and all the short people are in East Town, or if all the tall people are in the West Town and the other way around.

学生：如果所有高个子都在西镇，所有矮个子都在东镇，或者如果所有高个子都在西镇，反之亦然。

Professor Ben Polak: So Greg is saying there's two ways to be segregated and each of them seems like an equilibrium.

本·波拉克教授：格雷格说有两种隔离方式，每种方式似乎都是一种平衡。

Let's just spell it out.

让我们简单地拼写一下。

So all the tall people being in the East is one equilibrium, and all the tall people being in the West is the other equilibrium.

因此，所有高个子都在东方是一种均衡，所有高个子都在西方是另一种均衡。

Both of those, I claim--or Greg claims a bit more carefully--Greg claims both of those are Nash Equilibrium.

我声称，或者格雷格更仔细地声称，格雷格声称这两者都是纳什均衡。

How do we check that they're Nash Equilibrium?

我们如何检查它们是否是纳什均衡？

I mean they are, you're right, but how do we check that they are in fact equilibrium?

我的意思是它们是，你是对的，但是我们如何检查它们实际上是平衡的呢？

How do we go about checking that they're equilibrium?

我们如何检查它们是否平衡？

The guessing part is easy, checking requires a bit of thought, so how do we check?

猜测部分很简单，检查需要花点心思，那么我们如何检查呢？

Yeah.

是的。

Professor Ben Polak: You check for profitable deviations.

Ben Polak 教授：您检查是否存在有利可图的偏差。

So what's a deviation here?

那么这里有什么偏差呢？

If all the short people are at East Town and all the tall people are at West Town, what's a deviation?

如果矮的人都在东镇，高的人都在西镇，有什么偏差？

A deviation--let's ignore the capacity constraints for a minute--a deviation is for one of the short guys to move to West Town.

偏差——让我们暂时忽略容量限制——偏差是指其中一个矮个子搬到西镇。

Is that right?

是这样吗？

So at the equilibrium that short guy was getting a payoff of what?

那么在均衡状态下，那个矮个子得到的回报是什么呢？

What was his or her payoff in equilibrium?

他或她的均衡收益是多少？

One half.

一半。

If he or she deviates and moves to West Town, again ignoring the capacity constraints, let's assume that they can, if that short person moves to West Town what's their payoff going to be?

如果他或她偏离并搬到西镇，再次忽略容量限制，让我们假设他们可以，如果那个矮个子搬到西镇，他们的回报会是多少？

Zero.

零。

So that's not a profitable deviation.

所以这不是一个有利可图的偏差。

Conversely, if we did the same from the other side for the tall people, we'd find the same thing.

相反，如果我们从另一侧对高个子做同样的事情，我们会发现同样的事情。

So we've just checked that that is an equilibrium because nobody can deviate profitably.

所以我们刚刚检查了这是否是一个均衡，因为没有人可以通过偏离来获利。

So we found two equilibria here and as people pointed out they are segregated.

所以我们在这里发现了两个平衡，正如人们指出的那样，它们是隔离的。

Now, I claim that there's another equilibrium here.

现在，我声称这里存在另一种平衡。

What's the other equilibrium here?

这里的另一个平衡是什么？

I should get the mike right there on the other side.

我应该把麦克风放在另一边。

So let me try and lean in here, lean my way.

所以让我试着靠在这里，靠我的方式。

Yeah can you--thanks.

是的，你可以吗——谢谢。

Student: If you were to put exactly 50/50 then that would be an equilibrium.

学生：如果你精确地输入 50/50，那么这就是一个均衡。

Professor Ben Polak: Good, so if the crowd had split 50/50 that would also be an equilibrium.

本·波拉克教授：很好，所以如果人群分成 50/50，那也将是一个平衡。

Right, but the key word there is what?

是的，但是关键词是什么？

The key--50/50 is the key word.

关键——50/50 是关键词。

What's the other key word there?

还有一个关键词是什么？

Exact; it really has to be exact for this to work.

精确的;它确实必须精确才能发挥作用。

And since I'm hand waving a bit because equilibrium are always exact statements, but can people see what's going on here?

因为平衡总是精确的陈述，所以我有点挥手，但人们能看到这里发生了什么吗？

So if people split exactly 50/50, right split the class down the middle--I start off by splitting it this way into tall people and short people--if I had split the town down the middle into East and West, then everyone would have been happy and they would have stayed put.

因此，如果人们按照 50/50 的比例将班级分成中间——我首先将其分为高个子和矮个子——如果我将小镇从中间分成东边和西边，那么每个人都会很高兴，他们也会留在原地。

That's this gentleman's claim and I think that's correct.

这是这位先生的说法，我认为这是正确的。

But what's suspicious if you like, what's worrying about that equilibrium?

但如果你愿意的话，有什么值得怀疑的呢？这种平衡有什么值得担心的呢？

That is an equilibrium.

这就是一种平衡。

What's worrying about that equilibrium, that mixed equilibrium, that integrated equilibrium?

这种均衡、混合均衡、综合均衡有什么值得担心的？

Can we get the guy here?

我们能把那家伙带到这里来吗？

Student: It's a weak Nash Equilibrium because while there's no real incentive for you to deviate, there's no incentive for you to not deviate either.

学生：这是一个弱纳什均衡，因为虽然没有真正的动机让你偏离，但也没有动机让你不偏离。

Professor Ben Polak: Good, so one thing that distinguishes--thank you that's very good.

本·波拉克教授：很好，有一点与众不同——谢谢，这非常好。

So one thing that distinguishes that equilibrium from the equilibria that we were just looking at is it--sometimes it's weak, whereas, the other ones are strict.

因此，这种均衡与我们刚刚看到的均衡的区别在于——有时它很弱，而其他均衡则很严格。

What do we mean by that?

我们这么说是什么意思？

If we deviate away from this exactly mixed equilibrium, roughly speaking, a little bit of hand waving here, but roughly speaking we're exactly indifferent.

如果我们偏离了这种完全混合的平衡，粗略地说，这里有点挥手，但粗略地说，我们完全无动于衷。

It's true, that's not strictly true because I guess by ourselves moving we're changing the balance a little bit, but nevertheless we know if I smooth out the top it'll we'll be exactly right.

这是真的，这并不完全正确，因为我想通过我们自己的移动，我们正在稍微改变平衡，但尽管如此，我们知道如果我平滑顶部，我们就会完全正确。

So at that mixed equilibrium--I don't want to call it a mixed equilibrium--at that integrated equilibrium, I'm exactly indifferent about where I live, both towns look the same to me.

因此，在那种混合均衡——我不想称其为混合均衡——在那种综合均衡中，我对自己住在哪里完全不关心，两个城镇对我来说看起来都是一样的。

They're called East and West but they have the same mixture of tall and short people in them.

他们被称为东方和西方，但他们都有同样的高矮人种。

Whereas, at the segregated equilibria, I strictly prefer to go to the town in which I'm the majority.

然而，在隔离均衡下，我更喜欢去我占多数的城镇。

I'm doing a strictly higher payoff, 1/2 versus 0, by being in the town in which I'm in the majority.

通过住在我占多数的城镇，我得到了严格更高的回报，1/2 比 0。

So there's this notion of strictness, there's also a notion of stability here, so again, I don't want to be too formal here.

所以这里有严格的概念，也有稳定的概念，所以我不想在这里太正式。

I want to give you an informal idea about why we might worry about stability.

我想向您提供一个非正式的想法，说明为什么我们可能会担心稳定性。

So what do I mean by stability here?

那么这里所说的稳定性是什么意思呢？

Why do I think that that integrated equilibrium in which we divide the towns equally might not be stable?

为什么我认为我们平均划分城镇的综合均衡可能不稳定？

What do we mean by that?

我们这么说是什么意思？

For the physicists this is an easy idea, but for everyone I think it's a fairly intuitive idea.

对于物理学家来说，这是一个简单的想法，但对于每个人来说，我认为这是一个相当直观的想法。

Why is that not likely to be stable?

为什么不太可能稳定？

Anybody?

有人吗？

Yeah here.

是的，在这里。

What's your name?

你叫什么名字？

Student: If even one person deviates and all of a sudden then everyone would prefer to deviate to the completely segregated position.

学生：如果一个人突然偏离，那么每个人都会宁愿偏离到完全隔离的位置。

Professor Ben Polak: Right, good.

Ben Polak 教授： 好的，很好。

So at that integrated equilibria, if we move away from it a little bit, if it turns out that, let's say, one town has 5% more short people and the other town has 5% more tall people, then in some sense we're in trouble already.

因此，在这种综合均衡下，如果我们稍微偏离一点，如果结果是，比方说，一个城镇的矮个子人口多了 5%，而另一个城镇的高个子人口多了 5%，那么从某种意义上说，我们已经陷入麻烦了。

We haven't gone very far from this nice equilibrium but already we're in trouble because now all of the short people are going to prefer East Town and all of the tall people are going to prefer West Town.

我们距离这种良好的平衡还没有太远，但我们已经陷入麻烦了，因为现在所有矮个子的人都会更喜欢东镇，而所有高个子的人都会更喜欢西镇。

And in a few moves we're very quickly going to be back at segregation again.

只需采取一些措施，我们很快就会再次回到种族隔离的状态。

It's not stable in the sense that if we were a little bit off we're going to go a long way off.

从某种意义上说，它是不稳定的，如果我们稍微偏离一点点，我们就会走得很远。

Again, I'm not being formal here but the informal idea I think is important.

再说一次，我在这里并不正式，但我认为非正式的想法很重要。

Conversely, those segregated equilibria, because they were strict equilibria, as the gentleman over there was pointing out, they're already pretty stable.

相反，那些分离的均衡，因为它们是严格的均衡，正如那边的那位先生指出的那样，它们已经相当稳定了。

If we start close to 100% short people in East and close to 100% tall people in West and then we move it a little bit away, so I shake you up a bit and I force you to--I reallocate a few people and then let you play, I claim you'll go back to that equilibrium.

如果我们一开始在东部接近100%矮的人，在西部接近100%高的人，然后我们把它移开一点，所以我稍微动一下你，我强迫你——我重新分配一些人，然后让你玩，我声称你会回到那个平衡。

Is that right?

是这样吗？

So, the equilibrium that is integrated here, it is an equilibrium, it's only a weak equilibrium and it's not very stable in some sense.

所以，这里整合的均衡，它是一个均衡，它只是一个弱均衡，从某种意义上来说它不是很稳定。

Whereas, the segregated equilibria, they're clearly strict equilibria and they are stable.

而分离平衡显然是严格平衡并且是稳定的。

Let's put some of this down on the board now, partly because it feels weird for me being out there, even if it doesn't for you.

现在让我们把其中的一些内容写在黑板上，部分原因是我在外面感觉很奇怪，即使它不适合你。

So, we've got this several equilibria here; we've got at least three and we'll come back and talk about whether there are others in a minute.

所以，我们在这里得到了几个平衡；我们至少有三个，我们稍后会回来讨论是否还有其他的。

We've got at least three equilibria, two of them are segregated.

我们至少有三个平衡，其中两个是分离的。

So the Nash Equilibrium in this game--we have two segregated Nash Equilibria and these correspond to tall in East and short in West and vice versa.

所以这个游戏中的纳什均衡——我们有两个分离的纳什均衡，它们对应于东部高和西部矮，反之亦然。

We had a separate one which was an integrated one which was roughly half of each in each town.

我们有一个单独的，是一个综合的，每个城镇大约有一半。

So there's at least three equilibria here, although the two segregated ones are kind of the same, and we argued that these segregated equilibria were in some sense stable and they were strict equilibria in the sense that you strictly preferred not to deviate.

因此，这里至少存在三个平衡，尽管两个分离的平衡在某种程度上是相同的，并且我们认为这些分离的平衡在某种意义上是稳定的，并且它们是严格的平衡，因为您严格选择不偏离。

Whereas, these integrated equilibria, these were actually perhaps not stable, and again I'm not being formal here, so I was being a bit careful putting it in inverted commas, and again, a little bit informal but a kind of weak equilibrium.

然而，这些综合均衡实际上可能并不稳定，而且我在这里也不太正式，所以我有点小心地将其放在引号中，再说一次，有点非正式，但却是一种弱均衡。

Now, I want to bring up one other concept here, which is the idea of a "tipping point." So this is a game that was introduced by a guy called Schelling.

现在，我想在这里提出另一个概念，那就是“临界点”的想法。这是一个叫谢林的人推出的游戏。

Schelling went on to win the Nobel Prize largely for this; certainly in large part for this idea.

谢林后来获得诺贝尔奖主要归功于此。当然很大程度上是为了这个想法。

This is a game that has a tipping point.

这是一个有转折点的游戏。

There are really two stable equilibria, the segregation of one way and the segregation the other way, and in between there's a tipping point beyond which if you--beyond which--if you go beyond which you go to the other equilibrium.

实际上存在两种稳定的平衡，一种方式的隔离和另一种方式的隔离，在两者之间有一个临界点，超过这个临界点，如果你超过这个临界点，你就会到达另一种平衡。

I said that very badly, but do people get the idea?

我说得很糟糕，但是人们明白了吗？

There are two strict equilibria and if we got beyond the points of a 50/50 mix going the other way, we can whiz off to the other equilibrium.

有两个严格的平衡，如果我们超出了 50/50 的混合点，那么我们就可以迅速达到另一个平衡。

We already saw a game that had a tipping point when we played the investment game, where there were two equilibria, all invest and no one invest, there was a natural tipping point in that game and the tipping point was having exactly 90% of you invest, the point at which you actually would want to flip over and go to the other equilibrium.

当我们玩投资游戏时，我们已经看到一个有临界点的游戏，其中有两个均衡，所有人投资和没有人投资，该游戏中有一个自然的临界点，而临界点恰好是你有 90% 的人投资，在这个点上你实际上想要翻转并进入另一个均衡。

So this is a game that has a tipping point.

所以这是一个有转折点的游戏。

We can push people away from the equilibrium and they'll go on coming back and they'll go on coming back and go on coming back, but if we just push them a little bit beyond a half, whoops, they'll go on off to the other equilibrium; very dramatic change.

我们可以把人们推离平衡，他们会继续回来，他们会继续回来，继续回来，但如果我们只是把他们推到一半以上，哎呀，他们会继续回到另一个平衡；非常戏剧性的变化。

That seems like a rather important idea in, for example, sociology.

在社会学等领域，这似乎是一个相当重要的想法。

So I've got these segregated equilibria and I've got these integrated equilibria and let's just make the other obvious remark.

所以我已经得到了这些分离的平衡，并且我已经得到了这些综合的平衡，让我们做另一个明显的评论。

Which of these equilibrium is preferred by the population?

人们更喜欢这些平衡中的哪一个？

Would they rather be in the integrated equilibrium, given that these are their payoffs, or the segregated equilibrium?

鉴于这些是他们的收益，他们宁愿处于综合均衡，还是处于分离均衡？

They'd rather be in the integrated one.

他们宁愿成为集成的一员。

So here's a world in which they'd like to be in the--everybody would like to be in the integrated equilibrium.

因此，这是一个他们希望处于——每个人都希望处于整体平衡的世界。

This is not a Prisoner's Dilemma.

这不是囚徒困境。

This is not a case that we've seen before but it turns out that you're likely to end up in these inefficient, less preferred by everybody, segregated equilibria.

这不是我们以前见过的情况，但事实证明，你很可能最终会陷入这些低效、不太受所有人欢迎的隔离均衡。

However, if we're subtle about this, you might notice there's actually a third equilibrium in this game.

然而，如果我们对此保持微妙的态度，您可能会注意到这个游戏实际上存在第三个平衡。

If we're a little bit nerdy, there's actually another equilibrium in this game.

如果我们有点书呆子的话，这个游戏实际上还有另一种平衡。

So, I'll need to give you your mike back, I'm sorry.

所以，我需要把麦克风还给你，抱歉。

There's a guy there in gray.

那里有一个穿灰色衣服的人。

Student: If everyone chooses one town or the other, regardless of whether they're tall or short, then they would be redistributed.

学生：如果每个人都选一个城镇，无论高矮，都会重新分配。

Professor Ben Polak: Nick; so Nick's pointing out there's actually hidden in here another equilibrium.

本·波拉克教授：尼克；所以尼克指出这里实际上隐藏着另一种平衡。

It doesn't sound like anything very realistic, but let's just focus on it a second because I think it's an important lesson here.

这听起来不太现实，但让我们先关注一下它，因为我认为这是一个重要的教训。

There's going to turn out to be important lessons.

将会有重要的教训。

The other equilibrium is--actually there's two of them--is if everybody, everyone in the room chose East Town, what would happen?

另一个平衡是——实际上有两个平衡——如果房间里的每个人都选择东镇，会发生什么？

Well, we'd have to find a way of assigning everybody, so what we would do is we'd essentially randomize over the room and half the room would be in East Town and half the room would be in West Town.

好吧，我们必须找到一种分配每个人的方法，所以我们要做的就是对房间进行随机分配，一半的房间在东镇，一半的房间在西镇。

So there's this kind of silly equilibrium in a sense, in which everyone does the same thing, and the way in which people actually get allocated is not by their choice but by this detail of the original game, which was if there was overcrowding we were going to randomize people.

因此，从某种意义上说，存在这种愚蠢的平衡，每个人都做同样的事情，而人们实际分配的方式不是由他们自己选择，而是由原始游戏的这个细节决定，也就是说，如果出现过度拥挤，我们将随机分配人员。

So, there is actually a third equilibrium which is all--there's two of these of course--so all choose the same town and get randomized.

所以，实际上存在第三种均衡——当然有两种均衡——所以所有人都选择同一个城镇并进行随机化。

To check that is an equilibrium, notice that if everyone else is choosing--if everyone else is playing this strategy of all choosing East Town and allowing the randomization device to place you, then you're completely happy to do that, completely happy to do the same thing.

为了检查这是一个均衡，请注意，如果其他人都在选择——如果其他人都在玩这个策略，所有人都选择东镇并允许随机化设备来放置你，那么你就完全乐意这样做，完全乐意做同样的事情。

So it is in fact an equilibrium.

所以这实际上是一种均衡。

So, this is a slightly odd equilibrium here, and there's immediately a Game Theory lesson here.

所以，这是一个有点奇怪的平衡，这里立即有博弈论课程。

This equilibrium, sounds like something we might see in society, and this one's certainly worth talking about, it seems a natural part of the game.

这种平衡，听起来像是我们在社会中可能看到的东西，这当然值得讨论，它似乎是游戏的自然组成部分。

This equilibrium seems to have nothing to do with anything that's really in the world.

这种平衡似乎与世界上真正存在的任何事物无关。

It's just arising from a particular detail I threw into the model at the end to make things add up.

它只是由我最后放入模型中的一个特定细节引起的，以使事情加起来。

Is that right?

是这样吗？

When I was setting up this model and describing reality, I threw this in at the end to say we better do something just to make things add up, otherwise towns are going to be overcrowded.

当我建立这个模型并描述现实时，我在最后加上了这一点，说我们最好做点什么，让事情加起来，否则城镇将变得拥挤不堪。

It wasn't that's really out there in the real world.

现实世界中的情况并非如此。

It was just to make it a game, sort of define things carefully.

只是为了让它成为一个游戏，仔细地定义事物。

This seemingly innocent detail of my modeling technique threw up another equilibrium.

我的建模技术中这个看似无辜的细节引发了另一种平衡。

So there's a sort of warning lesson here.

所以这里有一个警告教训。

The lesson is, seemingly irrelevant details of the game, things that aren't really attempts to capture reality, they're just trying to get things through in a hurry, can end up mattering--can matter.

我们的教训是，看似无关的游戏细节，那些并不是真正试图捕捉现实的东西，他们只是试图尽快完成事情，最终可能很重要——可能很重要。

They can lead your game to give you a prediction you really don't believe in.

他们可以引导你的游戏给出你真正不相信的预测。

Is that right?

是这样吗？

Well, that's a very general lesson for those of you who are going to go out and model things more widely.

嗯，对于那些想要出去更广泛地建模的人来说，这是一个非常普遍的教训。

There's a second lesson here, however.

然而，这里还有第二个教训。

The second lesson here is if in fact, if this randomization process was available, if in fact it was possible for everyone in the town to chose East Town and then have the local government randomize you, then if we use the law of large numbers, if there's 100,000--I guess 200,000 people in all and they're all being randomized, we're going to end up very, very close in the limit exactly at integration and everyone's going to be better off.

这里的第二个教训是，如果实际上，如果这个随机化过程可用，如果实际上镇上的每个人都可以选择东镇，然后让当地政府对你进行随机化，那么如果我们使用大数定律，如果总共有 100,000 人——我猜总共有 200,000 人，并且他们都被随机化，那么我们最终将非常非常接近整合时的极限，每个人都会过得更好。

Is that right?

是这样吗？

So the other kind of strange thing here is by randomizing--wll, I don't want to say randomizing, I want to say having society randomize for you--ended up being better than choosing--ended up better than, what you might want to call "active choice." Here's an example of a place where by abdicating the right to choose my town and simply by all choosing East, having society randomize for me, we ended up better off: a slightly surprising result.

所以这里的另一种奇怪的事情是通过随机化——呃，我不想说随机化，我想说的是让社会为你随机化——最终比选择更好——最终比你可能想要称之为“主动选择”的东西更好。这是一个地方的例子，通过放弃选择我的城镇的权利，而只是通过所有人选择东部，让社会为我随机化，我们最终过得更好：一个有点令人惊讶的结果。

Let's try and push this a little harder.

让我们尝试更努力地推动这一点。

Okay, so let's try and draw some lessons.

好吧，让我们尝试吸取一些教训。

We've drawn out some Game Theory lessons already from this about irrelevant details and about stability and so on.

我们已经从中汲取了一些博弈论教训，涉及不相关的细节和稳定性等。

Let's try and draw out some other lessons here.

让我们尝试从这里得出一些其他的教训。

So, one lesson might be a lesson in sociology.

因此，一课可能是社会学课。

I'm not a sociologist so I want to be careful.

我不是社会学家，所以我要小心。

I'll put it in inverted commas.

我会把它放在引号中。

The sociological lesson is what?

社会学课是什么？

In this game, segregation is what resulted, at least in the stable equilibria, and you might be tempted if you're an empirical sociologist, to go around the world and say, look I see segregation everywhere.

在这个游戏中，至少在稳定均衡中，种族隔离是结果，如果你是一位经验社会学家，你可能会忍不住去环游世界说，看，我到处都看到种族隔离。

I've gone from country to country, from society to society, and wherever I go I see segregation.

我从一个国家到另一个国家，从一个社会到另一个社会，无论我走到哪里，我都看到种族隔离。

You might be tempted to conclude that that's because people prefer segregation and that might be right.

您可能会想得出这样的结论：这是因为人们更喜欢隔离，这可能是正确的。

Nothing in this model disproves that.

这个模型中没有任何内容可以反驳这一点。

It might be the facts that--it might in fact be in the case that the reason you see segregation in virtually every society is because segregation is preferred.

事实可能是这样的——事实上，你在几乎每个社会中看到种族隔离的原因可能是因为种族偏好。

I'm not ruling that out here I'm just raising another possibility.

我并不排除这种可能性，我只是提出另一种可能性。

What's the other possibility?

还有什么可能呢？

The other possibility is it could be that preferences are like this, roughly speaking.

另一种可能性是，粗略地说，偏好可能是这样的。

People don't actually prefer segregation, but when all people act in their own interest you end up with segregation anyway.

人们实际上并不喜欢隔离，但当所有人都按照自己的利益行事时，无论如何，最终都会出现隔离。

So, the fact that that we're seeing segregation in this model does not imply that there's a preference for segregation.

因此，我们在这个模型中看到隔离这一事实并不意味着存在对隔离的偏好。

It doesn't rule it out, of course, but you can't conclude, just because you see segregation everywhere that necessarily people want segregation.

当然，这并不排除这种可能性，但你不能仅仅因为你在任何地方看到种族隔离就得出人们必然想要种族隔离的结论。

Let me just take that outside of the context of segregation, more generally.

让我把这个问题排除在种族隔离的背景之外，更笼统地说。

If you see a social phenomenon in society after society, after society whether they are anthropologists going across societies or a historian going through societies in historical time, and you see the same phenomenon in each of these societies which results from the choices of thousands of different people, you can't conclude from the fact that you see it in all of these societies that those people prefer it.

如果你在一个又一个社会、一个又一个社会中看到一种社会现象，无论是人类学家穿越各个社会，还是历史学家穿越历史时期的各个社会，并且你在每个社会中看到同样的现象，这是成千上万不同人的选择的结果，你不能从你在所有这些社会中看到的事实中得出结论，这些人更喜欢它。

All you know is that each of their individual choices add up to this social outcome that may or may not be something they prefer.

你所知道的是，他们的每个个人选择都会产生这种社会结果，这可能是他们喜欢的，也可能不是他们喜欢的。

In this case it's not.

在这种情况下，情况并非如此。

That was really Schelling's big idea.

这确实是谢林的伟大想法。

I don't want to write that all out but it's kind of a--it's a huge idea.

我不想把这些都写出来，但这是一个伟大的想法。

It got him the Nobel Prize.

这让他获得了诺贝尔奖。

So you don't want to conclude from observation straight back to preference in these strategic settings.

因此，您不想从观察中直接得出结论，得出这些战略环境中的偏好。

Now, this matters a little bit, obviously in our own society because we live in a rather segregated society.

现在，这很重要，显然在我们自己的社会中，因为我们生活在一个相当隔离的社会中。

I mean, we needn't be quite so coy as to talk about tall people and short people all the time, what we're really talking about mostly I guess in our society is ethnic segregation, and we see this very dramatically in Connecticut, for example.

我的意思是，我们不必总是羞涩地谈论高个子和矮个子，我们真正谈论的主要是我想在我们的社会中是种族隔离，例如，我们在康涅狄格州看到了这一点。

So where we live in New Haven, at least if you go a few feet outside of the university, and actually where people live in New Haven, we see that New Haven is a fairly integrated town.

因此，我们住在纽黑文的地方，至少如果你走出大学几英尺，实际上人们住在纽黑文，我们会看到纽黑文是一个相当一体化的城镇。

It's not a hundred-percent integrated town but it has a fairly wide array of ethnicities.

它不是一个百分百一体化的城镇，但它有相当广泛的种族。

But if you go upstate in Connecticut, if you go north, ignoring Hartford, if you go in the rural areas of Connecticut, you find something dramatically different.

但如果你去康涅狄格州北部，如果你向北走，忽略哈特福德，如果你去康涅狄格州的农村地区，你会发现一些截然不同的东西。

How many of you have traveled around in rural Connecticut?

你们中有多少人去过康涅狄格州的乡村旅行？

Some of you have.

你们中有些人有。

So it's quite shocking for me as a foreigner when I came here.

所以当我来到这里的时候，对于我这个外国人来说是相当震撼的。

So New Haven and the towns along the coast are fairly integrated but, if you drive into rural Connecticut, you see something quite different.

因此，纽黑文和沿海城镇相当一体化，但如果你开车到康涅狄格州乡村，你会看到截然不同的东西。

So this weekend, for example, I was up at the Durham Fair.

例如，这个周末我参加了达勒姆集市。

How many of you know what the Durham Fair is?

你们中有多少人知道达勒姆博览会是什么？

Some of you know.

你们中有些人知道。

The Durham Fair is actually very good.

达勒姆博览会实际上非常好。

It's a place where you can--it's the biggest agricultural fair in Connecticut and you can take your kids there, I have a four-year old and a two-year old, and they discover at this fair that in fact it is the case that cows say moo and sheep say baa.

这是一个你可以的地方——这是康涅狄格州最大的农业博览会，你可以带你的孩子去那里，我有一个四岁的孩子和一个两岁的孩子，他们在这个博览会上发现，事实上，牛会说“哞”，羊会说“咩”。

Who knew?

谁知道？

Right, that's a good thing.

没错，这是一件好事。

So I don't want to knock the Durham Fair.

所以我不想破坏达勒姆集市。

I think it's a good thing.

我认为这是一件好事。

But if you're wandering around the Durham Fair and you're a social scientist as I am I guess, you can't help but be struck by--how do I put this without getting in trouble?

但是，如果你在达勒姆博览会上闲逛，并且你像我一样是一名社会科学家，你会情不自禁地震惊——我该如何表达而不陷入麻烦？

You would find more ethnic diversity at a Klan rally.

在三K党集会上你会发现更多的种族多样性。

This is not--I mean this is--that's not going to get me out of trouble.

这不是——我的意思是这——不会让我摆脱麻烦。

There's got to be some way of saying that it doesn't get me in trouble.

一定有某种方式可以表明这不会给我带来麻烦。

Okay, but you know what I mean, right?

好吧，但你知道我的意思，对吧？

It is strikingly, strikingly white at the Durham Fair.

在达勒姆博览会上，它的白色非常引人注目。

We're only--what--it's about--I think it's 18 miles from New Haven.

我们只是——什么——它是关于——我认为它距离纽黑文 18 英里。

It's a 20 minute drive from here.

距离这里有20分钟车程。

It's a public event, 20 miles from New Haven.

这是一场公共活动，距纽黑文 20 英里。

So this is a phenomenon one might be tempted to say, this is because people choose, people want-- clearly, people are choosing to go to the Durham Fair; they're choosing to some extent where they live in Connecticut.

所以这是一种现象，人们可能会想说，这是因为人们选择，人们想要——显然，人们选择去达勒姆集市;他们在某种程度上选择了康涅狄格州的居住地。

And you might think the fact that we see this incredibly dramatic segregation between activities in New Haven and activities 20 miles away is evidence that people might actually prefer segregation.

您可能会认为，我们看到纽黑文的活动与 20 英里外的活动之间存在令人难以置信的巨大隔离，这一事实证明人们实际上可能更喜欢隔离。

And just to repeat, I can't prove that it isn't.

再说一遍，我无法证明事实并非如此。

But we have to be at least aware of the possibility that it's got nothing to do with the preference for segregation; it could be the preferences look like the preferences above.

但我们至少必须意识到，这可能与种族隔离的偏好无关。可能是与上面的偏好类似的偏好。

It could be that simply thousands of activities by thousands of individuals, who at least would prefer to be in the majority than the--they'd rather be in the majority and not the minority, but they'd like to be integrated--leads to incredible segregation in the aggregate.

可能只是成千上万个人的数千项活动，他们至少更愿意成为多数人——他们宁愿成为多数人而不是少数人，但他们希望被整合——导致总体上令人难以置信的隔离。

So let's talk about policy now.

那么现在我们来谈谈政策。

So as I go down here I'm getting more and more in trouble I'm sure as a foreigner.

因此，当我来到这里时，作为一个外国人，我确信我遇到了越来越多的麻烦。

This is a big policy question in the U.S. How many of you are unaware of the fact this is a big policy question in the U.S.?

这是美国的一个重大政策问题。你们中有多少人不知道这是美国的一个重大政策问题？

So at least since the 1960s this issue about segregation versus integration has been a hot-button issue, particularly in what part of social life?

因此，至少自 20 世纪 60 年代以来，关于隔离与融合的问题一直是一个热点问题，特别是在社会生活的哪个部分？

Professor Ben Polak: So Jessica's pointing out correctly that since the 1960s, this has been an incredibly hot issue in the U.S., and we're talking about the bussing debate.

Ben Polak 教授：杰西卡正确地指出，自 20 世纪 60 年代以来，这在美国一直是一个非常热门的问题，我们正在谈论关于公交车的争论。

How many of you have not heard of bussing?

你们中有多少人没有听说过公交车？

How many of you have heard of bussing?

有多少人听说过公交车？

So it's slightly a policy debate of the 1960s so people tend to forget it, but in the 1960s and early 70s, people were so worried about segregation in schools that they--that children were bussed from one neighborhood to another neighborhood to go to school.

所以这有点像 20 世纪 60 年代的政策辩论，所以人们往往会忘记它，但在 20 世纪 60 年代和 70 年代初，人们非常担心学校的种族隔离，以至于孩子们被公共汽车从一个社区送到另一个社区去上学。

This was an incredibly controversial issue in the U.S..

这在美国是一个极具争议性的问题。

And I'm not--I don't want to take a position on it here.

我不想在这里表明立场。

I just want to point out that you could arrive at that policy by thinking about this kind of model.

我只是想指出，你可以通过思考这种模型来得出该政策。

In fact, this is not just an issue of history.

事实上，这不仅仅是一个历史问题。

If you read the newspapers last week, including the Yale Daily , you'll see that it's an issue today in Connecticut.

如果您上周阅读了报纸，包括《耶鲁日报》，您就会发现这是今天康涅狄格州的一个问题。

There's a worry in Connecticut today, particularly around Hartford, that the schools are still illegally segregated.

今天，康涅狄格州，尤其是哈特福德周边地区，人们担心学校仍然存在非法隔离。

They're not in compliance with the law.

他们不遵守法律。

So this continues to be an issue.

所以这仍然是一个问题。

Let me take to a slightly less contentious area, since that's obviously so contentious, and talk about--and try and relate it back to the model a bit.

让我谈谈一个争议较少的领域，因为这显然是有争议的，并尝试将其与模型联系起来。

So in the model what we saw was if everyone chose to go to East Town or if everyone chose to go to West Town, there was a policy that was a little bit like bussing going on.

所以在模型中我们看到的是，如果每个人都选择去东镇或者如果每个人都选择去西镇，就会有一个有点像公交车的政策。

If everyone chose to go East Town, what we ended up with was randomized placement across the towns.

如果每个人都选择去东镇，我们最终会在各个城镇中随机分配。

Is that right?

是这样吗？

That's a little bit like the bussing policy.

这有点像公交车政策。

It says everybody--we take away people who we--take away people's active choice of their school and in place we just randomize everybody.

它说每个人——我们剥夺了人们对学校的主动选择，我们只是将每个人随机化。

So leaving aside what happened in Connecticut, let's think about a school, not a very good school, but a school that's a little bit north of here.

因此，抛开康涅狄格州发生的事情不谈，我们来考虑一所学校，不是一所很好的学校，而是一所位于这里以北的学校。

So there's this school called Harvard and so that's--Harvard is outside Boston and at least when I was there as a graduate student, this was a hot issue in Harvard.

有一所学校叫哈佛，哈佛位于波士顿郊外，至少当我在那里读研究生时，这是哈佛的一个热门问题。

So whereas Yale has colleges, Harvard has "houses," but they're the same thing.

因此，耶鲁大学有学院，哈佛有“学院”，但它们是同一回事。

They are the same thing; it's just the same name for they're--maybe they're not the same, maybe there's some subtle difference--but they're roughly speaking the same thing.

它们是同一件事；只是他们的名字相同——也许他们不一样，也许有一些细微的差别——但他们大致说的是同一件事。

I'm still getting hissed.

我仍然发出嘶嘶声。

So these houses at Harvard, at least until about 1990 I think, roughly, certainly when I was there as a graduate student, the way in which you ended up at a particular house was that you chose which house to be in.

所以哈佛的这些房子，至少直到 1990 年左右，我想，粗略地说，肯定是当我在那里作为一名研究生时，你最终进入某个特定房子的方式是你选择了要住的房子。

Harvard administrators started worrying.

哈佛的管理人员开始担心。

So these houses started looking--started taking on certain characteristics.

所以这些房子开始看起来——开始呈现出某些特征。

So, for example, there was a house--I guess there still is--called Elliot House, and if you went to, as I used to occasionally as a graduate student, go and eat a dinner at Elliot House, you'd meet all these people.

举个例子，有一座房子——我想现在仍然存在——叫做埃利奥特之家，如果你去，就像我以前作为研究生时偶尔去的那样，去埃利奥特之家吃晚饭，你会遇到所有这些人。

And it was kind of an odd experience because more than half of them had names that just happened to be the same as the swimming pool that had just been built at Harvard.

这是一种奇怪的经历，因为其中一半以上的名字恰好与哈佛刚刚建成的游泳池相同。

Then if you went to Adams House, where I was actually a tutor, it was a very different mix of people.

然后，如果你去亚当斯之家，我实际上是那里的一名导师，那是一个非常不同的人的组合。

And an extraordinary large portion of them were daughters of recently deposed Latin American dictators.

其中很大一部分是最近被废黜的拉丁美洲独裁者的女儿。

It's not exactly what you expect to meet at random.

这并不完全是你期望随机遇到的。

And then to give you a third example, there was a house called--it's wrong now, Kirkland House, is that right?

然后举第三个例子，有一座房子叫——现在错了，Kirkland House，是吗？

Kirkland House.

柯克兰之家。

And Kirkland House was known to be the jock house, the athlete house, and of course this being Harvard that meant that everyone there was Canadian.

众所周知，柯克兰之家是运动员之家，当然，这是哈佛大学，这意味着那里的每个人都是加拿大人。

So this was--this worried the Harvard administrators.

这让哈佛的管理人员感到担忧。

So they were worried about it and presumably they were worried that little Ms. Pinochet wasn't going to have a chance to meet Mr. Master Gretzky or something, whatever it was.

所以他们很担心，大概他们担心小皮诺切特女士不会有机会见到格雷茨基大师先生或其他什么东西，不管是什么。

So they did something about it, so what did they do?

所以他们做了一些事情，那么他们做了什么？

What did the Harvard administrator so to change this outcome?

哈佛管理者到底做了什么才能改变这个结果呢？

Anyone know?

有人知道吗？

Shout it out.

大声喊出来。

Student: They randomized it.

学生：他们随机化了。

So you go in as a group but you don't get to choose which one you're going to go into.

所以你作为一个团体进去，但你不能选择你要加入哪一个。

Professor Ben Polak: Right, so what happened was they imposed randomization.

本·波拉克教授：是的，所以发生的事情是他们强加了随机化。

They did exactly the policy that we described up here.

他们完全执行了我们在这里描述的政策。

This was essentially their policy and notice actually the policy that they adopted, the policy in which house allocation was random, was basically going to what the policy Yale had all along.

这本质上是他们的政策，实际上请注意，他们采用的政策，即随机分配住房的政策，基本上遵循耶鲁大学一直以来的政策。

So as happens in other cases in education, Harvard arrived at the Yale solution eventually.

正如教育领域的其他案例一样，哈佛最终找到了耶鲁大学的解决方案。

So randomization or more dramatic things like bussing are policies arrived at really because we know of the existence of these models.

因此，随机化或更戏剧性的事情（例如公交车）实际上是因为我们知道这些模型的存在而制定的。

I'm not saying these are right or wrong.; It's not my position to say that they're right and wrong.

我并不是说这些是对还是错。我没有立场说他们是对还是错。

I'm saying if you were going to argue for these policies, this might be a way in which you might argue for these policies.

我是说，如果你要为这些政策辩护，这可能是你为这些政策辩护的一种方式。

Now, I want to bring out the third lesson here, and the third lesson here is once--we've drawn--I've already said I won't repeat it, there's a Game Theory lesson here about irrelevant details; irrelevant details mattering, but I want to bring out one more Game Theory lesson here.

现在，我要在这里讲出第三课，这里的第三课是有一次——我们画过——我已经说过我不再重复了，这里有一个博弈论课，讲了一些不相关的细节；不相关的细节很重要，但我想在这里再讲一堂博弈论课。

So here we've been discussing randomization in these settings in the following form.

因此，我们在这里以下面的形式讨论这些设置中的随机化。

In the game, the local governments randomized where you lived.

在游戏中，地方政府随机分配了你居住的地方。

In the bussing experiment it was randomly chosen who went to which school.

在校车实验中，随机选择谁去哪所学校。

I guess it was done by social security number, I don't know.

我猜这是通过社会安全号码完成的，我不知道。

In Harvard and in Yale too, there's some randomization about which college house you live in.

在哈佛和耶鲁大学，你住在哪所大学宿舍也是随机的。

That's one way to achieve randomization: you could have it done centrally.

这是实现随机化的一种方法：您可以集中完成它。

The central administration, this local government, the central government, can randomly assign people.

中央政府，这个地方政府，中央政府，可以随机分配人。

But in principle, there's another way to achieve randomization.

但原则上，还有另一种方法可以实现随机化。

What's the other way you could achieve randomization?

实现随机化的其他方法是什么？

Both in this experiment and elsewhere.

无论是在这个实验中还是在其他地方。

There was a hand way in the back; can we get it?

背后有一只手；我们能得到它吗？

Professor Ben Polak: Right, so rather than having centralized randomization, in principle, you could get to the same outcome by having individual randomization.

Ben Polak 教授：是的，原则上，你可以通过个体随机化来获得相同的结果，而不是集中随机化。

So there's another possibility here, which is individual randomization.

所以这里还有另一种可能性，那就是个体随机化。

In principle, you could have everybody in the room decide that what they're going to do is throw a coin, a fair coin, each of them separately, and if it comes up heads they'll go to East Town and if it comes up tails they'll go to West Town, and if all of them do that and they all stick with it, again, by the law of large numbers we'll get pretty close to integrated towns.

原则上，你可以让房间里的每个人决定他们要做的就是扔一枚硬币，一枚公平的硬币，每个人都分开，如果正面朝上，他们会去东镇，如果反面朝上，他们会去西镇，如果他们都这样做，并且都坚持下去，再次，根据大数定律，我们将非常接近一体化城镇。

But it won't have been some central authority doing the randomization.

但不会是某个中央机构进行随机化。

it'll be each of you individually doing the randomization.

你们每个人都会单独进行随机化。

Is that right?

是这样吗？

So that's a little bit different, and notice that if everyone else is doing this, if everyone else is randomizing, the towns will in fact be, at least asymptotically, the towns will be equally mixed and you'll be happy to randomize.

所以这有点不同，请注意，如果其他人都这样做，如果其他人都随机化，那么城镇实际上将是，至少渐进地，城镇将同样混合，你会很乐意随机化。

Why?

为什么？

Because you'll be indifferent whether you end up in East Town or West Town, so you may as well toss a coin anyway.

因为无论你最终是在东镇还是西镇都无所谓，所以你还是抛硬币吧。

So this is actually another Nash Equilibirum.

所以这实际上是另一个纳什均衡。

I'm being a bit loose here but we'll be much more precise in a moment.

我在这里有点宽松，但我们稍后会更加精确。

There's another Nash Equilibrium, yet another Nash Equilibrium, in this model, and it isn't as silly in some sense as having everyone going to East Town and then having the government randomize.

在这个模型中，还有另一个纳什均衡，又一个纳什均衡，从某种意义上说，它并不像让每个人都去东镇然后让政府随机化那么愚蠢。

It's each person on their own tossing a coin and deciding where to go.

这是每个人自己抛硬币并决定去哪里。

It sounds a little bit less like central planning and more like choice.

这听起来有点不像中央计划，而更像是选择。

Now, however, as soon as we introduced the idea of individual randomization we've gone a little bit beyond where we got to in the class.

然而，现在，一旦我们引入了个体随机化的想法，我们就已经超出了课堂上的范围。

Why?

为什么？

Because so far in the class, we've talked about strategies as the choices that are available to you.

因为到目前为止，在课堂上，我们已经讨论了作为您可以选择的策略。

The choices available to you were go to the East Town, go to the West Town.

你可以选择去东镇，去西镇。

Or, in the numbers game we played, it was choose a number.

或者，在我们玩的数字游戏中，是选择一个数字。

Or, in the ά β game we played, it was choose ά or β.

或者说，我们玩的ββ游戏中，是选择β或β。

Or deciding whether to invest or not, it was invest or don't invest, and so on and so forth.

或者决定是否投资、投资还是不投资等等。

What we're seeing now is a new type of strategy.

我们现在看到的是一种新型战略。

And the new type of strategy is to randomize over your existing strategies.

新型策略是对现有策略进行随机化。

So we're introducing here a new notion that's going to occupy us for the rest of the week, and the new notion is a randomized, or as we're going to call them, "mixed strategies."

因此，我们在这里引入一个新概念，它将在本周剩下的时间里占据我们的注意力，这个新概念是随机的，或者我们将称之为“混合策略”。

So what is--I'll be more formal next time, but what is a mixed strategy?

那么什么是——下次我会更正式一些，但是什么是混合策略呢？

It is a randomization over your pure strategies.

这是对你的纯粹策略的随机化。

The strategies that we've dealt with in the course up to now, from now on, we're going to refer to those as pure strategies, they are your choices.

到目前为止我们在课程中讨论过的策略，从现在开始，我们将把它们称为纯粹的策略，它们是你的选择。

And we're going to expand your actual available choices to include all randomizations over those.

我们将扩展您实际可用的选择，以包括所有随机选择。

This may seem a little weird, so to make it seem a little bit less weird, let's move immediately to an example.

这可能看起来有点奇怪，所以为了让它看起来不那么奇怪，让我们立即看一个例子。

So, the example I want to talk about is a game that I think is familiar to a number of you, but I'll put up the payoffs and see.

所以，我想谈论的例子是一款我认为你们中的许多人都熟悉的游戏，但我将给出收益并看看。

So the payoffs look like this; each person has--there's two players--each of them has three strategies and the payoff matrix looks as follows: (0,0) is down the lead diagonal and going around it's going to be (1,-1), (-1,1), (-1,1), (1,-1), (1,-1), (-1,1).

所以回报看起来像这样；每个人有两个玩家，每个人都有三种策略，支付矩阵如下所示：(0,0) 沿着主对角线向下，绕一圈将是 (1,-1)、(-1,1)、(-1,1)、(1,-1)、(1,-1)、(-1,1)。

Now, without me putting anymore letters up there what is this game?

现在，不用我再写字母了，这个游戏是什么？

Right, so somebody shouted it out, this is "rock, paper, scissors." So I think, if I get this right, this better be rock, and this better be scissors actually and this better be paper.

对，所以有人喊出来，这就是“石头、剪刀、布”。所以我想，如果我做对了，这最好是石头，这最好是剪刀，这最好是布。

This is rock, paper, scissors.

这是石头、剪刀、布。

How many of you have not heard of rock, paper, scissors?

你们中有多少人没有听说过石头、剪刀、布？

Good; the other day I was flicking channels and ESPN had a rock, paper, scissors contest and I just want to know who watches that?

好的;有一天，我在浏览频道时，ESPN 举办了一场石头剪刀布比赛，我只是想知道谁在观看？

So this is quite a fun game and it occurs all over the place.

所以这是一个非常有趣的游戏，而且它随处可见。

It occurs even in episodes of the Simpsons.

它甚至出现在《辛普森一家》的剧集中。

So there's I guess by now famous episode of the Simpsons in which the kids who are I guess Bart and Lisa, is that right?

所以我想现在有一个著名的《辛普森一家》剧集，其中的孩子是巴特和丽莎，对吗？

Somebody help me out here, Bart and Lisa is that what they're called?

有人来帮帮我吗，巴特和丽莎就是他们的名字吗？

So Bart and Lisa are playing rock, paper, scissors for some purpose, to allocate some prize.

所以巴特和丽莎为了某种目的而玩石头剪刀布游戏，以分配一些奖品。

And Bart is seen to think (as you can do it in cartoons) seed to think, "Rock, paper, scissors, I love playing rock, paper, scissors.

巴特会思考（就像你在卡通片中那样）思考，“石头、剪刀、布，我喜欢玩石头、剪刀、布。

Rock rocks.

岩石岩石。

What could possibly beat rock?" Lisa is seen to think, "I love playing rock, paper, scissors; Bart always chooses rock."

有什么能打败摇滚呢？”丽莎在想：“我喜欢玩石头、剪刀、布；巴特总是选择石头。”

That's really a hint of where we're going here.

这确实是我们要去哪里的暗示。

That's a hint of where we're going.

这是我们要去哪里的暗示。

This is a very simple game "rock, paper, scissors," but it's pretty obvious, I hope, that pure strategies in this game are probably not enough to model this guy.

这是一个非常简单的游戏“石头、剪刀、布”，但我希望，很明显，这个游戏中的纯粹策略可能不足以模拟这个人。

Is that right?

是这样吗？

So, in particular, I claim and if necessary will prove that there is no Nash Equilibrium in pure strategies, in the strategies that we've been looking at up to now.

因此，我特别声称，如果有必要的话，我将证明，在纯策略中，在我们迄今为止一直在研究的策略中，不存在纳什均衡。

There's no Nash Equilibrium in which people choose pure strategies here.

这里不存在人们选择纯粹策略的纳什均衡。

So there's no Nash Equilibrium.

所以不存在纳什均衡。

From now on I'm going to use the term pure strategies to mean the strategies we've been looking at so far.

从现在开始，我将使用术语“纯策略”来表示我们迄今为止一直在研究的策略。

So by pure strategies here, the set of pure strategies is equal to the set rock, paper, and scissors; so what we've called strategies up to now in the class.

所以这里的纯策略，纯策略集合等于石头、剪刀、布；到目前为止，我们在课堂上称之为策略。

So now, can everyone see why there's not going to be a Nash Equilibrium in pure strategies?

那么现在大家能明白为什么纯策略中不会存在纳什均衡了吗？

Let's just talk it through.

我们就来说说吧。

So, if one person plays rock, the best response against rock is?

那么，如果一个人玩摇滚，对抗摇滚的最佳反应是什么？

Let's try that again; you all know.

让我们再试一次；你们都知道。

Best response to rock is?

对摇滚最好的回应是？

Paper; but if you played paper the best response to paper is?

纸;但如果你玩纸，对纸的最好反应是什么？

Scissors; and the best response to scissors is?

剪刀;对剪刀的最佳反应是？

Rock; so everyone knows how to play this game.

岩石;所以每个人都知道如何玩这个游戏。

So clearly, there's not going to be a pure strategies Nash Equilibrium, right?

很明显，不会有纯粹的策略纳什均衡，对吧？

Because any attempt to look for best responses that are best responses to each other can lead to a cycle.

因为任何寻找彼此最佳响应的最佳响应的尝试都可能导致循环。

Everyone see that?

大家看到了吗？

There's no hope of finding two pure strategies that are best responses to each other because of that cycle.

由于这个循环，我们不可能找到两种彼此最佳反应的纯粹策略。

It's also, I claim, pretty easy to figure out what must be the Nash Equilibrium in this game.

我认为，很容易弄清楚这个游戏中的纳什均衡是什么。

We're going to prove it in a second, more or less, but nevertheless I think we actually know it.

我们将或多或少地在一秒钟内证明这一点，但尽管如此，我认为我们实际上知道这一点。

So what is, in fact, the Nash Equilibrium strategy in this game?

那么，这场博弈中的纳什均衡策略实际上是什么呢？

Let's get some--you could probably cold call somebody; just cold call somebody at random.

让我们找一些人吧——你可能可以给某人打电话；随机给某人打电话。

I'm sure they all know it, so just cold call somebody.

我确信他们都知道这一点，所以就打电话给某人吧。

So what's the--yeah the gentleman in yellow--what's the Nash Equilibrium in this game?

那么，是的，黄衣绅士，这个游戏中的纳什均衡是什么？

I'm allowing you mixed strategies now; I'm allowing you to randomize--if I allow you to randomize over your pure strategies, I'm allowing you to play mixed strategies, what's the mixed strategy we think people are going to play?

我现在允许你使用混合策略；我允许你随机化——如果我允许你随机化你的纯策略，我允许你玩混合策略，我们认为人们会玩什么混合策略？

Professor Ben Polak: No idea; you should be on that--I want to play against you.

Ben Polak 教授：不知道；你应该这么做——我想和你比赛。

Let's have someone else picked out, yeah.

让我们再选一个人吧，是的。

What do we think is likely to be the mixed strategy people are going to play in this game in equilibrium?

我们认为人们在这个均衡博弈中可能会采取什么混合策略？

Professor Ben Polak: No idea?

本·波拉克教授：不知道吗？

Who here is regarded as a champion of "rock, paper, scissors" player?

这里谁被称为“石头、剪子、布”选手的冠军？

Professor Ben Polak: Thank you.

本·波拉克教授：谢谢。

I thought that would be okay.

我以为这样就可以了。

So I'm--people didn't play this with their siblings?

所以我——人们没有和他们的兄弟姐妹一起玩这个？

Maybe they did and they just lost a lot.

也许他们确实做到了，但他们只是失去了很多。

So I claim as a guess that the Nash Equilibrium is each player, both players, each player chooses--and I'm going to call it 1/3, 1/3, 1/3--in other words, each player is playing the mixed strategy 1/3, 1/3, 1/3 which I'm sorry I didn't catch your name, your name is?

所以我认为纳什均衡是每个玩家、两个玩家、每个玩家选择的——我将称之为 1/3、1/3、1/3——换句话说，每个玩家都在玩混合策略 1/3、1/3、1/3，很抱歉我没听清你的名字，你的名字是？

Professor Ben Polak: Which is Moses' recommended strategy.

Ben Polak 教授：这是摩西推荐的策略。

So each player is going to play 1/3, 1/3, 1/3.

所以每个玩家要玩 1/3、1/3、1/3。

So I'm actually amazed that people didn't realize that and again I'm sort of puzzled as what happened in your childhood, but never mind.

所以我真的很惊讶人们没有意识到这一点，我再次对你童年发生的事情感到困惑，但没关系。

Let's figure out what in fact is the payoff to each of the pure strategies against it.

让我们弄清楚每种针对它的纯策略的实际回报是什么。

So what I'm going to do is I'm going to show you that this will in fact be a Nash Equilibrium by showing you payoffs.

所以我要做的是通过向你展示收益来向你展示这实际上是一个纳什均衡。

So what's the expected payoff--to start off, what's the expected payoff of rock against 1/3, 1/3, 1/3?

那么预期收益是多少——首先，摇滚相对于 1/3、1/3、1/3 的预期收益是多少？

So the other person's randomizing equally over rock, paper, scissors and I'm going to choose rock.

所以另一个人在石头、布、剪刀上随机分配，我会选择石头。

What will be my payoff, my expected payoff?

我的回报是什么，我的预期回报是什么？

Well, I claim that with probability 1/3 I'll meet another rock and get 0, and with probability of 1/3 I'll meet a scissors and get -1 and with probability of 1/3 I'll meet paper--I said that wrong, let me start again.

好吧，我声称以 1/3 的概率我会遇到另一块石头并得到 0，以 1/3 的概率我会遇到剪刀并得到 -1，以 1/3 的概率我会遇到布 - 我说错了，让我重新开始。

The probability 1/3 I'll meet rock and get 0, probability 1/3 I'll meet scissors and get +1, and a probability 1/3 I'll meet paper and get -1.

概率 1/3 我会遇到石头并得到 0，概率 1/3 我会遇到剪刀并得到 +1，概率 1/3 我会遇到布并得到 -1。

Is that right this time?

这次是这样吗？

So my expected payoff is what?

那么我的预期回报是多少？

It's 1/3 of 0, 1/3 of +1, and 1/3 of -1 for a total of 0.

它是 0 的 1/3、+1 的 1/3 和 -1 的 1/3，总共 0。

It's not hard to check that the same is true if I chose scissors.

如果我选择剪刀，不难检查是否也是如此。

If I chose scissors and the other person is randomizing 1/3, 1/3, 1/3 then with probability 1/3 I will get -1, with probability 1/3 I will meet another scissors and get 0, and with probability 1/3 I'll meet paper and get 1, and once again it nets out at 0.

如果我选择剪刀，而另一个人随机选择 1/3、1/3、1/3，那么我将以 1/3 的概率得到 -1，以 1/3 的概率我会遇到另一把剪刀并得到 0，以 1/3 的概率我会遇到纸并得到 1，并且再次得到 0。

Finally, if I played paper, and once again I'm going to meet somebody who is in fact randomizing a 1/3, 1/3, 1/3 over rock, paper, and scissors then with probability 1/3 I'll meet a rock and get 1, with probability 1/3 I'll meet scissors and get -1, and with probability 1/3 I'll meet paper and get 0.

最后，如果我玩布，我将再次遇到一个实际上在石头、布和剪刀上随机化 1/3、1/3、1/3 的人，那么我将以 1/3 的概率遇到石头并得到 1，以 1/3 的概率我会遇到剪刀并得到 -1，以 1/3 的概率我会遇到布并得到 0。

Is that correct?

这是正确的吗？

Everyone happy with that?

大家对此满意吗？

So it will net out at 0.

所以它的净值将为 0。

So each of the pure strategies here rock, paper, or scissors–actually, I did rock, scissors, and paper, each of them when they play against 1/3, 1/3, 1/3 they yield an expected payoff of 0.

所以这里的每一种纯策略都是石头、剪刀、布——实际上，我做了石头、剪刀和布，每种策略在与 1/3、1/3、1/3 对抗时，它们的预期收益都是 0。

What about playing the mix itself?

播放混音本身怎么样？

What's the expected payoff of playing--myself playing 1/3, 1/3, 1/3 if I play against somebody who's playing 1/3, 1/3, 1/3?

如果我和玩 1/3、1/3、1/3 的人玩，我自己玩 1/3、1/3、1/3 的预期收益是多少？

Well, a 1/3 of the time I'm going to be playing rock, so a 1/3 of the time I'm going to be playing rock and then I'll get the expected payoff from playing rock against 1/3, 1/3, 1/3.

好吧，1/3 的时间我会玩摇滚，所以 1/3 的时间我会玩摇滚，然后我会从玩摇滚与 1/3、1/3、1/3 的比赛中获得预期的回报。

What was the expected payoff from playing rock against 1/3, 1/3, 1/3?

与 1/3、1/3、1/3 玩摇滚的预期回报是多少？

Zero.

零。

So, 1/3 of the time I'm going to play rock and I'm going to get this 0.

所以，1/3 的时间我会玩摇滚，而我会得到这个 0。

And 1/3 of the time I'm going to be playing scissors, and then I'm going to get the expected payoff from scissors against 1/3, 1/3, 1/3 and what was the expected payoff of scissors against 1/3, 1/3, 1/3?

1/3 的时间我会玩剪刀，然后我会得到剪刀对 1/3、1/3、1/3 的预期收益，剪刀对 1/3、1/3、1/3 的预期收益是多少？

Zero again.

又归零了。

So 1/3 of the time I'll play scissors and I'll get this 0.

所以 1/3 的时间我会玩剪刀，我会得到这个 0。

And 1/3 of the time I'll be playing paper, in which case I'll get the expected payoff of paper against 1/3, 1/3, 1/3 which once again is 0.

1/3 的时间我会玩纸牌，在这种情况下，我会得到纸牌对 1/3、1/3、1/3 的预期收益，这又是 0。

So my total expected payoff is 1/3 of 0, plus 1/3 of 0, plus 1/3 of 0 which, for the math phobics in the room comes, out as 0.

所以我的总预期收益是 0 的 1/3，加上 0 的 1/3，加上 0 的 1/3，对于房间里的数学恐惧症来说，结果是 0。

So notice what we've shown here; we've shown that if I play what I claim is the equilibrium strategy 1/3, 1/3, 1/3 against 1/3, 1/3, 1/3 I get 0 and if I played any other strategy I'd still get 0, is that right?

所以请注意我们在这里展示的内容；我们已经证明，如果我采用我所声称的均衡策略 1/3、1/3、1/3 对抗 1/3、1/3、1/3，我会得到 0，如果我采用任何其他策略，我仍然会得到 0，对吗？

So therefore playing 1/3, 1/3, 1/3 is indeed a best response, albeit weakly, it is indeed a best response and this is in fact a Nash Equilibrium.

因此，打 1/3、1/3、1/3 确实是一个最佳反应，尽管很弱，但它确实是一个最佳反应，这实际上是一个纳什均衡。

So what we've shown here is in "rock, paper, scissors" playing 1/3, 1/3, 1/3 against the other guy playing 1/3, 1/3, 1/3 is a best response, anything would be a best response, but in particular it's a best response.

所以我们在这里展示的是在“石头、剪刀、布”中，玩 1/3、1/3、1/3，而另一个玩 1/3、1/3、1/3 的人是最佳响应，任何内容都可以是最佳响应，但特别是它是最佳响应。

So if both people play this, one person plays a 1/3, 1/3, 1/3 and the other person plays 1/3, 1/3, 1/3, this is a Nash Equilibrium.

因此，如果两个人都玩这个，一个人玩 1/3、1/3、1/3，另一个人玩 1/3、1/3、1/3，这就是纳什均衡。

So we're belaboring a point that we all knew already: playing 1/3, 1/3, 1/3 against--if everyone plays a 1/3, 1/3, 1/3 that is a Nash Equilibrium.

因此，我们正在详细阐述我们都已经知道的一点：以 1/3、1/3、1/3 对抗——如果每个人都以 1/3、1/3、1/3 对抗，这就是纳什均衡。

It's a little harder to show but it's true, that that's the only equilibrium in the game.

展示起来有点困难，但这是事实，这是游戏中唯一的平衡。

Before you go, if any of you know the people that run ESPN, this is why watching this game on TV is unlikely to be exciting.

在你出发之前，如果你们中有人认识 ESPN 的运营者，这就是为什么在电视上观看这场比赛不太可能令人兴奋的原因。

We'll come back and look at other mixed strategies and more interesting games on Wednesday.

我们将在周三回来看看其他混合策略和更有趣的游戏。