耶鲁大学博弈论公开课 - 第11集 落后的感应-声誉和决斗

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]

Tanya Cushman Reviewer’s Name Reviewer’s Name Okay, so this is what we did last time. We looked at a game involving an entrance and an incumbent in a market. And the entrance had to decide whether to enter that market or not, And if they stayed out, the incumbent remained a monopolist, and the monopolist made three million in profit. All right? And if the entrant goes in, then the incumbent can decide whether to accommodate the entrant and just settle for duopoly profits, making a million each, or the incumbent can fight, in which case the incumbent makes no money at all, and the entrant loses a million dollars. And we pointed out a number of things about this game. was that when we analyzed it in a matrix form, we quickly found that there were two Nash equilibria. Nash equilibria were in and not fight, and out and fight. But we argued that backward induction tells us that the sensible answer is in and not fight Once the incumbent knows the entrant is in they not going to fight because one is bigger than zero and the entrant anticipating this will enter And we talked a little bit more, we said this other equilibrium, this out-fight equilibrium, it is an equilibrium because if the entrant believes the incumbent is going to fight, then the entrant is going to stay out, and it’s costless for the incumbent to, quote, fight if in fact the entrant does stay out because they never get called upon to fight anyway.

[译文 1]

Tanya Cushman 审稿人姓名 审稿人姓名 好的，这就是我们上次做的事。我们看了一个涉及市场中一个进入者和一个在位者的游戏。进入者必须决定是否进入那个市场，如果他们不进入，在位者就继续作为垄断者，垄断者获得三百万的利润。好吧？如果进入者进入，那么在位者可以决定是容纳进入者、接受双头垄断利润、每家获得一百万，还是在位者可以发起竞争，这种情况下在位者一分钱都赚不到，而进入者会损失一百万美元。我们指出了关于这个游戏的一些事情。当我们用矩阵形式分析它时，我们很快发现有两个纳什均衡。纳什均衡是"进入且不竞争"和"不进入且竞争"。但我们认为逆向归纳法告诉我们合理的答案是"进入且不竞争"。一旦在位者知道进入者已经进入，他们不会竞争，因为一大于零，而进入者预见到这一点就会进入。我们又讨论了一些，我们说另一个均衡，这个"不进入-竞争"的均衡，它是一个均衡，因为如果进入者相信在位者会竞争，那么进入者就会留在外面，而对于在位者来说，如果进入者确实留在外面，引用"竞争"其实是没有成本的，因为他们根本不会被要求去竞争。

[段 2]

So the idea of this was that for the incumbent to say they’re going to fight is an incredible threat. Or, that’s terrible English, it’s the way it’s always taught in the textbooks, it’s usually called a not-credible threat. And that not-credible threat is he’s not really going to fight if the entrant comes in, I’m sorry, he’s not really going to fight if the entrant comes in and therefore the entrant should come in and in fact the incumbent will accommodate him. So what we’ve shown here is if we believe this argument, then the entrant will come in and the incumbent is going to let him in. Alright? And at the end, we started talking about this in a slightly more elaborate setting. So let’s just remind you what that more elaborate setting is. The more elaborate setting is suppose that there is one firm one monopolist and that monopolist holds a monopoly in ten different markets So we have our monopolist be Ale alright here Ale here our monopolist and he owns pizzeria monopolies in ten different markets. Alright, and each of these ten different markets are separate, alright, they’re different towns, and in each of those ten markets, he thinks he’s going to face, he knows he’s going to face an entrant. And those entrants are going to come in order. So let’s talk about who those entrants are going to be.

[译文 2]

所以这个想法是，对于在位者来说，说他们要竞争是一个难以置信的威胁。或者，这英语很差，这是教科书里一直教的，通常称为不可置信威胁。这个不可置信威胁就是，如果进入者真的进入，他不会真的去竞争，对不起，如果进入者真的进入，他不会真的去竞争，因此进入者应该进入，而实际上在位者会容纳他。所以我们这里展示的是，如果我们相信这个论点，那么进入者会进入，在位者会让他进来。好吧？在最后，我们开始讨论这个在稍微复杂一些的环境中的情况。那么让我们提醒你们那个更复杂的环境是什么。那个更复杂的环境是，假设有一家公司，一个垄断者，而那个垄断者在十个不同的市场中拥有垄断地位。所以让我们的垄断者是阿里，好吧这里阿里这里我们的垄断者，他在这十个不同的市场中拥有披萨店垄断权。好吧，这十个不同的市场是分开的，好吧，它们是不同的城镇，在每个这样的市场中，他知道他将面临一个进入者。而那些进入者将按顺序进入。那么让我们谈谈那些进入者会是谁。

[段 3]

The entrants are going to be this person, this person, and so on. Let’s find out who they are. So your name is? Isabella. And where are you from? Miami. Miami. So Miami is one of the markets, and your name is? Scott. From where? Wisconsin. Where in Wisconsin? Where? Madison. Madison. We’ve got two towns. Let’s give you your towns now. My name is Lang. I’m from Bridgeport, Connecticut. Okay, you’ve got three towns. I’m from Miami, too. It’s about Yale diversity. Well, we’ll pretend you’re from somewhere else. Put them in New Orleans or something. All right? Rich from Boston. From Boston. From Orange, Connecticut. From Orange, Connecticut, just down the road. St. Louis, Missouri. All right, haven’t done 10 yet. I’m not quite at 10. One, two, three, four, five, six, seven. Suffern, New York. All right. Hong Kong. Hong Kong, that’s way, way in. Long Island. Long Island. I think I probably got 10 markets here okay So Ali owns a pizza shop He the monopoly pizza shop owner in each of these ten markets And what we going to see is we going to see what happens as sequentially these entrants try to enter And the way that this game’s going to work is that they’re lined up. We know the order in which the entrants are going to come. They’re going to start off.

[译文 3]

进入者将是这个人，这个人，以此类推。让我们找出他们是谁。你叫什么名字？Isabella。你从哪里来？Miami。Miami。所以Miami是一个市场，你叫什么名字？Scott。从哪里来？Wisconsin。Wisconsin哪里？哪里？Madison。Madison。我们有两个城镇。现在给你分配你的城镇。我的名字是Lang。我来自Bridgeport, Connecticut。好的，你有三个城镇。我也来自Miami。这是关于Yale的多样性。好吧，我们就假装你来自别的地方。把他们放在New Orleans之类的地方。好吧？Rich来自Boston。来自Boston。来自Orange, Connecticut。来自Orange, Connecticut，就在路边。St. Louis, Missouri。好吧，还没到10个。我还没到10个。一、二、三、四、五、六、七。Suffern, New York。好吧。Hong Kong。Hong Kong，这太远了。Long Island。Long Island。我想这里大概有10个市场了，好的。所以Ali拥有一家披萨店。他是这十个市场中每个市场的垄断披萨店老板。我们将看到的是，我们将看到当这些进入者按顺序尝试进入时会发生什么。这个游戏的方式是他们排队等着。我们知道进入者到来的顺序。他们将从。

[段 4]

The first person to have to make a decision is… Enter. It’s Isabella, right? It’s Isabella. And we’re going to see how our monopolist responds, okay? So let’s have a look at this. All right. who’s in which market again? Miami in Miami okay what are you going to do? enter enter what are you going to do? I will fight oh dear oh dear you owe me a million dollars okay alright so one person’s down a million dollars let’s see what happens next I’m going to stay out okay so next question which market was this? Madison Madison stayed out I’m going to stay out staying out Bridgeport Bridgeport stayed out yes I’ll stay out stayed out again stayed out out which market are we up to now? somewhere in somewhere in Orange County wasn’t it? Where are we? Orange, Connecticut. Orange, Connecticut. And you’re? Stay out. Stay out? I think I’ll stay in. You think you’ll come in? Okay. And which market is this? St. Louis, Missouri. St. Louis, Missouri. Oh, you owe me a million dollars as well. Okay. Okay. A couple of you owe me a million dollars. This is a good class. We’re going to have plenty of money for lunch. All right. I’m also going to fight. You’re also going to fight? And which market is that? Suffern, New York. Oh, you owe me a million dollars as well.

[译文 4]

第一个必须做出决定的人是…进入。是Isabella，对吧？是Isabella。我们将看看我们的垄断者如何回应，好吧？让我们看看这个。好吧，谁在哪个市场？Miami在Miami好的你要怎么做？进入进入你要怎么做？我会竞争哎呀哎呀你欠我一百万美元好的好的所以有一个人亏了一百万美元让我们看看接下来发生什么我要留在外面好的下一个问题这是哪个市场？Madison Madison留在外面我要留在外面留在外面Bridgeport Bridgeport留在外面是的我要留在外面留在外面再次留在外面我们在哪个市场？ somewhere in somewhere in Orange County不是吗？我们在哪里？Orange, Connecticut。Orange, Connecticut。你呢？留在外面。留在外面？我想我会进入。你想进来吗？好的。这是哪个市场？St. Louis, Missouri。St. Louis, Missouri。哦，你也欠我一百万美元。好的。好的。有几个人欠我一百万美元。这是一个很好的班级。我们将有足够的钱吃午饭。好吧。我也要竞争。你也要竞争？那是哪个市场？Suffern, New York。哦，你也欠我一百万美元。

[段 5]

Okay, okay. A couple of people owe me a million dollars. This is a good class. We’re going to have plenty of money for lunch. All right. I’m also going to fight. You’re also going to fight? And which market is that? Sulphur, New York. Whereabouts is it? Sulphur, New York. Sulphur, New York. Where are we at? One, two, three, four, five, six, seven, eight. Ali? I’ll fight. You’ll fight? Okay, so you owe me a million dollars too. All right, that was eight. Nine? Out. Out. And ten? I’ll stay out. Stays out. Okay. Okay, now let’s just notice something here. Which was the tenth market? What town were you? What about Long Island? Huntington. Huntington. If Huntington, Long Island, our last market, had come in, as soon as you said in, what would Ali have said? I would have not fought. Would not fought. Aha. Aha. Okay, so what happened here? What happened here? Well, we analysed this last time as an individual market. We argued that each entrant should come in, just as our first entrant came in, and our monopolist should not fight. That’s what we have up on the board, that’s what backward induction suggests. But in fact, Alley fought, a whole bunch of people came in, and a whole bunch of them stayed out. Is that right A whole bunch of them stayed out Now why Why was Alley fighting these guys and why were they staying out Let talk about why What market were you again Madison Wisconsin So why did Madison Wisconsin stay out Well, we talked about last time how he has an incentive to fight now because there’s more than just our analysis up there in terms of establishing that he’ll fight to keep people out.

[译文 5]

好的，好的。有几个人欠我一百万美元。这是一个很好的班级。我们将有足够的钱吃午饭。好吧。我也要竞争。你也要竞争？那是哪个市场？Sulphur, New York。大概在哪里？Sulphur, New York。Sulphur, New York。我们现在在哪？一、二、三、四、五、六、七、八。Ali？我会竞争。你会竞争？好的，所以你欠我一百万美元。好的，那是8个。9？不进入。不进入。10？我要留在外面。留在外面。好的。好的，现在让我们注意一些事情。哪个是第10个市场？你在哪个城市？Long Island怎么样？Huntington。Huntington。如果Huntington, Long Island，我们的最后一个市场，如果进来的话，当你刚说进入，Ali会说什么？我不会竞争。不会竞争。啊哈。啊哈。好的，这里发生了什么？这里发生了什么？好吧，我们上次把这个作为单个市场来分析。我们认为每个进入者都应该进来，就像我们的第一个进入者进来一样，而我们的垄断者不应该竞争。这是我们黑板上写的，这是逆向归纳法建议的。但实际上，Ali竞争了，一大群人进来了，还有一大群人留在外面。对吗？一大群人留在外面现在为什么为什么Ali要和这些人竞争，为什么他们要留在外面让我们谈谈为什么你又在哪个市场？Madison Wisconsin所以为什么Madison Wisconsin要留在外面好吧，我们上次讨论了他有动机去竞争现在因为不仅仅是我们上面的分析在建立他会竞争来阻止人们进入方面。

[段 6]

All right, so it looks like there might be some reason for fighting to keep you out. So why… let’s talk about it a bit more. So let’s go to the third guy. Which market are you again? Bridgeport. Bridgeport. So why did you stay out? Because I knew he was going to fight. You knew he was going to fight. Now, how did you know he was going to fight? Because he had an incentive to establish that he was going to fight for every single market. And so I was going to lose out. All right. So we know, we think we know, we think we know that Ale is a tough Italian pizzeria owner. And we think he’s going to try and establish, what, a reputation as being a tough pizzeria owner. by fighting these guys, perhaps fighting a few guys early on, in order to keep these guys out. In fact, he successfully hit the fight the first person, but he kept out 2, 3, 4, 5, 6, and this person came in, so 7 and 8 came in, but then 9 and 10 he kept out. So he kept a lot of people out of the market by fighting early on And this argument sounds right It seems to ring true It about establishing reputation But now I want to show you that there’s a worry with this argument.

[译文 6]

好的，所以看起来可能有一些理由去竞争以阻止你进入。所以为什么…让我们再谈谈这个。让我们找第三个人。你又是哪个市场？Bridgeport。Bridgeport。那么你为什么留在外面？因为我知道他会竞争。你知道他要竞争。你怎么知道他要竞争的？因为他有动机建立他要为每个市场竞争的声誉。所以我要输了。好的。所以我们知道，我们认为我们知道，我们认为Ali是一个强硬的意大利披萨店老板。我们认为他会试图建立什么，作为强硬披萨店老板的声誉。通过和这些人竞争，也许在早期和几个人竞争，为了让这些人留在外面。事实上，他成功地和第一个人竞争了，但他让2、3、4、5、6留在了外面，而这个人进来了，所以7和8进来了，但然后9和10他让他们留在了外面。所以通过早期竞争，他让很多人离开了市场。这个论点听起来是对的它似乎很合理它是关于建立声誉的但现在我想向你们展示这个论点有一个问题。

[段 7]

The worry is, this is a sequential game. This is a sequential game, and like all sequential games of perfect information we’ve seen in the class, we should analyze this game how? Now that wasn’t loud enough. How? Backward induction. So where’s the back? Where’s the back of this game? Way back here, all right, way back here, sorry for the guys on the balcony, way back here, we have the last market in town, which was the last market, all right? And if we look at this last market, we in fact saw that if the last market came in, Ali in fact gave in, right? Ali gave in. Now why did Ali give in on the last market? Let’s have a look back on the board, all right? So on the board, we can see what that last market looks like. With ten markets, this is a very complicated game. This would be the first market, and then there’s three versions of the second market, depending on what Ali did in the first market, and so there’s nine versions of the third market. The tree for this game is horrendous. But nonetheless, once we get to the end of the game, the tenth market, which was what? Bridgeport or something I forgotten where it was at now Wherever it was Once we get to that last market this tree pretty well describes that last market Is that correct Right There isn another market afterwards there only ten markets So in this last market, what do we know Ale’s going to do?

[译文 7]

问题是，这是一个序贯博弈。这是一个序贯博弈，就像我们在课堂上见过的所有完美信息序贯博弈一样，我们应该如何分析这个博弈？刚才声音不够大。如何分析？逆向归纳法。那么尾部在哪里？这个博弈的尾部在哪里？远在后面，好，远在后面，抱歉坐在楼上的各位，远在后面，我们有镇上的最后一个市场，也就是第十个市场，对吧？如果我们看这最后一个市场，我们实际上看到，如果第十个市场进入，Ali实际上会让步，对吧？Ali让步了。Ali为什么会在最后一个市场让步？让我们回头看看白板，好吧？在白板上，我们可以看到这最后一个市场是什么样子。有十个市场，这是一个非常复杂的博弈。这将是第一个市场，然后有三种版本的第二个市场，取决于Ali在第一个市场中做了什么，所以第三个市场有九个版本。这个博弈的树形图很可怕。但尽管如此，一旦我们到达游戏的末尾，第十个市场，是什么？Bridgeport还是什么地方我忘了在哪里。无论在哪里，一旦我们到达最后一个市场，这棵树相当好地描述了那个最后市场。对吗？那之后没有其他市场了，只有十个市场。那么在这最后一个市场中，我们知道Ale会做什么？

[段 8]

In this last market, if the entrant enters, Ale’s going to not fight, which is exactly what Ale did do. So Ale, is that right? So when, in fact, we discussed the tenth guy coming in, you chose to… I would have chosen not to fight. Would have chosen not to fight. And that’s exactly what the model predicts. Right? He has no incentive to establish a reputation for the eleventh market, because there isn’t an eleventh market. He’s down at ten. Is that right? So we know that in the last market, the tenth market, Ali actually is not going to fight. Right? And therefore the person who’s the entrance in the tenth market should know that they can safely enter and Ali won’t fight them. But now we’re in trouble. Why are we in trouble? Well, let’s go back to the ninth market, the second to last market. I’ve forgotten where it was. Raise your hand if it’s the last market. Okay. Second to last market is this guy? No, this guy. You’re the tenth market. So this guy who’s in the Hong Kong market, he should know he’s the second to last market. This guy? You’re the 10th market. So this guy who’s in the Hong Kong market, he should know he’s the second to last market. He knows that no matter what he does, the 10th market’s going to enter and Ali’s going to give in to the 10th market.

[译文 8]

在这最后一个市场中，如果进入者进入，Ale将不会进行战斗，这正是Ale所做的。那么Ale，对吗？当我们讨论第十个人进入时，你选择了……我会选择不战斗。会选择不战斗。这正是模型预测的结果。对吧？他没有动力为第十一个市场建立声誉，因为没有第十一个市场。他停在第十个。对吗？所以我们知道在最后一个市场，第十个市场，Ali实际上不会战斗。对吧？因此，第十个市场的进入者应该知道他们可以安全进入，而Ali不会与他们战斗。但现在我们遇到麻烦了。为什么遇到麻烦？好吧，让我们回到第九个市场，倒数第二个市场。我忘了那是哪里。如果这是最后一个市场，请举手。好。倒数第二个市场是这位同学吗？不是，这位同学。你是第十个市场的。所以这位在香港市场的同学，他应该知道自己是倒数第二个市场。这位同学？你是第十个市场的。所以这位在香港市场的同学，他应该知道自己是倒数第二个市场。他知道无论他做什么，第十个市场都会进入，而Ali会对第十个市场让步。

[段 9]

Ali’s going to let the 10th market, the 10th entrant in. Is that right? So the 9th market actually knows that nothing Ali’s going to do here is to establish a reputation to keep the 10th guy out. so therefore in fact he should what? he should come in, right? he should come in and in fact if he’d come in, Alley would have had to give in because there’s no way that Alley can keep the tenth guy out we’ve just argued the tenth guy is coming in by backward induction so since we know that the tenth guy is coming in anyway and in fact Alley is going to accede to them there’s no point Alley trying to scare off the tenth guy so in fact Alley is going to say no fight to the ninth guy I was going to say no fight to the ninth guy but now we go to the eighth guy we’ve just argued that the tenth guy is coming in anyway and Ali’s going to give in to him we’ve argued that the ninth guy is coming in so Ali’s going to give in to this guy as well because he can’t put off the tenth guy and therefore we know that once we get to the eighth guy once again he can safely come in because Ali knows by backward induction he can keep the ninth and the tenth guy out anyway and so this guy should come in as well And if we do this argument all the way back what do we get What do we get He lets everybody in Everybody should come in and he should let everybody in.

[译文 9]

Ali会让第十个市场、第十个进入者进入。是这样吗？所以第九个市场实际上知道Ali在这里所做的任何事都不是为了建立声誉以把第十个人挡在外面，所以他实际上应该做什么？他应该进来，对吧？他应该进来，而实际上如果他进来，Alley就不得不让步，因为没有办法让Alley把第十个人挡在外面——我们刚刚论证了第十个人会通过逆向归纳法进入，所以既然我们知道第十个人反正会进来，而且实际上Alley会向他们让步，Alley试图吓退第十个人就没有意义了，所以实际上Alley会对第九个人说不战斗。我本想说对第九个人说不战斗，但现在我们来看第八个人。我们刚刚论证了第十个人反正会进来，而Ali会向他让步。我们论证了第九个人会进来，所以Ali也会向这个人让步，因为他无法阻止第十个人。因此我们知道，一旦到了第八个人，他同样可以安全进入，因为Ali通过逆向归纳法知道他反正可以把第九个人和第十个人挡在外面，所以这个人也应该进来。如果我们把这个论证一直往后推，我们会得到什么？我们会得到什么？他让每个人都进来。每个人都应该进来，而他应该让每个人都进来。

[段 10]

So we have a problem here. We have a problem. Backward induction says, even with these 10 markets, Ali, in fact, should let everybody in, Everyone should know that, so they should come in. So there’s a disconnect here. There’s a disconnect between what the theory is telling us, backward induction is telling us, Allais can’t keep people out by threatening to fight by establishing a reputation, and what we actually just saw that would happen, which was Allais did fight and did keep people out. And we know that other monopolists do that as well. So how can we make rigorous this idea of reputation? It’s not captured by what we’ve done so far in the class, so how can we bring back what must be true in some sense, it’s intuition, that by fighting, Allais could keep people out, and that therefore will keep people out. So to make that idea work, I want to introduce a new idea, and the new idea is that with very small probability, let say 1 chance Allais is crazy All right so stand up a second All right so he looks like a normal kind of guy but there this 1 chance that he really bonkers Right, if he was here today, I’d say, there’s 1% chance that he’s actually Rahul. All right? All right, so now let’s redo the analysis.

[译文 10]

所以我们这里有一个问题。我们有一个问题。逆向归纳法说，即使有这十个市场，Ali实际上应该让每个人都进来。每个人都应该知道这一点，所以他们应该进来。所以这里有一个脱节。有一个脱节存在于理论告诉我们的——逆向归纳法告诉我们的——Allais不能通过威胁战斗或建立声誉来把人挡在外面——与我们实际看到的会发生的事情之间，那就是Allais确实战斗了，确实把人挡在了外面。我们知道其他垄断者也会这样做。那么我们如何使声誉这个概念变得严谨？课堂上到目前为止我们所做的工作还没有捕捉到这个概念，所以我们如何才能把某种直觉——必定成立的东西——带回来，那就是通过战斗，Allais可以把人挡在外面，因此也会把人挡在外面。为了让这个想法行得通，我想引入一个新想法，这个新想法是：以非常小的概率，比如说1%的机会，Allais是疯狂的。好，站起来的同学。好，他看起来像个正常人，但有1%的机会他真的是个疯子。对，如果今天他在这里，我会说有1%的机会他实际上是Rahul。好吧？好，现在让我们重新做分析。

[段 11]

What do I mean by bonkers? Bonkers, I mean, with 1%, Ale is the kind of guy who likes to fight. So with 1% chance, he’s actually not got these payoffs at all. at all, he’s actually got some different payoffs, which are the payoffs of somebody who, okay, he’ll lose money, but he so much enjoys a fight, you know, he gets plus 10 here. All right? That’s the bonkers guy’s payoff. But you know, only 1% chance he’s this bonkers guy. All right? So now what happens? Let’s just walk it through. All right? With 1% chance, if there was only one market, if there’s only one market, not the 10 markets, if there’s only one market, and this one market was, I’ve forgotten your name. Isabella. Isabella, who was in which market? in Miami, then she doesn’t really much care about the 1% chance that Ale is actually Rahul. It doesn’t really bother her very much. Why? Because with 99% chance, Ale’s going to give way, and that’s good enough odds, right? With 99 chance she was happy to come in So if there only one market here we be done But with 10 markets things are a little different Why Let see why So suppose in fact that Isabella in Miami thinks that Ale and everybody else thinks Ale’s a pretty sane guy with 99% probability he’s a sane guy, and Isabella enters, and everyone sees this, and everyone’s surprised, rather than doing the same thing, which is letting Isabella in and switching to a duopoly in Miami, with 1% chance.

[译文 11]

我说的疯子是什么意思？疯子，我的意思是，有1%的可能性，Ale是那种喜欢战斗的人。所以有1%的机会，他实际上没有这些收益。根本没有，他实际上有不同的收益，那是某种人的收益，好吧，他会亏钱，但他太喜欢打架了，你知道，他在这里得到正10。好吧？这是疯子这家伙的收益。但你知道，只有1%的机会他是这个疯子。好吧？那么现在会发生什么？让我们一步步来。好吧？有1%的机会，如果只有一个市场，不是这10个市场，如果只有一个市场，而这个市场是——我忘了你的名字。Isabella。Isabella，你在哪个市场？在Miami，那么她不太在乎这1%的机会Ale实际上是Rahul。她不太在意。为什么？因为有99%的机会，Ale会让步，而这已经是很不错的赔率了，对吧？有99%的机会她就愿意进来。所以如果这里只有一个市场，我们就完成了。但有10个市场，事情就有点不同了。为什么？让我们看看为什么。那么假设Isabella在Miami认为Ale——以及所有人都认为Ale——有99%的概率是个相当正常的人，Isabella进入，每个人都看到了，每个人都很惊讶，而不是做同样的事情——让Isabella进入并在Miami转变为双头垄断——有1%的机会。

[段 12]

What happens, in fact, after Isabella comes in, is that Alay fights. Right? Fights. Okay? So now, too late for Isabella. She’s lost her money. But our second market is, what’s your name again? Scott. Scott. Which market were you? Madison. So Scott in Madison has observed what happened in Miami, and initially he thought that Alay was Alay. 99% sure, the alley was this same nice calm Italian guy, all right, but on the other hand, he just saw this same calm Italian guy fight, as he shouldn’t have fought, couldn’t have backed induction, fought the market, fought the entrance in Miami. So now, I’ve got to do it again, Scott, Scott thinks to himself, hmm, I’m not so sure that alley is the same guy. Maybe I should update my belief in order, in the direction of thinking that alley might actually be the same guy. Scott thinks to himself, hmm, I’m not so sure that Ali is the sane guy. Maybe I should update my belief in the direction of thinking that Ali might actually be the insane guy. So maybe, we’re up to maybe probably a third, that Ali’s actually insane. So he thinks, okay, probably a third, that’s still not very much, I’ll still risk it. He comes in, and Ali fights him again. Right? Probably a third he’s sane, he’s going to give in to me.

[译文 12]

Isabella进来之后，实际上发生的是Alay进行了战斗。对吧？战斗。好吧？所以现在，对Isabella来说太晚了。她已经损失了她的钱。但我们的第二个市场是——你叫什么来着？Scott。Scott。你在哪个市场？Madison。所以Madison的Scott观察到了Miami发生的事情，最初他认为Alay就是Alay。99%确定，alley就是那个 nice calm Italian guy，好吧，但另一方面，他刚刚看到这个同样冷静的意大利人战斗了——按照逆向归纳法他不应该战斗——他战斗了，战斗了市场，战斗了Miami的进入者。所以现在，Scott，我得再说一遍，Scott心里想，嗯，我不太确定alley是同一个人。也许我应该更新我的信念，朝着认为alley可能是同一个人——朝着认为Ali可能是同一个人——的方向。Scott心里想，嗯，我不太确定Ali是那个正常人。也许我应该更新我的信念，朝着认为Ali可能是那个疯子——的方向。所以也许，我们上升到也许大概三分之一，Ali实际上是不正常的。所以他想，好吧，大概三分之一，那仍然不太多，我还是愿意冒险。他进来，而Ali再次和他战斗。对吧？大概三分之一他是正常的，他会向我让步。

[段 13]

He comes in, Alley fights him again. So now we’re in the third market, which was which market? Bridgeport. Bridgeport. And Bridgeport’s seen this horrible fight going on in Miami and this horrible fight going on in Madison, and now he’s getting pretty sure that this nice, calm-looking Alley is not nice, calm-looking Alley, he’s crazy Rahul. All right? I mean, there’s a lot of evidence he’s crazy Rahul. He’s fought the last few markets and making huge losses. It must be that Alley likes that fight. So what does he do? He says, I’m going to stay out of here. Right? I’m convinced that this guy may be crazy, so I’ll stay out. And all the subsequent markets, they think, oh well, he fought these first two markets, that means he must be a crazy guy or he hyper is a crazy guy so they all stay out which is exactly what happened until we got to here and even here when they came in Ale acted like crazy Rahul So what made that argument possible was what What made that argument possible was the small possibility the 1 possibility that Ale is bonkers But you know, how well do you know Ale? There’s a 1% chance he’s bonkers, right? How many of you think you’re really sure that he’s the same guy? Right? And he supports Italian football teams.

[译文 13]

他进来了，Alley又和他打了起来。那么现在我们在第三个市场，是哪个市场？Bridgeport。Bridgeport。Bridgeport已经看到了迈阿密那场可怕的战斗和麦迪逊那场可怕的战斗，现在他越来越确信这个看起来和善、冷静的Alley其实不是那个和善、冷静的Alley，他是疯狂的Rahul。懂吗？我的意思是，有很多证据表明他是疯狂的Rahul。他在前几个市场都打了，而且造成了巨大的损失。一定是Alley喜欢那种战斗。所以他会怎么做？他说，我要远离这里。懂吗？我确信这家伙可能是疯子，所以我不会进去。在所有后续的市场中，他们都这么想，哦，他在这前两个市场打了，这意味着他一定是个疯子，或者他超疯狂，所以他们全都退出了——这正是发生的事情，直到我们来到这里，甚至在这里，当他们进来时，Ale表现得像疯狂的Rahul。是什么让这个论证成立？是什么让这个论证成立？是那个小的可能性，那1%的可能性，Ale是疯子。但是，你知道，你对Ale了解多少？他有1%的可能是疯子，对吧？你们中有多少人真的确信他是同一个人？懂吧？而且他支持意大利足球队。

[段 14]

He’s got to be pretty crazy, right? Right? Right? So what happened here? The small possibility that Alley is crazy allowed him to build up a reputation that kept all these guys out. But actually the argument is stronger than that. Let’s try and push this argument harder. Suppose, in fact, that Alley is not crazy. Suppose that Alley is the sane Alley, the nice, calm Alley we all know and love. All right? But we’ve just argued that if Ale acts as if he’s the crazy guy, then you’re going to be convinced that he is the crazy guy. So by acting crazy, he might be able to convince you that he is crazy and therefore keep you out. All right? So we argued before that if there’s some chance that Ale crazy by acting crazy early on he going to deter these late entrants from entering the market because they think they fighting Rahul and they don want to fight Rahul But we said these early guys they had probability 0 that he was sane and 0 that he was sane and maybe even 0.5 he was sane here, so they thought of coming in. But now we’re arguing that even if Ali is sane, even if he’s a sane guy, a rational guy, he should behave as if he’s crazy in order to keep these late guys out and these early players knowing that even the sane version of Alley is going to fight them should also stay out.

[译文 14]

他一定相当疯狂，对吧？对吧？对吧？那么这里发生了什么？Alley可能疯狂的那一小点可能性让他能够建立一种声誉，让所有这些人都不敢进来。但实际上这个论证比这更强。让我们试着把这个论证推得更深入。假设事实上Alley并不疯狂。假设Alley是我们都认识和喜欢的那个理智的、友善的、冷静的Alley。懂吧？但我们刚刚论证过，如果Ale表现得像疯子一样，你就会相信他就是疯子。所以，通过表现得疯狂，他也许能够让你相信他是疯子，从而把你挡在外面。懂吧？因此我们之前说，如果有某种可能的Ale是疯子，通过早期表现得疯狂，他会阻止这些晚期进入者进入市场，因为他们认为他们在和Rahul战斗，他们不想和Rahul战斗。但我们说这些早期的人，他们有概率0认为他是理智的，概率0认为他是理智的，也许这里甚至有0.5的概率他是理智的，所以他们考虑进来。但现在我们论证，即使Ali是理智的，即使他是个理智的人，理性的人，他也应该表现得像个疯子，以阻止这些晚期的人进来，而早期的玩家知道即使是理智版的Alley也会和他们打，所以也应该退出。

[段 15]

And notice something’s happened here. They’re not staying out because they think Alley’s crazy they’re staying out because they know that even the sane version of Alley is going to fight them in order to seem crazy. Is that right? I would say that’s a strong argument. So now even these early guys are going to stay out of the market. Alright? Now, we’re almost there. What we’ve argued, let’s just make sure we get the two pieces of this on the board. We’ve argued that if there’s an epsilon chance, a very small chance it got a 1 chance that Ale is crazy then he can deter entry by fighting i seeming crazy And we argue that what really makes this argument strong is once we realize that the same person’s person’s going to act crazy, we really know that everyone’s going to act crazy, and therefore we should stay out. Now that argument can’t quite be right. So that’s enough of the argument I want you to have for the purpose of the exam, but let me just point out that that argument isn’t quite correct. That can’t quite be an equilibrium. Now why can’t that be an equilibrium? We’ve just argued that even the same version of Alley, this is a slightly more subtle argument, just pay attention a second, we’ve argued that even the same version of Alley is going to act like a crazy guy.

[译文 15]

注意这里发生了什么。他们不退出的原因不是因为他们认为Alley是疯子——他们不退出的原因是他们知道即使是理智版的Alley也会和他们打，为了表现得像个疯子。对吗？我会说这是一个强有力的论证。所以现在甚至这些早期的人也会退出市场。懂吧？现在，我们快到了。我们论证的内容，让我们确保把这两部分内容放在黑板上。我们论证，如果有epsilon的机会，一个非常小的机会，有1%的机会Ale是疯子，那么他可以通过表现得疯狂来阻止进入。而且我们论证，真正使这个论证有力的是，一旦我们意识到同一个人会表现得疯狂，我们真的知道每个人都会表现得疯狂，因此我们应该退出。现在那个论证不可能完全正确。所以这就足够作为考试目的的论证了，但让我指出那个论证并不完全正确。这不可能完全是一个均衡。那么为什么这不可能是一个均衡？我们刚刚论证，即使是同一版本的Alley，这是一个稍微更微妙的论证，注意一下，我们论证即使是同一版本的Alley也会表现得像个疯子。

[段 16]

All right? All right, so everyone coming, if anyone came in, he’s going to act crazy anyway, so you’re not going to update your belief as to whether he’s crazy or not, because you know that the crazy guy is going to fight because he likes fighting, and the same guy is going to fight because he wants to seem like a crazy guy. So you’re really learning nothing whether you observe him fighting or not. But now let’s go back to our tenth market. Way back in our tenth market, our tenth market participant, whose name was, I don’t know, So you’re really learning nothing whether you observe him fighting or not. But now let’s go back to our 10th market. Way back in our 10th market, our 10th market participant, whose name was Andy, hasn’t learned anything about Ali. He hasn’t learned anything because whether Ali was sane or crazy, he’s going to fight. So observing what his actions early on, if that was really in equilibrium, our 10th guy wouldn’t have updated his belief at all and therefore would still believe with probability 0.99 that Alley was sane, in which case our tenth guy would enter. And once again, that argument would unravel from the back. So what I described before can’t quite be an equilibrium. It can’t be just as simple as all sane guys are going to act crazy, because then you wouldn’t learn the thing.

[译文 16]

懂吧？懂吧，所以每个人进来，如果有人进来，他无论如何都会表现得疯狂，所以你不会更新你对他是否疯狂的信念，因为你知道疯子会因为喜欢打架而打架，而同一个人会打架，因为他想表现得像个疯子。所以你实际上学不到任何东西，无论你是否观察到他在打架。但现在让我们回到第十个市场。很久以前，我们的第十个市场的参与者，他的名字是——我不知道——你真的学不到任何东西，无论你是否观察到他在打架。但现在让我们回到第十个市场。很久以前，我们的第十个市场的参与者，他的名字是Andy，他没有对Ali学到任何东西。他什么都没学到，因为无论Ali是理智还是疯狂，他都会打。所以观察他早期的行为，如果那真的是在均衡中，我们的第十个人根本不会更新他的信念，因此仍然会以0.99的概率相信Alley是理智的，在这种情况下我们的第十个人会进入。一旦再次，这个论证会从后往前瓦解。所以我之前描述的不可能完全是一个均衡。不可能简单到所有理智的人都会表现得疯狂，因为那样你就学不到那个东西了。

[段 17]

So it turns out that the equilibrium in this model is actually very subtle, and it involves mixed strategies, and mixed strategies are something we did in the first half of the semester, so I don’t want to go back to it now. So trust me, you can solve this out with mixed strategies, the basic idea I gave you is right. The basic idea is sane guys are occasionally going to seem like, act like crazy guys in order to establish a reputation and that reputation helps them down the tree All right So this idea this idea that even when there a chain store people will enter even when Ali has ten monopolies people will enter this is a famous idea it called the chain store paradox. It’s due to a guy called Selson who actually won the Nobel Prize. All right? This is the chain store paradox. And this idea of establishing reputation is a big idea. The idea is, once again, you might want to behave as if you’re someone else in order to deter people’s actions, in order to affect people’s actions down the tree. All right? Okay. So what have we learned here? Let’s just work it out. Thank you. All right. So the first thing we learned was kind of a nerdy point, but let me make it anyway. introducing just a very, very small probability, just a tiny probability, that Ali might be someone else.

[译文 17]

所以事实证明这个模型中的均衡实际上非常微妙，它涉及混合策略，而混合策略是我们在学期前半部分讲过的，所以我现在不想回到那个。所以相信我，你可以用混合策略来解决它，我给你的基本思路是对的。基本思路是理智的人偶尔会表现得像疯子，以建立声誉，而这种声誉帮助他们在博弈树的后续阶段。懂吧。所以这个想法——即使有连锁店人们也会进入，即使Ali有十个垄断企业人们也会进入——这是一个著名的想法，叫做连锁店悖论。它是由一个叫Selson的人提出的，他实际上获得了诺贝尔奖。懂吧。这就是连锁店悖论。建立声誉这个想法是一个重要的想法。这个想法是，再说一次，你可能想要表现得像另一个人，以阻止人们的行动，以影响人们在博弈树后续阶段的行为。懂吧？好的。那么我们在这里学到了什么？让我们算一下。谢谢。好吧。我们学到的第一件事是一个有点迂腐的观点，但让我还是说一下。仅仅引入一个非常非常小的概率，一个微小的概率，Ali可能是另一个人。

[段 18]

He might be a rat hole, he might be crazy, he must like fights. That very small probability radically changes the outcome of the game. If we were all 100% sure he was sane, we’d be tied by backward induction, an entry would follow he wouldn be able to keep anybody out but that small probability allows us to get a very different outcome That the first point I want to draw out of this And the second point I want to get out of this is really this idea of reputation. There are lots of settings in society where reputation matters. And one of them is a reputation to fight. How many of you have friends who have somewhat short fuses? You know people who have short fuses, right? Right? And when you’re going out on, you know, choosing some movie to go to with these guys who have short fuses, or trying to decide who’s going to order something at a restaurant, or who’s going to get to be which side when you’re playing some game, some video game, right? I claim, is this true, that the people who have slightly short fuses slightly more often get their way. Is that right? Right? If you have a sibling who has a short fuse, they slightly more often get their way. And that’s exactly this idea. They’re a little bit, they’re short fused, the fact that they tend to blow up and get angry at you gives them a little bit of an advantage, and notice that maybe they don’t have a short fuse at all.

[译文 18]

他可能是个卑鄙小人，他可能是疯了，他一定喜欢打架。那个非常小的概率从根本上改变了博弈的结果。如果我们都100%确定他是理智的，我们就会被后向归纳束缚，进入会随之而来，他不可能把任何人挡在外面。但那个小概率让我们得到一个非常不同的结果。这是第一个我想从这里面得出的观点。而我想从中得出的第二个观点是声誉这个想法。在社会中有很多情况下声誉很重要。其中之一就是战斗的声誉。你们中有多少人有一些相当容易发火的朋友？你认识一些容易发火的人，对吧？对吧？当你要和他们出去的时候，你知道，选一部电影一起去看，或者决定谁在餐厅点什么菜，或者谁在玩某个游戏、某个电子游戏时站在哪一边，对吧？我声称，这是真的吗，那些稍微容易发火的人稍微更经常能得到他们想要的东西。这是真的吗？如果你的兄弟姐妹中有一个容易发火的，他们稍微更经常能得到他们想要的东西。而这正是这个想法。他们有点容易发火，他们会对你发火这个事实给了他们一点优势，而且注意，也许他们根本就不是真的容易发火。

[段 19]

Maybe they’re just pretending to have a short fuse, because they know they’re going to get their way over you softies more often. Right? None of you have short fuses, you’re all the same people, right? So this idea should be a familiar idea to you but it not just an idea in this sort of trivial world of bargaining Notice this idea of reputation occurs all over the place So another place it occurs a somewhat grim place it occurs is in the subject of hostage negotiations. In the subject of hostage negotiations, when some other country has seized some hostages from the US or maybe some criminal organization has seized some members of your family or some members of your community and is holding the hostages, there’s a well-known idea which is what? which is that you should never negotiate with hostage takers is that right? everyone’s heard that idea? you should never negotiate with hostage takers you should never give in just because they have hostages and why? why? it’s the same idea because you want to have a reputation for being somebody who doesn’t give in to hostage takers in order to deter future potential hostage takers from taking hostages this has grim consequences but sometimes it’s worth bearing the cost of having your relatives come back in pieces in order to deter future relatives from being taken.

[译文 19]

也许他们只是在装出一副脾气暴躁的样子，因为他们知道自己能比你们这些软心肠的人更容易达到目的。对吧？你们中间没有人脾气暴躁，你们都是一样的人，对吧？所以这个想法对你们来说应该很熟悉，但这不仅仅是讨价还价这个琐碎世界中的一个想法。注意这个关于声誉的想法到处出现。所以另一个它出现的地方，是一个有些沉重的领域，那就是人质谈判。人质谈判这个领域，当其他国家扣押了美国的一些人质，或者犯罪组织绑架了你的家人或社区的成员并扣押这些人质，有一个众所周知的想法，是什么？那就是永远不要与劫持人质者谈判，对吗？每个人都听说过这个想法？你永远不应该与劫持人质者谈判，你永远不应该因为他们有人质就屈服。为什么？为什么？因为这是同一个想法，因为你想要拥有一种声誉，就是不会向劫持人质者让步的人，这样才能阻止未来潜在的劫持人质者。这有沉重的后果，但有时值得承担让你的亲人以碎片形式回来的代价，以阻止未来的亲人被绑架。

[段 20]

That’s a somewhat macabre version of this. Let me give you one other version. There are whole areas of the economy where reputation is crucial, where if people played according to their backward induction, sane incentives, we’d have a disaster. But having a reputation here isn’t necessarily a reputation as a tough guy. It could be a reputation for somebody who’s a nice guy. You want to have a reputation for being the sort of person who derives pleasure and utility from, quote, doing the right thing, from acting honestly. So think about certain professions where the reputation of the person in the profession is crucial. Doctors, for example. It’s crucial for a doctor that he or she has the reputation of someone who tells the truth, otherwise you’d stop going to that doctor. Accountants, accounting firms, rely on having a reputation for being honest and not cheating the book. When they stop having that reputation for being honest, think of Arthur Anderson after the events in Enron, they pretty quickly ceased to be in business. I gave that example a couple of years ago. It was very embarrassing because it turned out Arthur Anderson was in the class. Literally, Arthur Anderson III was in the class. These things happen at Yale. But nevertheless Arthur Anderson relied on his reputation the firm ran his reputation as an honest firm And it was worth behaving honestly to maintain that reputation for future business Alright?

[译文 20]

这是这个想法的一个有些可怕的版本。让我再给你们一个版本。在经济的某些领域，声誉是至关重要的，如果人们按照他们的逆向归纳和理智的激励来行事，我们会遭遇灾难。但在这里，拥有声誉不一定是要成为一个强硬的人。它可能是一个好人的声誉。你想要拥有一种声誉，就是那种从做正确的事情、从诚实行事中获得乐趣和效用的人。所以想想某些职业，在那里从业者的声誉至关重要。例如医生。对医生来说，他或她拥有说实话的声誉至关重要，否则你会停止去看那个医生。会计师、会计事务所，依靠拥有诚实和不在账目上做假的声誉。当他们失去这种诚实的声誉时，想想安达信在安然事件之后，他们很快就不再经营了。我几年前举过这个例子。非常尴尬，因为安达信三世正好在班上。这些事情发生在耶鲁。但安达信依靠他的声誉，这家事务所经营他的声誉是一家诚实的公司，而且值得通过诚实行为来维持这种声誉以获取未来的业务。好的？

[段 21]

So that’s it. I mean, reputation is a huge topic, and my guess is that the next time there’s a Nobel Prize in game theory, it’ll be for this idea. Alright? So, that’s my prediction there. Now, having said that, I want to spend the whole of the rest of today playing one game and analyzing one game. All right, we’re going to play this game, and for this game I need a couple of volunteers. So I’m going to pull out some volunteers. Well, anyone want to volunteer? I need two volunteers for this game. How about my guy from the football team? Was that a raised hand? It wasn’t a raised hand. How about my guy from the football team? Is it football team? Baseball team. That may be unfair in this particular game. this particular game, maybe I’ll take someone who isn’t in the baseball team but anyone who’s in the football team? these guys, these guys in the football team, ok great ok, so you two guys, right? I need you at the front and your names are? Shebby Shebby Patrick Shebby and Patrick so Shebby and Patrick are going to be our volunteers now the idea of this game is you guys provided the volunteers wait down here a second you guys provided the volunteers this game involves two volunteers who you just provided and two wet sponges I will provide the wet sponges Well I have here a couple of sponges, and in a minute I’m going to wet them, and I’ll tell you what the rules are in a second.

[译文 21]

所以就是这些。我的意思是，声誉是一个很大的话题，我猜测下次有诺贝尔经济学奖授予博弈论时，将会是因为这个想法。好的？这是我的预测。现在，话虽如此，我想用剩下的整节课时间玩一个游戏并分析一个游戏。好的，我们要玩这个游戏，这个游戏需要几个志愿者。所以我要抽出一些志愿者。有人想自告奋勇吗？我需要两个志愿者。有橄榄球队的人吗？那是不是举手了？那不是举手。橄榄球队的？有橄榄球队的人吗？这些家伙，橄榄球队的，好的好的，所以你们两个，对吧？我需要你们到前面来，你们叫什么名字？Shebby Shebby Patrick Shebby和Patrick，所以Shebby和Patrick将成为我们的志愿者。这个游戏的想法是你们这些人提供了志愿者，等一下，你们提供了志愿者，这个游戏涉及两个志愿者，就是你们刚刚提供的，还有两个湿海绵。我会提供湿海绵。我这里有几个海绵，待会儿我会把它们弄湿，我会告诉你们规则。

[段 22]

Okay, so I’m going to give one of these sponges each to Shebie and Patrick. And then I’m going to position Shebie and Patrick at either end of this central aisle, of this aisle here. Alright, and the game is going to be as follows. It’s important that everyone listens to the rules here, because I’m going to pick two more volunteers in a moment. Alright? So the game they’re going to play is as follows. Each of them has one sponge. It’s crucial they only have one sponge. And they’re going to take turns. And when it’s your turn, as long as you still have your sponge in your hand, you face a choice. You can either throw your sponge at your opponent, alright? And if you hit your opponent, if you hit your opponent, you win the game. Or you have to take a step forward Either you throw the sponge or you take a step forward All right Now there a crucial rule here Each player only has one sponge and once they’ve thrown that sponge, they do not get the sponge back. Everyone understand that? Once you’ve thrown the sponge, you do not get the sponge back.

[译文 22]

好的，我要给Shebie和Patrick每人一个海绵。然后我会把Shebie和Patrick安排在这个中央过道的两端，这个过道这里。好的，游戏的规则如下。每个人仔细听规则很重要，因为我待会儿要再选两个志愿者。好的？所以他们要玩的游戏规则如下。每个人有一个海绵。关键是他们只有一块海绵。然后他们轮流进行。当轮到你的时候，只要你手里还有海绵，你面临一个选择。你可以向对手扔海绵，对吧？如果你击中对手，如果你击中对手，你就赢了游戏。或者你必须向前迈一步。你要么扔海绵，要么向前迈一步。现在这里有一个关键规则，每个玩家只有一块海绵，一旦他们扔出那块海绵，就再也拿不回海绵了。大家都明白吗？一旦你扔出海绵，你就拿不回海绵了。

[段 23]

So once again, if you throw your sponge at your opponent and you hit your opponent, then you’ve won the game. but if you throw your sponge at your opponent and you miss the game continues all right right so let’s make sure we understand that if you throw your sponge and miss the game continues you still have to step forward so what’s your opponent going to do at that point what’s your opponent going to do is make sure that our football players understand this all right all right so no I don’t want to be meant that way and it could have been soccer players come on I didn’t appreciate that very much. I didn’t mean it that way. I didn’t mean it that way. If your opponent, whose name is Patrick, throws and misses, what are you going to do? I will walk forward until I slam the sponge in his face. Great, you’ll walk forward until you politely put it on his head. Everyone understand that if in fact you throw and miss, you’ve lost the game because the other guy can wait until he’s standing right on top of you and just place the sponge in his face. Let’s just politely put it on his head. Okay, everyone understand that if in fact you throw a miss, you’ve lost the game because the other guy can wait until he’s standing right on top of you and just place the sponge gently on his head.

[译文 23]

再重复一遍，如果你向对手扔海绵并击中对手，你就赢了游戏。但如果你向对手扔海绵并错过，游戏继续。好的，好的，确保我们理解这一点，如果你扔海绵并错过，游戏继续，你仍然必须向前迈步。那么你的对手会怎么做？在这一点上你的对手会做什么？就是确保我们的橄榄球队员理解这一点。好的，好的，所以不，我不想那个意思，那可能是足球运动员，拜托，我不太欣赏那个。我不是那个意思。我不是那个意思。如果你的对手，他的名字叫Patrick，扔了并错过了，你要怎么做？我会向前走直到我把海绵拍在他脸上。很好，你会向前走直到你礼貌地把它放在他头上。每个人都明白吗？事实上如果你扔了并错过了，你就输掉了游戏，因为另一个人可以等他站在你正上方，然后把海绵放在他脸上。让我们礼貌地把它放在他头上。好的，每个人都明白吗？事实上如果你扔了并错过了，你就输掉了游戏，因为另一个人可以等他站在你正上方，然后把海绵轻轻地放在你头上。

[段 24]

All right. All right, now for fairness sake, it’s important that these sponges are equally weighted and I’m going to put water in them now and since it’s nothing but the best for Yale students, I’m going to provide Yale University spring water. Who knew that Yale University had a spring? It’s kind of a phrase, huh? If it makes you feel better, you can think of this as American beer. All right. Sorry, sorry. All right. And I’m not going to make these too heavy, partly because it makes it too easy, and partly because I don’t want to get sued. All right. So I’m going to squeeze these out somewhere, away from the wires. All right. All right. And we’re going to get our judge here to weigh them. Oh, I need a mic here. Let me get a mic. So, okay, so I want you to hold those sponges in your hands, and tell me if they’re equally weighted. They’re pretty equal. Pretty equal okay they pretty equal everyone agrees All right so how is this gonna work I gonna give the blue sponge to Shabby is it And the green sponge to Patrick And Shabby going to stand here And Patrick going to stand as far back as I can get him on camera Which I’m going to be told how far back I can go.

[译文 24]

好的，好的，为了公平起见，重要的是这些海绵重量相等，我现在要往里面加水，既然对耶鲁学生只有最好的，我就要提供耶鲁大学泉水。谁知道耶鲁大学有一口泉呢？这有点像一句话，对吧？如果这让你感觉好一点，你可以把它想成美国啤酒。好的。抱歉，抱歉。好的。我不会让这些太重，部分原因是这太容易了，部分原因是我不想被起诉。好的。所以我要把这些在远离电线的地方挤一下。好的，好的，我们要让我们的裁判来称重。哦，我这里需要麦克风。让我拿个麦克风。好的，所以我想让你拿着那些海绵，告诉我它们重量是否相等。它们相当相等。相当相等，好的，它们相当相等，大家都同意。好的，那么这要怎么进行？我要把蓝色海绵给Shabby，是吧？绿色海绵给Patrick。Shabby要站在这里。Patrick要站到我在相机上能让他到的最远的地方，而我会得到指示能退多远。

[段 25]

Don’t go too far. Okay, come back, come back, Patrick. You’re too ambitious. Come back, come back, come back, come back, come back, come back, come back, come back, come back. Keep up, keep coming, keep coming. Keep coming, keep coming, keep coming. Stop! Yeah, okay, we’re going to start here. It’s not quite close actually, alright? Alright, everyone understand how we’re going to play this, alright? So, Shebby is player one, Patrick is player two. Shebby has to decide whether to throw or to step. Oh, step. Okay, let’s just hold the game a second, alright? Now it’s Patrick’s turn. Does anyone have any advice for Patrick at this point? Who thinks? Okay, wait, wait, wait. If you think throw, raise your hand. If you think step, raise your hand. There’s a lot more steps than throw. I thought the Yale property was good this year. Okay. Alright. Your choice, step or throw? Alright. I should announce two other rules that are important. I should have said this First a step has to be a proper step like a yard long And second I think this will work in America gentlemen never duck All right No dodging the sponge okay All right Sebi your turn Let me put the mic on you, right? I don’t really trust my arm. I’m going to step. All right, so you’re stepping again.

[译文 25]

别走太远。好，回来，回来，Patrick。你太激进了。回来，回来，回来，回来，回来，回来，回来，回来，回来。跟上，继续过来，继续过来。继续过来，继续过来，继续过来。停！好的，我们要从这里开始。实际上还没那么近，对吧？好，大家都明白我们要怎么玩这个游戏，对吧？那么，Shebby是一号玩家，Patrick是二号玩家。Shebby要决定是投掷还是迈步。哦，迈步。好，我们先暂停一下，好吗？现在轮到Patrick了。有人在这个时候给Patrick一些建议吗？谁有想法？好，等一下，等一下，等一下。如果你觉得应该投掷，举手。如果你觉得应该迈步，举手。迈步的人比投掷的多很多。我以为Yale的物业今年应该表现不错。好吧。好。你的选择，迈步还是投掷？好吧。我应该宣布另外两条重要规则。我应该先说明这一点：第一，迈步必须是一个正规的步幅，大约一码长；第二，我觉得这在美国应该行得通，绅士从不躲避。好。不许躲海绵，明白。好吧，Sebi，轮到你了。让我把麦克风递给你，好吗？我不太相信我的手臂。我要迈步。好，所以你又要迈步了。

[段 26]

Let me go to Patrick, all right? I feel like I’m in the line of fire here. Patrick, what are you going to do? I’m going to throw. Patrick’s going to throw. I’m really going to get out of the way then. All right? We’ll see this in slow motion. What? Ahhh! Continue, continue. Alright. You have to take a step forward. So, Sebi’s going to take a step forward, I assume. And Patrick’s going to take a step forward. And Sebi’s going to take a step forward. And Patrick’s going to take a step forward. And you… Alright. Alright. Good. So, a round of applause for our players. Thank you. What happened to my… there we are. So I think we have time to do this once more and then we’re going to analyse it. So I want to get two women involved, this is too sexist otherwise. So can we have two women in this class please? Two volunteers come on You can volunteer ah there a volunteer thank volunteer Thank you great Great okay great thank you So your name is Jessica and your name is Let me get the mic over to you Clara Leith. Clara Leith and Jessica. All right, we’ll start at the same positions, we’ll use the same sponges. And I need to remind you what that position was, so just give me a thumbs up when I’m in the right position.

[译文 26]

让我转向Patrick，好吗？我感觉自己处于火力范围内。Patrick，你要做什么？我要投掷。Patrick要投掷。那我真得闪开了，好吧？我们来看看这个慢动作。什么？啊！继续，继续。好。你必须向前迈一步。所以，我假设Sebi要向前迈一步。然后Patrick要向前迈一步。Sebi要向前迈一步。Patrick要向前迈一步。而你…好。好。很好。所以，让我们为我们的选手鼓掌。谢谢。我的…怎么了…好了。所以我想我们还有时间再来一次，然后我们要分析它。我想让两个女生参与，否则这太性别歧视了。我们班能有两个女生吗？两位志愿者请上来你们可以自愿…啊，有一位志愿者，谢谢志愿者谢谢很好好非常好谢谢那么你叫Jessica，你叫Clara Leith。Clara Leith和Jessica。好，我们会从相同的位置开始，用相同的海绵。我需要提醒你们那个位置是什么，所以当我处于正确的位置时给我竖个大拇指。

[段 27]

All right, good, good. All right, same rules. Clara Leith and Jessica. We’ll let Jessica be player one. So Jessica, you can step or throw. What do you want to do? I’m going to step. You’re going to step, okay. You know what might be a good idea? Why don’t you put the mic on Claralee so we have a mic on either end and we just run to and fro. That’s good. Alright, so Claralee, what are you going to do? I’m going to step. You’re going to step. And Jessica, what are you going to do? I’m going to step. You’re going to step. Allie and I are in danger here, but never mind. I’m going to step. Any votes now? Do people think that Jessica should throw? If you think she should throw, raise your hand. There’s a large majority for step, so up to you. What are you going to do? I’m going to step. Going to step, okay. Clearly, any decisions? I mean, this is pretty light. I don’t think it’s going to go anywhere. It’s a pretty light sponge. It’s pretty hard to throw this sponge. We’ve seen that. Okay. Stepping again. There’s some baby steps. I’m going to throw. It’s pretty light. I don’t think it’s going to go anywhere. It’s a pretty light sponge. It’s pretty hard to throw this sponge. We’ve seen that.

[译文 27]

好的，很好。好，同样的规则。Clara Leith和Jessica。我们让Jessica当一号玩家。所以Jessica，你可以选择迈步或投掷。你想怎么做？我要迈步。你要迈步，明白。你知道什么可能是个好主意吗？你为什么不给Clara Leith一个麦克风，这样我们两边都有麦克风，我们就来回跑。这样好。好，那么Clara Leith，你要怎么做？我要迈步。你要迈步。Jessica，你要怎么做？我要迈步。你要迈步。Allie和我正处于危险中，但没关系。我要迈步。现在有人投票吗？大家觉得Jessica应该投掷吗？如果你觉得她应该投掷，举手。大多数人选择迈步，由你决定。你要怎么做？我要迈步。迈步，明白。显然，有任何决定吗？我的意思是，这个海绵很轻。我觉得它飞不了多远。这是一个很轻的海绵。要把这个海绵投出去相当难。我们已经见识过了。好。又迈步了。有人在小心翼翼地迈步。我要投掷。它很轻。我觉得它飞不了多远。这是一个很轻的海绵。要把这个海绵投出去相当难。

[段 28]

Stepping again. There’s some baby steps. I’m going to throw. You’re going to throw. Okay. Here we go. Out of the way. Oh, okay. Continue. Continue, please. Clarice, it’s your turn. Your step. Jessica has to step. Clarice has to step. I’ll throw. Alright. Good. Good. Alright. Alright, so we’ve seen how the game works, everyone understands how the game works, I want to spend the rest of today analysing this game. Alright, and before I do so, before I do so, we should just talk about what this game is. Let me get some new boards down here. alright, so one quick announcement I’m going to analyse this and we’re going to spend the rest of today analysing this but I’m going to prove, this is going to be quite hard so I’m going to provide you a handout that I’ll put on the web probably tomorrow that goes over this argument you don’t have to take detailed notes now you’ll pay attention and see if we can follow the argument ok so this game is called Duel not surprisingly and you may wonder what are we doing as my colleagues are here what are we doing having a duel in class? And of course one answer to that is, it’s kind of fun watching the future leaders of America throw wet sponges at each other. That’s probably a reason in itself.

[译文 28]

又迈步了。有人小心翼翼地迈步。我要投掷。你要投掷。好。我们开始。让开。哦，好。继续。请继续。Clara Leith，轮到你了。你迈步。Jessica必须迈步。Clara Leith必须迈步。我来投掷。好。很好。很好。好，好，所以我们已经看到这个游戏是怎么运作的，大家都明白游戏是怎么运作的，我想用今天剩下的时间来分析这个游戏。好，在我这样做之前，在我们这样做之前，我们应该先谈谈这个游戏是什么。让我拿几块新白板过来。好，有一个简短的通知我要分析这个，我们要花今天剩下的时间来分析这个，但我要证明这一点，这会相当困难，所以我将给你们一份讲义，我可能会在明天放到网上，上面有这整个论证你们现在不需要做详细的笔记，你们只要注意听，看我们能不能跟上论证好所以这个游戏叫做"决斗"，这并不令人意外，你们可能会想，为什么我的同事们在这里，我们在课堂上进行决斗？当然，答案是，看着美国的未来领袖们互相投掷湿海绵，这很有趣。这本身就是一个理由。

[段 29]

But there are other reasons. Duels are real games. So those of you who are well versed in Russian literature, those of you who are well versed in Russian literature would have seen duels before or at least read of duels there are some famous duels in Russian literature anyone want to tell me some famous duels in Russian literature? any Russian mages here? no? nobody want to give me a shot at this? really nobody? this is Yale, come on well how about in War and Peace there’s a duel like this in War and Peace and in War and Peace without giving away the ending actually it’s in the middle of the book and it’s 800 pages long so it isn’t exactly the ending right but in War and Peace I think we’re led to believe that the hero the protagonist Pierre throws his shoots his gun it’s a gun in War and Peace it’s a gun not a sponge no surprise he shoots his gun too early we’re led to believe all right there’s a famous one in the Neagin right Pushkin’s Ugi de Neagin and there are lots of others actually so there lots in literature all right there are also settings which aren exactly out of literature so one example would be in a bike race How many of you ever watched the Tour de France Everyone know what I mean by the Tour de France?

[译文 29]

但还有其他原因。决斗是真实的游戏。你们中那些精通俄罗斯文学的人，你们中那些精通俄罗斯文学的人以前见过决斗，或者至少读到过决斗，俄罗斯文学中有些著名的决斗，有没有人想告诉我俄罗斯文学中一些著名的决斗？有俄罗斯大师吗？没有？没有人想试试吗？真的没人？这是Yale，加油嘛，那《战争与和平》中有一个这样的决斗，《战争与和平》中，在不剧透结局的情况下，它其实在书的中间部分，那本书有800页长，所以它不是结尾，对吧，但在《战争与和平》中，我认为我们被引导相信英雄主人公Pierre投掷射击他的枪，在《战争与和平》中是枪不是海绵，不出所料，他开枪太早了，我们被引导相信，好，在《叶甫盖尼·奥涅金》中有一个著名的，普希金的《叶甫盖尼·奥涅金》，实际上还有很多其他的，所以在文学中有很多，好，也有不完全来自文学的场景，一个例子是在自行车赛中。你们中有多少人看过环法自行车赛？大家都明白我说的环法自行车赛是什么意思吗？

[段 30]

Yeah, you’ve probably not watched it. So this is a bike race that goes around France, it takes stages, and in the Tour de France there’s a key decision, there’s a game within the game, and the game within the game, I’m looking at Jake who’s a real cyclist here, but the game within the game is when do you try to break away from the pack, which is called the peloton. And if you break away too early from the peloton, it turns out that you’re going to get reeled in. It turns out that over the long haul, the peloton can go much faster than you. So if you break too early, they’re going to catch you up. On the other hand, if you break too late, then you’re going to lose, because there are going to be some people in the peloton who are just excellent sprinters. So if you break too late, the sprinters are going to win the race. You have to decide when to break from the peloton. This is the second most important game game within a game in the Tour de France. The most important game within a game is where to hide your steroids. Alright? So, let me give you one other example. So imagine, I’ll give you a more economic example, it meant to be an economics class Imagine there two firms and these two firms are both engaged in R research and development and they trying to develop a new product and they going to launch this new product onto the market But the nature of this market is maybe it a network good the nature of this market is there’s only going to be one successful good out there.

[译文 30]

是的，你们可能没看过。所以这是一场环绕法国的自行车赛，分阶段进行，在环法自行车赛中有一个关键决策，这是比赛中的比赛，这个比赛中的比赛，我在看Jake，他是个真正的自行车手，但这个比赛中的比赛是：你什么时候尝试从车队中突围，这叫做主集团。如果你在太早的时候从主集团突围，事实证明你会被拉回来。事实证明，从长远来看，主集团可以比你快得多。所以如果你突围太早，他们会追上你。另一方面，如果你突围太晚，你就会输，因为主集团中会有一些人是非常出色的冲刺手。所以如果你突围太晚，冲刺手会赢得比赛。你必须决定什么时候从主集团突围。这是环法自行车赛中第二重要的比赛中的比赛。最重要的比赛中的比赛是藏类固醇的地方。明白吗？所以，让我再给你们一个例子。想象一下，我给你们一个更经济学的例子，这应该是一门经济学课，想象有两家公司，这两家公司都从事研发，它们试图开发一种新产品，它们要把这个新产品推向市场，但这个市场的本质可能是网络产品，这个市场的本质是只会有一种成功的产品。

[段 31]

Essentially there’s only one standard, let’s say, of a software or a technological standard, and only one of them is going to survive. The problem is you haven’t perfected your product yet. If you launch your products too early, it may not work, and then consumers are never going to trust you again. But if you launch it too late, the other side will have launched already, they will have got a toehold in the market, and you’re toast. All right? So that game, that game about product launch, is like Duel, except you’re launching a product rather than launching a sponge. Is that right? All right? Now, all of these games have a common feature, and it’s a new feature for us. It’s about the nature of the strategic decision. In most of the games we’ve looked at in the course so far, the strategic decision has been of the form, where should I locate? How much should I do? What price should I set? Should I stand for election or not? Right, here, the question of the strategic decision is not of the form, what should I do? It’s of the form what? When? All right, here, the question of the strategic decision is not of the form, what should I do? It’s of the form, what? When? It’s of the form, when am I going to launch the sponge?

[译文 31]

本质上，软件或技术标准只有一个，而且只有一个会存活下来。问题在于你还没有完善你的产品。如果太早推出产品，它可能不管用，消费者就再也不会信任你了。但如果推出得太晚，另一方就已经推出了，他们已经在市场上站稳了脚跟，你就完蛋了。好吧？所以那个游戏，那个关于产品发布的游戏，就像《对决》一样，只不过你推出的是产品而不是抛出海绵。对吧？好吧？所有这些游戏都有一个共同特征，这对我们来说是一个新特征。它涉及战略决策的本质。在我们迄今为止所看的绝大多数游戏中，战略决策的形式是：我应该定位在哪里？我应该做多少？我应该设定什么价格？我应该参选吗？对吧？但在这里，战略决策的问题不是“我应该做什么”这种形式，而是“什么时候”？好吧，在这里，战略决策的问题不是“我应该做什么”这种形式，而是“什么时候”？它的形式是：我什么时候抛出海绵？

[段 32]

We know perfectly well what you’re going to do. You’re going to throw the sponge. The strategic issue in question is when. All right, so the issue here is when. All right, so to analyze this, I’m going to need a little bit of notation, and let me put that notation up now. Alright, so in particular I want to use the notation P I of D to be what? Let P I of D be player I’s probability probability of hitting if I shoot at distance d. So p i of d is the probability that I will hit if he or she shoots at distance d I won have to do that This is the only notation I going to today Alright And I going to make some assumptions about the nature of this probability but they not two of the assumptions are pretty innocent Let’s draw a picture. Alright, so the picture’s going to look like this. Here’s a graph, and on the horizontal axis, I’m going to put d. This is the distance apart of the two players. and on the vertical axis I’m going to put P, which is the probability, or PR, the probability. All right? So here they’re at distance 0, and I’m going to make an assumption about what the probability of hitting is if you’re at distance 0. What’s the sensible assumption?

[译文 32]

我们完全清楚你要做什么。你要扔出海绵。战略问题在于什么时候。好吧，所以这里的问题是什么时候。好吧，要分析这个问题，我需要一点符号标记，让我现在把符号标记写出来。好吧，具体来说我想用PI(D)这个符号来表示什么？让PI(D)表示选手I在距离d射击时的命中概率。所以p_i(d)是我在距离d射击时命中对方的概率。我得把它写出来。这是我今天唯一会用到的符号。好吧，我将要对这个概率的性质做一些假设，但这些假设中有两个是相当合理的。让我们画个图。好吧，图大概是这样的。这是一张图，横轴上放d，这是两位选手之间的距离。纵轴上放P，也就是概率，或者PR，概率。好吧？他们距离为0的时候，我来做个假设，如果你距离为0，命中概率是多少？

[段 33]

What’s the probability of hitting somebody with your sponge if you’re 0 distance away? 1. Okay, I agree. So 1. All right, so the first assumption I’m going to make is if they’re right on top of each other, they’re going to hit with probability 1. And the second assumption I’m going to make is, as you get further away, this probability decreases. It doesn’t have to look exactly like this, but something like that. And that also, I think, is that an okay assumption? As you’re further away, there’s a lower probability of hitting. Now I not going to assume that these two players have equal abilities For example I don know I didn ask them but one of our two football players might be a quarterback and the other one might be a linebacker or a running back and I’m assuming the quarterback is probably more accurate. Is that right? So I’m not going to assume that they’re equally good. So it could be that their abilities look like this. This could be P1 of D, and this could be P2 of D. Everyone okay with that? All right? So, shout this out. In this picture, who is the better shot and who is the less good shot? Who’s the better shot? One, right? One is the better shot, because at every distance, if player one were to throw, player one’s probability of hitting is higher than player two’s probability of hitting, as drawn.

[译文 33]

如果你距离为0，用海绵击中对方的概率是多少？1。好，我同意。所以第一个假设是，如果他们正好重叠在一起，他们命中概率为1。第二个假设是，随着距离变远，这个概率会下降。它不必完全像这样，但差不多是这样的。这也行吧？随着距离变远，命中概率更低。我不会假设这两个选手能力相等。比如我不知道我没问过他们，但我们两个足球运动员中一个可能是四分卫，另一个可能是线卫或跑卫，我假设四分卫可能更准。对吧？所以我不会假设他们同样优秀。所以可能是他们的能力曲线像这样。这可能是P1(D)，这可能是P2(D)。大家都同意吗？好吧，大声说出来。在这张图里，谁是更准的射手，谁是相对较差的射手？谁是更准的射手？一，对吧？一号是更准的射手，因为在每个距离上，如果一号选手投掷，一号选手的命中概率都高于二号选手的命中概率，如图所示。

[段 34]

All right? Now, I don’t even need to assume this. It could well be that these probabilities cross. It could be that these curves cross, it could be that player 1 is better at close distances, but player 2 is better at far distances. That’s fine. We’ll assume it’s like this today, but I’m not going to use that. I could do away with that. As drawn, player 1 is the better shot. Now I going to make one assumption that matters and it really a critical assumption I going to make this assumption that keeps the math simple for today but we have enough math to do anyway I going to assume that these abilities are known I’m going to assume that not only do you know your own ability of hitting your opponent at any distance, I’m going to assume you also know the ability of your opponent to hit you. All right. All right. alright now let’s look at this a second we’ve got a bit of notation on the board let’s discuss this a second what do we think is going to happen here in this particular example we have a good shot and a less good shot who do we think is going to shoot first who do we think is going to shoot first let’s try and cold call some people alright so you sir, what’s your name Frank Frank, so who do you think is going to shoot first the better shot or the worse shot It’s a better shot, but it also depends on who’s step first as well, I think.

[译文 34]

好吧？我甚至不需要假设这样。也可能出现这些概率交叉的情况。也可能出现曲线交叉，可能一号选手在近距离时更准，但二号选手在远距离时更准。这没问题。我们今天假设它像图上这样，但我不会利用这一点。我可以不用这个假设。如图所示，一号选手是更准的射手。现在我要做一个重要的假设，而这确实是一个关键假设。我要做这个假设来让今天的数学简化，但我们已经有足够的数学要做了。我要假设这些能力是已知的。我要假设不仅你知道自己在任何距离上击中对手的能力，我还假设你也知道你对手击中你的能力。好吧。好吧，现在让我们再看一眼黑板上的符号标记，让我们讨论一下。在这个特定的例子中我们认为会发生什么？我们有一个好射手和一个较差的射手，我们认为谁会先开枪？我们认为谁会先开枪？让我们试着随机点名几个人。好吧先生，你叫什么名字？Frank？Frank，你觉得谁会先开枪？是更准的射手还是较差的射手？是更准的射手，但也要看谁先出手，我觉得。

[段 35]

Well, let’s assume player one is going to shoot first. Then player one. So Frank thinks player one is going to shoot first, because player one is a better shot. Who thinks… let’s see, what’s your name? Nick. Nick, who do you think is going to shoot first? I think player two will shoot first. All right. so let’s talk why. Why do you think player one is going to shoot first? Let’s do a poll. How many people think the better shot is going to shoot first? How many people think the worse shot is going to shoot first? How many of you are being chickened and abstaining? Quite a few, right? Okay, okay, all right. so why do we think the better shot might shoot first? well because at equal distance he has a better shot of because he has a better chance of hitting but why do you think that the less good shot might shoot first? he knows that if P1 gets too close he’s going to win anyway so he might as well take a shot with lower chance before P1 is guaranteed to hit him ok so we have two arguments here the first argument is maybe the better shot will shoot first because after all he has a better, a higher chance of hitting, and the other argument says maybe the worst shot will shoot first to what?

[译文 35]

好吧，让我们假设一号选手会先开枪。那就是一号选手。所以Frank认为一号选手会先开枪，因为一号是更准的射手。谁觉得……让我看看，你叫什么名字？Nick？Nick，你觉得谁会先开枪？我觉得二号选手会先开枪。好吧，让我们讨论一下为什么。你为什么认为一号选手会先开枪？让我们做个投票。有多少人认为更准的射手会先开枪？有多少人认为较差的射手会先开枪？你们有多少人在犹豫不决而弃权？不少人，对吧？好吧，好吧。那为什么我们认为更准的射手可能会先开枪？嗯，因为在同等距离上他有更好的机会命中，因为他有更高的命中概率。但为什么你认为较差的射手可能会先开枪？他知道如果一号选手靠得太近，他反正都会赢，所以他不妨在一号选手保证能命中他之前，先用一个较低的概率来尝试。好吧，所以我们这里有两个论点。第一个论点可能是更准的射手会先开枪，因为毕竟他有更高的命中概率，另一个论点说可能是较差的射手会先开枪，为什么？

[段 36]

To preempt the better shot from shooting him. Alright But now we get more complicated because after all if you the better shot and you know that the worst shot may be going to try and shoot first to try and preempt you from shooting him you might be tempted to shoot before the worse shot shoots to preempt the worse shot from preempting you from shooting him. And if you’re the worse shot, maybe you’re going to try and shoot first even earlier to preempt the better shot from preempting the worse shot from shooting the worse shot, and so on. So it’s clear, what’s clear is that this game has a lot to do with preemption. preemption’s a big idea here but I claim it’s not at all obvious it’s not at all obvious who’s going to shoot first the better shot or the worse shot right? so those people who abstain raise your hands again those people who abstain before it’s a pretty sensible time to abstain it’s not obvious at all to me who’s going to shoot first here alright? are people convinced that at least it’s a hard problem? yeah? yes or no? people convinced? Yeah, okay, good. It’s a hard problem. Okay, good. So what I want to do is, as a class, as a group, what I want us to do is solve this game.

[译文 36]

为了抢先于更准的射手开枪。好吧，但现在变得更复杂了，因为毕竟如果你是更准的射手，你知道较差的射手可能会试图先开枪来抢在你之前，你可能就忍不住想比那个较差的射手更早开枪，抢在他抢在你之前。而如果你是较差的射手，也许你会试图更早地先开枪，抢在更准的射手抢在你之前先向你开枪，如此等等。所以很清楚，清楚的是这个游戏与抢先密切相关。抢先在这里是一个重要概念，但我声称这绝非显而易见。谁会先开枪，更准的射手还是较差的射手，这绝非显而易见，对吧？所以那些弃权的人，再举一下手。那些弃权的人，这是一个相当合理的弃权时机，谁先开枪在这里一点也不明显，对吧？大家相信这至少是一个难题吗？嗯？信还是不信？大家相信吗？嗯，好吧，很好。这是个难题。好的。那么我想做的是，作为一个班级，作为一个团队，我想让我们来解这个游戏。

[段 37]

And I want to solve not just who is going to shoot first, I want to figure out exactly when they going to shoot Alright So we going to do this in the next half hour And we going to do it as a class right So you going to do it So we going to nail this problem basically all right And we’re going to do it using two kinds of arguments. One kind of argument is an argument we learned the very first day of the class, and that’s a dominance argument. And the second kind of argument is an argument we’ve been using a little bit recently, and what kind of argument is that? What is it? backward induction. So we’re going to use dominance arguments and backward induction, and we’re going to figure out not just whether the better shot or the worse shot is going to shoot, but exactly who’s going to shoot when. All right. Let’s keep our picture handy. Get rid of… well, I can get this down, I guess. And we’ll… Can we still see the picture? Yeah. All right. Let’s proceed with this argument. Alright, to do this argument, I first of all want to establish a couple of facts. Alright, my page and my notes. I want to establish two facts and we call the first fact fact A Alright let go back to our two players we had before In fact maybe it would be helpful to have our Can I use our first two players as props Do I have you guys on stage?

[译文 37]

我想解决的不仅是谁先开枪的问题，我要精确算出他们什么时候开枪。好的，我们将在接下来的半小时内完成这件事，我们要全班一起做。所以你们要参与，我们基本上要把这个问题彻底解决。我们要用两种论证方法。一种论证是我们上课第一天就学到的东西，即占优论证。第二种论证是我们最近稍微用过一点的论证，那是什么？逆向归纳法。所以我们要用占优论证和逆向归纳法，我们要不仅算出好枪手还是差枪手会开枪，还要精确算出谁在什么时候开枪。好的，让我们把图放在手边。删掉……好吧，我觉得可以把它放下来。我们……还能看到图吗？能。好，让我们继续这个论证。好的，为了进行这个论证，我首先要建立几个事实。好的，我的页面和笔记。我要建立两个事实，我们称第一个事实为事实A。好的，让我们回到之前的两个玩家。事实上，也许让我们的前两个玩家来做道具会有帮助，能让他们上台吗？

[段 38]

While they’re coming up? All right. Sorry, guys. I’m exploiting you a bit today. All right. Hope you both find your legal release forms. All right. All right. Why don’t you guys sit here a second so I can use you as props. So imagine, imagine that these two guys still have their sponges, all right, let’s actually set this up, so suppose that Shebi, Shebi? Shebi has a sponge, and Patrick still has his sponge, all right, and suppose it’s Shebi’s turn, all right, and suppose that Shebi is trying to decide whether he should throw his sponge or not. Whether he should throw his sponge or not. Have the mic. Let me give you a mic each, so you have a future reference. All right, so Shebi is trying to decide whether to throw his sponge or not. And suppose that Shebby knows that Patrick is not going to shoot next turn when it’s his turn. So Shebby’s trying to decide whether to shoot, and he knows that Patrick is not going to shoot next turn when it’s his turn. What should Shebby do? Shebby, what should you do? Take it. So Chevy’s trying to decide whether to shoot, and he knows that Patrick is not going to shoot next turn when it’s his turn. What should Chevy do? Chevy, what should you do? Take a step. Take a step.

[译文 38]

在他们上来的过程中？好的。抱歉，伙计们。今天我稍微利用你们一下。好的，希望你们俩都找到了法律免责书。好的，好的。你们坐这儿一会儿，这样我可以把你们当道具用。所以想象一下，想象这两个人还拿着海绵，好的，让我们实际设置一下，假设Shebi，Shebi？Shebi有海绵，Patrick还有他的海绵，好的，假设现在是Shebi的回合，好的，假设Shebi正在决定是否要扔他的海绵。是否要扔他的海绵。拿着麦克风。让我给你们每人一个麦克风，这样你们以后可以参考。好的，所以Shebi正在决定是否要扔他的海绵。假设Shebby知道Patrick在下一回合轮到他时不会开枪。所以Shebby正在决定是否要开枪，他知道Patrick在下一回合轮到他时不会开枪。Shebby应该怎么做？Shebby，你应该怎么做？走一步。所以Chevy正在决定是否要开枪，他知道Patrick在下一回合轮到他时不会开枪。Chevy应该怎么做？Chevy，你应该怎么做？走一步。走一步。

[段 39]

Chevy says take a step. That’s right. Why should he take a step? What’s the argument why he should take a step? Well, let’s find out. What’s the argument? Because I’ll get another… I’ll just be one step closer and I’ll have another chance to decide anyway. Shout it out so you can be heard. That’s right. That’s right. Shout it out so you can be heard. I’ll be one step closer and I’ll be able to make the same choice next time anyway. Good, good. Hold the mic up to you. You’re a rock star now. Okay? All right, good. All right? All right? He’s correctly saying he should wait. Why should he wait? Because he’s going to be closer next time. He’s going to be closer next time. So the first fact is, assuming no one has thrown yet, assuming no one has thrown, if player i knows at say distance d that j will not will not shoot let me call it tomorrow and tomorrow he be closer he be at distance d minus 1 then, as Sebi correctly says, I should not shoot today. And again, recall the argument, the argument is you’ll get a better shot, a closer shot, the day after tomorrow. alright now let’s turn things around suppose conversely once again we’re picking on Shebby a second so Shebby has his sponge, no one has thrown yet and suppose Shebby knows suppose Shebby knows that Patrick is going to throw tomorrow so he knows that Patrick is going to throw tomorrow now what should Shebby do now what should Shebby do now that’s a harder decision What should he do?

[译文 39]

Chevy说走一步。没错。为什么他应该走一步？为什么他应该走一步的论证是什么？好的，让我们找出答案。论证是什么？因为我会再……我会离得更近一步，而且我反正还会有另一次决定的机会。大声喊出来这样大家能听到。没错。没错。大声喊出来这样大家能听到。我会离得更近一步，而且我反正还能下次做同样的选择。好的，好的。把麦克风对着你。你现在是个摇滚明星了。好吗？好的，好的。好吗？好吗？他正确地说他应该等待。为什么要等？因为下次他会离得更近。他下次会离得更近。所以第一个事实是，假设还没有人扔过，假设还没有人扔过，如果玩家i比如说在距离d时知道j不会……不会在明天开枪，让我称之为明天，而明天他会更近，他会处在距离d减1的位置，那么，正如Sebi正确说的，我今天不应该开枪。再回顾一下这个论证，论证是你会在后天得到一个更好的机会，一个更近的机会。好的，现在让我们反过来，假设反过来我们再次针对Shebby一会儿，所以Shebby有他的海绵，还没有人扔过，假设Shebby知道假设Shebby知道Patrick明天要扔，所以他知道Patrick明天要扔，那么现在Shebby应该怎么做？现在Shebby应该怎么做？这是一个更难的决定。他应该怎么做？

[段 40]

He knows Patrick’s going to shoot tomorrow. What should he do? Should he shoot or what? What’s the answer this time? What do you reckon? It depends. It depends. I think that’s the right answer. All right, it depends. Good. So the question is someone else I don want to pick entirely on these guys So what does it depend on It right that it depends What does it depend on Yeah If the other guy chance is greater than or less than 50 of getting it Alright, so it might depend on the other’s chance of being greater than or less than 50%. It certainly depends on the other guy’s ability and on my ability. Everyone clear on that? But how exactly, everyone agree that whether I should shoot now, if I know the other guy’s going to shoot tomorrow, depends on our abilities. but how exactly does it depend on our abilities? Go ahead. It depends on if… Shout out. It depends on if your probability to hit is greater than his probability to miss. Good. Good. Your name is? Osman. Osman. So Osman is saying, let’s be careful here, it depends on whether my probability of hitting, if I throw now, is bigger than his probability of missing tomorrow. Now why is that the right comparison? That’s the right comparison because if I throw now, my probability of winning the game is the probability that I hit my opponent.

[译文 40]

他知道Patrick明天要开枪。他应该怎么做？应该开枪还是怎样？这次答案是什么？你们觉得呢？视情况而定。视情况而定。我认为这是正确答案。好的，视情况而定。好的。所以问题是别人我不想完全针对这些人。所以这取决于什么？取决于是对的。取决于什么？是的，如果另一个人的机会大于或小于50%。好的，所以这可能取决于另一个人成功的机会是大于还是小于50%。当然这取决于另一个人的能力和我的能力。大家都清楚吗？但是具体怎么，每个人都同意，如果我知道另一个人明天要开枪，我是否应该现在开枪，取决于我们的能力。但是具体怎么取决于我们的能力？说吧。大声说出来。取决于你的命中率是否大于他失误的概率。好的，好的。你叫什么？Osman。Osman。所以Osman在说，让我们仔细一点，取决于如果我现在扔，我的命中率是否大于他明天失误的概率。为什么这是正确的比较？这是正确的比较，因为如果我现在扔，我赢得游戏的概率是我击中对手的概率。

[段 41]

And if I wait and take a step then my probability of winning the game is the probability that he misses me tomorrow Right So I have to compare winning probabilities with winning probabilities I have to compare apples with apples not apples with oranges Everyone see that Okay, so let’s put that up. Right? So the same assumption, assuming no one has thrown, if I know at D that J will shoot, will shoot tomorrow at D minus 1, then I should shoot if, I need a gap here, if I’s probability of hitting at B, and let me leave a gap here, is bigger than, let’s make it bigger than or equal to, it doesn’t really matter about the equal case, It’s greater than or equal to J’s probability of missing tomorrow. Because this is the probability that you’ll win if you throw, and this is the probability that you’ll win if you wait. Okay? So let’s put in what those things are. if you throw, and this is the probability that you’ll win if you wait. Okay? Let’s put in what those things are. So the probability that I will hit at distance d, that’s not that hard, that’s PID. Everyone happy with that? What’s the probability that J will miss tomorrow if J throws? What’s the probability that J will miss? Somebody? How did it happen?

[译文 41]

如果我等待并走一步，那么我赢得游戏的概率是他明天失误的概率。对，所以我必须用获胜概率和获胜概率比较，我必须拿苹果和苹果比而不是苹果和橘子比。大家都明白吗？好的，让我们把它写出来。对吗？所以同样的假设，假设还没有人扔过，如果我在D处知道J会开枪，会在明天的D减1处开枪，那么我应该开枪如果，我需要留个空，如果I在B点的命中率，让我留个空，比……大于或等于，明天的……大于或等于J明天失误的概率。因为这是如果你扔你会赢的概率，而这是如果你等待你会赢的概率。好吗？让我们写出这些东西是什么。如果你扔，而这是如果你等待你会赢的概率。好吗？让我们写出这些东西是什么。所以我在距离d时命中的概率，这不太难，是pID。大家都明白吗？如果J扔，他明天失误的概率是多少？J失误的概率是多少？有人吗？是怎么算的？

[段 42]

Yep. Right. 1, let’s be careful. So it’s 1 minus pj, but what distance will they be at? d minus 1. So it’s 1 minus pj d minus 1. All right? All right. So this is the key rule. So, if Chebby knows that Patrick’s going to shoot tomorrow, then Chebby should shoot if his probability of hitting PID is bigger than Patrick probability of missing 1 minus P Patrick D minus 1 All right Now I want to do one piece of math This is the only math in this proof All right so everyone who math which I know there a lot of you can you just hold on to your seats don panic there a little bit of math coming okay This is the math. I want to add, I want to add pj d minus 1 to both sides of this inequality. All right, that’s it, okay? All right, so what does that tell me? if I add pj d minus 1 to this side, I get plus pj d minus 1. Everyone happy with that? And on the other side, if I add pj d minus 1, I get just 1. Everyone happy with that? All right. So here’s our rule.

[译文 42]

是的。对。1，让我们仔细一点。所以是1减pj，但他们会在什么距离？d减1。所以是1减pj，d减1。对吧？对的。所以这是关键规则。所以，如果Chebby知道Patrick明天要开枪，那么Chebby应该开枪如果他的命中率pID大于Patrick失误的概率1减P Patrick d减1。好的，现在我想做一点数学。这是这个证明中唯一的数学。好的，所以每个会数学的人，我知道你们很多人都懂，你们能坐稳不要慌吗？一点数学要来了好的这是数学。我想加，我想在这个不等式两边都加上pj d减1。好的，就这样，好吗？好的，所以这告诉我什么？如果我在这边加上pj d减1，我得到加上pj d减1。大家都明白吗？在另一边，如果我加上pj d减1，我得到1。大家都明白吗？好的。这就是我们的规则。

[段 43]

Our rule is, let’s flip it around. if Patrick hasn’t thrown yet and thinks that Chevy’s going to shoot tomorrow then Patrick should shoot now if his probability of hitting now plus Chevy’s probability of hitting tomorrow is bigger than once right let’s call this star let’s call this star and let’s put this stuff up somewhere where we can use it for future reference alright sorry guys I use it again in a minute I know you feeling self up there Believe me I self up here too All right Okay so let look at that star inequality up there Now, way out here, is that star inequality met or not met? It’s not met. It’s not met, right? Because way out here, these two probabilities are small, so the sum is less than one. and in here is the star inequality met or not met? Let me pick on you guys. Patrick, is it met or not met in here? Sheldon, do you have a microphone? It’s met. It’s met, thank you. Okay, alright. So in here the inequality is met. The sum is bigger than one and out here it’s less than one. If we put in all the steps here here they are getting closer and closer together. Here’s the steps. They get closer and closer together. We put these steps in, there’s going to be some step where, for the first time, the star inequality is met.

[译文 43]

我们的规则是，让我们反过来。如果帕特里克还没有出手，并且认为雪佛兰明天会出手，那么帕特里克现在就应该出手，如果他现在命中的概率加上雪佛兰明天命中的概率大于一。好的，让我们把这个称为星号，让我们把这个称为星号，让我们把这个放在某个地方，这样我们以后可以参考。好了，抱歉各位，我稍后还会再用到它。我知道你们感觉很好。相信我，我也感觉很好。好吧，好吧，让我们看看那个星号不等式。在这边很远的地方，星号不等式是满足还是不满足？它不满足。它不满足，对吧？因为在这边很远的地方，这两个概率都很小，所以和小于一。而在这边，星号不等式是满足还是不满足？让我点你们来回答。帕特里克，在这里是满足还是不满足？谢尔顿，你有麦克风吗？满足。满足，谢谢。好吧，好吧。所以在这种情况下不等式是满足的。和大于一，而在这边小于一。如果我们在所有这些步骤中插入，它们越来越接近。这是步骤。它们越来越接近。我们把这些步骤放进去，会有某个步骤，第一次满足星号不等式。

[段 44]

So notice that they start out here. They get closer and closer. It’s not met, not met, not met, not met, not met, not met, not met, and suddenly it’s going to be met. Maybe around here. Let’s just try and pick it out. Maybe it here So this might be the first time that this star inequality is met Let call it d star Everyone understand what d star is At every one of these steps to the right of d star when we take the sum of the probability at id plus pj d minus 1, we get something less than 1. But to the left of d star, or closer in than d star, the game is proceeding this way, it gets between right to left here, they’re getting closer and closer. To the left of d star, the sum of those probabilities is bigger than 1. right, so again, d star is the first step at which the sum of those two probabilities exceeds one right, everyone okay about what d star is, anyone want to ask me a question okay, people should feel free to ask questions on this, I want to make sure everyone’s following is everyone following so far, yeah, I need to see all your eyes you don’t look like you’re stuck in the headlamps like you were on Monday, alright, we’re better off than we were on Monday good, okay, okay, so here’s our picture and I actually want a bit more space.

[译文 44]

注意它们从这里开始，越来越接近。不满足，不满足，不满足，不满足，不满足，不满足，不满足，然后突然就满足了。大概在这里。让我们试着找出来。也许在这里。这可能是第一次满足这个星号不等式。让我们称之为d星。大家都明白d星是什么吗？在d星右边每一个步骤，当我们取id处的概率加上pj在d减1处的概率之和，我们得到的结果小于一。但在d星的左边，或者比d星更近的地方，游戏是这样进行的，它从右到左进行，它们越来越接近。在d星的左边，这些概率之和大于一。对，所以同样，d星是这两个概率之和第一次超过一的步骤。大家都明白d星是什么，有人想问问题吗？好的，人们应该自由提问，我确保每个人都在跟上。大家都跟上了吗？是的，我需要看到你们所有人的眼睛，你们看起来不像周一那样困在车灯里，好的，我们比周一好。好吧，好吧，这是我们的图，我实际上想要更多一点的空间。

[段 45]

I’m not going to have much more space. All right. All right, so now I’m going to tell you the solution. The solution to this game is this. I claim that the first shot… should occur at D star. That’s my claim. The first shot should occur at D star. No one should shoot until you get to D star, and whoever’s turn it is, whether it’s Chevy’s turn or Patrick’s turn at D star, that person should shoot. That’s my claim, and that’s what we’re going to prove. Okay, everyone understand the claim? It says, nobody shoots, nobody shoots, nobody shoots, nobody shoots, nobody shoots, nobody shoots, shoot. Okay? Okay? All right, let’s prove it. Everyone ready to prove it? Yeah? People to be awake. If your neighbor’s not awake, nudge them hard. All right, good. All right, let’s start this analysis way out here. Well, let’s start way out here, miles apart. All right, these guys are miles apart. Now I want to use those props So I got these guys let put them say we are Patrick but stand up and they be over here somewhere but that black line is So maybe they even further than this they really far apart And here they are miles away and let say it Shebby turn It’s Shebby’s turn, he’s way out here. That’s the first step of the game.

[译文 45]

我不会有多少空间了。好的。好的，现在我来告诉你们这个游戏的解法。这个游戏的解法是这样的。我声称第一枪应该在d星发生。这是我的声称。第一枪应该在d星发生。在到达d星之前没有人应该出手，无论轮到谁的回合，无论是雪佛兰的回合还是帕特里克的回合在d星，那个人就应该出手。这是我的声称，这就是我们要证明的。大家都明白这个声称吗？它说的是，没有人出手，没有人出手，没有人出手，没有人出手，没有人出手，没有人出手，然后出手。好吗？好吗？好的，让我们来证明它。大家都准备好证明了吗？是的？人们要保持清醒。如果你的邻居没醒，用力推他们一下。好的，好的，让我们从很远的地方开始这个分析。好的，让我们从很远的地方开始，遥遥相距。好的，这些家伙遥遥相距。现在我想用那些道具，我让这些家伙站起来，他们在这里某处，但那条黑线是，所以也许他们比这更远，他们真的很远。在这里他们遥遥相距，让我说轮到谢比了。轮到谢比了，他在这里远远的地方。那是游戏的第一步。

[段 46]

Imagine it’s even further, because it was even further. And let’s think through what should be going on in Shebby’s head. There are two possible things going on. He’s going to think about what Patrick’s going to do. So this is Shebby’s turn, here he is. and he should think tomorrow it’s going to be Patrick’s turn and there’s two possibilities one possibility is that Patrick is not going to shoot tomorrow and if we think that Patrick is not going to shoot tomorrow which fact should Shebby use? should he use fact A or fact B? so which fact do you use? fact A using fact A he should not shoot All right? Alternatively, he could think that Patrick’s going to shoot tomorrow. Is that right? He could think Patrick’s going to shoot tomorrow. If he thinks Patrick’s going to shoot tomorrow, which factor does he use? B. B. He uses factor B, in which case he has to look at this inequality up here.

[译文 46]

想象它甚至更远，因为它确实更远。让我们想想谢比脑子里应该在想什么。有两种可能的情况。他会考虑帕特里克将要做什么。这是谢比的回合，他在这里。他应该想明天将是帕特里克的回合，有两种可能。一种可能是帕特里克明天不打算出手，如果我们认为帕特里克明天不打算出手，谢比应该用事实A还是事实B？那么他用哪个事实？事实A，用事实A他不应该出手。好的，或者另一方面，他可能认为帕特里克明天会出手。对吗？他可能认为帕特里克明天会出手。如果他认为帕特里克明天会出手，他用哪个因素？B。B。他用因素B，在这种情况下他必须看这里这个不等式。

[段 47]

Look at this inequality and say I shoot if my probability of hitting now plus his probability of hitting tomorrow is bigger than 1 Well let have a look This is Patrick’s probability of hitting today, and this is Shebby’s probability of hitting today, and this is Patrick’s probability of hitting tomorrow, and is the sum of them bigger than 1 or not? is it bigger than one or not? it’s not bigger than one, so what should Shebby do? he should step right, he should step so he’d step now it’s Patrick’s turn and once again, imagine this distance is still pretty large and there’s two things Patrick could think Patrick could think that Shebby’s not going to shoot tomorrow here’s Patrick he’s looking forward to Shebby tomorrow and he could think that Patrick, he could think that Shebby’s not going to shoot tomorrow. If he thinks Shebby’s not going to shoot tomorrow, which fact should he use? A. Should use fact A, okay? And if he’s using fact A, he should not shoot. Right? Alternatively, he could think that Shebby is going to shoot tomorrow, in which case he uses fact… B. B. Alright and what did he do He adds up his probability Patrick probability of hitting today plus Shebby probability of hitting tomorrow he asks is that sum bigger than one and he concludes he concludes he says no.

[译文 47]

看这个不等式，然后说如果我现在命中的概率加上他明天命中的概率大于一，我就出手。好的，让我们看看。这是帕特里克今天命中的概率，这是谢比今天命中的概率，这是帕特里克明天命中的概率，它们的和大于一还是不大于一？它大于一还是不大于一？它不大于一，那么谢比应该做什么？他应该向右迈步，他应该迈步。所以他现在迈步，轮到帕特里克了，再想象这个距离仍然相当大，帕特里克可能想到两件事。帕特里克可能认为谢比明天不打算出手。帕特里克在这里，他展望明天谢比，他可能认为帕特里克可能认为谢比明天不打算出手。如果他认为谢比明天不打算出手，他应该用哪个事实？A。应该用事实A，好的？如果他用事实A，他不应该出手。对吗？或者另一方面，他可能认为谢比明天会出手，在这种情况下他用事实……B。B。好的，他做了什么，他把他的概率加起来，帕特里克今天命中的概率加上谢比明天命中的概率，他问那个和是否大于一，他得出结论他得出结论他说不。

[段 48]

So we have no shot here, and no shot here. And notice that both of those arguments were dominance arguments. In each case, whether Shebby thought that Patrick was going to shoot was going to shoot tomorrow or not, in either case, he concluded he should not shoot today. Right? And it was Patrick’s turn, whether Patrick thought that Chevy was going to shoot tomorrow or not, in either case, he concluded he should not shoot today. So he takes a step forward. All right? And this argument continues, it’ll be Chevy’s turn next, and once again, he’ll look at these two possibilities, if he thinks Patrick’s not shooting tomorrow, he wants to step, if he thinks Patrick is going to shoot tomorrow, he’s again going I’m going to want to step the way it’s drawn, and once again we’ll conclude step. Alright? And we’ll go on doing this argument, and everyone see that in each case, this dominance argument will apply. It won’t matter whether I think you should shoot tomorrow or not, in either case, it’ll turn out that I should step forward, whether fact A applies, or whether fact B applies. Alright? So we’ll go on going forward, and we’ll have no shot, no shot, no shot, no shot, no shot. Whether fact A applies or whether fact B applies. So we’ll go on going forward and we’ll have no shot, no shot, no shot, no shot, no shot.

[译文 48]

所以我们这里没有出手，这里也没有出手。注意这两个论证都是占优论证。在每种情况下，无论谢比认为帕特里克明天会出手还是不会出手，在任何一种情况下，他都得出结论他今天不应该出手。对吗？轮到帕特里克了，无论帕特里克认为雪佛兰明天会出手还是不会出手，在任何一种情况下，他都得出结论他今天不应该出手。所以他向前迈一步。好的？这个论证会继续下去，下一个轮到雪佛兰了，再一次，他会看这两种可能，如果他认为帕特里克明天不出手，他想迈步，如果他认为帕特里克明天会出手，他再次会想以我所画的方式迈步，再一次我们会得出迈步的结论。好的？我们会继续做这个论证，大家看到在每种情况下，这个占优论证都会适用。无论我认为你明天应该出手还是不应该出手，在任何一种情况下，结果都会是我应该向前迈步，无论事实A适用还是事实B适用。好的？我们会继续向前，我们会没有出手，没有出手，没有出手，没有出手，没有出手。无论事实A适用还是事实B适用。所以我们会继续向前，我们会没有出手，没有出手，没有出手，没有出手，没有出手。

[段 49]

And we’ll arrive at D star. 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1. So it turns out that D star is going to be Chebet’s turn again. And at D star we try exactly the same reasoning. at d star he says if I think Patrick is not going to shoot tomorrow what should I do? what should I do if I think Patrick is not going to shoot tomorrow? not shoot but now something different occurs now he says if I think Patrick is going to shoot tomorrow if I think Patrick is going to shoot tomorrow then when I look at my inequality up there my star inequality and add up my probability of hitting today which is this line here plus Patrick’s probability of hitting tomorrow, which is this line here, suddenly he finds it is bigger than 1. He finds it is bigger than 1. So now, if Shebby thinks that Patrick is going to shoot tomorrow, what should Shebby do He should shoot He should shoot So up until this point a dominance argument has told us no one should shoot But suddenly we have a dilemma The dilemma is, if Chevy thinks Patrick’s not shooting, he should step, and if Chevy thinks Patrick is shooting, he should shoot. Everyone with me so far? So what have we shown so far?

[译文 49]

我们会到达 D 星。2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1。因此 D 星又轮到 Chebet 出击。在 D 星我们用完全相同的推理。他说如果我认为 Patrick 明天不会射击，我该怎么办？如果我认为 Patrick 明天不会射击，我该怎么办？不射击，但现在出现了不同的情况，现在他说如果我认为 Patrick 明天会射击，如果我认为 Patrick 明天会射击，当我查看我上面的星号不等式，把我今天击中的概率相加，即这条线，加上 Patrick 明天击中的概率，即这条线，他突然发现这大于 1。他发现这大于 1。所以现在，如果 Shebby 认为 Patrick 明天会射击，Shebby 应该怎么做？他应该射击。他应该射击。到目前为止，优势策略论证告诉我们没有人应该射击。但突然我们陷入了困境。困境是，如果 Chevy 认为 Patrick 不会射击，他应该不出击，而如果 Chevy 认为 Patrick 会射击，他应该射击。大家都跟上了吗？那么到目前为止我们证明了什么？

[段 50]

We’ve shown that no one should shoot until D-star, but we’re stuck, because we don’t know what to do at D-star, because we don’t know what Chevy should believe at D-star. We don’t know whether Shebby should believe that Patrick’s going to shoot or whether Shebby should believe that Patrick’s not going to shoot. So how do we figure out what Shebby should believe Patrick’s going to do? What? Wait, wait, wait, wait, wait. Wake up the guy in orange there. Right? The guy with the ginger hair, that’s right. What’s the answer to that question? Good, the answer is this backward induction, right? Good. Round of applause for remembering the answer. Alright, good, good. Backward induction is the answer to all questions, especially when you’re asleep, right? Okay, so now we’re gonna use backward induction. But where does backward induction start here? Backward induction starts at the back of the game, and what the back of the game here The back of the game is where It when these two guys neither of them have thrown their sponge and they’ve reached here. All right, so come in a second. Step, step, step, step, step, step, step, step. All right, and let’s just say it’s Patrick’s turn, and they’re absurdly close. They’re uncomfortably close. All right, if they had longer noses, they’d be touching, right? They’re at distance zero. They’re at distance zero.

[译文 50]

我们已经证明在 D 星之前没有人应该射击，但我们陷入了困境，因为我们不知道在 D 星应该怎么做，因为我们不知道 Chevy 在 D 星应该相信什么。我们不知道 Shebby 应该相信 Patrick 会射击，还是应该相信 Patrick 不会射击。那么我们如何弄清楚 Shebby 应该相信 Patrick 会做什么？什么？等等，等等，等等。叫醒那边穿橙色衣服的人好吗？对，就是那个红头发的人。正确答案是什么？好的，答案是逆向归纳，对吧？好的，为记住答案鼓个掌。好的，好的。逆向归纳是所有问题的答案，尤其是在你睡着的时候，对吧？好的，现在我们要用逆向归纳。但是逆向归纳从哪里开始呢？逆向归纳从游戏的末尾开始，这里的游戏末尾是什么？游戏的末尾是当这两个人都没有投出他们的海绵，他们已经到达这里。好，让我们进来一下。出击，出击，出击，出击，出击，出击，出击，出击。好的，我们就说轮到 Patrick 了，他们离得非常近。近得令人不安。好，如果他们的鼻子再长一点，就会碰到了，对吧？他们处于零距离。他们处于零距离。

[段 51]

And at distance zero, at d equals zero, let’s suppose it’s Patrick’s turn, right? At d equals zero, no one is shot, it’s Patrick’s turn, he’s got the sponge what should Patrick do? Shout it out Patrick. He should shoot. Patrick should shoot, right? At d equals zero, say it’s two’s turn, and the answer is he should shoot because the probability of hitting at distance zero is one. Is one. Alright? Let’s just move you to the side a bit so people can see the board. I know it’s an awkward dance, but here you are, right? OK, that’s good, that’s good, that’s good, that’s good. OK. So at distance 0, they should certainly shoot. All right? So now let’s go back a step in the backward induction. So at distance 1, let’s take a step back. All right. So take a step back. It’s Shebi’s turn at distance 1. So it Shebi turn And what does Shebi know What does Shebi know at distance one Sheb what do you know Shout it out That Patrick will shoot next turn Right. So now Shebby knows that Patrick’s going to shoot tomorrow. So which fact should Shebby use in deciding whether he should shoot today? B. B. He should use fact B. And that tells us, right, so one knows that 2 will shoot tomorrow.

[译文 51]

在零距离，在 d 等于零时，假设轮到 Patrick，对吧？在 d 等于零时，没有人被击中，轮到 Patrick 了，他拿着海绵，Patrick 应该怎么做？大声喊出来 Patrick。他应该射击。Patrick 应该射击，对吧？在 d 等于零时，假设轮到二号，答案是他应该射击，因为在零距离击中的概率是 1。是 1。好吗？让我们把你移到一边一点，这样大家能看到黑板。我知道这很尴尬，但你就这样站着，好吧？好的，这样好，这样好，这样好，这样好。好的。所以在零距离，他们当然应该射击。好吗？现在让我们在逆向归纳中后退一步。所以在距离 1，让我们退一步。好。退一步。在距离 1 轮到 Shebi。所以轮到 Shebi 了，Shebi 知道什么？Shebi 在距离 1 时知道什么？Sheb，大声喊出来。Patrick 下回合会射击，对吧。所以现在 Shebby 知道 Patrick 明天会射击。那么 Shebby 在决定今天是否应该射击时应该使用哪个事实？B。B。他应该使用事实 B。这告诉我们，对吧，所以 1 知道 2 明天会射击。

[段 52]

So, by B, by B, 1 should shoot if his probability of hitting at distance 1 plus Patrick’s probability of hitting at distance 0, if that is bigger than 1. well is it bigger than one all right let’s have a look we we had a shot in here already we put a shot in here and we’re looking at distance one all right and we’re looking at this distance here plus this distance here is it bigger than one yeah it’s bigger than one here’s a bit more math I lied to you before one plus something is bigger than one okay Here, plus this distance here, is it bigger than 1? Yeah, it’s bigger than 1. Here’s a bit more math, I lied to you before. 1 plus something is bigger than 1, okay? All right? All right? So this is bigger than 1, so shoot. Let’s put it on our chart and shoot. All right? Let’s go back a step. Sorry, I’ll have to have Chevy go, we’re going backwards. Right, now we’re at distance what? We’re at distance 2.

[译文 52]

所以，根据 B，根据 B，1 应该射击，如果他在距离 1 击中的概率加上 Patrick 在距离 0 击中的概率，如果这大于 1。好的，它大于 1 吗？让我们看一下，我们在这里已经投过一次了，我们在这里投了一次，我们看距离 1，好的，我们看这段距离加上这段距离，它大于 1 吗？是的，它大于 1。这里有更多数学，我之前对你撒了谎。1 加某个数大于 1，好的。这里，加上这段距离，它大于 1 吗？是的，它大于 1。这里有更多数学，我之前对你撒了谎。1 加某个数大于 1，好吗？好的。所以这大于 1，所以射击。让我们把它写在图表上然后射击。好吗？让我们后退一步。对不起，我得让 Chevy 走，我们要往回走了。对，现在我们在距离几？我们现在在距离 2。

[段 53]

We’re at d equals 2. and it’s Patrick’s turn 2’s turn and what does Patrick know? shout it out Patrick I know that Chevy’s going to shoot next turn Patrick knows that Chevy’s going to shoot next turn so Patrick therefore should use which fact? fact B and fact B tells him that he should put this in so 2 knows that 1 will shoot tomorrow so by B it’s all the same thing we know that 2 should shoot if p2 of 2 plus p1 of 1 is bigger than 1 So if we look at it on the board here we are at 2 turn he looking at this distance plus this distance and is it bigger than 1 It is bigger than 1, so he should shoot. And we can go on doing this argument backwards. We’ll find that Shebby should shoot here, because this plus this is bigger than 1, and we’ll know that here, Patrick, once again, will know that Chevy’s going to shoot tomorrow, so he should use fact B, so he should shoot, provided this plus this is bigger than 1, but it is. And now we’re back at D star, and the question we had at D star, the question we’d left hanging at D star was what? At D star, we knew already that Chevy would not shoot if he thought Patrick was not going to shoot tomorrow, but he should shoot if he thinks Patrick is going to shoot tomorrow.

[译文 53]

我们在 d 等于 2。轮到 Patrick，轮到 2 号了，Patrick 知道什么？大声喊出来 Patrick。我知道 Chevy 下回合会射击。Patrick 知道 Chevy 下回合会射击，所以 Patrick 因此应该使用哪个事实？事实 B，事实 B 告诉他他应该把这个放进去，所以 2 知道 1 明天会射击，所以根据 B，同样的事情，我们知道 2 应该射击如果 p2 的 2 加上 p1 的 1 大于 1。所以如果我们在黑板上看，我们在 2 的回合，他看着这段距离加上这段距离，它大于 1 吗？它大于 1，所以他应该射击。我们可以一直用这个论证往回推。我们会发现 Shebby 应该在这里射击，因为这个加上这个大于 1，我们会知道在这里，Patrick 再一次，会知道 Chevy 明天会射击，所以他应该使用事实 B，所以他应该射击，只要这个加上这个大于 1，但它确实大于 1。现在我们回到 D 星，我们在 D 星留下的问题是 D 星时，我们已经知道如果 Chevy 认为 Patrick 明天不会射击，他不会射击，但如果他认为 Patrick 明天会射击，他应该射击。

[段 54]

But what does Chevy know at D star? I know Patrick’s going to shoot. He knows Patrick’s going to shoot, so he should shoot. Is that right? He knows Patrick’s going to shoot by backward induction, so he should shoot. So we just solved it. What did we actually show? We showed, I’ll have to keep you up here, we know that prior to d star, no one will shoot, not shoot. And we know that at d star and in fact at any point further on we should shoot That horrible writing but it says shoot All right So we shown more than we claimed We claimed that the first shot should occur at d star, but we’ve actually shown more than that. We’ve shown that even if you went beyond d star, and if somebody had forgotten to shoot at d star, at least you should shoot now. And give me like two more minutes, or three more minutes to finish up this. This is up because we’re at a high point now, okay? All right, everyone okay to wait a couple minutes? Okay. Okay. So, what do we prove here? We proved that the first shot occurs at D star whoever’s turn it is at D star, whoever’s turn it is at D star. It wasn’t that the best guy, the better shot should shoot first, or the worst shot should shoot first.

[译文 54]

但 Chevy 在 D 星知道什么？我知道 Patrick 会射击。他知道 Patrick 会射击，所以他应该射击。对吗？他通过逆向归纳知道 Patrick 会射击，所以他应该射击。所以我们解决了。实际证明了什么？我们证明了，我得让你留在这里，我们知道在 D 星之前，没有人会射击，不会射击。我们知道在 D 星及之后的任何时候都应该射击。那字迹很糟糕但写的是射击，好的。我们证明的比我们声称的更多。我们声称第一枪应该发生在 D 星，但我们实际上证明的更多。我们证明了即使你走到 D 星之后，如果有人在 D 星忘记了射击，至少现在你应该射击。再给我大约两分钟或三分钟来完成这个。我们现在处于高潮，好吗？好的，大家都准备好等几分钟吗？好的。好的。那么，我们证明了什么？我们证明了第一枪发生在 D 星，无论轮到谁在 D 星出击，无论轮到谁在 D 星出击。不是那个最好的射手，应该最好的射手先射击，或者最差的射手先射击。

[段 55]

It turned out that given their abilities, there was a critical distance at which they should shoot. If you go back to the 18th century military strategy, you should shoot when you see the whites of their eyes, which is at D star. All right? All right? But we learned something else on the way. I claim we learned that if you’re patient and you go through things carefully, that the arguments we’ve learned in the course so far, dominance arguments and backward induction arguments, can solve out a really quite hard problem This is hard Right It would be useful for the guy in War and Peace or in J or the guy cycling in the Tour de France or you guys, your funders, to know this, and we can solve this exactly using backward induction. And everyone in the room can do it. Let me just push the argument a tiny bit further. One thing we’ve always asked in this class is, OK, that’s fine if everyone knows what’s going on in the game. Here we have our smart Yale football players, and they know how to play this game, so they’re going to shoot at the right time. But what happens if instead of playing another smart Yale football player, they’re playing some uneducated, probably simple-minded football player from, say, Harvard? All right? Now that changes things a bit, doesn’t it?

[译文 55]

结果表明，根据他们的能力，有一个关键的射击距离。回顾18世纪的军事战略，你应该在看到敌人的眼白时射击，那就是在D星位置。好的？好的？但我们在过程中还学到了别的东西。我主张我们学到的是，如果你有耐心并且仔细地处理事情，我们到目前为止在课程中学到的论证——占优论证和逆向归纳论证——可以解决一个相当困难的问题。这很难，对吧。它对《战争与和平》中的人物或者环法自行车赛的车手或者你们这些出资人来说是有用的，我们可以用逆向归纳精确地解决这个问题。房间里的每个人都可以做到。让我把论证再稍微推进一点。我们在这门课上一直问的一个问题是，如果每个人都知道游戏里发生了什么，那没问题。这里我们有聪明的耶鲁橄榄球运动员，他们知道怎么玩这个游戏，所以他们会在正确的时间射击。但如果他们不是和另一个聪明的耶鲁橄榄球运动员比赛，而是和一个没受过教育的、可能头脑简单的橄榄球运动员比赛，比如哈佛的，会怎么样呢？好的？现在这改变了一些事情，不是吗？

[段 56]

Because we know that the Yale football player is sophisticated, has taken my class, and knows that he should shoot at D-star. But the Harvard guy doesn’t know anything anymore, right? So they’re stuck. So if you’re the Yale guy playing the Harvard guy, how does that change your decision? Should you shoot earlier than D-star when you’re playing against the Harvard guy, or later than D-star when you’re playing against the Harvard guy? Let’s try our Yale guy and see what they think. What do you think, shall we? Microphone up. Definitely not earlier. Definitely not earlier. Yeah, that’s. when you’re playing against the Harvard guy. Let’s try our Yale guy and see what they think. What do you think, Shobie? Microphone up. Definitely not earlier. Definitely not earlier. That’s the key thing, right? Now, why? Why definitely not earlier? Because you… If you miss, the other person has a probability of one. All right. You have a higher chance of missing. All right. All right. So I claim it’s definitely… Shobie’s right. It’s good because I’m just playing the Yale football players and it’s sophisticated. So he’s right, that even if you’re playing against a Harvard guy, you shouldn’t shoot before D-star because it was a dominant strategy not to shoot before D-star. It doesn’t matter whether you think the Harvard guy is going to be dumb enough to shoot early or not.

[译文 56]

因为我们知道耶鲁橄榄球运动员是老练的，上过我的课，知道他应该在D星射击。但哈佛的家伙什么都不懂了，对吧？所以他们陷入僵局了。所以如果你是耶鲁的家伙对哈佛的家伙，这如何改变你的决定？当你和哈佛的家伙比赛时，你应该比D星更早射击，还是比D星更晚射击？让我们试试我们的耶鲁家伙看看他们怎么想。你怎么想，Shobie？麦克风递上来。绝不应该更早。绝不应该更早。是的，那就是当你和哈佛的家伙比赛时。让我们试试我们的耶鲁家伙看看他们怎么想。你怎么想，Shobie？麦克风递上来。绝不应该更早。绝不应该更早。这就是关键，对吧？为什么绝不应该更早？因为你……如果你没打中，另一个人有概率为1。好的。你错过的机会更高。好的。好的。所以我主张绝对是……Shobie是对的，这很好，因为我只是在和耶鲁橄榄球运动员们玩，这很老练。所以他是对的，即使你在和一个哈佛家伙比赛，你也不应该在D星之前射击，因为不在D星之前射击是一个占优策略。你认为哈佛家伙是否会笨到提前射击并不重要。

[段 57]

If he is dumb enough to shoot early, so much the better. You should wait until D-star. Notice, that argument doesn’t depend on you playing against somebody who’s sophisticated, or someone who’s less sophisticated, like a Harvard football player, or somebody who’s basically a chair, like a Harvard football player. Right? Right? Right? You shouldn’t shoot before D star because it’s a dominant strategy not to shoot before D star. Now you might want to wait a little to see if they’re not going to shoot early. To see if they’re not going to shoot. But you certainly shouldn’t shoot early. And let me finish with one other thing. Every time when we play this game in class, whether it’s here or up in SOM, people shoot too early. People shoot too early. They miss. Right? You can do a kind of econometrics on this. you could figure out that on average, on average abilities, I’m sometimes getting the better shots, sometimes getting the worst shots, on average I should see people hitting about half of the time over a large sample But here I tend to see people miss as we did today almost all the time Why do we see so many misses Why do we see so many misses So one problem may be that people are just overconfident They’re overconfident on their ability to thrive. There’s a large literature in economics about how people tend to be overconfident.

[译文 57]

如果他笨到提前射击，那就更好了。你应该等到D星。注意，这个论证不取决于你是在和一个老练的人还是一个不太老练的人比赛，比如哈佛橄榄球运动员，或者一个基本上就是摆设的人，比如哈佛橄榄球运动员。对吧？对吧？对吧？你不应该在D星之前射击，因为不在D星之前射击是一个占优策略。现在你可能想等一会儿看看他们是否不会提前射击。看看他们是否不会射击。但你当然不应该提前射击。让我用另一件事来结束。每次我们在课堂上玩这个游戏，无论是在这里还是在SOM，人们都射击得太早了。人们射击得太早了。他们错过了。对吧？你可以做某种计量经济学分析，你可以计算出，平均而言，在平均能力下，我有时会得到更好的射击，有时会得到最差的射击，平均而言我应该看到人们在大样本中大约有一半的时间命中。但在这里，我倾向于看到人们几乎总是错过，就像我们今天所做的。为什么我们看到这么多失误？为什么我们看到这么多失误？一个可能的问题是，人们只是过于自信。他们对自己的能力过于自信。经济学中有一个关于人们如何倾向于过于自信的大量文献。

[段 58]

But there’s another possible explanation, and let me just push it past you as the last thing for today. I think Americans, I think doesn’t go for the Brits. Americans have what I call a proactive bias. You guys are brought up since you’re in kindergarten, maybe before, and you’re told you have to be proactive, you have to make the world come to you, right? And my evidence for this is based on sophisticated empirical work watching SportsCenter. So on SportsCenter, when they interview these sweaty athletes after the game, the sweaty athletes say, it’s great, I now control my own destiny? Well I’m a Brit, I think controlling my own destiny sounds kind of scary to me, it’s not a good thing at all. In fact, if I wanted to control my own destiny, I wouldn’t have got married. But, the point I want to make is this, every time I play this game and I ask people why they shoot early, I hear the same thing and it’s evidence this proactive bias. People say, well, at least I went down swinging. And the problem is, in life, the aim in life is not to go down swinging, it’s not to go down. All right? So one lesson to get from this lecture is, sometimes waiting is a good strategy. All right, well, we’ll come back to it on Monday.

[译文 58]

但还有另一个可能的解释，让我把它作为今天的最后一点推给你们。我认为美国人，我认为这不适用于英国人。美国人有我所说的主动偏见。你们从小就在幼儿园，也许更早，就被告知你必须主动，你必须让世界向你靠拢，对吧？对此我的证据基于观察SportsCenter的复杂实证工作。所以在SportsCenter上，当他们赛后采访这些汗流浃背的运动员时，这些汗流浃背的运动员说，太棒了，我现在掌控自己的命运？嗯，我是个英国人，我认为掌控自己的命运听起来对我来说有点可怕，这根本不是一件好事。事实上，如果我想掌控自己的命运，我就不会结婚了。但是，我想说的重点是，每次我玩这个游戏问人们为什么他们提前射击，我都听到同样的事情，这就是这种主动偏见的证据。人们说，嗯，至少我是在战斗中倒下的。问题是，在生活中，生活的目标不是战斗中倒下，不是倒下。对吧？所以从这个讲座中可以学到的一个教训是，有时候等待是一个好的策略。好的，我们星期一再来讨论。