耶鲁大学博弈论公开课 - 第20集 纳什均衡之约会游戏与古诺模型

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]

All right, so last time we were talking about the investor game, and this was a coordination game. And we learned some things, I just want to recall some of the things we learned so I can highlight them a bit. one thing we learnt was that communication can help in a coordination game so I forget who it was but someone down here who’s disappeared was our Jimmy Stewart character and helped coordinate you on a better equilibrium simply by suggesting what you should do and there’s two lessons here one is this is very different from the prisoner’s dilemma So we learnt the very first time that in the prisoner’s dilemma communication per se won’t help but in a coordination problem which could be just as serious socially as a prisoner dilemma in a coordination problem it may well help and the reason it helps is you trying to coordinate onto a Nash equilibrium And one way to think about Nash equilibria is that they are self-enforcing agreements, so provided everyone believes that everyone else is going to go along with this agreement, then everyone in fact will. I want to make another slightly more philosophical remark associated with this, and it’s to do with the idea of leadership.

[译文 1]

好的，上次我们讨论了投资者博弈，这是一个协调博弈。我们学到了一些东西，我只想回忆一下我们学到的一些东西，这样我可以强调一下。我们学到的一件事是，沟通可以帮助协调博弈。我忘记是谁了，但这里有个人消失了，他是我们的Jimmy Stewart角色，通过简单地建议你应该做什么来帮助你更好地协调到一个更好的均衡。这里有两个教训，首先这与囚徒困境非常不同。我们第一次学到，在囚徒困境中，沟通本身不会有所帮助，但在协调问题中——这可能在社会上与囚徒困境一样严重——在协调问题中，沟通可能会很有帮助。它有帮助的原因是，你试图协调到一个纳什均衡。思考纳什均衡的一种方式是，它们是自我执行的协议，因此，只要每个人都相信其他人会遵守这个协议，那么实际上每个人都会遵守。我想对此做另一个稍微更哲学性的评论，这与领导力的理念有关。

[段 2]

So leadership is kind of a big word that you see written probably too often these days in too many newspaper articles and it comes up in too many Yale classes and I don’t claim to know anything about leadership all right, but and then the game theory is going to contribute anything to understanding about leadership, but one thing we can do is tell you where leadership may help All right And in a coordination game where the idea is to try and get people to coordinate on a particular equilibria rather than another equilibria, or worse still, to be uncoordinated entirely, in those kind of games, leadership can help tremendously. A little bit of leadership can help tremendously. So these games these coordination games are games where there is a scope for leadership And just to see that in a very simple example again we don need such a complicated example as the one we looked at last time, you could imagine a game, a really trivial coordination game, which looked like this. All right, 1-1-0-0-0-0-1-1. And clearly, in this game, what matters is coordinating. You either want to coordinate on up left or you want to coordinate on down right. You don’t want to end up uncoordinated on down left or up right. Everyone see that? So in this game, if you just played it, it’s quite likely you’re going to end up uncoordinated.

[译文 2]

领导力是一个你在报纸文章和Yale课程中经常看到的大词。我不声称对领导力有任何了解，好吧，但博弈论将对理解领导力有所贡献。我们可以做的一件事是告诉你领导力可能在哪些地方有帮助。好吧，在协调博弈中，其想法是试图让人们协调到一个特定的均衡而不是另一个均衡，或者更糟糕的是，完全不协调，在这类游戏中，领导力可以非常有帮助。一点点领导力就可以非常有帮助。所以这些博弈，这些协调博弈，是有领导力发挥空间的博弈。为了在另一个非常简单的例子中看到这一点，我们不需要像上次那样复杂的例子，你可以想象一个游戏，一个非常简单的协调博弈，它看起来像这样。好吧，上左上右下右下左上右下右下左上左。在这个世界里，关键在于协调。你要么想在上左协调，要么想在下右协调。你不想最终不协调地落在左下或右上。大家都明白了吗？所以在这个游戏中，如果你只是玩它，你很可能会以不协调告终。

[段 3]

But if a little bit of leadership can say, okay, let’s make sure this is where we coordinate. Or let’s make sure this is where we coordinate. So this matters a lot. I don’t want to overplay the social importance of this, but go back a couple of years to what was happening in the aftermath of Katrina and realize how important, how bad things get when things fail to be coordinated. All right. One other remark before we leave this in the game we played last time in the investment game one feature of that game was that the more you thought other people were going to invest the more you wanted to invest Is that right The more you thought other people were going to invest the more likely you were going to invest. And if you go back to the time before that, we talked about this partnership game. And the partnership game, you were contributing effort to a joint project. It could have been a law firm, it could have been working in a study group on a homework assignment. And that game, if you remember what the best responses looked like. They looked like this, where this was the effort of player one and this was the effort of player two. This was the best response of player one and this was the best response of player two.

[译文 3]

但如果一点点领导力可以说，好吧，让我们确保这是我们协调的地方。或者让我们确保这是我们协调的地方。这很重要。我不想过分强调这在社会上的重要性，但回想几年前卡特里娜飓风的后果发生的事情，意识到当事情协调失败时，事情会变得多么糟糕，这是多么重要。好吧。在我们离开这个话题之前，还有另一个评论，在我们上次玩的投资游戏中，这个游戏的一个特点是，你认为其他人投资的越多，你就越想投资。对吗？你认为其他人投资的越多，你就越有可能投资。如果你回到之前的时间，我们讨论了这个合作关系游戏。在合作关系游戏中，你为一个 joint 项目贡献努力。这可能是一家律师事务所，也可能是在学习小组做作业。而那个游戏，如果你记得最佳反应是什么样子的。它们看起来像这样，这是玩家一的努力，这是玩家二的努力。这是玩家一的最佳反应，这是玩家二的最佳反应。

[段 4]

This game also has the feature that the more effort the other person, the more the other person does, the more you want to do. The more effort your partner provides into this project, the more effort you want to provide in this project. And I just want to introduce a bit of jargon here. These games in which the more the other person does, the more I want to do, these are called games of strategic compliments. These are games of strategic compliments. Strategic compliments. So both the investment game and the game of Complements. All right, so both the investment game and the game with the partnership firm game are games of strategic compliments. We can call the strategies strategic compliments. And we’ll come back to this later on today. All right, so before we leave coordination games, I want to look at another one, a little bit more complicated one perhaps, that we mentioned briefly last time. So we’ll look at and play another game, and we’ll call this game Going to the Movies. All right, so I always regard one purpose of this class to help hapless Yale students in their dating strategies. That seems like a good thing to do. And how many of you just… We didn’t need to show the camera on this, but how many of you are econ majors? There’s a lot of people who probably need a lot of help with their dating strategies, right?

[译文 4]

这个游戏还有一个特点，另一个人付出的努力越多，另一个人做得越多，你就越想做。伴侣在这个项目中投入的努力越多，你就越想在这个项目中投入更多。我在这里想引入一些术语。在这些游戏中，另一方做得越多，我就想做越多，这些被称为战略互补游戏。这些是战略互补游戏。战略互补。所以投资游戏和合作游戏都是战略互补的游戏。好吧，投资游戏和合作关系游戏都是战略互补游戏。我们可以把这些策略称为战略互补。我们今天稍后会再讨论这个。好吧，所以在离开协调游戏之前，我想看看另一个，一个可能更复杂一点的游戏，这是我们上次简要提到的。所以我们来看另一个游戏，我们把它叫做"去看电影"。好吧，我一直认为这门课的一个目的是帮助可怜的Yale学生约会策略。这似乎是一件好事。有多少你们……我们不需要把摄像机对准这个，但有多少人是经济学专业的？有很多人可能需要在约会策略方面提供很多帮助，对吧？

[段 5]

Right? Okay. So the idea of this game is a couple is going to meet up at the movies and let have a look at the movies concerned here and we draw just three possible movies All right, well the movies we’re going to look at are The Bourne Ultimatum, and the movie called Good Shepherd, and a movie called Snow White. And I’ll explain the game in more detail in a second. It used to be in the old days, before I had kids, I could list off 15 current movies and help you a bit more with your dating strategy by giving you instant movie reviews, but now I have kids, I get to see precisely two movies a year, and the two movies I got to see this year were The Bourne Ultimatum and The Good Shepherd, and now I think about it, both of those have Matt Damon in it, and my wife chose both, so maybe I need to work out some more. So just to give you some idea about this movie, how many of you saw Born Out Ultimatum? Quite a lot of you. This is a movie, it has pretty good action, virtually no plot. Basic lesson of this movie is if you take home lesson it that all spies are psychos or something And then how many of you saw Good Shepherd A lot of you saw that as well That actually pretty good It has a lot of plot and no action so it the other way around And the basic lesson of this movie is, and you probably knew this already, everyone at Yale, take place at Yale, everyone at Yale is a spy.

[译文 5]

对吗？好吧。那么这个游戏的想法是，一对情侣要在电影院见面，让我们看看这里涉及的电影，我们画出三种可能的电影。好吧，我们要看的电影是《伯恩的最后通牒》，还有一部叫《好牧人》的电影，以及一部叫《白雪公主》的电影。我会马上更详细地解释这个游戏。在过去的日子里，在我有了孩子之前，我可以列出15部当前的电影，并通过给你即时电影评论来帮助你更好地制定约会策略，但现在我有了孩子，我每年只能看两部电影，今年我看的两部电影是《伯恩的最后通牒》和《好牧人》，现在我想想，这两部都有Matt Damon主演，我妻子选了这两部，所以我可能需要再想想。让我给你一些关于这部电影的概念，有多少人看了《伯恩的最后通牒》？你们很多人。这是一部动作片，相当不错，几乎没有情节。这部电影的基本教训是，如果你在家里学到什么教训的话，那就是所有的间谍都是疯子或者其他什么。然后有多少人看了《好牧人》？你们很多人也看了。那部实际上相当不错。它有很多情节，没有动作，所以是另一种方式。这部电影的基本教训是，你们可能已经知道了，每个人在Yale，在Yale发生的事，每个人在Yale都是间谍。

[段 6]

And by the way, also a psycho. And the third movie is Snow White, which I haven’t gone out to see, but my four-year-old daughter has seen on 24 of the last 27 lights on video. And this movie, I don’t know if I’d recommend it that much. It’s, as I’m being too PC, but I’m not convinced that for the modern woman, hanging around waiting for your prince to come is really a good strategy. By the way, for those of you who are, I don’t know if you are, but if you are doing that strategy, I’m going to get in trouble, never mind. If you are doing that strategy, take it from a Brit, Most princes are as dumb as toast. It’s not worth waiting for. All right, so let’s put the payoffs in here. So the idea here is that these people are going to meet at the movie, they’re going to go out to a movie, and they’ve decided to go to the Criterion, or the local movie house, and there are these three movies showing, and they’re all excited about going to this movie, except being economics majors and not very good at dating they have forgotten to tell each other which movie they actually going to go to They going to meet in there on the back row probably but they’re not telling us which movie.

[译文 6]

顺便说一句，也是个疯子。第三部电影是《白雪公主》，我没有出去看过，但我四岁的女儿在过去27天里看了24次录像。我不知道我是否会那么推荐它。鉴于我表现得过于政治正确，但我不相信对于现代女性来说，四处闲坐等待王子到来真的是一个好的策略。顺便说一句，对于你们中正在这样做的人，我不知道你们是不是，但如果你们正在使用那种策略，我会遇到麻烦的，不要介意。如果你正在使用那种策略，听一个英国人的忠告，大多数王子都笨得像烤面包。不值得等待。好吧，让我们把收益放进去。这里的想法是，这些人要去电影院见面，他们要去看电影，他们决定去Criterion或者当地的电影院，有这三部电影正在上映，他们都很兴奋去看这部电影，只是因为他们是经济学专业的，不擅长约会，他们忘记告诉对方他们实际要去哪部电影。他们可能要在后排在里面见面，但他们不告诉我们是哪部电影。

[段 7]

So it’s a problem, and let’s put in the payoffs, and we’ll put in the payoffs that roughly, I think, would correspond to my preferences, and we’ll talk about what these preferences mean in a second. So here we go. So minus 1, minus 1, 0, minus 1, 0, minus 2, minus 2. So here are the preferences for these movies of player 1 and player 2. And you can see from these preferences, from these payoffs, that the best thing for player 1 is for both players, both people, to meet and go to the Bourne Automata. This is the action movie. All right, so there’s not to be sexiness, let’s assume this is the she and this is the he. So she would like to go and see Matt Damon beat people up and coordinate there, that’s her favorite thing, and she gets nothing if they fail to coordinate, and so on and so forth. Her second choice is to manage to coordinate at Matt Damon being a It’s nothing if they fail to coordinate, and so on and so forth. Her second choice is to manage to coordinate at Matt Damon being a Yale spy, and then she really doesn’t want to go and see Snow White, and the worst thing of all, actually this is for both players, the worst thing of all is to coordinate at Snow White, because then you’ve got to talk about it afterwards at coffee.

[译文 7]

所以这是个问题，我们来列出收益，我们列出的收益大致对应我的偏好，我们稍后会讨论这些偏好意味着什么。好，开始。负1，负1，0，负1，0，负2，负2。这就是玩家1和玩家2对这些电影的偏好。从这些偏好，从这些收益中，你可以看到，对玩家1来说，最好的情况是两个人都见面并去看《伯恩的最后身份》。这是动作片。好，所以这里没有色情内容，假设这是女主角这是男主角。所以她想去看马特·达蒙揍人并在那里碰头，这是她最喜欢的事，如果他们没能协调好她什么都得不到，等等。她的第二选择是能协调好去看马特·达蒙当耶鲁间谍的电影，然后她真的不想去看《白雪公主》，而最糟糕的事情，实际上对两个玩家来说，最糟糕的事情是在《白雪公主》那里碰头，因为那样你事后喝咖啡时还得讨论这部电影。

[段 8]

And similarly for the other person, same thing exactly, except that he would rather meet at the Good Shepherd, since he’s a Yale guy, he probably thinks of himself as the Matt Damon character there, or perhaps not, and failing that, he’d like to coordinate at the Bourne Ultimatum, but again, the disaster is to fail to coordinate and go to see different movies. All right? So these are the preferences. What can we tell immediately, if you’re stuck in this situation, what can you tell immediately about this game? What’s an easy first step? Let’s get the guy in red here. Snow White is dominated by both players. Good. So for both players, for both players, Snow White is a dominated strategy. So both players, having taken this class, I told you it would help you with your dating, but all right so both players will realize that they should at least not go to Snow White Is that right All right great animation but not perhaps a good date movie So it gone All right everyone see that So that leaves us just with these two options, the two Matt Damon movies, Born Ultimatum and Good Shepherd, and here are the remaining payoffs. And let’s see how we play this out. So let’s pick on some people. So, Ali, do you want to pick on somebody? I’ll get Kai to pick on somebody as well.

[译文 8]

同样对另一个人来说，完全一样的事情，只是他更愿意在《好牧人》碰头，因为他是个耶鲁人，他可能把自己想象成那里马特·达蒙的角色，或者也许不是，退而求其次，他想要在《伯恩的最后身份》协调，但同样，灾难是没能协调去看不同的电影。好吧？所以这就是偏好。如果你被困在这种情况下，你能立刻告诉我们这个游戏的什么信息？什么是简单的第一步？让我们找穿红衣服的人。《白雪公主》被两个玩家支配。好。所以对两个玩家来说，对两个玩家来说，《白雪公主》是一个被支配的策略。所以两个玩家，既然你们上过这门课，我告诉过你们这会帮助你们约会，但好吧所以两个玩家都会意识到他们至少不应该去看《白雪公主》对吗？好的很棒的动画但也许不是约会的好电影所以它走了好吧大家看到了吗？所以这让我们只剩下这两个选项，两部马特·达蒙电影，《伯恩的最后身份》和《好牧人》，这是剩余的收益。让我们看看我们怎么玩这个游戏。让我们挑几个人。所以，阿里，你想挑一个人吗？我也让凯来挑一个人。

[段 9]

All right. All right, so, okay, okay, so you picked on somebody there. When you stand, the person who Ali picked on, do you want to stand up? Okay, and your name is? Nina. Say again loudly. Nina. Aina, Aina, Aina. Just shout it out so everyone hears. Aina. And Kai, did you pick on a guy? Nina. Nina, okay Nina, I’m sorry, Nina. And Kai, get your guy to stand up there. And your name is? David. David, so do you know each other? No. No? Okay, so you’re about to go on a date, so this is a bit, you know, here’s a. All right all right so Nina preferences Nina preferences are Player One preferences here So she her best thing now is to successfully meet David at the Bourne Ultimatum and David preferences are Player 2 preferences so his favourite thing is to successfully meet at the Good Shepherd And why don’t you write down, a second, on the corner of your notepad what it is you’re going to choose to do, with my TAs watching, as you really can’t cheat. alright, both of them written down alright should we have a good look at them so we just embarrass them a bit we’ve looked at them already I guess, okay so Nina shout out to the crowd where it is you’ve chosen to go Bourne Ultimatum Bourne Ultimatum, and David where do you choose to go I also put down Bourne Ultimatum they’re going to coordinate, that’s very good alright I don’t know what this tells us about David and Nina, but don’t let them off the hook quite yet.

[译文 9]

好吧。好吧，所以，好，好，所以你挑了一个人。当阿里挑的人站起来，你想要站起来吗？好的，你叫什么名字？妮娜。再大声说一遍。妮娜。艾娜，艾娜，艾娜。就喊出来让大家都听到。艾娜。凯，你挑了一个男生吗？妮娜。妮娜，好吧妮娜，抱歉妮娜。凯，让你的男生也站起来。叫什么名字？大卫。大卫，你们认识吗？不？认识吗？不？好吧，所以你们要约会，这有点，你们知道，这是一个。好好好吧所以妮娜的偏好妮娜的偏好是玩家一偏好这里所以她最好的事情现在是成功地和大卫在《伯恩的最后身份》见面而大卫的偏好是玩家二偏好所以他最喜欢的事情是在《好牧人》成功地见面为什么不你们写下，第二个，在你们笔记本的角落里写下你们打算选择做什么，有我的助教看着，你们真的没法作弊。好吧，两人都写下来了我们应该好好看看所以我们让他们尴尬一下我们已经看过了我猜，好吧所以妮娜大声喊出来你选择了去哪里《伯恩的最后身份》《伯恩的最后身份》，大卫你选择去哪里我也写了《伯恩的最后身份》他们要协调了，这非常好好吧我不知道这告诉我们关于大卫和妮娜什么，但先别让他们下台。

[段 10]

So now, that was good. We managed to coordinate, although I think we can all see that we might not have done. We can all see we might not have done. Let’s try some communication in this game. We said that coordination games are helped by communication. So let imagine we playing the game again Once again you going to the movies Once again the first time didn happen so for some reason it was cancelled that night Someone had a bad cold or something and you are about to go off on Friday night to see these movies, and let’s allow some communication ahead of time. So David, you realize you’ve got this coordination problem, so you phone Nina up, and you can say whatever you like. So what would you say to Nina? Looks like you want to go to Bourne Ultimatum. I’d rather go and, if you’re going to be stubborn, I’d rather go and see Born Ultimatum with you than not go on a date at all. So here’s another clue. Stubborn is not a good first date word. All right? Okay. So Nina, what would you say in response to that? I’ll go to Born Ultimatum. All right. All right. So write down what you’re going to do this time. All right, and Nina, what did you do? Born ultimatum. David? Born ultimatum. So you’re still managing to coordinate, but you can, okay, so thanks for this couple.

[译文 10]

所以现在，这很好。我们成功协调了，虽然我认为我们都能看到我们可能没有做到。我们都能看到我们可能没有做到。让我们在这个游戏中尝试一些沟通。我们说过协调游戏有沟通会更容易。所以让我们想象我们再玩一次这个游戏又一次你们要去看电影又一次第一次没有发生所以出于某种原因那天晚上取消了有人得了重感冒或者什么你们准备周五晚上去看这些电影，让我们允许一些事先沟通。所以大卫，你意识到你有这个协调问题，所以你打电话给妮娜，你可以说任何你想说的话。所以你会对妮娜说什么？你想去《伯恩的最后身份》。我宁愿，如果你是要固执的话，我宁愿和你去看《伯恩的最后身份》而不是完全不约会。所以这是另一个线索。固执不是约会第一天的好词。好吧？好吧。所以妮娜，你会怎么回应？我去看《伯恩的最后身份》。好的。好的。所以写下你这次要做什么。好吧，妮娜，你做了什么？《伯恩的最后身份》。大卫？《伯恩的最后身份》。所以你们仍然成功协调了，但是你们可以，好的，所以谢谢这对情侣。

[段 11]

Let’s give them a round of applause. Thank you. We may come back and pick on you later on in the course, but we’ll leave it for now. for this couple. Let’s give them a round of applause. We may come back and pick on you later on in the course, but we’ll leave it for now. So in this case, the communication worked, but am I right in thinking that the communication isn’t such an instant solution as it was in the game we saw last time? This game’s a little bit harder to get communication to work. Now why? I mean, is that right, first of all? So what’s the problem here? Why is this a slightly more difficult game? Let’s keep the mics handy a second. Why is this a more difficult game to attain coordination in? Yeah, let’s… Can you… Neither of the best responses is the Nash equilibrium. Well, no, hang on a second. So let’s slow down a second. That’s a good one. Let’s start with that. What are the Nash equilibria in this game? What are the Nash equilibria in this game? Somebody, let’s just pick on somebody. Just pick on anybody, okay? Yeah, what are the Nash equilibria in this game? Both players doing Bourne Ultimatum and both players doing Good Shepherd. All right. So, in fact, if we check the Nash Equilibria in this game, the Nash Equilibria you can check that they are in fact best responses are both people doing the Born or both people doing the Good Shepherd And the reason for that is if the other person is going to go to Born I want to do that.

[译文 11]

让我们给他们一轮掌声。谢谢。我们后面可能还会挑你们，但这会儿先这样。对这对情侣。让我们给他们一轮掌声。我们后面可能还会挑你们，但这会儿先这样。所以在这种情况下，沟通起作用了，但我的想法对吗，沟通不像上次游戏中那样是即时解决方案？这个游戏更难让沟通起作用。现在为什么？我是说，首先这是对的吗？这里的问题是什么？为什么这是一个更难协调的游戏？让我们把麦克风准备好。为什么这是一个更难达到协调的游戏？是的，让我们…你能…两个最佳反应都不是纳什均衡。嗯，不，等一下。让我们慢下来。这是一个好的。让我们从这里开始。这个游戏中的纳什均衡是什么？这个游戏中的纳什均衡是什么？有人，让我们挑个人。就挑个人，好吧？是的，这个游戏中的纳什均衡是什么？两个玩家都看《伯恩的最后身份》和两个玩家都看《好牧人》。好吧。所以，实际上，如果我们检查这个游戏中的纳什均衡，你可以检查它们实际上是最佳反应，两个人都看《伯恩》或者两个人都看《好牧人》，原因是如果另一个人要去《伯恩》，我想去看《伯恩》。

[段 12]

And if the other person is going to go to Good Shepherd, I want to do that. So these are both Nash Equilibria. That isn’t so much the problem. That was true last time as well. There were two well-coordinated Nash Equilibria, although one of them was better than the other. what’s the tricky thing here? What’s the extra trick here? Somebody Is there a camera? Do we have a mic on this side? Sorry. There was a hand up there. Yeah. Yep. Go, go, go. There you go. Stand up. Great. Each player prefers a different Nash equilibrium. Alright, so unlike the other game the two examples of coordination games we saw so far were really pure coordination problems. Is that right? They were pure coordination problems. There was no conflict at all. In the case of the investor game last week, every player preferred one equilibria to the other. Right, in the game I just put up, this is a trivial example, it didn’t really matter which equilibrium we played here, we just wanted to play an equilibrium Is that right Is that right But here there a potential source of conflict here Both people would rather be at an equilibrium than to be malcoordinated or uncoordinated, but player one wants to go to born ultimatum and player two wants to go to good shepherd. And actually, I thought Nina’s strategy there was pretty good.

[译文 12]

如果另一个人要去《好牧人》，我想去看《好牧人》。所以这都是纳什均衡。这不是问题。那上次也是这样。有两个协调良好的纳什均衡，虽然其中一个比另一个好。这里棘手的事情是什么？这里额外的技巧是什么？有人有摄像机吗？我们这边有麦克风吗？抱歉。那里有人举手了。是的。好的，去，去，去。站起来。好。每个玩家偏好不同的纳什均衡。好吧，所以不像我们之前看到的协调游戏的两个例子，实际上是纯粹的协调问题。对吗？它们是纯粹的协调问题。根本没有冲突。在上周的投资者游戏中，每个玩家都偏好一个均衡而不是另一个均衡。对，在我刚放上去的游戏中，这是一个微不足道的例子，实际上选哪个均衡并不重要，我们只是想玩一个均衡，对吗？是对的吗？但这里有一个潜在的冲突来源。两个人都宁愿处于均衡状态也不愿失调或不协调，但玩家一想去《伯恩的最后身份》，玩家二想去《好牧人》。而且实际上，我认为妮娜的策略相当不错。

[段 13]

Nina’s version of communication was, I’m going to born ultimatum, right? Right? That basically fixed things, right? But you could imagine that communication could break down here. There could be a negotiation going on. So think about, I don’t want to push this too hard, but think about other games where people are communicating to avoid a failure of coordination. There’s a strike negotiation going on as we speak. In fact, there’s a critical moment this morning up in Detroit between General Motors and the United Auto Workers. And I don’t want to say that strikes are only about miscoordination, but clearly everyone’s better off if they come to some agreement rather than strike. There’s at least some agreement that’s going to be better for everybody than a strike. And yet, because there are conflicting interests there, basically in that case conflicting interests about health and pension payments, it could well be that you end up failing to coordinate Alright so this game has a name and actually somebody mentioned it last time Who the person who mentioned this last time Somebody mentioned this What’s this game called? Shout it out. Battle of the Sexes. That’s good. So this game’s called the Battle of the Sexes. And we’ll see it in various forms over the course of the semester. It’s actually a very interesting game. games like it, they are coordination games, but different people disagree about where you’d like to coordinate.

[译文 13]

Nina版本的沟通是，我要把这个变成最后通牒，对吧？对吧？这基本上能解决问题，对吧？但你可以想象沟通在这里可能会破裂。可能会有一场谈判正在进行。所以想想看，我不想把这一点推得太远，但想想其他一些游戏，人们在其中进行沟通以避免协调失败。此刻正在进行一场罢工谈判。事实上，今天早上在底特律，通用汽车和全美汽车工人联合会之间有一个关键时刻。我不想说罢工仅仅是因为协调失败，但很明显，如果他们能达成某种协议而不是罢工，对每个人都有好处。至少存在某种协议比罢工对每个人都更有利。然而，由于存在利益冲突——基本上在这个案例中，冲突涉及健康和养老金支付——很可能最终你会无法协调。好吧，这个游戏有一个名字，实际上上节课有人提到过。谁在上节课提到过这个？有人提到过。这个游戏叫什么？大声说出来。性别之战。很好。所以这个游戏叫做性别之战。我们将在整个学期中以各种形式看到它。这实际上是一个非常有趣的游戏。像这样的游戏，它们是协调游戏，但不同的人对于你想在哪里协调存在分歧。

[段 14]

All right, so so much for talking about coordination games and helping you with your dating strategy. This being a game theory class, there’s at least one game we have to discuss and we’ll come to it now. So we’re going to spend most of the rest of today talking about Cornode duopoly and before I do that let me just check how many of you have seen cornode duopoly before Let’s raise your hand this is just for me All right, so maybe about half of you so those of you have don’t worry I mean this will be a bit review But we’ll see it more through the eyes of game theory this time and for those you haven’t don’t worry We’re going to go through it. Okay, okay, so this is a classic game. Perhaps. It’s the only review, but we’ll see it more through the eyes of game theory this time. And for those of you who haven’t, don’t worry, we’re going to go through it. So this is a classic game, perhaps it’s one of the most famous games, and therefore worth studying in the class. Now as a purely game theory exercise, one reason for studying Cournot duopoly is that so far we’ve discussed how to find Nash equilibrium when there are few players, each of whom has few strategies, and we’ve discussed how to find Nash equilibrium where there are many players, each of whom has few strategies, and this is a game where there are few players, maybe two, but they have many strategies.

[译文 14]

好了，关于协调游戏和帮助你约会策略的话题就说到这里。既然这是一门博弈论课，至少有一个游戏我们必须讨论，我们现在就来讲。所以我们今天剩下的时间大部分会用来讨论古诺双寡头，在开始之前让我先看看你们中有多少人之前看过古诺双寡头。请举手，这只是为了让我了解情况。好吧，大概有一半的你们，所以你们这些看过的不用担心，我的意思是这会有点复习。但这次我们会更多地用博弈论的眼光来看待它，而对于你们没看过的，也不用担心，我们会过一遍。好吧，好吧，这是一个经典游戏。或许。这是一个唯一的复习，但这次我们会更多地用博弈论的眼光来看待它。而对于你们没看过的，也不用担心，我们会过一遍。所以这是一个经典游戏，也许它是最著名的游戏之一，因此在课堂上值得研究。作为一个纯粹的博弈论练习，研究古诺双寡头的一个原因是，到目前为止我们已经讨论了当玩家少、每个玩家策略少时如何找到纳什均衡，我们也讨论了当有很多玩家、每个玩家策略少时如何找到纳什均衡，而这是一个玩家少、也许两个，但他们有很多策略的游戏。

[段 15]

They actually have a continuum of strategies. For those people who aren’t familiar with this and are worried about the economics of it in particular, it’s gone over in considerable detail in chapter 6 of the textbook. In chapter 6 of the textbook. Of the data book. All right So let just motivate this a little bit So there two reasons why this game is interesting So this is a game in which there are two firms who are competing in the same market And we’ll give a bit more detail in a second. And one reason this game is interesting from the point of view of economics is this game lies between the two extreme cases that you learn about in your introeconomics class. One extreme case is perfect competition, and the other extreme case is monopoly. So this is really the first attempt, way back in the 19th century, to study a market that’s somewhere in the middle, where in fact most markets are. There are two firms. And we’re interested in two things here. We’re interested in what’s going to happen in these markets. And then from a welfare point of view, from a policy point of view, we’re interested about whether this is good for consumers or good for producers or what. How this relates to profits and consumer surplus. All right, so with that in mind, let’s just set the game up.

[译文 15]

他们实际上有无穷多个策略。对于那些不熟悉这个、特别担心其经济学原理的人，教科书第6章已经相当详细地介绍过了。教科书第6章。资料书的。好吧，让我们先稍微激发一下兴趣。所以这个游戏有趣有两个原因。这是一个有两个企业在同一个市场中竞争的游戏，我们稍后会给出更多细节。这个游戏从经济学角度有趣的一个原因是，它位于你在初级经济学课上学到的两个极端情况之间。一个极端是完全竞争，另一个极端是垄断。所以这实际上是19世纪第一次尝试研究一个处于中间位置的市场，而实际上大多数市场都是如此。有两个企业。我们这里感兴趣的是两件事。我们感兴趣的是这些市场将会发生什么。然后从福利角度、从政策角度，我们感兴趣的是这对消费者是好还是对生产者是好还是怎样。这如何与利润和消费者剩余相关。好吧，带着这个想法，让我们来设置这个游戏。

[段 16]

So the players in this game are two firms. Two firms. And the strategies in this game for the firms, and this is going to turn out to be important the strategies are the quantities that they produce of an identical product So they are the quantities they produce or each of them produces of an identical product So as far as the consumers are concerned, these two products are perfect substitutes. You could think of these as two companies producing bottled water. And now I’m going to get hundreds of letters saying not all bottled waters are the same, especially from Italians and the French, but never mind. Let’s pretend that they are. So just to emphasize that the strategies are quantities, rather than using S, let me use Q today to be the strategies. So QI and Q minus I, or Q1 and Q2, will be the strategies. I need to give a little bit more structure on the payoffs before I get to the payoffs so in particular I need to tell you what is the cost of production and the cost of production in this game is simply going to be C times Q so if I produce one unit it costs me C if I produce two units it costs me 2C if I produce 100 units it costs me 100C And if I produce 0 units it costs me 0 All right So for those people who took introeconomics, which is most of you, this is a game, this is a setting in which we have constant marginal costs.

[译文 16]

所以这个游戏中的玩家是两个企业。两个企业。而这个游戏中企业的策略——这将被证明是很重要的——这些策略是他们生产的同质产品的数量。所以他们生产的是每个企业生产的同质产品的数量。就消费者而言，这两个产品是完全替代品。你可以把它想象成两家公司生产瓶装水。现在我要收到几百封信说并非所有瓶装水都是一样的，特别是来自意大利人和法国人的，但没关系。让我们假设它们是一样的。所以为了强调策略是数量，而不是用S，让我今天用Q来表示策略。所以Q_I和Q_-I，或者Q1和Q2，将是策略。在讲收益之前，我需要给收益多一点结构，所以我特别需要告诉你生产成本是多少，而这个游戏中的生产成本简单地就是C乘以Q，所以如果我生产一个单位，成本是C；如果我生产两个单位，成本是2C；如果我生产100个单位，成本是100C；如果我生产0个单位，成本是0。好吧，所以对于那些学过初级经济学的——你们大多数人都学过——这是一个游戏，这是一个我们有常数边际成本的设定。

[段 17]

Constant marginal costs. The constant marginal cost is C. Alright? And I’ll need to tell you about how prices are determined in this market. So prices are determined as follows. Prices depend on some parameters I’m just going to call A and B. And let me write the equation and then we’ll see what it looks like. So basically the idea is that the more these firms produce, so the more is the total quantity produced, Q1 plus Q2, the lower is the price in the marketplace for this product. And let’s just draw a picture of that. We’ll come back to this in a minute. Let’s draw a picture which we can keep for later. Actually, let’s save myself some time and bring this down. … we can keep for later. Actually, let’s save myself some time and bring this down. All right. So this equation I’ve just written, you can imagine Q1 plus Q2, total quantity on the horizontal axis, and you can imagine price on the vertical axis, and basically what we’re saying is that prices depend on total quantities as follows, where the slope of this line is minus B. All right? Now, for those of you who took Economics 115 or equivalence, what is the name of this line I’ve just drawn? That was way fewer responses than I wanted. All right, let’s try again.

[译文 17]

常数边际成本。常数边际成本是C。好吗？我还需要告诉你这个市场中价格是如何决定的。所以价格决定如下。价格取决于一些参数，我就叫它们A和B吧。让我写出这个方程，然后我们看看它是什么样的。所以基本上这个想法是，这些企业生产得越多，总产量Q1加Q2越大，这个产品在市场上的价格就越低。让我们画个图。我们一会儿会回来讲这个。让我们画一个图，我们可以留到后面用。实际上，让我节省点时间把它放下来。……我们可以留到后面用。实际上，让我节省点时间把它放下来。好吧。所以这个我刚写的方程，你可以想象Q1加Q2，总产量在横轴上，你可以想象价格在纵轴上，基本上我们说的是价格按如下方式取决于总产量，这条线的斜率是负B。好吗？现在，对于那些学过经济学115或同等课程的人，我刚画的这条线叫什么？那比我想要的反应少多了。好吧，让我们再试一次。

[段 18]

What’s the name of this downward sloping, hint, hint, line that I’ve just drawn? It’s a demand curve. Thank you. It’s a demand curve. All right? This is the demand. It tells me, if I look at it the other way around and look at how prices correspond to quantities it tells me the quantity demanded at any given price All right We come back to that in a second But meanwhile let just finish up what we doing here and put in payoffs. So payoffs for these firms are going to depend on profits, the firms aim to maximise profit, and profit is going to be given by P times Q minus, sorry, let’s do this for firm 1, all right? Let’s be careful here. Let’s do this for firm 1. We’ll do firm 2 in a second. So the payoff for firm one, as it depends on the quantity that she produces and the quantity that the other firm produces, is going to be prices times the quantity that firm one produces minus costs times the quantity that she produces. All right? So this term here is revenues This term here is revenues and this term here is total costs Revenues minus costs makes profits. And again, for those of you who are less familiar with economics, I’m hoping this is not going to be too hard, but you can read up on it otherwise.

[译文 18]

我刚画的这条向下倾斜的——提示，提示——线叫什么？这是一条需求曲线。谢谢。这是一条需求曲线。好吗？这是需求。它告诉我，如果我反过来看，看价格如何对应数量，它告诉我在任意给定价格下的需求量。好吧，我们一会儿会回到这个。但与此同时让我们先完成我们正在做的事情，放进收益。所以这些企业的收益将取决于利润，企业的目标是最大化利润，而利润将等于P乘以Q减去——抱歉，让我们来算企业1，好吗？让我们仔细一点。让我们来算企业1。我们一会儿再算企业2。所以企业1的收益，取决于她生产的数量和其他企业生产的数量，将等于价格乘以企业1生产的数量减去成本乘以她生产的数量。好吗？所以这个项是收益，这个项是收益，而这个项是总成本。收益减成本等于利润。同样，对于那些不太熟悉经济学的人，我希望这不会太难，但你也可以另外阅读相关资料。

[段 19]

All right? And what I’m going to do here is I’m going to substitute in for prices. alright, so I’ve got this expression for prices here here it is and I want to plug it into that P there okay, alright so I’m going to rewrite it now with that P expanded out alright, so I’m going to get, and I’m going to multiply it out at the same time, so I’m going to have AQ1 minus minus bq1 squared minus bq1q2 minus cq1. So all I’m doing here, nothing particularly exciting, I simply plugging this expression in for p and then I multiplied the whole thing out because it was all multiplied by Q1 And I should warn you again that I very likely to make mistakes when I’m doing this kind of thing, so please catch me if I do. All right? So here I have a new expression for firm 1’s profit, and I could do the same for firm 2, but I’m not going to because that would get boring. Okay? All right? So this is just firm 1’s profit. Okay. And with this in hand, what I want to do is, and firm two have a similar expression, what I want to do is, I want to figure out the Nash equilibrium of this game. Right, now I will need the other board. I want to figure out the Nash equilibrium of this game.

[译文 19]

好的，我在这里要做的就是把价格代换进去。好，这里有这个价格的表达式，就是这个，我想把它代入那个P那里，好的，我现在要把它重写一下，把那个P展开，好的，我要同时把它展开，所以我会得到AQ1减去bq1平方减去bq1q2减去cq1。我在这里所做的没有什么特别激动人心的，就是把这个表达式代入p，然后把它全部展开，因为它本来就乘以了Q1。我要再次提醒你们，当我在做这种运算的时候很可能会出错，所以如果我出错了请一定要指出来。好的。现在我有了一个企业1利润的新表达式，我可以对企业2做同样的事情，但我不会，因为那样太无聊了。好的。这是企业1的利润。好的。有了这个之后，我要做的是，企业2也有一个类似的表达式，我要做的是，我想找出这个博弈的纳什均衡。对，我现在需要另一块白板。我想找出这个博弈的纳什均衡。

[段 20]

Both firms are producing quantities, both firms are trying to maximize profit, and I want to find out the Nash equilibrium. how do I go about that exercise what’s the what do I need to do to find each firm’s Nash equilibrium how do we always find Nash equilibrium when we’re not guessing what do we do anybody so you way over there is gonna be way off camera but perhaps we get a mic in there quick we get him sorry he’s way in the What do we do? Anybody? Somebody way over there who’s going to be way off camera, but perhaps we can get a mic in there. Sorry, he’s way in the corner there. Come this way a little bit closer to the aisle. There you go. Well, we would just delete the, we would iteratively delete dominated strategies. Ah, okay, we could do that. We could do that. We could try iteratively deleting dominated strategies and see if that and see that process converged. I have something else in mind here, something that’s more straightforward. What else? This other guy in red. Identify the best responses of each player as a function of the others and find out where they intersect. Okay, that was correct. Let’s say it more loudly. Shout. Identify the best responses of each player as a function of the others and find out where they intersect.

[译文 20]

两家企业都在生产产量，两家企业都在试图最大化利润，我想找出纳什均衡。我该怎么做这个练习，找每个企业的纳什均衡需要做什么，当我们不猜测的时候我们怎么总是能找到纳什均衡，有人知道吗？有个人在那边的会完全离开镜头，但或许我们能快点把麦克风递过去。抱歉，他在角落里。往这边来一点，离过道近一点。好了。我们可以删除，我们可以迭代删除被支配的策略。啊，好的，我们可以那样做。我们可以那样做。我们可以尝试迭代删除被支配的策略，看看那个过程是否收敛。我这里有别的方法，更直接一些。那个穿红衣服的还有别的吗。识别每个参与人的最佳反应作为其他人的函数，然后找出它们的交点。好的，那是对的。说大声点。大声喊。识别每个参与人的最佳反应作为其他人的函数，然后找出它们的交点。

[段 21]

Okay, good. So what we’re going to do is we’re going to figure out player one’s best response quantity to each possible choice of player two, and then we’re going to flip it around and figure out player 2’s best response quantity to each possible choice of player 1, and then we’re going to see where those coincide, where they cross. So just like we’ve done several times before. So what I need to do then is I need to figure out what is player 1 best response for each possible choice Q2 of player 2 Alright, so how do I do that? How do I do that? I want to find the best response, how do I do that? Everyone’s looking like this is… I bet you all know really. Anybody? Do you want me to cold call somebody? No. somebody yeah sorry somebody way at the back there sorry stand up and shout find the best response function by differentiating this and maximizing the function for every Q2 ok good, so what we’re going to do is alright good, we’re going to we’re trying to maximize, the best response is going to maximize these profits, alright So in particular, what we’re going to do to find out what quantity Q1 maximizes this profit for each choice of Q2, we’re going to differentiate this with respect to Q1, and then what?

[译文 21]

好的，很好。我们要做的就是找出参与人1对参与人2每个可能选择的最佳反应数量，然后反过来找出参与人2对参与人1每个可能选择的最佳反应数量，然后看看它们在哪里一致，在哪里相交。就像我们之前做过好几次的那样。所以我需要做的是，我需要找出对于参与人2的每个可能选择Q2，参与人1的最佳反应是什么。好的，我该怎么做？我想找出最佳反应，我该怎么做？大家看起来都……我打赌你们其实都知道。有人吗？要我随便点一个人吗？不用了。有人在后面那里，抱歉，站起来喊。对每个Q2，通过对这个函数求导并最大化来找到最佳反应函数。好的，我们要做的就是最大化，最佳反应就是要最大化这些利润。好的，具体来说，对于参与人2的每个选择Q2，我们要找出哪个Q1能最大化这个利润，我们要对这个关于Q1求导，然后呢？

[段 22]

And then set it equal to zero. Okay? So again, let me just take a look out there. How many of you remember this We did this once before already this class How many of you remember this from your high school calculus days or from 112 Quite a number of you actually remember it Okay good So what we going to do is we going to differentiate this thing to find a first order condition We’re going to differentiate with respect to q1, right, the thing we’re trying, our control variable that we’re trying to maximize the thing with respect to, and set it equal to 0. So what I’m going to get, so this term here, aq1, is going to become an a, and this term here, minus bq2, is going to become a 2bq1, and this term here, minus bq1q2, is going to become a minus bq2, and the last term, minus cq1, is going to become a minus c. All right, everyone happy with that? What I did was I differentiated this fairly simple function with respect to q1. And since I want to find a maximum, what I’m going to do is I’m going to set this thing equal to 0. All right at my maximum I put a hat over it to indicate this is the argmax At my maximum I going to set this thing equal to zero Now since we being nerdy in this class despite our attempts to go dating we’re being nerdy, let’s actually be a little bit careful.

[译文 22]

然后令它等于零。好的？再看一下那边。你们有多少人记得这个？我们在这门课上已经做过一次了。你们有多少人从高中微积分或者112课程还记得这个？实际上相当多的人记得。好的，很好。我们要做的就是求导找出一阶条件。我们要关于q1求导，对吧，就是我们试图最大化的控制变量，然后令它等于0。所以我会得到的，这个项aq1会变成a，这个项负bq2会变成2bq1，这个项负bq1q2会变成负bq2，最后一项负cq1会变成负c。好的，大家都明白了吗？我所做的就是对这个相当简单的函数关于q1求导。既然我想找出最大值，我会令它等于0。在我的最大值处我加了一个帽子表示这是argmax。在我的最大值处我要令它等于零。既然我们在这门课上很书呆子气，尽管我们试图去约会，我们在书呆子，让我们实际上稍微小心一点。

[段 23]

This was a first order condition, or a first order necessary condition, I actually need to check the second order condition. So how do I check the second order condition? I differentiate again, right? I differentiate a second time and check the sign. So the second order condition, I differentiate this expression again with respect to Q1. The only place Q1 appears here is here. So when I differentiate again, I’m just going to get minus 2B and sure enough that’s negative, which is what I wanted to know. That’s just to check that when I’m finding this thing, I’m finding a maximum and not a minimum. Good. So this is our key expression and I’m going to use this expression I’m going to solve it for q1 so the best response for player 1 as a function of what player 2 chooses q2 is just equal to the q1 hat in this expression and if I solve that out carefully I will no doubt make a mistake but let’s try it I’m going to have a minus c over 2 And if I solve that out carefully, I will no doubt make a mistake, but let’s try it. I’m going to have a minus c over 2b minus q2 over 2. All right, so try that at home. But I think I did that right, so I took this to the other side, and I divided through by 2B and the B here cancels.

[译文 23]

这是一个一阶条件，或者说一阶必要条件，我实际上需要检查二阶条件。我怎么检查二阶条件？我再求一次导，对吧？我再关于Q1求导一次，检查符号。所以二阶条件，我再关于Q1对这个表达式求导一次。Q1只出现在这里一个地方。所以当我再求导一次时，我只得到负2B，果然它是负的，这正是我想要知道的。这只是要检查当我找出这个东西时，我找的是最大值而不是最小值。好的。这是我们的关键表达式，我要用这个表达式，我要把它解出q1，所以参与人1作为参与人2选择q2的函数的最佳反应就等于这个表达式中的q1帽子，如果我仔细解的话，我肯定会犯错，但让我们试试。我会得到a减c除以2b减q2除以2。好的，回家试试这个。但我觉得我做得对，我把这个移到另一边，然后除以2B，这里的B抵消了。

[段 24]

Alright, everyone just stare at that a second, make sure I get it right. I’m very, very capable of getting these things wrong. Alright, good. So what I have here is an equation that tells me player 1’s best response for each possible choice of player 2. Alright? And I could do the same thing for Player 2 to find Player 2’s best response for every possible choice of Player 1. On the other hand, we know it’s, you know, the math is going to be the same. So let’s cheat. I know that Player 2’s best response for every possible choice of Player 1 which if we done it would be Q2 hat is going to equal a minus c over 2b minus q1 over 2 I’m just flipping around the 2s and the 1s. So at this point, I found player 1’s best response as a function of q2. I found player 2’s best response as a function of q1. And the way I did it, just remember, the way he did it was I applied a little bit of 112 and or high school calculus. Right? Essentially single variable calculus. All right. Now, we could just solve at this point, but let’s not. Let’s draw these. Let’s draw these. Okay? So what I’m going to do is I’m going to go back to my picture and draw these functions.

[译文 24]

好的，大家盯着看一秒，确保我做得对。我非常有可能把这些东西搞错。好的，很好。我这里有的是一个方程，它告诉我参与人1对参与人2每个可能选择的最佳反应。好的？我可以对参与人2做同样的事情，找出参与人2对参与人1每个可能选择的最佳反应。另一方面，我们知道数学是一样的。所以让我们偷个懒。我知道参与人2对参与人1每个可能选择的最佳反应，如果我们做了的话会是Q2帽子等于a减c除以2b减q1除以2。我只是把2和1换了一下。所以现在，我找到了参与人1作为q2函数的最佳反应。我找到了参与人2作为q1函数的最佳反应。我所做的就是记住，他所做的是我应用了一点112或者高中微积分。对，本质上就是单变量微积分。好的。现在我们可以就此求解，但让我们不要。让我们画这些。让我们画这些。好的？我要回到我的图上把这些函数画出来。

[段 25]

I’ll just remind ourselves what these functions are. It’s a minus c over 2b minus q2 over 2 and equivalently for player 2. All right, that’s the thing, it is hidden by the board now. What I want to do is I want to draw a picture a little bit like we did for the partnership game. So in the partnership game we put efforts on these axes and now I going to put quantities on these axes So this is going to be the choice of player 1 and this is going to be the choice of player 2. And what I want to do is I want to figure out what this looks like. For each Q2 that you give me, or that player 2 chooses, I want to find out and draw what is player 1’s best response. alright everyone happy with what I’m doing here alright so let’s start somewhere and what we’re going to do is we’ll refer back to this other picture which is why I left it here alright so in particular what would be player 1’s best response if player 2 didn’t produce at all how would we find that without worrying about the economics of it a second how would we find that this is a math exercise So suppose player two doesn’t produce at all, all right? What is player one’s best response?

[译文 25]

让我提醒一下这些函数是什么。是 a 减 c 除以 2b 减 Q2 除以 2，对玩家 2 也是类似的。好了，这就是那个东西，现在被黑板挡住了。我要做的是画一幅图，有点像我们在合伙博弈中做的那样。所以在合伙博弈中，我们在这些轴上放努力，而现在我要在这些轴上放产量。所以这将是玩家 1 的选择，而这将是玩家 2 的选择。我想弄清楚这个图长什么样。对于你给我的每个 Q2，或者玩家 2 选择的每个 Q2，我想找出并画出玩家 1 的最佳反应是什么。大家对我正在做的都清楚吗？好，让我们从某处开始，我们要做的是我们会参考另一幅图，这就是为什么我把它留在这里。所以具体来说，如果玩家 2 完全不生产，玩家 1 的最佳反应是什么？我们怎么找到它，先不考虑经济学，再想想怎么找到它，这是一个数学练习。假设玩家 2 一点都不生产，好吗？玩家 1 的最佳反应是什么？

[段 26]

Someone read it off for me? Why don’t we cold call somebody out? You can just. a minus c over 2b All right so shout that out a minus c over 2B Good so if I plug Q2 equals 0 into here this term disappears right And I just get A minus C over 2B. So what’s that telling me? Let me put the 45-degree line in here. What’s that telling me? It says if player 2 chooses not to produce, then player 1’s best response is a minus c over 2b now I claim that that quantity a minus c over 2b actually has another name what’s the name of that quantity what’s the name of the quantity a minus c over 2b once we escape the algebra and just think about it in terms of economics Well, let’s think about it a bit. Let’s go back to the previous picture. If player 2 is producing nothing, player 2 is producing nothing, then what is player 1, effectively? He’s a monopolist, or she’s a monopolist, all right? So we know how to figure out monopoly quantities from what we did in 115, or equivalent courses. Let me finish the picture here. Here’s my demand curve. Here’s my constant marginal cost at C. or equivalent courses. Let me just finish the picture here. Here’s my demand curve. Here’s my constant marginal cost at C.

[译文 26]

谁能给我念出来？我们为什么不随便叫个人？你可以。a 减 c 除以 2B 好吧喊出来 a 减 c 除以 2B 好，如果我把 Q2 等于 0 代入这里，这一项就消失了，对吧？我就得到 A 减 C 除以 2B。所以这告诉我什么？让我在这里画一条 45 度线。这告诉我什么？它说如果玩家 2 选择不生产，那么玩家 1 的最佳反应是 a 减 c 除以 2b，现在我声称那个量 a 减 c 除以 2b 实际上有另一个名字，那个量的名字是什么？一旦我们脱离代数只用经济学来思考，那个量 a 减 c 除以 2b 的名字是什么？好吧，让我们想想。回到之前的图。如果玩家 2 一点都不生产，玩家 2 一点都不生产，那么玩家 1 实际上是什么？他是一个垄断者，或者她是一个垄断者，对吧？所以我们知道怎么从我们在 115 或者同类课程中学到的来算出垄断产量。让我在这里把图画完。这是我的需求曲线。这是我的恒定边际成本 C。

[段 27]

All right, I want to use this picture on the left now to figure out what the monopoly quantity is. Can anyone tell me, let’s cold call somebody again, well, raise your hands if you took 115 or 110 or 150. Oh no, no, no, no, I know more of you took than that. It’s a prereq for the class, right? Raise your hands if you didn’t take any of those things. Right? So, Alec, keep your hands up. Cole Coulthard, you haven’t got their hand up. Anybody, anybody? Go ahead. We’re passing the buck here. Do you want to tell me what the… Where’s the monopoly quantity on this picture? Try the woman there. the woman one in where you are, the woman here I really don’t remember don’t remember? don’t remember? who else doesn’t remember? somebody must remember don remember is a good reply if you auditioning for it to be the Attorney General but it not such a good reply Anybody Somebody help me out here. Where’s the monopoly quantity on this picture? It’s a good review exercise, this. Yeah? It’s where the marginal revenue is equal to the marginal cost. Ah, it’s where marginal revenue equals marginal cost. I should have picked on one of the SOM people, by the way. It’s where marginal revenue equals marginal cost, right? The reason I should pick on the SOM people is their ambition in life is to run a monopoly, right?

[译文 27]

好，我现在要用左边这幅图来算出垄断产量是多少。有谁能告诉我，我们再随便叫个人，好吧，选过 115 或 110 或 150 的请举手。哦不，不，不，不，我知道你们更多的人选过。这是上课的先修要求，对吧？没选过这些任何一门课的请举手。对吧？Alec，手举着别放下。Cole Coulthard，你没举手。有没有人，有没有？说吧。我们在这里推卸责任呢。你想告诉我什么……这幅图上垄断产量在哪里？试试那边那位女士。你在哪里，那位女士，我真的不记得了不记得？不记得？还有谁不记得？肯定有人记得。不记得是个不错的回答，如果你是在应聘司法部长，但那不是个好的回答。有人，有人帮帮我。这幅图上垄断产量在哪里？这是个很好的复习练习。是的。就是边际收益等于边际成本的地方。啊，就是边际收益等于边际成本的地方。我本来应该挑一个 SOM 的人的，话说。就是边际收益等于边际成本，对吧？我应该挑 SOM 的人的原因是，他们的人生抱负就是经营一个垄断，对吧？

[段 28]

All right? Trouble is, I haven’t drawn the marginal revenue curve yet. Right? You can’t actually see the marginal revenue here. What does the marginal revenue look like in this picture? I don’t want to go to the same guy, the guy in brown. Where’s the marginal revenue look? What’s the marginal revenue look like in this picture on the left? It’s one half the slope of the price. Yeah, or it’s, I guess it’s twice the slope, but that’s fair enough. I know what you mean. All right? So the marginal revenue here, let’s put it in a different color. The marginal revenue looks something like this, right? And as, I don’t know what your name is, as the guy said, the monopoly quantity, the monopoly quantity is when marginal revenue equals marginal cost all right And as we know this line here has twice the slope of the original line of the demand curve so the slope of this thing is minus 2b All right, now I’m going to claim that this monopoly quantity I claimed before, our intuition says, if the other firm isn’t producing, my best response must be to produce my monopoly quantity, which I can see on this picture, and I also claim that the math is telling me, just a kind of nerdy math is telling me, that that quantity is a minus c over 2b, and I claim that they’re the same thing.

[译文 28]

好吧？问题是，我还没画边际收益曲线呢。对吧？你在这里实际上看不到边际收益。这幅图中边际收益是什么样的？我不想找同一个人，那个棕色衣服的人。边际收益在哪里？左边这幅图中边际收益是什么样的？它是价格斜率的一半。是的，或者说它是两倍斜率，但意思差不多。我懂你的意思。好吧？所以这里的边际收益，让我们用不同的颜色画。边际收益看起来像这样，对吧？而且正如，我不知道你叫什么名字，正如那个人说的，垄断产量，垄断产量是边际收益等于边际成本的时候，对吧。我们知道，这里的这条线是需求曲线原始线的两倍斜率，所以这条线的斜率是负 2b。好吧，现在我要声称这个垄断产量，我之前声称的，我们的直觉说，如果另一家企业不生产，我的最佳反应一定是生产我的垄断产量，我在这幅图上能看到，而且我也声称，数学告诉我们，就一种有点书呆子气的数学告诉我们，那个量是 a 减 c 除以 2b，我声称它们是同一个东西。

[段 29]

How can I see they’re the same thing? Well, look, here I have a line of slope minus 2. How far down does it have to go? It has to go a minus c down, this pink line. It has to drop off from a all the way to c. How far along do I have to go to drop off a minus c when I have slope minus 2b? Answer, a minus c over 2b. Right? That’s not even calculus, something you really all do in high school. All right? Okay, so this monopoly quantity is indeed a minus c over 2b. All right? Now I’m going to ask again, is some of this being, coming out of the fog? I mean, you may not have seen it with the algebra before, but some of these sort of pictures have you seen before It a good review exercise for some of you who have taken 150 and are about to take 150 or are there now Alright so we found precisely one point on this best response picture, and there’s a lot of points to find, and it’s 20 past 12, so we’re going to get going. Alright, so let’s try and find another point. Alright, it’s not a different question. How much quantity would firm 2 have to produce in order to induce firm 1 not to produce at all?

[译文 29]

我怎么看出来它们是同一个东西？好吧，看，这里我有一条斜率为负 2 的线。它需要向下走多远？它需要向下走 a 减 c，这个粉色线。它需要从 a 一直下降到 c。当斜率是负 2b 时，我需要走多远才能下降 a 减 c？答案，a 减 c 除以 2b。对吧？那甚至不是微积分，是你们大家在高中就做过的东西。好吧，所以这个垄断产量确实是 a 减 c 除以 2b。好吧？现在我要再问一次，有没有一些东西正从迷雾中走出来？我是说你可能之前没在代数中见过它，但这些图你以前见过吧。对于你们一些选过 150、将要选 150 或者现在正在上 150 的人来说，这是个很好的复习练习。好吧，所以我们在最佳反应图上精确地找到了一个点，还有很多点要找，现在已经 12 点 20 了，所以我们继续。好吧，所以让我们试着找另一个点。好吧，这不是一个不同的问题。企业 2 需要生产多少产量才能导致企业 1 完全不生产？

[段 30]

Say that again. How much quantity would firm 2 have to produce in order to induce firm 1’s best response to be 0? All right. Ali, help me out. It’s Katie, right? Is that right? The woman in green? Let’s get a mic to Katie. Oh, there’s a mic close to the mic. Oh, never mind. Sorry, Ali. Yeah. A minus C over 2B. All right, so that’s just solving out the algebra. We’re saying what Q solves A minus C over 2B minus Q2 over 2 equals 0. What Q2 makes this equal to 0? And Katie’s answer is, solving out the algebra here, Q2 over 2 equals 0, right? What Q2 makes this equal to 0? And Katie’s answer is, solving out the algebra here, is that Q2 that solves this must be A minus C over B. All right? That’s right. Where can I see that one on the picture, though? Let’s go back to the economics. All right, so I’m claiming that the sum quantity up here, I claim this quantity is A minus C over B, such that if firm 2 produces that much quantity, firm 1 will just produce nothing. But that’s just kind of algebra and math. Why can’t I see it on this picture? What quantity, if produced by firm 2, would cause firm 1 to shut down on this picture? Yeah, it’s going to make it in here.

[译文 30]

再说一遍。企业 2 需要生产多少产量才能导致企业 1 的最佳反应是 0？好吧。Ali，帮我一下。是 Katie，对吧？是吗？穿绿色衣服的那位女士？给 Katie 递个麦克风。哦，麦克风在麦克风附近。哦，没关系。抱歉，Ali。是的。A 减 C 除以 2B。好吧，那只是求解代数。我们在说什么 Q 能让 A 减 C 除以 2B 减 Q2 除以 2 等于 0。什么 Q2 能让这个等于 0？而 Katie 的回答是，求解这里的代数，Q2 除以 2 等于 0，对吧？什么 Q2 能让这个等于 0？而 Katie 的回答是，求解这里的代数，能解出这个的 Q2 一定是 A 减 C 除以 B。对吧？那是正确的。不过我能在图上哪里看到那个呢？让我们回到经济学。好吧，所以我声称上面这个总产量，我声称这个产量是 A 减 C 除以 B，这样如果企业 2 生产那么多产量，企业 1 就一点都不生产。但那只是代数和数学。我为什么在这幅图上看不到它？如果企业 2 生产多少产量，会导致企业 1 在这幅图上停产？是的，它会在这里画出来。

[段 31]

Sorry again. That’s where the marginal cost and the demand thing, demand lines intersect. All right so it here right It here It where it where marginal it where the marginal cost and the demand curve intersects And what the intuition Well, let’s work it out. Suppose Firm 2 has produced all of this quantity up to here. So Firm 2 is already producing all of this quantity. So already, just from Firm 2’s production, the price has been driven all the way down to costs. Alright? So if I, I’m firm one, if I produce any more quantity, what’s going to happen to the price? If I produce any quantity at all, in addition to what’s out there already, what’s going to happen to the price? It’s going to be pushed below costs. Alright? So I’ll be producing this stuff, this water, at cost C, and only getting P, which is now going to be less than C, in return. So I’ll be losing money on any product that I produce. Everyone see that? I’ll say it again. So firm two, if firm two produces all the way up to here, then any product produced by firm one is gonna put prices even lower. In particular they be lower here than costs and so you make losses on that product Alright What the name going back to Economics 115 what the name for this quantity The quantity where demand and marginal cost equal That’s the competitive quantity.

[译文 31]

再次抱歉。这就是边际成本和需求曲线相交的点。好的，这里就是边际成本与需求曲线相交的位置。直觉是什么呢？让我们来算一算。假设 Firm 2 已经生产了这么多量到这里。那么 Firm 2 已经生产了全部的量。于是，仅凭 Firm 2 的生产，价格已经被压到成本水平了，对吧？如果我是 Firm 1，再多生产一点，价格会怎样？如果我在已经存在的产量之外再生产任何数量，价格会怎样？它会被压到成本以下。好吧，那我就在成本 C 上生产这个东西（这里的水），而只能得到现在的 P，P 小于 C。于是我在任何生产的产品上都会亏损。大家明白了吗？我再重复一遍。如果 Firm 2 生产到这里，那么 Firm 1 生产的任何产品都会把价格压得更低。特别是它们会低于成本，所以你在这部分产品上会亏损。好的，回到 Economics 115，这个量叫什么？需求和边际成本相等的那个量叫什么？那是竞争性数量。

[段 32]

All right? So this is the perfect competition quantity. In a perfectly competitive market, that’s exactly where prices are going to end up. This is not a perfectly competitive market, but if it were, that’s where prices are going to end up. All right? So we have the monopoly quantity here, and we have the competitive quantity here, and in between, what does this best response curve look like? Somebody? It’s a straight line. It’s a straight line. Thank you. All right? So in between, this is just a straight line. All right? So here is the best response of firm one to each possible choice of firm two. All right? Everyone’s looking really like this was hard, but it can’t be that hard. You’ve seen it, most of you have seen this before. All right? So don’t, if you haven’t seen it before, don’t panic. All I did was I did a little bit of calculus a little bit of algebra and then I drew the thing I just drew the thing Okay so this is firm one best response as a function of q2 What is firm two best response as a function of q1? Just before we get there, just to remind you, the way we read this is you give me a quantity of firm two, I find firm one’s best response by going across to the pink decline and dropping down.

[译文 32]

好的？这就是完全竞争的数量。在完全竞争的市场中，价格正好会落在这里。这不是完全竞争的市场，但如果它是的话，价格也会落在这里。好的？那么这里我们有垄断数量，这里是竞争数量，两者之间，最佳反应曲线是什么样的？有人知道吗？是一条直线。是一条直线。谢谢。好的？在两者之间，这只是一条直线。好的？这里是对 Firm 2 每个可能的选择，Firm 1 的最佳反应。好的？大家看起来都觉得这很难，但它不可能那么难。你们大多数人都见过这个。好的？如果没见过的，别慌。我只做了一点微积分、一点代数，然后画了这个图——我就是画了这个图。好的，这是 Firm 1 的最佳反应，是 q2 的函数。Firm 2 的最佳反应是 q1 的函数是什么？在继续之前，提醒一下我们读图的方式：给我 Firm 2 的产量，我沿着粉红色的曲线走过去，向下看，就得到 Firm 1 的最佳反应。

[段 33]

So if firm 2 produces this amount, then firm 1’s best response is this amount. If firm 2 produces this amount, then firm 1’s best response is this amount. All right? Now conversely, how do we find firm 2’s best response as a function of Q1? What does that look like? Yeah, so you don’t get the two guys in front of you? It’s symmetrical. It’s It’s symmetrical across the 45 degree axis. All right. It’s reflected across the 45 degree line. So let’s reflect these two points. This will be here. And this will be… I’ll be a bit more careful. I missed by a mile. Let me just try again. Okay. I missed by a mile, let me just try again. I missed by a mile, which is a problem in drawing my lines, but never mind. All right, so this will be the monopoly quantity for firm 2, and this is the competitive quantity for firm 1. All right? So if I’d drawn that better, it would look as if this was half of this distance. Right? But I didn’t draw it well. Try and draw it better in your notes. All right? All right, so this is a minus c over b, and this is a minus c over 2b, so this distance and this distance are meant to be the same, and the way I’ve drawn it, these don’t look the same at all.

[译文 33]

如果 Firm 2 产量是这个，那么 Firm 1 的最佳反应就是那个量。如果 Firm 2 产量是那个，那么 Firm 1 的最佳反应就是这个。好的？反过来，我们怎么求 Firm 2 的最佳反应，它是 Q1 的函数？看起来是什么样？是的，你看不到两个人在你面前？它是对称的。它在 45 度轴上对称。好的。它关于 45 度线反射。让我们把这两个点反射回去。这里会是这个，那里会是……我稍微再仔细一点。我差得很远。再试一次。好，我差得很远，再试一次。我差得很远，这说明我画线有点问题，不过没关系。好的，这里将是 Firm 2 的垄断数量，这里是 Firm 1 的竞争数量。好的？如果我画得更好，看起来就好像这个是那个距离的一半。对吧？但我没画好。在你的笔记里试着画得更好。好的？好的，这里是 a 减 c 除以 b，这里是 a 减 c 除以 2b，这两个距离应该是相等的，但我画出来它们看起来完全不一样。

[段 34]

All right. And the way we read this green graph is you give me a choice of firm 1 Q1 I go up to the green line and go across and this tells me the best response for firm 2 All right So at this point how many of you have seen this picture before A good many of you have seen it before. All right. So I think we can really cold call somebody for this one. All right. So somebody tell me, pick someone to cold call, anybody, Anybody? Can you close by? There we go. Okay, okay. So here’s the tough question. What’s the Nash equilibrium on this picture? Where the green and the pink line intersect. All right. Where the green and the pink line intersect, all right? All right. So this must be the Nash equilibrium. That wasn’t hard, right? Okay, okay. All right. Now, okay, so why, let’s go back again. Why is this the Nash equilibrium? Because at this point, as in the partnership game, a similar thing. As in the partnership game, where the best responses intersect is where player 1 is playing a best response to player 2, and player 2 is playing a best response to player 1. So this is the Nash equilibrium in this corno game. Alright? Let’s do a little bit of algebra, just to make sure we can find this thing.

[译文 34]

好的。我们读这张绿色图的方式是：你给我 Firm 1 的选择 Q1，我向上走到绿线，再横过去，这告诉我 Firm 2 的最佳反应。好的？到这一步，有多少人之前见过这张图？你们很多人已经见过了。好的。我认为我们可以真的点名提问这个。好的？谁来告诉我，挑一个人来点名，随便谁？能不能靠过去？好，行。好的，这儿有个难题。这张图上的 Nash 均衡在哪里？绿色线和粉红线相交的地方。好的。绿色线和粉红线相交的地方，是吧？好的。那么这一定是 Nash 均衡。这并不难，对吧？好的，好的。好的，现在，为什么它是 Nash 均衡？让我们再回顾一下。为什么这是 Nash 均衡？因为在这个点上，和合作博弈一样，最佳反应相交的地方正是玩家 1 对玩家 2 的最佳反应，而玩家 2 对玩家 1 也是最佳反应。所以这就是这个 Cournot 博弈中的 Nash 均衡。好的？我们来做一点代数，确保我们能求出这个点。

[段 35]

Alright? So what we going to do is set these two things equal to each other put in stars here So at the Nash equilibrium quantities, I shouldn’t really do this, but never mind, at the Nash equilibrium quantities, I’m going to have q1 star is equal to a minus c over 2b q2 star, etc., and Q2 star is equal to A minus C over 2 B, Q1 star over 2. All right, and now I’m going to solve this out by setting Q1 equal to Q2, since I know the game is symmetric, and I know we’re on the 45 degree line. All right, so what I have to do here to do this algebraically is to solve out these equations. So let’s try and do that. so putting Q1 in here if I substitute Q1 in here what will it give me? It’ll give me Q1 star is equal to A minus C over 2B minus Q1 star over 2 let’s multiply both sides by 2 I get 2Q1 star is equal to A minus c over b minus q1 star Take that to the other side I have 3q1 star is equal to a minus c over b. And finally, divide by 3q1 star is equal to a minus c over 3b. All right? So that crossing point actually occurs at A minus C over 3B.

[译文 35]

好的？我们要做的是把这两个式子相等，并在上面加上星号。于是，在 Nash 均衡的产量上，我不该真的这么做，但没关系，在 Nash 均衡的产量上，我有 q1* = (a - c) / (2b) * q2*，等等，Q2* = (A - C) / (2B) * Q1* / 2。好的，现在我要通过设 Q1 = Q2 来求解，因为我知道博弈是对称的，而且我们在 45 度线上。好的，在这里我需要代数求解这些方程。让我们试试。把 Q1 代入这里，如果我把 Q1 代入这里会得到什么？它会给我 Q1* = (A - C) / (2B) - Q1* / 2。两边同乘以 2，得到 2Q1* = (A - c) / b - q1*。把 q1* 移到左边，得到 3q1* = (a - c) / b。最后，除以 3，q1* = (a - c) / (3b)。好的？于是这个交点实际上出现在 A - C / (3B)。

[段 36]

And again, I’m going to look desperately at my TAs, check I didn’t screw that up. I think I didn’t screw that up. All right? So this is sometimes called the Cournot quantity. Now this game, this game of imperfect competition between two firms competing in quantities was thought up and studied by a French economist called Cournot almost a hundred years before Nash. So a hundred years really before the invention of game theory. Someone had figured out this answer for this game. Okay, so what do we know so far? I’ll leave the algebra there. Transfer it up. All right. Okay, so what do we know so far? I’ll leave the algebra there, transfer it up here. Q1 star equals Q2 star equals A minus C over 3B. All right. So, so far we’ve been working pretty hard and we haven’t, I guess, learned a lot, we’re just going to solve the thing out. Now we get to draw some lessons out of this thing. All right? So everybody who’s feeling a little bit shell-shocked from having been doing algebra and calculus and drawing pictures and feeling like they’ve been cheated into taking a class that looks far too much like economics, calm down, we’re going to actually talk about it now. Okay? All right. So one thing to remark about this game, one thing we’ve learned immediately, is that this This game is different in a significant way.

[译文 36]

我再次忍不住去看我的 TAs，检查我没有搞错。我觉得我没有搞错。好的？这有时被称为 Cournot 数量。这个两个企业在数量上竞争的不完全竞争博弈，是一位名叫 Cournot 的法国经济学家在 Nash 之前大约一百年提出的并研究的。所以在博弈论出现之前的一百年，就已经有人为这个博弈找到了答案。好的，到目前为止我们知道了什么？我先停在这里。把代数搬上来。好的？好的，到目前为止我们知道了什么？我先停在这里，把代数搬上来。Q1* = Q2* = (A - C) / (3B)。好的。到目前为止我们一直在很努力地工作，但我们大概没有学到太多，我们只是在求解这个问题。现在我们可以从中提取一些教训。好的？如果有人因为刚才做了代数、微积分、画图，觉得自己被骗上了一门看起来太像经济学的课，有点惊魂未定，冷静下来，我们现在要来真正讨论它。好的？好的。关于这个博弈，有一点值得注意的是，这个博弈在某个重要方面是不同的。

[段 37]

It’s different from the partnership game. I mean, obviously it’s different because it’s about something different, but I mean in terms of the game theory it’s different. It different from the partnership game It different from the investment game What is it that different about this game from the part What the obvious thing just looking at this picture that makes it different from the investment game What is it that different about this game from the part what the obvious thing I was looking at this picture that makes it different from the partnership game In the partnership game the best response lines sloped up is that right The more I did of my strategy, the more the other player did as a best response. In the investment game, the more likely I was to invest, the more likely you, the more you wanted to invest, but in this game we have the exact opposite, the more player 1 produces the more player 1 produces the less player 2 wants to produce and the more player 2 produces the less player 1 wants to produce alright, so this game is a game not of strategic compliments but of strategic substitutes I want to be careful here. It’s not that these goods are substitutes. I mean, clearly that’s also the case, right? If both these firms are producing bottled water and it identical right then the goods themselves are substitutes That not the point I making here Strategic substitutes is a statement about the nature of the game So strategic substitutes is telling me that my strategy is a strategic substitute of your strategy if the more I do of my strategy, the less you want to do of yours, and conversely, the more you want to do of your strategy, the less I want to do of mine.

[译文 37]

这与合作博弈不同。我是说，显然是不同的，因为它涉及不同的内容，但从博弈论的角度来看它是不同的。它不同于合作博弈，它不同于投资博弈。这个博弈与合作博弈有什么不同？看看这张图，什么是明显的东西使它不同于投资博弈？这个博弈与合作博弈有什么不同？我看这张图，什么是明显的东西使它不同于合作博弈？在合作博弈中，最佳反应线是向上倾斜的，对吗？我越采取我的策略，对方作为最佳反应就越采取更多策略。在投资博弈中，我越倾向于投资，你就越倾向于投资，但在这个博弈中，我们恰恰相反，玩家1生产得越多，玩家1生产得越多，玩家2就越不想生产；玩家2生产得越多，玩家1就越不想生产。好吧，所以这个博弈不是战略互补的博弈，而是战略替代的博弈。我想在这里谨慎说明。这不是说这些商品是替代品。我的意思是，显然也是那样，对吧？如果这两家公司都生产瓶装水，而且是完全相同的，那么这些商品本身就是替代品。这不是我这里要说的要点。战略替代品是对博弈性质的一种陈述。所以战略替代品告诉我，如果我越采取我的策略，你就越不想采取你的策略，反之亦然，你越想采取你的策略，我就越不想采取我的策略。

[段 38]

All right? Okay. Now let’s come back to the question we started with, which is to do with profits and so on in society. We know that if these players play these games, at least if we believe in national equilibrium, then they’re going to produce these quantities. They’re going to produce here. But let’s ask a different question. Each of these firms is trying to maximize profit. But how about the total profit in the industry? industry. Each of these firms, we know by the fact they’re playing a best response, is maximizing their profits, taking as given what the other firm’s doing. But how about total industry profits? Are they being maximized here? Who thinks total industry profits are being maximized Who thinks total industry profits are not being maximized All right good good So so shout out okay so total industry profits are not being maximized Where on the picture let me get another bit of cold calling in here where on the picture maximizes total industry profits? Where on the picture maximizes total industry profits? We had it before, somewhere else. Yeah, no, no idea, no idea? people taking the fifth here yeah the guy in there yeah monopoly exactly so clearly if we produce the monopoly quantity by definition the monopoly quantity maximizes total industry profits and for example if firm to shut down and firm one produced its monopoly profits its monopoly quantity that would maximize firm profits and conversely if firm one shut down and firm two produced its monopoly quantity, that would maximize industry profits.

[译文 38]

好吗？好。现在让我们回到我们开始时的问题，这与利润和社会有关。我们知道，如果这些玩家玩这些博弈，至少如果我们相信纳什均衡，他们就会生产这些产量。他们会在这里生产。但让我们问一个不同的问题。这些公司中的每一个都在试图最大化利润。但整个产业的总利润呢？整个产业。每个公司，我们知道通过他们玩最佳反应这一事实，正在最大化他们的利润，假设其他公司的行为是既定的。但总产业利润呢？它们在这里被最大化了吗？谁认为总产业利润正在被最大化？谁认为总产业利润没有被最大化？好的，好的。那么，大声说出来，总产业利润没有被最大化。在这个图上，哪里使总产业利润最大化？在这个图上，哪里使总产业利润最大化？我们之前有过，在某个地方。是的，不，不知道，不知道？人们在这里引用第五修正案，是的，那里面的人，对，垄断，正是如此，所以显然如果我们生产垄断数量，垄断数量顾名思义就是使总产业利润最大化的数量，例如，如果公司二关闭，公司一生产其垄断利润即垄断数量，那将使公司利润最大化，反之亦然，如果公司一关闭，公司二生产其垄断数量，那将使产业利润最大化。

[段 39]

Where else in this picture maximizes industry profits? So this point maximizes industry profits and this point maximizes industry profits. Where else maximizes industry profits? Anybody? Well, how about all the points in between? Here, firm 1 produced nothing, and firm 2 produced the monopoly quantity. And here, firm 2 produced nothing, and firm 1 produced the monopoly quantity. But you could also just split the monopoly quantity, for example, half-half, at this point here. All right? So if the firm wanted to make more money, one thing they could do is they could sign an agreement saying, why don’t each of us produce not our corno quantity, but produce half monopoly profits. Sorry, half monopoly quantity. And that would produce this much each. All right, so firm one would be producing half its monopoly quantity, and firm two would be producing half its monopoly quantity. Alright, so what’s wrong with that agreement? So here they are they two water companies so they Poland Spring and Coca I guess these days have signed this agreement saying each one going to produce half a monopoly quantity and what going to go wrong with that agreement Yeah, the guy in red. First of all, it’s illegal. It’s illegal, it’s illegal, right? So even without there being an attorney general, somebody might be awake in the Justice Department and notice that that’s actually illegal.

[译文 39]

在这个图上还有什么地方使产业利润最大化？所以这个点使产业利润最大化，这个点也使产业利润最大化。还有什么其他地方使产业利润最大化？谁？嗯，它们之间的所有点怎么样？这里，公司一什么都不生产，公司二生产垄断数量。这里，公司二什么都不生产，公司一生产垄断数量。但你也可以简单地分割垄断数量，例如，一半一半，在这里的这个点。好吧，所以如果公司想赚更多钱，他们可以做的一件事就是签订协议，说为什么我们不各自生产不是我们的古诺数量，而是生产一半垄断利润。对不起，一半垄断数量。那样的话每家会生产这么多。好吧，所以公司一将生产其垄断数量的一半，公司二将生产其垄断数量的一半。好吧，那么这个协议有什么问题？所以他们这两家矿泉水公司，波兰弹簧泉和可口可乐，我猜这些日子已经签署了这个协议，说每家将生产一半垄断数量，这个协议会出现什么问题？是的，穿红色衣服的那位。首先，它是违法的。它是非法的，它是违法的，对吧？所以即使没有总检察官，司法部可能也有人醒着，注意到这实际上是违法的。

[段 40]

It’s illegal to sign an agreement, to sign a contract saying you’re going to restrict quantities. It’s illegal, so they can’t sign that as a contract, but hey, they can just agree to do it. Right? So Coca-Cola and Poland Spring could just agree to produce half the quantity of water each. All right? What’s going to go, what might go wrong in that agreement? Let’s just suppose that that game’s just being played once for now. And suppose that this is Coca-Cola’s quantity, and this is Poland Springs’ quantity. And suppose you’re the manager of Poland Springs, so water is coming out of the ground in Maine for you, and you know that this Coca-Cola guy is going to produce this quantity here. So this is Qm over 2. So let pick on our two managers let have two people who spoken before So the guy in the red shirt name is Steve Steve and I guess in red he should be the Coca guy right All right and let pick on our Jimmy Stewart from last week whose name is Patrick right Can I get a mic on Patrick, all right? All right, so Steve and Patrick are respectively the managers of Coca-Cola and Poland Spring, and here he is, right? All right, and Patrick believes that the Coca-Cola manager is going to, Maybe we should just make this Coke and Pepsi.

[译文 40]

签署协议，签署合同说你将限制产量是违法的。它是非法的，所以他们不能签署那样的合同，但嘿，他们可以仅仅同意这么做。对吧？所以可口可乐和波兰弹簧泉可以仅仅同意每家生产一半的水量。好吧，这个协议会出什么问题？让我们假设那个博弈现在只进行一次。假设这是可口可乐的产量，这是波兰弹簧泉的产量。假设你是波兰弹簧泉的经理，所以水在缅因州从地下涌出给你，你知道这个可口可乐的家伙将生产这个数量。所以这是Qm除以2。让我挑选我们的两位经理，让我们请两位之前说过话的人。所以穿红衬衫的那位叫史蒂夫，史蒂夫，他应该穿红色的，所以应该是可口可乐的人，对吧。让我挑选我们上周的吉米·斯图尔特，他的名字是帕特里克，对吧。我能给帕特里克一个麦克风吗，好吧？好吧，所以史蒂夫和帕特里克分别是可口可乐和波兰弹簧泉的经理，他在这里，对吧？好吧，帕特里克相信可口可乐经理将……也许我们应该只让这个成为可口可乐和百事可乐。

[段 41]

It’s too identical. They’re near enough identical. So these are Coke and Pepsi. That’ll confuse me less. So Patrick, our manager of Pepsi, believes that Coca-Cola is going to produce this quantity, and he’s agreed to produce this quantity. What is Patrick actually going to do? Patrick, what are you going to do? So I should cheat and pick the quantity on my best response line that’s much farther out, so I’ll produce more than that quantity. All right, so Patrick, being Jimmy Stewart or not, when he’s playing the manager of Pepsi, it’s going to produce more of this undrinkable liquid and produce this quantity here. All right, so what is this? This is Patrick best response to the other guy producing the monopoly quantity All right So Patrick response in Pepsi is to overproduce relative to the monopoly quantity, actually overproduce even relative to the Cournot quantity, and produce all the way out here. Okay, so Patrick’s producing here, but what about Mr. Coca-Cola guy? Now Mr. Coca-Cola guy knows Patrick pretty well, they’ve been in the same industry for a while, and Mr. Kukulga, whose name is Stephen, Stephen presumably knows that Patrick is actually going to produce this quantity. So Stephen, what should you produce? Increase my own production to match my best response line. Good. So Stephen, anticipating that Patrick is going to think that Stephen’s a sucker, but Patrick’s going to cheat on him and produce too much, Stephen’s going to produce the best response to that, and he’s going to produce this quantity here.

[译文 41]

它太相同了。它们足够相同。所以这些是可口可乐和百事可乐。那样我不会那么困惑。所以帕特里克，我们的百事可乐经理，相信可口可乐将生产这个数量，他已经同意生产这个数量。帕特里克实际上要做什么？帕特里克，你要做什么？我应该作弊，在他的最佳反应线上选择更远的数量，所以我会生产比那个数量更多。好吧，所以帕特里克，不管他是不是吉米·斯图尔特，当他扮演百事可乐经理时，他将生产更多这种不可饮用的液体，并生产这个数量。好吧，那么这是什么？这是帕特里克对对方生产垄断数量的最佳反应。好吧，所以帕特里克在百事可乐的反应是相对于垄断数量过度生产，实际上是相对于古诺数量也过度生产，一直生产到这里。所以帕特里克在这里生产，但可口可乐先生呢？现在可口可乐先生很了解帕特里克，他们在这个行业已经有一段时间了，库库尔加先生，他的名字是斯蒂芬，斯蒂芬大概知道帕特里克实际上将生产这个数量。所以斯蒂芬，你应该生产什么？增加我自己的产量以匹配我的最佳反应线。好的。所以斯蒂芬，预期帕特里克会认为斯蒂芬是个冤大头，但帕特里克会欺骗他并生产太多，斯蒂芬将对这个做出最佳反应，他将生产这里的这个数量。

[段 42]

So what is this? This is player two’s best response to player one’s best response to player two producing half a million. 2’s best response to player 1’s best response to player 2 producing half monopoly output. All right? But Patrick knows Stephen pretty well. He knows those guys who run Coca-Cola are pretty smart guys. So Patrick knows that the guy who runs Coca-Cola is going to anticipate that the guy who runs Pepsi is going to cheat on the guy who runs Coca-Cola. And hence, the guy who runs Coca-Cola is going to play a best response to Patrick’s cheating quantity. So what quantity is Patrick going to produce? So I reduce my fraction then and I’ll end up pretty close to the Cornell equilibrium. Alright, so you’re going to go to here so this point here is I don’t even write this, this is the best response of player one to the best response of player two to the best response of player one to player two producing half There’s lots of monopoly output and there’s lots of brackets here. All right? Can anyone see where this process is going? Can anyone see where this process is going? All right, it’s going back to the Nash equilibrium.

[译文 42]

这是什么？这是玩家二对玩家一在玩家二生产五十万时的最佳反应的应对。玩家二对玩家一在玩家二生产一半垄断产量时的最佳反应的应对。明白吗？但帕特里克非常了解斯蒂芬。他知道那些经营可口可乐的人都是相当聪明的人。所以帕特里克知道，可口可乐的经营者会预见到百事可乐的经营者会欺骗可口可乐的经营者。因此，可口可乐的经营者会对帕特里克欺骗的产量做出最佳反应。那么帕特里克会生产多少呢？所以我减少我的比例，然后我将非常接近古诺均衡。好，那么你会走到这里，所以这个点在这里，我甚至不写这个，这是玩家一对玩家二对玩家一对玩家二生产一半时的最佳反应的应对。有很多垄断产量，也有很多括号在这里。明白吗？有人能看到这个过程会走向哪里吗？有人能看到这个过程会走向哪里吗？好，它会回到纳什均衡。

[段 43]

Each of them is trying to play a best response to what they anticipate the other person going to do And in this game if they keep on doing that it going to drag them back to national equilibrium I want to be careful here This won be true of all games but in this game playing best response to each other figuring out that in fact the other guy is going to cheat on me, or the other guy is going to anticipate that I’m going to cheat on him, or the other guy is going to anticipate that I’m going to anticipate that he’s going to cheat on me, etc., etc., will eventually drag us back to the Cournot quantity. So it’s going to be pretty difficult for us to sustain this joint monopoly output, this collusive agreement. We can’t let the courts enforce it. We’re Coca-Cola and Pepsi, so we really don’t want to bring the mafia in, although maybe we do, I don’t know. So basically, we’re stuck with this verbal agreement, and we both have an incentive to cheat and produce more whatever it is, sugar water. Okay. Now, in practice, this is not the only problem facing two firms who are trying to produce the monopoly output. In practice, when firms try and have these agreements, which are not contracts, to try and produce joint monopoly output, what else goes wrong?

[译文 43]

他们每个人都在试图对另一个人预期会做的事情做出最佳反应。在这个游戏中，如果他们一直这样做，它会拖回全国均衡。我想在这里谨慎一点。这不会对所有游戏都成立，但在这个游戏中，相互做出最佳反应，弄清楚实际上另一个人会欺骗我，或者另一个人会预期我欺骗他，或者另一个人会预期我预期他会欺骗我，等等等等，最终会把我们拖回到古诺产量。所以我们要维持这个联合垄断产量、这个串谋协议将相当困难。我们不能让法院强制执行它。我们是可口可乐和百事可乐，所以我们真的不想把黑手党拉进来，虽然也许我们可以，我不知道。所以基本上，我们被困在这个口头协议上，我们都有欺骗和生产更多的动机，不管它是什么，糖水。好吧。现在，实际上，这不是两个试图生产垄断产量的企业面临的唯一问题。在实践中，当企业试图达成这些不是合同的协议，试图生产联合垄断产量时，还有什么会出问题？

[段 44]

What else goes wrong? So you could imagine some firms trying to have an agreement. We know they can sign a legal agreement but let take that off the table So for example in around 1900 in America or a little bit earlier let say 1880 in America it really isn clear that it was illegal to write agreements to say well restrict quantities So firms did used to write that kind of agreement, but still something went wrong. What went wrong? So we’re getting a little bit further away from the game and more back into the real world now. What kind of things went wrong? I want to get someone who’s nearer a mic. Can I get the woman here, the Yale shirt? Without a contract, there’s no means of enforcement. All right. So it might be that, again, there’s a problem with enforcement, although actually in 1880 in America, it’s not clear that they couldn’t have enforced that contract. All right. So one problem is unenforceability, right? But that might, far enough ago, that might have been a problem. What else? The guy next to her. Yeah. If they’re not at the competitive outcome, they could undercut price and take the entire market? Let’s be careful. We’ll come back to that next time. But right now, we’re competing in quantities. So we’re not fixing prices here. We’re not even naming prices.

[译文 44]

还有什么会出问题？所以你可以想象一些企业试图达成协议。我们知道他们可以签署合法协议，但让我们把那个放在一边。例如，在大约1900年的美国或更早一点，比如1880年的美国，确实不清楚签订限制产量的协议是否违法。所以企业确实曾经写过那种协议，但仍然出了问题。出了什么问题？所以我们正在离游戏稍微远一点，更多地回到现实世界。什么样的事情出了问题？我想让靠近麦克风的人回答。我能让那位穿耶鲁衬衫的女士来回答吗？没有合同，就没有执行手段。好。所以可能再次出现执行问题，虽然实际上在1880年的美国，他们是否无法强制执行那份合同并不清楚。好。一个问题是不具可执行性，对吧？但那可能，足够久远以前，可能是个问题。还有什么？在她旁边的那个男士。是的。如果他们不在竞争均衡，他们可以降低价格并夺取整个市场？让我们谨慎一点。我们下次会回到那个。但现在，我们在数量上竞争。所以我们不是在固定价格。我们甚至没有报价。

[段 45]

We’re just producing Coca-Cola and Pepsi and, for want of a better word, spewing it out into the market. What else goes wrong here Well one thing that goes wrong is suppose suppose Coke and Pepsi were able to either deride a contract or whatever to sustain joint monopoly outputs at this high price, this price which is, well, sorry, producing positive profits in the industry. It is going to turn out, therefore, that prices are going to be above costs. These firms are making a lot of profits. What do you think is going to happen in this industry? This industry that’s got a lot of profit around it. What’s going to happen in this industry? Yeah, the guy in the blue shirt here. Ali, this guy here. Another competitor could join the market. All right, another firm is going to come in and produce a similar product. So Dr. Pepper or something, is that the third one? Someone else is going to come in and produce cola in this industry. And that’s exactly what happened in the US at the turn of the last century. Companies did produce agreements to restrict, for example, the production of paper and the production of rubber and the production of steel and the production of iron and the production of railway lines, actually, quite complicated things. And sure enough, what you see very quickly emerging is new entrants entering the market to say, hey, these firms are restricting their quantities, we can get in there and make money.

[译文 45]

我们只是生产可口可乐和百事可乐，并且，为了用一个更好的词，把它们倾倒到市场上。这里还有什么会出问题？好吧，一个会出问题的事情是，假设可口可乐和百事可乐能够签订合同或什么的来维持联合垄断产量，在这个高价，这个价格，生产行业内的正利润。因此，价格将高于成本。这些企业正在赚取大量利润。你认为这个行业会发生什么？这个周围有很多利润的行业。这个行业会发生什么？是的，这里穿蓝色衬衫的这位。阿里，这个人。另一个竞争对手可以加入市场。好，另一家企业将进入并生产类似产品。所以胡椒博士或什么的，那是第三个？其他什么人将进入这个行业生产可乐。恰恰这就是在美国在上个世纪之交发生的事情。企业确实签订了协议来限制，例如，纸张的生产，橡胶的生产，钢铁的生产，铁的生产和铁路线路的生产，实际上是相当复杂的事情。果然，你很快就看到新进入者进入市场说，嘿，这些企业在限制它们的产量，我们可以进去赚钱。

[段 46]

So a competitive fringe of firms is going to enter and drive prices down. Now I talked about that in 1900 America, whereas that happened in the 20th century more dramatically. So we know the early example in sort of golden age America, but where else did we see a competitive fringe enter when there was a collusive agreement to keep quantities down? Airlines. So airlines, okay. good, so airlines is an example. We’ve seen that. Where else? Oil is probably the classic example. Airlines is a good example. So in oil, we all know that in the late 60s, early 70s, OPEC was formed precisely to restrict quantity among the major oil producing countries. And very quickly, we see other countries take advantage of this to start producing oil profitably. So who were the competitive fringe against OPEC? Anybody? Well, the Brits were, for a start. The Brits, you know, England started producing well Scotland I should say Britain anyway started producing oil in the North Sea Lots of countries in Latin America started looking and finding oil And of course Russian oil started being very very profitable All right. So there are various reasons why it’s hard to maintain this collusive agreement. One is there’s an incentive to cheat, and another is that other firms are going to enter.

[译文 46]

所以一个竞争的边缘企业群体将进入并将价格拉低。我谈到过1900年的美国，那发生在20世纪更加戏剧性地发生。所以我们知道在美国镀金时代的早期例子，但当我们看到当有一个串谋协议来保持产量低时，竞争边缘进入的还有哪里？航空公司。好的，航空公司，这是一个例子。我们已经看到了。还有哪里？石油可能是经典例子。航空公司是一个很好的例子。所以在石油领域，我们都知道在60年代末，70年代初，欧佩克成立正是为了限制主要石油生产国的产量。而且很快，我们看到其他国家利用这一点开始盈利地生产石油。那么谁是相对于欧佩克的竞争边缘？有人知道吗？好吧，英国人首先是这样。英国，你知道，英格兰开始生产，呃，我应该说苏格兰，英国开始北海生产石油。拉丁美洲的许多国家开始寻找和发现石油。当然俄罗斯石油开始变得非常非常有利可图。好，所以有各种原因为什么很难维持这个串谋协议。一个是有欺骗的动机，另一个是其他企业会进入。

[段 47]

But before we leave this, let’s just pose the question we started with. suppose in fact we do end up back at the corner quantity so here we are back at the corner quantity, how does this quantity compare and how do therefore the prices compare and how do profits compare with monopoly prices on the one hand and competitive prices on the other right so we know we know the quantity that’s going to be produced we figured that out here it is we know that the quantity is going to be produced each firm is going to produce A minus C over 3B, and there are two such firms, so the total quantity produced in the market, the total quantity will be 2a minus c over 3b How does that compare to the monopoly quantity? Well, the monopoly quantity was a minus c over 2b, and the competitive quantity was a minus c over b. Just to remind you of those, they’re on our picture. here’s a minus c over 2b and here’s a minus c over b so this is the total quantity produced in this equilibrium in this corner Nash equilibrium how does it compare to the monopoly quantity and to the competitive quantity anybody? a little bit of just staring at the board which of these is bigger so I claim that this total quantity being produced is less than the competitive quantity, but more in total than the monopoly quantity.

[译文 47]

但在我们离开这个之前，让我们提出我们开始时的问题。如果实际上我们确实最终回到角落产量，那么我们回到角落产量，这个产量如何比较，因此价格如何比较，利润如何比较，一方面是垄断价格，另一方面是竞争价格，对吧，所以我们知道我们知道将要生产的产量，我们已经算出来了，在这里，每家企业将生产A减C除以3B，有两家这样的企业，所以市场上生产的总产量，总产量将是2a减c除以3B。这与垄断产量相比如何？好吧，垄断产量是a减c除以2B，竞争产量是a减c除以B。只是提醒你那些，它们在我们的图上。这里是a减c除以2B，这里是a减c除以B，所以这是在这个均衡中在这个角落纳什均衡中生产的总产量，它与垄断产量和竞争产量相比如何？有人吗？只是盯着黑板看一点，这些中哪个更大，所以我声称这个生产的总产量小于竞争产量，但总量大于垄断产量。

[段 48]

Alright? I was about to get that the wrong way around That right The total quantity being produced is less than would be produced under perfect competition but more than would be produced under monopoly And consequently, prices are going to go the other way. And consequently, prices are going to be highest under monopoly, lowest under competition, and somewhere in between in this Cournot situation. So from the point of view of the producers, this corner equilibrium is worse than monopoly but better than perfect competition and from the point of view of the rest of us, the consumers this corner quantity is worse than perfect competition but better than monopoly. The only rider on that being if they’re producing Coca-Cola and you have any kind of concern for your teeth maybe you shouldn’t produce so much anyway. Alright, so far what we’ve done is we’ve looked at this classic, I admit, not the most exciting game in the world, but a classic application of game theory to imperfect competition. Next time, I’m actually going to take this a bit further, but I’m going to leave this game behind and look at other ways in which we can study imperfect competition using game theory.

[译文 48]

好吗？我差点把它弄反了 对吧 产量低于完全竞争下的产量，但高于垄断下的产量。因此，价格会往相反的方向走。因此，垄断下价格最高，完全竞争下价格最低，Cournot模型下价格处于两者之间。所以从生产者的角度看，这个角点均衡比垄断差，但比完全竞争好；从我们其余的人，也就是消费者的角度看，这个角点产量比完全竞争差，但比垄断好。唯一的附加条件是，如果他们在生产Coca-Cola，而你对自己的牙齿有任何担忧，那你可能无论如何都不应该生产这么多。好，到目前为止，我们所讨论的是这个经典应用，我承认，这不是世界上最有趣的游戏，但它是博弈论应用于不完全竞争的经典案例。下一次，我会把它进一步拓展，但我会放下这个游戏，看看我们还能用博弈论研究不完全竞争的其他方式。