耶鲁大学博弈论公开课 - 第5集 战争的消耗

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]

Last time we looked at how to apply our new idea, sub-game perfect equilibrium, to a whole bunch of games. And our general idea for how to solve for sub-game perfect equilibrium was as follows. We looked at each sub-game. We solved for the Nash equilibrium in the sub-game, something we learned to do long ago. And then we rolled back the payoffs. We rolled them back up the tree. and towards the end we learned something interesting I’m not going to go back to it today I just want to emphasize it we learned that strategic effects matter so in that investment game we looked at last time when you were considering whether to rent a new piece of machinery it made a very big difference whether you considered how this action would affect the actions of the other side in this case how it affects your competition and this is a very general idea very general point So just to give you a couple of more examples, when you’re designing tax systems, I mentioned this last time, when you’re designing a tax system, make some changes in the U.S. tax system, it’s not good enough to look at how people are behaving in the old tax system and just calculate in an accounting manner how much more money you’re going to raise or how much money it’s going to cost you.

[译文 1]

上次我们研究了如何将我们的新概念——子-game perfect equilibrium（子博弈完美均衡）——应用到一大堆游戏中。我们求解子博弈完美均衡的一般思路如下：审视每个子博弈，在子博弈中求解Nash equilibrium（纳什均衡），这是我们很久以前就学会做的事情。然后我们回推payoffs（收益），沿着树向上回推。在接近尾声时，我们学到了一些有趣的东西——我今天不打算回头再讲它，我只想强调这一点——我们学到strategic effects matter（战略效应很重要）。所以在我们上次看的那个投资游戏中，当你考虑是否rent（租用）一台新机器时，你是否考虑到这个行动会如何影响对方的行动——在这个例子中如何影响你的competition（竞争者）——这一点造成了很大的差异。这是一个非常general idea（普遍的想法），非常general point（普遍的要点）。所以我再给你们举几个例子，当你在设计tax systems（税制）时，我上次提到过，当你在设计税制时，对U.S.（美国）税制做一些改变，仅仅看人们在old tax system（旧税制）中的行为，然后用会计方式计算你会多raise（征收）多少钱或者会cost（花费）你多少钱是不够的。

[段 2]

You have to take into account how that’s going to lead to changes in behavior. And once again, that’s a strategic effect. And in the homework that you’re handing in today, all of you would have had a nice example of that in the toll booth problem.A nice example of that in the toll booth problem. So in the toll booth problem, when you’re putting tolls on roads, or more generally, when you’re building new roads, new bridges, new flyovers, new bypasses, you need to take into account how those new tolls, how those new roads, will affect all of traffic flow. Traffic flow down the tree will form a new equilibrium, and you need to consider that in designing your tolls and designing your road system. All right? So that’s another example of SPE. So today I want to do something quite different, a little bit like what we did with Duel. I want to play a game today and probably spend the whole of today analyzing this one game. So it’s quite a complicated game, but it’s quite a fun game. All right, so what’s the game we’re going to look at? All right. The game’s going to involve two players. and in each player, in each period, each period, they choose, or each chooses, I should say, each chooses, each chooses whether to fight or to quit.

[译文 2]

你必须考虑这将如何导致行为的变化。而再一次，这是一个战略效应。在你们今天提交的作业中，你们所有人都会在收费亭问题中看到一个很好的例子。在收费亭问题中有一个很好的例子。所以在收费亭问题中，当你在道路上收取通行费，或者更一般地说，当你建造新的道路、新的桥梁、新的立交桥、新的绕行道路时，你需要考虑这些新的通行费、这些新的道路将如何影响整个交通流。沿着树的交通流将形成一个新的均衡，你需要在设计通行费和你设计道路系统时考虑到这一点。好吗？这是SPE的另一个例子。所以今天我想做一件完全不同的事情，有点像我们在Duel中做的。我想今天玩一个游戏，可能整个今天都在分析这一个游戏。所以这是一个相当复杂的游戏，但这是一个相当有趣的游戏。好吧，我们要看的游戏是什么？好吧。这个游戏将涉及两个玩家，在每个玩家，在每个回合中，每个回合，他们选择，或者我应该说，每个选择，每个选择是战斗还是退出。

[段 3]

All right, so F means fight and Q means quit. And they make this choice simultaneously. All right, and the game ends, the game ends as soon as someone quits. Someone quits. Alright, so there’s good news and bad news for this game. Let’s do the good news first. The good news is that if the other player quits first, if the other player quits first, you win a prize. you win a prize. And generally we’ll call this prize V, but we’ll play for some cash in a minute. The bad news is, the bad news is, each period in which both fight, so each period in which both players choose to fight, each player pays a cost. So they pay minus C. Alright? And just to keep things interesting, let’s fill in the other thing here, which is if both quit at once. So if both quit at once, then they get zero that period alrightAll right. So this is a game we’ve seen a little bit before. We saw it a little bit under the auspices of Hawk Dove. Those people in the NBA class saw a game a lot like this. But we’re going to analyze this in much more detail than we did before, as we can spend the whole of today talking about it. So to start this out, let’s actually play this game.

[译文 3]

好的，F代表战斗，Q代表放弃。他们同时做出这个选择。好的，游戏结束，有人放弃时游戏立即结束。有人放弃。好的，这个游戏有好消息和坏消息。我们先说好消息。好消息是，如果对方先放弃，如果对方先放弃，你就赢得奖品。你赢得奖品。一般我们把这个奖品称为V，但过一会儿我们用现金来玩。坏消息是，坏消息是，在每个双方都战斗的时期，在每个双方都选择战斗的时期，每个玩家都要付出代价。所以他们支付负C。好吗？为了让事情更有趣，我们把另一件事也填上，就是如果双方同时放弃。如果双方同时放弃，那他们该时期得到零。好吧。这是一个我们之前稍微见过的游戏。我们在鹰鸽游戏的框架下见过一点。那些NBA课上的同学看到了一个与之非常相似的游戏。但我们将比以前更详细地分析这个游戏，因为我们可以花整整今天来讨论它。所以让我们开始玩这个游戏吧。

[段 4]

All right. So I want two volunteers. Let me just, since this is college football season, let me see if I can play off the rivalries. So do I have anybody here from the great state of Texas? Whole bunch of Texans, all right. All right, keep your hands around, I want to use you a second. And I guess the rivalry here is Oklahoma. Anybody from Oklahoma? No Oklahomans? All right, what we’ll do is we’ll pick two Texans then. We’ll assume this is Texas and Texas A&M. All right, so Texans raise their hands again. All right, we’re going to pick out two Texans. All right. And I’m going to give you a mic-ish. So your name is? Nick. Nick. Why don’t you keep hold of the mic and just point it towards you when you speak, but still shout because everyone here wants to hear you. All right. So this is Nick. Where was my other Texan back here? There was one here. Why don’t I go for the closer one? All right. Okay. We’ll start here. And your name is? Alec. Alec. So shout it out. Alec. That’s better. Okay. All right. So the game is this. All right. They’re going to have to write down for the first period whether they choose…for the first period, whether they choose fight or quit. Each player will have a referee, so the person behind Alec is going to be Alec’s referee to make sure that Alec is actually saying what he says he’s going to do.

[译文 4]

好的。我需要两个志愿者。让我想想，现在是大学橄榄球赛季，我想利用一下这种竞争关系。所以这里有人来自伟大的 Texas 州吗？好多人举手了，好的。好的，先保持举手，我等一下需要你们。我猜这里的竞争关系是 Oklahoma。有来自 Oklahoma 的人吗？没有 Oklahoma 人？好的，那我们就把两个 Texas 人都选上。我们假设这是 Texas 和 Texas A&M 的对决。好的，Texas 人再次举手。好的，我们要选两个 Texas 人。好的，我要给你一个麦克风之类的东西。你叫什么名字？Nick。Nick。你拿着麦克风，说话的时候对着它，但还是要大声喊，因为这里所有人都想听到你的声音。好的，这是 Nick。我的另一个 Texas 人在后面那边？这里有一个。我选近一点的那个怎么样？好的，我们从这里开始。你叫什么名字？Alec。Alec。所以大声喊出来。Alec。这样好多了。好的，游戏的规则是这样的。好的，他们要在第一个阶段写下他们选择的是战斗还是放弃。每个玩家都会有一个裁判，所以 Alec 后面的人会成为他的裁判，确保 Alec 真正说出他要做什么。

[段 5]

And what happened to my other Texan? I’ve lost my Alec. There he is. And your name again was? Nick. And Nick is going to write down fight or quit. And to make this real, let’s play for some real cash. So we’ll make the prize, why don’t we make the prize equal to a dollar? and the cost equal to 75 cents. All right, so I’ve got some dollars here. Here we go. This is a, what do you call this? In Texas, we call this a fistful of dollars. Is that right? Is that right? All right, so okay. So where am I playing? Why don’t you stand up, you guys? So everyone can see you. All right. I made this difficult, because now you’re going to have to write down your strategy, so you’re going to have to grab a pen. I didn’t make that easy for you. Let me come down to make it easier for the camera person. All right. All right. So why don’t you write down what your strategy is going to be and tell your neighbor what it’s going to be. And we’ll see what happened. All right. Show your referee. Have you shown your referee? All right. And, again, Nick, speaking into the microphone, what did you do? I quit. He quit? I quit as well. So much for the – what happened to remember the Alamo?

[译文 5]

我的另一个德州人怎么了？我失去了我的Alec。他在那儿。再说一次你的名字？Nick。Nick要写下“战斗还是退出”。为了让这更真实，我们来赌点真钱。我们来设奖金，为什么不设成一美元？成本设为75美分。好，我这里有一些美元。开始了。这是什么，你们怎么称呼这个？在德州，我们叫它“一把美元”。对吗？这样对吗？好，所以我要在哪里玩？你们站起来吧。这样大家都能看到你们。好。我把这个弄得困难了，因为现在你得写下你的策略，所以你得拿支笔。我没让你轻松。让我下来，方便摄影人员拍摄。好。好。你为什么不写下你的策略是什么，然后告诉你的邻居你的策略是什么。我们看看会发生什么。好。给你的裁判看一下。你的裁判看了吗？好。还有，Nick，对着麦克风说，你做了什么？我退出了。他退出了？我也退出了。对记住阿拉莫的人来说——发生了什么？

[段 6]

I mean, this is… All right.What happened to remember the Alamo? I mean, this is… All right, so Texas didn’t work very well. Okay, fine. Let’s try a different state. I have to say, my wife’s from Texas, and my wife’s family’s from Texas, and I thought they had more fight in them than that. Maybe that’s why they’re sliding in the polls. Let’s try somebody from Ohio. Anyone from Ohio? Nobody from Ohio in the whole class. That’s no good. That’s no good. I was going to pick Ohio against Michigan. How about some people from some of our own teams? Are there any players in the football team other than the two I picked on before? There we go. Now I get something. Here we go. So I need a different team. Anybody from the hockey team? Anybody from the baseball team? Ah, okay. Good, good, good. All right. So a friend from the baseball team, your name is? Chris. Chris. And our new football team player is? Ryland. Ryland. Rylan, okay. Okay, so Rylan and Chris are going to play this. We’ll see. Neither of you is from Texas, I take it. All right, so we have some hope of something happening here. All right, so write down what it is you’re going to choose. All right, have you both written something down? Yeah? All right, and Rylan, what did you choose?

[译文 6]

我的意思是，这……好吧。记住阿拉莫是怎么回事？我的意思是，这……好吧，德州不管用。好吧，好吧。我们换另一个州试试。我得说，我妻子是德州人，我妻子的家人也是德州人，我原以为他们更有斗志。也许这就是为什么他们的民调在下滑。我们试试俄亥俄州的人。有没有人来自俄亥俄州？全班没有人来自俄亥俄州。那可不行。那可不行。我本来要选俄亥俄对抗密歇根。我们自己的队伍里有没有人？除了之前我挑的那两个，橄榄球队还有没有其他球员？好了。现在有收获了。开始了。所以我需要另一支队伍。有没有人来自曲棍球队？有没有人来自棒球队？啊，好。好的，好的，好的。好吧，棒球队的这位朋友，你叫什么名字？Chris。Chris。那我们橄榄球队的新球员是？Ryland。Ryland。Rylan，好的。好的，Rylan和Chris要来玩这个。我们拭目以待。我猜你们两个都不是德州人。好吧，那我们还有希望看到点事情发生。好吧，把你要选的东西写下来。好吧，你们两个都写了吗？写了吗？好吧，Rylan，你选了什么？

[段 7]

Fight. And Chris? I’m going to quit. Oh, well, that was easy too. Okay. Okay. So the football team is looking pretty good here. All right. All right So we’re getting not getting much in the way of of actionAlright, so we’re not getting much in the way of action going here. Anyone else want to try here? Another little state rivalry here. I don’t suppose I’ve got anyone from Oregon. That’s asking too much. Anyone from Oregon? You guys must be from somewhere. There’s got to be a state where at least one of you is from. Alright, well let’s try something like New Jersey. How about that? Alright, there’s some players from New Jersey. That’s good. Alright, there we go. So, all right, and we’ll try New Jersey and New York. That seems like there’s a bit of a rivalry there. Are you from New York? Excellent. Here we go. Here we go. So, and your name is? Gerson. Gerson, and your name is? Andy. Andy, okay. So, Gerson and Andy, so stand up so everyone can see you are. Let’s see if there’s any fight in New York and New Jersey. All right, all right. Okay, so write down your strategies.

[译文 7]

战斗。克里斯呢？我要退出。哦，那也很容易。好吧。好吧。橄榄球队看起来相当不错。好吧。好吧，我们的比赛进展得不太顺利，没有看到太多动作。好的，这里的动作不多。其他人想试试吗？又一场州之间的 rivalry。我这里有来自俄勒冈的人吗？那要求太高了。有人来自俄勒冈吗？你们一定来自某个地方。肯定有一个州，你们中至少有一个人来自那里。好吧，那我们试试新泽西怎么样？有来自新泽西的选手。很好。好，我们开始了。好吧，我们会进行新泽西对纽约的比赛。这看起来有点 rivalry。你们来自纽约吗？太好了。我们开始。我们开始。你叫什么名字？Gerson。Gerson，你呢？Andy。Andy，好的。所以Gerson和Andy，站起来让大家看看你们。请写下你们的策略。

[段 8]

All right, good. now so all right so Andy Andy what did you choose I’m gonna fight fight ah here we go this is better now whoo getting worried there for a second I know I know it’s near the Thanksgiving but there has to be some sort of spark left in the glass that’s okay all right so we have we have both people fighting which means right now they’re down 75 cents but the prize is still a dollar so the game goes on so write down again what you’re going to do second period the 75 cents is gone So now we’re just looking at the this game for a dollar all right, and let’s let’s go to New York What does New York?Let’s go to New York. What does New York? I’m going to fight. Fight. Fight, okay. You have to stay on the East Coast to get life. There’s no point going West, is there? That makes sense. All right, so write down again what you’re going to do. Okay, and let’s go the other way around to New Jersey. Fight. Fight. Fight again. All right, so right now we’re down 3.75 cents, whatever that is. And it’s still this prize of a dollar out there, plus perhaps a bit of pride here.

[译文 8]

好的，干得好。Andy，Andy，你选了什么？我要战斗。战斗。啊，我们开始了，这好多了。呜，我刚才有点担心，我知道快到感恩节了，但玻璃杯里肯定还有一些火花。没关系。好的，所以我们两个人都在战斗，这意味着目前他们已经亏损了75美分，但奖金仍然是1美元，所以游戏继续。请再次写下你们第二轮要做什么。那75美分已经没了，所以我们现在只是为了这1美元而战。好，让我们看看纽约。纽约选什么？让我们看看纽约。纽约选什么？我要战斗。战斗。战斗，好的。你必须留在东海岸才能获得活力。往西走没有意义，对吧？这有道理。好的，请再次写下你们要做什么。好的，让我们反过来看看新泽西。战斗。战斗。又战斗了。好的，所以目前我们亏损了3.75美分，不管是多少。而且那个1美元的奖品还在那里，也许还有一点骄傲。

[段 9]

All right, so write down. again all right and let’s try again so let’s go with New York this time I’m gonna fight fight okay I’m guessing I’m guessing we could keep this going for quite a while is that right I’m guessing it’s good going quite a while now it might make a difference by the way if there are to talk to each other here so let’s see if it does okay so let’s allow New Jersey and New York to talk to each other you can’t insult each other about bridges and tunnels just just just regular talk all right so anything you want to say to your friend from New Jersey here? I can’t let New Jersey win. It’s just New York pride. You guys are just worse than every realm, so I’m sorry. All right, anything you want to say in reply? Well, I’m going to keep fighting, so your best choice is to give up. All right, let’s see if that works. Let’s see if we get anything out of that. So choose your strategies again. New York? I just can’t let Jersey win. Fight. Bring it on. Fight. All right, all right. So it’s clear that if we kept this going for a while, it would pay for my love.It’s clear that if we kept this going for a while, it would pay for my lunch.

[译文 9]

好的，请再次写下。好的，让我们再试一次。让我们选择纽约这一次。我要战斗。战斗。好的，我猜我们可以把这个持续很久，是吗？我猜可以持续很久。不过，顺便说一下，如果他们能互相交流的话，可能会有所不同。让我们看看是否有效。好的，让我们允许新泽西和纽约互相交流。你们不能互相侮辱对方的大桥和隧道，只能只能只能普通的交流。好的，有什么想对新泽西的朋友说的吗？我不能让新泽西赢。这只是纽约的骄傲。你们就是比每个州都差，所以对不起。好的，有什么要回应的吗？好吧，我会继续战斗，所以你最好的选择是放弃。好的，让我们看看这是否有效。让我们看看我们能不能从这得到什么。请再次选择你们的策略。纽约呢？我就是不能让泽西赢。战斗。来吧。战斗。好的，好的。很显然，如果我们继续这样下去，可以支付我的午餐费。很显然，如果我们把这个继续下去，可以支付我的午餐费。

[段 10]

Is that right? Is that right? Well, we’ll hold it here for a second. We’ll talk about it a bit. But thank you. A round of applause for our two feistier players. All right. So what’s going on here? So clearly we can see what can happen in this game. You can get people quitting early. All right. and it could be that one side quits and the other side doesn’t quit. That’s also something that can happen. That can happen pretty quickly. But it’s possible, and we just saw it happen, it’s possible that a fight could go on quite a while. A fight could go on quite a while here. Now why? What’s going on here? I mean, the prize here was what? It was a dollar, and the cost was 75 cents. I could have raised the stakes maybe on these guys and see if that made a difference, But I think $1.75 will do. And by the time they’d fought the second time, they’d exhausted the possible prize of $1. Is that right? So it’s true that if you won this in the first period, then that’s fine, because you just get $1 and it wouldn’t have cost you anything. And even if you won in the second period, you’d be okay. You’d only cost yourself $0.75 for fighting in the first period, but you’re getting $1, so that’s good. but there on and there on went on for plenty of time in this case there on you’re just accumulating losses so what’s going on soYou’re just accumulating losses.

[译文 10]

是吗？是这样吗？好吧，我们在这里暂停一下。我们来谈谈这个。但谢谢你们。请为我们的两位最激烈的选手鼓掌。好的。这里发生了什么？很明显我们可以看到这个游戏中会发生什么。你可以让人们提前退出。好的，而且可能是一方退出而另一方不退出。这也是可能发生的事。这可能发生得很快。但这是可能的，而且我们刚才看到了，这是可能的，战斗可以持续相当长的时间。战斗可以在这里持续相当长的时间。为什么？这是怎么回事？我的意思是，这里的奖品是什么？是1美元，成本是75美分。我本可以提高这些人的赌注，看看这是否会有区别，但我认为1.75美元就够了。当他们第二次战斗时，他们已经耗尽了可能的1美元奖品。是吗？所以这是真的，如果你第一轮赢了，那就没问题，因为你得到了1美元，而且不会花你任何钱。即使你第二轮赢了，你也会没事。你在第一轮战斗只花了0.75美元，但你得到了1美元，所以这很好。但战斗继续进行了相当长的时间，在这种情况下，你只是在累积损失。发生了什么？你只是在累积损失。

[段 11]

So what’s going on? There are various conclusions possible here. One is that people from New York and New Jersey are crazy. That’s a possible thing. But what else? Why do we get involved in this fight? What happened here? Why do we tend to see fights like this emerging? I’m claiming this isn’t such an implausible situation. Why do we see it emerging? Let’s talk to our friend from New York. Shout out. By the time she fought with me on the second round, I knew I was going to be losing money anyway, so why not just keep going? All right. There was no reason I wasn’t going to win anyway, so I might as well just keep fighting until she quit. All right. All right. So I think there’s two things. There’s two parts to that answer. All right. Part of the answer is I’m going to be, I’m sorry, I have lost some money anyway. Right. Let’s hold that piece of it. And the other part of it is what? The other part of it is I’m really determined to win this thing. All right. All right. So there’s two things going on there and they’re quite different. Let’s take the second one first. It’s possible that the reason these fights emerge and can go on for quite a while is that even though the prize is only a dollar in money, it could be that the actual thing that the players care about is what?

[译文 11]

发生了什么？这里有各种可能的结论。一个是纽约和新泽西的人是疯子。这是一个可能的结论。但还有什么？为什么我们会卷入这场战斗？这里发生了什么？为什么我们倾向于看到这样的战斗出现？我声称这不是一个不可能的情况。为什么我们看到它出现？让我们和纽约的朋友谈谈。请喊出来。到她第二轮和我战斗时，我知道我无论如何都会亏钱，所以我为什么不继续下去？好的。没有任何理由我会赢，所以我不如继续战斗直到她退出。好的。好的。我认为这有两个答案。有两个部分。第一部分回答是，我会，我已经亏了一些钱了。对，让我们先记住这部分。另一部分是什么？另一部分是，我真的决心要赢这场比赛。好的。好的。这里有两个事情在起作用，它们是相当不同的。让我们先看第二个。即使奖品只有1美元，但可能玩家真正关心的东西是什么，这可能是这些战斗出现并能持续很长时间的原因。

[段 12]

What do the players actually care about here? Somebody? Just raise your hand, I’ll put you on the mic. What do people tend to care about in these situations? Winning. They care about winning or they care about prize. Is that right? Is that right? That’s why I started with Texas.Is that right? That’s why I started with Texas, but I couldn’t find any pride in Texas, so we had to go to New York. Right? All right, so people care about people care about winning per se. It’s a pride thing. All right, so it could be that a dollar simply isn’t a good description of the true payoffs here. It could be that the payoffs are actually about winning. All right, it could also be that both of these guys know that they’re going to be interacting with you at other times in the class or other times at Yale, and they want to establish a reputation, both of them, as being the kind of guys who fight. In particular, when they got to talk about it, both of them said, look, I’m a fighter. Something we’ve seen before with Ali and his pizza shop. Both of them said, I’m a fighter, you better back out. Is that right? So both of them tried to signal the fact that they were going to fight to try and get the other side to quit.

[译文 12]

玩家在这里真正关心的是什么？有人吗？只要举手，我会把麦克风给你。在这种情境下人们通常关心什么？赢。他们关心赢或关心奖品。是这样吗？是这样吗？这就是为什么我从德克萨斯开始。是这样吗？这就是为什么我从德克萨斯开始，但我找不到德克萨斯的骄傲，所以我们只能去纽约。对吗？好的，人们关心，人们关心赢本身。这是一个关于骄傲的事情。好的，所以1美元可能并不能很好地描述这里的真实收益。可能收益实际上是关于赢的。好的，也可能这两个人都知道他们将在课堂上的其他时间或其他时间在耶鲁与你们互动，他们想建立一种声誉，两个人都想表现出自己是那种会战斗的人。特别是，当他们谈论这件事时，两个人都说，看，我是一个 fighter。我们之前在Ali和他的披萨店见过类似的情况。两个人都说，我是一个 fighter，你最好退出。是这样吗？所以两个人都试图发出信号，表明他们会战斗，试图让对方退出。

[段 13]

So that’s about reputation. And that reputation could extend beyond this game. It could be that they’re going to be involved in this kind of conflict later on in life. So both of those things are around. There’s another element to this, and it’s the other part of what our friend from New York said, which is about the costs. What’s true about the costs in this game as we move from period to period? What’s true? Somebody said it. Say it loudly. It’s a sunk cost. So all of those costs that you accumulate as the game goes on.that you accumulate as the game goes on, they’re irrelevant looking forward because they’re sunk. The fact I’ve played this game for 10 periods and hence lost 10 times 75 cents, which even I can do, that’s 750, right? The fact I’ve lost 750 is irrelevant because I’ve lost it anyway. I can’t get that back. That’s a sunk cost. So the game 10 periods through looks exactly the same as the game did at the beginning when fighting seemed a good option. Is that all right? Is that all right? So 10 periods through the game, you have the same view about fighting as you did at the beginning. Now, that’s not quite true, because at some point, you’re going to run out of money. All right? But if we ignore that, basically, those sunk costs are irrelevant.

[译文 13]

这就是关于声誉的问题。而这种声誉可以延伸到这个游戏之外。可能会是他们今后一生都会卷入这种冲突。所以这两件事都是存在的。这还有另一个要素，也就是我们那位来自纽约的朋友所说的另一个部分，即关于成本。随着我们从一个时期进入另一个时期，这个游戏中的成本有什么特点？有什么特点？有人说过了。大声说出来。是沉没成本。所以你在游戏进行过程中积累的所有成本……随着游戏进行而积累的所有成本……向前看它们是无关紧要的，因为它们已经沉没了。我已经玩了这个游戏10个回合，因此损失了10次75美分，这我也能算出来，是750，对吧？我损失了750这件事是无关紧要的，因为我反正已经损失了。我无法收回那些钱。那就是沉没成本。所以游戏进行了10个回合后，看起来和游戏开始时完全一样，当时战斗看起来是个不错的选择。是不是这样？是不是这样？所以游戏进行了10个回合后，你对战斗的看法和你在开始时的看法是一样的。现在，这并不完全正确，因为在某个时候，你会耗尽资金。好吧？但如果我们忽略这一点，基本上，这些沉没成本是无关紧要的。

[段 14]

All right? So what we’re seeing here is reasons why people fight. And some of these reasons seem to be for standard economic reasons like sunk costs, and some of them seem to be about things that are outside the game, like pride or possibly reputation. And it’s certainly the case that in the real world, we do see fights like this. So let’s just spell out what the key feature of this is. The key feature of this is in these fights, over a period of time, even though you may only be losing a little piece in each period, over a period of time, you could lose a lot. In fact, you could lose far more than the prize that was originally at stake. So the losses you could accumulate, and our friendSo the losses you could accumulate, and our friend from New Jersey and New York, the losses that they accumulated vastly outweighed the prize that was at stake after a while. All right? That’s a worry. That’s a worry, right? Okay? So this can occur in real life, not just in the classroom. What do we call these kind of fights? These fights where they’re holding out for this possibly small prize and incurring possibly small but they accumulate into being large costs each period. What do we call those fights? Nobody? Well, let’s think about some examples.

[译文 14]

好吧？所以我们在这里看到的是人们战斗的原因。其中一些原因似乎是出于标准的经济原因，比如沉没成本，而其中一些似乎与游戏之外的因素有关，比如骄傲，或者可能是声誉。在现实世界中，我们确实会看到这样的战斗，这是确定无疑的。让我们来阐述一下这件事的关键特征。这些战斗的关键特征是，随着时间的推移，尽管你可能每个时期只损失一小块，但随着时间的推移，你可能会损失很多。事实上，你可能会损失远远超过最初赌注的奖品。所以你们可能会积累的损失，而我们那位来自新泽西和纽约的朋友，他们积累的损失在一段时间后远远超过了赌注。好吧？这是个问题。这是个问题，对吧？好的？所以这可能发生在现实生活中，而不仅仅是在课堂上。我们把这些战斗称为什么？这些为了可能是很小的奖品而坚持却承受着可能很小但每个时期累积起来就变成大成本的战斗。我们怎么称呼那些战斗？没人知道吗？好吧，让我们想一些例子。

[段 15]

See if the word comes out of examples. So one example, let’s do some examples here. One example is what happened in World War I. So in World War I, as I’m assuming most of you know, on the Western Front at least, the German and Allies to Germany armies faced off with the British and French and Allied armies for an extraordinarily long time, fighting over extraordinarily small patches of land, little pieces of northern France and Belgium. And you could argue that these pieces of northern France and Belgium, I don’t wish to offend anyone French or Belgian here, but you could argue that those few acres of northern France and Germany weren’tfew acres of northern France and Germany weren’t worth a whole lot anyway. Nevertheless, the troops, the two sides kept on fighting for from what 1914 to 1918 and enormous losses of life were accumulated over that period. So that was a really costly long-term battle. Neither side would quit. Each year huge numbers of lives were lost. If you doubt that, go and look at the war memorial in Yale that shows how many Yale American lives were lost, and America was only in that war for about a year. Okay, so that’s an example. Another example, a more business example, an example we talked about in our MBA class, but not in this class so far, is examples in business where there’s a market, and that market is really only going to hold one firm.

[译文 15]

看看例子能不能引出这个词。所以一个例子，让我们在这里举一些例子。一个例子是一战中发生的事。在第一次世界大战中，我想你们大多数人都知道，至少在西线，德国及其盟国的军队与英国、法国及其盟国的军队对峙了非常长的时间，为了非常小片的土地而战斗，即法国北部和比利时的小块土地。你可能会说这些法国北部和比利时的小块土地，我在这里不想冒犯任何法国人或比利时人，但你可能会说那些法国北部和德国的几英亩地反正也不值多少钱。尽管如此，双方的军队从1914年到1918年一直在战斗，在那段时间里积累了巨大的生命损失。那是一场真正代价高昂的长期战斗。双方都不肯放弃。每年都有大量生命丧失。如果你怀疑这一点，去看看耶鲁的战争纪念馆，那里显示了有多少耶鲁美国人的生命丧失，而美国参战大约只有一年。好的，所以那是一个例子。另一个例子，一个更商业化的例子，一个我们在MBA课上讨论过但在这个课上还没讨论过的例子，是商业中的例子，其中有一个市场，而那个市场实际上只能容纳一家公司。

[段 16]

That market’s only going to hold one firm. you can end up with an extremely long fight about who’s going to end up being the one firm in that market. So a famous example, actually it’s a famous business school case, is the fight that incurred to control satellite broadcasting in Europe. So there was a fight between Sky Television and the British satellite broadcasting company that went on for a number of years. And these companies were doing things like charging zero prices and giving away satellite dishes and this, that, and the other. And over the course of the fight, they accumulated so many losses that if you didn’tthey accumulated so many losses that if you did the accounting, the entire future projected profit flow of winning this fight vastly outweighed the amount of money that they’d lost during the fight. That was a fight that involved on one side Rupert Murdoch, and you could argue maybe Rupert Murdoch is a little crazy and had reputation to keep up. But still, it looks like another example of this. So that example was British satellite broadcasting versus Sky. So with those two examples there, anyone think of a general term we call these fights? How do people refer to the method of fighting in World War I, or for that matter during the American Civil War? Yes, somebody at the back, shout it out.

[译文 16]

那个市场只能容纳一家公司。你最终可能会得到一场关于谁将成为那个市场中的那一家公司的非常漫长的战斗。所以一个著名的例子，实际上这是一个著名的商学院案例，是发生在欧洲控制卫星广播的斗争。所以Sky Television和英国卫星广播公司之间有一场持续了数年的斗争。这些公司做的事情包括收取零价格、赠送卫星天线等等。在战斗过程中，他们积累了如此多的损失，如果你做账的话，赢得这场战斗的全部未来预期利润流远远超过了他们在战斗期间损失的钱。那是一场一边有Rupert Murdoch参与的战斗，你可能会说Rupert Murdoch可能有点疯狂，需要维持声誉。但仍然，这看起来是另一个例子。所以那个例子是英国卫星广播对阵Sky。有了这两个例子，有人能想到我们称这些战斗的通用术语吗？人们怎么称呼第一次世界大战中，或者顺便说一句，在美国内战中的战斗方式？是的，后面有人，大声说出来。

[段 17]

It’s a war of attrition. So these are wars of attrition. These are wars of attrition. And what we know about wars of attrition is that they can go on a long time and a lot can be lost. A lot of life can be lost in the case of real wars, a lot of money can be lost in the case of business wars. And actually it turns out a lot of games have this structure of a war of attrition. Let me give you one more example. Suppose that two companies are competing for a market, not in the manner of BSB and Sky by simply advertising or whatever, but in the form of paying bribes. Alright, so suppose there’s a company, let’s say in France, and a company, let’s say in Britain, and these two companiesFrance and a company, let’s say, in Britain. And these two companies are trying to control, are trying to win a contract in some country where paying bribes is a successful strategy. And here I’m going to be careful about the film and not mention any real countries. Let’s call this imaginary country Fredonia, which comes from a Marx Brothers film. All right, so here’s this French company and this British company, and they both want this contract to build a bridge in Fredonia. And they start paying bribes to the general who controls Fredonia.

[译文 17]

这是消耗战。所以这些是消耗战。这些是消耗战。我们对消耗战的了解是，它们可以持续很长时间，可能会失去很多东西。在真正的战争中可能会失去很多生命，在商业战中可能会损失很多钱。事实上，事实证明很多游戏都有这种消耗战的结构。让我再给你一个例子。假设两家公司正在争夺一个市场，不是像BSB和Sky那样通过简单的广告或其他方式，而是以支付贿赂的形式。好吧，假设有一家公司，我们就说在法国吧，还有一家公司，我们就说在英国吧。这两家公司……法国的一家公司和英国的一家公司。这两家公司正试图控制，正在试图在某个支付贿赂是成功策略的国家赢得一份合同。在这个问题上我要小心电影中的内容，不提任何真实的国家。让我们把这个虚构的国家叫做Fredonia，它来自一部Marx Brothers电影。好吧，所以这里有一家法国公司和一家英国公司，他们都想在Fredonia获得一份建造桥梁的合同。他们开始向控制Fredonia的将军支付贿赂。

[段 18]

All right, And what happens? Well, you’re not going to get the bribe back. So both sides pay a few thousand dollars to this general. And then the general comes back and says, well, you’ve both paid a thousand dollars. Which of you wants to pay the bridge? So they go on and they put more money in and more money in. And you can see once again this is a war of attrition. Those bribes that they’ve paid, they’re never getting back. But once you’ve paid them, they’re a sunk cost. Once you’ve paid that bribe, good luck saying, I paid you this bribe. You didn’t let me build the bridge. Give me my money back. There isn’t a court in the world that’s going to enforce that. So these bribery contests look a lot like this. There’s a technical name for these bribe contests. They’re sometimes called all-pay auctions. So what do we want to establish today? We want to talk about why fighting occurs here. And we wanted to do so in some detail. So there may be informal reasons why fightingin some detail. So there may be informal reasons why fighting occurs, and we’ve talked about that. It could be that one side is crazy. It could be that both sides are crazy. It could be that national or regional pride is at stake. All of these things could affect why we get fighting.

[译文 18]

好吧，然后会发生什么？嗯，你是拿不回贿赂的。所以双方都给这个将军支付了几千美元。然后将军回来说，好吧，你们都付了一千美元。你们谁想付建桥的钱？于是他们继续投入越来越多的钱。你可以再次看到这是一场消耗战。那些他们支付的贿赂，他们永远不会拿回来。但一旦你支付了，那就是沉没成本。一旦你支付了那笔贿赂，想要说"我付了你这笔贿赂，你没让我建桥，把钱还给我"可没那么容易。世界上没有哪个法院会强制执行那个。所以这些贿赂竞赛看起来很像这样。这些贿赂竞赛有一个技术名称。它们有时被称为全付拍卖。那么我们今天想要确立什么？我们想要谈谈为什么在这里会发生战斗。我们想要相当详细地讨论这个问题。所以可能有非正式的原因导致战斗，我们已经讨论过了。可能是一方疯了。也可能是双方都疯了。可能国家或地区的骄傲正处于危险之中。所有这些因素都可能影响我们为什么会有战斗。

[段 19]

But what I want to try and establish today is that you can get long fights emerging in potential wars of attrition, even if everybody is rational, even if the payoff is just that one dollar, and even if there’s no reputation at stake. So again, my goal today is to try and convince you that you can get huge loss of life in World War I or huge loss of money in these business contexts without having to argue something outside the model like irrationality or reputation. Even rational players can get themselves in trouble in wars of attrition. So for the rest of today, I want to try and analyze this. And to get us started, I want to look at a version of this game, at a simplified version, which only lasts for two periods. We’ll do a two-period version of the game we played just now. And eventually, by the end of today, I want to look at the infinite version. We’ll start small. We’ll start with a two-period version. All right, and the trick in analyzing these things is to be able to come up with a truetrick in analyzing these things is to be able to come up with a tree and be able to come up with payoffs and be able to apply the analysis that we know about to get us to where we want to be.

[译文 19]

但我今天想要论证的是，即使每个人都是理性的，即使收益只是一美元，即使没有声誉受损，你仍然可以得到持久战中的漫长斗争。所以再次强调，我今天的目标是说服你们，一战中巨大的人员伤亡或商业领域中的巨额资金损失，不必借助模型之外的因素来解释，比如非理性或声誉。即使是理性的参与者也可能在消耗战中陷入困境。所以今天剩下的时间，我想来分析这个问题。首先，我想看一个简化版的游戏，它只持续两个时期。我们来做刚才那个游戏的两个时期版本。到今天结束时，我想看一下无限期版本。我们从小处着手，从两个时期开始。分析这些问题的关键在于能够构建一棵博弈树，能够写出收益，然后能够应用我们已知的分析方法到达我们想去的地方。

[段 20]

All right, so here’s the game I claim. I claim that it has the following tree. So first of all, player A chooses, and player A can either fight or quit. And let me put a one in brackets. We’ll see what the one is in a second. Then we’re going to model player two, but of course this is a simultaneous move game. This is a simultaneous move, so this is an information set. It’s not called player two, it’s called player B. So player B doesn’t know what A has done that first period when B is making her choice. This is a simultaneous move. And B is choosing between fighting or quitting. And just to distinguish them, let me use small letters for B. So once again, fight or quit. And fight or quit. All right. Now, if both sides fight, then the game continues. And everyone knows that both sides fought at that stage. So at this point on, we’re actually at a singleton information node, and it’s player A’s turn again. Thank you. Thank you. Sorry. All right. Thank you. All right. So here we go again. So in the second period,All right, so here we go again. So in the second period, once again, we’ve got A choosing whether to fight or quit. And this time we’ll put a 2 to indicate we’re in the second period.

[译文 20]

好的，这里是我认为的博弈结构。首先，玩家A做出选择，A可以选择战斗或退出。让我写一个括号里的1，我们稍后会看到1是什么。然后我们要模拟第二个玩家，但这是一个同时行动博弈。这是同时行动，所以这是一个信息集。它不叫玩家二，它叫玩家B。所以当B做出选择时，B不知道A在第一个时期做了什么。这是同时行动。B在战斗和退出之间选择。为了区分它们，让我用小写字母表示B。再说一遍，战斗或退出。战斗或退出。好的。如果双方都战斗，那么博弈继续进行。每个人都知道双方在该阶段都战斗了。所以在这一点上，我们实际上处于一个单点信息节点，轮到玩家A再次行动了。谢谢。谢谢。抱歉。好的。谢谢。好的，那我们继续。在第二个时期，再一次，A选择是战斗还是退出。这次我们加上2来表示我们处于第二个时期。

[段 21]

And after player A is moved, once again, player B is moving. It’s a simultaneous move. And once again, they’re choosing fight or quit. and I’ll put 2’s to indicate that we’re in the second stage. So that’s the structure of this two-period game. And let’s write down what the payoffs are, starting with the easy payoffs. So if both people quit in the first stage, they get nothing. If A quits and B fights, then A gets nothing and B gets V. if A fights and B quits, then A gets V and B gets nothing. And if they both fight, we go into the second stage. So let’s write down the payoffs in the second stage. So in the second stage, if they both quit in the second stage, then their payoffs are going to be for A minus C, the cost they accumulated in the first stage, plus zero. and for B minus C plus 0. If A quits and B fights in the second stage, then the payoffs are minus C plus 0.then the payoffs are minus c plus 0 and minus c plus v. If a fights and b quits, then the payoffs are minus c plus v and minus c plus 0. And if they both fight for two periods, we have a decision to make about how we’re going to end the game in this two-period game.

[译文 21]

在玩家A行动之后，再一次轮到玩家B行动。这是同时行动。再一次，他们在战斗或退出之间选择，我会加上2来表示我们处于第二个阶段。这就是这个两时期博弈的结构。让我们写出收益，从简单的收益开始。如果双方在第一阶段都退出，他们什么都得不到。如果A退出而B战斗，那么A得到0，B得到V。如果A战斗而B退出，那么A得到V，B得到0。如果双方都战斗，我们就进入第二阶段。让我们写出第二阶段的收益。在第二阶段，如果双方都在第二阶段退出，那么A的收益是负C，即他们在第一阶段累积的成本，加上0，B的收益是负C加0。如果A退出而B在第二阶段战斗，那么收益分别是负C加0和负C加V。如果A战斗而B退出，那么收益是负C加V和负C加0。如果双方战斗两个时期，我们需要做一个决定，关于如何结束这个两时期博弈中的游戏。

[段 22]

But let’s just assume that what we’ll get here is minus c minus c and minus c. minus C. All right, we’ll assume if they both fight, the game ends and no one gets the prize. All right, just to make life simple. All right, so this is a two-period version of the game, and the only change I’ve made, other than making it two periods, is I had to put in a payoff, I had to see what happens if the game didn’t resolve, and I’ve assumed that if the game didn’t resolve, no one got the prize. All right? Now there’s another assumption I’m going to have to make before we analyze this. there are two possible cases to consider here. There’s the case when v is bigger than c, which is the case we just played in the class, and there’s also the converse case when the cost is bigger than v. So today, we’ll focus on the case v bigger than c, which is the case we just played in class. I’m going to leave you to analyze the other case, the case when the cost is bigger than the prize, as a homework assignment. So V bigger than C is our assumption. All right, is everyone okay with the tree? This tree, I hope, describes the game, at least the two-period version of the game.

[译文 22]

但我们假设如果双方都战斗，博弈结束，没有人获得奖励。好的，为了简化。就这样，这是游戏的两时期版本，除了把它变成两个时期之外，我做的唯一改变是我必须给出一个收益，我必须看看如果博弈没有解决会发生什么，我假设如果博弈没有解决，没有人会得到奖励。好的？在分析之前，我还需要做另一个假设。这里有两种可能的情况需要考虑。一种是当V大于C时，也就是我们刚才在课堂上玩的情况，还有一种相反的情况是成本大于收益。所以今天我们将关注V大于C的情况，也就是我们刚才在课堂上玩的情况。我会让你们自己分析另一种情况，即成本大于奖励的情况，作为家庭作业。所以V大于C是我们的假设。好的，大家对这个博弈树没有疑问吧？我希望这个树描述了这个博弈，至少是两时期版本。

[段 23]

Now the first thing I want to point out here is if we look at the payoffs that are incurred at the end of the second period game, we notice that they all contain a minus C. There’s a minus C everywhere. What is that minus C? It’s the cost that you accumulated from having fought in the first stage. But the observation I want to make straight away is that this cost, so here it is, here it is, here it is, here it is, here it is, here it is, and here it is, and here it is, this cost is sunk. These objects here are sunk costs. there’s nothing you can do once you’re in the second period of the game to get these sunk costs back. They’ve just gone. They’re there, but the fact that they enter everywhere is going to make them strategically irrelevant. All right. Okay, so what we want to do here is we want to find all of the sub-game perfect equilibria of this little game. All right, so let me push this up quite high. We’ll see.and let me get rid of the rules, we all know the rules by now so our goal here is to use our solution concept which is sub game perfect equilibrium to try and analyze this game. So how are we going to analyze this game in terms of sub-game perfect equilibria?

[译文 23]

现在我想指出的第一件事是，如果我们看第二时期博弈结束时产生的收益，我们会注意到它们都包含一个负C。到处都是负C。那个负C是什么？它是你在第一阶段战斗时累积的成本。但我立即要提出的观察是，这个成本，在这里，在这里，在这里，在这里，在这里，在这里，在这里，在这里，这个成本是沉没成本。一旦你进入博弈的第二时期，你没有办法收回这些沉没成本。它们就在那里，但它们出现在所有地方这一事实将使它们在战略上变得无关紧要。好的。我们要做的是找出这个小博弈的所有子博弈完美均衡。好的，让我把这个推得很高。我们会看到的。让我去掉规则，我们都知道了，所以我们的目标是使用我们的解概念即子博弈完美均衡来尝试分析这个博弈。我们将如何用子博弈完美均衡来分析这个博弈？

[段 24]

How are we going to start that discussion? I’ve got a lot of work to do here. Where are we going to start in finding sub-game perfect equilibria? What’s the first thing we should do? Well, I claim the first thing we should do is just figure out what the sub-games are. Let’s start with that. So having just pushed it far away, I’m going to need to use the pointer. but I claim that the obvious subgame to analyze first is this subgame. It’s the subgame if you should end up in period 2. And notice that is a subgame. It starts from a singleton node. It doesn’t break up any information set, and it contains all the descendants of the node from which it starts. All right? So that is genuinely a subgame. All right? So we’re going to start our analysis by considering the second sub-game. The second sub-game. All right? So let’s write down the matrix that corresponds to that second sub-game. And I’m going to write it down in the following way. So I claim in this second sub-game, each player has two choices. They can fight or quit. Fight or quit. And I’m going to write the payoffs in a particular way. I’m going to write the payoffs as minus C plus this thing. So rather than keep that minus C in all the boxes, which is going to get boring after a while, I’m going to pull that minus C out and just put it in front.

[译文 24]

我们如何开始讨论？我有很多工作要做。我们从哪里开始寻找子博弈完美均衡？我们应该做的第一件事是什么？好吧，我声称我们应该做的第一件事是找出子博弈有哪些。让我们从那里开始。由于刚刚把它推得很远，我需要用指针。但我声称首先要分析的明显子博弈是这个子博弈。这就是如果你最终进入第二时期时的子博弈。注意这是一个子博弈。它从一个单点节点开始。它没有打破任何信息集，并且它包含从它开始的节点的所有后继。好的？那确实是一个子博弈。好的？我们将通过考虑第二个子博弈来开始我们的分析。第二个子博弈。好的？让我们写出对应于第二个子博弈的收益矩阵。我将用以下方式写出它。我声称在这个第二个子博弈中，每个玩家有两个选择。他们可以战斗或退出。战斗或退出。我将以一种特殊的方式写出收益。我把收益写成负C加上这个东西。所以与其在所有格子里都保留那个负C，那样很快就会变得无聊，我把它提出来，放在前面。

[段 25]

Is that okay? We had this sunk cost box everywhere, and I’m going to pull out this sunk cost box and put it in front. So here it is. If you get into the second period of the game, you’ve incurred this sunk cost. And your payoffs in this game, if you both fight, then you incur minus C for the second time. if A fights and B quits, then A is going to win the prize, so they’ll get V and player 2 will get nothing. If B gets nothing, if B fights and A quits,If player B gets nothing. If B fights and A quits, then conversely, A gets nothing and player B gets the prize. And if they both quit, they just get nothing. So just notice what I did here. I could have written out the box with minus C, minus C here, minus C, minus C here, minus C plus B here, minus C plus 0 here, minus C plus 0 here, etc. But I just pulled out that minus C because it’s just distracting everything. All right, so I pull out that minus C and put it in front. All right. Okay, so now we can analyze this little game. And let’s start off by talking about pure strategy equilibria in this game. So again, our goal is to find subgame perfect equilibria. So the way in which we find subgame equilibria is what?

[译文 25]

这样可以吗？我们有一个沉没成本方框到处都是，我要把这个沉没成本方框拿出来放到前面。就是这个。如果你进入游戏的第二个阶段，你就产生了这个沉没成本。在这个游戏中，如果你俩都战斗，那么你会第二次承受负C。如果A战斗而B退出，那么A会赢得奖品，所以他们会得到V，而玩家2什么都得不到。如果B什么都得不到，如果B战斗而A退出，如果玩家B什么都得不到。如果B战斗而A退出，那么反过来，A什么都得不到而玩家B得到奖品。如果他们都退出，他们就什么都得不到。注意我这里做了什么。我本来可以把这个方框写出来，这里是负C，这里是负C，这里是负C，这里是负C，这里是负C加B，这里是负C加0，这里是负C加0，等等。但我只是把那个负C抽出来了，因为它让一切都变得混乱。好吧，我把这个负C抽出来放到前面。好吧。现在我们可以分析这个小游戏了。让我们先从谈论这个游戏中的纯策略均衡开始。再次，我们的目标是找到子博弈完美均衡。那么我们找到子博弈均衡的方法是什么？

[段 26]

We start at the last subgames. We look for Nash equilibria in those last subgames. And then eventually we’re going to roll those back. All right, so we, there’s our last sub game, there’s the matrix for it. Let’s just find the, let’s find the Nash Equilibrium. So, if player B, if player B is fighting, then player A’s best response is to quit. And if player A is fighting, then player B’s best response is to quit. Conversely, if player B is quitting, player A’s best response is to fight. And if B is fighting, sorry, ifIf A is quitting, then player B’s best response is to fight. I didn’t say that right. Let me try again. So if A is fighting, if the other side is fighting, you want to quit. If the other side is quitting, you want to fight. All right, is that clear? So there are actually two, there are two, be careful here, pure strategy. there are two pure strategy Nash Equilibria in this sub-game. And what are they? They are fight, quit, and quit, fight. So if we get into the second sub-game, and if we know we’re going to play a pure strategy in the second sub-game, this is what’s going to happen. That’s our claim. All right. Now notice that it didn’t matter, the sunk cost didn’t matter there. I could have included the sunk cost in the payoffs, but I would have found exactly the same thing with or without the sunk cost.

[译文 26]

我们从最后的子博弈开始。我们在那些最后的子博弈中寻找纳什均衡。然后最终我们会把它们回推。好吧，这是我们的最后一个子博弈，这是它的收益矩阵。让我们找出纳什均衡。所以，如果玩家B正在战斗，那么玩家A的最佳反应是退出。如果玩家A正在战斗，那么玩家B的最佳反应是退出。反过来，如果玩家B退出，玩家A的最佳反应是战斗。如果B在战斗，抱歉，如果A在退出，那么玩家B的最佳反应是战斗。我说得不对。让我再试一次。所以如果A在战斗，如果对方在战斗，你想退出。如果对方在退出，你想战斗。好的，这清楚了吗？所以实际上有两个，要注意这里，纯策略。在这个子博弈中有两个纯策略纳什均衡。它们是什么？它们是战斗、退出，和退出、战斗。所以如果我们进入第二个子博弈，并且如果我们知道我们将在第二个子博弈中玩纯策略，这就是将要发生的事情。这是我们的主张。好的。现在注意，沉没成本在那里并不重要。我本来可以在收益中包括沉没成本，但有没有沉没成本，我都会找到完全相同的结果。

[段 27]

So as we expect, the sunk cost is irrelevant. All right. So we’ve got both the equilibria and the sub game. Let’s roll this back. Let’s roll these back into the first stage of the game.Before I do that, the payoffs associated with these, the payoffs associated with this one are v and 0, and the payoff associated with this one is 0 and v. Everyone okay with that? Okay, let’s revisit the first stage of this game. And now things get a little bit more complicated. So what I’m going to do is I’m going to redraw the first stage of this game. So here it is. A can fight or quit, just as before. And following this, B can fight or quit, just as before. So this is a picture of the first stage of the game, but I’m going to chop off the second stage. Let’s put the payoffs in. So the payoffs down here are the same as they were before. 0, 0, working up. 0, V. If A fights and B quits, then it’s V, 0. But what about the payoff if they both fight? So what I want to do now is I want to look at the payoff of the movement.So what I want to do now is I want to look at the payoff when they both fight by considering what they would get if they both fight in the second period of the game.

[译文 27]

所以如我们所料，沉没成本是无关的。好的。我们在子博弈中有了两个均衡。让我们把它们回推到游戏的第一阶段。在我做那件事之前，与这个相关的收益是v和0，与这个相关的收益是0和v。大家对那没问题吧？好的，让我们回到这个游戏的第一阶段。现在事情变得更复杂一些了。所以我要做的是重新画一下这个游戏的第一阶段。就是这个。A可以战斗或退出，和之前一样。然后在此之后，B可以战斗或退出，和之前一样。所以这是游戏第一阶段的图，但我把第二阶段切掉了。让我们把收益放进去。所以这里的收益和之前一样。0，0，往上走。0，V。如果A战斗而B退出，那就是V，0。但是如果他们都战斗的收益呢？所以我现在要做的是我想看一下移动的收益。所以我现在要做的是我想通过考虑如果他们在游戏的第二阶段都战斗会得到什么，来看看当他们都战斗时的收益。

[段 28]

Our idea is find the Nash equilibrium in the second period of the game and roll back these possible payoffs. So here, the payoffs are going to be minus C plus stage 2 Nash equilibrium payoffs for player 1, and minus C plus the same thing, stage 2 Nash equilibrium payoffs for player B. All right, just understand what I’ve written here then. I’ve got the same payoff as before, but I’ve replaced, can I reach it there, I’ve replaced that enormous thing that was the second stage of the game just with the payoffs that we know that we’re going to get in the second stage of the game if we play Nash equilibrium. in the second stage. So these objects have a name, and the name is continuation payoffs.These objects are the continuation payoffs, so the payoffs I’m going to get tomorrow and possibly forward in a more complicated case if, in this case, if I fight, if we both fight in the first period. All right. Now what we want to do is we want to draw up the matrix that corresponds to this first stage game, and we’re going to have to do so twice. We’re going to have to do so once for the case where the continuation payoffs are v0, where the equilibrium we’re playing tomorrow is fight-quit. And we’re going to have to do so again for the case where the continuation payoffs tomorrow are 0v, namely the continuation play is quit-fight.

[译文 28]

我们的想法是在游戏的第二阶段找到纳什均衡，然后回推这些可能的收益。所以这里的收益将是负C加第二阶段纳什均衡收益给玩家1，以及负C加同样的东西，第二阶段纳什均衡收益给玩家B。好的，理解我这里写的是什么。然后。我有和之前一样的收益，但我替换了，能达到那里，我用我们知道在游戏的第二阶段如果我们在第二阶段玩纳什均衡会得到的收益替换了那个巨大的代表游戏第二阶段的东西。所以这些对象有一个名字，名字叫续延收益。这些对象是续延收益，所以如果在这个例子中，如果我战斗，如果我们在第一阶段都战斗的话，这些是我明天以及可能在更复杂的情况下会得到的收益。好的。现在我们要做的是画出对应于这个第一阶段游戏的矩阵，而且我们必须做两次。我们必须为续延收益是v0的情况做一次，也就是我们明天要玩的均衡是战斗-退出。我们还必须为明天续延收益是0v的情况再做一次，也就是续延玩法是退出-战斗。

[段 29]

All right, I’ve done it twice. All right, so you’ve got this continuation payoff down, so let me give you a little bit of room. All right, so the matrix is going to look as follows. this big matrix, 2 by 2. Player A is choosing fight or quit. Player 2 is choosing fight or quit. That much is easy. It’s what we put in here that matters. So let’s do all the easy cases first. So quit, quit is 0, 0. Quit, fight is 0, V. Fight quit is V0.fight-quit is v0, and in here it’s going to depend which of these two games, which of these two equilibria is being played tomorrow. So what we’re going to do here is we’re going to do the case for the equilibrium fight-quit in stage two. We’re going to write up the matrix for the case, we’re going to play fight-quit tomorrow. So if we play fight… Okay, thank you, thank you. Thank you. Thank you. It’s a good idea. Try and keep me consistent today because it’s very easy to slow. So that’s good. Once again, I’m going to use capital letters for player A and small letters for player B. Thank you. All right? So what happens if we both fight? We both incur costs of C from fighting. And tomorrow, we’re going to get the payoffs from the equilibrium.

[译文 29]

好的，我做了两次。好的，你把这个续延收益放下，所以让我给你留点空间。好的，所以矩阵将如下所示。这个大矩阵，2乘2。玩家A在选择战斗或退出。玩家2在选择战斗或退出。这很容易理解。重要的是我们在这里放什么。让我们先处理所有简单的情况。所以退出，退出是0，0。退出，战斗是0，V。战斗，退出是V，0。战斗，退出是v0，而在这里将取决于这两个游戏中的哪一个，这两个均衡中的哪一个是明天在玩。所以我们在这里要做的是，我们要为第二阶段战斗-退出均衡的情况写出矩阵。我们要写出矩阵的情况，我们明天要玩战斗-退出。所以如果我们玩战斗……好的，谢谢，谢谢。谢谢。谢谢。这是个好主意。尽量让我今天保持一致，因为很容易走神。所以这很好。再说一次，我用大写字母表示玩家A，小写字母表示玩家B。谢谢。好的？如果我们都战斗会发生什么？我们都会从战斗中承受成本C。而明天，我们将从均衡中得到收益。

[段 30]

Fight quit, that’s this equilibrium. So we’re going to get V and 0 tomorrow. Let’s add those in. So this will be plus V, and this will be plus 0. All right? And let’s just put some short crown here to indicate that these are going to be continuation payoffs. So this is the matrix we’re going to use to analyze the first stage of the game in the case where the equilibrium we’re playing tomorrow is fight-quit. And as I promised, we have to do this twice. So the other case, of course, is if we’re playing the other equilibrium tomorrow. So let’s just do that. so once again we have fight 1 quit 1 fight 1 quit 1 and the payoffs are same as we had 0 0 0 and in here this time we’re going to have minus c plus 0 and minus c plus v and the reason for the change is that we’re now looking at the continuation game the continuation play where it’s player A who quits in the second stage. So this is for the case quit 2, fight 2 in period 2. Alright? So let’s just pause and make sure everyone’s got that down. Everyone okay? So what we’ve done here is we started off by analyzing what’s going to happen in period 2, and that really wasn’t very hard. Is that right?

[译文 30]

战斗，退出，这是这个均衡。所以我们明天会得到V和0。让我们把它们加进去。所以这将是加V，而这将是加0。好的？让我们在这里放个简短的标记来表示这些将是续延收益。所以这是我们要用来分析第一阶段游戏的矩阵，在这个情况下我们明天要玩的均衡是战斗-退出。正如我所承诺的，我们必须做两次。所以另一种情况是，如果我们明天在玩另一个均衡。让我们来做那个。所以又一次我们有战斗1退出1战斗1退出1，收益和我们之前一样0，0，0，而在这里这次我们将有负C加0和负C加V，而变化的原因是我们现在在看续延游戏，续延玩法是玩家A在第二阶段退出。所以这是第二阶段退出2，战斗2的情况。好的？让我们暂停一下确保每个人都理解了。大家都还好吗？所以我们在这里做的是，我们首先分析了第二阶段会发生什么，而这真的不太难。对吧？

[段 31]

That was a pretty simple game to analyze. Pretty easy to find the equilibrium. Then what we did was we rolled back the equilibrium payoffs from period 2, and we plonked them on top of the relevant payoffs in period 1. So in particular, if you both fight, and you know you’re going to play the fight-quit equilibrium tomorrow, then your payoffs will be minus c plus v and minus c plus 0. If you both fight, and you know you’re going to play the quit-fight equilibrium tomorrow, then your payoffs will be c plus 0, so minus c plus 0, and minus c plus v. And just to emphasize, once again, these four boxes we created, these four boxes we created correspond to the stage 2 Nash equilibrium payoffs, or the continuation payoffs of the game. All right? Okay, so now we’re ready to analyze each of these games. All right, so this isn’t going to be too hard. All right, let’s try and find out the Nash equilibrium of this game. So let’s start with the left-hand one. This is the case where player A is going to fight and win in period 2. So if player B is going to quit in period 1, then if player A fights, she gets V. If she quits, she gets 0. So she’s going to want to fight. Everyone okay with that? and if prayer be fightand if player B fights in period 2, then if pair A fights, she gets minus C plus V, and if she quits, she gets 0, and here’s where our assumption’s going to help us.

[译文 31]

这是一个非常简单的博弈分析。很容易找到均衡。然后我们做的，是把第二时期的均衡收益回推到第一时期的相关收益上。具体来说，如果你们双方都选择战斗，并且知道明天将进行战斗-退出博弈均衡，那么你们的收益分别是负C加V和负C加0。如果你们双方都选择战斗，并且知道明天将进行退出-战斗博弈均衡，那么你们的收益分别是负C加0和负C加V。再强调一下，我们构建的这四个格子，这四个格子对应的是第二阶段纳什均衡收益，或者是博弈的续存收益。明白吗？好的，现在我们准备分析每一个这样的博弈。这不会太难。让我们试着找出这个博弈的纳什均衡。让我们从左边这个开始。这是玩家A在第二时期将要战斗并获胜的情况。因此，如果玩家B在第一时期将要退出，那么如果玩家A战斗，她得到V。如果她退出，她得到0。所以她会想要战斗。大家同意吗？如果玩家B在第二时期战斗，那么如果玩家A战斗，她得到负C加V，如果她退出，她得到0，这就是我们的假设起作用的地方。

[段 32]

We’ve assumed, what do we assume? We assume V is bigger than C. Is that right? Just like we played in class. So because V is bigger than C, this is going to be the best response again. Fighting’s going to be the best response. Alright, so we know that in fact player A here has a dominant strategy. The dominant strategy is to fight in period one in this analysis of the game. And since A is fighting, not surprisingly, we’re going to find that B is going to quit. Alright, so B’s best response is to quit. Alright, so here is our Nash equilibrium in this sub-game. So this game has a Nash equilibrium, only has one, and the equilibrium is fight one, quit one. And let’s just talk about it intuitively for a second. Intuitively, if I know, if I’m playing Jake, and I know that Jake is going to fight tomorrow, and I know, sorry, all the way around. I know that Jake’s going to quit tomorrow. I know Jake’s going to quit tomorrow, and I’m going to fight tomorrow. I know that tomorrow I’m going to win. Right? So thatI know that tomorrow I’m going to win. So that prize is there for me tomorrow. So why would I want to quit today? I’m going to get a dollar tomorrow if I just fight in this period.

[译文 32]

我们假设了什么？我们假设V大于C。对吗？就像我们在课堂上玩的那样。因为V大于C，这又会是最佳反应。战斗会是最佳反应。好的，所以我们知道在这种情况下玩家A实际上有一个占优策略。在这个博弈分析中，占优策略是在第一时期战斗。既然A选择战斗，毫不奇怪，我们会发现B会选择退出。所以B的最佳反应是退出。好的，这就是这个子博弈中的纳什均衡。所以这个博弈有一个纳什均衡，只有一个，均衡是战斗一、退出一。让我们直观地解释一下。直观地说，如果我知道，如果我在和Jake玩，我知道Jake明天将要战斗，对不起，反过来说。我知道Jake明天将要退出。我知道Jake明天将要退出，而我明天将要战斗。我知道明天我会赢。对吧？所以那个奖励明天就在那里等着我。我为什么要今天退出？如果我在这期战斗中，我明天会得到一美元。

[段 33]

So why would I want to quit today when at worst case scenario, I’m only going to lose 75 cents today? So if I know Jake is quitting tomorrow, I’m going to stay and fight now. And conversely, if Jake knows he’s quitting tomorrow, and he knows I’m going to fight now, he may as well just quit. So what we’re learning here is in this particular example, in this particular example, if we know that tomorrow I’m going to win the war, I’m actually going to win it today. Say that again. If we know that tomorrow I’m going to win the war, I’m actually going to win it today. And the converse is true for the case where I’m the quitter and Jake’s the fighter tomorrow. So once again, it’s pretty quick to see that if I’m going to, from Jake’s point of view, If I’m going to fight, he’s going to want to fight. If I’m going to quit, he’s going to want to fight. So in either case, he’s going to want to fight, so I’m going to want to quit. So the Nash equilibrium in this game is quit 1, fight 2. I surely can write that at a better angle. Let’s try that again. Yes. Quick one, fight two. All right? Okay.fight two. Alright? So at this stage, we’ve found all of the pure strategy sub-game perfect equilibria in the game.

[译文 33]

那我为什么要今天退出，最坏的情况下我今天只会损失75美分？如果我知道Jake明天会退出，我现在就留下来战斗。相反，如果Jake知道他明天会退出，并且我知道我现在要战斗，他还不如就退出。所以我们在这里学到的是，在这个特定的例子中，如果 我们知道明天我将赢得这场战争，我今天实际上就会赢它。再说一遍。如果 我们知道明天我将赢得这场战争，我今天实际上就会赢它。对于我是退出者而Jake是明天战斗者的情况，反过来也是对的。所以再一次，很容易看出，如果我要战斗，他就会想要战斗。如果我要退出，他就会想要战斗。所以无论哪种情况，他都会想要战斗，所以我就会想要退出。所以这个博弈中的纳什均衡是退出一、战斗二。我当然可以写得更好看一些。让我们再试一次。是的。很快，战斗二。好的？那么在这个阶段，我们已经找到了这个博弈中所有的纯策略子博弈完美均衡。

[段 34]

Let’s describe them before I write them up. Right? One pure strategy Nash equilibrium has me fighting in period one, and Jake quitting in period one, and if we got to period two, which in fact we won’t, then I fight again, and he quits again. So let’s write that equilibrium up. And we’ll do it here. We’ll do it on the top board, actually. Thank you. Thank you. Let me get it right when I write it up as a whole equilibrium. So I claim that I’ve now found all of the pure strategy SBE in this game. One of them involves my fighting in the first period and fighting in the second period, and Jake quitting in the first period and quitting in the second period. And the other one, just flips it around, I quit in the first period, and if I got there I would quit in the secondin the first period, and if I got there, I would quit in the second period. And Jake fights in the first period, and if we got there, he would also fight in the second period. So these are perfectly natural equilibria to think about. If you want to get intuitively, each of these equilibria involves a fighter and a quitter. The fighter always fights, the quitter always quits. If I know I’m playing a quitter, I’m always going to fight, so that’s the best response.

[译文 34]

让我在写出来之前先描述一下。对吧？ 一个纯策略纳什均衡是，我在第一时期战斗，Jake在第一时期退出，如果进入第二时期——实际上我们不会进入第二时期——那么我再次战斗，他再次退出。让我们把这个均衡写出来。我们在这里做。我们实际上要在上面的黑板上做。谢谢。谢谢。让我在写整体均衡时把它写对。所以我声称我已经找到了这个博弈中所有的纯策略子博弈完美均衡。其中一个涉及我在第一时期战斗并在第二时期战斗，Jake在第一时期退出并在第二时期退出。另一个则反过来，我在第一时期退出，如果进入第二时期，我也会在第二时期退出。Jake在第一时期战斗，如果进入第二时期，他也会在第二时期战斗。这些都是非常自然的均衡。想直观理解的话，每个均衡都涉及一个战斗者 和一个退出者。战斗者总是战斗，退出者总是退出。如果我知道我在和一个退出者玩，我总是会战斗，所以这是最佳反应。

[段 35]

If I know I’m facing a fighter, I’m going to want to quit, so that’s the best response. And those are two very simple equilibria. That’s the good news. What’s the bad news here? The bad news is we haven’t achieved our goal. Our goal was to argue that rational players might get involved in a fight. And notice that in each of these two pure strategy sub-game perfect equilibria, in each of them, no real fight occurs. Is that right? In each of them, one person fights the first period, but the other person just runs away. That isn’t much of a fight. So again, in each of these equilibria, one side is willing to fight, but the other side isn’t, so no fight occurs. In particular, no costs are incurred in either of these equilibria. But I claimed at the beginning I wanted to explain how we can have costs occur in equilibrium. Rational players are going to incur costs. So what am I missing here? What should I do to try and find a more costly equilibrium? I claim I’m still missing some equilibria here. What kind of equilibria am I missing? I’m missing the mixed strategy equilibria. So far, all we’ve done is solve out the pure strategy equilibria. But we need to go back and reanalyze the whole game, looking now for mixed strategy equilibrium.

[译文 35]

如果我知道我在和一个战斗者对抗，我就会想要退出，所以这是最佳反应。这是两个非常简单的均衡。这是好消息。这里的坏消息是什么？坏消息是我们没有达到我们的目标。我们的目标是论证理性参与者可能会卷入战斗。请注意，在这三个纯策略子博弈完美均衡中，每一个均衡中都没有发生真正的战斗。对吗？在每一个均衡中，有一个人在第一时期战斗，但另一个人就跑掉了。那算不上什么战斗。所以再次，在每个这些均衡中，一方愿意战斗，但另一方不愿意，所以没有战斗发生。具体来说，在这些均衡中都没有产生任何成本。但我在开头声称我想解释我们如何在均衡中产生成本。理性参与者会产生成本。那我这里遗漏了什么？我应该怎么做才能找到一个成本更高的均衡？我声称我这里还遗漏了一些均衡。我遗漏了什么样的均衡？我遗漏了混合策略均衡。到目前为止，我们所做的只是求解纯策略均衡。但我们需要回去重新分析整个博弈，现在寻找混合策略均衡。

[段 36]

All right. So we’re going to do the entire, take a deep breath, because we’re going to take the whole analysis we just did, we’re going to repeat the entire analysis we just did, but this time we’re going to look at mixed strategy equilibrium. All right, everyone happy with what we’re doing? So first of all, we’re going to go back to the second sub-game. Here’s the second sub-game. And we already found the pure strategy equilibria, so let me get rid of them. And in your notes, you probably want to rewrite this matrix, but I’m not going to rewrite it here because we’re a little bit short of time. This is exactly the same payoff matrix we saw before, But now I want to look for a mixed strategy equationNow, I want to look for a mixed strategy equilibrium in this game. How do I go about finding, this is a good review, how do I go about finding a mixed strategy equilibrium in a game like this? What’s the trick for finding mixed strategy equilibrium? Should we try our guys from New Jersey and New York? Let’s try our guys from New Jersey and New York. Where’s my New Yorker? We’ll have the true battle here between New York and New Jersey. How do we find a mixed strategy equilibrium? Use the P’s and Q’s and set them equal to one another.

[译文 36]

好的。我们要做全部的，深吸一口气，因为我们要把刚才做的整个分析再做一遍，但这次我们要寻找混合策略均衡。好的，大家对我们要做什么满意吗？所以首先，我们要回到第二个子博弈。这是第二个子博弈。我们已经找到了纯策略均衡，所以让我把它们去掉。在你们的笔记中，你们可能想重写这个收益矩阵，但我不会在这里重写，因为我们时间有点紧。这正是我们之前看到的同一个收益矩阵。但现在我想寻找一个混合策略方程。现在，我想在这个博弈中寻找混合策略均衡。我如何着手寻找，这是很好的复习，我如何在这个博弈中寻找混合策略均衡？寻找混合策略均衡有什么诀窍？我们要用来自新泽西和纽约的家伙们吗？让我们试试来自新泽西和纽约的家伙们。我的纽约人在哪里？这将是纽约和新泽西之间的真正战斗。我们如何找到混合策略均衡？用P和Q，把它们设为相等。

[段 37]

That’s a crude explanation. Crude thing, okay. So the answer was we find the P’s and Q’s and, quote, set them equal to one another. What is it we’re actually setting equal to what? What are we setting equal to what? Let’s try and get some response on this. Did our New Jersey pipe flee? Where’s our New Jersey person? They fled. We could give our Texans another chance. There was a Texan down here somewhere. What is it I said equal to what I? Guess the chances that one that one would fail and the other that one would quit the other way to fight Not quite not quite That the remark about using P’s and Q’s was right, but what is it? This is good review for the final What is I’m gonna do with those P’s and Q’s yeah, you have it out you use the other players payoffs use the other players payoffs and And? And make them indifferent between their strategies. Good. Good. I’m going to choose player B.Good. I’m going to choose player B’s mix in such a way as to make player A indifferent between choosing fight and quit, so as to make it plausible that A is actually mixing. So again, the intuition is for A to be mixing, they must be indifferent between fight and quit. So I’m going to choose the mix of B to make A indifferent.

[译文 37]

这是一个粗糙的解释。粗糙的东西，嗯。好，答案是我们找出 P 和 Q，然后把它们设为相等。我们实际上是把什么设为相等？把什么设为相等？让我们试着得到一些回应。我们新泽西的管道跑了？我们新泽西的人呢？他们跑了。我们可以再给德州人一次机会。这里有位德州人。我说了什么相等？猜一猜机会，一个会失败，另一个会以另一种方式退出战斗，不太对，不太对。关于使用 P 和 Q 的说法是对的，但那是什么？这是期末复习的好内容，我要怎么处理这些 P 和 Q，嗯，你们把它们拿出来，用其他玩家的收益，用其他玩家的收益，然后呢？让他们对自己的策略保持无所谓。好好。我要选择 player B。我要选择 player B 的混合方式，使得 player A 在选择战斗和退出之间保持无所谓，使 A 实际上进行混合显得合理。再说一遍，直觉是，要让 A 进行混合，必须在战斗和退出之间保持无所谓。所以我要选择 B 的混合方式，使 A 保持无所谓。

[段 38]

So that’s good review. Let’s do that. So here I’ve usually used the letter Q, but to avoid confusion here, let me use the letter P. We’re going to choose P to make player A indifferent. So if A fights, then their payoff is what? Let’s have a look. It’s minus C with probability P and, all right, so minus C with probability P and V with probability 1 minus P. All right, this should be coming back now. This is before the midterm. But you guys were alive before the midterm, right? So you should remember this. So they fight, they get minus c times p plus v 1 minus p. If they quit, if they quit, then they get 0 with probability p and 0 again with probability 1 minus p. So we know that if A is mixing, B must be mixing in such a way as to make these two numbers equal. So we know these two must be equal to one another. Since they’re equal, we can now solve for p. All right, so what’s that going to give us? it’s going to give us v1 minus p is equal to pc. Okay. And that, I think, is p is equal to v over v plus c. Is that right? So I’ll just check my algebra. And if you remember the game of Hawk-Dove that we saw just before the midterm, the game involving two, it was a game we looked at when we were looking at evolution, This is essentially the same game, more or less, as that game, and you’ll notice it’s the same kind of mixture we’ve got here.

[译文 38]

这是很好的复习。让我们来做。这里我通常用字母 Q，但为了避免混淆，我用字母 P。我们将选择 P 使 player A 保持无所谓。如果 A 战斗，那么他的收益是什么？让我们看看。它是负 C 的概率为 P，嗯，所以负 C 的概率为 P，且 V 的概率为 1 减 P。好的，这应该回来了。这是在期中考试之前。但你们在期中考试之前就已经在这里了，对吧？所以你们应该记得这件事。于是他们战斗，他们得到负 c 乘以 p 加上 v 乘以 1 减 p。如果他们退出，如果他们退出，则得到 0 的概率为 p，且 0 的概率为 1 减 p。我们知道，如果 A 在混合，B 必须以这样的方式混合，使这两个数值相等。因此我们知道这两个必须相等。由于它们相等，我们现在可以求解 p。好，那么这会给我们什么？它会给我们 v 乘以 1 减 p 等于 p 乘以 c。好。我想这就是 p 等于 v 除以 v 加 c。对吗？让我检查一下我的代数。如果你记得我们在期中考试前看到的 Hawk‑Dove 游戏，涉及两方的游戏，这是我们在研究进化时看过的游戏，这本质上和那个游戏差不多，你会注意到它正是我们这里得到的同类混合。

[段 39]

So P is equal to V over V plus C, which means 1 minus P is equal to C over V plus C. I’m leaving it up there a bit, hoping that one of the TAs is going to just do my algebra for me. I think that’s right, though. I think that’s right. So this game is symmetric, so both players are going to… We could do the same for B, but we’ll find the same thing. It’s a symmetric game. alright, so the mixed stratificationthing, right? It’s a symmetric game. All right? So the mixed strategy equilibrium, the mixed Nash equilibrium in this game has both mix, both fight with probability equal to v over v plus c. All right? Now this is good news because at least we’re getting some fighting going on, but we need to do something. We need to take this Nash equilibrium we’ve just found, which is a Nash equilibrium in the second sub-game. It’s a Nash equilibrium in the sub-game way up here, and we need to roll back the payoffs from this sub-game, the equilibrium payoffs from this sub-game, into the first stage. That’s our method. how are we going to do that? Well we better first of all figure out what those payoffs are. So what are the payoffs in this equilibrium? The payoffs in this mixed Nash equilibrium are what?

[译文 39]

于是 P 等于 V 除以 V 加 C，这意味着 1 减 P 等于 C 除以 V 加 C。我把它留在那儿，希望某位 TA 能帮我做代数。不过我想那是对的。我想那是对的。这个游戏是对称的，所以两位玩家都会……我们也可以对 B 做同样的事，但会得到同样的结果。这是一个对称游戏。好的，那么混合分层的东西，对吧？这是一个对称游戏。好的，所以混合策略均衡，这个游戏中的混合 Nash 均衡让双方都混合，双方都以概率 v 除以 v 加 c 进行战斗。好的？这是个好消息，因为至少我们会有一些战斗在进行，但我们还需要做点什么。我们需要把我们刚刚找到的 Nash 均衡——这是第二个子游戏中的 Nash 均衡——它是子游戏中上方的 Nash 均衡，我们需要把这个子游戏的收益回滚到第一阶段。这是我们的方法。我们该怎么做呢？嗯，我们最好先弄清楚这些收益是多少。那么，这个均衡中的收益是多少？这个混合 Nash 均衡中的收益是多少？

[段 40]

Anyone see what the payoffs are going to be if they’re both playing this mix? Well presumably the payoff from fight and the payoff from quit must be the same. Is that right? So we may as well choose the easier one. So I claim that the payoff from quit is 0 times Pis 0 times p plus 0 times 1 minus p, which is 0 v over v plus c plus 0 c over v plus c, but that’s equal to what? 0. Okay, good. So it’s got to be the case, kind of conveniently, that if they do play this mixed strategy equilibrium in stage 2, the payoff they’ll get from playing it is 0. That’s going to make life a little bit easier later on. All right, that’s our new equilibrium in the second sub game. Now let’s roll that back to the first game. Here’s our first game again. All right, and everything about this is correct. Everything about it is correct except what’s down here. So let’s get rid of what’s down here. our analysis from before is more or less still intact. It’s still the case that if they both quit, they’ll get zero. If they quit fight, they’ll get zero, V or V zero. And it’s still the case that if they both fight, they’ll both incur costs of C, and they’ll both then get stage two continuation Nash payoffs.

[译文 40]

有人看出如果他们两个都采用这个混合，收益会是多少吗？嗯，大概战斗的收益和退出的收益必须相同。对吧？因此我们不妨选择更容易的那个。于是我宣称退出的收益是 0 乘以 P，即 0 乘以 p 加上 0 乘以 1 减 p，结果是 0 乘以 v 除以 v 加 c 加上 0 乘以 c 除以 v 加 c，但那等于什么？0。好，不错。于是出现了一个很方便的情况：如果他们在第二阶段真的采用这个混合策略均衡，他们获得的收益是 0。这在后面会让事情稍微容易一点。好的，这就是第二子游戏中的新均衡。现在让我们把它回滚到第一游戏。这是我们的第一游戏再次。好的，关于这点的所有内容都是正确的。所有内容都是正确的，除了这里下面的部分。让我们去掉下面的部分。我们之前的分析或多或少仍然成立。如果他们两个都退出，他们会得到零。如果他们退出战斗，他们会得到零、V 或 V 零。如果他们两个都战斗，他们会各自承担 C 的成本，然后各自获得第二阶段继续的 Nash 收益。

[段 41]

Is that right? But now, instead of those continuation Nash payoffs being V00V,payoffs being v0 or 0v, those continuation Nash payoffs are going to be what? They’re going to be zero. They’re going to be zero. So what we’re going to do here is we’re going to backward induct or roll back those zero payoffs and come up with the corresponding matrix to describe the first stage of the game. And here it is. Fight, quit, little f. trying to get it right without Jake having to correct me this time little f, little q, this is a this is b and the payoffs here are 0, 0 here, 0, v v, 0, just as before and in this box now we’ve got minus c plus 0 and minus c plus 0. So it’s exactly the same box we saw before, but now the continuation payoffs are just zero. And again, what is this? This is for the Nash equilibrium, as I say, for the mixed Nash equilibrium in period 2. all right and what I want to do is I want to find the mixed equationAnd what I want to do is, I want to find the mixed equilibrium in period 1. We found the mixed equilibrium in period 2, now I want to find the mixed equilibrium in period 1. So what I could do here, is I could spend a lot of time.

[译文 41]

是这样吗？但现在，那些继续的 Nash 收益不再是 V00V，收益是 v0 或 0v，那些继续的 Nash 收益会是什么？它们将是零。它们将是零。于是我们要做的是在这里进行逆向归纳或回滚这些零收益，并写出对应的矩阵来描述游戏的第一阶段。它是这样的。战斗，退出，little f. 这次努力把 little f, little q 写对，免得 Jake 来纠正我，这是 a 这是 b，这里的收益是 0,0 这里，0, v v, 0，和之前一样，而在这个格子里我们现在得到的是负 c 加 0 和负 c 加 0。因此这和我们之前看到的格子完全一样，只是现在的继续收益只是零。再说一遍，这是什么？这就是我在说的 Nash 均衡，混合 Nash 均衡在第 2 期的均衡。好的，而我想做的是，我想找到混合方程，我想找到第 1 期的混合均衡。我们已经在第 2 期找到了混合均衡，现在我想在第 1 期找到混合均衡。于是我在这里可以花很多时间。

[段 42]

I could put in a p and a 1 minus p, and I could work out what mix of player B will make A indifferent, I could work out what mix of player A will make B indifferent, but has anybody noticed something about this matrix? What do you notice about this matrix? Somebody? Somebody help me out? Somebody’s got to help me out here. Tell me something about this matrix. What’s true about this matrix? It’s the same as the one above. It’s the same as the one above, right? The matrix I just drew when I rolled back the payoffs, when I rolled back the payoffs, is exactly the same matrix that I had here. It’s exactly the same matrix. So we already know what the mixed strategy equilibrium is in this. the mixed Nash equilibrium in this matrix is to fight, is both fight with probability P equals V over V plus C. So we’re now ready to show our new sub-game perfect equilibrium. Let’s write it down. Here’s our whole game. We found the pure SPEs, but now we’re ready to find the mixed SPE. the mixed sub-game perfect equilibrium has player A well before I do that let me just give this P a name let me call this P P star so V over V plus C let’s call it P star so the mixed sub-game perfect equilibrium has player 1 mixing fighting with probability P star in the first stage and in the second stage again mixing fighting with probability P star this is player 1 and player 2 does exactly the same thing.

[译文 42]

我可以放进一个 p 和 1 减 p，我可以算出 player B 的何种混合会使 A 保持无所谓，我可以算出 player A 的何种混合会使 B 保持无所谓，但有人注意到这个矩阵有什么特别之处吗？你们注意到这个矩阵有什么？有人吗？有人帮我一下吗？这里一定要有人帮我。告诉我关于这个矩阵的什么。它和上面的那个一样。它和上面的那个一样，对吧？我刚才在回滚收益时画出的矩阵，和我在这里的矩阵完全一样。它完全一样。因此我们已经知道在这个矩阵中的混合策略均衡是什么。这个矩阵中的混合 Nash 均衡是战斗，双方都以概率 P 等于 V 除以 V 加 C 进行战斗。于是我们现在准备好展示新的子游戏完美均衡。让我们把它写下来。这是我们的完整游戏。我们已经找到了纯策略 SPE，但现在我们准备找到混合 SPE。混合子游戏完美均衡有 player A，嗯，在我做那之前，让我先给这个 P 起个名字，叫它 P star，即 V 除以 V 加 C。于是混合子游戏完美均衡有 player 1 在第一阶段以概率 P star 混合战斗，在第二阶段再次以概率 P star 混合战斗，这是 player 1，而 player 2 也做完全一样的事情。

[段 43]

And while we’re here, what’s the expected payoff for each player if they’re playing this mixed sub-game perfect equilibrium? What is it? It’s zero. It’s zero. The payoff from this, the expected payoff, is 0. So now we’re actually getting somewhere. Now we’re really getting somewhere.So now we’re actually getting somewhere. Now we’re really getting somewhere. So let’s just take a deep breath and see where we are. We broke this game down, this complicated game we played in class, that it conceivably, for example, when New York is playing New Jersey, conceivably could go on all night. Apparently not when the Texans are playing each other, or the football team is playing the baseball team, but when we have New York and New Jersey, it could go on all night. We curtailed it to a two-period game, but in a minute we’re going to go back to the infinite game. In this two-period game, I’m trying to convince you that you could get fighting occurring just in equilibrium with absolutely standard rational players. Nothing to do with pride, nothing to do with reputation, nothing to do with the fact that these guys are crazy guys who’ve drunk the water in New York and New Jersey. God help them. You can get fighting with ordinary people in equilibrium. And what we’ve shown is the way in which you can get fighting is in a mixed strategy equilibrium.

[译文 43]

那么当他们玩这个混合子博弈完美均衡时，每个玩家的预期收益是多少？多少？是零。是零。这个的预期收益是零。所以现在我们确实有所进展了。现在我们真的有所进展了。那么让我们深呼吸一下，看看我们处于什么位置。我们把这个游戏分解了，这个我们在课堂上玩的复杂游戏，比如当纽约对上新泽西时，理论上可能会打一整晚。显然当德州人对阵彼此，或者橄榄球队对棒球队时不会，但当我们有纽约对阵新泽西时，可能会持续一整晚。我们把它缩减为一个两期游戏，但一会儿我们会回到无限游戏。在这个两期游戏中，我试图说服你，在均衡中你也可以看到打架发生，而参与者是完全标准的理性人。与骄傲无关，与声誉无关，与这些人是因为喝了纽约和新泽西的水而变疯了这件事无关。愿上帝保佑他们。普通人也可以在均衡中打架。而我们所要展示的是，你可以得到打架的方式是在混合策略均衡中。

[段 44]

In each period of the game, people fight with probability P. That’s just enough fights to give the other side an incentive to quit, and just enough probability of the other side quitting to give the other side an incentive to fight. Just exactly enough. If they play that equilibrium in every period, there’s some chance of the game ending, but with probability P, the game goes on.But with probability P, the game goes forward to the next period. So you could potentially have fights for two periods. By the way, with what probability would there be a fight in both periods? That’s a good homework question. I won’t answer it here. You can work it out at home. All right? Do we do it here? Anybody want to tell me? Okay, with what probability do we get a fight in the first period? A real fight, a fight involving both players? We need both players to fight. Each are fighting with probability P, so the probability of both of them fighting is what? P squared. So to get a fight in the first period, the probability is P squared. To get a fight in both periods, it’s what then? P to the fourth. But we get a fight with probability P to the fourth going through. We get fighting in equilibrium. Moreover, we get some very intuitive things that we already learned in the hawk-dove game.

[译文 44]

在游戏的每一期，人们以概率P打架。这恰好足够的打架次数给另一方退出激励，也恰好足够的另一方退出的概率给这一方打架的激励。刚刚好。如果他们在每一期都玩那个均衡，游戏有一定概率结束，但以概率P，游戏继续进行。所以你可能会有两期的打架。顺便问一下，两个期都有打架的概率是多少？这是一个很好的作业题。我不会在这里回答。你可以在家算出来。好吗？我们要在这里算吗？有人想告诉我吗？好，在第一期有打架的概率是多少？真正的打架，两个玩家都参与的打架？我们需要两个玩家都打架。每个以概率P打架，所以两个都打架的概率是多少？P的平方。所以要在第一期有打架，概率是P的平方。要在两期都有打架，那是什么？P的四次方。但我们以概率P的四次方得到打架。我们得到均衡中的打架。而且，我们还得到了一些我们在鹰鸽博弈中已经学到的非常直观的结论。

[段 45]

The probability of fights, so in this equilibria, the probability of fights occurring goes up as v goes up. So the prize gets bigger, you’re more likely to see fights occur. That seems right. And it goes down in c. So the probability ofin C. So the probability of fights occurring goes up in the size of the prize, that seems intuitively right, and down in the cost of fighting. All right. Now, okay, that’s reasonable, but I claimed I could show you this not in a two-period game, but in an infinite period game. So let me spend the last five minutes taking us to infinite period games. All right, so everybody take a deep breath, we’re now going to consider something we’ve never done before. We’re going to consider a game that could go on forever. It could go on forever. And the way we’re going to do that is to use the following idea, the following picture. So I can’t really draw a true tree for the infinite period game. And the reason I can’t draw a true tree for the infinite period game is one, I would run out of chalk, and two, I’d run into lunchtime. But you could imagine what it looks like. It looks like this, roughly speaking. Try to get this right. Start again. The infinite period game looks something like this.

[译文 45]

打架的概率，在这个均衡中，打架发生的概率随着v增加而上升。奖金越大，你越可能看到打架发生。这看起来是对的。而且随着c下降。所以打架发生的概率随着奖品规模增加而上升，这看起来直觉上是对的，随着打架成本的增加而下降。好。那么，这个是合理的，但我声称我可以向你展示的不是在两期游戏中，而是在无限期游戏中。所以让我花最后五分钟带我们到无限期游戏。好，每个人深呼吸一下，我们现在要考虑我们从未做过的事情。我们要考虑一个可以永远进行下去的游戏。它可以永远进行下去。我们要做的是使用以下想法，以下图示。所以我无法真正为无限期游戏画一个真正的树。原因是，第一，我会用完粉笔，第二，我会遇到午餐时间。但你可以想象它长什么样。它看起来像这样，大致来说。试着把它画对。再来一次。无限期游戏看起来像这样。

[段 46]

Thank you.and then it goes again alright, and then it goes again and so on, alright and it would go right the way through the board and right the way across whatever street that is right across campus, alright that’s what the infinite tree would look like so I clearly can’t really analyze that object but what I want to show you is that we can still solve this game even though it’s an infinite period game How are we going to do that? Let’s look at a particular stage. Let’s call the stage stage 4,503. Whatever that number was, right? 4,503, whatever it was, right? So here is the stage, this arbitrary stage, and the tree for this arbitrary stage looks like this. All right? And what I’m going to do is, this is stage, what was it, 5,000, was it 4,503? All right, this is the stage. All right, and what I’m going to add to this is that before you get into this stage, you’re going to incur sunk costs. All right, and if you go on playing after this stage, then you’re going to get… continuation values. Continuation values.So going into the beginning of the game, you’ve incurred some sunk costs. And if you come out on the other side and go on playing, you’re going to play some equilibrium and get continuation values. But otherwise, everything else is the same.

[译文 46]

谢谢。然后它再次进行，好，然后它再次进行，以此类推，好，它会一直穿过整个黑板，一直穿过那条街，一直穿过整个校园，好，这就是无限期树的样子。所以我显然无法真正分析那个东西，但我想向你们展示的是，尽管这是一个无限期游戏，我们仍然可以求解这个游戏。我们要怎么做？让我们看一个特定的阶段。让我们称这个阶段为阶段4,503。不管那个数字是多少，对吧？4,503，不管它是什么，对吧？这是这个阶段，这个任意阶段，而这个任意阶段的树看起来像这样。好吗？我要做的，这是阶段，是多少，5,000，是4,503吗？好，这是这个阶段。好，我要补充的是，在进入这个阶段之前，你将承担沉没成本。好，如果你在这个阶段之后继续玩，那么你将得到延续值。延续值。所以在进入游戏开始时，你已经承担了一些沉没成本。如果你从另一边出来继续玩，你将玩某个均衡并得到延续值。但除此之外，其他一切都是一样的。

[段 47]

We still have 0, 0 here. We still have 0, V here. We still have V, 0 here. And here we still have minus C plus continuation values and minus C plus continuation. values. Alright? And this is something we’ve seen before. Right? This little box is something we’ve seen before. Essentially, we’ve got some cost in front, but they’re irrelevant. We’ve got continuation values at the end, but we know how to handle them. We just put them into the payoffs. Alright? So suppose now, suppose that in the continuation game, people play the mixed strategy that we just found. Suppose that in the continuation game, people mix with probability P, so they fight with probability P star and quit with probability 1 minus P star. Suppose in the continuation game, they’re playing a mixed strategy. In that case, what is the continuation value of the game? Let me say it again. What is it’s zero right if they’re mixing in the future they always had the option tothey’re mixing in the future, they always had the option to quit, so it must be that the continuation value is zero. So if they mix in the future, then the continuation value is zero, zero. All right, so now let’s go back to this board. All right, to make this board equivalent to the board above, all I need to do is one thing.

[译文 47]

我们这里仍然是0,0。我们这里仍然是0,V。我们这里仍然是V,0。而这里我们仍然是负C加延续值和负C加延续值。好吗？这东西我们之前见过。对吧？这个小方框是我们之前见过的。基本上，我们在前面有一些成本，但它们是无关的。我们在末尾有延续值，但我们知道如何处理它们。我们只是把它们放进收益中。好吗？所以假设现在，假设在延续游戏中，人们玩我们刚刚发现的混合策略。假设在延续游戏中，人们以概率P混合，所以他们以概率P星打架，以概率1减P星退出。假设在延续游戏中，他们正在玩混合策略。在这种情况下，游戏的延续值是多少？让我再说一遍。它是零，对吧，如果他们在未来混合，他们总是有退出的选项，所以延续值一定是零。所以如果他们在未来混合，那么延续值是零，零。好，现在让我们回到这块黑板。好，为了使这块黑板等价于上面的黑板，我只需要做一件事。

[段 48]

I need to add on some sunk costs at the front. I’ve got sunk costs at the front. I’m going to play the game, and then I’m going to get, instead of stage 2 values, I’m going to get stage, what was it, 4,503 in all stages in the future values in here and here. But otherwise, it’s the same thing. And what’s convenient is, all of those are zero anyway. Since they’re all zero anyway, this matrix is still correct. The continuation values are zero and zero. These are now the continuation values. And these are the continuation values. And so if I look for a mixed strategy equilibrium in this game, it’s something I’ve solved already. What’s the mixed strategy equilibrium in this game? Anybody? Exactly.It’s exactly what we found before. Just as before, I’m going to mix with probability V over V plus C. All right. Now let’s summarize. We did something today, just now, that we’ve never done before. We’ve looked at an infinite, or at least potentially infinite, period game. A game that could go on forever. and the way in which we handled the game that could go on forever was what? We noticed two things. We noticed that part of the game that comes before, that part of the game that’s passed already, anything that happened there is just a sunk cost. It’s irrelevant.

[译文 48]

我需要在前面加上一些沉没成本。我有前面的沉没成本。我要玩这个游戏，然后我得到的是，不是第2期的值，而是所有未来期中的阶段，是多少，4,503的值。但除此之外，它是一样的。方便的是，所有这些都是零。既然它们都是零，这个矩阵仍然是正确的。延续值是零和零。这些现在是延续值。这些是延续值。所以如果我在这个游戏中寻找混合策略均衡，它是我已经解过的东西。这个游戏中的混合策略均衡是什么？有人知道吗？正是如此。恰恰是我们之前发现的。就像之前一样，我要以概率V除以V加C混合。好，现在让我们总结一下。我们今天刚才做了一件我们从未做过的事情。我们研究了一个无限的，或者至少是潜在无限的期游戏。一个可以永远进行下去的游戏。我们处理这个可以永远进行下去的游戏的方式是什么？我们注意到两件事。我们注意到游戏在之前发生的部分，已经过去的部分，任何发生在那里的事情都只是沉没成本。是无关的。

[段 49]

I mean, it hurts if it’s a cost. It’s nice if it’s a game, but it’s sunk. You can’t affect it now. Anything in the future, anything in the future can be summarized by the value, the payoff I’m going to get in the future by playing the equilibrium in the future. In this case, the future meant mixing. Mixing gave me a value of zero. So here I am getting zero from the future. And then I can just analyze the game in the stage in which I’m in just as if it was an ordinary bog-standard simultaneous move game. When we did so in this particular example, we’re going to see more examples like this after the break, but in this particularexample, we’re going to see more examples like this after the break, but in this particular example we found out something quite surprising. The thing we found out was in these war of attrition settings, there is an equilibrium with rational players. Everyone, more than that, common knowledge of rationality, everyone is rational, everyone knows everyone else is rational, there are equilibria in which not only people fight, but they could fight forever. In every period, they fight with some probability. and we got an extra prediction out, a prediction that we weren’t expecting. And let me just give you that prediction and then we’ll leave the class with that.

[译文 49]

我的意思是，如果是成本，那会让人痛苦。如果是一场游戏，那还不错，但它已经沉没了。你现在无法影响它。未来任何事情，任何未来事情都可以用价值来概括，即通过在未来进行均衡博弈我将获得的回报。在这种情况下，未来意味着混合策略。混合策略给了我零的价值。所以我从未来得到的是零。然后我就可以像分析一个普通的同时行动博弈那样分析我所处的阶段。当我们在这个特定例子中这样做时，我们会看到更多这样的例子，休息之后我们会看到更多这样的例子，但在这个特定例子中，我们发现了一些相当令人惊讶的事情。我们发现的是在这些消耗战设定中，存在理性参与人的均衡。不仅如此，是理性的公共知识，每个人都是理性的，每个人都知道其他每个人是理性的，存在这样的均衡：不仅人们会战斗，而且他们可以永远战斗下去。在每个时期，他们以一定的概率战斗。我们得到了一个额外的预测，一个我们没有预料到的预测。让我把这个预测告诉你们，然后我们就带着这个离开课堂。

[段 50]

The extra prediction is this. If we look at these wars of attrition and we keep track of the time in which the… Hang on guys, don’t rustle the bags yet. One more thing. If we look at the time in which the games have gone on and keep track of the probability that a war will end at that time. So imagine this is World War I. You could imagine World War I going for one year or two years or three years or 20 years or whatever. The probability distribution in this war of attrition is going to look like this. In every period, the probability of continuing is just p star squared. So in every period, as you go further into the future, there’s a greater chance the war will end. You can get very long, very costly wars. That’s the bad news. The good news is it doesn’t happen very often.The good news is it doesn’t happen very often. I guess we’re all involved in a rather large and costly war right now, so I’ll leave you with that pleasant thought over Thanksgiving. Have a good break, and we’ll see you

[译文 50]

这个额外的预测是这样的。如果我们观察这些消耗战，并追踪游戏持续的时间……等一下，同学们，先别急着弄响袋子。还有一件事。如果我们观察游戏持续的时间，并追踪战争在那时结束的概率。假设这是第一次世界大战。你可以想象第一次世界大战持续一年、两年、三年、20年或任何时间。这个消耗战中的概率分布会呈现这样的形态。在每个时期，继续的概率就是p星的平方。所以在每个时期，随着你更深入地未来，战争结束的可能性就越大。你可以得到非常漫长、代价非常高的战争。这是坏消息。好消息是这种情况不经常发生。好消息是这种情况不经常发生。我想我们所有人现在都卷入了一场相当大规模且代价高昂的战争中，所以我把这个令人愉快的想法留给你们过感恩节。休息愉快，我们下次见。