Game Theory Lecture 7 | Game Theory
Chapter 1.
第一章。
Bertrand Duopoly: Standard Model [00:00:00] Professor Ben Polak: All right, so last time we started to study imperfect competition.
Bertrand 双寡头：标准模型 [00:00:00] Ben Polak 教授：好的，上次我们开始研究不完全竞争。
We looked at the Cournot model, but 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.
我们研究了古诺模型，但更大的主题是研究企业如何在 115 中研究的那种简单案例之外进行竞争：因此，在垄断的例子之外，只有一家企业，因此没有太多竞争；在有如此多企业的完全竞争情况之外，就好像每个企业都是价格接受者。
We're going to continue with that today, and in fact you're going to continue with it on into the homework.
今天我们将继续这个话题，事实上，你将在家庭作业中继续这个话题。
So last time, we looked at Cournot.
上次我们研究了古诺。
It was the model we started with.
这是我们开始时使用的模型。
We looked at two firms competing in quantities.
我们研究了两家公司在数量上的竞争。
And just to review a little bit what we did last time, the kind of exercise there--so I think, is this right, I think probably went a little fast last time.
只是回顾一下我们上次所做的事情，那里的练习 - 所以我认为，这是对的，我认为上次可能进展得有点快。
I was kicking myself afterwards.
后来我就在踢自己。
People can nod or shake their heads.
人们可以点头或摇头。
I think I went a bit fast, so I'm sorry for that.
我想我走得太快了，所以我对此感到抱歉。
If there was a time to go too fast it was probably last time, because it turns out this is something that's covered well in the textbook, so it's covered in chapter six of the textbook.
如果有一次走得太快，那可能是最后一次，因为事实证明这是教科书上很好地介绍的内容，所以它在教科书的第六章中介绍了。
But the kind of exercise we did last time is a useful exercise for those of you who are Econ majors.
但我们上次做的练习对于经济学专业的学生来说是一个有用的练习。
The way we solved that Cournot model was we did three kinds of things.
我们解决古诺模型的方法是做了三件事。
We did something very nerdy, namely, we just played with calculus and algebra.
我们做了一些非常书呆子的事情，即我们只是玩微积分和代数。
That was sort of mathy.
这有点数学。
We did something a little bit less nerdy, that is, we drew some pictures that represented what we'd found.
我们做了一些不那么书呆子气的事情，那就是，我们画了一些代表我们发现的东西的图片。
And we did a third thing, which we tried to match this up to the economic intuition about monopoly, and perfect competition, and demand curves, and so on.
我们做了第三件事，我们试图将其与有关垄断、完全竞争和需求曲线等的经济直觉相匹配。
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, just to be able to translate easily between those.
这种能够在这三种不同模式下工作的练习，经济直觉、图表和一种书呆子高中数学是我们希望让经济专业的你习惯的，只是为了能够在这些模式之间轻松转换。
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.
再次，我为速度太快而道歉，但我认为这仍然是一个有用的练习，所以请在书中看看。
Now back to the lessons and away from the nerdiness a second, what did we learn?
现在回到课程上来，暂时抛开书呆子，我们学到了什么？
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.
最终我们了解到，在古诺均衡中，事情正如我们可能预料到的那样——事情自然地处于极端情况之间。
So the amount of output produced by the industry was somewhere between the case that would be under monopoly and under perfect competition.
因此，该行业的产出量介于垄断和完全竞争的情况之间。
It was more than under monopoly, less than perfect competition.
这不仅仅是垄断，也不是完全竞争。
Prices in the industry, if we'd gone back and checked, would sit between.
如果我们回去检查一下，该行业的价格将介于两者之间。
They would be lower than the monopoly prices, but higher than the perfect competitive prices.
它们将低于垄断价格，但高于完全竞争价格。
Industry profits would be in between.
行业利润介于两者之间。
Industry profits would be less than monopoly.
行业利润将低于垄断企业。
They must be less than monopoly, since that's the highest they could ever be.
它们必须小于垄断，因为这是它们所能达到的最高水平。
And they're higher than perfect competition, which of course, has zero profits in this case.
而且它们比完全竞争要高，当然，在这种情况下，完全竞争的利润为零。
Consumer surplus, the benefits flowing to consumers lie in between.
消费者剩余，即流向消费者的利益介于两者之间。
So in fact the model, in addition to the sort of nerdiness of the model, it ended up with a result we kind of believe in.
所以事实上这个模型，除了模型的那种书呆子气之外，它最终得到了我们有点相信的结果。
If you have imperfect competition, it's somewhere between perfect competition and no competition.
如果存在不完全竞争，那么它就介于完全竞争和无竞争之间。
But, what's the "but"?
但是，什么是“但是”呢？
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.
“但是”是我们还有其他方法可以模拟不完全竞争，而且正如我们今天将看到的，它们会产生不同的答案。
So the first thing we're going to do today is look at a different form of competition which is called Bertrand competition.
因此，我们今天要做的第一件事就是研究一种不同形式的竞争，称为伯特兰竞争。
The Cournot was competing in quantities and Bertrand is competing in prices.
古诺在数量上竞争，伯特兰在价格上竞争。
Just a quick check and this is not for the camera, how many of you have seen Bertrand competition before?
快速检查一下，这不是针对相机的，你们中有多少人以前看过 Bertrand 比赛？
So, a good many of you.
所以，你们很多人。
So this is going to be review still again, and for those of you for whom it isn't review don't worry.
所以这将再次被审查，对于那些没有审查的人不要担心。
We'll do this relatively quickly but I hope I won't rush it like I did last time.
我们会相对较快地完成这件事，但我希望我不会像上次那样匆忙。
If it's any consolation, when I feel I've rushed a lecture, I have a sleepless night.
如果有什么安慰的话，当我觉得我的讲座很匆忙时，我会度过一个不眠之夜。
So if you had a sleepless night because you're worried about it, I'm having a sleepless night too about that.
所以，如果你因为担心而度过了一个不眠之夜，那么我也会为此而度过一个不眠之夜。
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.
所以有两家公司，就像以前一样，就像以前一样——所以参与者就是那两家公司，就像上次一样，他们生产相同的产品。
So the products we mentioned last time were Coke and Pepsi, but you could think of other products that are pretty much identical.
所以我们上次提到的产品是可口可乐和百事可乐，但你可以想到其他几乎相同的产品。
Just as last time, we're going to assume that costs are constant marginal costs.
就像上次一样，我们假设成本是恒定的边际成本。
We'll assume that constant marginal cost just as it was last time is equal to C.
我们假设恒定边际成本与上次一样等于 C。
Just to remind those people who are not Economics Majors what that means, it means if I produce 1 unit then it costs me C to produce, if I produce two units it costs me 2C, 100 units a 100C and so on.
只是为了提醒那些不是经济学专业的人这意味着什么，这意味着如果我生产 1 个单位，那么生产成本为 C，如果我生产两个单位，则生产成本为 2C，100 个单位则为 100C，依此类推。
Now what's different here from last time?
现在和上次有什么不同？
I've already told you but I'll repeat it.
我已经告诉过你了，但我会重复一遍。
This time, however, instead of setting quantities, instead of just deciding how much Coke and Pepsi to produce and spewing it out in 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.
然而，这一次，公司不再设定数量，也不再只是决定生产多少可口可乐和百事可乐，然后将其投放到市场上，让价格自行决定，而是公司将设定价格，让数量自行决定。
So the strategy sets this time are prices.
所以这次制定的策略是价格。
Again, so as we don't get confused, normally we use S for strategy, but let's use P since they're prices.
再次强调，为了避免混淆，通常我们使用 S 来表示策略，但我们使用 P，因为它们是价格。
So they're going to P1, will the price of Firm 1, and P2 will be the price of Firm 2.
因此，他们将选择 P1，即公司 1 的价格，P2 将是公司 2 的价格。
This strategy set--formally, let's just simplify it here--let's assume that for each Firm i they can set their price anything bigger than 0 and anything less than 1, just to keep life simple.
这个策略集——正式地，我们在这里简化一下——假设对于每个公司 i，他们可以将其价格设置为大于 0 和小于 1 的任何值，只是为了保持简单。
So this is a little bit more realistic perhaps.
所以这也许更现实一些。
Perhaps more realistic to think of firms actually thinking about what prices to set and then adjusting the quantities they actually ship based on demand.
也许更现实的想法是，公司实际上考虑设定价格，然后根据需求调整实际发货的数量。
Depends on--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.
取决于——我不想对此过于严格，因为你可以想象相反的方式，但如果你与经理交谈，这听起来更现实，至少对于可口可乐和百事可乐这样的零售商品来说。
So we know how prices are set, they're set by the firms, so the next question is where do the quantities come from?
所以我们知道价格是如何设定的，它们是由公司设定的，所以下一个问题是数量从哪里来？
So how do--where do the quantities from?
那么这些数量是如何——从哪里来的呢？
And they're going to come from demand.
它们将来自需求。
And let me use a big Q(P) to be the total quantity produced--sorry--the total quantity demanded in the market.
让我用一个大的 Q(P) 来表示生产总量——抱歉——市场需求总量。
So this will be total quantity of goods produced by Firm 1 and goods produced by Firm 2 that are consumed.
因此，这将是企业 1 生产的商品和企业 2 生产的商品的消耗总量。
So this is the total quantity of Coke plus Pepsi.
这是可口可乐加百事可乐的总量。
Now we're going to--notice there's no subscript on this P and you'll see why in a second.
现在我们要注意这个 P 上没有下标，您很快就会明白为什么。
So we'll assume just to make life simple, that the total quantity produced is 1 –P.
因此，为了简单起见，我们假设总产量为 1 –P。
And I'll say this and then write it more carefully, where P is the lower of the two prices.
我会这么说，然后写得更仔细，其中 P 是两个价格中较低的一个。
So total quantity demand is 1 -P where P is the lower of the two prices.
因此总需求量为 1 -P，其中 P 是两个价格中较低的一个。
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.
现在，为了使两个价格中较低者的评论更加严格，让我们弄清楚这些公司的实际需求是什么。
So what's going on here?
那么这是怎么回事呢？
So let's look at the demand for Firm 1, which is going to end up being the quantity that they sell.
因此，让我们看看对公司 1 的需求，这最终将成为他们销售的数量。
So that's going to be Q1 and what's that going to be?
那么这将是第一季度，那会是什么？
Well it's going to 1 -P1 if they're the low price firm.
好吧，如果他们是低价公司，那就是 1 -P1。
So if P1 is less than P2--so if they're the low price firm--if Coke is the low price firm, only Coke sells in the market.
因此，如果 P1 小于 P2——如果它们是低价公司——如果可口可乐是低价公司，则只有可口可乐在市场上销售。
There's no Pepsi in the market and 1 -P1 quantity of Coke is sold.
市场上没有百事可乐，出售 1 -P1 数量的可乐。
It's going to be 0 if P1 is bigger than P2.
如果 P1 大于 P2，则为 0。
So if Pepsi is the low price drink in the market then no Coke is sold at all.
因此，如果百事可乐是市场上的低价饮料，那么根本就没有可口可乐出售。
It's going to be (1 -P1)/2 in the case where Coke and Pepsi cost exactly the same.
如果可口可乐和百事可乐的成本完全相同，则为 (1 -P1)/2。
These should really be--realize that this is not exactly realistic.
这些确实应该是——意识到这并不完全现实。
We're making a lot of very strong assumptions here, but nevertheless, it's going to be instructive to look at this model.
我们在这里做出了很多非常有力的假设，但尽管如此，研究这个模型还是有启发性的。
So we'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, and I still need to tell you what payoffs are.
所以我们有公司，他们是参与者，我知道他们的策略是什么，我对市场结构了解一点，而且我仍然需要告诉你回报是什么。
And just as last time the firms are going to try and maximize profits.
就像上次一样，公司将努力实现利润最大化。
So what's the profit for Firm 1 going to be?
那么公司 1 的利润是多少？
I won't bother to write it separately for Firm 2, 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.
我不会费心为公司 2 单独写它，它将是它销售的数量乘以该数量的价格，减去它销售的数量乘以生产该数量所产生的成本。
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 his cost.
同样，对于公司 1 来说，它是它的销售数量乘以价格，这是它的收入，减去它销售的数量乘以 C，成本，这是他的成本。
I can rewrite that a little bit more simply as Q1[P1 -C].
我可以将其更简单地重写为 Q1[P1 -C]。
Where Q1, let's keep it in square brackets, this Q1 is this object here.
其中Q1，我们把它放在方括号里，这个Q1就是这里的这个对象。
Does everyone understand what's going on?
大家都明白这是怎么回事吗？
I mean the algebra doesn't really help the intuition here.
我的意思是代数在这里并不能真正帮助直觉。
The intuition is pretty straightforward.
直觉非常简单。
Coke and Pepsi are producing this stuff.
可口可乐和百事可乐正在生产这种东西。
They're setting prices.
他们正在定价。
If Pepsi is the low price then Coke sells nothing.
如果百事可乐是低价，那么可口可乐就什么也卖不了。
If Coke is the low price it sells this amount and its profits are basically given by this equation.
如果可口可乐是低价的话，它就会卖出这个数量，它的利润基本上是由这个方程给出的。
It's basically just revenues minus costs.
基本上就是收入减去成本。
So what we want to do is we want to figure out what the Nash Equilibrium looks like in this market.
所以我们想做的是弄清楚这个市场的纳什均衡是什么样的。
Just before we go any further, notice that really this is the same basic market we looked at before.
在我们进一步讨论之前，请注意，这实际上与我们之前看到的基本市场相同。
Both times there's a demand curve out there.
两次都存在需求曲线。
In one model we thought of the firms setting quantities and the market determining prices, and here we have the firms setting prices and the market determining quantities, but the basic underlying economic structure of this is very, very similar.
在一个模型中，我们认为企业设定数量，市场决定价格，这里我们有企业设定价格，市场决定数量，但其基本经济结构非常非常相似。
So to find the Nash Equilibrium we're going to have to find best responses.
因此，为了找到纳什均衡，我们必须找到最佳响应。
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.
因此，我们要做的就是弄清楚这些最佳反应是什么样的，然后让我们尝试这样做。
So I want to figure out the best response of Firm 1 as a function of the price chosen by Firm 2.
因此，我想根据公司 2 选择的价格计算出公司 1 的最佳响应。
Best price for Firm 1 to choose given that Firm 2 is choosing P2.
鉴于公司 2 选择 P2，公司 1 选择的最佳价格。
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.
我想小心翼翼地、慢慢地做这件事，在开始之前，让我顺便指出，微积分在这里对我们没有帮助。
So for those of you who are nerdy enough to understand the next sentence let me just say it.
因此，对于那些足够书呆子理解下一句话的人，让我说一下。
This is a discontinuous payoff function, so differentiating isn't going to get you very far.
这是一个不连续的收益函数，因此微分不会让你走得太远。
For people whom that didn't mean anything, it doesn't matter.
对于那些没有任何意义的人来说，这并不重要。
We're not going to calculus today.
我们今天不学习微积分。
Okay, so there were different cases to think about.
好的，所以有不同的情况需要考虑。
So one case to think about is what if the other firm, you're Coke, what if Pepsi is pricing below cost?
因此，需要考虑的一个情况是，如果另一家公司，你是可口可乐，百事可乐的定价低于成本怎么办？
You're Coke, what should you price as if you're trying to maximize your profits?
如果您是可口可乐，如果您想实现利润最大化，您应该定价多少？
Somebody try and wave their hands in the air if they know the answer.
如果有人知道答案，他们会尝试在空中挥动双手。
Pepsi's pricing below cost, what's your best response?
百事可乐的定价低于成本，您的最佳回应是什么？
The guy in there, do you want to shout out?
里面的那个人，你想大声喊出来吗？
Student: Just get out of the market because it's not worth it for you to sell anything.
学生：退出市场吧，因为你卖掉任何东西都不值得。
Professor Ben Polak: Right, I've forgotten your name, I should know it by now.
Ben Polak 教授：对了，我忘记了你的名字，我现在应该知道了。
Your name is?
你的名字是？
Professor Ben Polak: Sudipta says get out of the market.
本·波拉克教授：苏迪普塔说退出市场。
If the other guy is pricing below costs, you don't want to sell anything, that's the right intuition.
如果其他人的定价低于成本，你就不想出售任何东西，这是正确的直觉。
How in this game do I "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 cost.
在这个游戏中我如何“退出市场”？你说得完全正确，如果其他人的售价低于成本，我不想参与这个市场。
How do I operationalize that in this game?
我如何在这个游戏中运用它？
How do I actually manage?
我实际上如何管理？
Ale is there somebody here, just wait for the mike a second and now shout it out.
艾尔有人在这里，等一下麦克风，然后大声喊出来。
Student: You could set your price above his price and then no one will buy.
学生：你可以把你的价格定得比他的价格高，然后就没人买了。
Professor Ben Polak: The answer was set your price above his price.
本·波拉克教授：答案是将你的价格定得高于他的价格。
So the other guy is pricing below costs, the way which I avoid making losses is to set my price above his price.
因此，另一个人的定价低于成本，我避免亏损的方法是将我的价格定为高于他的价格。
Let's 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.
让我们确保我们理解这一点，如果另一个人以低于成本的价格出售，我可以进行任何销售的唯一方法就是以低于他的价格定价。
I will then make sales but each of those sales I'm making a loss on.
然后我会进行销售，但每次销售我都会亏损。
I don't want to make losses, so I just "get out of the market," and the way I "get out of the market," is by pricing above the other guy.
我不想亏损，所以我只是“退出市场”，而我“退出市场”的方式就是定价高于其他人。
Everyone understand that?
大家明白了吗？
Okay, so now more interestingly, we leave a little gap here, what if the other guy is pricing above costs?
好吧，现在更有趣的是，我们在这里留了一点差距，如果其他人的定价高于成本怎么办？
What do I want to do if the other guy is pricing above costs?
如果对方的定价高于成本，我该怎么办？
Can I get the guy with the beard here, shout it out.
我能把那个留胡子的家伙叫过来吗？大声喊出来。
Student: You set price below his but above or equal to cost.
学生：你设定的价格低于他但高于或等于成本。
Professor Ben Polak: 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.
本·波拉克教授：好吧，如果其他人的定价高于成本，我想将价格定为低于他的价格，这样我就可以抢占整个市场并从这些销售中获利。
So where is--Suppose he's pricing--We'll say the prices are between 0 and 1, suppose he's pricing at .8, what would be a good price for me to set?
那么，假设他定价，我们会说价格在 0 到 1 之间，假设他定价为 0.8，那么我设定什么价格比较合适？
Same guy, yeah.
同一个人，是的。
Professor Ben Polak: As small an amount below his price as I can.
Ben Polak 教授：我尽可能比他的价格低一点。
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 ε to mean just a little bit.
所以基本上我要做的就是将我的价格设置为等于他的价格，减去一点点，并且我将使用字母 ε 来表示一点点。
I'll just undercut him a little bit 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.
我只要比他便宜一点，通过比他便宜一点，我就能得到整个市场，我就能在这些销售中赚到尽可能多的钱。
Now that's almost right.
现在这几乎是正确的。
That's kind of ninety percent of right, but it's not quite right.
这是百分之九十正确的说法，但又不完全正确。
I need to be a little bit careful.
我需要小心一点。
Now why do I need to be a little bit careful?
现在为什么我需要小心一点？
This is almost right but it's not quite right.
这几乎是正确的，但又不完全正确。
Yeah, same guy.
是的，同一个人。
Professor Ben Polak: That's certainly true, okay.
Ben Polak 教授：这当然是真的，好吧。
So that's true, fair enough.
所以这是真的，很公平。
So I want to make sure this ε is not so big, so as it pushes me below cost, that's certainly true.
所以我想确保这个 ε 不是那么大，因此当它使我低于成本时，这当然是正确的。
That's correct.
这是正确的。
There's another little issue here.
这里还有一个小问题。
So here's the other issue, there's a price in this market, we might want to think of as the kind of focal price as an interesting price, and that's the price at which, the price I would choose were I the monopolist.
所以这是另一个问题，这个市场上有一个价格，我们可能想把焦点价格看作是一个有趣的价格，这就是如果我是垄断者我会选择的价格。
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.
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，那么我们就可以解决这个问题。
We would solve out what the monopoly price is.
我们会解决垄断价格是多少。
That's an exercise you did in 115.
这是你在 115 中做过的练习。
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.
请注意，如果百事可乐的定价高于垄断价格，假设百事可乐将这种商品定价如此之高，以至于高于垄断价格，那么，正如这位先生所说，我可以通过定价略低于百事可乐来占领整个市场。
However, then I'll 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.
但是，然后我将拥有整个市场，就好像我是垄断者，但突然我的定价高于垄断价格，这是不对的。
I never want to price above the monopoly price, because I know the monopoly price, when I'm the monopolist, is the most profit I could ever make.
我从来不想定价高于垄断价格，因为我知道，当我是垄断者时，垄断价格是我能赚到的最大利润。
Is that right?
是这样吗？
So we need to be a little bit careful.
所以我们需要小心一点。
This answer that says I just undercut Pepsi, that's true provided that Pepsi hasn't gone above the monopoly price.
这个答案说我只是削弱了百事可乐，只要百事可乐没有超出垄断价格，这是真的。
If he's gone above the monopoly price we need to be a little bit more careful.
如果他的价格超出了垄断价格，我们就需要更加小心了。
If P2 is actually above the monopoly price, then what's my best response in that case?
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？
What's my best response in that case?
在这种情况下我最好的反应是什么？
Yeah, can we get Katie who's in green there, shout out.
是啊，我们能把穿绿衣服的凯蒂叫出来吗，大声喊出来。
Professor Ben Polak: Pricing at the monopoly price.
Ben Polak 教授：按垄断价格定价。
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.
因此，如果百事可乐愚蠢到将价格高于垄断价格，我当然会削弱百事可乐，但我不会削弱百事可乐一分钱，我会一直削弱百事可乐直到垄断价格并赚取垄断利润。
I'll be very happy.
我会很高兴。
I'll be wondering what on earth is Pepsi up to, but that's fine.
我会想知道百事可乐到底在做什么，但没关系。
So here my best response is to price at the monopoly price.
所以在这里我最好的回应是按照垄断价格定价。
There's one other possibility here I've missed which is what if Pepsi prices at marginal cost itself?
我错过了另一种可能性，那就是如果百事可乐本身按边际成本定价怎么办？
That's the one missing case here.
这就是这里失踪的一例。
What if Pepsi prices at marginal cost?
如果百事可乐按边际成本定价怎么办？
What's my best response in this case?
在这种情况下我最好的回应是什么？
Pepsi's pricing exactly at marginal cost, what's my best response?
百事可乐的定价完全按照边际成本，我的最佳回应是什么？
Yeah, wait for the mike.
是的，等一下麦克风。
This is Henry I think, shout out.
我想这是亨利，大声喊出来。
Professor Ben Polak: I could price at marginal cost as well.
Ben Polak 教授：我也可以按边际成本定价。
How much profits will I make if I price at marginal cost as well?
如果我也按边际成本定价，我会获得多少利润？
Zero profits, I'll make zero profits, so that certainly is a best response, pricing at marginal cost as well.
零利润，我会赚零利润，所以这当然是最好的回应，也是按边际成本定价。
What else would be a best response?
还有什么是最好的回应？
That's correct, but what else would be a best response?
这是正确的，但是还有什么是最好的回应呢？
Price above that.
价格在上面。
It doesn't really matter, as long as I don't price below it, I'm not going to make any money anyway.
其实没关系，只要我的价格不低于它，我就赚不到钱。
If I price below it I'm going to lose money.
如果我的价格低于它，我就会赔钱。
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.
所以这个最佳响应实际上是一个相当复杂的对象，如果我们要像上次那样，我们可以花时间画这个东西，但它有点乱，所以我现在不担心这样做。
So the best response is actually pretty complicated.
所以最好的回应实际上是相当复杂的。
Even though the best response is pretty complicated--and by the way, obviously the things are symmetric for Player II.
尽管最佳反应相当复杂——顺便说一句，对于玩家 II 来说，显然事情是对称的。
Even though these best responses are pretty complicated it turns out that there's only one Nash Equilibrium in this game.
尽管这些最佳反应相当复杂，但事实证明这场游戏中只有一个纳什均衡。
What's the Nash Equilibrium?
什么是纳什均衡？
Somebody, what's the Nash Equilibrium?
有人问什么是纳什均衡吗？
Raise your hands.
举起你们的手。
All you guys, there's a huge number of hands who said they had seen it before.
你们所有人，有很多人说他们以前见过它。
Who saw this model before?
谁以前见过这个模型？
Fewer hands go up when they know there's a question coming.
当他们知道有问题出现时，很少有人举手。
There's a different incentive.
有一个不同的激励。
Who remembers what the Nash Equilibrium is?
谁还记得纳什均衡是什么？
Yeah, there's a guy, Tae there's a guy right by you there.
是的，有一个人，Tae，有一个人就在你旁边。
Professor Ben Polak: At C okay, so the Nash Equilibrium here, the Nash Equilibrium is for both firms to set their prices equal to marginal cost.
Ben Polak 教授：在 C 处，好的，所以这里的纳什均衡，纳什均衡是两家公司将其价格设定为等于边际成本。
You can check that both firms are then playing a best response, so that's all right.
你可以检查两家公司当时是否都做出了最佳反应，这样就可以了。
If Firm II is charging C, then a best response for Firm 1 is to set price equal to marginal costs, and if Firm 1 is pricing at marginal cost, then conversely a best response for Firm 2 is to price at marginal cost.
如果公司 II 收取 C 费用，那么公司 1 的最佳反应是将价格设定为等于边际成本，如果公司 1 按边际成本定价，则相反，公司 2 的最佳反应是按边际成本定价。
Slightly harder exercise is to check that nothing else is a Nash Equilibrium.
稍微困难一点的练习是检查是否没有其他东西是纳什均衡。
Well let's think about that for a second.
好吧，让我们考虑一下。
Suppose Firm 1 is pricing at marginal cost and Firm 2 is pricing at something higher, at C + 3ε, a little bit higher.
假设公司 1 以边际成本定价，而公司 2 以更高的价格定价，即 C + 3ε，略高一点。
Suppose Firm 1 is pricing at marginal cost and Firm 2 is pricing at C + 3ε, is this a Nash Equilibrium?
假设公司 1 以边际成本定价，公司 2 以 C + 3ε 定价，这是纳什均衡吗？
Let's think about Firm 2, first of all, is Firm 2 playing a best response to Firm 1?
让我们考虑一下公司 2，首先，公司 2 是否是对公司 1 的最佳回应？
So I claim it is, let's just check carefully.
所以我说是的，我们仔细检查一下。
Firm 1 is pricing at marginal cost, the best response if the other guy is pricing at marginal cost is to price at marginal cost or above.
公司 1 按边际成本定价，如果其他公司按边际成本定价，最好的反应就是按边际成本或更高定价。
Now this is marginal cost or above so this is the best response.
现在这是边际成本或以上，所以这是最好的回应。
So if Firm 2 is playing a best response, this is a best response for Firm 2, given that P1 is at C.
因此，如果公司 2 正在发挥最佳响应，则考虑到 P1 位于 C，这就是公司 2 的最佳响应。
What's the problem here?
这里有什么问题？
Nevertheless I claim this is not a Nash Equilibrium.
尽管如此，我声称这不是纳什均衡。
Why is it not a Nash Equilibrium?
为什么它不是纳什均衡？
Who has an incentive to deviate?
谁有偏离的动机？
Can we get the guy way at the back by the door.
我们能把那个人叫到后面靠门的地方吗？
Go ahead; shout out.
前进;喊出来。
Professor Ben Polak: Exactly, exactly.
本·波拉克教授：完全正确。
This is not a best response.
这不是最好的回应。
This is not a best response.
这不是最好的回应。
The reason is that the best response for Firm 1, if Firm 2 is charging C + 3ε is to price at C + 2ε, thank you.
原因是，如果公司 2 收取 C + 3ε，公司 1 的最佳反应就是定价为 C + 2ε，谢谢。
So it turns out it's pretty to check that this is the only Nash Equilibrium in the game.
所以事实证明，检查这是游戏中唯一的纳什均衡是很不错的。
Now what's the lesson we're going to draw from this?
现在我们要从中吸取什么教训？
Well, as an exercise in Game Theory, that really wasn't very hard.
嗯，作为博弈论的练习，这确实不是很难。
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.
作为寻找纳什均衡的练习，在这个阶段你们大多数人都会看到这句话，这并不难。
So what's the point here?
那么这里有什么意义呢？
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.
关键在于，在这场企业进行价格竞争的博弈中，即使市场上只有两家企业，只有一家企业超过垄断地位，我们得到的结果却与上次截然不同。
In particular, we find that prices in the market are equal to marginal cost.
特别是，我们发现市场价格等于边际成本。
We find that profit in equilibrium is equal to what?
我们发现均衡利润等于什么？
Zero.
零。
And we find there's lots of consumer surplus because the prices are really as low as they ever could be.
我们发现有大量的消费者剩余，因为价格确实已经很低了。
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.
事实上，这里的结果，这里的均衡是出于所有意图和目的，如果市场上有数千家公司并且这是一个完全竞争的市场，我们也会得到同样的均衡。
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.
因此，即使这里只有两家公司，进行价格竞争，提供相同的产品，我们最终得到的结果看起来与完全竞争完全一样，只是事实上只有两家公司。
The outcome is like perfect competition even though there's only two firms.
尽管只有两家公司，但结果就像是完全竞争。
That's a pretty surprising result.
这是一个非常令人惊讶的结果。
It would suggest, think of this as a policy thing.
这表明，将其视为一项政策。
So if you believe this model, if you think 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 one competitor in each market, two firms in each market and you'd be done.
因此，如果你相信这个模型，如果你认为这确实是一个准确的社会模型，并且你是司法部的监管者，或者你是试图审判某些垄断案件的法官，或者你是欧洲法院的专员或任何试图审判某些竞争案件的人，那么你所担心的就是每个市场都有一个竞争对手，每个市场有两家公司，你就完成了。
You wouldn't worry about entry beyond two.
您不必担心超过两个人的进入。
Now my guess is we don't believe that.
现在我的猜测是我们不相信这一点。
We'll come back to that in a second, but let me make a different remark before we get there.
我们稍后会回到这个话题，但在我们到达那里之前让我发表不同的评论。
Is it me doing that.
是我干的吗。
Let me move this down.
让我把这个移下来。
Still doing it okay.
做起来还是没问题的
It's probably the wire.
应该是电线吧
Let me just remove it, I'll shout.
让我把它去掉，我会喊的。
Okay, I'm going to shout now, can people still hear me?
好吧，我现在要喊，人们还能听到我吗？
Yep, okay thank you.
是的，好的，谢谢。
So we'll have to turn down the other mike a bit.
所以我们必须把另一个麦克风的声音调低一点。
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't make any really significant change--we just said instead of thinking of them pricing and setting quantities, think of them setting prices (happens to be just thought experiments)--we ended up with a radically different outcome.
因此，这里要指出的另一件事是，尽管我们所关注的市场与上次所关注的市场基本相同，而且我们没有做出任何真正重大的改变——我们只是说，不要考虑它们定价和设定数量，而是考虑它们设定价格（恰好只是思想实验）——但我们最终得到了完全不同的结果。
So another lesson here is: the same setting 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.
因此，这里的另一个教训是：与古诺上次一样的设置，但采用不同的策略集，以不同的方式思考他们正在做的事情，导致了非常不同的结果。
That's worrying.
这令人担忧。
It's worrying because you can't go away feeling comfortable about this.
这是令人担忧的，因为你不能对此感到舒服地离开。
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.
你不能认为它真的可以在现实世界中产生如此大的影响，价格、数量、福利和利润将如何取决于一些关于我如何看待我的策略集的思想实验。
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.
因此，不仅对这个例子，而且对课程，对学习博弈论的整个主题都有些担心。
So what's going to save us here?
那么什么能拯救我们呢？
What's 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.
拯救我们的是，如果我们在模型中注入更多的现实，我们将得到一个更合理的结果。
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.
所以我们现在要做的是放宽 Bertrand 模型的一些假设，它将为我们做两件事。
First, it's going to give us an outcome that we believe.
首先，它将给我们一个我们相信的结果。
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.
我们认为我认为的结果是，不完全竞争应该看起来介于垄断和完全竞争之间，而不应该看起来像完全竞争。
I think most of us don't believe that two firms is enough to make it a perfect competition, that the regulator shouldn't worry about the third firm.
我认为我们大多数人都不相信两家公司足以使其成为完美竞争，监管机构不应该担心第三家公司。
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 things aren't quite as bad we thought.
从更理论的角度来看，它要做的第二件事是向我们表明，实际上事情并没有我们想象的那么糟糕。
In fact, if we model this more carefully, we'll get roughly the same prediction either way.
事实上，如果我们更仔细地建模，无论哪种方式我们都会得到大致相同的预测。
So what's the assumption we're going to change?
那么我们要改变的假设是什么？
Well before I say that, I should say I'm not going to change it, you are.
在我这么说之前，我应该说我不会改变它，你会。
So rather than go through this again a whole third time, I've gone through it once in quantities and once in prices, I'm going to get you guys to do it this time by having you do it on a homework assignment.
因此，与其再重复一遍整个第三次，我已经在数量上和价格上重复了一次，这次我将通过让你们在家庭作业中完成它来让你们这样做。
Well why?
那么为什么呢？
Partly because I think it's a good exercise, but also I don't want this class to be a class where you sit there, 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.
部分原因是我认为这是一个很好的练习，但我也不希望这堂课成为一堂你坐在那里，如果需要保持清醒的话，喝一杯威洛比咖啡，然后看我解决模型的课程，因为这不是你学习的方式。
What I want this class to be is a class where at least sometimes, like on this homework assignment, I set up a model for you, 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?
我希望这门课至少有时，就像在这个家庭作业中，我为你建立一个模型，为你设置故事，然后你必须真正弄清楚我如何正确地设置它以及如何解决它？
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 able to go from the story to the model.
因为归根结底，如果博弈论对你以后的生活有用，无论是在约会、经营公司还是其他任何事情上，你都需要能够从故事到模型。
Having said that, I'll tell you what the homework assignment's going to be about.
话虽如此，我会告诉你家庭作业的内容。
The assumption we're going to change is the idea that products are identical.
我们要改变的假设是产品是相同的。
So we're going to look at a case, I'll use a new board, where products are differentiated.
所以我们要看一个案例，我将使用一个新的板，其中产品是差异化的。
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.
我想说，假设产品不相同，对于世界上的大多数人来说，对于你将在世界上看到的大多数事物来说，这是一个相当安全的假设。
So we're going to look at what are called differentiated products.
所以我们要看看什么是所谓的差异化产品。
In particular, we're going to look at a particular version of this that we're going to call the linear city model.
特别是，我们将研究它的一个特定版本，我们将其称为线性城市模型。
So what do I mean by products being differentiated?
那么我所说的产品差异化是什么意思呢？
So I started off with an example that's pretty bad for this story, namely, Coke and Pepsi.
所以我从一个对这个故事来说非常糟糕的例子开始，即可口可乐和百事可乐。
Now be honest, how many of you in a blind taste test can taste the difference between ordinary Coke and ordinary Pepsi?
现在说实话，在盲味测试中，有多少人能尝出普通可口可乐和普通百事可乐的区别？
Quite a few.
相当多。
That's good for my case, so a number of you think those are different products.
这对我的情况有好处，所以你们中的许多人认为这些是不同的产品。
So a number of you, even if the 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?
因此，即使价格稍有不同，你们中的许多人也可能会偏爱其中一种而不是另一种，因此，如果不会因为将相机放在您身上而陷入麻烦，你们中有多少人会偏爱，如果价格基本相同，你们中有多少人会严格偏爱百事可乐？
How many would have a strict preference for Coke?
有多少人会严格偏爱可口可乐？
How many would have a strict preference for Diet Coke?
有多少人会严格偏爱健怡可乐？
That's amazing, that stuff's--oh never mind.
太神奇了，那东西——哦，没关系。
Okay but you're making my point.
好吧，但你说的是我的观点。
The point I'm trying to make is, products are not identical.
我想说的一点是，产品并不完全相同。
For the most part there's a little bit of difference between products and that's actually going to--that is, if we inject that little bit of realism into the world--it's actually going to help us.
在大多数情况下，产品之间存在一些差异，这实际上会 - 也就是说，如果我们向世界注入一点点现实主义 - 它实际上会对我们有所帮助。
So what we would like, what would we like?
那么我们想要什么，我们想要什么？
We would 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.
我们想要一个由企业设定价格的模型，因为在大多数情况下，我们认为企业确实设定价格而不是数量：并不总是如此，但在大多数情况下。
But we'd like a model that yields an outcome that looks--that when you only have two firms looks somewhere between monopoly and perfect competition.
但我们想要一个模型，它能产生这样的结果——当你只有两家公司时，它看起来介于垄断和完全竞争之间。
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.
因此，我们想要一个看起来有点像古诺的结果，但我们希望将策略设置为价格，这样就可以解决问题。
This is going to turn out to do the trick as you'll find out in your homework assignment.
正如您在家庭作业中会发现的那样，这将会起到作用。
So how are we going to model this on the homework?
那么我们如何在家庭作业中对此进行建模呢？
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.
我们对差异化产品建模的方法是想象，举一个简单的例子，想象一个城市，这个城市有一条长直的道路穿过它。
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.
所以这个城市不是新英格兰的城市，而是中西部的城市，那里一切都很平坦，道路完全笔直。
You can think of this as being a mile long.
您可以将其视为一英里长。
It doesn't really matter, let's just think of it as being a mile long.
这并不重要，我们只要把它想象成一英里长就可以了。
We're going to assume that consumers are evenly spread along this city.
我们假设消费者均匀分布在这座城市。
So there's basically consumers everywhere, they're evenly distributed and we're going to assume that one of these firms, let's call it Firm 1 sits at point 0 and the other firm, Firm 2, sits at point 1.
因此基本上到处都有消费者，他们分布均匀，我们假设其中一家公司（我们称之为公司 1）位于点 0，而另一家公司（公司 2）位于点 1。
Now, this is what you're going to do in the homework assignment, but I'm just going to make the argument that you could also imagine firms sitting somewhere between 0 and 1.
现在，这就是你在家庭作业中要做的事情，但我只是想说你也可以想象公司位于 0 和 1 之间。
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.
如果我们愿意的话，我们可以做一项更一般的工作，但现在，我们假设其中一家公司的商店在城镇的一端，另一家公司的商店在城镇的另一端。
So let's think about a particular consumer, so suppose this consumer is here, at point Y say.
让我们考虑一个特定的消费者，假设这个消费者在这里，在 Y 点。
So notice that this consumer is a distance Y away from Firm 1.
因此请注意，该消费者与公司 1 的距离为 Y。
So if she consumes from Firm 1 she has to walk a distance of Y.
因此，如果她从公司 1 消费，她必须步行 Y 的距离。
She's a distance of 1 -Y from Firm 2, is that right?
她与公司 2 的距离为 1 -Y，对吗？
So if she consumes from Firm 2 she has to walk 1 -Y.
因此，如果她从公司 2 消费，她必须步行 1 -Y。
That's going to turn out to be key in our model as we'll see in a second.
正如我们稍后将看到的，这将成为我们模型中的关键。
So firms, as before, are going to set prices.
因此，公司将像以前一样制定价格。
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.
我不会把所有的东西都写下来，因为它们都写在网上已经有的作业上，但事实上，公司将利润最大化，目标是利润最大化，公司将有恒定的边际成本。
We'll make one other assumption to keep life simple.
为了让生活变得简单，我们将做出另一个假设。
We'll assume that each consumer buys one and only one product.
我们假设每个消费者只购买一种产品。
Each consumer is going to buy one product, either from Firm 1 or from Firm 2.
每个消费者都会从公司 1 或公司 2 购买一种产品。
So the issue is going to be which firm does each consumer go and buy their product from.
因此，问题将是每个消费者从哪家公司购买产品。
Which firm does each consumer choose?
每个消费者都会选择哪家公司？
We'll assume that each consumer chooses the product whose, and I'll be careful here, whose total cost to her is smaller.
我们假设每个消费者选择的产品，我在这里会小心，她的总成本较小。
What do we mean by total cost?
总成本是什么意思？
Well let's look at the consumer at point Y.
好吧，让我们看看 Y 点的消费者。
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.
例如，对于 Y 点的消费者来说，如果他们从企业 1 购买商品，那么他们会支付企业 1 设定的价格 P1，但他们还必须支付交通费用，即他们必须一路走到那里并一路走回来的成本。
We'll give a name to that transport cost.
我们将为该运输成本命名。
We'll say it's TY².
我们会说它是 TY²。
So Y is the distance they have to travel and TY² is their transport cost.
因此，Y 是他们必须行驶的距离，TY² 是他们的运输成本。
So this object here we could think of as a transport cost.
所以我们可以将这里的这个对象视为运输成本。
If the same consumer buys from Firm 2 she pays P2 + T times again the distance squared, so that's going to be (1 –Y)² and once again this last term is a transport cost.
如果同一个消费者从公司 2 购买产品，她将再次支付 P2 + T 乘以距离的平方，因此这将是 (1 –Y)2，最后一项又是运输成本。
Notice that these transport costs go up in the distance you have to travel and they go up pretty fast.
请注意，这些交通费用会随着您旅行距离的增加而增加，而且上涨得相当快。
They go up at rates squared.
它们以利率的平方上升。
So what you're going to do on the homework assignment is solve out this market.
所以你要做的家庭作业就是解决这个市场。
You're going to assume that firms set prices to maximize profits.
您将假设公司设定价格是为了实现利润最大化。
You're going to know what firm's costs are.
您将了解公司的成本是多少。
You're going to work out what firm's demands are going to look for each possible price they could set.
您将计算出公司对他们可能设定的每个可能价格的需求。
And you're going to solve out the whole Nash Equilibrium.
你将解决整个纳什均衡。
And then we're going to look at that Nash Equilibrium and you're going to think, how does that compare to what I saw in the Cournot case we solved.
然后我们将看看纳什均衡，你会想，这与我在我们解决的古诺案例中看到的情况相比如何。
So that's on the homework assignment.
这就是家庭作业。
But before I leave this, let me just point out that this is a little bit more general than it might appear.
但在结束之前，我想指出的是，这比看起来更笼统一些。
So here I've treated what makes products different as being where the shops are located.
因此，在这里我将产品的不同之处视为商店所在的位置。
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 of the end of the town, and the other one is selling at the whitehead end of town.
因此，在这里，我将这些术语解释为运输成本，并且我将这些产品的不同之处解释为其中一种产品在城镇的一端销售，另一种在城镇的怀特海德一端销售。
But actually, we could consider this model more generally, and let's just do so briefly here.
但实际上，我们可以更普遍地考虑这个模型，我们在这里简单地做一下。
So let me just redraw the town; here's my town again.
那么让我重新绘制这个城镇吧；这里又是我的城镇。
I don't have to regard this product, this line as being distance along the high street of the town.
我不必认为这个产品，这条线是沿着小镇的大街的距离。
It could be something else about the product.
可能是产品的其他原因。
So, for example, in whatever it is you guys imagine makes Coke and Pepsi different, it could be that thing.
所以，举例来说，无论你们想象的是什么让可口可乐和百事可乐有所不同，它都可能是那样的。
Don't quite know what that is, but whatever that is.
不太清楚那是什么，但不管那是什么。
Let me take an example that I understand better than I understand Coke and Pepsi.
举个我比可口可乐、百事可乐更理解的例子。
So imagine we're talking about beer, think about beer, so this is the beer market.
想象一下我们正在谈论啤酒，想想啤酒，这就是啤酒市场。
This distance here could be something like the alcohol content or the flavor of the beer.
这里的距离可能是酒精含量或啤酒的味道。
You can imagine different products therefore, that are on the market, positioning themselves, or being positioned at different points on the line.
因此，您可以想象市场上不同的产品，它们各自定位，或者定位在生产线上的不同点。
So, for example, up here if you want beer flavor this might be Guinness, which I can't even spell, but you know what I mean.
所以，举例来说，如果你想要啤酒口味，这可能是吉尼斯黑啤酒，我什至无法拼写，但你知道我的意思。
If we think about the drinks industry more generally here, rather than just beer, this would be Guinness, this would be Poland Spring, this is water and everything else would be in between here.
如果我们更广泛地考虑这里的饮料行业，而不仅仅是啤酒，这将是吉尼斯啤酒，这将是波兰泉，这将是水，其他一切都将在此之间。
So if we just go a tiny distance in here, this is Bud Light and so on.
所以如果我们在这里走一小段距离，这是百威淡啤等等。
You could put everything on this line, I think there may be, I'm not really allowed to do this on this model, the one thing I can't do is, part of the truth is that Bud Light might be down here somewhere but I'm not allowed to do that.
你可以把所有东西都放在这条线上，我想可能有，我真的不允许在这个模型上这样做，我不能做的一件事是，部分事实是百威淡啤可能在这里的某个地方，但我不被允许这样做。
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 to see how competition is going to work in that market.
因此，抛开啤酒的具体例子，你可以考虑一些在某些方面有所不同的产品，我们可以使用这个模型来看看竞争将如何在该市场中发挥作用。
But now notice that this transport cost is going to be a different interpretation.
但现在请注意，这种运输成本将有不同的解释。
Now, instead of being the cost of traveling that distance to go and buy the product, what's it going to be?
现在，不再是长途跋涉去购买产品的成本，而是什么呢？
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.
如果我最喜欢的啤酒口味在这里，比如说，这就是我，我最喜欢的啤酒口味在这里，那么如果我最终不得不喝吉尼斯黑啤酒，我就必须支付吉尼斯黑啤酒的价格，而且我还要承担一些费用，因为吉尼斯黑啤酒对我来说不是完美的啤酒，它有点太烈了。
If I consume Bud Light, then I have to pay for the Bud Light, I think it typically costs less, the price, the raw price is less than Guinness but I also pay an additional cost from the fact that Bud Light, when I drink it, causes me "disutility": we'll put it politely like that.
如果我喝百威淡啤，那么我必须为百威淡啤付费，我认为它通常成本更低，价格，原价低于吉尼斯啤酒，但我还支付额外的费用，因为当我喝百威淡啤时，它会给我带来“负效用”：我们会这样礼貌地表达。
So basically, that transport cost is now the lack of pleasure caused by drinking a product that isn't perfect for me.
所以基本上，现在的交通成本是因为喝了不适合我的产品而导致的缺乏乐趣。
Does everyone understand the story here?
看到这里，大家都明白这个故事了吗？
So you're going to figure out what happens in this story very generally, well not very generally, but 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.
因此，你将非常普遍地弄清楚这个故事中发生的事情，不是非常普遍，而是在家庭作业中实际的特殊情况下，但我希望你明白，这背后有一个更普遍的故事，这是很多市场的一个非常好的模型。
It's the kind of model that you'd see again if you went onto graduate school.
如果你进入研究生院，你会再次看到这种模型。
Now I want to spend the rest of today doing something quite different.
现在我想用今天剩下的时间做一些完全不同的事情。
But you'll see it isn't all that different.
但你会发现它并没有那么不同。
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.
我将不再研究不完全竞争，而是回去参观我们几乎第一天或可能第二天研究的东西，那就是选举。
I want to go back to elections and why do I want to go back at this point?
我想回去选举，为什么我现在想回去？
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.
我想回去的一个原因是你会注意到我已经在董事会上提出了这些路线，当我们第二天访问选举时，我们说你可以认为董事会上的这条路线不仅是左翼、右翼政治，而且还包括产品的某些方面。
So I want to go back again, now we're 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.
所以我想再回去，现在我们正在考虑黑板上的这条线是啤酒的味道，还是城镇的位置，我现在想回到政治，回到我们开始时的解释，这样左翼和右翼最终将成为左翼政治与右翼政治的对抗。
So I'm going to take the same basic idea back to the politics model.
因此，我将把同样的基本思想带回到政治模型中。
We'll spend the rest of today in politics.
今天剩下的时间我们将讨论政治。
So we're going to be doing Political Science or Political Science as it meets Game Theory for the rest of the day.
因此，今天剩下的时间我们将学习政治学或政治学与博弈论的结合。
We're going to study something called The Candidate-Voter Model.
我们将研究一种称为“候选人选民模型”的东西。
This model is going to look a lot like the models we looked at already a few weeks ago.
这个模型看起来很像我们几周前看过的模型。
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.
特别是，正如所宣传的，会有一条线，这条线将从 0 到 1，这条线的左侧将代表左翼，该线的右侧将代表右翼，所以与之前相同。
As before, as in the Downs or Hoteling model, we discussed already a few weeks ago, we're going to assume that voters are evenly spread along the line.
与之前一样，正如我们几周前讨论过的唐斯或酒店模型一样，我们假设选民沿线均匀分布。
So we're going to assume even distribution of voters on the line.
所以我们假设线上选民的分布是均匀的。
We're also going to assume, just as we did last time, that voters are going to vote for the closest candidate.
正如我们上次所做的那样，我们还将假设选民将投票给最接近的候选人。
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.
因此，所有这些假设都是我们两周前或两周半前研究的相同假设，也是您在第一份作业中实际研究的假设。
Is that right?
是这样吗？
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.
但我想回到那里，因为我现在想做的是，我想改变该模型的一些关键假设，并看到通过进行这些改变，我们将得到一些非常不同的结果。
So, in particular, the new things here, two things, one the number of candidates is not fixed.
所以，特别是这里的新事物，有两件事，一是候选人的数量不固定。
So the number of candidates in this model is going to be endogenous.
因此，该模型中的候选人数量将是内生的。
Previously, we looked at models where there were two candidates or on your homework assignment, three candidates.
之前，我们研究了有两名候选人的模型，或者在你的家庭作业中，有三名候选人的模型。
Now we're going to allow the number of candidates to adjust itself.
现在我们将允许候选人数量自行调整。
The second assumption we're going to make which is new is that we're going to assume that candidates cannot choose their position.
我们要做的第二个新假设是，我们假设候选人无法选择自己的职位。
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.
所以这个想法是，任何参加这次选举的候选人，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其他人。
So candidates cannot choose their position.
因此候选人无法选择自己的职位。
So this is a subtle thing because you think about the current election, there's a debate about whether Hilary 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.
所以这是一件微妙的事情，因为你想想当前的选举，考虑到希拉里·克林顿(Hilary Clinton)过去的投票历史，例如在伊拉克战争上的投票历史，存在一场争论，例如希拉里·克林顿(Hilary Clinton)是否可以选择现在成为民主党的中心。
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 republic party, given he has a record of governance, as being Governor of Massachusetts when he provided state healthcare for example.
或者在另一方面，关于米特·罗姆尼是否可以选择存在争议，我猜他试图选择自己成为共和党的右翼，因为他有治理记录，例如在提供州医疗保健时担任马萨诸塞州州长。
So it's quite difficult in the real world for candidates to position themselves.
因此，在现实世界中，候选人很难给自己定位。
Voters tend to know that those candidates have track records.
选民往往知道这些候选人有过往记录。
That's something we've 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.
这是我们三周前、两周前就已经提到过的事情，我们现在就要采取修复措施，我们假设你知道这些候选人是谁。
So what are we actually going to assume?
那么我们实际上要假设什么呢？
We're going to assume that each voter in the model is a potential candidate.
我们假设模型中的每个选民都是潜在的候选人。
So this is, I think, a nice idealized version of American politics.
所以，我认为，这是美国政治的一个很好的理想化版本。
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.
你们每个人，只要年满 18 岁，是美国公民，出生在美国（也许你们必须年满 18 岁，无论是什么，无论规则是什么，无论宪法如何规定），你们每个人现在都有可能站起来，说你们要竞选总统。
I'm guessing all of you are ruled out by age actually.
我猜你们所有人实际上都被年龄排除在外了。
There's some rule in the Constitution, but never mind, let's pretend that's not there.
宪法中有一些规则，但没关系，让我们假设它不存在。
So the idea of the model is each voter is a potential candidate.
所以该模型的想法是每个选民都是潜在的候选人。
That's a very appealing assumption I think in a democracy.
我认为在民主国家这是一个非常有吸引力的假设。
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.
所以我想做的是，我想把它作为一个游戏更正式一些，然后，由于我们已经有一周左右没有做过这样的事情了，我们将玩这个游戏。
So you're about to play this game, so pay attention please.
所以你即将玩这个游戏，所以请注意。
So who are the voters, who are the players?
那么谁是选民，谁是参与者？
The players are the voters.
球员就是选民。
The voters, or candidates, whatever you want to call them, depending on where they stand, in this game they're going to be you.
选民或候选人，无论你想怎么称呼他们，取决于他们的立场，在这个游戏中他们将是你。
The key strategy here, essentially the strategy is going to be very simple.
这里的关键策略，本质上策略将非常简单。
The strategy is: do you run or not?
策略是：你跑还是不跑？
The reason that's going to be the strategy is that voting's not going to be difficult.
采取这一策略的原因是投票并不困难。
You're always going to end up voting for the candidate who's closest.
你最终总会投票给最接近的候选人。
So the only really relevant strategy is to run or not to run, to stand or not to stand.
因此，唯一真正相关的策略是跑或不跑，站或不站。
So just to make that clear, voters vote for the closest running candidate first.
因此，为了明确这一点，选民首先投票给最接近的竞选候选人。
And second, what did it mean to win?
其次，获胜意味着什么？
Well assume that we're in a plurality election here, so the winner is the person who gets a plurality.
假设我们正在进行多数选举，那么获胜者就是获得多数票的人。
You win if you get the most votes, in other words.
换句话说，如果你获得最多选票，你就赢了。
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, whichever.
我们假设，如果出现平局，我们会掷一枚公平的硬币或最高法院法官，无论你想采取什么，都可以。
Flip if tie.
如果打结则翻转。
The payoffs in this game are as follows.
该游戏的收益如下。
We'll assume that there's a prize for winning.
我们假设获胜会有奖品。
So if you win the election you get a prize equal to B.
因此，如果您赢得选举，您将获得相当于 B 的奖金。
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, let's just keep things simple, and assume it's actually equal to 2C.
我们还将假设存在运行成本，因此，如果您参加这次选举，无论获胜与否，您都会产生 C 的成本，并且我们将假设 B 大于或等于 2C，实际上今天，让我们保持简单，并假设它实际上等于 2C。
But that's not the only part of the payoffs.
但这并不是回报的唯一部分。
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.
还有一部分回报类似于强迫我喝百威淡啤或强迫喝百事可乐的人喝可口可乐。
What's that?
那是什么？
That's if, regardless of whether I run or not, if some other candidate is elected other than me, then they're not going to have my ideal policies.
也就是说，无论我是否参选，如果除了我之外还有其他候选人当选，那么他们就不会拥有我理想的政策。
So it's going to cause me unhappiness, it's going to cause me disutility having a candidate win, who's far away from me.
所以这会让我不高兴，让我因为一个离我很远的候选人获胜而感到无用。
So there's going to be an extra thing and, so if you're position is X, if you are at X on that line and the winner of the election is at Y, than you pay a cost of –|X –Y|, the absolute distance between you and the winning candidate.
所以会有额外的事情，所以如果你的位置是 X，如果你在那条线上的 X 处，而选举的获胜者在 Y 处，那么你需要支付 –|X –Y| 的成本，即你和获胜候选人之间的绝对距离。
So again, if you're at X and the winner is at Y, it hurts you minus the distance between X and Y, in terms of your unhappiness, about having a winner who's far away from you, winning.
再说一次，如果你在 X 处，而获胜者在 Y 处，那么你会因为距离你很远而获胜而感到不高兴，从而减去 X 和 Y 之间的距离，这会伤害你。
So let's do an example, just want to make sure we understand the payoffs.
让我们举个例子，只是想确保我们理解收益。
So Example 1, if you're at X, let's just call you Mr. X, just to make it clear, so Mr. X is the person at position X.
所以示例1，如果你在X，我们就称你为X先生，只是为了清楚起见，所以X先生就是位置X的人。
If he enters the election and he wins the election then his payoff is B because he entered -C,--sorry B because he won, -C because he had to pay his election expense.
如果他参加选举并赢得选举，那么他的收益是 B，因为他参加了 -C，--对不起 B，因为他赢了，-C 因为他必须支付选举费用。
But the winning candidate is him so he gets no disutility from the winning candidate being someone else.
但获胜的候选人是他，因此他不会因为获胜的候选人是其他人而受到损害。
Second possibility, if Mr. X enters but Mr. Y wins, then X's payoff is what?
第二种可能，如果X先生进入，但Y先生获胜，那么X的收益是多少？
He still incurred -C because he ran and he also has a cost of |X-Y| because he doesn't like Mr. Y winning.
他仍然产生了-C，因为他跑了，而且他还有|X-Y|的成本。因为他不喜欢Y先生获胜。
Third, if Mr. X stays out but Mr. Y wins then Mr. X's payoff, he doesn't win so he doesn't get B, he doesn't lose C because he didn't run, but he still has this disutility -|X –Y| because he doesn't like Y winning.
第三，如果X先生不参加，但Y先生赢了，那么X先生的回报是，他没有赢，所以他没有得到B，他也没有因为没有竞选而失去C，但他仍然有这个负效用-|X –Y|因为他不喜欢Y获胜。
Does everyone understand this game?
大家都懂这个游戏吗？
Everyone understand the game?
大家都懂这个游戏吗？
Anyone not understand the game at this point?
现在还有人看不懂这个游戏吗？
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, well I'll grab it in a minute.
原则上，我们可以与全班同学一起玩这个游戏，但是让我们挑出班级中的某一行，这样我就会来到这里，我想最终，我会在一分钟内抓住它。
So I'm going to use this row, I think, everyone in this row stand up a second, and this row stand up, this is the row of potential voters.
所以我要用这一排，我想，这一排的每个人都站起来，这一排站起来，这就是潜在选民的排。
They're the voters.
他们是选民。
They're also the potential candidates.
他们也是潜在的候选人。
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 the people in front of them.
所以我们这里需要开一个大会，无论是站在后面的人的角度来看，还是站在前面的人的角度来看，我们都站在前面的人的角度来看。
So I got an R, I'll do it the other way around.
所以我得了 R，我会反过来做。
People from behind that have the perspective.
背后的人有观点。
So this gentleman here who's name is Andy.
这位先生的名字叫安迪。
So Andy is our crazy right-wing guy and Sudipta is our crazy left-wing guy, everyone else is in between.
所以安迪是我们疯狂的右翼分子，苏迪普塔是我们疯狂的左翼分子，其他人都介于两者之间。
There's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 people here.
这里有 1、2、3、4、5、6、7、8、9、10、11、12、13、14、15、16 人。
Let me add, let me make it 17 people so we have an odd number, so go and sit in that row.
让我补充一下，让我把人数定为 17 人，这样我们的人数就是奇数，所以去坐在那一排。
Okay, now we have 17 people it's a bit easier.
好吧，现在我们有 17 个人了，这就容易一些了。
Just want to avoid obvious ties.
只是想避免明显的联系。
So how is this going to work?
那么这将如何运作呢？
Everybody sit down again but don't put your--keep your books handy because this is going to be pretty aerobic.
每个人都重新坐下，但不要把书放在手边，因为这将是相当有氧的活动。
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's deciding whether to run or not.
这些都是潜在选民，他们每个人都处于弗雷德·汤普森两周前共和党初选时所处的位置，当时他正在决定是否参选。
But in the real world that's a sequential game, people decide to run in some order, we're going to assume this is all simultaneous.
但在现实世界中，这是一个顺序游戏，人们决定按某种顺序运行，我们假设这一切都是同时发生的。
So what we're going to do is we're going to hold essentially this game.
所以我们要做的就是继续举办这场比赛。
We're going to ask everybody in this row whether they want to enter or not.
我们将询问这一排的每个人是否愿意进入。
Let's make some real numbers.
让我们做一些实数。
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.
假设 B 等于 2 美元，那么政府的战利品是 2 美元，而进入的成本是 1 美元，所以 C 是 1 美元。
And we'll assume that the amount of disutility you get from being far away from the winner is 1/17th of the distance, each of these, each place is worth 1/17th of a $1.
我们假设远离获胜者而获得的负效用是距离的 1/17，每个位置的价值是 1 美元的 1/17。
Everyone understand?
大家明白了吗？
So whatever 1/17th of a $1, roughly $.05 for each position away.
因此，无论 1 美元的 1/17，每个位置大约为 0.05 美元。
Everyone understand in this row what we're doing?
这一排大家都明白我们在做什么吗？
I'm not getting any nods, you guys over here you understand what we're doing?
我没有得到任何点头，你们明白我们在做什么吗？
So on the count of three, anybody who's entering this election is going to stand up.
因此，数到三后，任何参加这次选举的人都会站起来。
Ready?
准备好？
One, two, okay what?
一，二，好么？
That's the opposite of what's happening with the Florida Primary.
这与佛罗里达州初选的情况正好相反。
Yeah question?
是的问题？
Professor Ben Polak: You guys have to decide whether you, okay.
本·波拉克教授：你们必须决定是否要这样做，好吗？
So if you're going to, if you're running you stand up.
所以，如果你愿意，如果你正在跑步，你就站起来。
If you're standing you stand.
如果你站着，你就站着。
Let's try again one, two, three.
让我们再尝试一、二、三。
Stay standing and I'll come around a second.
站着，我一会儿就过来。
So here's our election, and it's between a, which way do I want to do it?
这是我们的选举，我想以哪种方式进行？
This is our left wing candidate, this is our moderate left wing candidate whose name is?
这是我们的左翼候选人，这是我们的温和左翼候选人，他的名字是什么？
Professor Ben Polak: Our kind of centrist candidate, maybe even this centrist candidate who's name is?
本·波拉克教授：我们这种中间派候选人，甚至可能是这个中间派候选人，他的名字叫什么？
Professor Ben Polak: Beatrice.
本·波拉克教授：比阿特丽斯。
So this is the election, who's going to win this election?
这就是选举，谁会赢得这次选举？
Beatrice is going to win this election.
比阿特丽斯将赢得这次选举。
Let's just check that.
让我们检查一下。
Beatrice is going to get all of these right wing votes.
比阿特丽斯将获得所有这些右翼选票。
Beatrice is also going to get one and a half of the votes in the middle, so this vote and half the Pierson guy's vote and, I've forgotten your name already, again?
比阿特丽斯也将获得中间的一票半，所以这一票和皮尔森那家伙的一半选票，我又忘了你的名字了？
Alex is going to get all the left wing votes, but there aren't enough left wing votes here, right?
亚历克斯将获得所有左翼选票，但这里没有足够的左翼选票，对吧？
So Beatrice is going to win.
所以比阿特丽斯会赢。
So Beatrice won this election, is this array a Nash Equilibrium?
那么Beatrice赢得了这次选举，这个数组是纳什均衡吗？
Is this an equilibrium?
这是一个平衡吗？
How do we know it's not an equilibrium?
我们怎么知道这不是平衡？
Katie, how do we know this is not an equilibrium?
凯蒂，我们怎么知道这不是均衡？
Student: Because Alex's best response to her running is to not run.
学生：因为亚历克斯对跑步的最佳反应就是不跑。
Professor Ben Polak: Absolutely.
本·波拉克教授：当然。
So Alex here, Alex ends up with a payoff of what?
那么亚历克斯在这里，亚历克斯最终得到了什么回报？
Alex ended up with a payoff of -C, actually more than that, -C minus the distance between him and Beatrice, so that's another $1, $.05, $.10 or even $.20 whatever it is, and he would have done better to have not run.
Alex 最终得到了 -C 的回报，实际上比这个还要多，-C 减去他和 Beatrice 之间的距离，所以这又是 1 美元、0.05 美元、0.10 美元甚至 0.20 美元，无论是什么，他如果不逃跑的话会做得更好。
If he hadn't run, Beatrice would still have won, but it would still have cost him the $.20 of unhappiness at Beatrice winning, but at least he would have saved himself the C of running.
如果他没有参选，Beatrice 仍然会获胜，但这仍然会让他因 Beatrice 获胜而损失 0.20 美元的不快感，但至少他会节省自己参选的 C 分。
Everyone see that?
大家看到了吗？
Let's try it again.
让我们再试一次。
Okay, same row, but don't hesitate so much this time.
好吧，同一排，但这次别犹豫了。
Count of three, you're just going to go or not.
数到三，你就决定去还是不去。
Everybody in this row close their eyes so you can't look around you.
这一排的每个人都闭上眼睛，这样你就无法环顾四周。
Close your eyes, you can do all sorts of nasty things, close your eyes.
闭上眼睛，你可以做各种肮脏的事情，闭上眼睛。
Myrto you're still in this row okay.
米尔托，你还在这一排好吗。
So one, two, three.
所以一、二、三。
You can open your eyes again, so now what happened, we ended up with two centrist candidates.
你可以再睁开眼睛，所以现在发生了什么，我们最终有了两位中间派候选人。
Let's see what happened, so we have Beatrice and your name is Claire Elise.
让我们看看发生了什么，我们有比阿特丽斯，你的名字是克莱尔·埃莉斯。
Professor Ben Polak: Claire Elise is going to get her own vote plus 1, 2, 3, 4, 5, 6, 7, 8, 8 votes and Beatrice is getting 1, 2, 3, 4, 5, 6, 7, how did I end up with eight, I thought I had an odd number.
本·波拉克教授：克莱尔·埃莉斯将获得自己的选票加上 1, 2, 3, 4, 5, 6, 7, 8, 8 票，而比阿特丽斯将获得 1, 2, 3, 4, 5, 6, 7，我怎么最终得到了 8 票，我以为我的票数是奇数。
1, 2, 3, 4, 5, 6, 7 there must have been an odd number before.
1, 2, 3, 4, 5, 6, 7 之前一定是奇数。
So Claire Elise is going to win this election because she got one more vote.
所以克莱尔·埃莉斯将赢得这次选举，因为她多了一张选票。
Now here we ended up with two centrist candidates, which is a result pretty close to what we saw in the Hoteling model, is that right?
现在我们最终有了两位中间派候选人，这个结果非常接近我们在酒店模式中看到的结果，对吗？
Is this an equilibrium?
这是一个平衡吗？
So check that I counted right, 1, 2, 3, 4, 5, 6, 7, 8 and 1, 2, 3, 4, 5, 6, 7.
检查一下我数是否正确：1、2、3、4、5、6、7、8 和 1、2、3、4、5、6、7。
I guess I did count right, so Claire Elise turned out to win this.
我想我确实算对了，所以克莱尔·埃莉斯最终赢得了这场胜利。
I counted that correctly.
我算对了。
There's another vote hidden behind there, I didn't see you.
后面还藏着另一张选票，我没看到你。
Let's try again.
让我们再试一次。
So 1, 2, 3, 4, 5, 6, 7, 8.
所以 1、2、3、4、5、6、7、8。
I apologize Beatrice won this election, I apologize.
我很抱歉比阿特丽斯赢得了这次选举，我很抱歉。
Beatrice won this election.
比阿特丽斯赢得了这次选举。
Is this an equilibrium?
这是一个平衡吗？
How do we know this isn't an equilibrium?
我们怎么知道这不是平衡？
Because Claire Elise would rather not have run, is that right?
因为克莱尔·埃莉斯宁愿不参选，对吗？
Let's try once more, down again, close your eyes again, just to keep life interesting, let's switch to this row.
让我们再试一次，再下来，再闭上眼睛，只是为了让生活保持有趣，让我们切换到这一行。
Can we switch to this row okay?
我们可以切换到这一行吗？
So this is our row now, everyone understand this.
这就是我们现在的争吵，大家都明白这一点。
Close your computer so you can get up without destroying thousands of dollars of technology.
关闭你的电脑，这样你就可以在不破坏数千美元技术的情况下起床。
This is our row and at the count of three we're going to see who's going to run.
这是我们的排，数到三后我们就看谁先跑。
Everyone understand the game?
大家都懂这个游戏吗？
Close your eyes, 1,2,3.
闭上眼睛，1、2、3。
It's obvious he's going to win this election, is that right?
很明显他将赢得这次选举，对吗？
Just to ask the question, is this an equilibrium?
只是想问一下，这是一个均衡吗？
Let's count.
我们来算一下。
1, 2, 3, 4, 5, 6, 7, 8, 9 he's going to get 10 votes, including himself there are 10 votes here.
1、2、3、4、5、6、7、8、9 他将获得 10 票，包括他自己在内，这里有 10 票。
I can't count correctly, 1, 2, 3, 4, 5, 6, is he actually the center guy here?
我数不清了，1、2、3、4、5、6，他真的是这里的中心人物吗？
I don't think he is the center guy here.
我不认为他是这里的中心人物。
So I claim, if I've counted correctly, that this actually isn't an equilibrium.
所以我声称，如果我计算正确的话，这实际上不是一个均衡。
Who can deviate and do better here?
谁能在这里偏离并做得更好？
This guy, stand up a second, whose name is?
这个家伙，站起来，谁的名字？
Professor Ben Polak: Had John entered, if I've counted correctly, let's pretend I have, John would actually have won this election because John is actually further to the center.
本·波拉克教授：如果约翰参加了，如果我数得正确的话，让我们假装我有，约翰实际上会赢得这次选举，因为约翰实际上更接近中间。
Okay, so let's, thanks for a second guys, I'll come back to this in a second.
好的，那么让我们感谢第二位朋友，我稍后会回到这个问题。
Let's try and figure out how to analyze this game.
让我们尝试找出如何分析这场比赛。
We've played it three times, let's try and figure out what the equilibria looked like in this game.
我们已经玩了三遍了，让我们尝试找出这个游戏中的平衡是什么样的。
This game that's going on more or less right now in the primaries, well I guess the entry stage of it is gone now.
这场比赛现在或多或少正在进行初选，我想它的入门阶段现在已经过去了。
So first of all, is there a Nash Equilibrium in which no candidates stand?
那么首先，是否存在一个没有候选人站立的纳什均衡？
Is there a Nash Equilibrium in which nobody stands?
是否存在无人站立的纳什均衡？
No. How do we know that's not a Nash Equilibrium?
不，我们怎么知道这不是纳什均衡？
Somebody?
有人吗？
Yeah, how do we know that's not a Nash Equilibrium?
是的，我们怎么知道这不是纳什均衡？
Shout it out.
大声喊出来。
Student: Because then somebody's definitely better off by standing up.
学生：因为那么站起来肯定会更好。
Professor Ben Polak: Right, if nobody stands, each and every possible candidate would do better individually, so any particular voter would do better standing.
本·波拉克教授：是的，如果没有人参选，那么每一位可能的候选人都会表现得更好，因此任何特定的选民都会获得更好的表现。
They would win the election.
他们会赢得选举。
Is that right?
是这样吗？
So clearly it's not an equilibrium for nobody to stand.
很明显，这不是一个没有人能够站立的平衡。
That's good because we don't see that very often.
这很好，因为我们不经常看到这种情况。
Is there a Nash Equilibrium in which one and only one candidate stands?
是否存在一个且只有一名候选人的纳什均衡？
Professor Ben Polak: 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.
Ben Polak 教授：好吧，考虑到我们在这一排中选择了奇数人，如果恰好有一个候选人站立并且该候选人是中间候选人，那么这就是均衡。
How do we know that's an equilibrium?
我们怎么知道这是一个平衡？
Somebody give me the argument why that's an equilibrium?
有人给我论证为什么这是均衡吗？
It's a good, in fact, it's a correct guess.
这很好，事实上，这是一个正确的猜测。
But how do we know that's an equilibrium?
但我们怎么知道这是一个平衡呢？
That's the guess part.
这就是猜测部分。
How do we check that's an equilibrium?
我们如何检查这是一个平衡？
Student: If anybody on either side of them chose to run, they would lose.
学生：如果他们任何一方选择竞选，他们都会输。
Professor Ben Polak: They would lose, exactly.
本·波拉克教授：他们确实会输。
So if anybody on either side of them runs, that person would simply lose.
因此，如果他们任何一方的任何人逃跑，那个人就会输。
It wouldn't make any difference to the outcome.
这不会对结果产生任何影响。
If a person on the right stood, so let's just try it again.
如果右边的人站起来了，那我们就再试一次。
So we figured out that, Beatrice stand up a second.
所以我们发现，比阿特丽斯站起来了。
So here's our potential equilibrium with just Beatrice standing, and if somebody to the, which way do we do it, somebody to the right of Beatrice was to stand, so ma'am if you stand a second, and your name is?
所以这是我们的潜在平衡，只有比阿特丽斯站着，如果有人，我们该怎么做，比阿特丽斯右边的人要站起来，所以女士，如果你站一下，你的名字是？
Professor Ben Polak: So if Stacy stands, it doesn't make any difference she just loses.
本·波拉克教授：所以如果史黛西坚持下去，她只是输了，这并没有什么区别。
Beatrice wins anyway; it just costs Beatrice some money.
不管怎样，比阿特丽斯赢了；只是花了比阿特丽斯一些钱。
Conversely, if Beatrice doesn't stand and the woman who's one in from the side there, sit down a second Stacy and if this person stands, and your name is?
相反，如果比阿特丽斯没有站起来，而坐在旁边的那个女人，请让第二个史黛西坐下，如果这个人站起来，你的名字是？
If Sarah stands, it doesn't help Sarah at all, she just loses.
如果莎拉站起来，那对莎拉一点帮助也没有，她只是输了。
Is that right?
是这样吗？
So there is an equilibrium with exactly one candidate, the center candidate.
因此，存在一个均衡，只有一个候选人，即中心候选人。
Now that's looking a little bit like the median voter model.
现在看起来有点像中位选民模型。
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.
它表示只有一名候选人处于平衡状态，该候选人将是中间候选人，就像我们在家庭作业和课堂上看到的模型中一样。
However, we're not done yet, there could be other equilibria.
然而，我们还没有完成，可能还有其他平衡。
So let's just try and think about whether there are other equilibria.
因此，让我们尝试思考是否存在其他均衡。
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.
因此，举例来说，现在假设，为了让生活变得更有趣一点，假设选民实际上是两排，并且让我暂时停止怀疑这些排中的人数相同，我知道事实并非如此。
So now what I want you to think of is that at every political position there are two voters, and hence two possible candidates.
所以现在我想让你们想到的是，每个政治立场都有两个选民，因此有两个可能的候选人。
There's two people at every position, these two people at this position, these two people at this position.
每个位置都有两个人，这两个人在这个位置，这两个人在这个位置。
Everyone understand that?
大家明白了吗？
Let's just assume that the rows are the same even though they're not and let's examine the following thing.
我们假设行是相同的，即使它们不是相同的，然后让我们检查以下内容。
Suppose Beatrice stands again, sorry Beatrice, and the gentleman in front of her whose name is?
假设比阿特丽斯再次站起来，对不起比阿特丽斯，而她面前的那位先生叫什么名字？
Robert stands as well, so stand a second.
罗伯特也站了起来，所以请站起来。
So now we have two candidates standing who are identical, politically, they're identical.
现在我们有两位候选人，他们在政治上是相同的。
They're right on top of each other.
他们就在彼此之上。
Now is that an equilibrium?
现在这是一个平衡吗？
After all that looks a lot like the Downs-Hoteling model, we've got two candidates exactly at the middle, is that an equilibrium?
毕竟这看起来很像 Downs-Hoteling 模型，我们有两个候选人正好处于中间，这是一个均衡吗？
Let me try the guy up here?
让我试试上面那个人吗？
Student: No, because if someone on either side of them were to stand up that person on the side would win.
学生：不，因为如果两边有人站起来，那一边的人就赢了。
Professor Ben Polak: Good, this can't be an equilibrium because look what happens if Claire Elise stands a second.
本·波拉克教授：很好，这不可能是平衡，因为看看如果克莱尔·埃莉斯站第二会发生什么。
Claire Elise is going to win all of the right wing votes, ranging from crazy to moderate, and these guys are going to split the left wing votes.
克莱尔·埃莉斯将赢得所有右翼选票，从疯狂到温和，这些人将瓜分左翼选票。
So the total sum of right wing votes is going to beat out half of the left wing votes.
因此，右翼选票总数将超过左翼选票的一半。
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.
所以这不可能是均衡，唐斯模型的精确预测，两个人叠在一起不是均衡。
Sit down again guys.
伙计们，再坐下吧。
Let's go back to one row again because it's easy to work with.
让我们再次回到一行，因为它很容易使用。
Let's try a slightly different pattern.
让我们尝试一个稍微不同的模式。
So suppose, assuming I counted right, suppose that the following candidates enter.
所以假设，假设我数对了，假设以下候选人进入。
So Claire Elise enters and I'm sorry I don't know your name, Jean enters so let's have a look at this array.
Claire Elise 进来了，很抱歉我不知道你的名字，Jean 进来了，让我们看看这个数组。
Assume I got it right and Beatrice is actually the center.
假设我猜对了，比阿特丽斯实际上是中心。
Now is that an equilibrium?
现在这是一个平衡吗？
Who thinks that is an equilibrium?
谁认为这是一种均衡？
Who thinks it's not an equilibrium?
谁认为这不是一个平衡？
Who's waiting to see how other people vote?
谁在等着看其他人如何投票？
Well let's check.
好吧，让我们检查一下。
So there's three possible types of deviation here that we need to check.
因此，我们需要检查三种可能的偏差类型。
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 the third type of deviation?
我们需要从外侧检查另一个进入者，左翼或右翼，我们需要检查中间的另一个进入者，那里只有一种可能，还有第三种偏差我们应该检查，第三种偏差是什么？
One of these might choose not to run, one of you might choose not to run.
其中一位可能选择不跑步，你们中的一位可能选择不跑步。
Let's do them in turn.
让我们依次做。
So suppose that we consider a deviation in which, and again I've forgotten your name in which Stacy stands as well.
因此，假设我们考虑了一种偏差，我再次忘记了史黛西所代表的你的名字。
Stacy, just stand a second.
史黛西，稍等一下。
So is this a profitable deviation for Stacy?
那么这对于史黛西来说是有利可图的偏差吗？
No it's not.
不，不是。
Why isn't it a profitable deviation for Stacy?
为什么这对史黛西来说不是一个有利可图的偏差？
So two reasons: one is that she loses, she's not going to win this election by standing.
所以有两个原因：一是她输了，她不会通过竞选赢得这次选举。
But there's a second reason why this is a really bad deviation.
但还有第二个原因可以解释为什么这是一个非常糟糕的偏差。
Why is it, what's the second reason?
为什么会这样，第二个原因是什么？
Yes sir.
是的，先生。
Professor Ben Polak: 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.
本·波拉克教授：很好，所以，通过站立，她不仅没有赢得选举，而且实际上使离她较远的吉恩肯定赢得了选举。
So Stacy's going to pick up the right wing votes, she's going to split.
所以史黛西会赢得右翼选票，她会分裂。
The crazy right wing votes she's going to pick up.
她将获得疯狂的右翼选票。
The moderate right wing votes she's going to split with Claire Elise, and Jean's going to have all the left wing votes, extreme and moderate, and so from Stacy's point of view this is a double bad.
她将与克莱尔·埃莉斯（Claire Elise）平分温和右翼选票，而让（Jean）将获得所有左翼选票，无论是极端的还是温和的，因此从史黛西的角度来看，这是双重坏事。
She doesn't get to be President, sorry and you end up with a left wing President, which you didn't like.
她没有成为总统，抱歉，你最终得到了一位你不喜欢的左翼总统。
Everyone see that?
大家看到了吗？
So clearly, that's not, thank you, that's not a profitable deviation and it's also not a profitable deviation for people on the left.
很明显，这不是，谢谢，这不是一个有利可图的偏差，对于左翼人士来说也不是一个有利可图的偏差。
Hang on a second.
等一下。
So I think it's probably obvious why it's not a good strategy for Beatrice to enter here.
所以我认为对于碧翠斯来说进入这里不是一个好的策略可能是显而易见的。
If Beatrice enters she loses.
如果比阿特丽斯进来，她就输了。
Everyone can see that?
大家都看得出来吗？
The person in the middle if they enter here loses, so they're not going to want to enter.
中间的人如果进入这里就输了，所以他们不会想进入。
So the other possible deviation we have to consider, is a deviation of the form of one of these guys dropping out.
因此，我们必须考虑的另一个可能的偏差是其中一个退出的人的形式偏差。
So let's consider it, so come back to Claire Elise.
让我们考虑一下，然后回到克莱尔·埃莉斯。
Is Claire Elise doing better in or out?
克莱尔·埃莉斯 (Claire Elise) 是室内表现更好还是室外表现更好？
Is Claire Elise doing better in this election or out?
克莱尔·埃莉斯（Claire Elise）在这次选举中表现更好还是在选举中表现更好？
Let's think it through a second, if she stays in what's her payoff?
让我们想一想，如果她留下来，她的回报是什么？
She wins B with probability a half.
她以一半的概率赢得 B。
She costs C for sure, that's a wash because B, the way we've worked things out, B was $2, C was $1, so B/2-C is a wash.
她肯定要花 C 钱，这是一次清洗，因为 B，我们解决问题的方式，B 是 2 美元，C 是 1 美元，所以 B/2-C 是一次清洗。
In terms of the benefits of--the spoils of government versus the cost of running, it's a wash for Claire Elise, whether she enters or not.
就政府的收益与运行成本而言，无论克莱尔·埃莉斯是否加入，这对她来说都是一次洗礼。
But by entering she has a half chance of winning the election.
但如果她加入，她就有一半的机会赢得选举。
So one way to think about it is, is the expected distance from her of the winning candidates, is: with probability of a half it's herself so that's nothing, no distance, and with probability of a half it's two places away.
因此，一种思考方式是，获胜候选人与她的预期距离是：有一半的概率是她自己，所以没什么，没有距离，有一半的概率是两个位置。
So in expectation it's one place away.
所以预计它会差一个地方。
If she drops out, Jean wins for sure, so the expected distance away from her of the winning candidate is two away.
如果她退出，吉恩肯定会获胜，因此获胜候选人与她的预期距离是两分。
Just to say it again, so the cost and direct benefits of government are a wash for her.
再说一遍，所以政府的成本和直接收益对她来说是一笔洗礼。
But by dropping out she insures that somebody further away from her wins for sure.
但通过退出，她确保了距离她较远的人肯定会获胜。
That's bad so she's going to stay in.
情况很糟糕，所以她要留下来。
The same argument goes the other way, so this is an equilibrium.
同样的论点反之亦然，所以这是一个均衡。
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?
现在我们还没有完成，当然我们已经没有时间了，那么下周回来时我们要做什么呢？
Don't pack up yet, what are we going to do?
还不收拾行李，我们要做什么？
We're going to come back to this row and we're going to figure out not just this equilibrium, but all equilibria.
我们将回到这一行，我们不仅要找出这个均衡，还要找出所有的均衡。
Before you go though, let's just think about it a second, 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 Claire Elise, who both stood very close to each other.
不过，在你走之前，让我们先想一想，这样你就可以在家里想一想并谈论它，所以我们刚刚看到的是一场选举，我们看到了两位候选人，Jean 和 Claire Elise，他们的立场都非常接近。
But a different question is--before you go: stand up my very left-wing guy, Sudipta, way at the end, and stand up my very right-wing guy.
但一个不同的问题是——在你走之前：站起来，我的左翼人士苏迪普塔，最后，站起来，我的右翼人士。
Here's a candidate--here's an outcome with two entrants in it, an extreme right-wing guy and an extreme left-wing guy.
这是一个候选人——这是一个有两个候选人的结果，一个极右翼的人和一个极左翼的人。
Is that an equilibrium?
这是一个平衡吗？
Think about it until we come back on Monday.
考虑一下，直到我们周一回来。