Game Theory Lecture 3 | Game Theory

Chapter 1.

第一章。

Iterative Deletion of Dominated Strategies: The Median Voter Theorem [00:00:00] Professor Ben Polak: Last time we talked a lot about the idea--I didn't mention the name but here's some jargon for you--we talked about the idea of deleting dominated strategies: looking at a game; figuring out which strategies are dominated; deleting them; looking at the game again; looking at which strategies are now dominated; deleting those; and so on and so forth.

迭代删除主导策略：中位选民定理 [00:00:00] Ben Polak 教授：上次我们谈论了很多关于这个想法的内容——我没有提到名字，但这里有一些行话给你——我们讨论了删除主导策略的想法：看一场比赛；找出哪些策略占主导地位；删除它们；再看比赛；看看哪些策略现在占主导地位；删除那些；等等。

This process is called the "iterative deletion of dominated strategies." This is not a great title for Barry's book.

这个过程被称为“主导策略的迭代删除”。对于巴里的书来说，这个标题不太好。

You need to do better than that to get the two hundred and fifty dollars.

你需要做得更好才能拿到两百五十美元。

This is a boring title, but never mind.

这是一个无聊的标题，但没关系。

Iterative Deletion of Dominated Strategies.

迭代删除主导策略。

The idea is--It embodies the idea of putting yourself in someone else's shoes and trying to figure out what they're going to do, and then think about them putting themselves in your shoes, figuring out what you're going to do, and so on and so forth.

这个想法是——它体现了这样一种想法：把自己放在别人的立场上，试图弄清楚他们要做什么，然后想想他们把自己放在你的立场上，弄清楚你要做什么，等等。

Last time, we saw already, that this is a very powerful idea, in that game last time.

上次，我们在上次的游戏中已经看到，这是一个非常强大的想法。

But we also saw it's a dangerous idea to take too literally: that sometimes this can get you to over-think the problem and actually, as in that numbers game last time, the best choice, the winning choice, might not involve so many rounds.

但我们也看到，过于字面理解是一个危险的想法：有时这会让你过度思考问题，实际上，就像上次的数字游戏一样，最好的选择，获胜的选择，可能不会涉及那么多回合。

We also saw in that numbers game last time that in some games, but by no means all games, in some games this process actually converges to a single choice.

我们在上次的数字游戏中也看到，在某些游戏中，但绝不是所有游戏中，在某些游戏中，这个过程实际上收敛到单一选择。

In that numbers game it converged to 1.

在那个数字游戏中它收敛到 1。

So once again what this idea is--this idea is--you yourself should not play a dominated strategy.

因此，这个想法是什么——这个想法是——你自己不应该采取占优策略。

Delete those.

删除那些。

Delete those for everyone else, because everyone else is not going to play a dominated strategy.

为其他人删除这些，因为其他人不会采取主导策略。

Delete those.

删除那些。

Look at the game with all those dominated strategies deleted.

看看游戏中所有那些占主导地位的策略都被删除了。

See if there are any strategies that are now dominated.

看看现在有没有什么占主导地位的策略。

Delete those.

删除那些。

Look again, etc., etc. I could write that all up.

再看一遍，等等，等等。我可以把这些都写下来。

We saw it in practice last time.

我们上次在实践中看到了这一点。

It's probably just easier to have the idea there.

在那里可能更容易产生想法。

One tip about this, try to identify all the dominated strategies of all players before you delete, then delete.

关于这一点的一个提示是，在删除之前尝试识别所有玩家的所有主导策略，然后删除。

Then look again.

然后再看一下。

Try to identify all the dominated strategies of all players again, and then delete.

尝试再次识别所有玩家的所有占优策略，然后删除。

That process will prevent you from getting into trouble.

这个过程将防止您陷入麻烦。

So today, I want to be a little bit less abstract, if you like, and I want to look at an application.

所以今天，如果你愿意的话，我想讲得不那么抽象，我想看一个应用程序。

I want to look at a famous application from politics.

我想看一个著名的政治应用程序。

So we're going to look at a model of politics.

所以我们要研究一个政治模型。

The idea is going to be this.

这个想法是这样的。

We're going to imagine that there are two candidates, and what these candidates are doing is they're choosing their political positions for an election.

我们假设有两名候选人，这些候选人正在做的是他们正在为选举选择政治立场。

So these are the players, and the strategies are going to be - they're going to choose positions on a spectrum, on a political spectrum.

所以这些是参与者，策略将是——他们将在一定的政治范围内选择立场。

To make life easy, we're going to assume that this political spectrum has ten positions.

为了方便起见，我们假设这个政治光谱有十个立场。

So here are the positions.

这是职位。

We'll call them 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

我们将它们称为 1、2、3、4、5、6、7、8、9 和 10。

Very difficult.

非常困难。

What's the idea here?

这是什么想法？

We haven't finished describing the game.

我们还没有完成对游戏的描述。

What's the idea?

有什么想法吗？

The idea is that these positions are left wing positions and these positions are right wing positions.

这个想法是，这些位置是左翼位置，这些位置是右翼位置。

(Can people not see past the podium?

（难道人们看不到讲台后面的东西吗？

Let's get it out of the way.)

让我们把它移开。）

So you could think of these extreme left wing positions.

所以你可以想到这些极左翼立场。

These are people who, I guess, they don't eat anything except for fruit and they think that trees should have the vote.

我猜这些人除了水果之外不吃任何东西，他们认为树木应该有投票权。

These guys out here, these are the extreme right wing positions.

这里的这些人是极右翼立场。

So they think the poor shouldn't have the vote and they eat immigrants.

所以他们认为穷人不应该有投票权，他们吃移民。

I'm an immigrant: better be careful.

我是移民：最好小心点。

The candidates here are going to try and choose positions.

这里的候选人将尝试选择职位。

We're going to assume--this is not realistic--we're going to assume for now that there are 10% of the voters at each of these positions.

我们假设——这是不现实的——我们现在假设每个职位都有 10% 的选民。

So there's 10% of the voters at each position.

所以每个位置都有 10% 的选民。

So they are uniformly distributed.

所以它们是均匀分布的。

We're going to assume that voters will eventually vote for the closest candidate.

我们假设选民最终会投票给最接近的候选人。

So voters vote for the closest candidate: the candidate whose position is closest to their own.

因此，选民投票给最接近的候选人：立场最接近自己的候选人。

And we'll need a tiebreaking assumption and we'll do the obvious tiebreak.

我们需要一个决胜局假设，并且我们将进行明显的决胜局。

If there's a tie then the voters split.

如果出现平局，那么选民就会分裂。

The voters of that position split evenly.

支持这一立场的选民人数各占一半。

If there's a tie, half the voters at that position go for one of the candidates and half of them go for the other.

如果出现平局，则该位置的一半选民会支持其中一名候选人，另一半则支持另一名候选人。

So here's a game, I've got the players, that's the candidates.

这是一场游戏，我有球员，那就是候选人。

I've got the strategies, that's the political positions.

我有策略，那就是政治立场。

What am I missing?

我缺少什么？

I'm missing payoffs, right?

我错过了回报，对吗？

I'm missing payoffs.

我缺少回报。

It matters in this game a lot how we specify the payoffs but we're going to assume that the payoffs are that the candidates aim to maximize their share of the vote.

在这个游戏中，我们如何指定收益非常重要，但我们假设收益是候选人的目标是最大化他们的选票份额。

But that's not the only thing we could have assumed.

但这并不是我们唯一可以假设的事情。

We could have assumed that all they really care about is winning and that winning gave them a high payoff and that losing gave them nothing.

我们可以假设他们真正关心的是胜利，胜利给他们带来了高额回报，而失败则没有给他们带来任何好处。

I'm going to assume a little bit more and I'm going to assume that--I haven't spelled maximized right, max will do--they want to max their share of the vote.

我将假设更多一点，我将假设——我还没有拼写“最大化”，“最大”就可以了——他们想要最大化他们的选票份额。

The reason this isn't a terrible assumption is you could think that getting a higher share of the vote gives you a mandate.

这并不是一个可怕的假设，因为你可能会认为获得更高比例的选票会给你带来授权。

Or, if this is a primary election, a larger share of the vote gives you a bigger push for the next primary or whatever.

或者，如果这是一次初选，那么更大的选票份额会给你带来下一次初选或其他什么的更大推动力。

So we'll assume that they're trying to maximize their share of the vote.

因此，我们假设他们正在努力最大化自己的选票份额。

So we want to know what's going to happen in this game.

所以我们想知道这场比赛会发生什么。

It seems like a pretty natural, pretty important game.

这似乎是一个非常自然、非常重要的游戏。

Okay, so given what we've learned so far in the class, a natural first question to ask is: are any of the strategies here?

好的，鉴于我们到目前为止在课堂上学到的知识，自然要问的第一个问题是：这里有什么策略吗？

There are ten strategies 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, are any of the strategies dominated?

有十种策略1、2、3、4、5、6、7、8、9、10，有哪一种策略占优？

Are any strategies dominated?

是否有某种策略占主导地位？

Let's get some--Let me get my microphones up and ready a little bit.

让我们开始吧——让我把麦克风举起来，准备一下。

So any strategies dominated?

那么有什么策略占主导地位吗？

How about this gentleman in blue?

这位蓝衣绅士怎么样？

So stand up and shout, yeah.

所以站起来喊，是的。

Professor Ben Polak: Steven says 1 and 10.

本·波拉克教授：史蒂文说的是 1 和 10。

We'll come back to 10.

我们会回到10点。

You're right.

你是对的。

We'll come back to 10.

我们会回到10点。

So he says that position 1, strategy 1, choosing the most extreme left wing position is a dominated strategy.

所以他说位置1，策略1，选择最极端的左翼位置是占优策略。

What dominates it by the way?

到底是什么在主导它？

Steven, you want to shout it out?

史蒂文，你想大声喊出来吗？

Professor Ben Polak: 2.

本·波拉克教授：2。

So for example--So our conjecture here is that 2 dominates 1.

举例来说，我们的猜想是 2 支配 1。

That's the claim.

这就是主张。

Let's just be careful here, what does it mean to say that 2 dominates 1?

这里我们要小心一点，2 支配 1 是什么意思？

It means that choosing position 2 always gives me a higher share of the vote than choosing position 1, no matter where the other candidate positions herself.

这意味着选择位置 2 总是比选择位置 1 给我更高的选票份额，无论其他候选人的立场如何。

It does not mean 2 beats 1.

这并不意味着2拍1。

So let's take Steven's conjecture and see if it's true.

那么我们就来看看史蒂文的猜想是否正确。

So, in particular, let's start working out what share of the votes you'd get if you chose position 1 or position 2, against different positions the other guy can choose.

因此，特别是，让我们开始计算，如果您选择位置 1 或位置 2，相对于其他人可以选择的不同位置，您将获得多少选票份额。

So, for example, what we're going to do is we're going to test does 2 dominate 1?

例如，我们要做的是测试 2 是否支配 1？

While we're doing this we'll figure out how these payoffs work as well.

当我们这样做时，我们也会弄清楚这些回报是如何运作的。

Well, how about versus 1.

好吧，对比 1 怎么样？

So suppose the other candidate has chosen position 1.

因此，假设另一位候选人选择了位置 1。

So the other candidate has chosen position 1.

所以另一位候选人选择了位置 1。

Then we're going to compare my payoff from choosing 1 against the other candidate's payoff from choosing 1.

然后我们将比较我选择 1 的收益与其他候选人选择 1 的收益。

We're going to compare this with my payoff from choosing 2 against the other candidate choosing 1.

我们将把这个与我选择 2 和其他候选人选择 1 的收益进行比较。

Let's figure out what that is.

让我们弄清楚那是什么。

So what's my--somebody can shout this out--what's my share of the vote if I choose 1 and the other candidate chooses 1?

那么我的——有人可以大声喊出来——如果我选择 1 而其他候选人选择 1，我的选票份额是多少？

50%.

50%。

That was pretty easy, right?

那很容易，对吧？

It must be a tie for everybody, so I'll get 50% of the vote.

大家肯定都是平局，所以我会得到 50% 的选票。

What's my share of the vote if I choose 2 and the other candidate chooses 1?

如果我选择 2 而另一位候选人选择 1，我的选票份额是多少？

90%.

90%。

He will get or she will get all the voters at 1.

他将获得或她将获得 1 的所有选民。

I'll get everyone else.

我会去找其他人。

So I get 90%.

所以我得到了90%。

So in this case choosing 2 is better than choosing 1.

所以在这种情况下选择 2 比选择 1 更好。

But of course we're not done yet.

当然，我们还没有完成。

We have to consider other possible positions of my opponent.

我们必须考虑对手的其他可能的立场。

So suppose my opponent chooses 2.

所以假设我的对手选择2。

So now we're comparing my choosing 1, when my opponent chooses 2, and the payoff I would get if I choose 2 when my opponent chooses 2.

所以现在我们比较我选择 1，当我的对手选择 2 时，以及当我的对手选择 2 时我选择 2 时我会得到的回报。

So what's my payoff if I choose 1 and my opponent chooses 2?

那么如果我选择 1 而我的对手选择 2，我的收益是多少？

I get all the voters right on top of me at position 1 and she gets everyone else, is that right?

我让所有选民都在我的位置 1 之上，而她则让其他人都获得，是这样吗？

So I get 10%.

所以我得到10%。

And what about if we both choose 2?

如果我们都选择2呢？

50%, everybody happy with that?

50%，大家满意吗？

So 50% in this case, and once again, 2 did better than 1.

所以在这种情况下是 50%，而且 2 的表现再一次比 1 好。

Everyone happy with that?

大家对此满意吗？

So we showed this choice was better and this choice was better, and we're going to keep on going.

所以我们证明了这个选择更好，这个选择更好，我们将继续前进。

Against 3: so now we're comparing my choosing 1 versus 3 and my choosing 2 versus 3.

与 3：所以现在我们正在比较我选择 1 与 3 以及我选择 2 与 3。

If I choose 1 against 3, what share of the vote do I get?

如果我选择 1 反对 3，我会得到多少票？

I'm going to get all the people at position 1 and half the people at position 2 for a total of 15.

我将收集位置 1 的所有人员和位置 2 的一半人员，总共 15 人。

And if I chose 2 against 3, what do I get?

如果我选择 2 对 3，我会得到什么？

I get 20, I get all the people at 1 and I get all the people 2, so I get 20%, so we're okay again.

我得到 20，我得到所有的人 1，我得到所有的人 2，所以我得到 20%，所以我们又好了。

Let's do one more just to see a pattern here.

让我们再做一次来看看这里的模式。

So if I choose 1 against 4, and comparing my choosing 2 against 4.

因此，如果我选择 1 和 4，然后比较我选择的 2 和 4。

In the former case if I choose 1 against 4, how many votes do I get?

在前一种情况下，如果我选 1 选 4，我会得到多少票？

I get all the people at 1, and all the people at 2, so I get 20% of the votes, she gets everyone else.

我得到了所有人的 1 票，所有的人都得到了 2 的票，所以我得到了 20% 的选票，她得到了其他人的选票。

And here if I choose 2 against 4, I get all the people at 1, all the people at 2, and what?

在这里，如果我选择 2 对 4，我会得到所有的人都为 1，所有的人都为 2，然后呢？

Half the people at 3, so that comes out as 25% and once again and so on.

一半的人在 3 岁，所以结果是 25%，再次依此类推。

Now, I could go on right the way through here, but I'm going to stop because it gets to be a bit boring after a while.

现在，我可以继续从这里开始，但我要停下来，因为过了一会儿就会变得有点无聊。

I'm guessing you can see a pattern emerging here.

我猜你可以看到这里出现的一种模式。

What's the pattern here?

这里的模式是什么？

What's the pattern here?

这里的模式是什么？

Somebody?

有人吗？

Where's that?

那是哪里？

Somebody raise their hand so we can get a micke out there.

有人举手，这样我们就可以得到一个麦克风。

Can we get a mike on this guy in white?

我们可以给这个白衣人拿个麦克风吗？

Stand up, stand up, yep, stand up and shout.

站起来，站起来，是的，站起来喊。

Professor Ben Polak: 2 is always better than 1, but we can see more than that in the pattern.

Ben Polak 教授：2 总是比 1 好，但我们可以在模式中看到更多。

That's right, but what's the pattern here?

没错，但是这里的模式是什么？

Mike back here.

迈克回到这里。

Student: Whatever's closer to 5 is better than whatever's farther away from 5.

学生：越接近 5 的越好，越远离 5 的越好。

Professor Ben Polak: That's not quite the pattern I'm looking for.

Ben Polak 教授：这并不是我正在寻找的模式。

Let's look at the numbers here.

让我们看看这里的数字。

We had 15, 20, then this one, up to 20, 25 and so on.

我们有 15 个、20 个，然后是这个，一直到 20 个、25 个，依此类推。

So what's the pattern here?

那么这里的模式是什么呢？

Yes, Steven again.

是的，又是史蒂文。

Student: If your opponent doesn't choose 1 or 2, then you're always going to be 5% better off if you choose 2 than 1.

学生：如果你的对手不选择 1 或 2，那么如果你选择 2，你总是比 1 好 5%。

Professor Ben Polak: Exactly, so ignoring these first two positions which were a bit weird, choosing 2 always gave me 5% more votes than choosing 1, regardless of what the other person chooses.

Ben Polak 教授：没错，所以忽略前两个有点奇怪的位置，选择 2 总是比选择 1 给我多 5% 的选票，无论其他人选择什么。

So I mean if you want, you can fill in the next--whatever it is--the next 6 positions and see.

所以我的意思是，如果你愿意，你可以填写下一个——无论它是什么——接下来的 6 个位置，然后看看。

But you'll see that very clearly that Steven's right: that choosing 2 will always get me 5% more of the votes than choosing 1 from here on down.

但你会非常清楚地看到史蒂文是对的：从现在开始，选择 2 总是比选择 1 多获得 5% 的选票。

And in fact, they'll go up in intervals of 5.

事实上，它们会以 5 为间隔上涨。

So, in fact, it is true, it is in fact true, so we can conclude that 2 dominates--in fact, we can do a bit better than that--we can say strictly dominates 1.

所以，事实上，这是真的，事实上也是真的，所以我们可以得出结论，2 占主导地位——事实上，我们可以做得更好一点——我们可以说严格占主导地位 1。

2 strictly dominates 1.

2严格支配1。

Is anything else dominated here, Steven already told us that 10 is dominated.

还有其他什么在这里占主导地位吗，史蒂文已经告诉我们，10 是占主导地位的。

I'm not going to go through that argument.

我不打算讨论这个争论。

Can everyone see that is symmetric?

大家能看出是对称的吗？

Hang on a second, everyone see that 10 is going to be dominated by 9 in exactly the same way?

等一下，大家看到 10 会以同样的方式被 9 支配吗？

Is that obvious?

这很明显吗？

It's just really the same argument coming in the other direction.

这实际上是同样的论点从另一个方向而来。

So similarly, 9 strictly dominates 10.

类似地，9 严格支配 10。

And again, we're not saying choosing 2 beats choosing 1, 2 wins against 1 or 9 wins against 10.

再说一次，我们并不是说选择 2 节拍选择 1、2 胜 1 或 9 胜 10。

Wwe're saying choosing 2 always does better than choosing 1, regardless of what the other person chooses.

Wwe 说选择 2 总是比选择 1 更好，无论其他人选择什么。

Choosing 9 always is better than choosing 10 regardless of what the other person chooses.

无论别人选择什么，选择 9 总是比选择 10 好。

There was a question, are we okay?

有人问，我们还好吗？

You want to get the mike in for the question?

你想用麦克风来回答问题吗？

Stand and shout out to everyone.

站起来向大家喊话。

Student: I think you may have defined it already, but I couldn't find a definition of strictly in the book we have.

学生：我想你可能已经定义了它，但是我在我们的书中找不到严格的定义。

Professor Ben Polak: Okay, well you have one on the handout.

本·波拉克教授：好的，讲义上有一份。

I think there probably is one in there somewhere, but just in case, we have had two in class, there's one on the handout.

我想那里可能有一个，但为了以防万一，我们课堂上有两个，讲义上也有一个。

All right we've concluded that 1 is dominated by 2, and 10 is dominated by 9.

好吧，我们已经得出结论：1 被 2 支配，10 被 9 支配。

Is anything else dominated here?

这里还有什么东西占主导地位吗？

Well what about thinking about whether 2 is dominated by 3, after all--I've forgotten your name--what's your name there?

好吧，考虑一下 2 是否被 3 支配怎么样，毕竟——我忘记了你的名字——你在那里叫什么名字？

Yes?

是的？

Professor Ben Polak: Sudiptha had suggested earlier on, you want to be close to 5.

Ben Polak 教授：Sudiptha 早些时候曾建议，你希望接近 5。

So 3 is closer to 5 than 2.

所以 3 比 2 更接近 5。

Is 2 dominated by 3?

2 是否被 3 支配？

Anyone else?

还有其他人吗？

Any other hands?

还有其他手吗？

There's a hand here.

这里有一只手。

Can you get this lady here?

你能把这位女士叫到这里来吗？

Student: Once you delete the dominated strategies, then you kind of go through it again and then 2 is dominated by 3.

学生：一旦你删除了占主导地位的策略，那么你就可以再回顾一遍，然后2就被3占主导地位了。

Professor Ben Polak: Good, so you're getting ahead of us, good.

Ben Polak 教授：很好，所以你领先于我们了，很好。

So we'll get there, hold that thought.

所以我们会到达那里，保持这个想法。

But what about right now before we delete anything?

但是在我们删除任何内容之前现在该怎么办？

So let's check it out.

那么让我们来看看吧。

Let's check it out.

我们来看看吧。

So, in particular, let's just try this process.

所以，特别是，让我们尝试一下这个过程。

So how about my payoff versus 1, for example.

例如，我的收益与 1 相比怎么样？

So 2 against 1 gives me 90% of the votes, 3 against 1 gives me what?

那么 2 对 1 给了我 90% 的选票，3 对 1 给了我什么？

85% of the votes which is actually lower, so it's clear that 3 doesn't dominate 2.

85% 的得票率实际上更低，所以很明显 3 并不压倒 2。

In particular, if the other candidate were to choose position 1, I would get a higher share of the vote choosing 2 than I would have done if I had chosen 3.

特别是，如果其他候选人选择位置 1，我将获得比选择位置 3 更高的选票份额。

So, in fact, it's not the case that 2 is dominated by 3.

所以，其实并不是2被3支配的情况。

But the woman who's name I either didn't get or forgot.

但那个女人的名字我要么不知道，要么忘记了。

Professor Ben Polak: So Christine has pointed something out.

本·波拉克教授：克里斯汀指出了一些事情。

So Christine, why don't you wait for the mike and point that out again.

那么克里斯汀，你为什么不等一下麦克风并再次指出这一点呢？

So we know that 2 is not dominated, and particularly not dominated by 3, but …

所以我们知道 2 不被支配，特别是不被 3 支配，但是……

what's the "but"?

什么是“但是”？

Student: When you delete the dominated strategy of 2 dominating 1, or 1 being dominated, when you delete that and 10, then it is.

学生：当你删除2支配1的支配策略，或者1被支配时，当你删除那个和10时，那就是了。

Professor Ben Polak: So what Christine is arguing is, even though it's the case that 2 is not a dominated strategy, if we do the process of iterative deletion of dominated strategies and we delete the dominated strategies, then maybe we should look again and see if it's dominated now.

Ben Polak教授：所以Christine争论的是，即使2不是占优策略，如果我们进行迭代删除占优策略的过程，并且我们删除了占优策略，那么也许我们应该再看看它现在是否是占优。

So let's be careful here, we're not saying that we're going to delete the voters at 1, or delete the voters at 10 (though we might wish to).

所以我们在这里要小心，我们并不是说我们要删除 1 处的选民，或者删除 10 处的选民（尽管我们可能希望这样做）。

We're just saying we want--What we're saying is, we know that the candidates aren't going to position themselves at 1 and aren't going to position themselves at 10.

我们只是说我们想要——我们想说的是，我们知道候选人不会将自己定位在 1 也不会将自己定位在 10。

So in some sense we're deleting those strategies.

因此，从某种意义上说，我们正在删除这些策略。

The voters are still there.

选民还在那里。

So let's try that.

那么让我们尝试一下吧。

So if we--there's a but here I guess--if we delete the strategies 1 and 10, which were dominated, then does 3 dominate 2?

因此，如果我们——我猜这里有一个但是——如果我们删除占主导地位的策略 1 和 10，那么 3 是否会主导 2？

Once again, we can try it out.

再次，我们可以尝试一下。

So let's try it out.

那么我们来尝试一下吧。

So the payoff from choosing 1, let's start against 2.

所以选择 1 的回报，让我们从 2 开始。

So the payoff from choosing 2 against 2 is 50%.

所以选择 2 对抗 2 的回报是 50%。

The payoff from choosing 3 against 2 is what?

选择 3 而不是 2 的回报是多少？

What's the payoff from choosing 3?

选择3的回报是什么？

Someone can shout out.

有人可以喊出来。

What's the payoff from choosing 3 against 2?

选择 3 而不是 2 的回报是什么？

80%: the person at 2 is going to get all the ones at 1, and all the ones at 2; and 3 will get everything else.

80%：2的人会得到1的所有东西，2的人会得到所有的东西； 3将得到其他一切。

So this is 80% versus 3.

所以这是 80% 对 3。

The payoff of 2 against 3 is all the people at 1, all the people at 2, so it's 20%.

2 对 3 的收益是所有处于 1 的人，以及所有处于 2 的人，所以是 20%。

The payoff from choosing 3 against 3, is of course 50%, so, so far so good.

选择 3 对 3 的回报当然是 50%，所以，到目前为止还不错。

So far choosing 3 is better.

目前来看选择3比较好。

Against 4, the payoff from choosing 2 against 4 is what?

针对 4，选择 2 对抗 4 的收益是多少？

What's the payoff from choosing 2 against 4?

选择 2 而不是 4 的回报是什么？

Again, all the people at 1, all the people at 2, and half the people at 3, so that's 25%.

同样，所有的人都为 1，所有的人都为 2，一半的人都为 3，所以这是 25%。

The payoff from choosing 3 against 4 is going to be all the people at 1, all the people at 2, and all the people at 3, for a total of 30%, which again is bigger.

选择 3 而不是 4 的回报将是所有处于 1 的人、所有处于 2 的人以及所有处于 3 的人，总共 30%，这又更大。

Let's do one more.

我们再做一个吧。

Versus 5, the payoff from choosing 2--(I can afford to raise this a bit)--against 5, is all the people at 1, all the people at 2, and all the people at 3; 30%.

相对于 5，选择 2（我可以稍微提高一点）相对于 5 的回报是所有处于 1 的人、所有处于 2 的人以及所有处于 3 的人； 30%。

Whereas the payoff from choosing 3 against 5 is all the people at 1, all the people at 2, all the people at 3 and half the people at 4, 35%.

而选择 3 和 5 的回报是全体人员为 1，全体人员为 2，全体人员为 3，半数人员为 4，即 35%。

And again, you can see exactly the same pattern is going to emerge here.

再次，您可以看到这里将出现完全相同的模式。

From here on down, if I bothered to do it, we'd find that choosing 3 always gets me 5% more of the vote than I would have had from choosing 2, against any of these higher numbers.

从现在开始，如果我愿意这样做，我们会发现选择 3 总是比选择 2 获得的选票多 5%，与任何更高的数字相比。

The same pattern Steven pointed out before.

史蒂文之前指出过同样的模式。

So, I've forgotten your name again, I'm sorry.

所以，我又忘记了你的名字，抱歉。

Professor Ben Polak: So Christine is correct in saying that once we delete the strategies 1 and 10--once we realize that those positions are not going to be chosen by our sophisticated candidates--then we realize that probably choosing 2 (and of course 9) isn't a good idea either.

Ben Polak 教授：克里斯汀的说法是正确的，一旦我们删除策略 1 和 10，一旦我们意识到我们成熟的候选人不会选择这些职位，那么我们就会意识到选择 2（当然还有 9）可能也不是一个好主意。

So 2 and 9 are not dominated but they are dominated once we realize 1 and 10 won't be played, or won't be chosen.

因此，2 和 9 不是占优的，但一旦我们意识到 1 和 10 不会被播放或不会被选择，它们就会被占优。

Okay, so where are we going here?

好吧，那么我们要去哪里呢？

Where are we going?

我们要去哪里？

Where is this discussion going to end up?

这场讨论最终会走向何方？

Let's come back to our original game.

让我们回到原来的游戏。

We've argued that 1 and 10 shouldn't be chosen because they're dominated.

我们认为不应该选择 1 和 10，因为它们是占优的。

We've argued that once we realize those aren't going to be played, that 2 and 9 aren't going to be played.

我们认为，一旦我们意识到这些不会被播放，那么 2 和 9 也不会被播放。

What's going to end up being played?

最后会演什么？

Well let's just go slower.

好吧，让我们走慢一点。

So we were able to dominate 1 and 10 in the first round, so we know this won't be played, and we know this won't be played.

所以我们能够在第一轮中统治1号和10号，所以我们知道这不会被打，我们也知道这不会被打。

And remember the voters are still there: we're just deleting the strategies.

请记住，选民仍然存在：我们只是删除策略。

Then we dominated–Then we ruled out--(I'll put double crosses for "in the second round")--2 and 9.

然后我们占主导地位——然后我们排除了——（我会在“第二轮”中打上双叉）——2和9。

Then we ruled out 3 and 8, and then we ruled out, we would have done 4 and 7 and that leaves just 5 and 6.

然后我们排除了 3 和 8，然后我们排除了，我们本来可以做 4 和 7，只剩下 5 和 6。

So if we do the procedure of iteratively deleting dominated strategies, going back again, looking what's dominated, doing it again, doing it again …

因此，如果我们执行迭代删除主导策略的过程，再次返回，查看主导策略，再次执行，再次执行......

all that's left is 5 and 6.

剩下的就是5和6了。

So this procedure leads us to conclude that the candidates will choose positions 5 and 6.

因此，这个过程使我们得出结论，候选人将选择位置 5 和 6。

Does 5 dominate 6, or 6 dominate 5, or anything like that?

是 5 支配 6，还是 6 支配 5，还是类似的情况？

No, at that point it's just a tie, right?

不，到那时就只是平局了，对吧？

So in the end, the procedure, so iterative deletion--(let's write " it del ")--leads to deleting all except 5 and 6.

所以最后，这个过程，所以迭代删除——（让我们写“ it del ”）——导致删除除 5 和 6 之外的所有内容。

So this was just a simple exercise, I think.

所以我认为这只是一个简单的练习。

I mean I'm hoping that was a simple exercise.

我的意思是我希望这是一个简单的练习。

Let me just look out there, is everyone following that okay?

我看看大家都在关注吗？

That really wasn't meant to be hard, but what's relevant about this exercise is not the fact that it's hard.

这确实并不意味着很难，但与此练习相关的并不是它很难这一事实。

In fact, it isn't hard.

事实上，这并不难。

What's relevant is the fact that it's relevant.

相关的是它相关的事实。

It's about the real world.

这是关于现实世界的。

In particular, this is a famous model in political science.

尤其是，这是政治学中著名的模型。

Anyone know the name of this model?

有谁知道这个型号的名字吗？

What's the prediction here?

这里的预测是什么？

The prediction here is that the candidates are going to be squeezed towards the middle.

这里的预测是候选人将被挤向中间。

They're going to choose positions very close to each other and very close to the center.

他们将选择彼此非常接近并且非常接近中心的位置。

What's that called?

那叫什么？

Anyone know?

有人知道吗？

Hold the thought on that.

保持这个想法。

Someone at the back, shout out.

后面有人喊道。

This is the " Median Voter Theorem "; thank you.

这就是“中位选民定理”；谢谢。

So let's have a new board, this one will do fine.

因此，让我们拥有一块新板，这个就可以了。

So the prediction is candidates crowd the center and this is called, in Political Science, the Median Voter Theorem.

因此，预测候选人会聚集在中间，这在政治学中被称为中位选民定理。

It's called the Median Voter Theorem because the voters at the center, in this case 5 and 6, actually get to decide not just the election, but to decide therefore, what policies are put in place.

这被称为中位选民定理，因为中间的选民（在本例中为 5 号和 6 号）实际上不仅可以决定选举，还可以决定制定哪些政策。

How does this do as a prediction of the real world?

作为对现实世界的预测，这有何作用？

Well there are examples in American History in which the Median Voter Theorem looks pretty good.

美国历史上有一些例子，中位选民定理看起来相当不错。

They're pretty old examples.

它们是很古老的例子。

They're going to be before you guys were born, but let me mention them anyway.

它们会在你们出生之前出现，但无论如何让我提一下它们。

So I think a lot of people--if you go back today and you look at tapes of the Kennedy Nixon Election in 1960 (some American help me out here, '60 right?)--I think it was '60.

所以我想很多人——如果你今天回去看 1960 年肯尼迪尼克松选举的磁带（一些美国人在这里帮助我，60 年吧？）——我认为那是 60 年。

So in the 1960's actually, between Kennedy and Nixon, if you ever go over and look at those tapes, it's amazing how conservative Kennedy looks when you think about his reputation today.

事实上，在 1960 年代，在肯尼迪和尼克松之间，如果你仔细看一下那些录音带，当你想到肯尼迪今天的声誉时，你会发现他的保守态度令人惊讶。

They really were crowding the center.

他们确实挤满了中心。

And eight years later Nixon seemed to have learned his lesson.

八年后，尼克松似乎已经吸取了教训。

If you look at the '68 election, which Nixon won, Nixon sounds, on those tapes of his policies being put forth in that election, he looks extremely liberal for a Republican.

如果你看看尼克松赢得的 68 年选举，尼克松听起来，在他在那次选举中提出的政策的录音带上，你会发现他对于共和党人来说是极其自由派的。

So again, they were crowding that center space.

如此一来，他们又挤满了中心空间。

Some people argue that that's exactly what Clinton did in '92.

有些人认为这正是克林顿在 92 年所做的。

He took the Democratic Party to the right, i.e., towards the center in order to pick up those central voters and win.

他将民主党推向右翼，即走向中间，以吸引中间选民并获胜。

So there are examples in American History of elections that seem to--in which the Median Voter Theorem seems to do well.

因此，美国选举史上似乎有一些例子，其中中位选民定理似乎表现良好。

We'll come back and discuss it more in a second, but before I do, let me mention that the same model, exactly the same idea, has an application in Economics.

我们稍后会回来详细讨论这个问题，但在此之前，让我提一下，相同的模型、完全相同的想法在经济学中有应用。

Somebody mentioned it already over there.

那边已经有人提到了。

The application in Economics is to do with product placement, something that Jake's working on.

经济学中的应用与植入式广告有关，这是杰克正在研究的内容。

In product placement, you might think when you're placing a gas station, you might think it would be nice if gas stations spread themselves evenly out over the town or out over the road So wherever you happen to be, when you run out gas, there was a gas station close by.

在植入式广告中，您可能会认为，当您放置加油站时，您可能会认为，如果加油站均匀地分布在城镇或道路上，那就太好了。因此，无论您身在何处，当您用完汽油时，附近都会有一个加油站。

That would be convenient.

这样就方便了。

But unfortunately, gas stations, as we all know, they tend to crowd into the same corners, they tend to be on the same road junctions, right?

但不幸的是，众所周知，加油站往往挤在同一个角落，它们往往位于同一个路口，对吧？

Why?

为什么？

Because they're trying to, they're competing for voters who happen to be close, or running out of gas at those moments, and by crowding together they avoid being out competed by each other in terms of position.

因为他们正在努力，所以他们正在争夺那些恰好接近的选民，或者在那些时刻耗尽了汽油，并且通过拥挤在一起，他们避免在位置上相互竞争。

I said that a bit quickly, but we'll come back and look at that idea in more detail later on in the class.

我说得有点快，但我们稍后会在课堂上更详细地讨论这个想法。

So in politics, this is about candidates crowding close together towards the center, to try and get as many voters who are close to them.

因此，在政治上，候选人们紧密地向中间派靠拢，试图获得尽可能多的与他们关系密切的选民。

And in Economics, this is about firm's crowding together to try and get shoppers who are close to them.

在经济学中，这是关于公司聚集在一起试图吸引与他们关系密切的购物者。

The people associated with this are a guy called Anthony Downs, who did this in Political Science, in a book in 1957.

与此相关的人是一个叫安东尼·唐斯 (Anthony Downs) 的人，他在 1957 年的一本《政治学》书中做了这件事。

And in Economics, a guy called Hotelling, who wrote a paper about this in 1929.

在经济学领域，一个叫霍特林的人于 1929 年写了一篇关于这个问题的论文。

In Political Science, they call this "simultaneous discovery." I think 1929 is a bit earlier, but never mind.

在政治学中，他们称之为“同时发现”。我认为 1929 年有点早，但没关系。

Now, what I want to do now, what I want to is, periodically in the class and we have a model up there and we've got to analyze it.

现在，我现在想做的是，定期在课堂上做，我们在那里有一个模型，我们必须分析它。

We've used a bit of Game Theory to analyze it.

我们使用了一些博弈论来分析它。

I want us to talk about whether we think it's right.

我想让我们谈谈我们认为这是否正确。

What do we think of the strengths and weaknesses of this model?

我们如何看待这个模型的优点和缺点？

So what I want you to start thinking about is: do you believe this model?

所以我想让你开始思考的是：你相信这个模型吗？

Is this a good model of politics?

这是一个好的政治模式吗？

What's it missing?

它缺少什么？

Let me come off the stage a second, so I can encourage you to get involved in the discussion.

让我下台一下，这样我可以鼓励你们参与讨论。

How many of you are Poly Sci majors?

你们中有多少人是理工科专业的？

Raise your hands you Poly Sci majors.

理工科专业的同学请举手。

By the way, how many of you are Poly Sci majors who have seen this model before?

顺便问一下，你们中有多少理工科专业的人见过这个模型？

Some of you have seen it before.

你们中有些人以前见过它。

So what's this model missing?

那么这个模型缺少什么呢？

You all live in America.

你们都住在美国。

You all live in a democracy.

你们都生活在民主国家。

You don't all live in America, but a lot of you live in a democracy.

你们并不都生活在美国，但很多人生活在民主国家。

Do you have some idea about this?

你对此有什么想法吗？

So what are we missing here?

那么我们在这里缺少什么？

So can we get the woman here Ale?

那么我们能把那个女人叫到这里来吗？

Student: The voters are not evenly distributed 10%, 10%, 10%.

学生：选民分布不均10%、10%、10%。

Professor Ben Polak: So one thing that seems odd about the way we set up this model is that the voters are not evenly distributed.

本·波拉克教授：因此，我们建立这个模型的方式似乎奇怪的一件事是选民分布不均匀。

Let me go and put that up--it'll be tiring but let me go to and fro a bit.

让我去把它放上去——虽然会很累，但让我来回走动一下。

So one thing that's missing, or wrong if you like, but the one thing is the voters are not evenly distributed in the real world.

所以，如果你愿意的话，有一件事是缺失的，或者是错误的，但一件事是选民在现实世界中的分布并不均匀。

Good, anything else?

很好，还有别的吗？

Good we'll talk about each of these in turn.

很好，我们将依次讨论这些内容。

Can I get the guy over here?

我可以叫那家伙过来吗？

Sorry I'm not making this easy for Jude, let me just stay down here.

抱歉，我不会让裘德轻松，让我留在这儿吧。

I'll collect a few and then go back to the board.

我会收集一些，然后回到董事会。

Student: The model doesn't take into account differences between the groups in voter turnout, based on the candidates.

学生：该模型没有根据候选人考虑不同群体之间选民投票率的差异。

Professor Ben Polak: Good, so there's an issue of turnout.

本·波拉克教授：好的，所以存在投票率问题。

We assumed here that everyone votes, is that right?

我们在这里假设每个人都投票，对吗？

But in fact, not everyone votes, there's always the option of not voting.

但事实上，并不是每个人都投票，总有不投票的选择。

Remind me to put that one up, "not voting," you'll have to remind me of these.

提醒我把那句话“不投票”，你必须提醒我这些。

How about the gentleman here in the yellow.

这里那位穿黄色衣服的绅士怎么样？

Student: There's a difference between the primary and general election.

学生：初选和大选是有区别的。

Professor Ben Polak: Good, so we assumed, we kind of jumped to just one election and at least in the American system, for better or for worse--I could voice an opinion on that--there are lots of little elections on the way.

本·波拉克教授：很好，所以我们假设，我们有点跳到只有一次选举，至少在美国的体系中，无论好坏——我可以对此发表意见——即将举行很多小型选举。

There are primaries in all these states that you never, you probably have heard of, states you probably have heard of along the way.

在所有这些州中都有一些你从未听说过的初选，或者你可能一路上听说过的州。

Anything else?

还要别的吗？

Can we get the--Where's the other mike?

我们能找到——另一个麦克风在哪里吗？

Can we get the woman in green here?

我们能把那个绿衣女人叫来吗？

Student: American voters often vote based on character rather than position.

学生：美国选民通常根据性格而不是立场来投票。

Professor Ben Polak: So there's more than just positions.

Ben Polak 教授：所以不仅仅是职位。

Another way of saying that, stop me if this is wrong, but one way to say that is we've been assuming that politics is one dimensional, left/right and there might be other things involved: character or even among issues, even among political issues it could be more than one dimension.

另一种说法是，如果这是错误的，请阻止我，但一种说法是，我们一直假设政治是一个维度，左/右，可能还涉及其他因素：性格，甚至问题之间，甚至政治问题之间也可能不止一个维度。

It could be that there's a dimension about the environment and a dimension about foreign policy, and maybe another dimension about redistributive politics.

可能有一个关于环境的维度和一个关于外交政策的维度，也许还有另一个关于再分配政治的维度。

So that's an important idea.

所以这是一个重要的想法。

Let me get a couple more, can we get?

让我再拿几个，可以吗？

Student: The model might not apply with more than two candidates.

学生：该模型可能不适用于两名以上的候选人。

Professor Ben Polak: So another issue here is, I just assumed that there were two candidates.

本·波拉克教授：所以这里的另一个问题是，我只是假设有两个候选人。

We know in some elections there was third candidate.

我们知道在一些选举中存在第三位候选人。

For example, there was Nader in the election in--well I guess he was there in the other one as well--but he was particularly there in 2000.

例如，纳德参加了 2000 年的选举——我猜他也参加了另一场选举——但他的表现尤其突出。

So, by the way, I don't mean this as a joke, in some sense there's always a third candidate and that comes from the problem we had earlier, because not voting is always an option.

所以，顺便说一句，我并不是说这是一个笑话，从某种意义上说，总是有第三个候选人，这来自我们之前遇到的问题，因为不投票总是一种选择。

So in principle, in that election in 2000, there was at least four candidates, as it were, four things you could do with your vote.

因此，原则上，在 2000 年的那次选举中，至少有四名候选人，可以说，你可以用选票做四件事。

You could vote for, as it was in that time, Gore, Bush, Nader, or not voting--and I'll leave it to you to decide which of those is equivalent to not voting.

你可以像当时那样投票给戈尔、布什、纳德，也可以不投票——我将让你来决定哪一个相当于不投票。

Anyone else?

还有其他人吗？

One other thing I think that's really, maybe it's a small thing, there's one other glaring thing here.

我认为另一件事确实是，也许这是一件小事，这里还有另一件引人注目的事情。

Someone we haven't had yet, Ale help me out here.

我们还没有找到一个人，艾尔来帮我吧。

Let me get this guy here.

让我把这个人带到这里来。

Student: Extrapolating from the fact that there are different things which people take into account, not everyone closest to 6 might actually vote for 6, it might be coincidence.

学生：从人们考虑的因素不同这一事实推断，并不是所有最接近 6 的人实际上都会投票给 6，这可能是巧合。

Professor Ben Polak: That's true, although that may be a question of dimension.

Ben Polak 教授：确实如此，尽管这可能是一个维度问题。

I had a different thing in mind.

我有不同的想法。

Let me say it.

让我说一下。

So I think this is important.

所以我认为这很重要。

A candidate could say during an election that "I'm a moderate candidate, I'm at position 5" but you might not believe him or her.

候选人可能在选举期间说“我是一名温和派候选人，我处于第五位”，但你可能不相信他或她。

You've got the candidates track record, unless that candidate can commit to policies at position 5, you might know full well that the candidate is in fact a position 10 or a position 1 candidate, is that right?

你已经掌握了候选人的往绩记录，除非该候选人能够承诺执行排名 5 的政策，否则你可能完全清楚该候选人实际上是排名 10 或排名 1 的候选人，对吗？

So unless the candidate can actually commit to a position, there may be a problem in them simply announcing, "my position is moderate."

因此，除非候选人能够真正承诺某个职位，否则他们简单地宣布“我的立场是温和的”可能会出现问题。

So again, let's think of an example here, a couple of examples, in the Bush Gore election, we saw President Bush, candidate Bush as was, saying "I'm not a right wing candidate, I'm a compassionate conservative." And you guys believed him.

所以，让我们再想一个例子，几个例子，在布什·戈尔的选举中，我们看到布什总统，候选人布什，说“我不是右翼候选人，我是一个富有同情心的保守派。”你们相信他。

I don't know, maybe he was, I don't know.

我不知道，也许他是，我不知道。

His track record in Texas probably did actually look a lot like that.

他在德克萨斯州的记录实际上可能确实很像那样。

Or today, if you look at both primaries going on now, you're seeing various candidates on both the Republican side and the Democratic side, seek to position themselves.

或者今天，如果你看看现在正在进行的两场初选，你会看到共和党和民主党方面的各种候选人都在寻求自己的立场。

But as you all know, those candidates almost all have track records, and not everybody believes that Clinton, that Hilary Clinton, is quite so centrist as she now seems.

但众所周知，这些候选人几乎都有过往记录，并不是每个人都相信克林顿，即希拉里·克林顿，像她现在看起来那么中间派。

And not everyone believes that the former Governor of Massachusetts is quite so conservative as he now seems.

并不是所有人都认为这位前马萨诸塞州州长像现在这样保守。

So it's not clear that you can just choose your position willy nilly.

所以不清楚你是否可以随意选择你的立场。

So let's put some of these up.

让我们把其中一些放出来。

So I'm going to forget remember, so you're going to have to help me out a bit.

所以我会忘记记住，所以你必须帮我一点忙。

So we have many candidates, more than just two; or we have not voting, which is related to that actually; and we have choosing your position, the inability to commit to a position.

所以我们有很多候选人，不仅仅是两个；或者我们没有投票，这实际上与此有关；我们选择了你的立场，无法承诺一个立场。

So your position may be not believed, and I want to claim that that's connected to the idea that you can't commit yourself to your policies.

所以你的立场可能不被相信，我想说这与你不能遵守你的政策的想法有关。

We had some others.

我们还有其他一些。

We had primaries; and we had other dimensions, so we had higher dimensions.

我们进行了初选；我们有其他维度，所以我们有更高的维度。

Have I got the main ones now?

我现在已经拿到主要的了吗？

I think I got most of the main ones.

我想我已经掌握了大部分主要内容。

So there's lots of things--when we look at this model--there's lots of things that are missing.

因此，当我们查看这个模型时，有很多东西缺失。

A lot of things are not seeming to be captured by it.

很多东西似乎都没有被它捕捉到。

And here's where I want to get a little bit more religious, if you like.

如果你愿意的话，我想在这里变得更加宗教化一点。

It's tempting to say, look, we wrote up this model.

人们很容易会说，看，我们编写了这个模型。

It's missing all this stuff, so it must be useless.

它缺少所有这些东西，所以它一定是无用的。

You know we shouldn't model it.

你知道我们不应该对其进行建模。

You might even go further, you might say modeling it is always going to miss stuff out, so let's just not bother.

您甚至可能会更进一步，您可能会说建模总是会错过一些东西，所以我们不要打扰。

That, I think, is the wrong conclusion here.

我认为这是错误的结论。

The right conclusion, I think, is: the reason we write down these models is to try and capture and test our intuitions.

我认为正确的结论是：我们写下这些模型的原因是尝试捕捉和测试我们的直觉。

In this case, the intuition about crowding towards the center to get votes.

在这种情况下，是为了获得选票而挤向中间的直觉。

But, of course, these models abstract from very important parts of reality, and the next step to do is what?

但是，当然，这些模型是从现实中非常重要的部分抽象出来的，下一步要做的是什么？

Is to say, okay now let's try to enrich the model, to add more into the model, and see if you get a different result, and if so why?

也就是说，现在让我们尝试丰富模型，向模型中添加更多内容，看看是否会得到不同的结果，如果是，为什么？

So you start with your basic model, then you add in, you enrich the model, and you see if the results change, and that'll help you explain why you're getting different results in different settings.

因此，您从基本模型开始，然后添加内容，丰富模型，然后查看结果是否发生变化，这将帮助您解释为什么在不同的设置下会得到不同的结果。

So the sort of religious part of today's class is models are always abstractions.

因此，当今课堂的宗教部分是模型始终是抽象的。

You want to use them to see what's missing, and then add things back in, to see if they make a difference, and if so how.

您想使用它们来查看缺少的内容，然后将内容添加回来，看看它们是否会产生影响，如果有的话，如何影响。

Let's go through these a little bit.

让我们稍微回顾一下这些。

So it turns out that the very first one, voters are not evenly distributed is certainly true, it's undoubtedly true.

所以事实证明，第一个，选民分布不均肯定是真的，毫无疑问是真的。

But it turns out that if you make the voters distributed more realistically, let's say like on a bell-curve shape, it makes actually no difference to the result at all.

但事实证明，如果你让选民的分布更加现实，比如说像钟形曲线一样，那么实际上对结果没有任何影响。

So this is true, but it doesn't actually change the result.

所以这是真的，但它实际上并没有改变结果。

So on this one we're okay.

所以在这一点上我们没问题。

The result survives this change.

结果在这次改变中仍然存在。

What about the one about many candidates?

那么关于许多候选人的情况又如何呢？

Well clearly that matters a lot.

显然这很重要。

We know that there were many candidates in the '92 election and the 2000 election, and we know that "not voting" is important.

我们知道92年选举和2000年选举有很多候选人，我们知道“不投票”很重要。

So this one I'm going to let you see whether it makes a difference.

因此，我将让您看看这是否会有所不同。

This one's going to be on the problem set.

这个将出现在问题集上。

So I'm actually going to get you to do this.

所以我实际上会让你这么做。

Let's go back to the model, add in an extra candidate, and see what happens.

让我们回到模型，添加一个额外的候选者，看看会发生什么。

You can learn whether it makes a difference.

您可以了解它是否有所作为。

I'm not going to give it away.

我不会放弃它。

The one about committing to policy: this is important.

关于对政策的承诺：这一点很重要。

It's really hard for candidates to convince you, that they really are a left, or a right, or a centrist candidate.

候选人真的很难让你相信他们确实是左翼、右翼或中间派候选人。

Let me give you a historical example outside of America, but in the real world, in England.

让我给你举一个美国以外的历史例子，但在现实世界，在英国。

In the '97--In the 80's and 90's, the Labor Party in England had lost election, after election, after election.

97年——80、90年代，英国工党在选举中屡败屡战。

And in each election, they would come forward and say, "we are not really a left wing party, we're a centrist party these days" And people would say, we don't believe you.

在每次选举中，他们都会站出来说，“我们并不是真正的左翼政党，我们现在是中间派政党”，人们会说，我们不相信你。

So what Tony Blair, the guy who won the '97 election, did with "New Labor policy was: he managed to commit to a centrist policy by literally committing to it.

因此，赢得 97 年大选的托尼·布莱尔 (Tony Blair) 对“新工党政策”的做法是：他通过字面意义上的承诺，成功地推行了中间派政策。

He took the--In England you have to publish your economic and financial plans of the Government five years in ahead.

他提出——在英国，你必须提前五年公布政府的经济和财政计划。

So he took the financial plans of the conservative Government that was then in power, five years ahead, and he said these are my financial plans, I'm going to announce these today as my financial plans five years ahead.

所以他拿了当时执政的保守派政府未来五年的财务计划，他说这些是我的财务计划，我今天将宣布这些作为我未来五年的财务计划。

They're exactly the same policies as the Conservative Government has.

它们与保守党政府的政策完全相同。

So not only was he choosing a position close to the Conservatives: he was choosing exactly their position.

因此，他不仅选择了接近保守党的立场：他选择的正是他们的立场。

And it came down to what somebody else said here: that left the election on the other dimension, which might have been character or whatever.

归根结底是其他人在这里所说的话：这将选举留在了另一个维度，这可能是性格或其他什么。

That won; that won by a big way.

那赢了；那赢得了很大的胜利。

So nevertheless, even having given you that historical example, we're going to come back and we'll look later on in the class at a model in which politicians cannot choose their positions, but, rather, you know their positions ahead of time.

因此，尽管如此，即使给了你们这个历史例子，我们还是会回来，稍后我们会在课堂上看到一个模型，在这个模型中，政治家无法选择他们的立场，但相反，你可以提前知道他们的立场。

So we'll do this one.

所以我们就做这个。

We'll do this one later.

我们稍后再做这个。

These other two, primaries and higher dimensions, they're both great points.

另外两个，初级维度和更高维度，它们都是很重要的点。

We're not going to have time to do them in this course, but if you take a class in the Political Science department on voting and on elections, on modeling elections, you will see that both of these have been modeled and discussed in great detail.

我们不会有时间在本课程中完成这些任务，但是如果您在政治学系选修一门关于投票、选举、模拟选举的课程，您会发现这两门课程都已被详细建模和讨论。

There's even some empirical work on just how high dimensions are really involved in elections.

甚至还有一些关于选举到底有多高的维度的实证研究。

So this is great stuff for this area of Political Science.

所以这对于政治学领域来说是很棒的东西。

I want to inspire you to take some more advanced courses albeit in another department.

我想激励你学习一些更高级的课程，尽管是在另一个系。

So have I convinced you that there's something you can do with iterative deletion?

那么我是否已经让您相信可以通过迭代删除来完成某些操作？

We used iterative deletion in a relatively abstract setting, or not abstract, rather a play setting last time in which were choosing numbers.

我们在一个相对抽象的环境中使用了迭代删除，或者说不是抽象的，而是上次选择数字的游戏环境。

And then we talked about it in terms of Hannibal invading Rome.

然后我们谈论了汉尼拔入侵罗马的事。

And now we've seen how it applies here in an election game.

现在我们已经看到了它如何在选举游戏中应用。

I think that's probably enough for iterative deletion.

我认为这对于迭代删除来说可能就足够了。

So what I want to do now is I want to switch course and introduce a new tool.

所以我现在想做的是改变方向并引入一个新工具。

This is what we'll do typically in the class, we'll have new tools.

这是我们在课堂上通常会做的事情，我们将拥有新工具。

We'll see how they apply.

我们将看看它们如何应用。

We'll see what we can learn with them.

我们将看看我们能从他们身上学到什么。

And we'll move onto a new tool.

我们将转向新工具。

So I'm going to switch to a new tool.

所以我要改用新工具。

Any questions at this point?

此时还有疑问吗？

Everyone happy at this point?

这下大家都高兴了吧？

Happy is too strong--are people satisfied at this point?

快乐太强烈了，此时人们满意了吗？

Okay, so we're going to move away from this example and I'm going to introduce a new subject, and I'll need several boards I think, so I might as well collect them.

好吧，我们将不再讨论这个例子，我将介绍一个新主题，我认为我需要几块板，所以我不妨收集它们。

That little semi-religious speech about modeling, I think economists, those of you who are Economics majors you're thinking, yeah, what's the problem?

关于建模的那个半宗教式的小演讲，我想经济学家，那些经济学专业的人，你们会想，是的，问题是什么？

For those of you who are History majors or Political Science majors, that's more of an issue.

对于历史专业或政治学专业的学生来说，这更是一个问题。

So you really want to think about why we're modeling, what do we get from this exercise?

所以你真的想想想我们为什么要建模，我们从这个练习中得到什么？

I want to move to a different approach now and a different approach to analyzing games is going to be about, what we're going to call, best response.

我现在想转向一种不同的方法，一种不同的游戏分析方法将是我们所说的“最佳反应”。

To kick this off, let's look at an example.

首先，让我们看一个例子。

So here's a game, a very simple game with two players.

这是一个游戏，一个非常简单的游戏，有两个玩家。

Player I is choosing Up, Middle, or Down.

玩家 I 选择向上、中间或向下。

And Player II is choosing Left or Right, and the payoffs are as follows.

玩家 II 选择向左或向右，收益如下。

They're nothing very interesting.

它们没什么特别有趣的。

They're just numbers thought up for the purpose: (4, 2) and (2, 3).

它们只是为此目的而想出的数字：(4, 2) 和 (2, 3)。

So we've got (5, 1), (0, 2), (1, 3), (4, 1), (4, 2), and (2, 3).

所以我们有 (5, 1)、(0, 2)、(1, 3)、(4, 1)、(4, 2) 和 (2, 3)。

So, first of all, let's try and analyze this game using the tools we've learned so far.

因此，首先，让我们尝试使用迄今为止学到的工具来分析这个游戏。

So does either player have a dominated strategy?

那么这两个玩家是否都有占优策略呢？

Any takers?

有接受者吗？

So I claim that neither player has a dominated strategy, let's just check.

所以我认为两个玩家都没有占优策略，让我们来检查一下。

So, for example, you might think Down is dominated, but if we look a bit carefully we see that Down actually does better than Up against Right, and Down does better than Middle against Left.

因此，举例来说，您可能认为“下”占主导地位，但如果我们仔细观察，我们会发现“下”实际上比“上”对抗“右”要好，“下”比“中”对抗“左”要好。

Similarly, we can see that Right is not dominated, because it does better against Up, than does Left.

同样，我们可以看到右不占优势，因为它比左更能对抗上。

But Left isn't dominated because it does better against Middle than does Right …

但左派并没有占主导地位，因为它在对抗中派时比右派表现更好......

so on and so forth.

等等。

You can work through this game and you can check pretty quickly that nothing is dominated.

你可以玩完这个游戏，很快就能检查到没有任何东西被控制。

So if the only tool I taught you in this class was dominated strategies and the iterative deletion of dominated strategies we'd be stuck.

因此，如果我在这堂课上教给你的唯一工具是主导策略和主导策略的迭代删除，我们就会陷入困境。

You wouldn't be able to get started on this game.

你将无法开始这个游戏。

Nevertheless, it's a pretty simple game, pretty straightforward game.

尽管如此，这是一个非常简单的游戏，非常简单的游戏。

Imagine you're Player I in this game, what do you think you'd do?

想象一下，您是这场游戏中的玩家 I，您认为您会做什么？

What would you do if you were Player I?

如果你是玩家 I，你会怎么做？

Let's have a show of hands actually.

让我们实际举手吧。

So let's just see, look at it a bit.

所以让我们看看，稍微看一下。

No abstentions.

没有弃权票。

How many of you would choose Up?

有多少人会选择Up？

Raise your hand for Up.

举手向上。

Raise your hands right now.

现在请举手。

Keep your hand down, you're an SOM student.

把手放下，你是 SOM 的学生。

You can keep your hand up.

你可以举起手来。

Oh, your math was wrong, okay.

哦，你的数学错了，好吧。

You did accounting last year.

你去年是做会计的。

No one's choosing Up?No one's choosing Up at all, okay.

没有人选择Up？根本没有人选择Up，好吧。

How about Middle?

中呢？

One person chose Middle.

一个人选择了中间。

How about Down?

唐呢？

Oh my goodness.

天哪。

Okay there's a massive majority for Down.

好吧，绝大多数人都支持唐氏。

So let's think about this a second.

让我们考虑一下。

What if you knew that Player II was going to choose Left?

如果您知道玩家 II 将选择左怎么办？

What would you choose if you just happened to know that Player II was going to choose Left?

如果你恰好知道玩家二要选择左，你会选择什么？

Everyone agree because 5 is bigger than 1, and 5 is bigger than 4, so Up is--and to use a technical term we'll define formally next time--Up is the best response to Left.

每个人都同意，因为 5 大于 1，5 大于 4，所以 Up 是——用一个我们下次将正式定义的技术术语来说——Up 是对 Left 的最佳回应。

Everyone agree with that?

大家都同意吗？

So Up, as I say, does best against left.

所以正如我所说，向上对抗左翼效果最好。

Now what does best against Right?

现在什么最能对抗右派呢？

Middle does best against Right, after all, 4 is bigger than 0 and 4 is bigger than 2, so Middle does best against Right.

中路对抗右路最好，毕竟 4 比 0 大，4 比 2 大，所以中路对抗右路最好。

So here's an interesting thing.

这是一件有趣的事情。

All of you chose Down, but Up does best against Left and Middle does best against Right.

你们所有人都选择了“下”，但“上”最适合对抗“左”，而“中”最适合对抗“右”。

Let's just push this a bit harder, suppose that you work for somebody.

让我们更努力一点，假设你为某人工作。

Some of you are going to go out of Yale and be your own boss, but most of you are going to, like me, are going to end up working for somebody.

你们中的一些人将离开耶鲁大学并成为自己的老板，但你们中的大多数人将像我一样最终为某人工作。

And suppose now this game came along in your line of work and you'd chosen up.

假设现在这款游戏出现在您的工作中，并且您选择了它。

Then your boss came to you, and looks rather fiercely at you--so this is Rick Levin or somebody--looking rather fiercely at me, and says "why did you choose Up?" What can I do?

然后你的老板走到你身边，相当凶狠地看着你——所以这是里克·莱文（Rick Levin）或其他人——相当凶狠地看着我，说“你为什么选择向上？”我能做什么？

I can say: "oh I chose Up because I thought the other guy was going to choose Left." I have a way of rationalizing choosing Up.

我可以说：“哦，我选择向上，因为我认为另一个人会选择向左。”我有一个合理化选择Up的方法。

I believe the other guy is going to choose Left, I can rationalize choosing Up.

我相信另一个人会选择左，我可以合理地选择上。

Is that right?

是这样吗？

Because that might save me from getting fired, at least for now.

因为这可能会让我免于被解雇，至少现在是这样。

Similarly, if I had chosen Middle and President Levin comes along to me and says "why did you choose Middle?" I can say: "okay I chose Middle because I believed that the other guy was going to choose Right and Middle was the best thing for me to do against Right." Once again, if I'm lucky I won't get fired.

同样，如果我选择了米德尔，莱文总统会过来对我说：“你为什么选择米德尔？”我可以说：“好吧，我选择了中间，因为我相信另一个人会选择右翼，而中翼是我对抗右翼的最佳选择。”再说一次，如果我幸运的话，我就不会被解雇。

It seems like I've managed to rationalize choosing middle.

看来我已经设法合理化选择中间。

But Down, all of you chose Down.

但是Down，你们所有人都选择了Down。

Pretty much all of you chose Down.

几乎所有人都选择了Down。

Down is going to be a little trickier.

向下会有点棘手。

How am I going to rationalize choosing down?

我该如何合理化选择？

Can I rationalize choosing Down?

我可以合理化选择 Down 吗？

You all chose it.

你们都选择了它。

I'm your boss now.

我现在是你的老板了。

Why did you choose down?

为什么选择下来？

Somebody who hasn't said anything yet.

一个还没有说话的人。

Can I get the mikes up and perhaps the person right in front of you.

我可以把麦克风和你面前的人都举起来吗？

Stand up and shout.

站起来大声喊。

Student: It's the safer answer if you can't decide whether or not your opponent's going to choose Left or Right, so you don't have to worry about having a payoff of 0.

学生：如果你不能决定你的对手是选择左还是右，这是更安全的答案，所以你不必担心收益为0。

Professor Ben Polak: All right, so in some sense it's safer, it avoids the 0, although--I mean it's true it avoids the 0--but it also doesn't get the 5 and the 4.

Ben Polak 教授：好吧，所以从某种意义上说它更安全，它避免了 0，尽管——我的意思是它确实避免了 0——但它也没有得到 5 和 4。

But I think the key part of your answer was I might not know.

但我认为你回答的关键部分是我可能不知道。

I might not be sure whether the opponent, whether my opponent, is choosing Left or Right.

我可能不确定对手，无论是我的对手，是选择左还是右。

For example, I might believe that it's equally likely that they choose Left and Right, is that right?

例如，我可能认为他们选择左和右的可能性是相同的，对吗？

So let's look at something.

那么让我们看看一些事情。

Let's look at my payoffs from choosing these three options Up, Middle, and Down, if I think it's equally likely that my opponent will choose Left and Right.

如果我认为我的对手选择左和右的可能性相同，那么让我们看看我选择上、中和下这三个选项的收益。

So what we're going to look at is the expected payoff of Up versus, let me call it a (½, ½): a (½, ½) being equally likely that my opponent chooses Left and Right.

因此，我们要考虑的是“上”与“上”的预期收益，让我称之为 a (½, ½)：a (½, ½) 是我的对手选择“左”和“右”的可能性相同。

So what's my expected payoff from choosing Up where I believe the other person's going to choose Left and Right, equally likely?

那么，当我相信其他人同样有可能选择“左”和“右”时，我选择“向上”的预期回报是多少？

It's what?

是啥啊？

It's a ½ times 5 plus ½ times 0 for a total of 2.5.

这是 1/2 乘以 5 再加上 1/2 乘以 0，总共为 2.5。

Is that right?

是这样吗？

What about my expected payoff from choosing Middle against (½, ½), so in this case where I think it's equally likely that my opponents going to choose Left or Right?

那么我选择中对 (½, ½) 的预期收益如何呢？在这种情况下，我认为我的对手选择左或右的可能性相同？

So in this case, my expected payoff is a ½ of 1 plus a ½ of 4, for a total of again 2.5.

因此，在这种情况下，我的预期收益是 1 的 1/2 加上 4 的 1/2，总共又是 2.5。

Everyone happy with that?

大家对此满意吗？

Is my math correct so far?

到目前为止我的数学正确吗？

How about my expected payoff from choosing down versus (½, ½)?

选择 down 与 (½, ½) 的预期收益如何？

This is going to equal a ½ of 4 plus a ½ of 2, which is a ½ of 6, so that's 3.

这等于 4 的 1/2 加 2 的 1/2，即 6 的 1/2，所以就是 3。

So it turns out that, as the gentleman out there said, if I thought it was equally likely that my opponent was going to choose Left or Right, then actually my best choice, my best response is to choose Down.

所以事实证明，正如那位先生所说，如果我认为我的对手选择左或右的可能性是相同的，那么实际上我最好的选择，我最好的反应就是选择下。

Not only is it a good compromise., Not only is it safe.

它不仅是一个很好的折衷方案，而且不仅是安全的。

It's actually the best I can do.

这实际上是我能做的最好的事情。

It maximizes my expected payoff.

它最大化了我的预期回报。

Now we're not quite there yet.

现在我们还没有完全做到这一点。

We know that Up does best against Left.

我们知道向上对抗左的效果最好。

We know that Middle does best against Right.

我们知道中路在对抗右路时表现最好。

And we know that Down does best if I think it was equally likely that the person was going to choose Left and Right.

我们知道，如果我认为这个人选择左和右的可能性相同，那么“向下”效果最好。

But that's not the only beliefs I could have.

但这并不是我唯一的信念。

For example, I could believe that it's twice as likely that the person's going to choose Left as Right.

例如，我可以相信这个人选择左的可能性是右的两倍。

It could be, they're twice as likely to choose Left as Right.

他们选择左派的可能性可能是右派的两倍。

So equally likely was probabilities of ½ and ½.

因此 1/2 和 1/2 的概率是相同的。

What probabilities are associated with thinking it's twice as likely they're going to choose Left than Right?

认为他们选择左派的可能性是选择右派的可能性的两倍，与此相关的概率是多少？

What probabilities are associated with that: (2/3, 1/3).

与之相关的概率是：(2/3, 1/3)。

If I thought there was 2/3 probability they'd choose Left and 1/3 probability they'd choose Right, then I think it's twice as likely they're going to choose Left, and I could redo this calculation.

如果我认为他们有 2/3 的概率选择左，1/3 的概率选择右，那么我认为他们选择左的可能性是他们的两倍，我可以重做这个计算。

And in principle, I could redo this calculation for every single possible probability you could think of.

原则上，我可以对你能想到的每一个可能的概率重新进行计算。

But that would get very boring, so rather than do that, let's draw a picture.

但这会变得很无聊，所以我们不这样做，而是画一张图。

What I want to do is I want to draw a picture here, in which on the horizontal axis, I'm going to put the probability of the other guy choosing Right.

我想做的是在这里画一张图，在横轴上，我将把其他人选择正确的概率放在其中。

It's my belief that the other guy's going to choose Right.

我相信另一个人会选择正确的。

Think of this as a belief.

将此视为一种信念。

On the vertical axis--I'll put two vertical axes in, just for the sake of things--on the vertical axis I'll put my expected payoffs.

在垂直轴上——我将放置两个垂直轴，只是为了方便起见——在垂直轴上我将放置我的预期收益。

And let's start by considering my payoffs on this picture, just to help myself a bit, let me put some points in here.

让我们首先考虑一下我在这张照片上的收益，只是为了帮助自己一点，让我在这里提出一些观点。

So I've got 1, 2, 3, 4, 5.

所以我有 1、2、3、4、5。

You should probably do the same in your notes.

您可能应该在笔记中做同样的事情。

You probably have lined paper, that makes this easier.

你可能有横格纸，这会让这更容易。

Let's start by plotting my expected payoff from choosing the strategy Up.

让我们首先绘制出选择“向上”策略后的预期收益。

So this isn't so hard, we know that if I choose Up and there's probability 0 that the other guy is going to choose Right, that's the same as saying I choose Up and the other guy chooses, let's try it again.

所以这并不难，我们知道如果我选择向上，而另一个人选择正确的概率为 0，这与我选择向上而另一个人选择相同，让我们再试一次。

If this probability is 0 that the other guy chooses Right, that's the same as saying that the other guy is going to choose Left.

如果另一个人选择右的概率为 0，那么就相当于说另一个人会选择左。

So the probability is 0 of the other guy choosing Left is, the same as, let's try it again.

所以另一个人选择Left的概率是0，同样的，我们再试一次。

Probability is 0 of their choosing Right is the same as saying they choose Left, so that my payoff from Up against that is given by the top box up there, I get 5.

他们选择“右”的概率为 0，这与说他们选择“左”是一样的，所以我从“向上”中得到的回报是由上面的顶部框给出的，我得到 5。

So this is my payoff from choosing Up against Left.

所以这就是我选择“上”而不是“左”的回报。

Conversely, if there was probability 1 that the other guy is going to choose Right, and I choose Up, then I get 0.

相反，如果另一个人选择“右”的概率为 1，而我选择“上”，那么我得到 0。

So this point here corresponds to my payoff if I choose Up and they choose Right.

所以这里的这一点对应于我选择“向上”而他们选择“正确”时我的回报。

And we already know if I look at the probability of a ½, which is here, that the payoff I get from choosing, the expected payoff I get from choosing up against a (½, ½) is 2.5.

我们已经知道，如果我看一下 1/2 的概率（这里就是），我从选择中获得的回报，我从选择 a (1/2, 1/2) 中获得的预期回报是 2.5。

So here's a third point we can use.

这是我们可以使用的第三点。

What is this graph going to look like other than these three points?

除了这三点之外，这张图还会是什么样子？

What's it going to look like other than these three points?

除了这三点之外，还会是什么样子呢？

It's going to be a straight line.

这将是一条直线。

So, you can confirm that for yourself at home, but in fact, if I draw a straight line, this will actually be correct.

所以，你可以自己在家确认一下，但事实上，如果我画一条直线，这实际上是正确的。

So what is this?

那么这是什么？

This is my expected payoff from choosing Up against the probability of the other person choosing Right, and what is it actually equal to, just to confirm the equation, it's: (1 minus the probability that they choose Right), in that case I'll get a payoff of 5 plus (probability that they choose Right), in which case I get a payoff of 0.

这是我选择向上相对于其他人选择正确的概率的预期收益，它实际上等于什么，只是为了确认等式，它是：（1减去他们选择正确的概率），在这种情况下我将得到5加（他们选择正确的概率）的收益，在这种情况下我得到的收益为0。

So it doesn't really matter, but that's actually the equation of that line.

所以这并不重要，但这实际上是该线的方程。

What about choosing Middle?

那选择中号呢？

Well once again, we can plot that point So if the other person's choosing Right with probability 0, that's the same as saying they're going to choose Left, and if I chose Middle against Left, I get what?

那么，我们可以再次绘制该点，因此，如果其他人以 0 的概率选择右，这相当于说他们将选择左，如果我选择中而不是左，我会得到什么？

I get 1, I should use this pointer.

我得到 1，我应该使用这个指针。

So that's here, and if I knew they were choosing Right for sure, that's the probability of 1 that they're choosing Right, and if I choose Middle I get 4, that's here.

所以这就是这里，如果我知道他们肯定选择了“右”，那么他们选择“右”的概率就是 1，如果我选择“中”，我得到 4，这就是这里。

And we've already agreed that if I think it's equally likely they're going to choose right and left, that there's a probability ½ of them choosing right I get 2.5 from choosing Middle.

我们已经同意，如果我认为他们选择右和左的可能性相同，那么他们中选择右的概率为 1/2，那么我选择中间的概率为 2.5。

So putting that all together, I know that it must go through here again, and once again, I get a straight line.

因此，将所有这些放在一起，我知道它必须再次经过这里，并且再一次，我得到了一条直线。

I think I might have missed slightly but it's more or less right.

我想我可能有点错过了，但或多或​​少是对的。

So what is this?

那么这是什么？

This is the payoff to Player I of choosing Middle against Left.

这是玩家 I 选择中路而不是左路的回报。

This is the payoff to Player I of choosing Middle against Right, and the line in between, this line here, is the expected payoff to Player I of choosing Middle as a function of the probability that other people choose Right.

这是玩家 I 选择“中”而不是“右”的回报，而中间的线（这里的这条线）是玩家 I 选择“中”的预期回报，作为其他人选择右的概率的函数。

Now again, we could write down the equation.

现在，我们可以再次写下方程式。

it's going to be: (1 minus the probability they'll choose Right) times 1, plus (the probability that they choose Right) times 4.

它将是：（1减去他们选择正确的概率）乘以1，加上（他们选择正确的概率）乘以4。

Again, don't worry too much about the equation, we're not really going to do any math here, I'm just putting it in for completeness.

再说一次，不要太担心这个方程，我们实际上不会在这里做任何数学计算，我只是为了完整性而将其放入。

Everyone following me so far?

到目前为止大家都在关注我吗？

So let's put in now, the third of these lines.

现在让我们输入第三行。

I really should have used a different color already.

我真的应该已经使用不同的颜色了。

Let me switch colors.

让我换个颜色吧

Let's put the line that corresponds to choosing Down.

让我们放置与选择“向下”相对应的行。

So that third line, not to belabor the point, is going to go through here, through 4 and through 2, so this is the payoff from choosing Down against Left, and this is the payoff from choosing Down against Right, and I got those from looking at these payoffs here and here.

因此，第三行（不赘述）将穿过这里，穿过 4 并穿过 2，因此这是选择“向下”而不是“左”的回报，这是选择“向下”而不是“右”的回报，我从此处和此处查看这些回报得到了这些回报。

In between, once again, it's a straight line.

在两者之间，再次是一条直线。

So here's the straight line, and this line is the expected payoff to Player I from choosing Down as it depends on the probability that the other person chooses Right, and then once again, we can write down the equation.

所以这是一条直线，这条线是玩家 I 选择向下的预期收益，因为它取决于其他人选择右侧的概率，然后我们可以再次写下方程。

It's (1 minus PR) times 4, plus PR times 2.

它是（1 减去 PR）乘以 4，加上 PR 乘以 2。

I claim that this picture really tells me everything I could possibly want to know when my boss comes along.

我声称这张照片确实告诉了我当我的老板出现时我可能想知道的一切。

When President Levin or Barry Nalebuff who is now walking to the back of the room, when my boss comes along and asks me why did you do that?

当莱文总统或巴里·内勒巴夫现在走到房间后面时，当我的老板走过来问我你为什么这么做时？

Why did you do that?

你为什么这么做？

I can now say.

我现在可以说。

So, for example--let's put in these crossing points here, for example--what I can say is, if I think it's very likely –let's call this point X--if I think that the probability that the other person's going to choose Right, is less than X, then the highest payoff I can get corresponds to this line.

所以，举个例子——让我们在这里输入这些交叉点，例如——我可以说的是，如果我认为很有可能——我们称这个点为X——如果我认为对方选择正确的概率小于X，那么我能得到的最高回报对应于这条线。

That's the highest line and that line corresponds to me choosing Up.

这是最高的线，该线对应于我选择向上。

If I think the probability that they're going to choose Right is less than X, then my best response is to choose Up.

如果我认为他们选择“右”的概率小于 X，那么我最好的反应是选择“上”。

Conversely, if I think it's more likely than this crossing point Y--I think it's very likely they're going to choose Right.

相反，如果我认为它比这个交叉点 Y 更有可能——我认为他们很可能会选择右。

It's more likely than Y--the probability that they're going to choose Right is more than Y, then the highest line of these three is this one, which corresponds to my choosing Middle.

比Y的可能性更大——他们选择右的概率比Y大，那么这三条线中最高的就是这一条，它对应着我选择的中间。

So over here, my best response is to choose Middle.

所以在这里，我最好的反应是选择中间。

Let's put that in.

让我们把它放进去。

So over here my best response is Middle and over here my best response is Up, and in between it's going to turn out that the highest lines, if my belief that the person's going to choose right is more than X and less than Y, then my highest expected payoff comes from that blue line, and in that case my best response is Down.

所以在这里我的最佳反应是中间，在这里我的最佳反应是向上，在这之间，如果我相信这个人会做出正确的选择，最高的线大于 X 且小于 Y，那么我的最高预期回报来自那条蓝线，在这种情况下，我的最佳反应是向下。

I can rationalize all three of these decisions and hope that Barry won't fire me.

我可以合理解释这三个决定，并希望巴里不会解雇我。

Now we can do more than that if we're being nerdy, but I'm not going to be this nerdy today.

现在，如果我们是书呆子，我们可以做更多的事情，但我今天不会这么书呆子。

What we can actually do is we could solve out for the X and for the Y.

我们实际上能做的是求解出 X 和 Y。

How would we go about, I mean I don't want to do it because I'll probably get it wrong, but if I wanted to solve out for this X and the Y, since this is a QR class, let's just talk about it a second.

我们该怎么做，我的意思是我不想这样做，因为我可能会弄错，但如果我想解决这个 X 和 Y，因为这是一个 QR 类，让我们再讨论一下。

How would I do it?

我该怎么做呢？

Somebody who's not a math major, tell me how I solve out for the--maybe the math majors can't do it actually, it's too simple.

非数学专业的人，告诉我如何解决这个问题——也许数学专业的人实际上做不到，太简单了。

Somebody who is a math major, tell me how to solve out for the X and the Y.

谁是数学专业的，请告诉我如何解出X和Y。

Let's get a mike on this person here.

让我们对这个人进行麦克风录音。

Student: Set the intersection points equal, the equations equal to each other and solve.

学生：设交点相等，方程组相等并求解。

Professor Ben Polak: Good, good to define--to solve out for X: X is when this blue line equals, sorry, this blue line equals this red line.

Ben Polak 教授：很好，很好定义——求解 X：X 是当这条蓝线等于这条蓝线时，抱歉，这条蓝线等于这条红线。

So what we're going to do is take the equations for those two lines, so here's one of those equations and here's the other one, set the P in those equations equal to X, I've got two equations in one unknown, I'm sorry, I've got one equation and one unknown.

所以我们要做的就是求出这两条线的方程，这是其中一个方程，这是另一个方程，将这些方程中的 P 设置为 X，我有两个方程，其中一个未知数，抱歉，我有一个方程和一个未知数。

I've set these two things equal to each other: I've got one equation and one unknown.

我将这两件事设置为彼此相等：我有一个方程和一个未知数。

So I can solve out for X.

这样我就可以解出X。

I think I did that at home.

我想我在家里就是这样做的。

If you do that at home I think, but don't trust me, you'll find out that X is equal to a third.

我想，如果你在家里这样做，但不要相信我，你会发现 X 等于三分之一。

Now, just to repeat, again I know I've got some math phobics in the audience, let me just slow down a second, all I'm doing here is I'm saying look at this equation of the pink line, look at this equation of the blue line, X is when they cross, i.e., they are equal to each other.

现在，重复一遍，我知道观众中有一些数学恐惧症，让我放慢速度，我在这里所做的就是说看看这个粉红色线的方程，看看这个蓝色线的方程，X 是它们交叉的时间，即它们彼此相等。

Take these two equations, put an equal sign between them, replace this PR throughout with X, I'm going to have one equation and one unknown and that even the math phobics in the audience did in high school.

拿这两个方程，在它们之间加上等号，用 X 替换这个 PR，我将得到一个方程和一个未知数，甚至连观众中的数学恐惧症在高中时也会这样做。

Is that right?

是这样吗？

Now we're going to--I'm going a bit slow here--but I'm going to use this method next time to introduce you to the most important class, sorry, the most important game we're going to see in the whole course.

现在我们要——我在这里说得有点慢——但下次我会用这个方法向你介绍最重要的课程，抱歉，我们将在整个课程中看到的最重要的游戏。

In fact, it's the most important game in the world.

事实上，这是世界上最重要的游戏。

What is the most important game in the world?

世界上最重要的游戏是什么？

Don't go yet.

还不走。

What's the most important game in the world?

世界上最重要的游戏是什么？

We're going to use this idea of best response.

我们将使用最佳响应的想法。

We're going to think about it, and we're going to learn from it in the most important game in the world.

我们会思考这个问题，我们会在世界上最重要的比赛中从中学习。

What is the most important game in the world?

世界上最重要的游戏是什么？

Football: soccer actually.

足球：实际上是足球。

Soccer is the most important game.

足球是最重要的运动。

So we're going to show how this technique is going to help us shoot penalty shots in soccer.

因此，我们将展示这项技术如何帮助我们在足球比赛中射出点球。

Before you go, that's Barry Nalebuff at the back, he has copies of the book, it's a great book, go buy it from him.

在你走之前，后面的是巴里·内勒巴夫（Barry Nalebuff），他有这本书的副本，这是一本很棒的书，去他那里买吧。