耶鲁大学博弈论公开课 - 第19集 纳什均衡之伯川德模型与选民投票

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 started to study imperfect competition. we looked at the Cournot model but the bigger theme the bigger theme is the study of how firms compete outside of the sort of easy cases that you study in 115 so outside of the example of monopoly where there is only one firm and so there’s not much competition and outside the case of perfect competition where there are so many firms it’s as if each firm is a price taker We’re going to continue with that today, and in fact you’re going to continue with it on into the homework. So last time we looked at Cournot, that was the model we started with. We looked at two firms competing in quantities, and just to review a little bit what we did last time kind of exercise there so I think is this right I think I probably went a little fast last time I was kicking myself afterwards I don know people can nod or shake their heads but I think I went a bit fast So I sorry for that but if there was a time to go too fast it was probably last time because it turns out this is something that’s covered very well in the textbook. So it’s covered in chapter 6 of the textbook.

[译文 1]

好的，上一次课我们开始研究不完美竞争。我们看了古诺模型，但更重要的主题是研究企业如何在一些简单情况之外进行竞争，这些简单情况是你们在115课程中学习的，即除了只有一个企业因此没有太多竞争的垄断案例，以及企业数量众多、每个企业都像是价格接受者的完全竞争案例。我们今天将继续这个主题，事实上你们在作业中也会继续研究。上次课我们看了古诺模型，这是我们开始研究的模型。我们回顾一下上次课做的练习，我想可能是这样，我认为上次课我可能讲得太快了一些，事后我一直在后悔，不知道大家能否点头或摇头表示，但我觉得确实讲得有点快。所以对此我很抱歉，但如果有一节课讲得太快的话，那可能就是上次课了，因为事实证明这在教材中有很好的覆盖。所以在教材的第6章中有详细的讲解。

[段 2]

But the kind of exercise we did last time is a useful exercise for those of you who are econ majors. the way we solved that Cournot model was we did three kinds of things we did something very nerdy namely we played with calculus and algebra alright, that was a little mathy we did something a little bit less nerdy that is we drew some pictures that represented what we’d found and we did a third thing which was we tried to match this up to the economic intuition about monopoly and perfect competition and demand curves and so on and this exercise of being able to work in these three different modes economic intuition, graphs, and kind of nerdy high school math, is a lot of what we want to get you used to as economic majors, to be able to translate easily between those. So again, I apologize for going too fast, but it’s still a useful exercise, I think, so have a look at it in the book. Now back to the lessons and away from the nerdiness a second, what do we learn? Well, at the end of the day we learned that in the Cournot equilibrium things were as we perhaps might have anticipated things sat naturally between the extreme cases So the amount of output produced by the industry was somewhere between the case that would be under monopoly and under perfect competition.

[译文 2]

但上次课我们做的这种练习对于你们中经济学专业的学生来说是一个有用的练习。我们解古诺模型的方法是做了三件事：我们做了一件非常书呆子的事，就是运用微积分和代数，好吧，那有点数学化；我们做了稍微不那么书呆子的事，就是画了一些图来表示我们发现的结果；我们还做了第三件事，就是尝试将这些与关于垄断、完全竞争和需求曲线的经济直觉联系起来。在这种三种不同模式之间工作的练习——经济直觉、图表和有点书呆子的中学数学——是我们想让你们作为经济学专业学生习惯的东西，能够轻松地在这些模式之间转换。所以再次抱歉讲得太快，但我认为这仍然是一个有用的练习，所以请大家看看书中的内容。现在回到课程内容，暂时离开书呆子的部分，我们学到了什么？嗯，归根结底我们学到的是，在古诺均衡中，情况正如我们可能预期的那样，自然地处于极端情况之间。所以行业生产的产量介于垄断情况下和完全竞争情况下之间。

[段 3]

It was more than under monopoly, less than perfect competition. Prices in the industry, if we’d gone back and checked, would sit between. They would be lower than the monopoly prices, but higher than the perfect competitive prices. industry profits would be in between industry profits would be less than monopoly they must be less than monopoly since that’s the highest they could ever be and they’re higher than perfect competition which of course has zero profits in this case and consumer surplus the benefits flowing to consumers lie in between so in fact the model in addition to the sort of nerdiness of the model it ended up with the result we kind of believe in if you have imperfect competition it’s somewhere between perfect competition and no competition but, what’s the but? The but is that there are other ways we could model imperfect competition, and as we’re going to see today, they yield different answers. So the first thing we’re going to do today is look at a different form of competition, which is called Bertrand competition. So Cournot was competing in quantities and Bertrand is competing in prices All right? And just a quick check, and this is not for the camera, how many of you have seen Bertrand competition before? So a good many of you. So this is going to be a review still.

[译文 3]

它比垄断情况下多，比完全竞争情况下少。如果回过头来检查的话，行业中的价格也会处于中间位置。它们会低于垄断价格，但高于完全竞争价格。行业利润会处于中间水平——行业利润会低于垄断利润，因为垄断利润是它们能达到的最高水平，而且会高于完全竞争利润，完全竞争在这种情况下当然利润为零。消费者剩余——流向消费者的利益——也处于中间水平。所以实际上，这个模型除了模型本身的书呆子气之外，最终得出的结果是我们大致相信的：如果存在不完美竞争，它就介于完全竞争和没有竞争之间。但是，但是什么呢？但是是，我们可以用其他方式来建模不完美竞争，正如我们今天将看到的，这些方式会产生不同的答案。所以今天我们要做的第一件事是看一种不同的竞争形式，叫做伯川德竞争。古诺是产量竞争，而伯川德是价格竞争。好吧？快速检查一下，这不是给摄像机看的，你们中有多少人之前见过伯川德竞争？所以你们中有很多人见过。所以这仍然会是一个复习。

[段 4]

Again, for those of you for whom it isn’t a review, don’t worry. Don’t worry. We’ll do this relatively quickly, but I hope I won’t rush it like I did last time. If there’s any consolation, when I feel I’ve rushed a lecture, I have a sleepless night. So if you had a sleepless night because you were worried about it, I’m having a sleepless night too about that. All right. So there’s two firms, just as there was before. And just as there was before, so the players are those two firms. And just as there was last time, they’re producing an identical product. All right, so the products we mentioned last time were Coke and Pepsi, but you could think of other products that are pretty much identical. All right, and just as last time, we’re going to assume that costs are constant marginal. that are pretty much identical. And just as last time, we’re going to assume that costs are constant marginal costs. We’ll assume that constant marginal cost, just as it was last time, is equal to C. Just to remind those people who are not economics majors what that means, it means if I produce one unit, then it costs me C to produce. produce 2 units, it costs me 2C, 100 units, 100C, and so on. All right? Now, what’s different here from last time?

[译文 4]

对于你们中不是复习的人来说，不要担心。不要担心。我们会相对较快地讲完，但我希望不会像上次课那样匆忙。如果有什么安慰的话，当我感觉自己一堂课讲得太赶时，我会彻夜难眠。所以如果你们因为担心而彻夜难眠，我也在为那件事彻夜难眠。好吧。有两个企业，就像之前一样。就像之前一样，参与者就是这两个企业。就像上次课一样，它们生产的是同质产品。好吧，我们上次课提到的产品是可口可乐和百事可乐，但你可以想到其他基本相同的产品。好吧，就像上次课一样，我们假设成本是常数边际成本。我们假设常数边际成本，就像上次课一样，等于C。只是提醒一下那些不是经济学专业的学生这意味着什么，这意味着如果我生产一个单位，那么生产它的成本是C。生产2个单位，成本是2C，100个单位，100C，以此类推。好吧？那么，这里与上次课有什么不同呢？

[段 5]

I’ve already told you, but I’ll repeat it. This time, however, instead of setting quantities, instead of just deciding how much Coke and Pepsi to produce and spewing it out from the market and letting prices take care of themselves, this time the firms are going to set prices and let quantities take care of themselves. All right? So the strategy set this time are prices. And again, so we don’t get confused, normally we use S for strategy, but let’s use P since they’re prices. So they going to be P1 will be the price of good one of firm one and P2 will be the price of firm two And the strategy set formally let just simplify here let assume that for each firm I they can set their price anything bigger than zero, and anything less than one, just to keep life simple. All right, so this is a little bit more realistic perhaps, it’s perhaps more realistic to think of firms actually thinking about what price is to set, and then adjusting the quantities they actually ship based on demand. I don’t want to be too rigorous about that, because you could imagine the other way around, but if you talk to managers this sounds more realistic, at least for retail goods like Coke and Pepsi. Alright, so we know how prices are set, they’re set by the firms, so the next question is, where do the quantities come from?

[译文 5]

我已经告诉你们了，但我要重复一下。然而这一次，不是设定产量，不是决定生产多少可口可乐和百事可乐，然后把它们从市场上推出去，让价格自己解决，这一次企业将设定价格，让产量自己解决。好吧？所以这一次的策略集是价格。同样，为了不让我们混淆，通常我们用S表示策略，但让我们用P因为它们是价格。所以它们将是P1将是企业1的产品1的价格，P2将是企业2的价格。从形式上讲策略集只是简化一下，假设对于每个企业i它们可以把价格设定为大于零且小于一的任何值，只是为了简化生活。好吧，这可能更现实一些，也许想到企业实际考虑设定什么价格，然后根据需求调整他们实际发货的数量，这可能更现实。我不想在这上面太严格，因为你可以想象另一种方式，但如果你和管理人员交谈，这听起来更现实，至少对于像可口可乐和百事可乐这样的零售商品。好吧，所以我们知道价格是如何设定的，它们是由企业设定的，那么下一个问题是，产量从哪里来？

[段 6]

Alright? So where do the quantities come from? They’re going to come from demand, and let me use a big Q of P, a big Q of P, to be the total quantity produced, sorry, the total quantity demanded in the market. So this will be the total quantity of goods produced by firm 1 and goods produced by firm 2 that are consumed Alright So is the total quantity of Coke plus Pepsi And notice there no subscript on this P and you see why in a second So we’ll assume, just to make life simple, that the total quantity produced is 1 minus P, and I’ll say this and then write it more carefully, where P is the lower of the two prices. So the total quantity demanded is 1 minus P, where P is the lower of the two prices. All right? Now, to make that lower of the two prices comment a little bit more rigorous, let’s figure out what the demand actually is for these firms. So what’s going on here? So let’s look at the demand for firm 1, which is going to end up being the quantity that they sell. So that’s going to be Q1. And what’s that going to be? Well, it’s going to be 1 minus P1 if they’re the low price firm. So if P1 is less than P2.

[译文 6]

好吧？那么产量从哪里来？它们将来自需求，让我用大Q对P，大Q对P，来表示总产量，抱歉，是市场上总需求量。所以这将是企业1生产的商品和企业2生产的商品中被消费的总数量。好吧，所以这是可口可乐加百事可乐的总产量。注意这个P没有下标，你马上就会知道为什么。所以我们假设，只是为了简化生活，总产量是1减P，我会说出来然后更仔细地写出来，其中P是两个价格中较低的那个。所以总需求量是1减P，其中P是两个价格中较低的那个。好吧？现在，为了使两个价格中较低的那个这个说法更严谨一些，让我们弄清楚这些企业的实际需求是什么。那么这里发生了什么？让我们看看企业1的需求，这将最终成为他们销售的数量。所以那是Q1。那会是什么？嗯，如果他们是低价企业，那就是1减P1。所以如果P1小于P2。

[段 7]

Right? So if they’re the low price firm, if Coke is the low price firm, only Coke sells in the market. There no Pepsi in the market and 1 minus P1 quantity of Coke is sold It going to be 0 if P1 is bigger than P2 So if Pepsi is the low-priced drink in the market, then no Coke is sold at all. And it’s going to be 1 minus P1 over 2 in the case, well, Coke and Pepsi cost exactly the same. You should already be realizing this is not exactly realistic. We’re making a lot of very strong assumptions here. But nevertheless, it’s going to be instructive to look at this model. Alright, so I’ve got the firms, they’re the players, I know what their strategies are, I know a little bit about the structure of the market, I still need to tell you what payoffs are, and just as last time, the firms are going to try and maximize profits. Alright, so what’s the profit for firm one going to be? I won’t bother to write it separately for firm two. it’s going to be the quantity it sells times the price it gets for that quantity minus that quantity it sells times the cost it incurs in producing that quantity. So again, for firm one, it’s the quantity it sells times the price, this is its revenue, minus… producing that quantity.

[译文 7]

明白吗？所以如果他们是低价公司，如果可口可乐是低价公司，那么只有可口可乐在市场上销售，百事可乐根本不在市场上销售。可乐的销量是1减P1。如果P1大于P2，销量就是0。所以如果百事可乐是市场上的低价饮料，那么可口可乐就根本卖不出去。而如果可口可乐和百事可乐价格完全相同，销量就是1减P1除以2。你们应该已经意识到这不完全现实。我们在这里做了很多非常强的假设。但尽管如此，研究这个模型还是很有启发性的。好，我有这些企业，它们是参与者，我知道它们的策略是什么，我也对市场结构也有一些了解，但我还需要告诉你们什么是收益，就像上次一样，这些企业会试图最大化利润。好，那么企业一的利润是多少？我就不单独为企业二写了。它等于销售数量乘以该数量获得的价格，减去销售数量乘以生产该数量所产生的成本。所以同样，对于企业一，它是销售数量乘以价格，这是它的收入，减去生产该数量的成本。

[段 8]

So again, for firm 1, it’s the quantity it sells times the price, this is its revenue, minus the quantity it sells times C, the cost, this is its cost. Alright, I can rewrite that a little bit more simply as Q1 brackets P1 minus C. where Q1, where this Q1, let’s keep it in square brackets, this Q1 is this object here. Do everyone understand what’s going on? I mean, the algebra doesn’t really help the intuition here. The intuition is pretty straightforward. Coke and Pepsi are producing this stuff. They’re setting prices. If Pepsi is the low price, then Coke sells nothing. If Coke is the low price, it sells this amount, and its profits are basically given by this equation. It’s basically just revenues minus costs. All right, so what we want to do is we want to figure out what the Nash equilibrium looks like in this market And just before we go any further notice that really this is the same basic market we looked at before right Both cut times there a demand curve out there In one model we thought of the firm setting quantities and the market determining prices And here we have the firm setting prices and the market determining quantities but the basic underlying economic structure of this is very very similar Alright, to define the Nash equilibrium, we’re going to have to find best responses.

[译文 8]

所以同样，对于企业一，它是销售数量乘以价格，这是它的收入，减去销售数量乘以C，也就是成本，这是它的成本。好，我可以把它改写得稍微简单一些，写作Q1乘以括号里的P1减C。其中Q1，这个Q1，我们把它留在方括号里，这个Q1就是这里的这个式子。大家都明白这是怎么回事吗？我的意思是，代数在这里对直觉理解并没有真正帮助。直觉相当直接。可口可乐和百事可乐在生产这种东西。他们在设定价格。如果百事可乐是低价，那么可口可乐就卖不出去。如果可口可乐是低价，它就卖这个量，而它的利润基本上由这个等式给出。它基本上就是收入减成本。好，那么我们要做的是找出这个市场中纳什均衡是什么样的。在我们继续之前，请注意，这实际上就是我们之前看过的同一个基本市场，对吧？两个模型中都有一个需求曲线。在一个模型中，我们认为企业设定产量，市场决定价格；而在这里，我们让企业设定价格，市场决定产量，但基本的潜在经济结构是非常非常相似的。好，要定义纳什均衡，我们必须找出最佳反应。

[段 9]

All right, so what we’re going to do together is to figure out what these best responses look like, and let’s try and do that. So I want to figure out the best response of firm 1 as a function of the price chosen by firm 2. The best price for firm 1 to choose, given that firm 2 is choosing P2. And I want to do this carefully and slowly, and before I even start, let me point out in passing that calculus is not going to help us here. So for those of you who are nerdy enough to understand the next sentence, let me just say it, this is a discontinuous payoff function, so differentiating isn’t going to get you very far. For people whom that didn’t mean the thing, it doesn’t matter, we’re not going to use calculus today. All right? Okay so there are different cases to think about So one case to think about is what if the other firm your Coke what if Pepsi is pricing below cost What if Pepsi is pricing below cost and your Coke What should you price as if you’re trying to maximize your profits? Somebody try and wave their hands in the air if they know the answer. Pepsi’s pricing below cost. What’s your best response? Yeah, there’s a guy in there. Do you want to shout out?

[译文 9]

好，那么我们要一起做的是找出这些最佳反应是什么样的，让我们试着做到这一点。我想找出企业一的最佳反应，作为企业二所选择价格的函数。在企业二选择P2的情况下，企业一应该选择的最佳价格。我想仔细地、慢慢地做这件事，在我开始之前，让我顺便指出，微积分在这里帮不了我们。对于那些足够书呆子气能够理解下一句话的人来说，我就直说了，这是一个不连续的收益函数，所以求导不会让你们走得很远。对于那些不理解这句话的人来说，这没关系，我们今天不会用到微积分。好吗？好，所以有不同的情况需要考虑。有一个情况需要考虑是：如果另一家企业，你说的是可口可乐，如果百事可乐定价低于成本呢？如果百事可乐定价低于成本，而你是可口可乐，你应该怎么定价才能最大化你的利润？如果有人知道答案的话，试着举手示意。百事可乐定价低于成本。你的最佳反应是什么？是的，那边有个人，你想大声说出来吗？

[段 10]

Just get out of the market because it’s not worth it for you to sell anything. Right, so I’ve forgotten your name, I should know it by now. Your name is? Sudipta. Sudipta says, get out of the market, right? If the other guy is pricing below costs, you don’t want to sell anything. That’s the right intuition. How in this game do I, quote, get out of the market? You’re absolutely right. I don’t want to be involved in this market if the other guy is selling below costs. How do I operationalize that in this game? How do I actually manage, I don’t know if there’s somebody here. Just wait for the mic a second, and I’ll shout it out. You could set your price above his price. The answer was set your price above his price. So the other guy is pricing below cost. The way which I avoid making losses is to set my price above his price All right let just make sure we understand this If the other guy is selling below cost the only way I can make any sales is to price below his price I will then make sales but each of those sales I making a loss on I don’t want to make losses, so I just, quote, get out of the market, and the way I, quote, get out of the market is by pricing above the other guy.

[译文 10]

就直接退出市场，因为不值得你去卖任何东西。对，我忘了你的名字，我应该现在已经知道了。你叫什么名字？Sudipta。Sudipta说，退出市场，对吧？如果另一个人定价低于成本，你就不想卖任何东西。这是正确的直觉。在这场博弈中，我如何"退出"市场？你完全正确。如果另一个人以低于成本的价格销售，我不想参与这个市场。我如何在这个博弈中具体实施这一点？我实际上如何处理，我不知道这里有没有人。等一下麦克风，我会大声说出来。你可以把自己的价格定得比他高。答案是把你的价格定得比他高。所以另一个人定价低于成本。我避免亏损的方式就是把价格定得比他的高。好，让我们确保理解这一点。如果另一个人以低于成本的价格销售，我唯一能做成任何销售的方法就是把价格定得比他的低。然后我会做成销售，但我每做一笔销售都在亏损。我不想亏损，所以我只是"退出"市场，而我"退出"市场的方式就是把价格定得比另一个人高。

[段 11]

All right, everyone understand that? All right? Okay, so now, more interestingly, let me leave a little gap here. What if the other guy is pricing above costs? What do I want to do if the other guy is pricing above costs? Can I get the guy with the beard here? Shout it out, because remember, that mic isn’t number one. You set price below his, but above or equal to cost. All right, so if the other guy is pricing above costs, I want to set prices below his so that I steal the whole of the market and make profits on those sales So where is it to suppose his pricing it? We said the price of it in 0 1 suppose. He’s pricing at 0.8 or be a good price for me to set Same same guy yeah as a small amount below his prices you can do a small amount as a small amount below his Same guy, yeah? As a small amount below his price as you can do. As a small amount below his price as I can. As small amount below his price as I can. So basically what I’m going to do here is I’m going to set my price to equal his price minus a little bit. And I’ll use the letter epsilon to mean just a little bit. I’ll just undercut him a little bit.

[译文 11]

好的，大家都明白了吗？好的？好，那么现在，更有趣的是，让我在这里留一点空白。如果另一个人定价高于成本呢？如果另一个人定价高于成本，我该怎么办？我能找一下那边留胡子的人吗？大声说出来，因为记住，那个麦克风不是一号。你把价格定得比他的低，但要高于或等于成本。好的，所以如果另一个人定价高于成本，我想把价格定得比他的低，这样我就能抢走整个市场，并在那些销售中获利。那么假设他定价多少呢？我们说它的价格是0.1。他定价0.8，或者0.7会是我设定的好价格。同一个人，是的？比他的价格低一点点，你可以做到。比他的价格低一点点。同一个人，是吗？比他的价格低一点点，你可以做到。比他的价格低一点点，我可以。所以基本上我在这里要做的是把我的价格定得等于他的价格减去一点点。我会用字母 epsilon 来表示只是一点点。我就稍微比他低一点点。

[段 12]

And by just undercutting him a little bit, I’m going to get the whole of the market and I’ll make as much money as I can on those sales. Now that’s almost right. That’s almost right. That’s kind of 90% of right, but it’s not quite right. I need to be a little bit careful. Now why do I need to be a little bit careful here? This is almost right, but it’s not quite right. Same guy? You want your price to be at or above cost. Okay, that’s certainly true. That’s true. Fair enough. So I want to I want to make sure this epsilon is not so big as it pushes me below cost. That’s certainly true That that’s that’s correct. There’s another little issue here So here the other issue there a price in this market We might want to think of as a kind of focal price There an interesting price and that the price at which the price I would choose were I the monopolist Suppose the other guy didn’t exist, suppose Pepsi didn’t exist, so Coke has the whole market, then we would solve out this problem, we would solve out what the monopoly price is, that’s an exercise we did in 115, and notice that if Pepsi has priced above the monopoly price, suppose Pepsi has priced this good so high that it’s above the monopoly price, then as the gentleman said, I can capture the whole market by pricing just below Pepsi.

[译文 12]

通过稍微比他低一点点，我就能获得整个市场，我将在那些销售中尽可能多地赚钱。现在这几乎是正确的。这几乎是正确的。这大概有90%是对的，但并不完全对。我需要稍微小心一点。为什么我需要在这里稍微小心一点？这几乎是正确的，但并不完全对。同一个人？你希望你的价格不低于成本。好的，这当然是正确的。这是正确的。好的。所以我想要确保这个 epsilon 不会大到把我推到成本以下。这当然是正确的，这是正确的。这里还有另一个小问题。所以这里另一个问题是，这个市场中有一个价格，我们可能想把它看作一种焦点价格，有一个有趣的价格，就是如果我是垄断者我会选择的价格。假设另一个人不存在，假设百事可乐不存在，所以可口可乐拥有整个市场，那么我们会解出这个问题，我们会解出垄断价格是什么，这是我们在115课程中做过的一个练习，并且注意到，如果百事可乐定价高于垄断价格，假设百事可乐把这个商品定得这么高，以至于高于垄断价格，那么就像那位先生说的，我可以通过定价只比百事低一点点来获得整个市场。

[段 13]

However, then I have the whole market. It’s as if I’m a monopolist, but suddenly I’m pricing above the monopoly price. That can’t be right. I never want to price above the monopoly price, because I know the monopoly price, where I’m the monopolist, is the most profit I could ever make. Is that right Right So we need to be a little bit careful This answer that says I just undercut Pepsi that true provided that Pepsi hasn gone above the monopoly price Right If he gone above the monopoly price we need to be a little bit more careful If P2 is actually above the monopoly price then what’s my best response in that case? What’s my best response in that case? Yeah, can we get Katie who’s in green there? Pricing at the monopoly price? Pricing at the monopoly price. So if Pepsi is dumb enough to price above Monopoly, sure, I will undercut Pepsi, but I won’t undercut Pepsi by a penny. I’ll undercut Pepsi all the way down to the Monopoly price and make Monopoly profits. I’ll be very happy. I’ll be wondering what on earth is Pepsi up to, but that’s fine. All right? So here, my best response is to price at the Monopoly price. All right? And there’s one other possibility here I’ve missed, which is what if Pepsi prices at marginal cost itself?

[译文 13]

然而，这样我就拥有了整个市场。就好像我是一个垄断者，但突然之间我的定价却高于垄断价格。这不可能是对的。我从来不想定价比垄断价格高，因为我清楚地知道，垄断价格——也就是我作为垄断者时的价格——是我能获得的最大利润。对吧？是的。所以我们需要稍微谨慎一点。回答说“我只是低于Pepsi的价格”，这在Pepsi没有超过垄断价格的条件下才是对的。如果Pepsi已经超过了垄断价格，我们就需要更加谨慎。如果P2实际上高于垄断价格，那么在这种情况下我的最佳反应是什么？在这种情况下我的最佳反应是什么？是的，能请穿绿衣服的Katie吗？定价在垄断价格？定价在垄断价格。所以如果Pepsi傻到定价高于垄断价格，当然我会低于Pepsi的价格，但我不会只低一分钱。我会把Pepsi的价格一路压低到垄断价格，然后赚取垄断利润。我会非常高兴。我会好奇Pepsi到底在干什么，但没关系。所以在这种情况下，我的最佳反应是定价在垄断价格。还有一种可能性我遗漏了，那就是如果Pepsi自己也定价在边际成本呢？

[段 14]

That’s the one missing case here. What if Pepsi prices at marginal cost? What’s my best response in this case? Pepsi is pricing exactly at marginal cost. What’s my best response? Yeah, wait for the mic. This is Henry I think Shout out To price at marginal cost as well I could price at marginal cost as well How much profit will I make if I price at marginal cost as well Zero profits I make zero profits So that certainly is a best response pricing at marginal cost as well. What else would be a best response? That’s correct, but what else would be a best response? Price above that, all right? It doesn’t really matter, as long as I don’t price below it, all right, as long as I don’t price below it, I’m not gonna make any money anyway. If I price pull low I’m not going to lose money. Alright? So this best response actually is a pretty complicated object and we could if we’re going to go like we did last time, we could take the time to draw this thing but it’s a bit of a mess so I won’t worry about doing that right now. Alright? So the best response is actually pretty complicated even though the best response is pretty complicated, and by the way obviously the things are symmetric for Player 2, even though these best responses are pretty complicated it turns out that there’s only one Nash equilibrium in this game.

[译文 14]

这就是那个遗漏的情况。如果Pepsi定价在边际成本呢？在这种情况下我的最佳反应是什么？Pepsi正好定价在边际成本。我的最佳反应是什么？是的，等一下麦克风。这是Henry，我想大声说出来，也定价在边际成本，我也可以定价在边际成本。如果我也定价在边际成本，我会获得多少利润？零利润，我获得零利润。所以这当然是一种最佳反应，也定价在边际成本。还有什么会是最佳反应？没错，但还有什么会是最佳反应？定价比那高，可以吗？只要我不定价比那低，就无所谓，对吧，只要我不定价比那低，反正我也赚不到钱。如果我定价很低，我也不会亏钱。所以这个最佳反应实际上是一个相当复杂的对象，如果我们像上次那样做，我们可以花时间来画这个图，但有点乱，所以我现在就不担心画它了。所以最佳反应实际上是相当复杂的，即使最佳反应相当复杂，顺便说一下，对于参与者2来说，情况显然是对称的，即使这些最佳反应相当复杂，但事实证明，在这个博弈中只有一个纳什均衡。

[段 15]

Watch the Nash equilibrium. Somebody watch the Nash equilibrium. Raise your hand. All you guys, there’s a huge number of hands who said they’ve seen it before. So who saw this model before? Fewer hands go up when they know there’s a question coming, right? There’s a different incentive, right? Who remembers what the Nash equilibrium is? Yeah, there’s a guy right by you there. Two. Fewer hands go up when they know there’s a question coming, right? There’s a different incentive, right? Who remembers what the Nash Equilibrium is? Yeah, there’s a guy right by you there. At C. Shout that out, even I can’t hear you. At C. At C, okay. So the Nash Equilibrium here, the Nash Equilibrium is for both firms to set their prices equal to marginal cost. and you can check that both firms are then playing a best response so that’s alright, if firm 2 is charging C then a best response for firm 1 is to set price equal to marginal cost and if firm 1 is pricing at marginal cost then conversely a best response for firm 2 is to price at marginal cost. Slightly harder exercise is to check that nothing else is in ash equilibrium Well, let’s think about that for a second. Suppose firm one is pricing at marginal cost, and firm two is pricing at something higher, at C plus, I don know three epsilon a little bit higher Suppose firm 1 is pricing at marginal cost and firm 2 is pricing at C plus 3 epsilon.

[译文 15]

看纳什均衡。有人知道纳什均衡吗？举手。你们当中有很多人之前说他们见过这个模型。那么谁之前见过这个模型？当他们知道有题目要问时，举手的人就少了，对吧？这是不同的激励，对吧？谁记得纳什均衡是什么？是的，就在你旁边那个人。两个人。当他们知道有题目要问时，举手的人就少了，对吧？这是不同的激励，对吧？谁记得纳什均衡是什么？是的，就在你旁边那个人。C。大声说出来，我都听不到你。C。C，好的。所以这里的纳什均衡是，两家公司都将价格设定为等于边际成本。你可以检验，两家公司随后都在进行最佳反应，所以这没问题，如果公司2收取C，那么公司1的最佳反应就是将价格设定为等于边际成本，如果公司1定价在边际成本，那么反过来公司2的最佳反应也是定价在边际成本。稍微难一点的练习是检验没有其他情况处于纳什均衡。好的，让我们想一下。假设公司1定价在边际成本，而公司2定价更高一些，在C加上，我不知道，3个epsilon，略高一点。假设公司1定价在边际成本，而公司2定价在C加上3个epsilon。

[段 16]

Is this an actual equilibrium? Let’s think about firm 2 first of all. Is firm 2 playing a best response to firm 1? Is firm 2 playing a best response to firm 1? so I claim it is right let’s just check carefully firm 1 is pricing at marginal cost right the best response if the other guy is pricing at marginal cost is to price at marginal cost or above and this is marginal cost or above so this is the best response so firm 2 is playing a best response this is the best response for firm 2 given that P1 is at C what’s the problem here why is it nevertheless I claim this is not a Nash equilibrium equilibrium, why is it not a Nash equilibrium? Who has an incentive to deviate? So can we get the guy way at the back, way, way, way at the back by the door? Go ahead. Shout out. Firm 1 is going to want to produce at C plus 2 epsilon. Exactly, exactly, exactly. This is not a best response This is not a best response And the reason is that the best response for firm 1 if firm 2 is charging c plus 3 little bits is to price at c plus 2 little bits. Thank you. All right? So it turns out, it’s pretty easy to check, that this is the only Nash equilibrium in the game.

[译文 16]

这是一个真正的均衡吗？让我们先想想公司2。公司2是否对公司1进行最佳反应？公司2是否对公司1进行最佳反应？所以我声称它是，让我们仔细检查一下，公司1定价在边际成本，对吧，如果对方定价在边际成本，最佳反应就是定价在边际成本或以上，而这一定价在边际成本或以上，所以这是最佳反应，所以公司2在进行最佳反应。这是公司2的最佳反应，给定P1在C。这里有什么问题，为什么尽管如此我声称这不是纳什均衡？为什么这不是纳什均衡？谁有激励偏离？所以能请后面那位，离门很远很远的那位吗？请说出来。公司1会想要定价比C高2个epsilon。没错，没错，没错。这不是最佳反应，这不是最佳反应。原因是如果公司2收取C加上3个小单位，公司1的最佳反应就是定价比C高2个小单位。谢谢。所以事实证明，这很容易检验，这是这个博弈中唯一的纳什均衡。

[段 17]

All right? now what’s the lesson we’re going to draw from this well as an exercise in game theory that really wasn’t very hard as an exercise in finding Nash equilibrium by this stage in the course most of you are looking at that saying that wasn’t hard so what’s the point here the point is that in this game in which firms competed in prices even though there were only two firms in the market only one firm more than monopoly, we get a dramatically different result than we had last time. In particular, we find that prices in the market are equal to marginal costs. We find that profit in equilibrium is equal to what Zero And we find there lots of consumer surplus because prices are really as low as they ever could be In fact the outcome here, this equilibrium here, is, for all intents and purposes, the same equilibrium we would have had, had there been thousands of firms in the market, and had this been a perfectly competitive market. So even though there’s only two firms here, with price competition, identical products, we end up with an outcome that looks exactly like perfect competition, except for the fact there’s only two firms. So the outcome is like perfect competition, even though there’s only two firms. That’s a pretty surprising result.

[译文 17]

好的，现在我们要从这个例子中得出什么教训？作为一个博弈论的练习，这真的不太难，作为一个寻找纳什均衡的练习，在这个课程的这个阶段，你们大多数人看着它说这个不难，所以这里的要点是什么？要点是在这个企业以价格进行竞争的博弈中，尽管市场上只有两家企头，只比垄断多一家企业，我们得到了与上次截然不同的结果。具体来说，我们发现市场价格等于边际成本。我们发现均衡中的利润等于多少？零。而且我们发现有很大的消费者剩余，因为价格真的已经低到不能再低了。事实上，这里的结果，这个均衡，从各方面来看，与如果有数千家企业在这个市场中，以及如果这是一个完全竞争的市场，我们会得到的均衡是一样的。所以尽管这里只有两家企业在进行价格竞争，产品相同，我们最终得到的结果看起来完全像完全竞争，除了只有两家企业的这个事实。所以结果就像完全竞争一样，尽管只有两家企业在竞争。这是一个相当令人惊讶的结果。

[段 18]

It would suggest, think about this as a policy thing, so if you believe this model, you think this is really an accurate model of society, and you’re a regulator working in the Department of Justice, or you’re a judge trying to judge some monopoly case, or you’re a commissioner on the European Court, or whatever, trying to judge some competition case, all you’d worry about is getting nothing of justice, or you’re a judge trying to judge some monopoly case, or you’re a commissioner on the European Court, or whatever, trying to judge some competition case, all you’d worry about is getting one competitor in each market, two firms in each market, and you’d be done. You wouldn’t worry about entry beyond two. Now, my guess is we don’t believe that. We don’t believe that. All right? We’ll come back to that in a second, but let me make a different remark before we get there. Is that me doing that? Let me move this down. Still doing it, okay. Probably the wire. Sorry. Let me just remove it and I’ll shout. Okay, I’m going to shout now. Can people still hear me? Yep, okay, thank you. All right. So we’ll have to turn down the other mic a bit. All right. So another thing to remark here is even though we looked at essentially the same market as we looked at last time and we didn make any really significant change we just said instead of thinking of them pricing and setting quantities think of them setting prices these were just thought experiments we ended up with a radically different outcome.

[译文 18]

这表明，把这作为一个政策问题来思考，如果你相信这个模型，你认为这真的是社会的准确模型，而你是在 Department of Justice 工作的监管者，或者你是一个法官试图审判某个垄断案件，或者你是 European Court 的委员，或者无论什么，试图审判某个竞争案件，你所担心的就是每个市场中只有一个竞争者，两家企业，你就完成了。你不用担心超过两家企业的进入。现在，我猜测我们不相信这一点。我们不相信这一点。好的。我们待会儿会回到这个问题，但让我在继续之前先说另一件事。是我在制造那个声音吗？让我把这个移下去。还在响，好的。可能是有线的问题。对不起。让我把它拿掉然后我喊。好的，我现在要喊了。各位还能听到我吗？好的，谢谢。好的，所以我们得把另一个麦克风调小一点。好的。所以在这里要指出的另一件事是，尽管我们看的市场与上次看的本质上相同，我们没有做任何真正重大的改变，我们只是说，不要想他们在定价和设定产量，而是想他们在设定价格，这些只是思想实验，我们最终得到了一个截然不同的结果。

[段 19]

Alright so another lesson here is the same setting, the same setting as as last time as Cournot but with a different strategy set a different way of thinking about what they’re doing led to a very different outcome And that’s worrying. It’s worrying because you can’t go away feeling comfortable about this. You can’t think that it really could make so much difference in the real world how prices and quantities and welfare and profit is going to work out depending on some thought experiment about how I think about my strategy set. So there’s a little worry going in about not just this example, but about the course about the whole theme of learning game theory All right So what going to save us here What going to save us is if we inject a little bit more reality back in the model we’re going to get back a more sensible result. So what we’re going to do now is we’re going to relax some of the assumptions of this Bertrand model, and it’s going to do two things for us. First, it’s going to give us an outcome that we believe. the outcome we believe I think is that imperfect competition should look something between monopoly and perfect competition, it shouldn’t look like perfect competition I think most of us don’t believe that two firms is enough to make the perfect competition, that the regulator shouldn’t worry about the third firm alright, and the second thing it’s going to do from a more theoretical point of view is it’s going to suggest to us that actually, actually things aren’t quite as bad as we thought, in fact if we model this more carefully we’ll get roughly the same prediction either way.

[译文 19]

好的，这里的另一个教训是，同样的设定，和上次古诺的设定一样，但不同的策略集、不同的思考方式导致了非常不同的结果。这令人担忧。令人担忧的是，你不能就此安然离去。你不能认为在现实世界中，价格、数量、福利和利润会如何运作，真的会因为我对策略集的想法实验而产生如此大的影响。所以有一点担忧，不仅关于这个例子，还关于这门课程，关于学习博弈论的整个主题。好的，那什么能拯救我们呢？如果我们往模型中注入一点更多的现实，我们会得到一个更合理的结果。现在我们要做的是放宽这个伯特兰模型的一些假设，这会为我们做两件事。首先，它会给我们一个我们相信的结果。我们相信的结果是，不完全竞争应该介于垄断和完全竞争之间，不应该看起来像完全竞争，我认为我们大多数人不相信两家企业就足以形成完全竞争，监管者不应该担心第三家企业，好的，第二件事从更理论的角度来看，它会告诉我们，实际上，实际上事情并没有我们想象的那么糟糕，事实上，如果我们更仔细地建模，两种方式会得到大致相同的预测。

[段 20]

All right? So what’s the assumption we’re going to change? Well, before I say that, I should say I’m not going to change it. You are. All right? So rather than go through this again a whole third time, I’ve gone through it once in quantities and once in prices, I going to get you guys to do it this time by having you do it on a homework assignment And well why Well partly because I think it a good exercise but also I don want this class to be a class where you sit there you know with your cup of Willoughby’s coffee, if needed to keep awake, and you watch me solve models, because that’s not how you learn. What I want this class to be is a class where at least sometimes, like in this homework assignment, you take, you know, I set up a model for you, or sort of set up the story for you, and then you have to actually figure out how do I set this up properly, and how do I solve it out? Because at the end of the day, if game theory is going to be useful for you in your later lives, whether it’s in dating or in running a company or whatever it happens to be, you need to be able to go from the story to the model.

[译文 20]

好吗？那么我们要改变的假设是什么？在我说出来之前，我应该说，我不打算改变它。你们来改。好的，所以与其再完整地过第三遍，我已经讲过一次关于数量的，一次关于价格的，这次我要让你们来做，通过让你们做作业的方式。为什么要这样做呢？部分是因为我认为这是一个很好的练习，但也因为我不希望这门课变成你们坐在那里，手里端着威洛比咖啡，如果需要的话保持清醒，然后看我解模型的课，因为那不是你们学习的方式。我希望这门课是什么样的呢？是至少有时候，比如在这个作业中，你们，我给你们设定一个模型，或者给你们设定一个情境，然后你们必须真正弄清楚我如何正确地设定它，以及如何求解它？因为归根结底，如果博弈论要在你们以后的生活中对你们有用，无论是在约会中还是在经营公司中，或者无论它是什么，你需要能够从情境走向模型。

[段 21]

All right. Having said that, I’ll tell you what the homework assignment is going to be about. The assumption we’re going to change is the idea that products are identical. So we’re going to look at a case, I need to move forward, where products are differentiated. and I’m going to claim that assuming that products are not identical is a pretty safe assumption for most of the world for most things that you’re going to see in the world so we’re going to look at what are called differentiated products. And in particular, we’re going to look at a particular version of this that we’re going to call the linear city model. alright so what do I mean by products being differentiated so I started off with an example that’s pretty bad for this story namely coke and pepsi now be honest how many of you in a blind taste test can taste the difference between ordinary coke and ordinary pepsi oh quite a few okay that’s good for my case right so a number of you think those are different products right so a number of you even if their price was just a tiny bit different might have a preference for one rather than the other so without getting in trouble by putting the camera on you how many of you would have a preference, if the prices were essentially the same, how many of you would have a strict preference for Pepsi?

[译文 21]

好的。话虽如此，我会告诉你们作业的内容是什么。我们将要改变的假设是产品是相同的。所以我们要看一个情况，我需要继续前进，其中产品是差异化的。我要说的是，假设产品不相同，对于世界上大多数你们会看到的大多数事情来说，这是一个相当安全的假设。所以我们要看所谓的差异化产品。特别是，我们要看看一个特定的版本，我们称之为线性城市模型。好的，我说的产品差异化是什么意思呢？我一开始就举了一个对这个故事相当不利的例子，那就是可口可乐和百事可乐。现在诚实地说，你们中有多少人能在盲品测试中尝出普通可口可乐和普通百事可乐之间的区别？哦，有不少人，好的，这对我的论证有好处。所以你们中的一些人认为这些是不同的产品，即使它们的价格只有一点点不同，也可能对其中一个有偏好。所以不要让摄像头对准你们，如果价格基本相同，你们中有多少人对百事可乐有严格偏好？

[段 22]

And how many of you would have a strict preference for Coke? And how many of you would have a strict preference for diet, diet Coke? That amazing that stuff oh never mind Okay Okay but you making my point The point I trying to make is products are not identical Alright? Products are not identical. Okay? For the most part, there’s a little bit of difference between products, and that’s actually going to… if we inject that little bit of realism into the world, it’s actually going to help us. So what we would like, what would we like? We’d like a model in which firms set prices. because for the most part we think firms do set prices, not quantities. Not always, but for the most part. But we’d like a model that yields an outcome that when you only have two firms looks somewhere between monopoly and perfect competition. So we’d like an outcome that looks a little bit like Cournot but we’d like the strategy set to be prices. And this is going to do the trick. This is going to turn out to do the trick as you’ll find out in your homework assignment. So how are we going to model this on the homework? the way we’re going to model differentiated products is to imagine, just to take a simple example, imagine a city, and this city has one long straight road through it.

[译文 22]

你们中有多少人对可口可乐有严格偏好？你们中有多少人对健怡可口可乐有严格偏好？这种东西太神奇了，算了算了，但是你在帮我论证我的观点。我想说的观点是产品不是相同的。产品不是相同的。好的？在大多数情况下，产品之间有一些微小的差异，而这实际上会……如果我们把这种微小的现实注入世界，它实际上会帮助我们。所以我们想要什么？我们想要一个企业设定价格的模型。因为大多数情况下我们认为企业确实设定价格，而不是数量。不总是，但大多数情况下。但我们想要一个模型，它产生的 Outcomes，当你只有两家企业时，看起来介于垄断和完全竞争之间。所以我们想要一个看起来有点像古诺的结果，但我们想要策略集是价格。这会起作用。在你们作业中你们会发现这会起作用。所以我们在作业中要如何建模呢？我们建模差异化产品的方式是想象，仅仅是举一个简单的例子，想象一座城市，这座城市有一条长长的直路穿过。

[段 23]

So this city is not a city in New England, it’s a city in the Midwest, where everything’s flat and the roads just go completely straight. And you can think of it as being a mile long it doesn really matter let think of it as being a mile long And we going to assume that consumers are evenly spread along this city Alright, consumers are evenly spread along this city. So there’s basically consumers everywhere, they’re evenly distributed. And we’re going to assume that one of these firms, one of these firms, let’s call it firm one, sits at point 0 and the other firm firm 2 sits at point 1 now this is what you’re going to do in the homework assignment but let me just make the argument that you could also imagine firms sitting somewhere between 0 and 1 we could do a more general job if we wanted to, but for now let’s assume that one of these firms has its shop at one end of the town and the other one has its shop at the other end of the town So let’s think about a particular consumer. So suppose this consumer is here, point Y. So notice that this consumer is a distance Y away from firm 1. So if she consumes from firm 1 she has to walk a distance of Y And she a distance of 1 minus y from firm 2 Is that right?

[译文 23]

所以这座城市不是新英格兰的城市，是中西部的城市，那里一切都是平的，道路完全笔直。你可以把它想象成一英里长，实际上不重要，假设它是一英里长。我们将假设消费者均匀分布在这座城市里。好的，消费者均匀分布在这座城市里。所以基本上到处都是消费者，均匀分布。我们将假设其中一家企业，其中一家企业，我们称它为企业1，位于0点，另一家企业企业2位于1点。这是你们在作业中要做的事情，但我只是论证一下，你们也可以想象企业位于0和1之间的某个地方，如果愿意的话我们可以做更一般的情况，但现在让我们假设其中一家企业在城镇的一端有它的商店，另一家企业在城镇的另一端有它的商店。所以让我们考虑一个特定的消费者。假设这个消费者在这里，Y点。所以注意这个消费者与企业1的距离是Y。所以如果她从企业1消费，她必须走距离Y。她与企业2的距离是1减Y。对吗？

[段 24]

So if she consumes from firm 2, she has to walk 1 minus y. That’s going to turn out to be key in our model, as we’ll see in a second. So firms, as before, are going to set prices. and I won’t write everything down because it’s all written down on the homework assignment which is already on the web, but in fact firms are going to maximize profits, aim to maximize profits, firms are going to have constant marginal costs, we’ll make one other assumption to keep life simple. We’ll assume that each consumer buys one and only one product. So each consumer is going to buy one product either from firm one or from firm two. so the issue is going to be which firm does each consumer go and buy their product from which firm does each consumer choose and we’ll assume that each consumer chooses the products that are chooses the product whose, and I’ll be careful here, whose total cost to her is smaller. And what do we mean by total cost? Well, let’s look for the consumer at point Y. For example, for the consumer at point Y, if they buy from firm 1, then they pay the price P1, which is set by firm 1, but they also have to pay a transport cost. the cost to them of having to walk all the way there and walk all the way back.

[译文 24]

所以如果她从企业2消费，她必须走1减Y。这将证明是我们模型中的关键，我们马上就会看到。所以企业，和之前一样，将设定价格。我不会把所有东西都写下来，因为都写在作业里了，作业已经在网上了，但事实上企业将最大化利润，致力于最大化利润，企业将有不变的边际成本，我们再做一个假设来简化生活。我们假设每个消费者只购买一种产品。所以每个消费者将购买一种产品，要么从企业1，要么从企业2。所以问题将是每个消费者去哪家买他们的产品，每个消费者选择哪家企业，我们将假设每个消费者选择那个总成本对她来说较小的产品。我会在这里仔细说。什么是总成本？好的，让我们看看位于Y点的消费者。例如，对于位于Y点的消费者，如果他们从企业1购买，那么他们支付企业1设定的价格P1，但他们还必须支付运输成本。走那么远去然后走回来的成本。

[段 25]

And we’ll give a name to that transport cost. We’ll say it’s T y squared. So y is the distance they have to travel, and T y squared is their transport cost So this object here we could think of as a transport cost Right? If the same consumer buys from firm 2, she pays p2 plus t times, again, the distance squared, so that’s going to be 1 minus y squared. And once again, this last term is a transport cost. Notice that these transport costs go up in the distance you have to travel, and they go up pretty fast. They go up at rates squared. So what you’re going to do on the homework assignment is solve out this market. You’re going to assume that firms set prices to maximize profits. You’re going to know what firms’ costs are. You’re going to work out what firms’ demands are going to look like for each possible prices they could set. And you’re going to solve out the whole Nash equilibrium. And then we’re going to look at that Nash equilibrium, and you going to think how does that compare to what I saw in the Cournot case and so on So that on the homework assignment But before I leave this, let me just point out that this is a little bit more general than it might appear.

[译文 25]

我们会给那个运输成本起个名字。我们把它记作 T y 平方。所以 y 是他们需要旅行的距离，而 T y 平方是他们的运输成本。这个对象我们可以理解为运输成本，对吧？如果同样的消费者从 firm 2 购买，她支付 p2 加上 t 乘以距离的平方，所以那是 1 减 y 平方。同样地，这最后一项是运输成本。注意这些运输成本随着你需要旅行的距离增加而增加，而且增加得非常快。它们以平方的速度增加。所以你们在作业中要做的就是求解这个市场。你们要假设 firms 设定价格以最大化利润。你们要知道 firms 的成本是什么。你们要找出 firms 的需求在不同可能价格下的形状。然后你们要求解出整个 Nash equilibrium。接着我们会来看这个 Nash equilibrium，你会思考它与我在 Cournot 情形中看到的情况相比如何等等。这就是作业要求。但在我离开之前，让我指出这比它看起来的要更 general 一些。

[段 26]

So here I’ve treated what makes products different as being where the shops are located. so here I’ve interpreted these terms here as transport costs and I’ve interpreted what makes the products different is the fact that one of them is selling at one end of the town and the other one is selling at the right end of town but actually we could consider this model more generally let’s just do so briefly here so let me just redraw the town, here’s my town again I don’t have to regard this product, this line as being distance along the high street of the town it could be something else about the product so for example, in whatever it is you guys imagine makes Coke and Pepsi different it could be that thing I don’t quite know what that is, but whatever that is let me take an example that I understand better than I understand Coke and Pepsi so imagine we talking about beer All right Think about beer This is the beer market All right? This distance here could be something like the alcohol content or the flavor of the beer. All right? And you can imagine different products, therefore, that are on the market positioning themselves or being positioned at different points on the line. So, for example, up here, if we’re talking about beer flavor, this might be Guinness. all right this might be Guinness but I can’t even spell but you know what I mean all right and if you think about the drinks industry more generally here rather than just beer this would be Guinness this would be Poland spring all right this is Warsaw right and everything else would be in between here all right so if we just go a tiny distance in here all right this is Bud Bud Light.

[译文 26]

所以在这里，我把使产品不同的因素视为商店所在的位置。所以在这里，我把这些术语理解为运输成本，我把产品差异化的原因理解为一个在城镇的一端销售，另一个在城镇的右端销售。但实际上我们可以更一般地考虑这个模型，让我们在这里简要地做一下。让我重新画一下城镇，这是我的城镇。我不必把这个产品、这条线视为城镇主街上的距离，它可能是关于产品的一些其他东西。例如，你们认为是什么让 Coke 和 Pepsi 不同，那可能就是那个东西。我不太清楚那是什么，但不管是什么。让我举一个我比了解 Coke 和 Pepsi 更好的例子——想象一下我们在谈论啤酒。这是啤酒市场。这个距离可能类似于酒精含量或啤酒的口味。因此，你可以想象市场上不同的产品在定位自己，或者说被定位在这条线的不同点上。例如，在这里，如果我们谈论啤酒口味，这可能是 Guinness。好的，这可能是 Guinness，但我连拼写都不会，但你懂我的意思。好的，如果你更广泛地考虑饮料行业而不只是啤酒，这会是 Guinness，这会是 Poland Spring。好的，这是 Warsaw，对，其他的都在这之间。好的，所以如果我们在这里走一小段距离，这是 Bud Bud Light。

[段 27]

Right? And so on, right? Right? You could put everything on this line, all right? I think there may be… I’m not really allowed to do this on this model. The one thing I can’t do is… The truth is Bud Light might be down here somewhere, but I’m not allowed to do that. All right? All right? So, leaving aside the specific example of beer, you think about some product that has some dimension on which it varies. Alright, alright. So leaving aside the specific example of beer, you think about some product that has some dimension on which it varies and we can use this model, we can use this model to see how competition is going to work in that market. But now notice that this transport cost is going to be a different interpretation. Now instead of being the cost of traveling that distance to go and buy the product, what’s it going to be? It’s going to be that if my preferred beer flavor is here, say, this is me, my preferred beer flavor is here, then if I end up having to consume Guinness, I have to pay the price of Guinness, and I also incur some costs because Guinness isn’t the perfect beer for me. It’s a little bit too strong. And if I consume Bud Light, then I have to pay for the Bud Light, I think typically cost less, the price, the raw price is less than Guinness, but I also pay an additional cost from the fact that the Bud Light, when I drink it, causes me disutility.

[译文 27]

对吧？以此类推，对吧？对吧？你可以把所有东西都放在这条线上，对吧？我想可能……在这个模型中我真的不能这样做。我唯一不能做的事情是……事实是Bud Light可能在下面某个位置，但我不允许那样做。好吗？好吗？所以，暂且不谈啤酒的具体例子，你想想某个在某些维度上有所不同的产品。好的，好的。所以，暂且不谈啤酒的具体例子，你想想某个在某些维度上有所不同的产品，我们可以使用这个模型，我们可以用这个模型来看竞争如何在那市场中运作。但现在注意，这个运输成本将会有不同的解释。现在它不再是走那段距离去买产品的成本，它会是什么？会是这样：如果我偏好的啤酒口味在这里，比如说，这是我，我偏好的啤酒口味在这里，那么如果我最终不得不喝Guinness，我必须支付Guinness的价格，而且我还会产生一些成本，因为Guinness对我来说不是完美的啤酒。它有点太浓了。如果我喝Bud Light，那么我必须为Bud Light付费，我想通常更便宜，价格，原价比Guinness低，但我也支付额外的成本，因为Bud Light当我喝它时，会给我带来负效用。

[段 28]

We put it lightly like that. All right? All right? All right? So basically that transport cost is now the lack of pleasure caused by drinking a product that isn perfect for me So everyone understand the story here So you’re going to figure out what happens in this story very generally, well not very generally, in the particular case actually, on the homework assignment, but I want you to understand that there’s this much more general story underlying this, and this is a pretty good model of a lot of markets. It’s the kind of model that you’d see again if you went on to graduate school. All right. Now I want to spend the rest of today doing something quite different. But you’ll see it isn’t all that different. I’m going to go away from studying imperfect competition and go back and visit something we studied almost the first day, or maybe the second day, and that’s elections. I want to go back to elections. And why do I want to go back at this point? Well, one reason I want to go back is you’ll notice I’ve been putting up these lines on the board, and when we visited elections on the second day, we said that you could think of that line on the board as being not just left-wing, right-wing politics, but also some dimension of products.

[译文 28]

我们轻描淡写地说一下。好吧？好吧？好吧？那么基本上，运输成本现在就是喝一个对我来说并不完美的产品所带来的不愉悦。大家都理解这个故事了吧。所以你们会发现，这个故事里发生了什么，在作业里你会看到非常具体的案例，但实际上我想让你们理解，这背后还有一个更加普遍的故事，这是一个非常好的市场模型。如果你们继续读研究生的话，你们会再次看到这种模型。好吧。现在我想在剩下的时间里做一些完全不同的事情。但你们会发现它并没有那么不同。我要离开不完全竞争的研究，回到我们几乎在第一天或者第二天研究过的东西，那就是选举。我想回到选举。为什么在这个时候我想回去呢？一个原因是你们会注意到我一直在黑板上画这些线，当我们第二天讨论选举时，我们说过你们可以把黑板上的那条线不仅仅理解为左翼、右翼政治，也可以理解为产品的一个维度。

[段 29]

So I want to go back again Now we considering the line on the board as being flavor in beer or location in a town And I want to go back to politics now and go back to the interpretation we started with so that left and right will end up being left-wing politics versus right-wing politics. So I’m going to take the same basic idea back to the politics model. And we’ll spend the rest of today in politics. So we’re going to be doing political science or political science as it meets game theory for the rest of the day. We’re going to study something called the candidate voter model. Candidate voter model. And this model is going to look a lot like the models we looked at already a few weeks ago. in particular as advertised there will be a line and this line will go from 0 to 1 and the left hand side of this line will represent the left wing and the right hand side of the line will represent the right wing so that’s the same as before and as before as in the downs or hotelling model we discussed already a few weeks ago we going to assume that voters are evenly spread along the line So we going to assume even distribution of voters on the line And we’re also going to assume, just as we did last time, that voters are going to vote for the closest candidate.

[译文 29]

所以我想回到之前的话题。现在我们把黑板上的这条线视为啤酒中的口味或者小镇中的位置。现在我想回到政治学，回到我们一开始的那种解释，这样左和右最终会变成左翼政治与右翼政治。所以我要把同样的基本概念带回政治模型。我们今天剩下的时间都会在政治学领域。我们将研究政治科学，或者说政治科学与博弈论的结合，今天剩余的时间都是这个内容。我们将学习一个叫做“候选人-选民模型”的东西。候选人-选民模型。这个模型看起来会非常像我们几周前看过的那些模型。特别是，按照之前预告的，将会有一条线，这条线从0到1，线的左侧代表左翼，线的右侧代表右翼，这和之前是一样的。和我们几周前讨论的Downs或Hotelling模型一样，我们假设选民均匀分布在这条线上。所以我们假设选民在直线上的分布是均匀的。我们还假设，就像上次一样，选民会投票给最接近的候选人。

[段 30]

So voters vote for the closest candidate. So all of these assumptions are the same assumptions we studied two weeks ago, or two and a half weeks ago, and which you actually studied in your first homework assignment. Is that right? But I want to go back there, because what I want to do now is I want to change some critical assumptions of that model, and see that by making those changes, we’re going to get some very different outcomes. So in particular, the new things here, two things. One, the number of candidates, the number of candidates is not… is not fixed. So the number of candidates in this model is going to be endogenous. Previously, we looked at models where there were two candidates, or on your homework assignment three candidates. Now we’re going to allow the number of candidates to adjust itself. And the second assumption we’re going to make, which is new, is that we’re going to assume that candidates cannot choose their position. So the idea is, any candidate who stands in this election, you know who that candidate is. You know whether they’re right-wing or left-wing. So they can’t tell you they’re something else. so candidates can not choose their position so this is a subtle thing because you think about the current election there a debate about whether Hillary Clinton for example can choose right now to be at the center of the Democratic Party given her past history of votes for example on the Iraq War Or on the other side, there’s a debate about whether Mitt Romney can choose to be, I guess he’s trying to choose himself to be on the right-hand wing of the Republican Party, given he has a record of government as being governor of Massachusetts when he provided state health care, for example.

[译文 30]

所以选民投票给最接近的候选人。所以所有这些假设都和我们两周前或两周半前研究的假设相同，也就是你们在第一次作业中研究的内容。对吗？但我想回到那个模型，因为我现在想做的是改变该模型的一些关键假设，看看通过做出这些改变，我们会得到一些非常不同的结果。所以特别是，这里的新内容有两件事。第一，候选人的数量，候选人的数量不是……不是固定的。所以这个模型中候选人的数量将是内生的。之前，我们研究了有两个候选人的模型，或者在你们的作业中有三个候选人的模型。现在我们将允许候选人数量自行调整。而我们要做的第二个假设，这是新的，是我们将假设候选人不能选择自己的立场。所以这个想法是，任何在这次选举中参选的候选人，你知道那个候选人是谁。你知道他们是右翼还是左翼。所以他们不能告诉你他们是别的什么人。所以候选人不能选择自己的立场，这是一件微妙的事情，因为你想一想当前的选举，有一个关于Hillary Clinton现在是否能选择站在民主党中间位置的争论，考虑到她过去的投票记录，例如关于Iraq War。或者在另一方面，有一个关于Mitt Romney是否能选择让自己站在共和党右翼的争论，考虑到他担任Massachusetts州长时的政府记录，当时他提供了州医疗保健，例如。

[段 31]

So it’s quite difficult in the real world for candidates to position themselves. Voters tend to know that those candidates have track records. That’s something we mentioned already three weeks ago, two weeks ago. We’re going to take that fix now. We’re going to assume you know who these candidates are. So what are we actually going to assume? We’re going to assume that each voter in the model is a potential candidate. So this is, I think, a nice idealized version of American politics. Each of you who is above 18 and is an American citizen and was born in America, maybe you have to be more than above 18, above whatever it is, whatever the rule is, whatever the Constitution says, each of you can potentially stand up now and say you’re going to run for president. I guessing all of you are ruled out by age actually there some rule in the Constitution But never mind let pretend that not there So the idea of the model is each voter is a potential candidate That’s a very appealing assumption, I think, in a democracy. So what I want to do is I want to lay this out a little bit more formally as a game, and then, since we haven’t done anything like this for a week or so, we’re going to play the game. All right, so you’re about to play this game, so pay attention, please.

[译文 31]

所以在现实世界中，候选人很难定位自己。选民往往知道这些候选人是有过往记录的。这是我们三周前、两周前已经提到的事情。我们现在要采取这个修正。我们假设你已经知道这些候选人是谁。那么我们实际要假设什么呢？我们假设模型中的每个选民都是潜在的候选人。我认为这是对美国政治的一个很好的理想化版本。每个年满18岁、是美国公民、且在美国出生的人，也许需要超过18岁，无论什么规定，无论宪法怎么说的，你们每个人现在都可以站出来说你要去竞选总统。我猜你们所有人都因为年龄被排除在外了，宪法里确实有一些规定。但没关系，就当我们假装没有这条规定。模型的理念是每个选民都是一个潜在的候选人。我认为在民主制度中这是一个非常有吸引力的假设。所以我想做的是把这个更正式地作为一个游戏来展开，然后，因为我们大约有一周左右没有做过类似的事情了，我们要玩这个游戏。好，你们即将玩这个游戏了，请集中注意力。

[段 32]

All right, so who are the voters, who are the players? The players are the voters. Right, the voters or candidates, whatever you want to call them. candidates, candidates, depending on where they stand, in this game they’re going to be you. And the key strategy here, essentially the strategy is going to be very simple, the strategy is do you run or not The reason that going to be the strategy is that voting is not going to be difficult You always going to end up voting for the candidate who closest All right So the only really relevant strategy is to run or not to run to stand or not to stand All right? So just to make that clear, voters vote for the closest running candidate first. And second, what does it mean to win? We’ll assume that we’re in a plurality election here. So the winner is the person who gets a plurality. you win if you get the most votes in the woods you win if you get the most votes and we’ll assume that if there’s a tie that we flip a fair coin or a supreme court judge whatever you want to take it flip if tie alright and the payoffs in this game and All right. And the payoffs in this game are as follows. We’ll assume that there’s a prize for winning.

[译文 32]

好，那么谁是选民，谁是参与者？参与者就是选民。对，选民或候选人，你怎么称呼他们都行。候选人，候选人，取决于他们在这个游戏中的立场，他们就是你们。而这里的关键策略，本质上策略会非常简单，策略就是你是否参选。之所以这会成为策略是因为投票不会困难，你总是会投票给最接近的候选人。好，所以唯一真正相关的策略就是参选或不参选，站出来或不站出来，好吗？所以说清楚，选民首先投票给最接近的参选候选人。其次，获胜意味着什么？我们假设这是一场简单多数制选举。所以获胜者是获得最多票数的人。如果你在林中获得最多票你就赢了，如果你获得最多票你就赢了，我们假设如果有平局，我们就抛硬币或让最高法院法官决定，随你怎么理解来处理平局，好，而这场游戏的收益是这样的。好，这场游戏的收益如下。我们假设获胜有一个奖励。

[段 33]

So if you win the election, you get a prize equal to B. All right. We’ll also assume that there’s a cost of running so if you enter this election win or not you incur a cost of C and we’ll assume that B is greater than or equal to 2C and actually for today for today let’s just keep it simple and assume it’s actually equal to 2C alright But that’s not the only part of the payoffs. There’s also a part of the payoffs that’s analogous to forcing me to drink Bud Light, or forcing the Pepsi drinkers to drink Coca-Cola. And what’s that? That if regardless of whether I run or not if some other candidate is elected other than me then they not going to have my ideal policies So it going to cause me unhappiness it going to cause me disutility having a candidate win who far away from me So there’s going to be an extra thing. and so if if your position is x if you are at x on that line and the winner of the election is at y then you pay a cost of minus x minus y, the absolute distance between you and the winning candidate it. So again, if you’re at x and the winner’s at y, it hurts you minus the distance between x and y in terms of your unhappiness at having a winner who is far away from you winning.

[译文 33]

所以如果你赢得选举，你会得到一个等于B的奖励。好。我们还假设参选有成本，所以无论你是否赢得选举，只要你参选就要承担C的成本，我们假设B大于等于2C，实际上今天，今天我们简单起见就假设它等于2C，好。但这不是收益的唯一部分。收益还有一部分类似于强迫我喝百威淡啤，或者强迫喝百事可乐的人喝可口可乐。那是什么？就是无论我是否参选，如果有其他候选人当选而不是我，那么他们不会实施我的理想政策。所以这会让我不开心，会给我带来负效用，让一个离我很远的候选人当选。所以还会有额外的东西。所以如果你的位置是x，如果你在那条线上的x位置，而选举获胜者在y位置，那么你会支付一个负的代价，即|x-y|，你与获胜候选人之间的绝对距离。所以同样，如果你在x位置，获胜者在y位置，获胜者离你很远会让你不开心，你会付出x和y之间距离的代价。

[段 34]

So let’s do an example. So make sure we understand the payoffs So example one if you at X let call you Mr X let call him Mr X make it clear so Mr. X, the person who’s at position X, if he enters the election and he wins the election, then his payoff is B because he entered, minus C, sorry, B because he won, minus C because he had to pay his election expense. But the winning candidate is him, so he gets no dissent from the winning candidate being someone else. All right? Second possibility, if Mr. X enters but Mr. Y wins, then X’s payoff is what? He’s still incurred minus C because he ran, and he also has a cost of X minus Y because he doesn’t like Mr. Y winning. Right? And third if Mr X stays out but Mr Y wins then Mr X payoff he doesn win he doesn get B he doesn’t lose C because he didn’t run, but he still has this disutility, minus X minus Y, because he doesn’t like Y winning. alright does everyone understand this game? everyone understand the game? everyone not understand the game at this point? alright in principle we could play this with the whole class but let’s single out a particular row of the class so I’m going to come down here and I guess eventually I’ll grab it in a minute Alright, so I’m going to use this row, I think, this row, okay, this row, does everyone in this row stand up a second?

[译文 34]

所以我们来举个例子。确保我们理解收益。好，例子一，如果你在X位置，我们叫他X先生，我们叫他X先生以示清楚，所以X先生，在X位置的那个人，如果他参选并且赢得选举，那么他的收益是多少？因为他赢了所以是B，减去C，对不起，因为他参选要支付选举费用所以是B减去C。但获胜者是他自己，所以他不会因为获胜者是别人而产生不满。好。第二种可能，如果X先生参选但Y先生赢了，那么X的收益是多少？他仍然要承担负C因为他参选了，而且他还要承担X减Y的成本，因为他不喜欢Y先生获胜。好？第三种，如果X先生不参选但Y先生赢了，那么X先生的收益是，他没有赢，他没有获得B，他没有损失C因为他没有参选，但他仍然有这种负效用，负X减Y，因为他不喜欢Y获胜。好，大家都理解这个游戏了吗？所有人都理解这个游戏了吗？目前还有人没理解这个游戏吗？好，原则上我们可以全班一起玩这个游戏，但我们先挑出一排来，所以我要到这边来，我想等会儿会拿到的。好，我要用这排，我看看，这排，好，这排，这排所有人都站起来一下好吗？

[段 35]

And this row stand up. Alright, this is the row of potential voters. Sorry, the voters, they’re also the potential candidates. Alright, so we need to have a convention here. This is the row of potential voters. Sorry, they’re the voters, they’re also the potential candidates. Alright? So, we need to have a convention here, whether we’re viewing this from the perspective of people behind them, or people in front of them. Let’s view it from the perspective of people in front of them. So, I got an R, okay, I’ll do the other way around. People from behind them have the perspective, okay. So, this gentleman here, whose name is… Andy. Andy, thank you. So, Andy is our crazy right-wing guy. all right and uh… Sudipto? Sudipto is our crazy left-wing guy. Everyone else is in between there’s one two three four five six seven eight nine ten eleven twelve thirteen fourteen fifteen sixteen people here let me add more, make it seventeen people so we have an odd number so go and sit in that row. Yeah, yeah, yeah, let’s go sit somewhere else. Okay now we have seventeen people, it’s a bit easier. Alright, just want to avoid obvious ties. Okay? Alright. So how’s this going to work? Everybody sit down again, but don’t put your… keep your books handy, because this is going to be pretty aerobic.

[译文 35]

这排站起来。好，这是潜在选民所在的那一排。对不起，选民，他们也是潜在的候选人。好？所以我们需要一个约定。我们是从站在他们后面的人的视角，还是从站在他们前面的人的视角来看？让我们从站在他们前面的人的视角来看。好，我得到一个R，好，我换另一种方式。从站在他们后面的人的角度，好。所以这位先生，他的名字是……Andy。谢谢你。所以Andy是我们的疯狂右翼家伙。好，Sudipto？Sudipto是我们的疯狂左翼家伙。其他所有人都在中间，这里有一二三四五六七八九十十一十二十三十四十五十六个人，让我再加一些人，凑成十七个人，所以去坐在那一排。好，好，好，去坐在那边的某个位置。好，现在我们有十七个人了，这样会更容易一点。好，我们只是想避免明显的平局。好？所以这要怎么进行？所有人再坐下，但不要把你们的……把书放在手边，因为这会相当消耗体力。

[段 36]

Alright So these are the potential voters and each of them is in the position that Fred Thompson was in two weeks ago in the Republican primary when he deciding whether to run or not But in the real world that a sequential game People decide to run in some order We’re going to assume this is all simultaneous. So what we’re going to do is, we’re going to hold essentially this game. We’re going to ask everybody in this row whether they want to enter or not. Let’s make some real numbers. Let’s assume that B is equal to $2. So the spoils of government are $2. and entering costs you $1 so C is $1 and we’ll assume that the amount of disutility you get from being far away from the winner is 1 17th of the each place is worth 1 17th of a dollar alright everyone understand? so whatever 1 17th of a dollar roughly 5 cents for each position away ok everyone understand in this row what we’re doing? I’m not getting any nods you guys over here you understand what we’re doing? All right, so on the count of three, on the count of three, anybody who’s entering this election is going to stand up. Ready? One… Two… Okay, what? That the opposite of what happening with the Florida primary Yeah question I guys have to decide whether you okay if you gonna if you running you stand up.

[译文 36]

好，所以这些是潜在选民，他们每个人都处于Fred Thompson两周前在共和党初选中的位置，当时他在决定是否参选。但在现实世界中这是一个序贯博弈，人们按某种顺序决定是否参选。我们将假设这一切都是同时发生的。所以我们要做的就是，我们将进行这个游戏。我们将问这一排的每个人他们是否要参选。让我们设定一些实际的数字。让我们假设B等于2美元。所以政府的好处值2美元，参选要花费你1美元所以C是1美元，我们假设你因为远离获胜者而产生的负效用是1/17，每个位置值1/17美元，好大家都理解了吗？所以每个位置大约是5美分，好每个人都理解我们在这排要做的事了吗？我没看到有人点头，你们这边的人理解我们在做什么吗？好，那么我们数到三，数到三的时候，任何要参选的人就站起来。准备好了吗？一……二……好，什么？这和佛罗里达初选发生的情况相反是的，问题是我大家要决定你们是否好如果你要参选你就站起来。

[段 37]

Okay, if you’re standing, you stand. All right, if you’re standing, you stand. All right, let’s try again. One, two, three. Come on. All right, stay standing, stay standing, Stay standing, stay standing. And I’ll come around a second. All right, so here’s our election. Here’s our election. And it’s between a… Which way do I do it? This is our left-wing candidate. This is our moderate left-wing candidate, whose name is… Alex. Alex. And our kind of centrist candidate, maybe even the centrist candidate, whose name is… Beatrice. Beatrice. All right, all right. So this is the election. Who’s going to win this election? Beatrice is going to win this election, right? Let’s just check that. Beatrice is going to get, for me too, Beatrice is going to get all of these right-wing votes, right? Beatrice is also going to get one and a half of the votes in the middle, right? So this vote and half the Pearson guys vote right And I forgotten your name already again And Alex is going to get all the left votes but there aren enough left votes here right So Beatrice is going to win So Beatrice won this election Is this array a Nash equilibrium Is this an equilibrium? How do we know it’s not an equilibrium? Katie, how do we know this is not an equilibrium? Wait for the mic, with the mic.

[译文 37]

好，如果你站着，就站着。好，如果你站着，就站着。好，我们再来一次。一、二、三。快点。好，保持站着，保持站着，保持站着，保持站着。我会再绕一圈。好，这里是我们的选举。这里是我们的选举。它在… 我该往哪边进行？这是我们的左翼候选人。这是我们温和左翼候选人，名字叫……Alex。Alex。还有我们那种中间派候选人，甚至可能是中间派候选人，名字叫……Beatrice。Beatrice。好，好。这是选举。谁会赢得这场选举？Beatrice会赢，对吧？让我们检查一下。Beatrice也会得到我这边所有的右翼选票，对吧？Beatrice还会得到中间票中的一票半，对吧？这一票和Pearson那一半的票… 我又忘了你的名字。Alex会得到所有的左翼选票，但这里左翼票不够。所以Beatrice会赢。所以Beatrice赢了这场比赛。这个阵型是纳什均衡吗？这是均衡吗？我们怎么知道这不是均衡？Katie，我们怎么知道这不是均衡？等一下麦克风，拿着麦克风。

[段 38]

How do we know this is not an equilibrium? Because Alex’s best response to her running is to not run. Absolutely. So Alex here, Alex ends up with a payoff of what? Alex ended up with a payoff of minus C, actually more than that, minus C minus the distance between him and Beatrice, so that’s another $1.05, $0.10, $0.20, whatever it is, right? And he would have done better not to run. If he hadn’t run, Beatrice would still have won. It would have still cost him the $0.20 of unhappiness at Beatrice winning, but at least he would have saved himself the C of running. Everyone see that? Let’s try it again. Let’s try it again, okay? Okay, same row, same row, but don’t hesitate so much this time, right? Count of three, you’ve got to just go or not. everybody in this row close their eyes you can’t look around you, close your eyes close your eyes, we can do all sorts of nasty things close your eyes, alright mostly you’re still in this row ok alright, alright, so ready, alright so, one two, three three All right, all right, so ready, all right, so one, two, three. All right, you can open your eyes again. All right, so now, all right, so now what happened? We end up with two centrist candidates. Let’s see what happened.

[译文 38]

我们怎么知道这不是均衡？因为Alex对她的参选的最佳反应是不参选。完全正确。那么Alex这里，Alex的收益是多少？Alex的收益是负C，实际上比这更多，负C再减去他与Beatrice之间的距离，也就是另外的$1.05、$0.10、$0.20，不管是多少，对吧？他不参选会表现更好。如果他没有参选，Beatrice仍然会赢。这仍然会让他损失$0.20因Beatrice赢而带来的不快，但至少他省下了参选的C费用。大家明白了吗？我们再来一次。好，再来一次，对吧？同一排，同一排，但这次不要犹豫太多，对吧？数到三，你要么去要么不去。这一排的每个人都闭上眼睛，你们不能四处张望，闭上眼睛，闭上眼睛，我们可以做各种讨厌的事，闭上眼睛，好，基本上你们仍然在这排里，好，好，准备好了，好，一、二、三、三。好，好，准备好了，好，一、二、三。好，你们可以睁开眼睛了。好，现在，好，现在发生了什么？我们最终得到两位中间派候选人。让我们看看发生了什么。

[段 39]

So we have Beatrice and your name is? Clara Lease. Clara Lease, all right, and Clara Lease is going to get her own vote plus one, well, one, two, three, four, five, six, seven, eight, eight votes. and Beatrice is getting 1, 2, 3, 4, 5, 6, 7 how do they end up with 8? I’ve not done an odd number 1, 2, 3, 4, 5, 6, 7 it must have been an odd number 4 so Clarice is going to win this election alright, alright because you’ve got one more vote now here we end up with two centrist candidates which is a result pretty close to what we saw in the hotelling model is that right? is this an equilibrium? let’s check out a counted right 1, 2, 3, 4, 5, 6, 7, 8 And so you, one, two, three, four, five, six, seven. I guess they do count right. So Claralee’s turned out to win this. I counted that correct. There’s another vote hidden behind that. I didn’t see you. I didn’t. You add the merger but you subtract the Yeah OK So let just try again So one two three four 5 6 7 8 I apologize Beatrice won this election I apologize Beatrice won this election All right. Is this an equilibrium? Is this an equilibrium? How do we know this isn’t an equilibrium? Because pluralists would rather not have run.

[译文 39]

于是我们有Beatrice和你的名字是？Clara Lease。Clara Lease，好，Clara Lease将得到她自己的票再加一票，嗯，一、二、三、四、五、六、七、八，八票。而Beatrice得到1、2、3、4、5、6、7，怎么会得到8？我没有用奇数，1、2、3、4、5、6、7，应该是奇数4，所以Clara会赢得这场选举，好，好，因为你现在多了一票。我们这里出现了两位中间派候选人，这个结果与我们在Hotelling模型中看到的相当接近，对吧？这是均衡吗？让我们检查一下计数，1、2、3、4、5、6、7、8。于是你，一、二、三、四、五、六、七。我觉得他们数得对。于是Claralee’s的票最终获胜。我数得正确。还有另一票藏在后面，我没看到你。我没看到。你把合并的票加进去但又把… 对，好，那我们再来一次，一、二、三、四、五、六、七、八。抱歉，Beatrice赢了这场比赛。抱歉，Beatrice赢了这场比赛。好。这是均衡吗？这是均衡吗？我们怎么知道这不是均衡？因为多元派宁愿不参选。

[段 40]

Is that right? All right. Let’s try once more. All right. Down again. close your eyes again let’s let’s just just to keep life interesting let’s switch to this row can we switch to this row okay alright so this is our row now everyone understand this close your computer so you can get up without destroying thousands of dollars of technology alright this is our row and at the count of three we’re going to see who’s going to run okay everyone understand the game alright close your eyes close your eyes madam one One, two, three. All right. Yeah, yeah. All right. All right. So it’s obvious he’s going to win this election. Is that right? All right. And let just ask the question is this an equilibrium Is this an equilibrium Let count One 2 3 4 5 6 7 8 9 He’s going to get 10 votes. Including himself, there are 10 votes here. I can’t count correctly. 1, 2, 3, 4, 5, 6. Is he actually the center guy here? I don’t think he is the center guy here. So I claim, if I’ve counted correctly, that this actually isn’t an equilibrium. Who can deviate and do better here? Yeah, this guy. Stand up a second. Whose name is… Had John entered, if I counted correctly, let’s pretend I have, John would actually have won this election because John is actually further to the center.

[译文 40]

对吧？好。再试一次。好。再来一次。再次闭上眼睛，让我们保持生活有趣，让我们换到这一排，可以吗？好，这就是我们的排了，大家都明白了吗？关闭电脑，这样你起身时不会毁掉价值数千美元的科技产品。好，这是我们的排，到了数到三时我们来看看谁会参选，好，大家都明白这个游戏了吗？好，闭上眼睛，闭上眼睛，女士。一、二、三。好。是的，是的。好。好。很明显他会赢得这场选举。对吧？好。我们来问这个问题，这是均衡吗？这是均衡吗？让我们数数，一、二、三、四、五、六、七、八、九。他会得到10票。包括他自己，这里有10票。我数得不对。1、2、3、4、5、6。他真的是这里的中心人物吗？我不认为他是中心人物。于是我主张，如果我数得正确的话，这实际上不是一个均衡。谁能偏离并在这里做得更好？是这个人。站起来一下。他叫什么名字？如果John加入的话，如果我数得正确，假设我数对了，John实际上会赢得这场选举，因为John实际上更靠近中心。

[段 41]

All right? Okay. Thanks for a second, guys. I’ll come back to you in a second. All right? Let’s try and figure out how to analyze this game. We’ve played it three times. Let’s try and figure out what the equilibria look like in this game. This game is going on more or less right now in the primaries. Well, I guess the entry stage of it is gone now. So first of all, is there a Nash equilibrium in which no candidates stand? Is there a Nash equilibrium in which nobody stands No How do we know that not a Nash equilibrium Somebody Somebody Yeah How do we know that not a Nash equilibrium Shout it out. Because then somebody’s definitely better off by standing up. Right. If nobody stands, each and every possible candidate would do better individually. So any particular voter would do better standing. They would win the election. Is that right? All right. so clearly it’s not an equilibrium for nobody to stand. That’s good because we don’t see that very often. Is there a Nash equilibrium in which one and only one candidate stands? Yeah, so… If they’re directly in the center. All right, so given that we chose an odd number of people in the row, if exactly one candidate stands and that candidate is the center candidate, then that’s an equilibrium. How do we know that’s an equilibrium?

[译文 41]

好？好。谢谢大家，稍等一下。好？我们来试着分析这个游戏。我们已经玩了三次。我们来试着找出这个游戏的均衡是什么样的。这个游戏现在或多或少正在初选阶段进行。嗯，我想它的进入阶段现在已经结束。那么，首先，有没有没有候选人站出来的纳什均衡？有没有没有人站出来的纳什均衡？没有人站出来。没有。我们怎么知道这不是纳什均衡？有人，有人喊出来。因为如果没有人站出来，那么每个可能的候选人单独站出来都会更好。于是任何特定的选民站出来都会更好。他们会赢得选举。对吧？好，所以显然没有人站出来的状态不是均衡。这很好，因为我们不常看到这种情况。那么，有没有只有一个候选人站出来的纳什均衡？有……如果他们正好在中心。好，既然我们在这一排选择了奇数个人，如果恰好只有一个候选人站出来，而该候选人是中心候选人，那么这就是均衡。我们怎么知道这是一个均衡？

[段 42]

Somebody give me the argument why that’s an equilibrium. It’s a good, in fact, a correct guess, but how do we know that’s an equilibrium? That’s the guess part. How do we check that’s an equilibrium? Yeah, Clarice. Shout out. If anybody on either side of them chose guest part, how do we check that’s an equilibrium? Yeah, there it is. Shout out. If anybody on either side of them chose to run, they would lose? They would lose, exactly. So if anybody on either side of them runs, right, that person would simply lose. It wouldn’t make any difference to the outcome, right? If a person on the right stood, let’s just try it again. So we figured out that, if Beatrice stands up a second, right, so here’s our potential equilibrium with just Beatrice standing, right? And if somebody to the, which way are we doing it? If somebody to the right of Beatrice was to stand, so Madeline, if you stand a second, and your name is? Stacey. So if Stacey stands, it doesn’t make any difference. She just loses. Beatrice wins anyway. It just costs Beatrice some money, right? And conversely, if Beatrice doesn’t stand, and, say it again a second, and the woman who’s running from the side there, sit down a second, Stacey, and if this person stands, and your name is? Sarah. If Sarah stands, it doesn’t help Sarah at all.

[译文 42]

有人给我解释一下为什么这是一个均衡。这是个好的，实际上是正确的猜测，但我们怎么知道这是一个均衡？这就是猜测的部分。我们怎么检验这是一个均衡？是的，Clarice。喊出来。如果他们任何一边的人选择参选，我们会怎么检验这是均衡？是的，就是这样。喊出来。如果他们任何一边的人选择参选，他们会输？他们会输，正是如此。于是如果任何一边的人参选，那个人只会输。对结果没有任何影响，对吧？如果右边的人站出来，我们再试一次。我们已经算出，如果Beatrice站出来，好，这就是我们唯一的潜在均衡，只有Beatrice站着，对吧？如果有人站在Beatrice的右边，Madeline，如果你站出来，你叫什么名字？Stacey。如果Stacey站出来，这没有任何区别。她只会输。Beatrice仍然会赢。这只会让Beatrice花一些钱，对吧？反过来，如果Beatrice不站出来，再说一次，那边的女候选人，Stacey，坐下，如果这个人站出来，你叫什么名字？Sarah。如果Sarah站出来，这对Sarah一点帮助都没有。

[段 43]

She just loses. Is that right? So there is an equilibrium with exactly one candidate, the center candidate. Now that’s looking a little bit like the median voter model. It says there is an equilibrium with only one candidate standing and that candidate would be the median candidate Just like it was in the model we saw on your homework assignment and also in class However, we’re not done yet. There could be other equilibria. There could be other equilibria. So let’s just try and think about whether there are other equilibria. So for example, suppose now, just to make life a little bit more interesting, suppose that the voters were actually two rows and just allow me the suspension of disbelief that these rows have the same number of people in them. I know they don’t really. So now what I want you to think of is that at every political position there are two voters and hence two possible candidates. There’s two people at every position. These two people are at this position. These two people are at this position. Everyone understand that? Let’s assume that the rows are the same even though they’re not. And let’s examine the following thing. Suppose Beatrice stands again, sorry Beatrice, and the gentleman in front of him, in front of her, whose name is Robert stands as well. So stand a second.

[译文 43]

她就是输。是这样吧？所以存在一个恰好只有一个候选人的均衡，就是中间位置的候选人。现在这看起来有点像中位选民模型。它说的是存在一个均衡，只有一个候选人站在那儿，而那个候选人就是中位选民。就像我们在作业和课堂上看到的模型那样。然而，我们还没有结束。可能还有其他均衡。可能还有其他均衡。那么让我们试着想想是否存在其他均衡。比如说，假设现在为了让情况更有趣一点，假设选民实际上是两排，请允许我先搁置一下怀疑，假设这两排的人数相同。我知道它们实际上并不相同。所以现在我想让你们思考的是，在每一个政治立场上都有两个选民，因此也就有两个可能的候选人。每个立场有两个人。这两个人在这个立场上。这两个人在这个立场上。大家都明白吗？假设这两排是一样的，即使它们实际上不是。让我们来检验以下情况。假设比阿特丽斯再次参选，对不起比阿特丽斯，还有她面前的那位先生，他叫罗伯特，也参选。所以再站出来一下。

[段 44]

All right, so now we have two candidates standing who are identical Politically they identical Right they right on top of each other Now is that an equilibrium Is that an equilibrium? After all, that looks a lot like the down to telling model, right? We’ve got two candidates exactly at the middle. Is that an equilibrium? Let me try the guy up here. No, because if someone on either side of them were to stand up, that person on the side would win. Good, good. This can’t be an equilibrium because look what happens if Clara Lees stands a second. Alright? Clara Lees is going to win all of the right-wing votes, right, ranging from crazy to moderate, alright, and these guys are going to split the left-wing votes, right? So the total sum of right-wing votes is going to beat out half of the left-wing votes. So it can’t be an equilibrium. The exact prediction of the Downs model, two guys right on top of each other, is not an equilibrium. Sit down again, guys. All right. Let’s go back to one row again, because it’s easy to work with. Let’s try a slightly different pattern. So suppose assuming I counted right suppose that the following candidates enter So Clara Lees enters and I sorry I don know your name Jean Jean enters So let’s have a look at this array.

[译文 44]

好的，现在我们有两位候选人，他们政治立场相同，完全重合。这是一个均衡吗？这是个均衡吗？毕竟，这看起来很像唐斯模型，对吧？我们有两位候选人恰好站在中间。这是一个均衡吗？让我试试上面这个人。不行，因为如果他们两边任何一边有人站出来，那边的人就会赢。好的。好的。这不可能是个均衡，因为看看如果克拉拉·利斯站出来会发生什么。好吧？克拉拉·利斯会赢得所有右翼选票，从激进到温和，好吧，而这些人会分走左翼选票，对吧？所以右翼选票的总数会超过左翼选票的一半。所以这不可能是个均衡。唐斯模型的确切预测——两个候选人完全重合——不是一个均衡。坐下吧，各位。好吧。让我们回到一排的情况，因为这样更容易处理。让我们试试一个稍有不同的情况。假设我数对了，假设以下候选人参选。克拉拉·利斯参选，还有——对不起我不知道你叫什么名字——让，让参选。让我们看看这个排列。

[段 45]

All right? And assume I got it right and Richard is actually the center. All right? Now, is that an equilibrium? Is that an equilibrium? Who thinks that is an equilibrium? Who thinks it’s not an equilibrium? room? Who’s waiting to see how other people vote? Well, let’s check. Let’s check. All right. So there’s three possible types of deviation here that we need to check. We need to check another entrant from the outside, left wing or right wing. We need to check another entrant in the middle. There is only one possible one there. And there’s a third kind of deviation we should check. What’s What’s the third type of deviation? One of these might choose not to run. One of you might choose not to run. Let’s do them in turn. So suppose that we consider a deviation in which, again I’ve forgotten your name, in which Stacy stands as well. Stacy doesn’t stand a second. So is this a profitable deviation for Stacy? No, it’s not, right? It’s not. Why isn’t it a profitable deviation for Stacy? So two reasons. So is this a profitable deviation for Stacey? No, it’s not, right? It’s not. Why isn’t it a profitable deviation for Stacey? So two reasons. One is that she loses. She’s not going to win this election by standing. But there’s a second reason why this is a really bad deviation.

[译文 45]

好吗？假设我数对了，理查德实际上是中间的那个。好吧？现在，这是一个均衡吗？这是一个均衡吗？谁认为这是一个均衡？谁认为这不是一个均衡？房间里谁在等着看别人怎么投票？好吧，让我们检查一下。让我们检查一下。好吧，所以这里有三种可能的偏离类型我们需要检查。我们需要检查来自外部的另一个参选者，左翼或右翼。我们需要检查中间的另一参选者。那里只有一个可能的。我们还应该检查第三种偏离类型。什么是第三种偏离类型？这些人中的一个可能选择不参选。你们中一个可能选择不参选。让我们依次检查。假设我们考虑这样一种偏离，我又忘了你名字了，斯泰西也参选。斯泰西不再参选第二次。这对斯泰西来说是一个有利可图的偏离吗？不，不是的，对吧？不是。为什么这不是一个有利可图的偏离？有两个原因。一个是她会输。她不会通过参选赢得这次选举。但还有第二个原因说明这是一个很糟糕的偏离。

[段 46]

What’s the second reason? Yes, sir? The winner is going to be farther away from her. Good. Good. So by standing, not only does she not win the election, but she actually causes the election to be won for sure by Jean who’s further away from her so Stacey’s going to pick up the right wing votes she’s going to split the crazy right wing votes she’s going to pick up the moderate right wing votes she’s going to split with Claralise and Jean’s going to have all the left wing votes extreme and moderate and so from Stacey’s point of view this is a double bad she doesn’t get to be president and you end up with a left-wing president which you didn’t like. Everyone see that? So clearly that’s not a profitable deviation. Hang on a minute, you guys. And it’s also not a profitable deviation for people on the left. Hang on a second. So I think it’s probably obvious why it’s not a good strategy for Beatrice to enter here. If Beatrice enters, she loses. Everyone can see that? The person in the middle, if they enter here, loses, so they’re not going to want to enter. So the other possible deviation we have to consider is a deviation of the form of one of these guys dropping out. All right? So let’s consider it. So come back to…

[译文 46]

第二个原因是什么？是的，先生？获胜者会离她更远。好的。好的。所以通过参选，她不仅没有赢得选举，而且实际上导致选举必定被让——那个离她更远的人——赢得，所以斯泰西会获得右翼选票她会分走激进的右翼选票她会获得温和的右翼选票她会与克拉丽斯分票，而让会获得所有的左翼选票，激进和温和的，所以从斯泰西的角度来看，这是双重的糟糕她没能成为总统，而你最终得到的是一个左翼总统，这是你不喜欢的。大家都明白吗？显然这不是一个有利可图的偏离。等一下，各位。这对左翼的人来说也不是一个有利可图的偏离。等一下。所以我认为为什么比阿特丽斯在这里参选不是一个好策略可能很明显。如果比阿特丽斯参选，她会输。大家都看到了吧？中间的那个人，如果他们在这里参选，会输，所以他们不会想参选。所以我们必须考虑的另一种可能的偏离是这些人中有一个退出。好吧？让我们考虑一下。让我们回到…

[段 47]

Claire Elise. Is Claire Elise doing better in or out? Is Claire Elise doing better in this election or out? All right, let’s think it through a second. If she stays in, what’s her payoff? She wins B with probability of half actually cost C for sure That a wash right Because B over the way we worked things out B was C was so B over 2 minus C is a wash All right? In terms of the benefits of the spoils of government versus the cost of running, it’s a wash for Clarice whether she enters or not. Right? But by entering, she has a half chance of winning the election, alright, so one way to think about it is the expected distance from her of the winning candidates is, with probability of half, it’s herself, that’s nothing, no distance, and with probability of half, it’s two places away. So on expectation, it’s one place away. If she drops out, Jean wins for sure. So the expected distance away from the winning candidate is two away. Alright, just to say it again, so the cost and direct benefits of government to rewash for her. But by dropping out, she ensures that somebody further away from her wins for sure. That’s bad, so she’s going to stay in. And the same argument goes the other way. So this is an equilibrium.

[译文 47]

克莱尔·伊莉丝。克莱尔·伊莉丝在里面还是外面表现更好？克莱尔·伊莉丝在这个选举中是参选还是退出更好？好吧，让我们仔细想一想。如果她留在里面，她的收益是什么？她有50%的概率赢得B，实际成本是C确定的。这是一个平手，对吧？因为根据我们计算的方式，B是C，所以B除以2减去C是平手。好吧？就政府好处的收益与参选的成本而言，对克莱里斯来说参选与否是一样的。对吧？但通过参选，她有50%的机会赢得选举，好吧，所以一种思考方式是，从她角度看获胜候选人的预期距离是，50%的概率是她自己，那是零，没有距离，50%的概率是两格之远。所以预期上是一格之远。如果她退出，让必定获胜。所以与获胜候选人的预期距离是两格之远。好吧，再说一遍，所以成本和政府的直接收益对她来说是平手。但如果退出，她确保了某个离她更远的人必定获胜。这很糟糕，所以她要留下。而同样的论证反过来也成立。所以这是一个均衡。

[段 48]

Now we’re not done, and of course we’re out of time. So what are we going to do when we come back next week? Don’t pick up yet. What are we going to do? We’re going to come back to this row, and we’re going to figure out not just this equilibrium, but all equilibrium. Before you go, though, let’s just think about a second, hang on, so you can think about it and talk about it at home. So what we just saw was an election in which we saw two candidates, Jean and Clarice, who both stood very close to each other. But a different question is, hang on before you go, before you go, stand up my very left-wing guy. Yeah, way at the end, way there. Yep, up, up. Yep, and stand up my very right-wing guy. here’s a candidate, here’s an outcome with two entrants in it an extreme right wing guy and an extreme left wing guy is that an equilibrium? think about it until we come back on Monday

[译文 48]

现在我们还没结束，当然时间到了。所以下周我们回来时我们要做什么？先别收拾。我们要做什么？我们会回到这一排，我们要弄清楚不仅仅是这一个均衡，而是所有的均衡。但在你们走之前，让我们再想一下，等一下，所以你们可以在家思考和讨论。我们刚才看到的是一个选举，我们看到两位候选人，让和克莱里斯，他们站得非常近。但一个不同的问题是，等一下，在你们走之前，在你们走之前，站起来我的极左翼的家伙。是的，在最边上，就那儿。是的，起来，起来。是的，还有站起来我的极右翼的家伙。这里有一个候选人，这里有一个结果，有两个参选者，一个极右翼的家伙和一个极左翼的家伙，这是一个均衡吗？思考一下直到我们周一回来。