耶鲁大学博弈论公开课 - 第18集 纳什均衡之立场选择、种族隔离与策略随机化

视频全文转录（中英双语）

以下为完整中英双语转录，已加标点。英文在前，中文在后，逐段对照。

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

[段 1]

So last time we left things in the middle of a model which was the candidate voter model. And what was, I don’t want to go over the whole model again, but just to iterate a little bit what was different about that model from what we saw before. And the main thing that was different was that the candidates cannot choose their positions. If you like, every voter is a potential candidate, but you know the positions of the voters. And let me just bring out two lessons that we left hanging last time. I want to put them on the board to make sure they’re in your notes. So the first lesson is, one thing we saw already last time, is there can be lots of different national equilibria in this model. There are multiple possible national equilibria in this model. and more to the point, not all of those equilibria have the candidates crowded at the center. Alright? We saw early on in the classic downed or median voter model that that model predicted crowding the center. This one doesn and we come back to that in a second And the second thing we saw last time was that entry can if you enter on the left one effect of entering on the left can be to make the candidate on the right more likely to win.

[译文 1]

上一次我们把讨论停留在一个模型上，那就是候选人选民模型。我不想再重复整个模型，但只是想稍微回顾一下这个模型与之前看到的有什么不同。最主要的区别是候选人不能选择自己的立场。可以说，每个选民都是潜在的候选人，但我们知道选民们的立场。让我把上次悬而未决的两个要点写出来，确保你们记在笔记里。第一个要点是，我们上次已经看到的一点，那就是这个模型中可以存在很多不同的全国均衡。这个模型有多个可能的全国均衡。更重要的是，并非所有这些均衡都让候选人聚集在中心。好吗？我们在早期的经典唐斯模型或中位选民模型中看到，那个模型预测候选人会在中心聚集。这个模型则不会，我们稍后会回到这个问题。第二个我们上次看到的要点是，进入可能——如果你在左侧进入——进入左侧的一个效应是让右侧的候选人更有可能获胜。

[段 2]

And 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 wins. It can lead to the winner being further away from your ideal position. and just to revisit this a little bit let me go back and just illustrate those two points again so I’ll take a row further back this time I’m getting a nice full row we can see the whole of 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 so let me choose this row this row good, ok I’m sorry, the people in the balcony are going to have to imagine this it’s a penalty being in the balcony so this row and here’s my left wing of this row for everyone who’s in front of it, it’s almost everybody and here’s my right wing and let’s try and illustrate some equilibrio we saw last time 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, sorry and let assume all the seats are filled for now just to help me out a little bit since I doing this on the fly this I going to claim is an equilibrium Notice that there are two candidates standing and notice that they are not particularly close to the centre We have our sort of middle of the Democratic Party left candidates.

[译文 2]

反过来，如果你在右侧进入，你是一个右翼选民候选人，你进入的话，这可能更可能导致左侧获胜。这可能导致获胜者离你的理想立场更远。让我再稍微回顾一下，我来举例说明这两个要点。这次我会退后一排，这样我们可以看到全景，至少几乎不会混淆什么是左翼和右翼。让我选这一排，这一排好，哦抱歉，楼上的人只能想象了，在楼上是一种惩罚。这一排，这是我的左翼，对所有坐在前面的人来说，几乎是所有人，这是我的右翼。让我们举例说明上次看到的一些均衡。特别是，如果我能让穿蓝色耶鲁衬衫的那位站起来一下，还有那位拿电脑的站起来一下，抱歉，假设现在所有座位都坐满了，只是帮我一点，因为我是在现场发挥。我要声称这是一个均衡。注意，这里有两个候选人站着，而且他们离中心并不特别近。我们有民主党中间偏左的候选人。

[段 3]

So I’m tempted to give you a name, but perhaps I would not. And here’s a middle of the Republican Party candidates. And if this is the election, they’re going to split the votes equally if I’d actually chosen correctly. So let’s observe some things here. First, for this to be an equilibrium, it better be the case that they’re symmetric on left and right. right, 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’ll deviate to drop out, is that correct? alright, let’s also illustrate this so we’ve already illustrated that they’re not particularly close to the centre, let’s illustrate what I meant by somebody on the left causing the right wing to win so if this was the election and 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 suppose this is Dennis Kucinich and he enters alright 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 candidate wins If these three guys are standing, Denis Kucinich is going to steal some votes from our centre-left candidate and cause our centre-right candidate to win.

[译文 3]

我很想给你们一个名字，但也许我不会。这是共和党中间偏右的候选人。如果这是选举的话，如果我选得正确的话，他们将平分选票。让我们观察一些事情。首先，要使这成为一个均衡，双方必须在左右两侧对称，对吧？如果他们不对称，那它就不是一个均衡，因为其中一个候选人肯定会输，按照我们建立模型的方式，这意味着其中一方会退出，他们会偏离退出，是这样吗？好，让我们也说明一下。我们已经说明了他们离中心并不特别近，让我们说明一下我说的"左侧的人导致右翼获胜"是什么意思。所以如果这是选举，我们越来越接近选举了，突然我们的一个左翼人士，比如 Dennis Kucinich 之类的，决定参选。假设这是 Dennis Kucinich，他参选了。如果 Dennis Kucinich 我们的左翼候选人参选，可能参选本身有一些道义上的胜利，但结果将是我们的右翼候选人获胜。如果这三个人站着，Denis Kucinich 将从我们的中间偏左候选人那里抢走一些选票，导致我们的中间偏右候选人获胜。

[段 4]

Now that may or may not be Mr. Kucinich’s intention, but we should be aware of it. alright? So this is a real effect, go back to the 2000 election and think about what happened when Nadia entered and do the math as it were, alright? Which people didn’t do at the time apparently alright? And if you were listening to the newspapers, or if you read the New York Times this morning or listen to the radio this morning, you’ll find exactly the same debate is going on now in the Republican Party some members, actually it’s the other way around, it’s the right wing this time, switch it around so the right wing, 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. And of course the debate is exactly in these terms. If they run a right-wing third-party candidate, so that would be our guy over here, this might have payoff in terms of other things, but in terms of the election, it’s going to result in Hillary winning. All right? All right? All right. Thank you. OK. So we’ve seen these two effects. They’re very real effects. We’re not necessarily getting people crowding the center, and we have to worry about, when we enter, causing the other wing to win.

[译文 4]

这可能不是 Kucinich 先生的意图，但我们应该知道。好吗？这是一个真实的效应。回到 2000 年选举，想想法斯滕伯格进入时发生了什么，做一下计算，好吗？当时人们显然没有这样做。好吧，如果你听报纸，或者如果你今天早上读了《纽约时报》或听广播，你会发现现在共和党内部正在进行完全相同的辩论。有些成员——其实是反过来，是右翼——反过来，所以共和党右翼的一些人说，如果结果 Giuliani——他目前在民调中领先——赢得共和党提名，他们将推出一个第三党候选人。当然这场辩论完全就是用这些术语。如果他们推出一个右翼第三党候选人，那就是我们这边这位，这可能在其他方面有回报，但在选举方面，这会导致希拉里获胜。好吗？好，谢谢。好的，我们已经看到了这两个效应。它们是非常真实的效应。我们不一定能看到人们聚集在中心，而且我们必须担心，当我们进入时，会导致另一翼获胜。

[段 5]

All right? I’m not making a left-wing, right-wing politics argument here. This is true for both wings, symmetrically. Alright, 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 centre can we be. 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, 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? I’ll get in trouble whoever I name, so whoever the most crazy right-wing candidate is you can think of. alright, and then we have the full spectrum represented with just the extremes standing, alright? Here we have, they’re symmetric around the center, right? But I claim this is not an equilibrium. So the people on the balcony I got the extreme right and the extreme left standing here Why is this not an equilibrium Yeah actually I should it my fault I should have brought the mic Can I have the mic Ale Sorry my fault Thanks. It’s this one. So Katie, why is that not an equilibrium? Because the person in the centre could stand up and win. Exactly, because the person in the centre could stand up and win. Actually, it doesn’t only have to be the person exactly in the centre.

[译文 5]

好吗？我不是在做一个左翼、右翼政治论点。这对两翼都是真的，对称地。好吧，让我们试着在这里引入另一个想法，这就是我们上次结束的地方，也就是候选人离中心能有多远。让这些人先坐下，让我们再让 Kucinich 先生站起来，我知道这不是你的真名，但没关系，Kucinich 先生在这边左侧，还有疯狂右翼先生，他的名字我——这些候选人中最疯狂右翼的人是谁？我不管提谁的名字都会惹麻烦，所以你可以想象最疯狂右翼的候选人。好吧，然后我们有完整的频谱，只有极端派站着，好吗？在这里，他们是围绕中心对称的，对吧？但我声称这不是一个均衡。所以楼上的朋友们，我让极右翼和极左翼站在这里。为什么这不是一个均衡？是的，其实这是我的错，我应该带麦克风的。我能把麦克风递过来吗？Ale 抱歉，我的错。谢谢。是这个。所以 Katie，为什么这不是一个均衡？因为中心的人可以站起来获胜。没错，因为中心的人可以站起来获胜。实际上，不一定必须是正好在中心的人。

[段 6]

A wide array of centre candidates could deviate and win at this point. Alright, so if this was the two candidates standing, and you imagine a third candidate standing, who for example is this gentleman, alright? If he was to stand, well, perhaps he needs to be a little bit close to the centre, let’s say this guy, the guy in grey, alright? Fairly clearly, he’s going to end up winning. Alright? So, this is the third lesson. If the candidates are too far apart, we’re going to see some centre entry, which is going to win. Thank you guys. Alright? So even though there isn’t this full downsian effect of pushing candidates towards the centre. Even though we don’t have the median voter theorem here, we still have part of the intuition surviving. The part of the intuition that survives is if the candidates are too far apart, then the centre will enter and win. So there is something pulling people to the centre still. Thank you Ah not much I can Sorry someone asking if I can move the podium I can do so a little bit Not much let me try Is that better Alright, so a reasonable question here is just how far apart in equilibrium can the candidates be? We’ve established that we can have two candidates and they needn’t both be at the centre.

[译文 6]

很大范围的中间候选人都可以偏离并获胜。好吧，所以如果这是两个候选人站着，你想象有第三个候选人站着，比如这位先生，好吗？如果他要站起来，也许他需要离中心近一点，比如这位，穿灰色衣服的这位，好吗？相当明显，他将最终获胜。好吗？所以，这是第三个要点。如果候选人离得太远，我们将看到中间候选人进入并获胜。谢谢各位。好吗？所以，尽管没有这种全面的唐斯效应把候选人推向中心。尽管我们这里没有中位选民定理，我们仍然保留了部分直觉。保留的那部分直觉是，如果候选人离得太远，那么中间将进入并获胜。所以仍然有一些力量把人们拉向中心。谢谢。啊，没什么我能做的。抱歉，有人问我能不能挪一下讲台。我可以稍微挪一下。不是很多，让我试试。这样更好吗？好的，这里一个合理的问题是，在均衡中候选人能离得多远？我们已经确定我们可以有两个候选人，而且他们不必都在中心。

[段 7]

We’ve established that they can’t be at the extremes. all right, how far apart can they be? How far apart can they be? Well this is really just a, it’s kind of a nerdy question to get it precisely, but let’s get it precise nevertheless. All right, so let’s have a look. Let’s use the other board. So here’s the full extent of our political spectrum from zero to one, and let me just try and illustrate how far apart these can be. What I’m going to do is I’m going to divide this into sixths. One sixth, two sixths, a half, four sixths, five sixths. Alright? So I claim, and I’ll show afterwards, I’ll claim that provided the two candidates aren’t outside of a sixth and five sixths that that will be an equilibrium Alright So in In particular if the candidates are just inside I put them just there a sixth and just inside five-sixths, or just more than a sixth and just less than five-sixths, right, so here’s one of these candidates who’s standing, and here’s the other one, then we’ll be okay. Now why? Why is that the right answer? Anyone want to try a guess? This was the question I sent you over the weekend. I’m sure you were doing other things over the weekend. But nevertheless, why is this the answer? Well, let’s see.

[译文 7]

我们已经确定他们不能处于极端位置。好吧，他们能相隔多远？他们能相隔多远？这个问题确实有点书呆子式的精确，但我们还是精确地来算一算。好吧，让我们来看一看。让我们用另一块白板。这是我们政治光谱从零到一的全部范围，让我试着说明这些位置能相隔多远。我要做的是把它分成六份。六分之一、六分之二、二分之一、六分之三、六分之四、六分之五。好吧？我声称，而且之后我会证明，只要两位候选人不在六分之一和六分之五的范围之外，那就会是一个均衡。好吧，特别是在这种情况下，候选人刚好在六分之一以内和刚好在六分之五以内，或者刚好超过六分之一和刚好低于六分之五。好的，这是一位候选人的位置，这是另一位候选人，那么就可以了。为什么？为什么这是正确答案？有人想猜一下吗？这是我周末发给你们的问题。我确定你们周末在做其他事情。但不管怎样，为什么这是答案呢？好吧，让我们来看看。

[段 8]

Let’s see. What are they vulnerable to? They’re vulnerable to deviation by somebody entering at the center. And if somebody enters at the center, what would make somebody enter at the center? They’re going to enter at the center if they can win, basically. All right? So if they enter at the center in this case, let’s see how many votes everyone gets. So if you enter at the center here, here’s our new candidate, who’s thinking about entering the centre, so he’s sort of thinking about it, right? So what’s his or her calculation going to be? Well, let’s look at what would have happened. So all of these voters are going to vote for our left-wing candidate, so these are going to vote for the left-wing candidate, All of these voters are going to vote for the right-wing candidate, they’re all closest to the right-wing candidate. 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 centre candidate. Is that right? So I’ve basically divided, the reason I divided it into sixths is I want to put everybody at the middle of a third. All right? 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 centre candidate is at the middle of the centre third, not surprisingly.

[译文 8]

让我看看。他们的弱点是什么？他们容易受到有人在中间位置参选而偏离的影响。如果有人在中间参选，什么会促使他们在中间参选呢？他们会在中间参选基本上是因为他们能够获胜。好吧？如果在这种情况下他们在中间参选，让我们看看每个人能获得多少票。如果你在这里的中间参选，这是我们的新候选人，他正在考虑在中间参选，所以他有点在考虑，对吧？那么他或她的计算会是什么呢？好吧，让我们看看原本会发生什么。所有这些选民会投票给我们的左翼候选人，所以这些会投票给左翼候选人。所有这些选民会投票给右翼候选人，他们最接近右翼候选人。而中间的三分之一，那是三分之一的选民，另一个三分之一，以及中间的三分之一，这些人会投票给中间候选人。是对的吗？所以我基本上做了分割，我把它分成六份的原因是我想把每个人都放在三分之一的中点。好的？那么在这里，左翼候选人处于左三分之一的中点，右翼候选人处于右三分之一的中点，而中间候选人处于中间三分之一的中点，这并不奇怪。

[段 9]

All right? So if this centre candidate enters, if they were exactly at a sixth and five-sixths, they would split the vote, and the centre candidate would win with probability one-third. Is that right? Is that right? 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 a sixth, and the right-wing candidate is just slightly to the left of five sixths, now this left candidate gets a few extra votes here and the right candidate gets a few extra votes here Let sort of put them in a bit harder And you can see now that the center candidate isn going to win Is that right? Because the left-wing candidate is getting slightly more than a third of the votes. The right-wing candidate is going to get slightly more than a third of the votes. So the center candidate is going to get squeezed out. All right? So just a little bit of thinking about it, we can tell that in this very simple model, the furthest apart the left and right wing candidates can be is as a sixth and five-sixths. They can’t go to the extremes, but they’re not pulled all the way to the center. All right? So that’s just a little bit of nerdy math to confirm it, or nerdy thinking about it.

[译文 9]

好吧？所以如果这个中间候选人参选，如果他们恰好在六分之一和六分之五的位置，他们会分割选票，而中间候选人将以三分之一的概率获胜。是对的吗？是对的吗？但不用担心那个精确的情况，假设像我说的那样，左翼候选人刚好稍微在六分之一右边，右翼候选人刚好稍微在六分之五左边，那么这位左翼候选人在这里获得一些额外选票，右翼候选人也在那里获得一些额外选票。让我更清楚地标出来，你可以看到现在中间候选人不会获胜。是对的吗？因为左翼候选人获得的选票略多于三分之一。右翼候选人获得的选票也略多于三分之一。所以中间候选人会被挤出局。好吧？只要稍微想一想，我们就能知道，在这个非常简单的模型中，左翼和右翼候选人能相隔最远的距离是六分之一和六分之五。他们不能走到极端，但也不会被拉到正中间。好吧？那这只是一些书呆子式的数学或书呆子式的思考来确认这一点。

[段 10]

But coming back to our lessons, which I guess is what we care about, coming back to our lessons, the third lesson here is if the candidates, if the two candidates candidates are too extreme, where too extreme in this model meant less than a sixth and more than five-sixths, but are too extreme someone in the center will enter And again if you look back in both American history and other countries history you see that when candidates are perceived to be too far apart there’s been tremendous temptation for centre parties, for third parties, to establish themselves in the centre. So, again, with some bias towards England, this is what happened in English history during the Thatcher period, for example. The Thatcher government was perceived to be quite far on the right, the Labour Party at the time was perceived to be quite far on the left, and we saw a centre party set up in between them. Okay, so these seem to be the three main politics lessons of this model. Is everyone happy with that? I’m rushing it slightly because we said that already last time. Everyone happy with how it works? Okay. There’s also a game theory lesson that I want to just keep in mind here. Well, let me say it. There’s a game theory lesson here. And the game theory lesson is that our method of finding equilibrium in this model, which was what?

[译文 10]

但回到我们的经验教训，我想这是我们关心的东西，回到我们的经验教训，第三个教训是：如果候选人，如果两位候选人太过极端，在这个模型中太过极端意味着低于六分之一和高于六分之五，但如果太过极端，中心就会有人参选。再说一次，如果你回顾美国历史和其他国家的历史，你会看到当候选人被认为相隔太远时，中间党派、第三党派在中间建立自己就有了极大的诱惑。所以，再带着一点对英国的偏见，举例来说，这在英国历史上撒切尔时期就发生过。撒切尔政府被认为相当偏右，工党当时被认为相当偏左，而我们看到中间党派在他们之间建立了起来。好吧，所以这些似乎是这个模型的三个主要政治教训。大家对此满意吗？我稍微加快一点因为上次我们已经说过了。大家对它的运作方式满意吗？好吧。这里还有一个博弈论教训我想记在心里。让我说一下。这里有一个博弈论教训。而这个博弈论教训是：我们在找到均衡的过程中采用了什么方法？

[段 11]

It was guess and check, is actually pretty effective. You might think that guess and check, since it doesn’t sound like advanced mathematics wouldn be such a great way of going about solving games and thinking about them thinking about the real world But actually guess and check did pretty well here We were able to make sensible guesses pretty quickly we were pretty quickly able to figure out what was going on and the key here, the key here is what? Without writing it necessarily, the key here is be systematic when you’re guessing. Make sure you’ve looked everywhere. And second, be systematic, be careful when you’re checking. The big error is to ignore certain types of deviation. In this particular model, people very quickly realize that one possible deviation is for someone else to enter, but perhaps they’re a little bit less good at spotting that another possible type of deviation is for somebody to drop out. You want to look at all possible types of deviation. But be careful, this is a very effective method. All right, so that’s all I want to say about this politics model and I want to move on to a model that perhaps has more to do with sociology. We’re doing a little tour of the social sciences here and showing how game theory can apply to each in turn. So to do that I’m going to…

[译文 11]

就是猜测和验证，这实际上相当有效。你可能会认为猜测和验证，因为它听起来不像高等数学，所以不会是什么解决游戏和思考现实世界的好方法。但实际上猜测和验证在这里表现得相当好。我们能够相当快速地做出合理的猜测，我们相当快速地能够弄清楚发生了什么，这里的关键是什么？不一定非要写下来，这里的关键是你在猜测时要系统化。确保你已经看遍了所有地方。第二，在你验证时要系统化，要仔细。大的错误是忽略某些类型的偏离。在这个特定的模型中，人们很快意识到一种可能的偏离是其他人参选，但也许他们在发现另一种可能的偏离类型，即有人退出方面，稍微不那么擅长。你要查看所有可能的偏离类型。但要小心，这是一种非常有效的方法。好吧，这就是关于这个政治模型我要说的一切，我想转向一个可能与社会学更相关的模型。我们在这里对社会科学做一次小巡礼，展示博弈论如何逐一应用到每个领域。为了做到这一点，我将…

[段 12]

I’ll put this up so you can still read it and work on this. So we’re going to play another game this morning and it’s going to be completely different from the games Still read it, and work on this one. 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. It’s going to be another location model, so that’s a connection to what we’ve done before. But the idea of this game is as follows. We’re going to imagine that there are two towns, two possible locations and we’ll call them East Town and West Town and we’re going to assume that there are two types of people in the world so there’s two types of people and these types of people are tall and short and deliberately this is arbitrary East and West seems kind of meaningless, almost meaningless nomenclature for the towns, except I guess it says something about where they are, and tall and short is a pretty meaningless nomenclature for the people, except it says something about how tall they are. The idea here is that people are going to choose where they live Let assume that there lots of people There 100 of each type of person And let’s assume that each town holds 100,000 people.

[译文 12]

我会把这个放上去，这样你们还能看到并研究这个。好了，我们今天早上要玩另一个游戏，这个游戏将与我们到目前为止玩过的游戏完全不同，但它仍然会有其中的教训。它将是另一个位置模型，这是与我们之前所做的一个联系。但这个游戏的想法如下。我们想象有两个城镇，两个可能的位置，我们称它们为东镇和西镇，我们假设世界上有两种类型的人，所以有两种类型的人，这些类型的人是高的和矮的，而且故意地这是随意的。东和西对城镇来说似乎有点毫无意义，几乎毫无意义的命名，除了我猜它说明了一些关于它们在哪里，而高和矮对这些人来说是一个相当毫无意义的命名，除了它说明了一些关于他们有多高。这里的想法是人们将选择他们住在哪里。假设那里有很多人。每种类型有 100 人。假设每个城镇能容纳 100,000 人。

[段 13]

These are fairly big towns. All right. 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 so actually you’re going to be the players of this game and the strategies are going to be a choice of whether you choose east or west so these are the players and these are going to end up being the strategies 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 So as usual, what’s missing is the payoffs. Alright? And to model the payoffs, let me first of all draw a picture, and then come back and explain it. Okay? So the picture’s going to look like this. Here’s the payoff picture. And it a little complicated so just bear with me for a second So on the horizontal axis I going to put the number of people of your type in the town you end up in So the horizontal axis is the number of your type in your town so the lowest it 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 is that right? and I’m going to draw this payoff function this is going to be your payoff this is going to be the utility or payoff of u.

[译文 13]

这些是相当大的城镇。好吧。这场游戏的参与者将是人们，200,000人，200,000个高个子和200,000个矮个子，但等一会儿我会告诉你们你是高个还是矮个，所以实际上你们将成为这场游戏的参与者，而策略将是选择东边还是西边。这些是参与者，而这些将成为你们每个人将要选择的策略——我是想住在东部城镇还是西部城镇。像往常一样，缺少的是收益。好吧？要建立收益模型，让我先画一张图，然后再回来解释。好吧？这张图看起来是这样的。这是收益图。它有点复杂，所以请耐心听我讲一下。在横轴上，我会放你最终所在城镇中你这类人的数量。所以横轴是你所在城镇中你这类人的数量。它最低可能是100,000，因为这意味着城镇中每个人都和你同类型；它最低也可能是0，因为这意味着城镇中除了你之外每个人都是另一种类型，对吗？我要画这个收益函数，这是你的收益，这是效用或收益u。

[段 14]

Let’s assume this is type x. It doesn’t really matter. Let’s assume this is the tall type payoff. It’s going to be symmetric for the other type as well. So the payoff’s going to look like this. Be careful. Let me draw it and then explain it. All right that goes like this and then like this alright, so the idea here the idea is more important than the picture the idea is, if you are a minority of one in your town, so everyone else is of the other type you get a payoff of zero alright 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 half. 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 are tall and half of your town are short, then you get a payoff of one. And I put a half here, that’s really the wrong thing to put here, this should be a half of 100,000, so I guess it’s 50,000. So 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. So we’re going to assume that each and every one of you has this payoff.

[译文 14]

让我们假设这是x类型。这真的不重要。让我们假设这是高个子类型的收益。它对另一种类型也是对称的。所以收益看起来是这样的。注意。让我画出来然后再解释。好吧，它走成这样然后像这样，好吧，这里的想法比图更重要。想法是，如果你是城镇中唯一的一个少数派，所以其他所有人都是另一种类型，你的收益是零。好吧如果你在多数中，实际上城镇中所有人都是同一种类型，那么你的收益是二分之一。而且如果情况是你们城镇中的每个人都——抱歉——如果情况是你的城镇正好是混合的，所以你的城镇中一半是高个子一半是矮个子，那么你的收益是一。我在这里放了一个二分之一，这真的是放错了，这应该是100,000的二分之一，所以我想是50,000。所以情况是城镇中有50,000人是你的类型，50,000人是另一种类型，那么你会得到最高的收益，也就是1。所以我们假设你们每个人都拥有这个收益。

[段 15]

Does that make sense? That makes sense? which is one. So this is, we’re going to assume that each and every one of you has this payoff. Does that make sense? Does that make sense? Alright, 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. Does that make sense? they’d like to live in mixed towns but if the town is not if they have a choice between two towns and the towns are not evenly mixed then they’d prefer to be in the majority town the town in which they’re the majority everyone understand this? since we’re about to play this game it’s important you’re all on board here so to play this game I need to put down a few more rules 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 whenever they happen to be moving. But for the moment let leave it as that And second I need to just say what happens if too many people choose the same town So if there no room in a town for example if too many people chose East Town then we allocate the surplus randomly.

[译文 15]

这有道理吗？这有道理吗？这是1。所以我们假设你们每个人都拥有这个收益。这有道理吗？这有道理吗？好吧，这些是人，让我们试着把想法用语言表达出来，这些人是喜欢住在混合城镇的人，但如果他们要住在一个不是混合的城镇，他们宁愿住在一个他们是多数的城镇。这有道理吗？他们喜欢住在混合城镇，但如果没有——如果他们要在两个城镇之间选择，而这两个城镇不是均匀混合的，那么他们宁愿在多数城镇中——他们所在的多数城镇——大家都懂这个吗？既然我们即将玩这个游戏，重要的是你们都能参与进来。所以玩这个游戏，我需要再设几条规则。第一条规则是选择是同时的。这有点不现实，因为在实际中人们当然是在搬家的过程中依次选择城镇。但现在让我们先保持这样。第二，我需要说明如果太多人选择了同一个城镇会发生什么。所以如果一个城镇没有空间了，比如说太多人选择了东部城镇，那么我们随机分配多余的人。

[段 16]

We ration the places randomly. alright, so then we then randomise to ration people understand that? so for example, there’s a hundred thousand places in East Town so if a hundred and fifty thousand people chose East Town, you’re going to have a two thirds chance of getting into East Town and the rest of it is going to be allocated to West Town alright, does that make sense? I just need something to make things add up okay 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. I’m going to grab that mic again. Alright, so let me just count backwards. I guess we’ll ignore this row. So 1, 2, 3, 4, 5, 6, 7, possibly 8. 1 2 3 4 5 6 7 8 Well maybe this one Okay so up to here this row and forwards you are short people You understand that Everyone forward of where I am now Rahul included is short Okay? And the rest of you guys, and the guys in the balcony too, can’t see me anymore, the guys in the balcony, you are tall. Alright? Short, tall. Okay? Okay, so how many rows of these do you have? One, two, three, four, five, seven rows here, right? 1, 2, 3, 4, 5, 6. We have seven rows of short people.

[译文 16]

我们随机分配名额。好吧，然后我们随机分配来配给，人们理解吗？举个例子，东部城镇有100,000个名额，所以如果有150,000人选择了东部城镇，你就有三分之二的机会进入东部城镇，剩下的会分配到西部城镇，好吧，这有道理吗？我只是需要让事情加起来。好吧，为了做到这一点，首先我需要决定班级里谁是高个子谁是小个子。我要再拿那个麦克风。好吧，让我倒着数。我猜我们忽略这排。所以1,2,3,4,5,6,7，可能8。1 2 3 4 5 6 7 8 嗯也许这一个。好吧，到这里这排，往前你是矮个子。你明白吗？我现在所在位置往前的人，Rahul也包括在内，是矮个子。好吧？你们剩下的人，还有阳台上的那些人，看不到我了，阳台上的那些人，你们是高个子。好吧？矮个子，高个子。好吧？好吧，你们有多少排是这样的？1,2,3,4,5,7排，是吧？1,2,3,4,5,6。我们有七排矮个子。

[段 17]

Okay? All right? And the rest of you are tall. I’m going to hope I’ve split that evenly. Okay? And 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. All right? 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. 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. Doesn’t matter, you can move, but your initial position is in East Town. So East East East East Everyone understand that East And these three rows this row this row and this row you in West Town 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 in East Town. Oh, I didn’t notice I had this one. You’re in East Town as well. Up to here is East Town. And the rest of you is in West Town. Everyone understand where they are now? Let’s do a test with the with the camera watching. So if you are currently in East Town, raise your hands.

[译文 17]

好吧？好吗？好，你们剩下的人是高个子。我希望我平分得很均匀。好吧？我要稍微作弊一下。等一会儿，你们要选择你们要住在哪个城镇。我们举手来做。好吧？但在我们做之前，我要稍微作弊一下，给你们每个人一个初始位置。这无关紧要，但我只是想要设置一下。所以我们要做的是，我们把这排，所以这排，1,2,3,4，这些排，你们的初始位置在东部城镇。没关系，你们可以搬，但你们的初始位置在东部城镇。所以东部、东部、东部、东部，大家都懂东部。这三排这排这排和这排，你们在西部城镇。这接下来的四排，所以你们你们，你们，你们，让我们再做一遍，这排、这排、这排、这排，你们在东部城镇。哦，我没注意到还有这一个。你们也在东部城镇。到这里为止是东部城镇。你们剩下的人在西部城镇。大家都懂自己现在在哪里吗？让我们对着摄像机测试一下。所以如果你现在在东部城镇，举起你的手。

[段 18]

And if you’re currently in West Town, raise your hands. Right, so I’ve set you up with a stripy pattern, so you have the stripy pattern, and notice that I’ve set things up so that slightly more of the short people were in East Town, right, and slightly more of the tall people, if I did that if I did that right, were in Westtown. Alright? 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 Easttown, and slightly more of my tall people started life in Westtown. Okay, so now, think about it a second. Remember, these are your payoffs, right? That’s a suspension of disbelief. These are your payoffs. And in a minute, I want all the people who are choosing Easttown to raise their hands. I’m going to do it at the count of three. And don’t cheat. Don’t look around you and see what’s going on. All right? So really, you all close your eyes. All right? Everyone close their eyes. everyone in the room close their eyes, alright? I could do anything now, I’d be straight naked alright, so all right, all right, all right your eyes closed, at the count of three, anyone who’s choosing East Town should raise their hands, one two three alright, so this is my East Town people can open their eyes now, look at the East Town East Town distribution so I’ve got this big block of East Town here I’ve got a little bit going on the wings, a little bit of splidge here, let’s just make, let’s just check it, so those of you who didn’t raise your hands just now So West Town people raise their hands.

[译文 18]

如果你现在在西部城镇，举起你的手。对，我已经给你们设定了一个条纹图案，所以你们有了这个条纹图案，注意我已经设定了使得稍微更多一些的矮个子在东部城镇，对吧，而且稍微更多一些的高个子——如果我做对了的话——在西部城镇。好吧？如果我做对了的话，我不确定我是否做对了，但如果我做对了的话，稍微更多一些的矮个子开始在东部城镇，稍微更多一些的高个子开始在西部城镇。好吧，现在，想一下。记住，这是你们的收益，对吧？这是悬疑时刻。这些是你们的收益。等一会儿，我想要所有选择东部城镇的人举手。我要在数到三的时候做。不要作弊。不要四处看看发生了什么。好吧？所以真的，你们都闭上眼睛。好吧？所有人都闭上眼睛。房间里所有人都闭上眼睛，好吧？我现在可以做任何事情，我就赤身裸体了，好吧，所以好吧，好吧，好吧，你们闭上眼睛了，在数到三的时候，任何选择东部城镇的人都应该举手，一二三好吧，所以这是我的东部城镇的人，现在可以睁开眼睛了，看看东部城镇的分布情况，所以我有这一大块东部城镇在这里，我有一些在边缘上，这里有一些斑块，让我们只是——让我们检查一下，所以你们中那些刚才没有举手的人——所以西部城镇的人举起你们的手。

[段 19]

Alright, so we’ve got a little bit of spillage here, but basically I’ve got West Town over here, alright? Alright, everyone understood what just happened? Everyone saw what happened? Alright, let’s do it again. That where you are now Let see what happens next time So once again close your eyes Alright close your eyes Don communicate Alright everybody who going to choose East Town raise their hands now Alright, so I’ve got pretty much all of these rows now are in East Town. I’ve got a little bit of spillage here, but basically I’ve got East Town in the front. I can’t see the balcony. What have I got in the balcony up there? No hands up in the balcony, that’s typical. Alright, so okay, okay, alright. They live in the balcony, right? so I’ve more or less got Easttown here where’s my Westtown? raise your hands if you’re in Westtown now and Westtown a little bit of spillage here but basically I’ve got Westtown in the back alright let’s do it one more time one more time everyone understand where things were just now? alright am I leaving the hands up long enough? are they up long enough? yep, okay so let’s try it again so Easttown count of three one, two, three alright so here’s my Easttown I’m pretty much among the Easttown dwellers right now West Town raise your hands, one, two, three.

[译文 19]

好，这里有一点溢出，但我基本上是得到了西城这边，好吧？好吧，大家都明白刚才发生了什么吗？大家都看到了吗？好吧，我们再来一次。现在你们在那个位置，看看下次会怎样。再来一次，闭上眼睛。好的，闭上眼睛。不要交流。好的，现在要选择东城的人举手。好吧，现在这些行几乎都在东城了。我这里有一点溢出，但我基本上是让东城在前排。我看不到阳台。阳台上我有什么？没有人在阳台举手，这很正常。好吧，好吧，好吧。他们住在阳台上，对吧？所以我或多或少把东城放在这里，我的西城在哪里？现在在西城的人举手，西城有一点溢出，但我基本上是让西城在后排。好的，我们再来一次，再来一次。大家都知道刚才的情况在哪里吧？好吧，我把手举得够久了吗？他们举得够久了吗？是的，好吧，那我们再试一次。东城倒计时三、二、一。好吧，这是我的东城，我现在几乎都是在东城的居民中。西城的人举手，一、二、三。

[段 20]

And I’m pretty much in the West Town dwellers right now and it’s not at all stripey. All right, we start off with a stripey pattern and we ended up, we ended up how? 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 The balcony was just flying over So everyone saw what happened Now, can anyone say what’s happened there? Why did we end up like that? How did we end up like that? We started off with a stripy pattern. 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. It wasn’t quite at 50-50, but it wasn’t far off, actually. alright and we pretty quickly ended up with all these short people, you guys are short, all living in East Town and all of you tall people living in West Town so all of you pretty much ended up with a payoff of a half some of you didn’t, there were a few deviants, who are my deviants? Who are my tall guys who end up in East Town? so these guys, it’s fine but they end up with a payoff close to zero right? and who are my short deviants who ended up in West Town?

[译文 20]

我现在基本上是在西城的居民中，而且一点都不条纹状。好的，我们从一个条纹状的模式开始，我们最后，我们最后怎么样了？我们最后几乎是整个房间的前排选择东城，整个房间的后排也许包括阳台选择西城。阳台就直接飞过去了。所以大家都看到了发生了什么。现在，有谁能说说发生了什么？为什么我们会变成那样？我们怎么会变成那样的？我们从一个条纹状的模式开始。它离完美的状态有点偏离，因为你们所有人，根据这些偏好，会更愿意在一个正好是50对50的镇上。它没有完全达到50对50，但实际上也差得不远。好的，我们非常快地就让所有矮个子，你们这些矮个子，都住在东城，所有高个子都住在西城，所以你们几乎所有人都得到了大约一半的收益，有些人没有，有一些离异者，谁是我的离异者？谁是我的在东城的高个子？所以这些人，没关系，但他们最终得到的收益接近零，对吧？谁是我的在西区城中的矮个子离异者？

[段 21]

There’s a few, not many of them at all actually, a few of them, they end up with a payoff of zero as well. But pretty much everyone else was ending up with a payoff close to a half, close to a half, and splitting down the middle of the room. Alright? Now why? What do we call that? What do we call that process? What the outcome here Segregation Yeah so this is say it again Segregation So this is segregation We ended up getting the class to segregate on tall and short It wasn’t the people wanted to segregate. I gave you your preferences, and I told you that you had to have these preferences, which actually were preferences that favoured being in a mixed town, but we very quickly settled down to a segregated distribution of the class why? what led to that? let’s just talk it through a bit who has an opinion of what it was that got us there? so Patrick, shout out for the class, remember this doesn’t speak to the class I mean I think what happened was basically because it was the first choice was simultaneous everyone in the slightly bad situation all changed to what was perceived to be a better situation so we sort of shot past the equilibrium where we were 50-50. Right, right. So I cheated a little bit by starting you off away from the 50-50 point, and you kind of went further very, very quickly.

[译文 21]

有一些，不是很多，实际上很少，他们最终得到的收益也是零。但几乎其他所有人都最终得到了接近一半的收益，接近一半，分割在房间的中间。对吧？为什么？我们叫这什么？我们叫这个过程什么？这里的结局是什么——隔离。是的，我们再说一遍——隔离。所以这是隔离。我们最终让班级按高矮分开了。这不是人们想要隔离的。我给了你们偏好，我告诉你们必须接受这些偏好，这些偏好实际上是有利于在一个混合的镇子里，但我们非常快地就稳定在了一个班级分离的分布上——为什么？是什么导致了这个？让我们来讨论一下，谁对这个有什么看法？帕特里克，大声说出来，记住这不是在跟全班说话，我的意思是，我认为发生的事情基本上是因为第一次选择是同时的，所有处于稍微不利处境的人都转向了被认为更好的处境，所以我们有点冲过了50对50的均衡点。对，对。所以我稍微作弊了，让你们从50对50点开始偏离，你们非常快地就走得更远了。

[段 22]

Now I want to say that I think with a longer time period to play with, I could have started you off just random, right, just to do whatever, and if we played enough times, my guess is we’d have ended up segregated anyway. It might not have been segregated with started you off just random, right, just choose whatever, and if we played enough times, my guess is we’d have ended up segregated anyway. It might not have been segregated with east here and west there, it might have been the other way around, but I’m pretty sure we’d have ended up segregated anyway. All right, I can’t prove that because I wouldn’t have enough time to do it. I cheated by pushing us that way. All right, so what happened was, is this right, people who started off in the minority, but not badly in the minority, who were short guys in West Town, they moved to East Town. That was largely the rows here. You guys moved to West Town. 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. Is that right? Is that right? Okay, we’ll talk about this more in a second, but let’s just get some ideas down on the board.

[译文 22]

现在我想说，我认为如果有更长的时间来玩，我可以让你们随机开始，对吧，随便怎么做，如果我们玩足够多次，我猜我们最终还是会分开的。它可能不是东边和西边这样的分离，可能会是反过来，但我相当确定我们最终还是会分开的。好的，我无法证明这一点，因为我没有足够的时间来做这件事。我通过把我们推向那个方向来作弊了。好的，所以发生的事情是，这是对的吗？一开始处于少数但不是严重少数的人，那些在西城的高个子，他们搬到了东城。这主要是这里的那几排。你们搬到了西城。还有那些最终的人，我一开始让他们处于少数，我甚至不确定那实际上是不是少数，那些在东城的高个子，你们漂移到了西城。对吗？是这样吗？好吧，我们等一下再讨论这个，但我们先把一些想法写到黑板上。

[段 23]

Well, let’s talk about… I don’t know, actually while I’m down here, let’s do a little bit more work. Alright? So before we leave this, I mean before we leave talking about it, let’s figure out what are the equilibria here. So what an equilibrium of this game It a pretty simple game it has lots of people two choices per person We going to be doing guess and check here right So what do people think the equilibrium is Let me get someone who hasn spoken yet I surely can’t have the cold call here. This is an easy one. What’s an equilibrium here? Somebody wants an equilibrium here. How about the gentleman here? You want to have a guess at what an equilibrium is? Everyone being segregated. Everyone being segregated. Let’s be a bit more precise. So I claim there’s two ways they can be segregated. So spell out the two ways. 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 West Town and there’s no way around it. Good, good. And your name is? Greg. So Greg is saying there’s two ways to be segregated and each of them seems like an equilibrium. Right, let’s just spell it out. So all the tall people being in the East is one equilibrium and all the poor people being in the West is the other equilibrium.

[译文 23]

好吧，让我们讨论……我不知道，实际上我在这里的时候，让我们再多做一点工作。好吧？在我们离开这个之前，我是说在我们停止讨论之前，让我们弄清楚这里的均衡是什么。所以这个游戏的均衡是什么——这是一个非常简单的游戏，它有很多人，每个人有两个选择。我们将在这里做猜测和检验，对吧？所以大家认为均衡是什么？让我找一个还没说过话的人。我肯定不能冷不丁叫人。这是个简单的问题。这里的均衡是什么？有人想要一个均衡。这边这位先生怎么样？你想猜一下什么是均衡吗？所有人都分离。所有人都分离。让我们更精确一点。我声称有两种方式可以分离。所以说明这两种方式。如果所有高个子都在西城，所有矮个子都在东城，或者如果所有高个子都在西城，没有办法绕过去。好，好。你叫什么名字？格雷格。所以格雷格说有两种方式可以分离，它们每一个看起来都是一个均衡。对，让我们来说清楚。所以所有高个子都在东城是一种均衡，所有穷人都在西城是另一种均衡。

[段 24]

Both of those, I claim, or Greg claims, but more carefully, Great claims, both of those are Nash equilibria. How do we check that they’re Nash equilibria? So they are, you’re right, but how do we check that they are in fact equilibria? How do we go about checking that they’re equilibria? The guessing part is easy, checking requires a bit of thought, so how do we check? Yeah You check for profitable deviations You check for profitable deviations So what a deviation here If all the short people are in East Town and all the tall people are in West Town what a deviation A deviation, let’s ignore the capacity constraint for a minute, a deviation is for one of the short guys to move to West Town. Is that right? So at the equilibrium, that short guy was getting a payoff of what? What was his payoff, his or her payoff in equilibrium? One half. If he or she deviates and moves to Westtown, again ignoring the capacity constraints, let’s assume that they can, if that short person moves to Westtown, what’s their payoff going to be? Zero. So that’s not a profitable deviation. And conversely, if we did the same from the other side, for the tall people, we’d find the same thing. So we’ve just checked that that is an equilibrium because nobody can deviate profitably.

[译文 24]

这两种，我声称，或者格雷格声称，但更仔细地说，格雷格声称，这两种都是纳什均衡。我们怎么检验它们是纳什均衡？是的，它们是，你说得对，但我们怎么检验它们实际上是均衡？我们如何来检验它们是均衡？猜测部分很容易，检验需要一点思考，所以我们怎么检验？是的，你检验有利可图的偏离。你检验有利可图的偏离。所以这里的偏离是什么？如果所有矮个子都在东城，所有高个子都在西城，什么是偏离？偏离，让我们忽略容量限制一分钟，偏离是一个矮个子搬到西城。对吗？所以在均衡时，那个矮个子得到的收益是多少？什么是他或她在均衡时的收益？一半。如果他或她偏离并搬到西城，再忽略容量限制，假设他们可以，如果那个矮个子搬到西城，他们的收益会是多少？零。所以这不是一个有利可图的偏离。反过来，如果我们从另一边做同样的事情，对于高个子，我们会发现同样的事情。所以我们刚刚检验了这是一个均衡，因为没有人能够有利可图地偏离。

[段 25]

So we’ve found two equilibria here, and as people have pointed out, they are segregated. Now I claim that there’s another equilibrium here. What’s the other equilibrium here? What’s the other equilibrium here? I should get the mic right the way on the other side. So let me try and lean in here. Lean my way. Yeah, can you? Thanks, thanks. If the tour split exactly 50 then that would be an equilibrium as well Good good So if the crowd had split exactly 50 that would also be an equilibrium but the key word there is what? the 50-50 is the key word what’s the other key word there? exact it really has to be exact for this to work and since I’m hand-waving a bit because equilibria are always exact statements but can people see what’s going on here? So if people split exactly 50-50, if I’d split the class down the middle, I start off by splitting it this way into tall people and short people, and if I’d split the town down the middle into east and west, then everyone would have been happy and they’d have stayed put. That’s this gentleman’s claim, and I think that’s correct. But what’s suspicious, if you like? What’s worrying about that equilibrium? That is an equilibrium. What’s worrying about that equilibrium? That mixed equilibrium, that integrated equilibrium? Can we get the guy here?

[译文 25]

所以我们找到了两个均衡，如大家所指出的，它们是隔离的。现在我声称这里还有另一个均衡。另一个均衡是什么？另一个均衡是什么？我应该把麦克风放到另一边。让我试着靠过去。靠我这边。是的，可以吗？谢谢，谢谢。如果人群刚好五五分成，那也是一种均衡。很好。如果人群刚好五五分成，那也是一种均衡，但这里的关键词是什么？五五分成是关键词，另一个关键词是什么？精确，必须是完全精确才能成立，因为我这里有点随意，但均衡总是精确的陈述，大家能看到这里的情况吗？如果人们刚好五五分成，如果我把班级对半分，我开始是按这种方式把它分成高个子和矮个子，如果我把镇对半分成东镇和西镇，那么每个人都会很高兴，他们会留在原地。这是那位先生的观点，我认为这是正确的。但令人怀疑的是什么？令人担忧的是什么？那个均衡令人担忧的是什么？那是一个均衡。那个均衡令人担忧的是什么？那个混合均衡，那个整合均衡？能把那位先生请过来吗？

[段 26]

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. Good, good. So one thing that distinguishes that equilibrium from the equilibria that we were just looking at is it incentives. Good, good. So one thing that distinguishes, thank you, that’s very good. So one thing that distinguishes that equilibrium from the equilibria that we were just looking at is in some sense it’s weak whereas the other ones are strict. What do we mean by that? 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. It’s true, it’s not strictly true because I guess by ourselves moving we’re changing the balance a little bit. But nevertheless, we know that if I smooth out the top, you’ll be exactly right. 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. They’re called East and West, but they have the same mixture of tall and short people in them. Whereas at the segregated equilibrium, I strictly prefer to go to the town in which I’m the majority. I’m doing a strictly higher payoff, half versus zero, by being in the town in which I’m in the majority.

[译文 26]

这是一个弱纳什均衡，因为虽然你没有真正的动机去偏离，但你也没有动机不去偏离。很好，很好。那么这个均衡与我们刚才看到的均衡的一个区别是它涉及激励。谢谢，非常好。那么这个均衡与我们刚才看到的均衡的一个区别是，在某种意义上它是弱的，而其他的均衡是强的。我们这是什么意思？如果我们偏离这个完全混合的均衡，粗略地说，这里稍微随意一点，但粗略地说，我们是完全无差异的。确实，这不完全正确，因为我想我们自己移动时会稍微改变平衡。但尽管如此，我们知道如果我把顶部抹平，你会完全正确。所以在那个混合均衡，我不把它叫做混合均衡，在那个整合均衡中，我对住在哪里完全无差异。两个镇对我来说看起来一样。它们叫东镇和西镇，但它们有相同的高矮人口混合。而在隔离均衡中，我严格偏好去我占多数的那个镇。通过在多数人群中，我的收益严格更高，一半对零。

[段 27]

So there’s this notion of strictness. There also a notion of stability here So again I don want to be too formal here I want to give you an informal idea about why we might worry about stability So what do I mean by stability here Why do I think that that integrated equilibrium in which we divide the towns equally might not be stable? What do I mean by that? For the physicists in the room this would be an easy idea, but for everyone I think it’s a fairly intuitive idea. Why is that not likely to be stable? Anybody? Yeah, here, what’s your name? Chris. If even one person deviates and all of a sudden everyone will prefer to deviate to the completely segregated position? Right, good. So at that integrated equilibrium, 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. 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 and in a few moves we’re very quickly going to be back at segregation again It not stable in the sense that if we were a little bit off we going to go a long way off And again I not being formal here but the informal idea I think is important Conversely those segregated equilibria because they were strict equilibria, as the gentleman over there was pointing out, they’re really pretty stable.

[译文 27]

所以有严格性的概念。这里也有稳定性的概念。所以我不想太正式，我想给你们一个非正式的想法，为什么我们可能会担心稳定性。我这里说的稳定性是什么意思？为什么我认为那个我们把镇对半分的整合均衡可能不稳定？我这是什么意思？对于在座的物理学家来说，这是一个简单的想法，但对每个人来说，我认为这是一个相当直观的 idea。为什么这可能不稳定？有人知道吗？是的，这位，你叫什么名字？克里斯。如果有一个人偏离，突然间每个人都会偏好完全隔离的位置？是的，很好。所以在那个整合均衡中，如果我们稍微偏离它，比如一个镇多了5%的矮个子，另一个镇多了5%的高个子，那么我们在某种意义上已经遇到麻烦了。我们还没有离这个好的均衡走得太远，但已经遇到麻烦了，因为现在所有的矮个子都会偏好东镇，所有的高个子都会偏好西镇，在几次移动后我们会很快回到隔离状态。它不稳定，因为如果我们稍微偏离，我们就会偏离很远。同样，我这里不是正式的，但这个非正式的想法我认为很重要。相反，那些隔离均衡，因为它们是严格的均衡，正如那边那位先生指出的，它们确实非常稳定。

[段 28]

If we start close to 100% short people in East, and close to 100% tall people in West, and then we move a little bit away, so I shake you off a bit and I force you I reallocate a few people and then let you play I claim you’ll go back to that equilibrium is that right? 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 whereas the segregated equilibria they’re clearly strict equilibria and they are stable 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 alright so we’ve got these several equilibria here we got at least three and we come back and talk about whether there are others in a minute we got at least three equilibria two of them are segregated so the Nash equilibria in this game, we have two segregated Nash equilibria, and these correspond to tall in east and short in west and vice versa and we had a separate one which was an integrated one which was roughly half of each in each town 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.

[译文 28]

如果我们从东镇接近100%的矮个子、西镇接近100%的高个子开始，然后我们稍微移动一点，所以我把你们甩开一点，我强迫你们重新分配一些人，然后让你们行动。我声称你们会回到那个均衡，对吗？所以这个整合均衡它是一个均衡，但它只是一个弱均衡，在某种意义上它不太稳定，而那些隔离均衡它们显然是严格的均衡，它们是稳定的。让我把这些写到白板上，部分是因为我站在那里感觉很奇怪，即使你们不觉得。好吧，我们这里有几个均衡，我们至少有三个，我们回头再讨论是否还有其他。至少有二个均衡，其中两个是隔离的。所以在这个博弈中的纳什均衡，我们有两个隔离的纳什均衡，它们对应于高个子在东镇矮个子在西镇，反过来也一样，我们还有一个单独的，是整合的，大约各一半在每个镇。所以这里至少有二个均衡，虽然两个隔离均衡在某种意义上是一样的。我们论证了这些隔离均衡在某种意义上是稳定的，它们是严格均衡，因为你严格偏好不偏离。

[段 29]

Alright? Whereas these integrated equilibria, these were actually perhaps not stable, and again, I’m not being formal here, so I have to be careful, I’ll put it in inverted commas. And, again, a little bit informal, they’re kind of weak equilibria. alright now I want to bring up one other concept here which is the idea of a tipping point alright so this is a game that was introduced by a guy called Schelling Schelling went on to win the Nobel Prize largely for this I think certainly in large part for this idea alright this is a game that has a tipping point there are really two stable equilibria the segregation one way and the segregation the other way and in between there’s a tipping point beyond which, if you go beyond which, you go to the other equilibrium. I said that very badly but if people get the idea that there are two stripped equilibria and if we got beyond the point of a half mix going the other way we can we off to the other equilibrium Right And we already saw a game that had a tipping point When we played the investment game, where there were two, 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.

[译文 29]

好吗？而这些整合均衡，实际上可能不稳定，同样，我不是正式的，所以我必须小心，我会用引号括起来。同样，有点非正式，它们可以说是弱均衡。好吧，现在我想在这里引入另一个概念，那就是临界点的概念。好的，所以这是一个由一个叫谢林的家伙引入的博弈。谢林后来获得了诺贝尔奖，主要是因为这个，我认为很大程度上是因为这个想法。好的，这是一个有临界点的博弈。有两个稳定的均衡，一个是这种方式的隔离，另一个是那种方式的隔离，中间有一个临界点，超过这个临界点，你就会走向另一个均衡。我说得很糟糕，但如果人们有这个想法，有两个极端均衡，如果我们超过了正好一半的混合，走向另一边，我们可以转向另一个均衡。对。我们在玩投资博弈时已经看到了一个有临界点的博弈，当时有两个均衡，全部投资和无人投资，那个博弈中有一个自然的临界点，临界点是正好有90%的你们投资。

[段 30]

Right? The point at which you actually would want to flip over and go to the other equilibrium. So this is a game that has a tipping point. 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 the half, whoops, they’ll go off to the other equilibrium. Very dramatic change. So that seems like a very important idea in, for example, sociology. So I’ve got these segregated equilibria and I’ve got these integrated equilibria, and let’s just make the other obvious remark. Which of these equilibria is preferred by the population? Would they rather be in the integrated equilibrium, given that these are their payoffs, or the segregated equilibrium? They’d rather be in the integrated one, right? So here a world in which they like to be in the everybody would like to be in the integrated equilibrium This is not a prisoner dilemma alright This is not a case that we seen before but it turns out that you likely to end up in these inefficient less preferred by everybody, segregated equilibrium. Alright? However, if we’re subtle about this, you might notice there’s actually a third equilibrium in this game. If we’re a little bit nerdy, there’s actually another equilibrium in this game.

[译文 30]

对？那就是你实际想要翻转并走向另一个均衡的点。所以这是一个有临界点的博弈。我们可以把人们推离均衡，他们会一次又一次地回来，一次又一次地回来，但如果我们把他们稍微推到超过一半，哎呀，他们就会转向另一个均衡。非常戏剧性的变化。所以这在社会学中似乎是一个非常重要的想法。我有这些隔离均衡，我有这些整合均衡，让我们只做一个另一个明显的评论。这些均衡中，哪一个是这个人群偏好的？考虑到这些是他们的收益，他们更愿意处于整合均衡，还是隔离均衡？他们更愿意处于整合的，对吧？所以在一个他们都喜欢处于整合均衡的世界里，每个人都愿意处于整合均衡。这不是囚徒困境。这不是我们之前看到的那种情况，但结果是，你很可能会最终处于这些低效的、对每个人来说都不太受欢迎的、隔离的均衡。好吗？然而，如果我们对此很敏锐，你可能会注意到这个博弈中实际上有第三个均衡。如果我们有点书呆子气，这个博弈中实际上还有另一个均衡。

[段 31]

So, I need to give you a mic back, I’m sorry. There’s a guy there in grey. If everyone chooses one town or the other, regardless of whether they’re tall or short, that would be redistributed. Good, good. So your name is? Nick. Nick. So Nick’s pointing out there’s actually hidden in here another equilibrium. It doesn’t sound like anything very realistic, but let’s just focus on it in a second, because I think it’s an important lesson here. It’s going to turn out to be important lessons. The other equilibrium is, actually there’s two of them, if everybody, everyone in the room, chose East Town, what would happen? Well, we’d have to find a way of assigning everybody, so what we do is we essentially randomize over the room and half the room would be in East Town and half the room will be in West Town Right so there 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 group this detail of the original game Which was if there was overcrowding we were going to randomize people 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.

[译文 31]

好的，我需要把话筒还给你，抱歉。那边有个穿灰色衣服的人。如果每个人都选择这个镇或那个镇，不管他们高还是矮，那就会被重新分配。好的，好的。你叫什么名字？尼克。尼克。所以尼克指出，实际上这里面还隐藏着另一个均衡。这听起来不太现实，但让我们先关注一下，因为我认为这是一个重要的教训。这些教训最终会变得重要。另一个均衡是，实际上有两个这样的均衡，如果房间里的每个人，每个在场的人，都选择了东镇，会发生什么？嗯，我们必须找到一种分配每个人的方法，所以我们所做的基本上是在整个房间进行随机分配，一半的人在东镇，一半的人在西镇。所以这是一种愚蠢的均衡，在某种意义上，每个人都做同样的事情，而人们实际被分配的方式不是通过他们的选择，而是通过这个群体的这个细节，这个原始游戏的细节，也就是如果有过度拥挤，我们就会随机分配人们。所以实际上存在第三种均衡，也就是所有人选择同一个镇并被随机分配。

[段 32]

And to check that it’s 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. You’re completely happy to do the same thing. So it is, in fact, an equilibrium. So this is a slightly odd equilibrium here, and there’s immediately a game theory lesson here. This equilibrium sounds like something we might see in society. This one’s certainly worth talking about. It seems a natural part of the game. This equilibrium seems to have nothing to do with Yeah. This equilibrium sounds like something we might see in society. This one’s certainly worth talking about. It seems a natural part of the game. This equilibrium seems to have nothing to do with anything that’s really in the world. It’s just arising from a particular detail I threw into the model at the end to make things add up. Is that right? When I was setting up this model and describing reality, I threw this in at the end to say, well, we better do something just to make things add up, otherwise towns are going to be overcrowded. alright, well it wasn’t that’s really out there in the real world, it was just to make it a game to define things carefully and this seemingly innocent detail of my modelling technique threw up another equilibrium alright, so there’s a sort of warning lesson here the lesson is seemingly irrelevant details of a game seemingly irrelevant details things that aren’t really attempts to capture they’re just trying to get things through in a hurry, can end up mattering, can matter.

[译文 32]

为了验证这是一个均衡，请注意，如果其他每个人都在选择，如果其他每个人都在玩这种全选东镇并允许随机装置来安置你的策略，那么你就完全乐意这样做。你完全乐意做同样的事情。所以这实际上确实是一个均衡。这是一个稍微奇怪的均衡，这里立即有一个博弈论教训。这个均衡听起来像我们可能在社会中看到的东西。这个确实值得讨论。它似乎是游戏的一个自然部分。这个均衡似乎与现实世界中任何真正的东西都无关。它只是产生于我在模型末尾为了使事情加起来而加入的一个特定细节。对吗？当我设置这个模型并描述现实时，我在末尾加入了这一点，说好吧，我们最好做点什么来使事情加起来，否则城镇会过度拥挤。好的，呃那不是真的存在于现实世界中，那只是为了使它成为一个游戏，仔细定义事物，而这个看似无害的建模技术细节产生了另一个均衡。好的，所以这里有一个警告性的教训，这个教训是，博弈中看似无关紧要的细节，这些并非真正试图捕捉现实的细节，这些只是为了赶时间让事情通过的东西，最终可能会很重要，可能会产生影响。

[段 33]

It can lead your game to give you a prediction you really don’t believe in. Is that right And that a very general lesson for those of you who are going to go out and model things more widely There a second lesson here however The second lesson here is if in fact this randomization process was available if in fact it was possible for everyone in the town to choose East Town and then have the local government randomize you, then, if we use the law of large numbers, if there’s a hundred thousand, I guess there’s two hundred thousand people in all, and they’re all being randomized, we’re going to end up very very close in the limit exactly at the at integration and everyone’s going to be better off is that right? so the other kind of strange thing here is by randomizing well I don’t want to say randomizing I want to say having society randomized for you ended up being better than choosing. Ended up better than what you might want to call an active choice. Here an example of a place where by abdicating the right to choose my town and simply by all choosing east having society randomized for me we ended up better off Slightly surprising result. Alright, let’s try and push this a little harder. Get rid of that. Nope, didn’t manage it.

[译文 33]

它可能导致你的博弈给你一个你真的不相信的预测。对吗？而这是你们中那些将要走出去更广泛地建模事物的人的一个非常普遍的教训。然而，这里还有第二个教训。这里的第二个教训是，如果这种随机化过程实际上可行，如果实际上镇上每个人都有可能选择东镇然后由当地政府进行随机分配，那么，如果我们使用大数定律，如果有十万，我猜总共有二十万人，而且他们都被随机分配，我们将非常非常接近极限，正好达到整合，每个人都会过得更好，是这样吗？所以这里另一个奇怪的事情是，通过良好的随机化，我不想说随机化，我想说的是让社会为你随机分配，结果比你主动选择要好。结果比你们可能称之为主动选择的东西要好。这里有一个例子，通过放弃选择我的城镇的权利，简单地通过全选东镇让社会为我随机分配，我们结果过得更好。稍微令人惊讶的结果。好的，让我们试着把这个推得更深入。去掉那个。不，没能做到。

[段 34]

Okay. So let’s try and draw some lessons. We’ve drawn out some game theory lessons already from this about irrelevant details and about stability and so on. Let’s try and draw out some other lessons here. So one lesson might be a lesson in sociology. I’m not a sociologist so I want to be careful, I’ll put it in inverted commas here. The sociological lesson is what? In this game, segregation is what resulted at least in the stable equilibrium and you might be tempted if you an empirical sociologist to go around the world and say look I see segregation everywhere I gone from country to country from society to society and wherever I go I see segregation you might be tempted to conclude that that’s because people prefer segregation and that might be right nothing in this model disproves that it might be the fact that it might in fact be the case that the reason you see segregation in virtually every society is because segregation is preferred. I’m not ruling that out here. I’m just raising another possibility. What’s the other possibility? The other possibility is it could be that preferences are like this, roughly speaking. People don’t actually prefer segregation, but when all people act in their own interest, you end up with segregation anyway. All right? So the fact that we’re seeing segregation, seeing segregation in this model, Therefore, seeing segregation does not imply that there’s a preference for segregation.

[译文 34]

好的。让我们试着总结一些教训。我们已经从这个博弈中提取了一些博弈论教训，关于无关紧要的细节，关于稳定性等等。让我们试着在这里提取一些其他的教训。所以一个教训可能是社会学方面的教训。我不是社会学家，所以我想谨慎一点，我在这里用引号把它括起来。社会学的教训是什么？在这个博弈中，隔离是至少在稳定均衡中产生的结果，如果你是一个实证社会学家，你可能会走遍全世界说，看，我所到的每个地方，从一个国家到另一个国家，从一个社会到另一个社会，我都看到隔离，无论我走到哪里我都看到隔离，你可能会得出结论认为那是因为人们偏好隔离，这可能是对的，这个模型没有反驳这一点，这可能是事实，这可能确实是事实，你几乎在每个社会看到隔离的原因是人们偏好隔离。我在这里不排除这种可能性。我只是在提出另一种可能性。另一种可能性是什么？另一种可能性是，偏好可能是这样的，大致来说。人们实际上并不偏好隔离，但当所有人按照自己的利益行事时，你最终还是会得到隔离。对吗？所以我们在这个模型中看到隔离的事实，因此，看到隔离并不必然意味着存在对隔离的偏好。

[段 35]

It doesn’t rule it out, of course, but you can’t conclude, just because you see segregation everywhere, that necessarily people want segregation. And let me just take that outside of the context of segregation more generally. If you see a social phenomenon in society after society after society, whether you’re an anthropologist 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. 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, in this case it’s not. That was really Schelling’s big idea. So I don’t want to write that all up, but it’s a huge idea. You’ve got it in the Noble Rights, right? All right? So you don’t want to conclude from observation straight back to preference in these strategic settings. All right? Now this matters a little bit obviously in our own society because we live in a rather segregated society OK So I mean we needn 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.

[译文 35]

它当然不能排除这种可能性，但你不能仅仅因为你在到处看到隔离就得出结论，认为人们必然想要隔离。让我把这一点带到隔离的背景之外，更一般地来说。如果你在一个又一个社会，一个社会接一个社会地看到一种社会现象，无论你是一个跨社会的人类学家，还是一个在历史时间中穿越社会的历史学家，你看到这些社会中每一个都有同样的现象，而这是由成千上万不同的人的选择产生的，你不能仅仅因为你看到所有这些社会都有这种现象就得出结论说那些人偏好它。你所知道的是，他们每个人的个人选择加总起来就是这个社会结果，而这个结果可能也可能不是他们偏好的，在这个例子中它不是。这确实是谢林的大思想。我不想把这些全部写下来，但这是一个巨大的思想。你在诺贝尔奖中得到了它，对吗？好的？所以在这些战略环境中，你不想从观察直接得出偏好的结论。好的？现在这在我们的社会中显然有点重要，因为我们生活在一个相当隔离的社会。好的，我是说，我们不必如此含蓄以至于一直谈论高的人矮的人。我们真正在谈论的，在我看来，在我们的社会中大部分是种族隔离。

[段 36]

And we see this very dramatically in Connecticut, for example. So where we live in New Haven, at least if you go a few feet outside of the university and actually look at where people live in New Haven, we see that New Haven is a fairly integrated town. It’s not 100% an integrated town, but it has a fairly wide array of ethnicities. But if you go upstate in Connecticut, if you go north, ignoring Hartford, if you go into the rural areas of Connecticut, you find something dramatically different. How many of you have travelled around in rural Connecticut a bit? Some of you have. So it’s quite shocking for me as a foreigner when I came here, So New Haven and the towns along the coast are fairly integrated, but if you drive into rural Connecticut, you see something quite different. So this weekend, for example, I was up at the Durham Fair. How many of you know what the Durham Fair is? Some of you know. The Durham Fair actually is very good. 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 bar Who knew right That a good thing So I don want to knock the Durham Fair I think it a good thing But if you wandering around the Durham Fair and you a social scientist as I am, I guess, you can’t help but be struck by, how will I put this without getting in trouble, you would find more ethnic diversity at a Klan rally.

[译文 36]

我们非常戏剧性地在康涅狄格州看到这一点，例如。我们住在纽黑文，至少如果你走出大学几英尺，真正看看纽黑文的人们住在哪里，我们看到纽黑文是一个相当融合的城镇。它不是100%的融合城镇，但它有相当广泛的种族。然而，如果你去康涅狄格州的上州，如果你往北走，忽略哈特福德，如果你进入康涅狄格州的农村地区，你会发现截然不同的情况。你们中有多少人在康涅狄格州的农村地区旅行过？你们有些人去过。所以当我这个外国人刚来这里时，这对我来说相当令人震惊。纽黑文和沿海的城镇相当融合，但如果你开车进入康涅狄格州的农村，你会看到相当不同的东西。例如，这个周末，我在杜伦博览会。有多少人知道杜伦博览会是什么？你们有些人知道。杜伦博览会实际上非常好。它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，实际上奶牛真的会哞哞叫，绵羊真的会咩咩叫。谁知道呢，那是一件好事。所以我不想批评杜伦博览会，我认为那是一件好事。但是如果你在杜伦博览会上闲逛，而你是像我一样的社会科学家，你不禁会被——我该怎么措辞才不会惹麻烦——你会发现，在三K党集会上会有更多的种族多样性。

[段 37]

This is not… I mean, that’s not going to get me out of trouble. There’s got to be some way of saying that, it doesn’t get me in trouble. But you know what I mean, right? It is strikingly, strikingly white at the Durham Fair. And we’re only, what, it’s about, I think it’s 18 miles from New Haven. It’s a 20 minute drive from New Haven. It’s a public event 20 miles from New Haven. So this is a phenomenon one might be tempted to say, ah, 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. 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. And just to repeat I can prove that it isn Right but we have to be at least aware of the possibility that it got nothing to do with the preference for segregation It could be that preferences look like the preferences above It could be that simply thousands of activities by thousands of individuals who at least would prefer to be in the majority 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.

[译文 37]

这不是……我的意思是，这不会让我摆脱麻烦。一定有什么方式可以说，这不会让我陷入麻烦。但你知道我的意思，对吧？达勒姆博览会上的人种构成惊人地、惊人地单一。而我们离那里只有……我认为是18英里。从纽黑文开车20分钟就到。这是一个公共活动，距离纽黑文20英里。所以这可能引出一个现象，人们可能会忍不住说，啊，这是因为人们的选择，人们想要的，显然人们选择去达勒姆博览会。他们在某种程度上选择住在康涅狄格州的什么地方。你可能会认为，我们在纽黑文的活动和在20英里外的活动之间看到的这种极端的种族隔离，证明了人们实际上可能更喜欢种族隔离。我要重复一下，我可以证明情况并非如此——但我们至少必须意识到这种可能性，即这与对种族隔离的偏好毫无关系。可能是偏好看起来就像上面的那些偏好。可能是成千上万的人做出的成千上万的选择，他们至少宁愿成为多数派而不是少数派，但他们想要整合，结果导致总体上出现令人难以置信的种族隔离现象。

[段 38]

All right. So let’s talk about policy now. So as I go down here I’m going to get more and more in trouble I’m sure as a foreigner. This is a big policy question in the US. How many of you are unaware of the fact that this is a big policy question in the US? 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? Schools, busing for segregating schools. Okay, so I don’t know your name, I’m sorry. So Jessica’s pointing out correctly that since the 1960s, this has been an incredibly hot issue in the US, and we’re talking about the busing debate. How many of you have not heard of busing? How many of you have heard of busing? All right, so it’s slightly a policy debate of the 1960s, so people tend to forget it, but in the 1960s and early 70s, How many of you have heard of passing? Alright, 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 children were bussed from one neighbourhood to another neighbourhood to go to school. Alright, and this is an incredibly controversial issue in the US, and I don’t want to take a position on it here, I just want to point out that you could arrive at that policy by thinking about this kind of model.

[译文 38]

好的，让我们现在谈谈政策。当我继续讲下去时，我相信作为一个外国人，我会越来越陷入麻烦。这是美国的一个重要政策问题。你们中有多少不知道这是美国的一个重要政策问题？至少从1960年代开始，这个种族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领域？学校，为种族隔离学校提供的校车接送。是的，我不知道你的名字，抱歉。所以杰西卡正确地指出，自1960年代以来，这一直是美国一个极其热门的问题，我们谈论的是校车接送的争论。你们中有多少人没有听说过校车接送？你们中有多少人听说过校车接送？好的，所以这略微是1960年代的政策辩论，所以人们往往会忘记它，但在1960年代和1970年代初，人们非常担心学校中的种族隔离问题，以至于孩子们被用校车从一个社区送到另一个社区去上学。是的，这在美国是一个非常具有争议的问题，我不想在这里表明立场，我只是想指出，你可以通过对这种模型的思考来得出那个政策。

[段 39]

In fact, this is not just an issue of history. If you read the newspapers last week, including the Yale Daily, you’ll see that it’s an issue today in Connecticut. There’s a worry in Connecticut today, particularly around Hartford, that the schools are still illegally segregated. They’re not in compliance with the law. All right, so this continues to be an issue. Let me take it 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. 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 busing going on If everyone chose to go to East Town what we ended up with was randomized placement across the towns Is that right And that’s a little bit like the busing policy. It says everybody, we take away people’s active choice of their school, and in place, we just randomize everybody. 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. So there’s this school called Harvard, so Harvard is outside Boston, and at least when I was there as a graduate student, this was a hot issue in Harvard.

[译文 39]

事实上，这不仅仅是一个历史问题。如果你上周读报纸，包括《耶鲁每日新闻》，你会发现这是今天康涅狄格州的一个问题。康涅狄格州今天有一个担忧，特别是在哈特福德周围，学校仍然是非法隔离的。它们不符合法律。好的，所以这继续是一个问题。让我把它带到一个稍微不那么有争议的领域，因为那显然是如此有争议，谈谈并尝试将其与模型联系起来。所以在模型中，我们看到的是，如果每个人都选择去东城或者如果每个人都选择去西城，有一种政策有点像校车接送正在进行——如果每个人都选择去东城，我们最终得到的是跨城镇的随机分配，对吗？这有点像校车接送政策。它说每个人，我们剥夺人们对学校的主动选择权，取而代之的是，我们只是随机分配每个人。所以先不考虑康涅狄格州发生了什么，让我们想想一所学校——不是一所很好的学校，但是一所在这稍微偏北一点的学校。所以有一所学校叫做哈佛，哈佛在波士顿外面，至少当我在那里读研究生时，这是哈佛的一个热点问题。

[段 40]

So whereas Yale has colleges, Harvard has houses, but they’re the same thing. And these, someone’s hissing at me, they are the same thing. It’s just the same name, maybe they’re not the same, Maybe there’s some subtle difference, but they’re roughly speaking the same thing. And I’m still getting it. And, uh, alright, so these houses at Harvard, at least until about 1990, I think, roughly, 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 All right And Harvard administrators started worrying So these houses started looking started taking on certain characteristics 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 at dinner at Elliot House, you’d meet all these people. 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. It was that, right? And then if you went to Adam’s house, where I was actually a tutor, it was a very different mix of people. And an extraordinarily large proportion of them were daughters of recently deposed Latin American dictators. It’s not exactly what you’d expect to meet at random.

[译文 40]

耶鲁有学院，哈佛有住宿楼，但它们是同一回事。有人对我发出嘘声，它们是同一回事。只是名字相同，可能它们不是同一回事，也许有一些微妙的区别，但大致来说是同一回事。我还在讲这个。好的，所以在哈佛的这些住宿楼，至少到大约1990年，我想，当我作为研究生在那里时，你最终进入特定住宿楼的方式是你选择要进入哪个住宿楼。好的，哈佛的管理人员开始担心。所以这些住宿楼开始呈现出，开始具有某些特征。例如，有一个住宿楼，我想现在仍然有，叫做艾略特楼。如果你去那里，作为我以前偶尔作为研究生所做的，去艾略特楼吃晚餐，你会遇到所有这些人。这是一种奇怪的体验，因为超过一半的人的名字恰好与哈佛刚建成的游泳池相同。是这样的，对。然后如果你去亚当斯楼，我实际上在那里担任导师，那是一个非常不同的人群。其中非常大比例的人是最近被推翻的拉丁美洲独裁者的女儿。这不是你在随机情况下会遇到的情况。

[段 41]

And then, to give you a third example, there was a house called, I think it’s wrong now, Kirkland House, is that right? Kirkland House, 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. Alright? Alright? Alright? So, this worried the Harvard administrators. Alright? So they were worried about it and presumably they were worried that you know little Ms Pinochet wasn going to have a chance to meet Mr Master Gretzky or something right Whatever it was So they did something about it So what did they do? What did the Harvard administrators do to change this outcome? Anyone know? Yeah, shout it out. They randomized it. So you go out into a group, you know, and you choose which one you’re going to go into. Right, right. So what happened was they imposed randomization. They did exactly the policy that we described up here. 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 the policy Yale had all along. So as happens in other cases in education, Harvard arrived at the Yale solution eventually. All right? So randomization, or more dramatic things like busing, our policies arrived at really because we know of the existence of these models.

[译文 41]

然后，给你们第三个例子，有一个住宿楼，我想现在不对了，叫做柯克兰楼，是吗？柯克兰楼，柯克兰楼被认为是运动员楼，运动员宿舍，当然这是在哈佛，这意味着那里的每个人都是加拿大人。好的？好的？好的？好的，这让哈佛的管理人员担心。好的？所以他们担心它， presumably 他们担心你知道小皮诺切特小姐没有机会见到冰球大帝格雷茨基先生或者其他什么人 对 所以他们做了一些事情。那么他们做了什么？哈佛的管理人员做了什么来改变这个结果？有人知道吗？是的，大声说出来。他们随机安排了。所以你走出去进入一个团体，你知道的，你选择你要进入哪个团体。好的，好的。所以发生的事情是他们实施了随机化。他们正是做了我们在这里描述的政策。这本质上是他们的政策。注意，实际上他们采取的政策，住宿分配是随机的政策，基本上就是耶鲁一直以来的政策。所以正如教育领域的其他情况一样，哈佛最终找到了耶鲁的解决方案。好的？所以随机化，或者像校车接送这样更激进的措施，我们的政策之所以出现，确实是因为我们知道这些模型的存在。

[段 42]

I’m not saying these are right or wrong. It’s not my position to say they’re right or wrong. 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. All right. Now, I want to bring out the third lesson here. And the third lesson here is once again… we’ve drawn, well, I already said, I won’t repeat it. There’s a game theory lesson here about irrelevant details. All right? Irrelevant details mattering. But I want to bring out one more game theory lesson here. So here, we’ve been discussing randomization in this settings in the following form. in the game, the local governments randomize where you lived. In the busing experiments, it was randomly chosen who went to which school. I guess it was done by a social security number, I don’t know. Alright? And in Harvard, it was, and in Yale too, there’s some randomization about which college stroke house you live in. That’s one way to achieve randomization. You can have it done centrally. The central administration, the local government, the central government can randomly assign people. but in principle there’s another way to achieve randomization what’s the other way you could achieve randomization? what’s another way you could achieve randomization? both in this experiment and elsewhere there a hand way at the back there can I get him Each player in the game could randomly choose Right, right.

[译文 42]

我不是说这些是对还是错。我没有资格说它们是对还是错。我是说，如果你要为这些政策辩护，这可能是一种你可能会为这些政策辩护的方式。好的。现在，我想在这里提出第三个教训。而第三个教训是再一次……我们已经画了，嗯，我已经说过了，我不会重复它。这里有一个关于无关细节的游戏理论教训。好的？无关细节很重要。但我想在这里提出另一个游戏理论教训。所以在这里，我们一直在以下形式讨论随机化。在游戏中，地方政府的随机分配决定了你的居住地。在校车实验中，是随机选择谁去哪个学校。我想它是按社会安全号码做的，我不知道。好的？在哈佛，而且在耶鲁也是，关于你住哪个学院或住宿楼有一些随机化。这是一种实现随机化的方式。你可以集中进行。中央政府可以随机分配人员。但原则上还有另一种实现随机化的方式。那是什么？你还能用什么其他方式实现随机化？在这项实验和其他地方，后面有人举手了，我能让他回答吗？游戏中的每个玩家可以随机选择。对，对。

[段 43]

So rather than having centralized randomization, in principle, you could get to the same outcome by having individual randomization. Alright? So there’s another possibility here, which is individual randomization. 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. But it won’t have been some central authority doing the randomization. it’ll be each of you individually doing the randomization. Is that right? 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 be happy to randomize Why Because you be indifferent whether you end up in East Town or West Town so you may as well toss a coin anyway So this is actually another Nash equilibrium I’m being a bit loose here, but we’ll be much more precise in a moment. There’s another Nash equilibrium, yet another Nash equilibrium in this model, and it isn’t as silly in some sense as having everyone go into the East Town and then having the government randomize.

[译文 43]

所以，与其进行集中随机化，原则上，你可以通过个人随机化达到相同的结果。明白吗？所以还有另一种可能性，即个人随机化。原则上，你可以让房间里的每个人都决定他们要做的事情是扔一枚硬币，一枚公平的硬币，每个人分别扔，如果正面朝上，他们就去东城，如果是反面朝上，就去西城。如果他们都这样做，而且都坚持下来，同样根据大数定律，我们会非常接近融合的城镇。但这不会是由某个中央机构进行随机化，而是你们每个人单独进行随机化。对吧？所以这有点不同。注意，如果其他每个人都在这样做，如果其他每个人都在随机化，城镇实际上至少在渐近意义上会是均等混合的，你会乐于随机化。为什么？因为你对你最终在东城还是西城是无差异的，所以你不妨还是抛硬币。所以这实际上是另一个纳什均衡我在这里说得有点松散，但我们稍后会更加精确。还有另一个纳什均衡，在这个模型中还有另一个纳什均衡，而且它在某种意义上并不像让所有人都进入东城然后让政府随机化那样愚蠢。

[段 44]

It’s each person on their own tossing a coin and deciding where they go. It sounds a little bit less like central planning and more like choice. Now, however, as soon as we introduce the idea of individual randomization, we’ve gone a little bit beyond where we’ve got to in the class. Why? Because so far in the class, we’ve talked about strategies as the choices that are available to you. And the choices available to you were go to the east town, go to the west town. Or in the numbers game we played, it was choose a number. Or in the alpha beta game we played, it was choose alpha or beta. Or in telling whether to invest or not, it was invest or don’t invest, and so on and so forth, right? All right? What we seeing now is a new type of strategy And the new type of strategy is to randomize over your existing strategies So we introducing here a new notion that 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. So what is, I’ll be more formal next time, but what is a mixed strategy? It is a randomization over your pure strategies. The strategies we’ve dealt with in the course up to now, from now on we’re going to refer to those as pure strategies, they’re your choices, and we’re going to expand your actual available choices to include all randomizations over those.

[译文 44]

每个人自己抛硬币决定去哪里。这听起来有点像中央计划，少一点更像选择。然而，现在一旦我们引入个人随机化的概念，我们就超越了我们课堂上已经讲到的内容。为什么？因为到目前为止在课堂上，我们把策略作为你可供选择的选项。而你可供选择的选项是去东城或去西城。或者在我们玩过的数字游戏中，是选择一个数字。或者在我们玩过的alpha beta游戏中，是选择alpha或beta。或者在决定是否投资时，是投资或不投资，等等等等，对吧？我们现在看到的是一种新型策略，这种新型策略是在你现有的策略上进行随机化。所以我们在这里引入一个新概念，它将在本周剩余时间里占据我们，这个新概念就是随机化的，或者我们将要称之为的混合策略。什么是混合策略？我下次会更正式地讲，但什么是混合策略？它是在你的纯策略上的随机化。我们这门课到目前为止处理的策略，从现在起我们将其称为纯策略，它们是你的选择，我们将把你的实际可用选择扩展到包括所有这些策略上的随机化。

[段 45]

This may seem a little weird, so to make it seem a little bit less weird, let’s move immediately to an example. I’m done with this, I can use it 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 in the next 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. So the payoffs look like this. Each person has 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 minus 1 minus 1 1 minus 1 1 1 minus 1 1 minus 1 1. Now without me putting any more letters up there, what is this game? Right, so somebody shouted out that this is rock, paper, scissors, right? 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. This is rock, paper, scissors. How many of you have not heard of rock, paper, scissors? Good. The other day I was flicking channels and ESPN had a rock, paper, scissors contest. I just want to know who watches that?

[译文 45]

这可能看起来有点奇怪，为了让它看起来不那么奇怪，让我们立即转向一个例子。我已经用完这个例子了，我可以用它，所以我想谈论的例子是一个我认为你们很多人都熟悉的游戏，但我会在下一个展示收益。我想要谈论的例子是一个我认为你们很多人都熟悉的游戏。但我会展示收益看看。收益看起来是这样的。每个参与者有两个玩家。每个人有三个策略。收益矩阵如下。主对角线上是0,0，绕着走的话是1,-1,-1,1,-1,1,1,-1,1,-1,1。现在不让我放更多字母上去，这是个什么游戏？对的，有人喊出这是石头、剪刀、布，对吧？石头、剪刀、布。所以我想如果我做得对的话，这个应该是石头，而这个应该是剪刀实际上而这个应该是布。这是石头、剪刀、布。你们有多少人没听说过石头、剪刀、布？好的。前几天我在切换频道，ESPN有一个石头、剪刀、布比赛。我只是想知道谁在看那个？

[段 46]

This is quite a fun game and it occurs all over the place It occurs even in episodes of The Simpsons So there a I guess by now famous episode of The Simpsons in which the kids who are I guess Bart and Lisa is that right Somebody help me out here? Bart and Lisa, is that what they’re called? Bart and Lisa are playing rock, paper, scissors for some purpose, to allocate some prize. And Bart is seen to think, as you can do in cartoons, is seen to think, rock, paper, scissors. I love playing rock, paper, scissors. Rock rocks. What could possibly beat rock? And Lisa is seen to think, I love playing rock, paper, scissors, but always chooses rock. And that’s really a hint of where we’re going here. That’s a hint of where we’re going. 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 game. Is that right? So in particular, I claim, and I, if necessary, will prove, that there is no Nash equilibrium in pure strategies, in the strategies we’ve been looking at up to now. There’s no Nash equilibrium in which people choose pure strategies here. So there no Nash equilibrium and from now on I going to use the term pure strategies to mean the strategies we been looking at so far So by pure strategies here the set of pure strategies is equal to the set rock paper and scissors All right?

[译文 46]

这是个相当有趣的游戏，它到处出现。它甚至出现在《辛普森一家》的剧集中。所以有一个我想现在应该很著名的《辛普森一家》剧集，其中孩子们，我想是Bart和Lisa对吧？有人帮我确认一下吗？Bart和Lisa，他们是这样叫的吗？Bart和Lisa为了某个目的在玩石头、剪刀、布，来分配某个奖品。Bart被看到在思考，正如你可以在卡通里做的那样，被看到在思考，石头、剪刀、布。我喜欢玩石头、剪刀、布。石头克石头。什么能打败石头？而Lisa被看到在思考，我喜欢玩石头、剪刀、布，但总是选择石头。这实际上是关于我们要去哪里的一个提示。这是一个非常简单的游戏，石头、剪刀、布，但很明显，我希望，纯策略在这个游戏中可能不足以建模这个游戏。对吗？因此，特别是，我声称，如果必要的话我会证明，在纯策略中，在我们现在一直在看的策略中，没有纳什均衡。没有纳什均衡，在人们选择纯策略的情况下这里没有纳什均衡。所以没有纳什均衡，从现在起我将使用纯策略这个术语来表示我们迄今为止一直在看的策略。所以这里的纯策略是石头、布和剪刀的集合，对吧？

[段 47]

So what we’ve called strategies up to now in the class. Now, can everyone see why there’s not going to be a Nash equilibrium in pure strategies? Let’s just talk it through. So if one person plays rock, the best response against rock is? Let’s try that again. You all know best response to rock is? Paper. All right, but if you played paper the best response to paper is? And the best response to scissors is? Okay, so everyone knows how to play this game, okay? So clearly there’s no there’s not going to be a pure strategy in Nash equilibrium, right? Because because any attempt to look for best responses that are best response to each other can lead to a cycle Right, everyone see that? There’s no hope of finding two pure strategies that are best response to each other because of that cycle. It’s also, I claim, pretty easy to figure out what must be the Nash equilibrium in this game. We’re going to prove it in a second, more or less, but nevertheless I think we actually know it. So what is, in fact, the Nash equilibrium strategy in this game Let get some we can probably cold call somebody Do you want to just cold call somebody at random Yeah just cold call somebody somebody I sure they all know it Just cold call somebody So what the yeah the gentleman in yellow What the Nash equilibrium in this game So I’m allowing you mixed strategies now.

[译文 47]

所以这是我们目前在课堂上称为策略的。现在，大家都能明白为什么纯策略中不会有纳什均衡吗？让我们只是讨论一下。如果一个人出石头，对石头的最佳反应是？让我们再试一次。你们都知道对石头的最佳反应是？布。好的，但如果你出布，对布的最佳反应是？对剪刀的最佳反应是？好的，所以每个人都知道怎么玩这个游戏，对吧？显然不会有不会有纯策略纳什均衡，对吧？因为因为任何寻找互为最佳反应的尝试都会导致一个循环。对吧，大家都看到了？没有希望找到两个互为最佳反应的纯策略，因为那个循环。我还声称，弄清楚这个游戏中必然是什么纳什均衡是相当容易的。我们稍后会证明它，差不多，但我认为我们实际上已经知道了。那么实际上，这个游戏中的纳什均衡策略是什么？让我们叫一个人来回答。我们可以随机点一个人的名。你想随便点一个人吗？是的，就随机点一个人。我确信他们都知道。只是随便点一个人。什么是的，那位穿黄色衣服的先生。这个游戏中的纳什均衡是什么？所以我现在允许你使用混合策略了。

[段 48]

So 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 here? No idea. No idea? You should be able to, I want to play against you. Let’s have someone else pick out. Yeah. What do we think is likely to be the mixed strategy people are going to play in this game in equilibrium? No idea. No idea? All right. Who here regards themselves as a champion rock, paper, scissors player? Playing each choice with one third probability. Thank you. Thank you. I thought that would be okay. So I’m amazed. People live and play this with their siblings or maybe they do and they’ve lost a lot. I don’t know. I don’t know. All right. So I claim, I claim as a guest, I claim that the Nash equilibrium is each player, both players, each player chooses, and I’m going to call it a third, a third, a third. Each player is playing the mixed strategy, a third, a third, a third, which I’m sorry I didn’t catch your name, your name is? Which is Moses. A third, a third, so each player is playing the mixed strategy, a third, a third, a third, which I’m sorry I didn’t catch your name, your name is?

[译文 48]

所以我允许你在你的纯策略上进行随机化。我允许你使用混合策略。我们认为人们在这个游戏中会玩的混合策略是什么？不知道。不知道？你应该能知道的，我想和你玩。让我找另一个人。是的。我们认为在均衡中人们可能会玩的混合策略是什么？不知道。不知道？好的。这里有多少人认为自己是石头、剪刀、布的高手？每个选择以三分之一的概率。谢谢。谢谢。我以为那会行的。所以我很惊讶。人们和他们的兄弟姐妹生活并玩这个，也许他们玩过而且输了很多。我不知道。我不知道。好的。所以我声称，作为一个客人，我声称纳什均衡是每个玩家，两个玩家，每个玩家选择，而且我要称之为三分之一，三分之一，三分之一。每个玩家都在玩混合策略，三分之一，三分之一，三分之一，我很抱歉我没听清你的名字，你的名字是？这是Moses。三分之一，三分之一，所以每个玩家都在玩混合策略，三分之一，三分之一，三分之一，我很抱歉我没听清你的名字，你的名字是？

[段 49]

Moses. Which is Moses’ recommended strategy, alright? So each player is going to play a third, a third, a third. So I’m actually amazed that people didn’t realise that, and again I’m sort of puzzled as to what happened in your childhood, but never mind. Let’s figure out what in fact is the payoff to each of the pure strategies against this. 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. All right? So what’s the expected payoff? So what’s the expected payoff of rock against a third, a third, a third? So the other person’s randomizing equally over rock, paper, scissors, and I’m going to choose rock. What would be my payoff, my expected payoff? Well, I claim that with probability of third, I’ll meet another rock and get 0. And with probability of third, I’ll meet a scissors and get minus 1. And with probability of third I meet paper and sorry I said that wrong Let me start again Probability a third, I’ll meet rock and get zero. Probability a third, I’ll meet scissors and get positive one. And probability a third, I’ll meet paper and get minus one. Is that right this time? That’s plus one and that’s minus one. So my expected payoff is what? It’s a third of 0, a third of plus 1, and a third of minus 1 for a total of 0.

[译文 49]

摩西。也就是摩西推荐的策略，明白吗？每个玩家各出三分之一、三分之一、三分之一。我真的很惊讶人们没有意识到这一点，我又一次感到困惑，你们的童年到底发生了什么，但没关系。让我们来算一下每个纯策略对抗这个策略的收益是多少。我要做的是，通过展示收益来向你们证明这将是一个纳什均衡。明白吗？那么期望收益是多少？当我选择石头，而对方以三分之一、三分之一、三分之一随机选择石头、剪刀、布时，我的期望收益是多少？我声称，以三分之一的概率会遇到另一个石头，得到0。以三分之一的概率会遇到剪刀，得到负1。以三分之一的概率会遇到布，抱歉我说错了，让我重新开始。以三分之一的概率遇到石头，得到0。以三分之一的概率遇到剪刀，得到正1。以三分之一的概率遇到布，得到负1。这次对了吗？那是正1，那是负1。所以我的期望收益是多少？是三分之一的0，三分之一的正1，和三分之一的负1，总和为0。

[段 50]

And it’s not hard to check that the same is true if I chose scissors. If I chose scissors and the other person is randomizing a third, a third, a third, then with probability of third, I will get minus 1. With probability of third, I’ll meet another scissors and get 0. And with probability of third, I’ll meet paper and get 1. and once again it nets out at 0. And finally, if I played paper, and once again I’m going to meet somebody who is in fact randomizing a third, a third, a third over rock, papers, and scissors, then with probability of thirds, I’ll meet a rock and get 1. With probability of thirds I meet scissors and get minus 1 And with probability of thirds I meet paper and get zero Is that correct Everyone happy with that And so it net out at zero 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 a third, a third, a third, they yield an expected payoff of zero. All right? What about playing the mix itself? what’s the expected payoff of playing, myself playing, a third, a third, a third, if I play against somebody who’s playing a third, a third, a third? Well, a third of the time, I’m going to be playing rock.

[译文 50]

不难验证，如果我选择剪刀，结果也是一样的。如果我选择剪刀，而对方以三分之一、三分之一、三分之一随机选择，那么以三分之一的概率我会得到负1。以三分之一的概率我会遇到另一个剪刀，得到0。以三分之一的概率我会遇到布，得到1。最终也是抵消为0。最后，如果我出布，而对方以三分之一、三分之一、三分之一随机选择石头、布和剪刀，那么以三分之一的概率我会遇到石头，得到1。以三分之一的概率我会遇到剪刀，得到负1。以三分之一的概率我会遇到布，得到0。这对吗？大家对这个满意吗？最终也是抵消为0。所以这里每个纯策略，石头、剪刀或布，实际上我算了石头、剪刀和布，每个策略，当它们对抗三分之一、三分之一、三分之一的策略时，它们的期望收益都是0。明白吗？那如果我自己玩这个混合策略呢？如果我玩三分之一、三分之一、三分之一，而对方也玩三分之一、三分之一、三分之一，我的期望收益是多少？三分之一的时间里，我会出石头。

[段 51]

Is that right? A third of the time, I’m going to be playing rock. So a third of the time I’m going to be playing rock, and then I’ll get the expected payoff from playing rock against a third, a third, a third. What was the expected payoff from playing rock against a third, a third, a third? Zero. So a third of the time I going to play rock and I going to get this zero and I wishing I had color Let me just do it without colours messily alright and a third of the time I going to be playing scissors and then I going to get the expected payoff from scissors against a third, a third, a third, and what was the expected payoff of scissors against a third, a third, a third? Zero again, alright, so a third of the time I’ll play scissors and I’ll get this zero, All right? And a third of the time, I’ll be playing paper, in which case I’ll get the payoff, the expected payoff of paper against a third of third of third, which once again was zero. So my total expected payoff is a third of zero plus a third of zero plus a third of zero, which for the math phobics in the room comes out as zero. Okay, a third of zero plus a third of zero plus a third of zero is zero.

[译文 51]

对吗？三分之一的时间里，我会出石头。所以三分之一的时间里我会出石头，然后我将从石头对抗三分之一、三分之一、三分之一的策略中获得期望收益。石头对抗三分之一、三分之一、三分之一的期望收益是多少？是0。所以三分之一的时间里我会出石头，我会得到这个0，我真希望我有彩色笔。让我不用颜色来做，乱一点也行。三分之一的时间里我会出剪刀，然后我会得到剪刀对抗三分之一、三分之一、三分之一的期望收益，剪刀对抗三分之一、三分之一、三分之一的期望收益是多少？还是0，好的，所以三分之一的时间里我会出剪刀，我会得到这个0。好的？三分之一的时间里，我会出布，在这种情况下我会得到布对抗三分之一、三分之一、三分之一的期望收益，这又会是0。所以我的总期望收益是三分之一的0加上三分之一的0加上三分之一的0，对于在场有数学恐惧症的人来说，结果是0。好的，三分之一的0加三分之一的0加三分之一的0等于0。

[段 52]

All right, so notice what we’ve shown here. we’ve shown that if I play what I claim is the equilibrium strategy, a third, a third, a third, against a third, a third, a third, I get zero. And if I played any other strategy, I’d still get zero. Is that right? So therefore playing a third, a third, a third is indeed a best response, albeit weakly, it is indeed a best response, and this is in fact a Nash equilibrium. So what we’ve shown here is, and we’ll stop with this, what we’ve shown is, in rock, paper, scissors, in rock, paper, scissors playing a third, a third, a third against the other guy playing a third, a third, a third is a best response. In fact, anything would be a best response, but in particular it’s a best response so if both people play this one person plays a third, a third, a third and the other person plays a third, a third, a third this is a Nash equilibrium so we’re belabouring a point that we all knew already playing a third, a third, a third if everyone plays a third, a third, a third that is a Nash equilibrium it’s a little harder to show but it’s true that that’s the only equilibrium in the game and before you go So if any of you know the people who run ESPN, this is why watching this game on TV is unlikely to be exciting.

[译文 52]

好的，注意我们在这里展示了什么。我们展示了，如果我玩我声称的均衡策略，三分之一、三分之一、三分之一，而对方也玩三分之一、三分之一、三分之一，我会得到0。如果我玩任何其他策略，我仍然会得到0。对吗？所以玩三分之一、三分之一、三分之一确实是一个最优反应，虽然是弱最优反应，它确实是一个最优反应，而这实际上是一个纳什均衡。我们在这里展示的是，我们在这里停下来，我们展示的是，在石头、剪刀、布中，在石头、剪刀、布中，一方玩三分之一、三分之一、三分之一对抗另一方玩三分之一、三分之一、三分之一是一个最优反应。实际上，任何策略都是最优反应，但特别是它是一个最优反应，所以如果两个人都玩这个，一个人玩三分之一、三分之一、三分之一，另一个人玩三分之一、三分之一、三分之一，这就是一个纳什均衡，所以我们在反复强调一个我们都已经知道的点，如果每个人都玩三分之一、三分之一、三分之一，那就是一个纳什均衡。这有点难证明，但确实这是这个游戏中唯一的均衡，在你们走之前。所以如果你们中有谁认识ESPN的人，这就是为什么在电视上观看这个游戏不太可能精彩。

[段 53]

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

[译文 53]

我们周三会回来，看看其他混合策略和更有趣的游戏。