耶鲁大学博弈论公开课 - 第16集 学会换位思考

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]

Okay, so last time we looked at and played this game. You had to choose grades, sorry, you had to choose alpha and beta, and this table told us what outcome would arise. In particular, what grade you would get and what grade your pair would get. So for example, if you’d chosen beta and your pair had chosen alpha, then you would get a C and your pair would get an A. And one of the first things we pointed out is that this is not quite a game yet. It’s missing something. This has outcomes in it. It’s an outcome matrix. but it isn’t a game because for a game we need to know payoffs. Okay? And then we looked at some possible payoffs, and now it is a game. So this is a game, just to give you some more jargon, this is a normal form game, and here we’ve assumed the payoffs are those that arise if players only care about their own grades which I think was true for a lot of you It wasn true for the gentleman who sitting there now but it was true for a lot of people Alright and we pointed out that in this game alpha strictly dominates beta What do we mean by that? We mean that if these are your payoffs, no matter what your pair does, you attain a higher payoff from choosing alpha than you do from choosing beta.

[译文 1]

好的，上次我们看了这个游戏并玩了它。你们需要选择成绩，对不起，你们需要选择 alpha 和 beta，这张表格告诉我们会产生什么结果。具体来说，就是你会得到什么成绩以及你的搭档会得到什么成绩。例如，如果你选择了 beta 而你的搭档选择了 alpha，那么你会得到 C 而你的搭档会得到 A。我们指出的第一件事是，这还不算一个完整的游戏。它缺少一些东西。这里面有结果，它是一个结果矩阵，但它不是一个游戏，因为对于一个游戏，我们需要知道收益。好吗？然后我们看了某些可能的收益，现在它才是一个游戏。所以这是一个游戏，用术语来说，这是一个标准式博弈，而且这里我们假设收益来自于玩家只关心自己的成绩，我想这对你们中的很多人来说是成立的，对坐在那里的那位先生来说不成立，但对很多人来说是成立的。好，我们指出在这个游戏中 alpha 严格支配 beta。我们这是什么意思？我们的意思是，如果这些是你的收益，无论你的搭档做什么，选择 alpha 都会比选择 beta 给你带来更高的收益。

[段 2]

Right, let’s focus a couple of lessons of the class before I come back to this. One lesson was, do not play a strictly dominated strategy. Everyone remember that lesson? And then much later on, when we looked at some more complicated payoffs, and a more complicated game, we looked at a different lesson, which was this. Put yourself in others’ shoes to try and figure out what they’re going to do. In fact, what we learned from that is it doesn’t just matter what your payoffs are, that that’s obviously important, it’s also important what other people’s payoffs are, because you want to try and figure out what they’re going to do and then respond appropriately. So we’re going to return to both of these lessons today. Both these lessons will reoccur today. Now, a lot of today is going to be fairly abstract, so I just want to remind you that game theory has some real world relevance And again still in the interest of recapping this particular game is called the Prisoner Dilemma It written there the Prisoner Dilemma Alright, notice it’s Prisoner’s plural. And we mentioned some examples last time. Let me just reiterate and mention some more examples which are actually written here, so they’ll find their way into your notes. So for example, if you have a joint project that you’re working on, Perhaps it’s a homework assignment or perhaps it’s a video project like these guys that can turn into a prisoner’s dilemma Why because each individual might have an incentive to shirk?

[译文 2]

好，在我回到这个话题之前，让我们先聚焦于这节课之前的几堂课。一条经验是，不要采用被严格支配的策略。大家还记得这条经验吗？然后在很久之后，当我们看了一些更复杂的收益和一个更复杂的游戏时，我们看到了另一条经验，是这样的。设身处地地为他人着想，试图弄清楚他们要做什么。事实上，我们从中了解到重要的不只是你的收益，那当然重要，但其他人的收益也很重要，因为你想试图弄清楚他们要做什么，然后做出适当的回应。所以今天我们要回到这两条经验。这两条经验今天都会再次出现。现在，今天的大部分内容会相当抽象，所以我只是想提醒你们，博弈论有一些现实世界的重要性。同样，为了回顾，这个特定的游戏叫做囚徒困境。上面写着囚徒困境。注意，囚徒是复数形式。我们上次提到了一些例子。让我重申并再提一些例子，它们实际上就在这里，所以会进入你们的笔记。例如，如果你有一个你们正在做的合作项目，也许是一份作业，也许是一个视频项目，就像这些人的那样，可能会变成一个囚徒困境。为什么？因为每个人可能都有偷懒的动机。

[段 3]

All right price competition two firms Competing with one another in prices can have a prisoner’s dilemma aspect about it Why because no matter how the other firm your competitor prices you might have an incentive to undercut them If both firms behave that way, prices will get driven down towards marginal cost and industry profits will suffer. In the first case, if everyone shirks, you end up with a bad product. In the second case, if both firms undercut each other, you end up with low prices that’s actually good for consumers but bad for firms. And let me mention a third example. Suppose there a common resource out there Maybe it a fish stock or maybe it the atmosphere There a prisoner dilemma aspect to this too You might have an incentive to overfish Why Because if the other countries, with this fish stock, let’s say the fish stock is the Atlantic, if the other countries are going to fish as normal, you may as well fish as normal too. And if the other countries aren’t going to cut down on their fishing, then you want to catch the fish now because there aren’t going to be any there tomorrow. Another example of this would be global warming and carbon emissions. Again, leading aside the science about which I’m sure some of you know more than me here, the issue of carbon emissions is a prisoner’s dilemma.

[译文 3]

好的，价格竞争，两个公司以价格相互竞争，可能会有囚徒困境的方面。为什么？因为无论另一家公司、你的竞争对手如何定价，你都可能会有压低价格与他们竞争的动机。如果两家公司都这样做，价格将被压低到边际成本，行业利润将受损。在第一种情况下，如果每个人都在偷懒，你最终会得到一个糟糕的产品。在第二种情况下，如果两家公司互相压价，你最终会得到低价格，那实际上对消费者有利但对公司不利。让我提第三个例子。假设那里有一个公共资源，也许是鱼群，也许是大气。这也有囚徒困境的方面。你可能会有过度捕捞的动机。为什么？因为如果其他国家，用这个鱼群来说，比如大西洋，如果其他国家正常捕鱼，你不妨也正常捕鱼。如果其他国家不打算减少捕鱼，那么你想现在就捕到鱼，因为明天那里就不会有任何鱼了。另一个例子是全球变暖和碳排放。同样，先不谈科学问题，我确定你们有些人比我懂得多，碳排放问题是一个囚徒困境。

[段 4]

Each of us individually has an incentive to emit carbons as usual. If everyone else is cutting down, I don’t have to. And if everyone else does cut down, I don’t have to. I end up using hot water and driving a big car and so on. In each of these cases, we end up with a bad outcome. So this is socially important. This is not just some abstract thing going on in a class in Yale. And we need to think about solutions to this right from the start of the class. And we already talked about something. We pointed out that this is not just a failure of communication. Alright? Communication per se will not get you out of a prisoner’s dilemma. You can talk out that this is not just a failure of communication. Communication per se will not get you out of a prisoner’s dilemma. You can talk about it as much as you like, but as long as you’re going to go home and still drive your Hummer and have 16 hot showers a day, we’re still going to have high carbon emissions. You can talk about working hard on your joint problem set, but as long as you go home and don’t work hard, it doesn’t help. In fact, if the other person is working hard or is cutting back on their carbon emissions, you have every bit more incentive to not work hard or keep high carbon emissions yourself.

[译文 4]

我们每个人在个人层面都有保持正常排放碳的动机。如果其他所有人都在减排，我就不用减排。如果其他所有人都在减排，我也不需要减排。我最终会使用热水、驾驶大排量汽车等等。在这些情况下，我们都得到了糟糕的结果。所以这在社会上是重要的。这不仅仅是耶鲁课堂上发生的一些抽象的事情。我们从一开始就需要考虑解决这个问题的方法。我们已经谈过一些事情。我们指出，这不仅仅是沟通失败。好吗？沟通本身不会让你摆脱囚徒困境。你可以尽情讨论这件事，但只要你回家后仍然开你的悍马车、每天洗 16 次热水澡，我们仍然会有高碳排放。你可以讨论在合作问题上努力学习，但只要你回家后不努力，那也没有帮助。事实上，如果另一个人正在努力或者正在减少碳排放，你有更多的动机不去努力或者保持高碳排放。

[段 5]

So we need something more, and kind of things we can see more. We can think about contracts. We can think about treaties between countries. We can think about regulation. All of these things work by changing the payoffs. Not just talking about it, but actually changing the outcomes, actually, and changing the payoffs. Changing the incentives. Another thing we can do, a very important thing, is we can think about changing the game into a game of repeated interaction and seeing how much that helps. And we’ll come back and revisit that later in the class. And one last thing we could think of doing, but we have to be very careful here, is we could think about changing the payoffs by education I think of that as the Maoist strategy Lock people up in classrooms and tell them they should be better people That may or may not work I not optimistic but at least it’s the same idea, we’re changing payoffs. Alright? So that’s enough for recap. And I want to move on now. And in particular, we left you hanging at the end last time.

[译文 5]

所以我们需要更多的东西，我们可以看到的各种东西。我们可以考虑合同。我们可以考虑国家之间的条约。我们可以考虑监管。所有这些东西都是通过改变收益来工作的。不只是谈论它，而是真正改变结果，实际上是改变收益。改变激励。我们可以做的另一件事，一件非常重要的事情，是我们可以考虑把这个游戏变成一个重复互动的游戏，看看这能有多大帮助。我们稍后会在课上回到这个问题。我们可以想到做的最后一件事，但我们必须非常小心，是通过教育来改变收益。我认为那是毛泽东策略，把人关在教室里，告诉他们应该成为更好的人。这可能有效也可能无效，我不太乐观，但至少是同样的想法，我们在改变收益。好吗？所以回顾部分就到这里。现在我想继续前进。尤其是上次结束时我们让你们悬在那里。

[段 6]

We played a game at the very end last time where each of you chose a number all right, all of you chose a number and we said the winner was going to be the person who gets closer to two thirds of the average in the class now we’ve figured that out we’ve figured out who the winner is and I know a lot of you are trying to see if you won, is that right? and I’m going to leave you in suspense okay, so I’m going to tell you today who won we did figure it out and we’ll get there, but I want to do a little bit of work first, all right, so I’m just going to leave it in suspense that’ll stop you walking out early if you want to win the prize all right So there’s going to be lots of times in this class when we get to play games, we get to have classroom discussions, and so on, but there’s going to be some times when we have to slow down and do some work. And the next 20 minutes are going to be that. All right? So with apologies for being a bit more boring for 20 minutes let do something we call formal stuff In particular I want to develop and make sure we all understand what are the ingredients of a game.

[译文 6]

上次在最后我们玩了一个游戏，你们每个人都选择了一个数字，好，你们每个人都选择了一个数字，我们说赢家将是那个最接近班级平均数三分之二的人。现在我们已经算出来了，我们已经算出了谁是赢家，我知道你们很多人在试图看看自己是否赢了，对吗？我要让你保持悬念，好吧，所以今天我要告诉你们谁赢了。我们确实算出来了，我们会知道的，但我想先做一点工作，好吧，所以我先保持悬念。这会让你如果想赢得奖品的话不会早退。在这门课上会有很多次我们玩游戏、进行课堂讨论等等，但也会有一些时候我们需要放慢脚步做一些工作。接下来的 20 分钟就是那个时候。好吗？所以对于接下来 20 分钟稍微无聊一点表示歉意，让我们做一些我们称之为正式的东西。尤其是我想发展并确保我们都理解一个游戏的要素是什么。

[段 7]

So in particular, we need to figure out what formally makes something into a game. And the formal parts of a game are this. we need players, alright, and while we’re here, let’s develop some notation, so for standard notation for players, I’m going to use things like little i and little j, alright, so in that numbers game, the game when all of you wrote down a number and handed it in at the end of last time, the players were who? Players were you, right, you all were the players, right, useful text and expression meaning you plural, right, In the numbers game, we’ll give something over here. In the numbers game, you all were the players. All right? Second ingredient of the game are strategies There a good clue here If I writing you should be writing Okay Strategies All right Notation All right So I’m going to use little si, little si, to be a particular strategy of player i. Particular strategy of player i. So an example in that game might have been choosing the number 13. All right, everyone understand that? Now I need to distinguish this from the set of possible strategies of player I. So I’m going to use capital SI to be what? to be the set of alternatives, the set of possible strategies of player I. So in that game we played at the end last time, what were the set of strategies?

[译文 7]

那么，特别是，我们需要弄清楚到底是什么正式地使某样东西成为一个游戏。游戏的正式组成部分是这样的。我们需要玩家，好吧，既然我们在讨论这个，让我们发展一些符号，所以对于玩家的标准符号，我将使用小写的i和小写的j，好吧，所以在那个数字游戏里，就是你们所有人都写下一个数字并在最后交上去的那个游戏，玩家是谁？玩家是你们，对，你们都是玩家，对，有用的文本和表达意思是你们复数，对，在数字游戏里，我们在这里给一些东西。在数字游戏里，你们都是玩家。明白了吗？游戏的第二个要素是策略。这里有一个很好的线索，如果我在写，你们应该也在写。好，策略。好的，符号。好的，所以我将用小写的si，小写的si，来表示玩家i的一个特定策略。玩家i的特定策略。所以在那个游戏中一个例子可能是选择了数字13。大家都明白吗？现在我需要把这个与玩家i的可能策略集合区分开来。所以我将用大写的SI来表示什么？表示备选方案集合，玩家i的可能策略集合。所以在那个我们上次结束时玩的游戏中，策略集合是什么？

[段 8]

They were the set 1, 2, 3, all the way up. To 100. Alright, so… I’m distinguishing a particular strategy from the set of possible strategies. And while we’re here, I want a third notation for strategy. I’m going to use little s without an i, alright, no subscripts, little s without an i, to mean a particular play of the game. a particular play of the game. So what do I mean by that? All of you at the end last time wrote down this number and handed them in. So we had one number, one strategy choice for each person in the class. So here they are. Here’s my collected in set of strategy choices. Here’s the bundle of bits of paper you handed in last time. this is a particular play of the game. Alright? I’ve got each person’s name, and I’ve got a number from each person, a strategy from each person. Alright? We actually have it on a spreadsheet as well, so here it is written out on a spreadsheet Each of you each of your name is on this spreadsheet and the number you chose Alright So that a particular player of the game and that has a different name We sometimes call this a strategy profile. So in the textbook, you’ll sometimes see the term a strategy profile, or a strategy vector, or a strategy list.

[译文 8]

它们是集合1，2，3，一直到100。好的，所以……我是在把特定策略与可能策略的集合区分开来。既然我们在讨论这个，我想要第三个策略符号。我将用小写的s不带i，好吧，没有下标，小写的s不带i，来表示游戏的一个特定行动。游戏的一个特定行动。这么说是什么意思？你们所有人在上次结束时写下了这个数字并提交了。所以我们每个人有一个数字，班上每个人有一个策略选择。所以这里就是。这是我的收集到的策略选择集合。这是你们上次提交的纸条束。这是一个特定的游戏行动。明白吗？我有每个人的名字，我有每个人给的一个数字，每个人给的一个策略。实际上我们也在电子表格上有这个，所以这里是电子表格上写出来的。你们每个人的名字都在这个电子表格上，你选择的数字。好的，所以这是游戏的特定行动，它有一个不同的名字。我们有时称之为策略组合。所以在教科书里，你们有时会看到术语策略组合，或策略向量，或策略列表。

[段 9]

It doesn’t really matter. What it’s saying is, one strategy for each player in the game. Alright? So in the numbers game, this is the spreadsheet. An example of this is the spreadsheet. So you can still see that. I’m going to pull down these boards. And let me clean something. So you might think we’re done, right? We’ve got players. We’ve got the choices they could make. That’s their strategy sets. We’ve got those individual strategies. And we’ve got the choices they actually did make. That’s the strategy profile. Seems like we’ve got everything you could possibly want to describe a game. What are we missing here Say it shout it out Payoffs We missing payoffs right So to complete the game we need payoffs All right? And again, I need notation for payoffs, so in this course I’ll try and use u for utile to be player i’s payoff. All right? So u i will depend on player 1’s choice all the way to player i’s own choice all the way up to player M’s choices. All right? So player I’s payoff, UI, depends on all the choices in the class, in this case, including her own choice. And of course, a shorter way of writing that would be UIS. It depends on the profile. All right? So in the numbers game, what is this? In the numbers game, UI of S can be two things.

[译文 9]

这其实不重要。它的意思是，游戏中的每个玩家各有一个策略。明白吗？所以在数字游戏中，这就是电子表格。这就是一个例子。所以你们还能看到那个。我要把这些板子拉下来。让我清理一下。所以你们可能认为我们完成了，对？我们有了玩家。我们有了他们可以做的选择。那是他们的策略集合。我们有了那些个体策略。我们有了他们实际做出的选择。那是策略组合。好像我们已经有了描述一个游戏可能需要的一切。这里我们漏掉了什么？说出来，大声喊出来。收益。我们漏掉了收益，对？所以要完成这个游戏我们需要收益，好吧？同样，我需要收益的符号，所以在这门课里我将尝试用u来表示utile，作为玩家i的收益。好的？所以ui将取决于玩家1的选择一直到玩家i自己的选择一直到玩家M的选择。好的？所以玩家I的收益，UI，取决于班上所有的选择，在这个案例中，包括她自己的选择。当然，更简短的写法是UIS。它取决于这个组合。好的？所以在数字游戏中，这是什么？在数字游戏中，UI of S可以是两种情况。

[段 10]

It can be $5 minus your error in pennies if you won. I guess it could be something if it was a tie. I won’t bother writing that now. And it’s going to be zero otherwise. All right So we now got all of the ingredients of the game Players strategies payoffs All right Now we going to make an assumption today and for the next ten weeks or so All right? So for almost all the class. We’re going to assume that these are known. We’re going to assume that everybody knows the possible strategies everyone else could choose. and everyone knows everyone else’s payoffs. Now that’s not a very realistic assumption and we are going to come back and challenge it at the end of the semester but this will be complicated enough to get us a lot of material in the next 10 weeks. I need one more piece of notation and then we can get back to having some fun. So one more piece of notation. I’m going to write S minus I to mean what? It’s going to mean a strategy choice for everybody except person I It’s going to be useful to have that notation around Alright? So this is a choice for all except I this is a choice for all except person i or player i. So in particular, if you’re person 1, then s minus i would be s2, s3, s4 up to sn, but it wouldn’t include s1.

[译文 10]

如果获胜的话，可以是5美元减去你的误差（以分为单位）。我想如果是平局的话可能也有点什么。现在我就不写那个了。否则就是零。好的所以我们现在有了游戏的所有要素。玩家、策略、收益。好的。现在我们要做一个假设，今天以及接下来的大约十周。好吧？所以几乎整个学期。我们将假设这些是已知的。我们将假设每个人都知道其他人可能选择的策略，而且每个人都知道其他人的收益。现在这不是一个非常现实的假设，我们将在学期末回来挑战它，但这将足够复杂，足以让我们在未来10周获得大量材料。我还需要一个符号，然后我们就可以回到找点乐子了。所以再来一个符号。我要写S减I来表示什么？它将表示除了人I之外所有人的策略选择。这将是有用的符号。好的？所以这是除了I之外所有人的选择，这是除了人i或玩家i之外所有人的选择。所以特别地，如果你是1号，那么s减i就是s2，s3，s4一直到sn，但不包括s1。

[段 11]

All right? And it’s useful. Why? Because sometimes it’s useful to think about the payoffs as coming from I’s own choice and everyone else’s choices. It’s just a useful way of thinking about things. All right? Now this one I want to stop for a second. And I know that some of you from past experience are somewhat math-phobic. You do not have to wave your hands in the air if you’re math-phobic. But since some of you are, let me just get you all to take a deep breath. This goes for people who are math-phobic at home too. All right? So everyone’s in a slight panic now. You know, you came here today, you thought everything was going to be fine, now I’m putting math on the board. Alright? Take a deep breath, it’s not that hard, and in particular, notice that all I doing here is writing down notation There actually no math going on here at all I just developing notation Now I don want anybody to quit this class because they worried about math or math notation So if you are in that category of somebody who might quit it because of that, come and talk to me, come and talk to the a’s, we will get you through it. Alright? It’s fine to be math-phobic, I’m phobic of all sorts of things. Alright? Not necessarily math, but all sorts of things.

[译文 11]

好的？而且这很有用。为什么？因为有时候把收益看作来自I自己的选择和其他所有人的选择是有用的。这只是一种有用的思维方式。好的？这里这个我想停一下。我知道你们中有些人在过去经验中有些害怕数学。你们不必在空中挥动手如果你害怕数学的话。但既然你们有些人会害怕，让我让你们都深呼吸一下。这对在家害怕数学的人也适用。好的？所以现在每个人都稍微恐慌了。你们知道，你们今天来的时候，以为一切都会很好，现在我在黑板上写数学。好的？深呼吸，这没那么难，特别是，注意我在这里所做的只是写下符号。这里实际上根本没有数学在进行。我只是在发展符号。所以我不想让任何人因为担心数学或数学符号而退出这门课。所以如果你属于可能因为这个原因而退课的那类人，来找我谈谈，来找助教们谈谈，我们会帮你度过的。好的？害怕数学没关系，我也有各种各样害怕的东西。好的？不一定害怕数学，但有各种各样害怕的东西。

[段 12]

Alright? So serious thing, a lot of people get put off by notation, it looks scarier than it is, There’s nothing going on here except for notation at this point. All right. So let’s have an example to help us fix some ideas. And again, I’ll have to clean the board, so give me a second. I think an example might help those people who are disturbed by the notation. so here’s a game which we’re going to discuss briefly it involves two players and we’ll call the players one and two and player one has two choices top and bottom and player two has three choices left centre and right right It just a very simple abstract example for now And let’s suppose the payoffs are like this, they’re not particularly interesting. We’re just going to do it for the purpose of illustration. So here are the payoffs. 5 minus 1, 11, 3, 0, 0. 6, 4, 0, 2, 2, 0. alright, let’s just map the notation we just developed into this game, okay, so first of all who are the players here, right, well there’s no secret there, the players are let’s just write it down, why don’t we, the players here in this game are player 1 and player 2 alright, what about the strategy sets what about the strategy sets, or the strategy alternatives All right?

[译文 12]

好的？那么说正经的，很多人被符号吓到，它看起来比实际更可怕，这里目前除了符号什么都没有。好的。让我们举一个例子来帮助我们巩固一些想法。同样，我得清理一下黑板，所以给我一秒钟。我想一个例子可能对那些被符号困扰的人有帮助。所以这里有一个游戏我们要简单讨论一下，它涉及两个玩家，我们将称玩家为1和2，玩家1有两个选择，上和下，玩家2有三个选择，左、中、右。对这只是现在一个非常简单的抽象例子。让我们假设收益是这样的，它们不是特别有趣。我们只是要把它作为说明的目的来做。所以这里是收益。5减1，11，3，0，0。6，4，0，2，2，0。好的，让我们把我们刚发展的符号映射到这个游戏中，好吧，那么首先这里的玩家是谁，对，没什么秘密，玩家是让我们写下来吧，我们不妨写下来，这个游戏中的玩家是玩家1和玩家2，好的，策略集合呢？策略集合，或者策略备选方案呢？好的？

[段 13]

So here, player one’s strategy set, she has two choices, top or bottom, represented by the rows, which are helpfully the top row and the bottom row. And player two has three choices This game is not symmetric so they have different number of choices that fine Player two has three choices left centre and right, represented by the left, centre, and right column in the matrix. And just to point out in passing, up to now we’ve been looking mostly at symmetric games. Notice this game is not symmetric in the payoffs or in the strategies. There’s no particular reason why games have to be symmetric. And payoffs alright so again this is not rocket science but let’s do it anyway so just an example of payoffs so player 1’s payoff if she chooses top and player 2 chooses centre we read by looking at the top row and the centre column and player 1’s payoff is the first of these payoffs so it’s 11 alright and player 2’s payoff from the same choices top for player 1 center for player 2 again we go along the top row and the center column but this time we choose player 2’s payoff which is the second payoff so it’s 3 ok so again I’m hoping this is calming down the math phobics in the room now how do we think this history.

[译文 13]

那么在这里，玩家一的策略集有两个选择，上或下，用行来表示，也就是上排和下排。玩家二有三个选择。这个博弈不是对称的，所以他们有不同的选择数量，这很正常。玩家二有三个选择，左、中、右，用矩阵中的左列、中列和右列来表示。顺便说一下，到目前为止我们主要看的是对称博弈。注意这个博弈在支付或策略上都不是对称的。博弈没有必须是对称的理由。支付也是一样的，所以再次说明，这不是什么高深的东西，但我们还是来做一下。举一个支付的例子，如果玩家一选择上，玩家二选择中，我们看第一行和中间列，玩家一的支付是这些支付中的第一个，所以是11。玩家二从同样的选择中得到的支付，玩家一选上，玩家二选中心，我们同样看第一行和中间列，但这次我们取玩家二的支付，也就是第二个支付，所以是3。所以再次希望这能让房间里对数学恐惧的人平静下来。我们怎么看待这段历史。

[段 14]

Okay. All right. So again, I’m hoping this is calming down the math phobics in the run. Now, how do we think this game is going to be played? This is not a particularly interesting game, but while we’re here, why don’t we just discuss it for a second? All right? So if the, if our mic guys get a little bit ready here, all right? So how do we think this game should be played? Well, let’s ask somebody at random, perhaps. So, Alley, do you want to ask this guy in the blue shirt here? Right. Does player one here, does player one have a dominated strategy? No, if player one doesn’t have a dominated strategy, for instance, if player two picks left and player one wants to pick bottom, but if player two picks center, player one wants to pick center. Good, excellent. Very good. I should get you to stand up. I forgot that. Never mind, but that was very clear, thank you. And was that loud enough people going to hear it? Do people hear that? People at the back, you hear it. So even that wasn’t loud enough, okay? So we really need to get people to do that. That was very clear. very nice. We need people to stand up and shout. All those people at the back can’t hear. So what’s, I’m going to know, your name is Patrick.

[译文 14]

好的。好的。所以再次，我希望这能让那些对数学恐惧的人平静下来。那么，我们认为这个博弈会如何进行呢？这不是一个特别有趣的博弈，但我们在这里，为什么不花一点时间来讨论一下呢？好吗？如果我们的麦克风工作人员准备好一点的话，好的？我们认为这个博弈应该怎么进行？好吧，让我们随机问一个人。也许吧。Alley，你想问这个穿蓝衬衫的人吗？好的。这个玩家一有没有被支配的策略？没有，如果玩家一没有被支配的策略，例如，如果玩家二选择左，玩家一想选下，但如果玩家二选择中，玩家一想选中。很好，非常好。我应该让你站起来。我忘了。没关系，但这非常清楚，谢谢。这个声音够大吗，大家能听到吗？后面的人能听到吗？后面的人，你们听到了吗？所以即使那个声音还不够大，好的？所以我们真的需要让人这样做。这非常清楚。非常好。我们需要人们站起来大声喊。后面那些人都听不到。那么，你叫什么名字来着，Patrick。

[段 15]

Patrick said was, no, player one does not have a dominated strategy. Top is better than bottom against left because, sorry, bottom is better than top against left because six is bigger than five. But top is better than bottom against center because 11 is bigger than zero. Everyone see that? Right? So it’s not the case that top always always does better than bottom, or that bottom always does better than top. Okay, what about, I’ve raised hands this time, what about player two, does player two have a dominated strategy? Everyone’s keeping their hands firmly down, so as not to get spotted here. Should we, yeah, should we try, uh, Ali, can we try this guy in white? Do you want to stand up and wait until Ali gets there? And really, but yell it out now. Louder. I believe right is a dominated strategy because if player one chooses top, then player two will choose center. And if, or I’m getting confused now, it looks better on my paper. But yeah, right is never the best choice. Okay, good, good. So let’s be a little bit careful here. So your name is Thomas. Thomas. So Thomas said something which was true, but it doesn’t quite match with the definition of a dominated strategy. What Thomas said was, right is never a best choice. That’s true. that’s true.

[译文 15]

Patrick说的是，没有，玩家一没有被支配的策略。上比下对左更好，因为，抱歉，下比上对左更好，因为6大于5。但上比下对中更好，因为11大于0。大家都明白了吗？好的？所以上并不总是比下好，或者下并不总是比上好。好的，那么，关于，我已经举手了，玩家二有没有被支配的策略？大家的手都紧紧放下，以免被注意到。我们要不要，呃，Ali，我们能不能试试这个穿白衣服的人？你想站起来等Ali过来吗？真的，现在大声喊出来。再大声一点。我认为右是一个被支配的策略，因为如果玩家一选择上，那么玩家二会选择中。或者我有点搞混了，在我纸上看起来更好。但是，是的，右从来不是最佳选择。好的，很好，很好。让我们仔细一点。你的名字是Thomas。Thomas。Thomas说的是某些真的东西，但这与被支配策略的定义不太完全一致。Thomas说的是，右从来不是最佳选择。那是对的。那是对的。

[段 16]

All right? But to be a dominated strategy, we need something else. We need that there’s another strategy of player two that always does better. Right? Now that turns out also be true in this game, but it’s a C. All right. So in this particular game, I claim that center dominates right. The center dominates right. So it’s to see that. If player one shows top, center yields three, right yields zero, three is bigger than zero. And if player one chooses bottom, then center yields two, right, yields zero, two is bigger than zero again. So in this game, center strictly dominates right. All right? What you said was true, but I wanted something specifically about domination here. All right. So what we know here, we know that player two should not choose, should not choose right. Now, in fact, that’s as far as we can get with dominance arguments in this particular. particular game. But nevertheless, let’s just stick with it a second. I gave you the I gave you the definition of strict dominance last time. And it’s also on the handout, by the way, the handout on the web. But let me write that definition again using or making use of the notation from the class. So definition, all right, so player I, player I’s strategy, s, little I prime is strictly dominated by player I’s strategy SI if, and now, if, and now we can use use annotation, if U.I.I. from choosing S-I when other people choose S-minus-I is strictly bigger than U-I-S-I-prime when other people choose S-minus-I.

[译文 16]

好吗？但要成为一个被支配的策略，我们需要别的东西。我们需要玩家二有另一个策略总是做得更好。对吗？现在，这在这个博弈中也被证明是真的，但它是一个C。好的。所以在这个特定的博弈中，我声称中支配右。中支配右。所以这是显而易见的。如果玩家一出示上，中产生3，右产生0，3大于0。如果玩家一选择下，那么中产生2，右产生0，2再次大于0。所以在这个博弈中，中严格支配右。好吗？你说的是对的，但我想要的是关于支配的具体内容。好的。我们在这里知道的，我们知道玩家二不应该选择，不应该选择右。现在，事实上，这就是我们在这个特定的特定博弈中用支配论证能得到的极限。但尽管如此，让我们再坚持一会儿。我给了你，我给了你严格支配的定义上次。还有它也在讲义上，顺便说一下，网络上的讲义。但让我用课堂上的符号再写一下这个定义。定义，好的，玩家I，玩家I的策略s，小I素被玩家I的策略SI严格支配，如果，而且现在，如果，而且现在我们可以使用使用注释，如果U.I.I.当其他人选择S-minus-I时选择S-I产生的严格大于U-I-S-I-素当其他人选择S-minus-I时产生的。

[段 17]

And the key part of the definition is for all S-minus-I. All right. All right. So say in words, player I strategy S-I-prime is strictly dominated by her strategy S-I-I-S-I, I, if S-I always does strictly better. words, player I strategy S.I. Prime is strictly dominated by her strategy S.I. If S.I. Always does strictly better, always yields a higher payoff for player I, no matter what the other people do. This is the same definition we saw last time, just being a little bit more nerdy and putting it in some notation. Are people panicking about that? People look like deer in the headlamps yet? No. You look alright, alright-ish. All right. Let’s have a look at another example. People okay? I can move this? All right. So it’s a slightly more exciting example now. So imagine the following example. An invader is thinking about invading a country. And there are two ways, there are two passes, if you like, through which he can lead his army. And you, you are the defender of this country and you have to decide which of these passes or which of these routes into the country you’re going to choose to defend. And the catch is you can only defend one of these two routes. All right? If you want a real-world example of this, think about the third century BC. Someone can correct me afterwards.

[译文 17]

定义的关键部分是针对所有S-minus-I。好的。好的。用语言来说，玩家I的策略S-I素被她的策略S-I-I-S-I严格支配，如果S-I总是做得更好。用语言来说，玩家I的策略S.I.素被她的策略S.I.严格支配，如果S.I.总是做得更好，总是为玩家I产生更高的收益，无论其他人做什么。这与我们上次看到的定义相同，只是更书呆子气一点，用一些符号表示。人们对此感到恐慌吗？人们看起来像车灯前的鹿吗？不。你看起来还好，还算正常。好的。让我们看另一个例子。人们还好吗？我可以移动这个吗？好的。所以这是一个稍微更刺激的例子。想象以下例子。一个入侵者正在考虑入侵一个国家。有两条路，有两个关隘，如果你愿意的话，可以通过它们来率领他的军队。而你，你是这个国家的防御者，你必须决定你要选择保卫这些关隘中的哪一个，或者这个国家的哪些路线。关键是你只能保卫这两条路线中的一条。好的？如果你想要一个真实世界的例子，想想公元前三世纪。有人之后可以纠正我。

[段 18]

I think it’s the third century BC when Hannibal is thinking of crossing the Alks. not Hannibal Lecter, Hannibal the general in the third century BC. The one with the elephants. All right. Okay. So the key here is going to be that there are two passes. One of these passes is a hard pass. It goes over the Alps. And the other one is an easy pass. It goes along the coast. All right. If the invader chooses the hard pass, he will lose one battalion of his army simply in getting over the mountains. simply going through the hard past. All right. And if he meets your army, whichever pass he chooses, if he meets your army defending a past, then he’ll lose another battalion. All right? Now, I haven’t given you roughly the choices. The choice is going to be for the attacker which pass to choose and for the defender which pass to defend. But let’s put down some payoffs so we can start talking about this. All right. So in this game, the payoffs for this game are going to be as follows. So it’s a simple two-by-two game. This is going to be the attacker. This is Hannibal. And this is going to be the defender. And I’ve forgotten which general was defending, and someone’s about to tell me they don’t. And there are two passes you could defend the easy pass or the hard pass.

[译文 18]

我认为是公元前三世纪，汉尼拔正在考虑穿越阿尔卑斯山。不是汉尼拔·莱克特，是公元前三世纪的将军汉尼拔。那个带着大象的。好的。好的。关键是有两个关隘。其中一个是艰难的关隘。它要翻越阿尔卑斯山。另一个是容易的关隘。它沿着海岸走。好的。如果入侵者选择艰难的关隘，他将损失一个营的军队，仅仅是在翻越高山的途中。仅仅是通过艰难的关隘。如果他遇到你的军队，无论他选择哪个关隘，如果他遇到你的军队防守一个关隘，他将再损失一个营。好的。现在，我没有粗略地给你选择。选择将是进攻者选择哪个关隘，以及防守者选择防守哪个关隘。但让我们放下一些收益，这样我们就可以开始谈论这个了。好的。在这个博弈中，这个博弈的收益将如下。这是一个简单的2x2博弈。这将是进攻者。这将是汉尼拔。这将是防守者。我忘了是哪位将军在防守，有人正要告诉我他们也不知道。有两条关隘你可以防守，容易的关隘或艰难的关隘。

[段 19]

And there’s two you could use to attack through, easy or hard. And again, easy pass here just means no mountains. We’re not talking about something on the New Jersey turnpike. So the payoffs here are as follows, and I’ll explain them in a second. All right. So his payoff, the attacker’s payoff, is how many battalions does he get to bring into your country? He only has two to start with. And for you, it’s how many battalions of his get destroyed. So just to give an example, if he goes through the hard pass and you defend the hard pass, he loses one of those battalions going over the mountains, and the other one because he meets you. So he has none left, and you’ve managed to destroy two of them. Conversely, if he goes on the hard pass and you defend the easy pass, he’s going to lose one of those battalions, who he’ll have one left. He lost it in the mountains, but that’s the only one he’s going to lose because you were defending the wrong past. All right, everyone understand the payoff of this game? All right. So now imagine yourself as a Roman general. This is going to be a little bit of a stretch for imagine. but imagine yourself as a Roman general, and let’s figure out what you’re going to do.

[译文 19]

这里有两个通道可以进攻，简单的或困难的。同样，这里说的简单通道只是指没有山。我们说的不是新泽西收费公路上的那种情况。收益如下，我稍后解释。好，攻击者的收益是他能有多少营进入你们国家？他一开始只有两个营。对你来说，收益是他有多少营被摧毁。举个例子，如果他走困难通道而你防守困难通道，他在翻山时会损失一个营，另一个营是因为遭遇你。所以他一个营都不剩，而你成功摧毁了两个。相反，如果他走困难通道而你防守简单通道，他会损失一个营，还剩一个。他损失在山里的那个营，但这是他唯一会损失的营，因为你在防守错误的通道。大家明白这个游戏的收益了吗？好，现在想象你自己是罗马将军。这想象起来可能有点勉强，但想象你自己是罗马将军，我们来想想你会怎么做。

[段 20]

You’re the defender, what are you going to do? So let’s have a show of hands. How many of you think you should defend the easy pass? Raise your hands, let’s raise your hands so as you can see them up, wave them in the air, bit of motion, raise them in the air, should bring you flags, okay? Because these are the Romans defending the easy pass, and how many of you think you’re going to defend the hard pass? We have a huge number of people who don’t want to be Roman generals here. All right? Let’s try it again. No extensions, right? I’m not going to penalize you getting the wrong answer. All right. So how many of you think you’re going to defend the easy pass? Raise your hands again. And how many think you’re going to defend the hard pass? All right. So we have a majority choosing easy pass. We’ve got a large majority. So what’s going on here? Is it the case that defending the easy pass dominates defending the hard pass? Is that the case, is the case that depending the easy pass dominates defending the hard pass? You can shout out? No, it’s not. defending the hard pass? Is that the case? Is the case that defending the easy pass dominates defending the hard pass? You can shout out? No, it’s not.

[译文 20]

你是防守方，你会怎么做？让我们举手表决。有多少人认为应该防守简单通道？举起手来，举高一点，让你们能看到，挥一挥，举高，如果能有旗帜就更好了，好吗？因为这些是防守简单通道的罗马人，有多少人认为你会防守困难通道？我们这里有大量的人不想当罗马将军。好吗？让我们再试一次。没有延长时间，对吧？我不会因为你答错而扣分。好，有多少人认为你会防守简单通道？再举起手来。有多少人认为你会防守困难通道？好，我们大多数人选择了简单通道。我们有非常大的多数。怎么回事？防守简单通道是否优于防守困难通道？是这样吗？是防守简单通道优于防守困难通道吗？谁能喊出来？不是的，防守困难通道是吗？是防守简单通道优于防守困难通道吗？谁能喊出来？不是的。

[段 21]

In fact, in fact, we can check that if the attacker attacks through the easy pass, not surprisingly, you do better if you defend the easy pass than the hard pass, one versus zero. But if the attacker was to attack through the hard pass, again, not surprisingly, you do better if you defend the hard pass than the easy pass. All right, so that’s not an unintended. intuitive finding, it isn’t the case that defending easy dominates defending hard. You just want to match with the attacker. Nevertheless, almost all of you chose easy. What’s going on? Can someone tell me what’s going on? Let’s get the mics going a second. So can we catch this guy with a beard? If we just wait for the mic to get there. And if you could stand up, stand up, stand up, and shout. There you go. Because you want to minimize the amount of enemy soldiers that reach. Rome or whatever location is. All right, you want to minimize the number of soldiers that reach Rome. That’s true. On the other hand, we’ve just argued that you don’t have a dominant strategy here. It’s not the case that easy dominates hard. What else could be going on? While we’ve got you up, why don’t we get the other guy who’s going to his hand up there in the middle? And again, stand up and shout in that mic.

[译文 21]

事实上，事实上，我们可以检查，如果攻击者从简单通道进攻，毫不奇怪，如果你防守简单通道比防守困难通道表现更好，一比零。但如果攻击者从困难通道进攻，同样毫不奇怪，如果你防守困难通道比防守简单通道表现更好。好，这不是什么出人意料的发现，防守简单优于防守困难并不是事实。你只是想和攻击者匹配。尽管如此，你们几乎所有人都选择了简单通道。怎么回事？谁能告诉我怎么回事？让我们把麦克风传一下。我们等那个有胡子的人吧。如果麦克风能传到他那里的话。如果你能站起来，站起来，站起来，大声说。就你了。因为你想尽量减少到达罗马或任何地点的敌军士兵数量。好，你想要尽量减少到达罗马的士兵数量。这是真的。另一方面，我们刚刚论证了你没有优势策略。简单优于困难并不是事实。还可能发生了什么？既然我们已经让你站起来了，为什么不让另一个在中间举起手的人过来呢？同样，站起来对着麦克风大声说。

[段 22]

Point your face towards Mike. Good. It seems as though, well, while you don’t have a dominating strategy, it seems like Hannibal is, is better off attacking through, it seems like he would attack through the easy pass. Good. Why does it seem that? I guess that’s right. We’re on the right lines now. Why does it seem like he’s going to attack through the easy pass? Well, if you’re not defending the easy pass, he gets, he doesn’t, he doesn’t lose anyone. And if he attacks through the hard pass, he’s going to lose at least one battalion. All right, all right. So let’s have a look at it from, let’s do the exercise, let’s do the second lesson by emphasizing the beginning. Let’s put ourselves in Hannibal’s shoes, that probably boots, something, I don’t know, whatever you do when you’re riding an elephant, whatever you wear, all right? Let’s put ourselves in Hannibal’s shoes and try and figure out what Hannibal’s going to do here. All right? So it could be, from Hannibal’s point of view, he doesn’t know which pass you’re going to defend, all right? But let’s have a look at his payoffs. If he, if you would defend, if you were to defend the easy pass and he goes through the easy pass, he will get into your country with one battalion, and that’s the same as he would have got if he went through the hard pass.

[译文 22]

把你的脸朝向麦克风。好的。看起来，似乎，虽然你没有优势策略，但汉尼拔似乎似乎更倾向于从简单通道进攻。好的。为什么看起来他会从简单通道进攻？嗯，如果你不防守简单通道，他不会损失任何人。如果他从困难通道进攻，他至少会损失一个营。好，好。让我们从另一个角度看看，让我们通过强调开头来做练习。让我们站在汉尼拔的立场上，那大概是靴子之类的，我不知道，你骑大象时穿的什么，好吧？让我们站在汉尼拔的立场上，试图弄清楚汉尼拔会怎么做。好？从汉尼拔的角度，他不知道你会防守哪个通道，好吧？但让我们看看他的收益。如果你会防守简单通道，而他走简单通道，他会有一个营进入你们国家，如果走困难通道，结果也是一样。

[段 23]

So if you defend the easy pass, from his point of view, it doesn’t matter whether he chooses the easy pass, he gets one in there, or the hard pass, he gets one in there. But if you were to defend the hard pass, if you would have to defend the mountains, then if he chooses the easy pass, he gets both battalions in, and if he chooses the hard pass, he gets no battalions in. All right? So in this case, easy is better. And we have to be a little bit careful. It’s not the case that for Hannibal, choosing the easy pass to attack through strictly dominates choosing the hard pass, but it is the case that there’s a weak notion of domination here. It is the case, and used in jargon, it is the case that the easy pass for the attacker weakly dominates the hard pass for the attacker. What do I mean by weakly dominate? It means by choosing the easy pass he does at least as well and sometimes better than he would have done had he chosen the hard pass. All right? So here we have a second definition, a new definition for today. And again, we can use our jargon. Definition. Player I’s strategy. S.I. Prime is weakly dominated by her strategy. S.I. if, and now we’re going to take advantage of annotation, if player I’s payoff from choosing S.I against S.

[译文 23]

所以如果你防守简单通道，从他的角度来看，他选择简单通道还是困难通道并不重要，他都有一个营进入。但如果你会防守困难通道，如果你必须防守山脉，那么如果他选择简单通道，他两个营都进入，如果他选择困难通道，他一个营都进不去。好？在这种情况下，简单更好。我们需要小心一点。对于汉尼拔来说，选择简单通道进攻并不严格优于选择困难通道，但确实存在一种弱的支配关系。确实，用术语来说，对于攻击者来说，简单通道弱支配困难通道。弱支配是什么意思？意思是通过选择简单通道，他至少表现得一样好，有时甚至比他选择困难通道更好。好？这里我们有第二个定义，今天的新定义。同样，我们可以使用我们的术语。定义。玩家I的策略。S.I. prime被她的策略S.I.弱支配，如果，玩家I选择S.I对抗S.

[段 24]

Minus I is always as big as or equal to her payoff from choosing s i prime against s minus i and this has to be true for all things that anyone else could do and in addition player i is payoff from choosing s i against s minus i is strictly better than her payoff from choosing s i prime against s minus i is strictly better than her payoff from choosing S.I. Prime against S minus I for at least one thing. strictly better than her payoff from choosing S-I-prime against S-minus I for at least one thing that everyone else could do. So you just check that exactly corresponds to the easy and hard thing we just had before. Say it again. Player I strategy S-I-prime is weakly dominated by her strategy S-I. if she always does at least as well by choosing S-I than choosing S-I prime, regardless of what everyone else does, and sometimes she does strictly better. And it seems a pretty powerful lesson, just as we said, you should never choose a strictly dominated strategy. You’re probably never going to choose a weakly dominated strategy either, but it’s a little more subtle. All right. Now that definition, if you’re worried about what I’ve written down here and you want to see it in words, on the handout I’ve already put on the way, that has the summary of the first class, I included this definition in words as well.

[译文 24]

Minus I的收益总是大于或等于她选择s i prime对抗s minus i的收益，而且这必须对其他人可能做的所有事情都成立，此外玩家i选择s i对抗s minus i的收益严格优于她选择s i prime对抗s minus i的收益，对于其他人可能做的至少一件事来说严格优于她选择S-I-prime对抗S-minus I的收益。所以你可以检查这正好对应我们之前关于简单和困难的例子。再说一遍。玩家I的策略S-I-prime被她的策略S-I弱支配，如果无论其他人做什么，她选择S-I总是至少和选择S-I prime一样好，有时甚至严格更好。这似乎是一个相当有说服力的教训，正如我们所说，你应该从不选择被严格支配的策略。你可能也永远不会选择被弱支配的策略，但这更微妙一些。好，如果你担心我写在这里的东西，你想看文字版的，在我已经在走的路上发的讲义里，有第一节课的总结，我也包含了用文字写的这一定义。

[段 25]

So compare the definition in words with what’s written here in the nerdy notation on the board. Now, since we think that Hannibal, the attacker, is not going to play a weekly dominated strategy, we think that Hannibal is not going to choose the hard pass. He’s going to attack on the easy pass. And given that, which should we defend? We should defend easy. most of you chose. So be honest now, was that why most of you chose easy? Yeah, probably was, right? You may be able to read this. All right? So by putting ourselves in Hannibal’s shoes, we could figure out that his hard attack strategy was weakly dominated. He’s going to choose easy, so we should defend easy. Having said that, of course, Hannibal went through the mountains, which kind of screws up the lesson, but too late now. All right. Now then, I promised you we’d get back to the the game from last time. All right. So where have we got to so far on this class? We know, we know from last time that you should not choose a dominated strategy. And we also know we probably aren’t going to choose a weekly dominated strategy. And we also know that you should put yourself in other people’s shoes and figure out that they’re not going to play a strictly or weakly dominated strategy.

[译文 25]

现在，把用文字描述的定义和黑板上用符号写的定义对比一下。既然我们认为汉尼拔，也就是攻击者，不会选择弱被支配的策略，我们就认为汉尼拔不会选择困难通道。他会选择简单通道发起攻击。既然如此，我们应该防守哪个？大多数人选了简单。好吧，现在诚实一点，大多数人选简单是因为这个原因吗？是的，大概是的，对吧？你应该能看懂这个。好吧。所以，通过把自己代入汉尼拔的立场，我们就能推断出他的强攻策略是弱被支配的。他会选择简单，所以我们应该防守简单。话虽如此，当然了，汉尼拔穿过了山脉，这多少让这堂课的效果打了折扣，但现在后悔也来不及了。好吧。现在，我答应过你们会回到上节课的那个博弈。我们在这堂课上到目前为止进行到哪里了？我们从上节课知道，不应该选择被支配的策略。我们也知道我们大概不会选择弱被支配的策略。我们还知道应该把自己代入他人的立场，算出他们不会选择严格被支配或弱被支配的策略。

[段 26]

That seems a pretty good way to predict how other people are going to play. So let’s take those ideas and go back to the numbers game from last time. Now, before I do that, I don’t need the people at home to see this, but how many of you were here last time? How many were not? I asked the wrong question. How many were not here last time? All right. So we handed out again that game. We handed out again the game with the numbers. But just in case, let me just read out the game you played. This was the game you played. Without showing your neighbor what you’re doing, put in the box below a whole number. between 1 and 100. We will, and in fact have, calculated the average number chosen in the class, and the winner of this game is the person who gets closest to 2 thirds times that average number, they will win $5 minus the difference in pennies. All right? So everybody filled that in last time, and I have their choices here. So let’s just, before we reveal who won, let’s discuss this a little bit, and that we come down hazardously off this stage, and figure out it’s got the mic’s up a bit a second. We can get some mics ready. All right. So let me find out some people here and see what people did a second.

[译文 26]

这似乎是一个预测别人行为的相当不错的方法。那么让我们带着这些想法回到上节课的数字游戏。在我开始之前，我不需要在家的人看到这个，但有多少人上节课来了？有多少人没来？我问错问题了。有多少人上节课没来？好吧。所以我们又发了那个游戏。又发了那个有数字的游戏。但以防万一，让我把你们玩的这个游戏读一下。这是你们玩的游戏。不要让你旁边的人看到你在做什么，在下面的格子里填一个1到100之间的整数。我们已经计算出了班上选出的数字的平均值，获胜者是那个数字最接近这个平均值的三分之二的 人，他将获得5美元减去差额美分的奖金。好吧？所以上节课每个人都填了这个，我有他们的选择。所以让我们在揭晓谁获胜之前先讨论一下，然后我们就从舞台上下来，看看是不是麦克风抬高了一点。我们可以准备一些麦克风。好吧。让我在这里找一些人，看看他们刚才做了什么。

[段 27]

You can be honest here since I’ve got everything in front in front of me. So how many of you chose some number like 32, 33, 34? Actually, I’ll tell you, nine of you did. So should I read out the name if I embarrass people? All right. So we’ve got Lynette, Lecousen, we’ve got Christian, Bargen. There’s nine of you here. Let’s try to get. How many of you chose numbers between 32 and 34? Yeah, okay, so a number of you, now we’re seeing some hands up. All right. So keep your hands up a second. Give a hands up a second. There’s people. All right. So let me ask people why. Can we, can you get your hand into the guy. What’s your name? If you get him to stand up? Stand up a second. And shout out to the class can hear you. What’s your name? Chris. Chris, you’re on this list somewhere. I mean, you’re not on this list somewhere. No, mind. What did you just? I think I chose 30. Okay, 30. That’s pretty close. Okay. So why did you choose 30? Because I thought everyone was going to be around like the 45 range, because 66 is two-thirds, or right around of 100. And then we’re going to go two-thirds less than that. And I did one less than that one. All right.

[译文 27]

你可以在这里诚实回答，因为我这里都有记录。你有多少人选了像32、33、34这样的数字？实际上，我告诉你，有九个人选了。那么如果我念出名字会让人尴尬吗？好吧。我们有Lynette、Lecousen、Christian、Bargen。有九个人在这里。让我们试着找一下。有多少人选了32到34之间的数字？是的，好吧，你们中有一些人选了这个数字，现在我们看到一些人举手了。好吧。所以请把手举一会儿。再举一下手。有一些人。好吧。所以让我问问大家为什么。能请你把麦克风递给他吗？你叫什么名字？如果你让他站起来的话？站起来一下。然后喊出来让全班听到。你叫什么名字？Chris。Chris，你在这份名单上的某个地方。我的意思是，你不在这份名单上的某个地方。没关系的。你刚才选了什么？我想我选了30。好的，30。相当接近。好的。所以你为什么选30？因为我想每个人都会在45左右，因为100的三分之二是66。然后我们再取三分之二。我想的是比那个数小一。好吧。

[段 28]

Okay. All right. So thank you. Let’s get one of the others. There was another one in here. Can you just raise your hands again? The people are sort of around 33, 34. There’s somebody in here. Can we get you to stand up and you’re between Mike. So let me, you Yep. Go ahead. Shout it out. What’s your name, first of all? Ryan. and you’re between Mike, so let me, yep, go ahead. Shout it out, what’s your name, first of all? Ryan. Ryan, I must have you here as well, and never mind. What did you choose? 33, I think. 33, yeah, you did. You are Ryan Lowe. Yeah. You are Ryan Lowe, okay. Good, go ahead. I thought similar to Chris, actually, and I also thought that if we got two-thirds and everyone was choosing numbers in between one and a hundred, end up 33 would be around the number. All right, so just to repeat the argument that we just heard, again, you have to shout it out more than that. I’m guessing people didn’t hear that in the room, so let’s repeat it to make sure everyone hears it. A reason for choosing a number like 33 might go as follows. If people in the room choose randomly between 100, then the average is going to be around 50, say, and two-thirds of 50 is around 33.

[译文 28]

好的。好的。谢谢。让我们找另一个人。这里还有一个人。你们这些人，手举起来一下。就是那些选33、34左右的。这里有一个人。你能站起来吗？你在Mike旁边。让我看看，你。好吧，说吧。喊出来。你叫什么名字，首先？Ryan。Ryan，你在Mike旁边，所以让我，好吧，说吧。喊出来，首先叫什么名字？Ryan。Ryan，我这里应该也有你的名字，没关系。你选了什么？我想是33。33，是的，你选了。你是Ryan Lowe。是的，你是Ryan Lowe，好。好的，我想的和Chris差不多，实际上，而且我还以为如果我们取三分之二，而每个人都选择1到100之间的数字，最后33就是这个数字。好吧。所以让我们重复一下我们刚才听到的论点，再说一遍，你要喊出来让大家听到。我在猜测人们在房间里没听到那个，所以让我们重复一下确保大家都听到。选像33这样的数字的一个理由可能是这样的。如果房间里的人随机选择1到100之间的数字，那么平均值会是50左右，而50的三分之二是33。

[段 29]

33 and a third, literally. All right? So that’s a pretty good piece of reasoning. All right? What’s wrong with that reasoning? What’s wrong with that? What’s wrong with that? Can we get the woman in the striped shirt here? Yeah, yeah, sorry. We have not a woman for a while, so that’s that old woman. Thank you. That even if everyone else had the same reasoning as you, it’s still going to be way too high. All right. So in particular, if everyone else had the same reasoning as you, it’s going to be way too high, all right? So if everyone else reasons that way, then everyone in the room would choose a number like 33 or 33. And in that case, the average would be what? Sorry, the average, I’ve just told it that, that two-thirds of the average would be what? Something like 22, all right? So the flaw in the argument that Chris and Ryan had, it isn’t a bad argument, it’s a good starting point, but the flaw in the argument, mistaken in the argument, it was the first sentence in the argument. The first sentence in the argument was, if the people in the room choose random, then they will choose. choose around 50. That’s true. The problem is, the people in the room aren’t going to choose as random. All right? Look around the room a second.

[译文 29]

33又三分之一，确切地说。好吧？所以这是一个相当好的推理。好吧？那这个推理有什么问题？有什么问题？有什么问题？能请穿条纹衬衫的那位女士吗？是的，是的，对不起。我们很久没有女士了，所以是那位老女士。谢谢。即使其他每个人都有和你一样的推理，这也仍然会太高了。好吧。所以在特定情况下，即使其他每个人都有和你一样的推理，这也仍然会太高了，好吧？所以如果其他所有人都有和你一样的推理，那么房间里的每个人都会选择像33这样的数字。在这种情况下，平均值会是多少？对不起，平均值，我已经说过了，三分之二的平均值会是多少？大概是22，好吧？所以Chris和Ryan的论点中的缺陷，这不是糟糕的论点，这是一个好的起点，但这个论点中的缺陷，论点中的错误，是论点的第一句话。论点的第一句话是，如果房间里的人随机选择，那么他们会选择。选择接近50。这是真的。问题是，房间里的人不会随机选择。好吧？看一下房间一会儿。

[段 30]

Look around yourself. Do any of you look like a random number generator? Actually, from here I can see something that do, I’m not going to, I’m not going to put, all right? All right. So, I’m actually looking at some of your answers, maybe some of you are. On the whole, Yale students are not random number generators. They’re trying to win the game. All right. So they’re unlikely to choose numbers at random. If they, we also a further argument, if in fact everyone thought that way and if you figured out everyone was going to figure out, was going to figure out, was going to think that way, then you would expect everyone to choose a number like 33, and in that case, you should choose a number like 22. How many if you, raise your hand a second, how many of you chose numbers in the range 21 to 23? There’s way more of you than that. I’ll start reading you out as well. There are actually about 12 of you, Raise your hands. There’s 12 hands going up somewhere. There’s two, three hands going up, four, five hands going. There’s actually 12 people who chose exactly 22. So considerably more if we include 23 and 21. All right. So those people, I’m guessing, I’m guessing, were thinking this way. Is that right? Let me get one of my 22s up again.

[译文 30]

看看你周围。你们中有任何人看起来像随机数生成器吗？实际上，从这里我能看到一些是的，我不会，我不会，好吧？好吧。实际上，我正在看你们的一些答案，也许你们中的一些人是。总的来说，耶鲁学生不是随机数生成器。他们试图赢得游戏。好吧。所以他们不太可能随机选择数字。如果事实上每个人都是这样想的，而且如果你算出每个人都会算出来，会算出来，会这样想，那么你就会期望每个人选择像33这样的数字，在这种情况下，你应该选择像22这样的数字。有多少人，如果你们，举手一会儿，有多少人选了21到23之间的数字？你们中有更多的人选了这个范围。我也开始念你们的名字。有大约12个人，举起手来。有12只手举起来。有两，三只手举起来，三，四，五只手举起来。实际上有12个人选了恰好22。所以如果我们包括23和21，那就更多了。好吧。所以这些人，我猜，我猜，是这样想的。对吗？让我再找一个选22的人。

[段 31]

Here’s a 22. Do you want to get this guy? So what’s your name, sir? Stand up, stand up, Michelle. Right. Ryan? Yep. And you chose 22? I chose 22 because I thought that most people would play the game dividing by two-thirds a couple of times and give numbers averaging around the low 30s. All right. All right. So if you think people are going to play a particular way, in particular if you think people are going to choose the strategy of Ryan and Chris and choose around 33, then 22 seems a great answer. answer. But you underestimate your Yale colleagues. You underestimate your Yale colleagues. In fact, 22 was way too high. It was way too high. Now, again, let’s just iterate the point here, let’s repeat the point here. The point here is when you’re playing a game, you want to think about what other people are trying to do to try and predict what they’re trying to do. And it’s not necessarily a great starting point to assume that the people around you are random number generators. They have aims. trying to win and they have strategies too. Let me take this back to the board a second. So in particular, are there any strategies here we can really rule out. We said already people are not random. Are there any choices we can just rule out?

[译文 31]

22，这是你的答案。你想选这个人吗？先生，你叫什么名字？站起来，站起来，Michelle。好的。Ryan？是的。你选22是因为我觉得大多数人玩这个游戏会除以三分之二几次，给出的数字平均在30出头。所以，如果你认为人们会以一种特定的方式玩游戏，特别是如果你认为人们会像Ryan和Chris那样选择大约33的策略，那么22似乎是一个很棒的答案。但你低估了你的耶鲁同事。你低估了你的耶鲁同事。事实上，22太高了。太高了。现在，让我们再重复一下这一点。在玩游戏时，你想要思考其他人试图做什么，试图预测他们试图做什么。假设你周围的人是随机数生成器并不是一个好的起点。他们有目标。他们试图获胜，他们也有策略。让我回到黑板这里。所以，有没有什么策略是我们真的可以排除的？我们已经说过人们不是随机的。有没有什么选择我们可以直接排除？

[段 32]

We know people are not going to choose those choices. Let’s have someone here. Can we have the guy in green? Wait, wait for LA. Let’s go. Stand up. Give me your name. My name is Nick. And shout it out to people here. Let’s have someone here. We have the guy in green, just wait, wait, Phalae, there’s go. Great, stand up. Give me your name. My name’s Nick. And shout it out to people going to hear. No one is going to choose number over 50. No one is going to choose number over 50, okay, I was gonna, that’s, that’s, that’s fair enough, some people did, I, let me, let me. It’s fair enough. I was thinking a little bit less, that’s fine, I was thinking a little bit less ambitious. Somebody said 66, somebody said 66, so let’s, let’s start analyzing this. All right, so in particular, there’s something about these strategy choices that are greater than 67, at any rate. All right, certainly, I mean, so it says 66, let’s, let’s go up a little bit. So these numbers bigger than 67, what’s wrong with numbers bigger than 67? What’s wrong? Raise your hands if you ever answered that. What’s wrong? Can we get the guy in red who’s right close to the mic from there? Stand up, give me your name, stand up, shout it out to the crowd.

[译文 32]

我们知道人们不会选择那些选择。让我们这里找一个人。我们能让穿绿色衣服的那个人吗？等等，等一下，LA。让我们开始。站起来。告诉我你的名字。我叫Nick。向大家大声喊出来。让我们这里找一个人。我们有穿绿色衣服的那个人，等一下，等一下，Phalae，来了。很棒，站起来。告诉我你的名字。我叫Nick。向大家大声喊出来。没人会选择超过50的数字。没人会选择超过50的数字，好的，我本来想说，没错，没错，这是合理的。有些人确实选了，我，让我，让我稍微不那么雄心勃勃一些。有人选了66，有人选了66，所以让我们开始分析这个。好的，所以，特别是，关于这些大于67的策略选择有一些问题。无论如何。是的，当然，我的意思是，所以有人选了66，让我们稍微往上一点。超过67的数字有什么问题？有什么问题？如果你知道答案，请举手。那边紧挨着麦克风穿红衣服的那个人，你能站起来吗？站起来，告诉我你的名字，站起来，向人群大声喊出来。

[段 33]

Peter? Yep. And if everyone chooses 100, it would be 67. All right, good. So even if everyone in the room didn’t choose randomly, but they all shows 100, a very unlikely circumstance. But even if everyone, everyone in the room, didn’t choose randomly, but they all shows 100, a very unlikely circumstance. Every one in shows 100, the highest the average could, sorry, the highest two-thirds of the average could possibly be is 66 and 2-thirds, hence 67 would be a pretty good choice in that case. So numbers bigger than 67 seem pretty crazy choices, but crazy isn’t the word I’m looking for here. What can we say about those choices, those strategies, bigger than 67, 68 and above? What can we say about those choices? There’s somebody right behind you, a woman right behind you. Shout it out? They have no pay. They have no payoffs? What’s the jargon here? Let’s use our jargon. Somebody shouted out. What’s the jargon about that? They’re dominated. So these strategies are dominated. Actually, they’re only weekly dominated, but that’s okay. They’re certainly dominated. In particular, a strategy like 80 is dominated by choosing 67. Right? You will always get a payoff from choosing 67 at least as high and sometimes higher than the payoff you would have got had you chosen 80 no matter what else happened in the room.

[译文 33]

Peter？是的。如果每个人都选择100，那就是67。好的，很好。所以即使房间里的每个人不是随机选择的，但他们都选了100，一个非常不可能的情况。但即使每个人，房间里的每个人，不是随机选择的，但他们都选了100，一个非常不可能的情况。每个人都选了100，平均值的三分之二最高能是66又三分之二，因此在这种情况下67会是一个相当好的选择。所以超过67的数字似乎是相当疯狂的选择，但疯狂不是我在这里要找的词。对于那些选择，那些大于67、68及以上的策略，我们能说些什么？你的正后方有个人，一位女士，在你身后。大声喊出来？他们没有收益。他们没有收益？这里的术语是什么？让我们用我们的术语。有人喊出来了。这方面的术语是什么？它们是被压制的。所以这些策略是被压制的。实际上，它们只是弱被压制，但没关系。它们当然是被压制的。特别是，像80这样的策略被选择67所压制。对吧？无论房间里发生了什么情况，选择67你得到的收益至少与选择80一样高，有时甚至更高。

[段 34]

So these strategies are dominated. We know from the very first lesson of the class last time that no one should choose these strategies. They’re dominated strategies. All right. So did anyone choose strategies bigger than 67? Okay, I’m not going to read out names here, but. Turns out four of you did. I’m not going to make you way of it. Okay. All right. So, okay, for the four of you did, never mind, never mind, but, well, yeah, mind, actually. Yeah. All right. All right. So once we’ve eliminated, once we’ve eliminated the possibility that anyone in the room is going to choose a strategy bigger than 67, it’s as if those numbers, 68 through 100, are irrelevant. It’s really as if the game is being played, are the only choices available on the table are 1 through 67. Is that right? Is that right? We know no one’s going to choose 68 and above, so we can just forget them. We can delete those strategies. And once we delete those strategies, all that’s left are choices 1 through 67. All right. So can somebody help me out now? What can I conclude? No, I’ve concluded that the strategy is 68 through 100 essentially don’t exist or have been deletes. What can I conclude? Let me see if I can get a mic in here. So stand up and wait for the mic.

[译文 34]

所以这些策略是被压制的。我们从上次课的第一个教训中知道，不应该有人选择这些策略。它们是被压制的策略。好的。那么有没有人选择了大于67的策略？好的，我不会在这里读出名字，但是。结果你们中有四个人选了。我不会让你们难堪的。好的。好的。那么，一旦我们排除了，一旦我们排除了房间里任何人会选择大于67的策略的可能性，就好像那些数字，68到100，是无关的。它实际上就好像游戏正在进行的，只有1到67的选择可用在桌上。是对的吗？是对的吗？我们知道没人会选68及以上，所以我们可以忘记它们。我们可以删除那些策略。一旦我们删除那些策略，剩下的就是1到67的选择。好的。那么谁能帮我一下？我能得出什么结论？不，我已经得出结论，策略68到100基本上不存在或已被删除。我能得出什么结论？让我看看能不能在这里拿到麦克风。站起来，等麦克风。

[段 35]

And he comes to the mic. Good. Shout out. Then all strategies 45 and above are this also ruled out. Good, good. So your name is… Henry. Okay. So Henry is saying, once we’ve figured out that no one should choose a strategy bigger than 67, then we can go another step and say, if those strategies never existed, then the same argument rules out, or a similar argument, rules out strategies bigger than 45. Let’s be careful here. The strategies that are less than 67, but bigger than 45, these strategies are not, they are not dominated strategies in the original game. In particular, we just argued that if everyone in the room chose 100, then 67 would be a winning strategy. So it’s not the case that the strategies between 45, and 67 are dominated strategies, but it is the case that… they’re dominated once we delete the dominated once we delete 67 and above. All right? So these strategies, to be careful here, the word weekly here, these strategies are not weekly dominated. in the original game, but they are dominated, they’re weakly dominated, once we delete 68 through 100. So all the strategies 45 through 67 are gone now. So, okay, let’s have a look. Did anyone choose, going to raise your hands, be brave here, did anyone choose the strategy between 45 and 67? Or between 46 and 67?

[译文 35]

他来到麦克风这里。很好。大声喊出来。那么所有45及以上的策略也被排除了。很好，很好。那么你叫……Henry。好的。所以Henry在说，一旦我们弄清楚不应该有人选择大于67的策略，我们就可以再走一步，说，如果那些策略从未存在过，那么同样的论证或类似的论证排除了大于45的策略。让我们仔细一点。小于67但大于45的这些策略，它们不是，在原始游戏中不是被压制的策略。特别是，我们刚刚论证过，如果房间里的每个人都选了100，那么67会是一个获胜策略。所以45到67之间的策略不是被压制的策略，但确实的情况是……一旦我们删除了被压制的，一旦我们删除了68到100，它们就是被压制的。好的？所以这些策略，仔细一点，这里的术语是弱的，这些策略在原始游戏中不是弱被压制的。但它们是被压制的，它们是弱被压制的，一旦我们删除68到100。所以所有45到67的策略现在都消失了。那么，让我们看看。有没有人选择，举起你们的手，要勇敢，有没有选择45到67之间的策略？或者46到67之间？

[段 36]

No one’s written there, but I know some of you did because I got it in front of me, right? So at least four of you did, and I won’t read out those names yet, but I might read them out next time. So four more people shows those strategies. All right. Now notice. there’s a different part of this. This argument, the argument that eliminates strategies 67 and above, or 68 upwards, that strategy just involves the first lesson of last time. Do not choose a dominated strategy. Admittedly weak here, but still. All right? But this second slice, strategy’s 45 through 67, getting rid of those strategies involves a little bit more. You’ve got to put yourself in the shoes of your fellow classmen and figure out they’re not going to choose 67 and above. And so the first argument, that’s a straightforward argument. The second argument says, I put myself in other people’s shoes, I realize they’re not going to play a dominated strategy, and therefore, having realized they’re not going to play a dominant strategy, I shouldn’t play a strategy between 45 and 67. So this argument is an in shoes argument. Now what? Where can we go now? Yes, so let’s have the guy in the beard, but let me let the mic get to him. All right, and yell out your name. Carter, you just repeat the same reasoning again and again, and you eventually get down to one.

[译文 36]

没人写出来，但我知道你们有些人选了，因为我有记录，对吧？所以你们至少有四个人选了，我还不读出那些名字，但我下次可能会读。所以又有四个人选了这些策略。好的。现在注意。这里有一个不同的部分。这个论证，这个消除67及以上或68以上的策略的论证，那个策略只涉及上次课的第一课。不要选择被压制的策略。诚然是弱的，但仍然如此。好的？但这第二步，策略45到67，摆脱那些策略需要更多一点。你必须设身处地为你的同学着想，弄清楚他们不会选择67及以上。所以第一个论证是一个 straightforward 的论证。第二个论证说，我设身处地为他人着想，我意识到他们不会玩被压制的策略，因此，意识到他们不会玩被压制的策略，我不应该玩45到67之间的策略。所以这个论证是一个设身处地的论证。现在怎么样？我们要往哪里走？是的，让我们找那个留胡子的人，但让我让麦克风到他那里。好的，大声喊出你的名字。Carter，你只是重复同样的推理一次又一次，你最终会得到一个。

[段 37]

All right, well, let’s, let’s do that, but let’s go one step at a time. So now we’ve ruled out the possibility that anyone’s going to choose the strategy 68 and above because they’re weakly dominated, and we’ve ruled out the possibility that anyone’s going to choose a strategy between 46 and 67, because those strategies are dominant. because those strategies are dominated once we’ve ruled out the dominated strategies. So we know no one’s choosing any strategies above 45, it’s as if the numbers 46 and above don’t exist, so we know that the highest anyone could ever choose is 45, and two-thirds of 45 is roughly, something helping out here? 30, right, roughly 30. All right? So we know that all the numbers between 45 and 30, these strategies, were not dominated, and they weren’t dominated even after deleting the dominated strategies, but they are dominated once we deleted not just the dominated strategies, but also the strategies that were dominated once we deleted the dominated strategies. All right? I’m not going to try and write that, but you should try and write it in your notes. All right. So without writing that argument down in detail, notice that we can rule out the strategies 30 through 45, not by just examining our own payoffs, not just by putting ourselves in other people’s shoes, and realizing they’re not going to choose a dominant strategy, but by putting ourselves in other people’s shoes, while they are putting themselves in someone else’s shoes and figuring out what they’re going to do.

[译文 37]

好的，让我们一步一步来。现在我们已经排除了任何人会选择68及以上的策略的可能性，因为它们被弱支配了，我们也排除了任何人会选择46到67之间的策略的可能性，因为那些策略是支配的。因为一旦我们排除了被支配的策略，这些策略就被支配了。所以我们知道没有人会选择45以上的任何策略，就好像46及以上不存在一样，所以我们知道任何人能选择的最高值是45，而45的三分之二大约是，帮忙算一下？30，对，大约30。好的。所以我们知道45到30之间的所有这些数字，这些策略，没有被支配过，即使在删除被支配的策略之后也没有被支配，但一旦我们不仅删除了被支配的策略，而且还删除了那些在被删除被支配策略后被支配的策略，它们就被支配了。好的，我不打算试图写下来，但你应该试着把它写在笔记里。好的。所以不详细写这个论证，注意我们可以排除30到45的策略，不是仅仅通过检查我们自己的收益，不是仅仅通过设身处地为别人着想，意识到他们不会选择支配策略，而是通过设身处地为别人着想，而他们正在设身处地为另一个人着想，并弄清楚他们会做什么。

[段 38]

All right? So this isn’t in shoes, we’re careful where we are here, this isn’t in shoes, in shoes arguments, at which point you might want to invent the sock. All right. Now, where’s this going? So we were told where it’s going. We were able to rule out 68, above then we’re able to rule out 46 and above, now we’re able to rule out 31 and above, all right, by the next slice down, we’ll be able to eliminate, what is it, about 20 and above, all right, so 30 down to above 20, and this will be an in shoes, in shoes, right? These patches aren’t dominated, nor are they dominated once they delete the dominates, strategies, nor we dominated the strategies that are dominated once we delete the shoes. These strategies aren’t dominated, nor are they dominated once we dominated once we delete the dominated strategies, nor are we dominated the strategies but they are dominated once we delete the strategies as the dominated in the next. You get what I’m doing, okay? All right. So where does this argument going to go? Where’s this argument going to go? It’s going to go all the way down to one. It’s going to go all the way down to one. We could repeat this argument or we’re down to one. Notice that once we’ve deleted the dominated strategies, we You know, I said before about four people chose this strategy.

[译文 38]

好的？所以这不是设身处地，我们要小心我们在这里的位置，这不是设身处地，设身处地的论证，你可能想发明袜子。好的。现在，这要去哪里？所以我们被告知它要去哪里。我们能够排除68以上，然后我们能够排除46及以上，现在我们能够排除31及以上，好的，下一个层次，我们将能够消除，大约是20及以上，好的，所以从30到20以上，这将是一个设身处地，设身处地，对吧？这些补丁没有被支配，一旦他们删除支配的策略，它们也没有被支配，我们也没有支配那些在被删除袜子后被支配的策略。这些策略没有被支配，一旦我们删除被支配的策略后它们也没有被支配，一旦我们删除那些在下一个被支配的策略后它们也没有被支配。你明白我在做什么吗？好的。所以这个论证要去哪里？这个论证要去哪里？它要一直下降到1。它要一直下降到1。我们可以重复这个论证，直到我们下降到1。注意，一旦我们删除了被支配的策略，你知道，我之前说过大约四个人选择了这个策略。

[段 39]

And in here, about four people chose this strategy. But in this range 30 through 45, I had lots of people. How many of you chose the number between 30 and 45? Well, more than that, I guarantee you more than that shows number between 1345. In fact, the people who chose, well, we started off, 33, shows in that range. And a lot more of you chose numbers between 20 and 30. So we’re really getting into the meat of the distribution here, and we’re seeing that these choices that perhaps are ruled out by this kind of reasoning. All right. Now, I’m still not going to quite reveal yet who won. I want to take just one step more abstract. All right? So I want to just discuss this a little bit more. I want to discuss the consequence of rationality in playing games. Slightly philosophical for a few minutes. All right? So I claim that if you are a rationality, player, by which I mean somebody who’s trying to maximize their payoffs by their player of the game, that simply being rational, just being a rational player, rules out playing these dominated strategies. All right? So the four of you who chose numbers bigger than 67, whose names are not going to read out, maybe they were making a mistake. All right? However, however. the next slice down requires more than just rationality.

[译文 39]

在这里，大约四个人选择了这个策略。但在这个30到45的范围内，我有很多人。你们有多少人选择了30到45之间的数字？嗯，比那个更多，我向你们保证比那更多的人选择了13到45之间的数字。事实上，选择了那个数字的人，嗯，我们开始，33，显示在那个范围内。而且更多你们选择了20到30之间的数字。所以我们真的开始进入分布的核心了，我们看到这些选择可能被这种推理所排除。好的。现在，我仍然不完全打算透露谁赢了。我想再抽象一步。好的？所以我想再讨论一下。我想要讨论一下理性在玩游戏中的后果。稍微哲学几分钟。好的？所以我声称，如果你是一个理性的参与者，我的意思是试图通过他们的游戏参与者来最大化他们的收益，仅仅是有理性的，仅仅是一个理性的参与者，就排除了玩这些被支配的策略。好的？所以你们四个选择了大于67的数字的人，我不会念出你们的名字，也许他们犯了一个错误。好的？但是，但是。下一个层次需要不仅仅是理性。

[段 40]

What else does it require? Yes, can I get this guy again? Sorry, shout out your name again, I’ve forgotten it? Nick. Shout to that? Nick? Yep. The assumption that your opponents are being rational as well. Good, good. So to rule out the second slice, I need to be rational myself, and I need to know that others are rational. All right, that’s illegible, but never mind what it says is rational and knowledge that other people are rational. Now, how about the next slice after that? Well, now I need to be rational. I need to know that other people are rational, and I need to know that other people know that other people are rational. All right? So to get this slice, let’s get this next slice. here, I need rationality. As some of you know, that’s widely criticized in the social scientists these days. I’ll be right to assume that people are rational. But to get this slice, I need rationality. I need knowledge of rationality. Let’s call that K-R. And I need knowledge of knowledge of rationality. All right? And as I go down further, I’m going to need rationality. I need to know people are rational. I need to know that people know that people are rational. And I need to know that people are rational. And I need to know that people know that people are rational.

[译文 40]

它还需要什么？是的，我能再叫这个人吗？抱歉，再喊出你的名字，我忘记了？Nick。喊一下？Nick？是的。假设你的对手也是理性的。好的，好的。所以为了排除第二层，我需要自己是理性的，我需要知道其他人是理性的。好的，这个字迹难以辨认，但不管它说什么，是理性和知道其他人是理性的知识。现在，那下一个层次呢？好吧，现在我需要是理性的。我需要知道其他人是理性的，我需要知道其他人知道其他人是理性的。好的？为了得到这一层，让我们得到下一层。在这里，我需要理性。正如你们一些人所知，这在当今的社会科学家中受到广泛批评。我有权假设人们是理性的。但为了得到这一层，我需要理性。我需要理性的知识。让我们称之为K-R。而且我需要理性的知识的知识。好的？当我进一步往下时，我需要理性。我需要知道人们是理性的。我需要知道人们知道人们是理性的。我需要知道人们是理性的。我需要知道人们知道人们是理性的。

[段 41]

All right. And let’s just make this more concrete for you. These people, these people, the four people who chose this, they made a mistake. What about the four people who chose numbers between 45 and 67? What can we conclude about those people? The numbers, the people who chose between 45 and 67. Should I read out their names? No, perhaps I better not. What can we conclude about these people? Yeah, we’re never going to get the mic. Let’s try and get the mic in there. Come forward as far as you can and then really share it. Yep. They think their classmates are pretty dumb. Right, right. It’s not necessarily that the four people who chose between 46 and 67 are themselves thick. It’s that they think the rest of you are thick. All right. And down here, this doesn’t require people to be thick or to think the rest of you are thick. They’re just people who think that you think, sorry, they’re just people. who think that you think that they’re thick. All right, and so on. But to get all the way to one, we’re going to need very, very many rounds of knowledge, of knowledge, of knowledge, of rationality. Does anyone know what we call it if we assume an infinite sequence of I know that you know, that I know, that I know, that you know, that you know, that I know, that you know, that you know, that I know that you know something?

[译文 41]

好的。让我们把这个给你们说得更具体一些。这些人，这些人，选择了这个的四个人，他们犯了一个错误。那四个选择了45到67之间数字的人呢？关于那些人我们能得出什么结论？那些数字，选择了45到67之间的人。我应该念出他们的名字吗？不，也许我最好不要。我们关于这些人能得出什么结论？是的，我们永远拿不到麦克风。让我们试着把麦克风递过去。尽量往前靠，然后真正分享。好吧。他们认为他们的同学相当笨。对，对。不一定选择46到67之间的四个人本身就很笨。而是他们认为你们其余的人很笨。好的。在下面，这里不需要人们很笨或者认为你们其余的人很笨。他们只是认为你认为是笨的人。好的，以此类推。但为了一直降到1，我们需要非常非常多的知识、知识、知识、理性的循环。有人知道如果我们假设一个无限序列：我知道你知道了，我知道我知道了，你知道你知道了，你知道我知道了，我知道你知道了，你知道我知道了，你知道你知道了，我知道你知道某事，我们称之为什么？

[段 42]

What’s the expression for that? Believe it or not, technical expression. Technical expression for that in philosophy is common knowledge. Common knowledge. Believe it not, technical expression. Technical expression for that in philosophy is common knowledge. Common knowledge. I can never spell, so I’m going to wing it. All right. Common knowledge. Common knowledge is, I know something. You know it. You know that I know it. I know that I know it. I know it, et cetera, et cetera, et cetera, et cetera, et cetera. An infinite sequence. All right. All right. But if we had common knowledge of rationality in this class, then the optimal choice would have been one. How many of you chose one? Look around the rubles. Let’s pan the room. Keep your hands up a second. How many you chose one? So actually, a lot of you chose one. One was the modal answer in this class. A lot of you chose one. All right. So those people did pretty well. They must have done, you know, they must be thinking they’re about to win. But they didn’t win. So it turns out that the average in this class, the average choice, was about 13 and a third, which means two-thirds of the average was nine. Two-thirds of the average was nine. And some of you chose nine. So if you were here, stand up.

[译文 42]

这的表述是什么？信不信由你，技术术语。哲学上这的技术术语是共同知识。共同知识。信不信由你，技术术语。哲学上这的技术术语是共同知识。共同知识。我永远拼不对，所以我就即兴发挥了。好的。共同知识。共同知识是，我知道某事。你知道它。你知道我知道了它。我知道我知道了它。我知道它，等等，等等，等等，等等。无限序列。好的。好的。但如果在这个班级我们有理性的共同知识，那么最优选择就会是1。你们有多少人选择了1？看看周围的人。让我们环顾房间。把手举一会儿。有多少人选择了1？所以实际上，你们很多人选择了1。1是这个班级的众数答案。你们很多人选择了1。好的。所以那些人做得相当好。他们一定在想他们即将获胜。但他们没有赢。所以事实证明这个班级的平均值，平均选择，大约是13又三分之一，这意味着平均值的三分之二是9。平均值的三分之二是9。你们有些人选择了9。所以如果你在这里，站起来。

[段 43]

The following people chose nine. That’s not right. Where are people who shows nine. I’ve gotten here somewhere. Sorry, there’s only pages of people. Here we go. The following people shows nine. So stand up if you’re here. And if you’re there, a person’s roommate, if they’re not here. All right, so Lee Ching Chang, is Lee Ching Chang here? Stand up if you’re here. And G. Christopher Berrero, you can stand up if you’re here. And William Fischel, are you here? I don’t know if he’s here. Jed Gleckstein, are you here? Stand up if you’re here. And Jeffrey Green, stand up if you’re here. And Alison Hoyt. Stand up if you’re here? No, Alison Hoyt? Okay. And there’s John Robinson. John Robinson. All right, so these people, these people, well… Can we get… Can we get… Say up a sense, the camera can see you. There you go. There you go. All the way around, all the round. Wave, wave to mum at home. All right. Can we get a round applause for our winners? So Jedd has trustworthy brought back the $5. I’ll go to focus for a second to get it. Here is the $5. We’re going to tear this into nine pieces that they get arrested and deported to that. So we’re going to find a way to break this into change later, okay? Come and come and claim it afterwards.

[译文 43]

以下这些人选择了9。不对。选9的人在哪里。我到某个地方了。对不起，这里有好几页的人。好了，以下这些人选了9。如果你在场就站起来。如果你在那里，你是那个人的室友，如果他不在场的话。好了，李清昌（Lee Ching Chang），你在吗？如果在的话就站起来。还有G. Christopher Berrero，你在的话可以站起来。还有William Fischel，你在吗？我不知道他在不在。Jed Gleckstein，你在吗？如果在的话就站起来。还有Jeffrey Green，站起来。还有Alison Hoyt。站起来？如果不在？好吧。Alison Hoyt？好的。还有John Robinson。John Robinson。好了，所以这些人，这些人，好吧……我们能……我们能……说上去，镜头能看到你。好了。就是那里。就是那里。绕一圈，绕一圈。向在家的妈妈挥挥手。好了。我们为获胜者来一轮掌声。Jedd已经把可靠的5美元拿回来了。我先集中注意力拿一下。5美元在这里。我们要把这撕成九份，然后他们会被逮捕并驱逐。等等，我们之后再想办法把它找开，好吗？之后过来领。

[段 44]

All right, but you’re all entitled to whatever a nine, whatever that fraction of, whatever that fraction of five dollars is. All right. Okay, so why was it, after all that work, why was it that won wasn’t the winning answer? Why wasn’t one the winning answers? That’s have someone we haven’t had before. Can we get a mic in way in the back there? Can we get a mic on there? mic and that’s on the row you’re on. See if you can point. Actually, yeah, good. Stand up, shout. Stand up, shout. There go. Shout away. One would have been the winning answer. Louder, louder, louder? One would have been the winning answer had everyone assumed that the average would have been constantly compounded down to one, but since a couple of people chose the, I mean, not incorrect answers, but the higher averages than it was pushed up to 13. So to get it all the way good. So to get all the way, thank you. So to get all the way to one, we need a lot, right? We need not just that you’re all rational players, not just that you know each other’s rational, but you know everyone else’s rational. I mean, I know you all know you all know each other because you’ve made at Yale, but you also know each other well enough to know that not everyone in the room is rational, and you’re pretty sure that not everyone knows that you’re rational and so on and so forth, right?

[译文 44]

好了，但你们都有权得到这九份，也就是这五美元的任何那一份。好了。为什么，为什么经过这么多努力之后，9不是正确答案？为什么1不是正确答案？让我们找个没上过台的人。后排那里能拿个麦克风吗？能放个麦克风吗？麦克风就在你们那排。看能不能指向。好的，是的，站起来，大声说。站起来，大声说。就这样。说吧。大声点，再大声点，再大声点？1本来会是正确答案，如果所有人都假设平均数会被持续复利到1的话，但因为有几个人选了，我是说，不是错误的答案，而是高于平均数的答案，所以被推高到了13。所以要想得到正确答案。所以要想全部，谢谢。所以要想得到1，我们需要很多，对吧？我们不仅需要你们都是理性的玩家，不仅需要你们知道彼此是理性的，而且需要你们知道其他所有人的理性。我是说，我知道你们都知道彼此是因为你们都在耶鲁，但你们也足够了解彼此，知道房间里不是每个人都是理性的，而且你们相当确定不是每个人都知道你是理性的，以此类推，对吧？

[段 45]

So it’s asking a lot to get to one here, and in fact we didn’t get to one. In previous years, we were even higher. This was a low this year. In 2003, the average was 18 and a half, and in 2004 it was 21 and a half, In 2005, we had a class that didn’t trust each other at all, I guess, because the average was 23. All right, and this year it was 13 in a third. Okay, we’re getting better there, I think. All right. One nice thing, by the way, this is just a chance, I think. The median answer in the class was nine, which is spot on. So the median hit this bang on. All right. Now, what I want you to do is I want you all to play again. We haven’t got time to this properly, even though I’ve given you the sheet. So write down, don’t talk to your neighbors, write down. down a number. Don’t talk among yourselves, that’s cheating. Write down a number. If you haven’t got a sheet in front, if you just write it on your notepad, write down a number. Has everyone written down a number? All right, I’m going to do a show of hands now. How many, right, we’ll get the camera on you. How many of you chose a number higher than 67?

[译文 45]

所以要想得到1需要很多，实际上我们没有得到1。在前几年，平均数甚至更高。今年是比较低的一年。2003年平均数是18.5，2004年是21.5，2005年我们有一个班完全不信任彼此，我猜，因为平均数是23。好了，今年是13又1/3。好了，我觉得我们在进步。顺便说一句，有一件事很好，这只是机会。我认为班里的中位数答案是9，正好命中目标。所以中位数完全准确。好了。现在我想让你们做的是我想让你们再玩一次。虽然我已经给了你们表格，但我们没有时间好好做这个。所以写下来，不要和邻居说话，写下。一个数字。不要互相交谈，那是作弊。写下一个数字。如果你面前没有表格，就写在笔记本上，写下一个数字。每个人都写下来了吗？好了，我现在举手表决。多少人，选的数字高于67？

[段 46]

Oh, there’s some spoil makers. How many of you chose a number higher than you? higher than 20. you chose a number higher than 67? Oh, this is a spoil maker’s not, okay. How many of you chose a number higher than 20? How many of you chose a number higher than 10? How many chose a number between 5 and 10? How many of you chose a number between 0, between 1 and 5? How many of you, excluding the people who chose 1 last time, how many of you chose a number that was lower than the number you chose last time? Keep your hands up a second? So almost all of you came down. Why? Why are we seeing this massive contraction? I’m guessing the average number in the class now is probably about three or four, maybe even lower. Why are we seeing this massive contraction in the numbers being chosen? The woman in greening, I forgot your name, I’m sorry. I’m just that lecture you’ve told us we’re not being rational if we pick a high number. All right. All right. So part of it is you yourselves have figured out, some of you, that you shouldn’t choose a high number, right? What else, though? What else? What else is going on here? Let’s get somebody, there’s a guy waving an arm out there. You want to stand up behind the hat there? yeah, yeah.

[译文 46]

哦，有几个搅局的。多少人选的数字高于你？高于67？哦，这不是搅局的，好吧。多少人选的数字高于20？多少人选的数字高于10？多少人选的数字在5到10之间？多少人选的数字在0，在1到5之间？多少人，除了上次选1的人，多少人选的数字比上次选的数字低？手先别放下？所以你们几乎都下降了。为什么？为什么我们看到这么大的收缩？我猜现在班里的平均数大概是3或4，也许甚至更低。为什么我们看到所选数字大幅收缩？穿绿衣服的那位女士，我忘了你的名字，对不起。我只是想说你们告诉我们如果我们选一个高的数字就不是理性的。好了。好了。部分原因是你们自己已经想明白了一些，你们不应该选一个高的数字，对吧？还有什么？还有什么？但这里还有什么在起作用？让我们找个人，有个家伙在外面挥手。你想站在帽子后面站起来吗？是的，是的。

[段 47]

That’s, yeah, you, yeah. Because we’ve repeated the game. It’s true we’ve repeated it, it’s true we’ve repeated it, but what is it about repeating? What is it about talking about this game that makes a difference. Let me hazard to get here. I think what makes the difference is not only do you yourselves know better how to play this game now, but you also know that everybody around you knows better how to play the game. Discussing this game raised not just each person’s sophistication, but it raised what you know about other people’s sophistication. And you know that other people now know that you understand how to play the game. So the main lesson I want you to get from this is that not only did it matter that you need to put yourself in other people’s shoes and think about what their payoffs are, you also need to put yourself into other people’s shoes and think about how sophisticated are they at playing games. And you need to think about how sophisticated do they think you are at playing games. And you need to think about how sophisticated do they think that you think that they are playing games and so on. Right? level of knowledge, these layers of knowledge, lead to very different play in the game. And to make this more concrete, if a firm is competing against a competitor, it can be pretty sure that that competitor is a pretty sophisticated game player and knows that the firm is itself.

[译文 47]

是的，就是你，是的。因为我们重复了这个游戏。确实我们重复了它，确实我们重复了它，但重复的意义是什么？讨论这个游戏是什么让它有所不同。让我试着说到这里。我认为不同的是不仅你们自己现在更知道怎么玩这个游戏，而且你们也知道你们周围的每个人更知道怎么玩这个游戏。讨论这个游戏不仅提高了每个人的认知水平，也提高了你们对其他人认知水平的了解。你们知道其他人现在知道你们理解怎么玩这个游戏。所以我想让你们从中学到的主要教训是，不仅把自己放在别人的立场、考虑他们的收益很重要，你也需要把自己放在别人的立场，考虑他们在玩游戏方面有多老练。你也需要考虑他们认为你在玩游戏方面有多老练。你也需要考虑他们认为你认为他们在玩游戏方面有多老练，以此类推。对吧？这种知识水平，这些知识层次，导致游戏中的玩法非常不同。为了使这更具体，如果一家公司正在与竞争对手竞争，它可以相当确定那个竞争对手是一个相当老练的博弈玩家，并且知道那家公司本身也是。

[段 48]

If a firm is competing against a customer, let’s say for a non-prime loan, perhaps that assumption is not quite so safe. It matters, it matters in how we take game through to the real world, and we’re going to see more of this as the term progresses. Now I’ve got five minutes. Do I have five minutes left? So I’ve got five minutes to just to take a little small aside here. We’ve been talking about knowledge and about common knowledge. I just want to do a very quick experiment. Everyone stay in their seat. I’m going to get two TAs up here. I want to get LA and Kai up here. I want to show that common knowledge is not such an obvious a concept as I’ve made it seem on the board. Come up on stage. a second. You can leave the mic. It’s okay. All right. Here we have two of our TAs, actually these are two head TAs, and I want you to face forward so you don’t see what I’m doing a second, okay? All right? I’m about to put on their heads a hat. All right? All right, here’s a hat on on Ali’s head. And here’s a hat on Caius head. Let’s move them this way so they’re in focus. All right. Now you can all see these hats and they can see that, and if they turn around to each other, they can see each other’s hat.

[译文 48]

如果一家公司正在与客户竞争，比如说为了非优质贷款，也许这个假设就不那么安全了。这很重要，这很重要在我们把博弈带到现实世界方面，随着学期推进我们会看到更多这样的情况。现在我还有五分钟。我还有五分钟吗？所以我还有五分钟时间做一个小的题外话。我们一直在谈论知识和共同知识。我只是想做一个非常简单的实验。每个人都坐在座位上。我要请两位助教上来。我要让LA和Kai上来。我想说明共同知识并不是像我在黑板上让它看起来的那么显而易见的概念。走上台来。等一下。你们可以把麦克风放下。没关系。好了。这里是我们的两位助教，实际上这是两位首席助教，我想让你们面朝前，这样你们看不到我等一下在做什么，好吗？好了。我要在他们头上放一顶帽子。好了？好了。这是Ali头上的帽子。这是Caius头上的帽子。让我们把他们移到这里来对焦。好了。现在你们都能看到这些帽子，他们也能看到，如果他们转过身来彼此看，他们能看到对方的帽子。

[段 49]

All right? Now I want to ask you the question here. Here is a fact. All right. So is it common knowledge that is it common knowledge that is it common knowledge that at least one of these people has a pink hat on their head? Is it common knowledge? So I claim it’s not common knowledge. What is known here? here? Well, I’ll reveal the fact I need now that in fact Alley knows that Kai has a pink hat in his head. So it’s true that Ali knows that at least one person in the room has a pink hat on their head. And it’s true that Kai knows that Alley has a pink hat on his head. They both look absurd, never mind. But notice that Ali doesn’t know the color of the hat on his own head. Right? So even though both people know, even though it is mutual knowledge, that there’s at least one pink hat. in the room, Alley doesn’t know what Kai is seeing. So Alley does not know that Kai knows that there’s a pink hat in the room. In fact, from Ali’s point of view, this could be a blue hat. So again, they both know that someone in the room has a pink hat on their head. It is mutual knowledge that there’s a pink hat on their head. that there’s a pink hat in the room.

[译文 49]

好的？现在我想问你们这个问题。这是一个事实。好的，那么这是共同知识吗？这是共同知识吗？这是共同知识吗，至少这两个人中有一个头上戴着粉色帽子？这是共同知识吗？我声称这不是共同知识。这里知道什么？好吧，我现在要揭示这个事实，实际上Ali知道Kai头上戴着一顶粉色帽子。所以这是真的，Ali知道房间里至少有一个人头上戴着粉色帽子。而且这是真的，Kai知道Alley头上戴着一顶粉色帽子。他们看起来都很荒谬，别在意。但注意Ali不知道他自己头上帽子的颜色。对吧？所以尽管两个人都知道，尽管这是相互知识，房间里至少有一顶粉色帽子，Alley不知道Kai看到的是什么。所以Alley不知道Kai知道房间里有一顶粉色帽子。实际上，从Ali的角度来看，这可能是一顶蓝色帽子。所以再次，他们两个都知道房间里有人头上戴着粉色帽子。有一顶粉色帽子在房间里，这是相互知识。

[段 50]

But Ali does not know that Kai knows that he’s wearing a pink hat. And Kai does not know that Ali knows that Kai is wearing a pink hat. Each of their hats, each of their own hats, might be blue. So notice that common knowledge, thanks guys, common knowledge is a rather subtle thing, thank you. Common knowledge is a subtle thing. Mutual knowledge doesn’t imply common knowledge. Common knowledge. is a statement about not what I know. It’s about what do I know the other person knows that I know that the other person and so on and so forth. And even in this simple example, where you might think it’s obviously common knowledge, it wasn’t common knowledge that there was a pink hat in the room. Does anybody have smaller siblings or children of their own they can have a pink hat at the end of the class? All right, we’ll see you on Wednesday.

[译文 50]

但Ali不知道Kai知道他戴着一顶粉色帽子。Kai也不知道Ali知道Kai戴着一顶粉色帽子。每个人的帽子，他们自己的帽子，可能是蓝色的。所以注意，共同知识，朋友们，共同知识是一件相当微妙的事情，谢谢。共同知识是一件微妙的事情。相互知识并不意味着共同知识。共同知识不是关于我知道什么。它是关于我知道别人知道我知道的什么，别人知道我知道的什么，如此往复。即使在这个简单的例子中，你可能认为这显然是共同知识，但房间里有一顶粉色帽子并不是共同知识。有人有弟弟妹妹或者自己的孩子可以在下课后戴一顶粉色帽子吗？好的，我们周三见。