Game Theory Lecture 4 | Game Theory

Chapter 1.

第一章。

Best Response: Penalty Kicks in Soccer [00:00:00] Professor Ben Polak: So last time we introduced a new idea and the new idea was that of best response.

最佳回应：足球中的点球 [00:00:00] Ben Polak 教授：上次我们介绍了一个新想法，这个新想法就是最佳回应。

And what was the idea?

想法是什么？

The idea was to think of a strategy that is the best you can do, given your belief about what the other people are doing: what your opponents are doing, what other players are doing.

这个想法是考虑到你能做的最好的策略，考虑到你对其他人在做什么的信念：你的对手在做什么，其他玩家在做什么。

And you could think of this ─ you could think of this belief as the belief that rationalizes that choice.

你可以这样想——你可以把这种信念视为使这种选择合理化的信念。

So if you have a boss you might want to ─ and he or she is going to ask you why you chose the action you did.

因此，如果你有一位老板，你可能会想要——他或她会问你为什么选择你所做的行动。

If you took a best response to some believe, you can say I took this action because I believed other people are going to do this.

如果您对某些信念做出了最佳回应，您可以说我采取此行动是因为我相信其他人也会这样做。

And since that was the best you could do under that belief, you'll hopefully keep your job.

既然这是你在这种信念下能做的最好的事情，你就有希望保住你的工作。

I promised that today we would look at the most important game in the world.

我承诺今天我们将观看世界上最重要的比赛。

And, as announced last time, the most important game is the penalty kick game.

而且，正如上次宣布的那样，最重要的比赛是点球比赛。

So this is a game that occurs in soccer and just to give an idea of how important it is for those people who are unfortunate enough not to be soccer fans here, the last World Cup was decided on penalty kicks.

所以这是一场发生在足球比赛中的比赛，只是为了让人们了解对于那些不幸不是这里足球迷的人来说有多么重要，上届世界杯​​是通过点球决定胜负的。

In England's case, England goes out of every single World Cup and every single European competition because it loses on penalty kicks, usually to Germany, it has to be said.

就英格兰队而言，不得不说的是，英格兰队在每届世界杯和每届欧洲比赛中都出局了，因为它在点球大战中输了，而且通常是输给了德国队。

And more immediately, this weekend, as all of you are thinking the most event in the world was whatever was happening in Congress to do with Iraq, actually, the most important event in the world was taking place in England where my favorite team, Portsmouth, were playing Kaj, the head TA's favorite team, Liverpool.

更直接的是，这个周末，当你们所有人都认为世界上最重要的事件是国会发生的与伊拉克有关的事情时，实际上，世界上最重要的事件正在英格兰举行，在那里我最喜欢的球队朴茨茅斯队对阵卡伊队，TA最喜欢的球队利物浦队。

And, about a third of a way through that game there was a penalty, and …

而且，那场比赛进行到三分之一左右时，出现了点球，而且……

I'll let you know what happened later.

稍后我会告诉你发生了什么。

So keep at the back of your mind that the real world example that matters here is Portsmouth versus Liverpool this weekend.

因此，请记住，现实世界中最重要的例子是本周末朴茨茅斯对阵利物浦。

(Kaj, the head TA is Scandinavian so I've got no idea why he's supporting Liverpool anyway, but I think maybe he spells it like this or something like this).

（Kaj，首席助教是斯堪的纳维亚人，所以我不知道他为什么支持利物浦，但我想也许他会这样拼写或类似的东西）。

So what we're going to do is we're going to look at some numbers that are approximately the probabilities of scoring when you kick the penalty kick in different directions.

因此，我们要做的是查看一些数字，这些数字大约是当您向不同方向踢点球时得分的概率。

But just make sure everyone ─ do I need to explain what's going on here?

但请大家确认一下──我需要解释一下这是怎么回事吗？

Is everyone familiar with this situation?

这种情况大家都熟悉吗？

There's one guy who's going to run up and kick the ball.

有一个人要跑上去踢球。

The goal keeper is standing at the goal.

守门员站在球门前。

And their aim is to kick it into the goal.

他们的目标是将球踢入球门。

That's probably enough.

这可能就足够了。

You've all seen this right?

大家都看到了吧？

If you haven't seen this, go see it.

如果你还没有看过这个，那就去看看吧。

I mean come on!

我的意思是来吧！

So things you should do in life: read Shakespeare and see a soccer game.

所以你在生活中应该做的事情：读莎士比亚和看一场足球比赛。

So the rough numbers for this are as follows ─ and actually later on in the class I'll give you some more accurate numbers, but these will do for now.

因此，粗略的数字如下 ─ 实际上，稍后在课堂上我会给你们一些更准确的数字，但现在就这些了。

There are three ways, the goal─ the attacker could kick the ball.

共有三种方式，球门──进攻者可以踢球。

He could kick the ball to the left, the middle, or the right.

他可以把球踢到左边、中间或右边。

And I shouldn't just say he here of course, I mean this is he or she but if I get that wrong going on, please forgive me for it.

当然，我不应该在这里只说他，我的意思是这是他或她，但如果我弄错了，请原谅我。

The goalie can dive to the left or the right.

守门员可以向左或向右扑救。

In principle the goalie could stay in the middle.

原则上守门员可以留在中间。

We'll come back and talk about that later.

我们稍后再回来讨论这个问题。

So this is the guy who is shooting, he's called the shooter and this is the goalie.

这就是射门的人，他被称为射手，这是守门员。

These are roughly ─ well, let me put up the payoffs for this game and then I'll explain them.

这些是粗略的──好吧，让我列出这个游戏的收益，然后我会解释它们。

So you'll notice that I'm just going to put in numbers here and then the negative of the number and the numbers are roughly like this: (4,-4).

所以你会注意到，我只是要在这里输入数字，然后输入该数字的负数，这些数字大致如下：(4,-4)。

So the numbers are (4, -4), (9, -9), (6, -6), (6, -6), (9, -9) and (4, -4).

所以数字是 (4, -4)、(9, -9)、(6, -6)、(6, -6)、(9, -9) 和 (4, -4)。

And the idea here is that the number 4 represents 40% chance of scoring if you shoot the ball to the left of the goal and the goal keeper dives to the left.

这里的想法是，如果你将球射向球门左侧并且守门员向左侧扑救，则数字 4 代表得分的机会为 40%。

So the payoff here is something like u1(left) if the goal keeper dives to the left is equal to 4, by which I mean there's a 40% chance of scoring.

因此，如果守门员向左扑救等于 4，则这里的收益类似于 u1(left)，我的意思是有 40% 的机会进球。

So the number for the--The payoff for the shooter is his probability of scoring and the payoff for the goal keeper is just the negative of that.

因此，射手的回报是他进球的概率，而守门员的回报只是其负数。

Let's keep things simple.

让我们把事情简单化。

As I said before, for now we'll ignore the possibility that the goal keeper could stay put.

正如我之前所说，现在我们将忽略守门员留在原地的可能性。

So how should we start analyzing this important game?

那么我们应该如何开始分析这场重要的比赛呢？

Well we start with the ideas we learned already several weeks ago now, or more than a week ago, which is the idea of dominant strategies.

好吧，我们从几周前或一周多前学到的想法开始，这就是主导策略的想法。

Does either player here have a dominated strategy?

这里的任何一个玩家都有占优策略吗？

Does either player have a dominated strategy?

任何一方都有占优策略吗？

No, it's kind of clear that they don't.

不，很明显他们不这样做。

Let's just look at the shooter, for example.

例如，让我们看看射手。

So you might think that maybe middle dominates left, but notice that middle has a higher payoff against left than shooting to the left.

所以你可能会认为也许中路主宰左路，但请注意中路对左路的回报比向左射击更高。

It has as lower payoff if the goalie dives to the right.

如果守门员向右扑救，则回报会较低。

So, not surprisingly in this game, it turns out, that if the goalie dived to the left you're best off shooting to the right, second best off shooting to the middle, and worst off shooting to the left.

因此，在这场比赛中，事实证明，如果守门员向左扑，你最好向右射门，其次是向中间射门，最差向左射门。

That's if the goalie dives to the left.

如果守门员向左扑球的话。

And if the goalie dives to the right, you're best off shooting to the left, second best off shooting to the middle, and worst off by shooting to the right; and that's kind of common sense.

如果守门员向右扑球，你最好向左射门，其次是向中间射门，最差的是向右侧射门；这是常识。

Okay, so if we had stopped the class after the first week where all we learned to do was to delete dominated strategies, we'd be stuck.

好吧，如果我们在第一周之后停止上课，我们所学到的就是删除主导策略，那么我们就会陷入困境。

We'd have nothing to say about this game and as I said before, this is the most important game, so that would be bad news for Game Theory.

我们对这场比赛没什么可说的，正如我之前所说，这是最重要的比赛，所以这对博弈论来说是个坏消息。

But luckily, we can do a little bit better than that.

但幸运的是，我们可以做得更好一点。

Before I do that, let's just take a poll of the class.

在此之前，我们先对班级进行一次民意调查。

How many of you, if you were playing for, I guess it's going to be America, which is a sad thing to start with, never mind.

你们中有多少人，如果你们为之效力，我想那将是美国，这从一开始就是一件悲伤的事情，没关系。

You guys are playing for America and you're taking this penalty kick and it's the last kick in the World Cup, how many of you, show of hands, how many of you would shoot to the left?

你们为美国队踢球，你们要罚这个点球，这是世界杯的最后一个球，你们中有多少人举手，有多少人会射向左边？

How many of you would shoot to the middle?

你们中有多少人会向中间射击？

How many of you would shoot to the right?

你们中有多少人会向右侧射击？

We've got kind of an even split there, pretty much an even split.

我们在那里有一种平等的分裂，几乎是平等的分裂。

We're going to assume these are the correct numbers and we're going to see if that even split is really a good idea or not.

我们将假设这些是正确的数字，然后我们将看看这种均匀的分割是否真的是一个好主意。

So how should we go about thinking about this?

那么我们应该如何去思考这个问题呢？

What I suggest we do is we do what we did last time and we start to draw a picture to figure out what my expected payoff is, depending on what I believe the goalie is going to do.

我建议我们做的是我们上次做的事情，我们开始画一张图来计算出我的预期回报是什么，这取决于我相信守门员会做什么。

So this is the same kind of picture we drew last time.

这和我们上次画的图是一样的。

So on the horizontal axis is my belief, and my belief is essentially the probability that the goalie dives to the right.

所以横轴上是我的信念，我的信念本质上就是守门员向右扑救的概率。

Now as I did last time, let me put in two axes to make the picture a little easier to draw.

现在，就像我上次所做的那样，让我放入两个轴以使图片更容易绘制。

So this is 0 and this is 1.

所以这是 0，这是 1。

And you probably have lines in your notes but I don't, so let me just help myself a bit.

你的笔记中可能有台词，但我没有，所以让我帮自己一点忙。

So this is 2, 4, 6, 8, 10, so this is going to be 2, 4, 6, 8, and 10 and over here 2, 4, 6, 8, and 10, 2, 4, 6, 8, and 10.

所以这是 2、4、6、8、10，所以这将是 2、4、6、8 和 10，这里是 2、4、6、8 和 10、2、4、6、8 和 10。

This would be the basis of my picture.

这将是我的照片的基础。

So it starts with a possibility of shooting to the left.

所以一开始就可以向左射击。

Let's do this in red.

让我们用红色来做这个。

So I shoot to the left and the goalie dives to the left, my payoff is what?

所以我向左射门，守门员向左扑救，我的回报是多少？

It's 4.

是 4。

If I shoot to the left and there's no probability of the goalie diving to the right, which means that they dive to the left, then my payoff is 4, meaning I score 40% of the time.

如果我向左射门，而守门员不可能向右扑救，这意味着他们会向左扑扑，那么我的收益是 4，这意味着我进球的概率是 40%。

If I shoot to the left and the goalie dived to the right, then I score 90% of the time, so my payoff is .9.

如果我向左射门而守门员向右扑救，那么我得分的概率为 90%，所以我的回报是 0.9。

By the way why is it 90% of the time and not 100% of the time?

顺便问一下，为什么是 90% 的时间而不是 100% 的时间？

I could miss; okay, I could miss.

我可能会错过；好吧，我可能会错过。

That happens rather often it turns out, well 10% of the time.

事实证明，这种情况经常发生，大约有 10%。

So we know this is going to be a straight line in between, so let's put this line in.

所以我们知道这将是两者之间的一条直线，所以让我们把这条线放进去。

So what's this?

那么这是什么？

It's the expected payoff to Player I of shooting to the left as it depends on the probability that the goal keeper dives to the right.

这是球员 I 向左射门的预期回报，因为它取决于守门员向右扑救的概率。

And conversely, we can put in …

相反，我们可以放入……

well let's do them in order.

好吧，让我们按顺序做。

So middle: so if I shoot to the middle and the goal keeper dives to the left, then my payoff is .6, is 6, or I score .6 of the time, and if I shoot to the middle and the goalie dives to the right I still score 60% of the time, so once again it's a straight line in between.

那么中间：所以如果我射向中间并且守门员跳向左侧，那么我的回报是 0.6，是 6，或者我得分 0.6，如​​果我射向中间并且守门员跳向右侧，我仍然有 60% 的机会得分，所以这又是一条介于两者之间的直线。

So this line represents the expected payoff of shooting to the middle as a function of the probability that the goal keeper dives to the right.

因此，这条线代表了向中间射门的预期收益，作为守门员向右扑救的概率的函数。

Finally ─ let's do it in green ─ let's look at the payoffs, expected payoffs, if I shoot to the right.

最后——让我们用绿色来做——让我们看看如果我向右射击的话，预期的回报。

So if I shoot to the right and the goalie dives to the left, then I score with probability .9, or my payoff is 9.

因此，如果我向右射门，而守门员向左扑救，那么我得分的概率为 0.9，或者我的回报为 9。

Conversely, if I shoot to the right and he or she dives to the right, then I score 40% of the time, so here's my payoff .4.

相反，如果我向右射门，而他或她向右扑救，那么我得分的概率为 40%，所以这是我的回报 0.4。

And here's my green line representing my expected payoff as the shooter, from shooting to the right, as a function of the probability that the goalie dives to the right.

这是我的绿线，代表我作为射手的预期收益，从向右射门，作为守门员向右扑救的概率的函数。

Did everyone understand how I constructed this picture?

大家明白我是如何构造这张图的吗？

Easier picture than the one we constructed last time.

比我们上次构建的图片更简单。

So what does everyone notice from this picture?

那么大家从这张照片中注意到什么呢？

What's the first thing that jumps out at you from this picture?

从这张照片中，你首先想到的是什么？

Assuming these numbers are true, what jumps out at you from this picture?

假设这些数字是真实的，您从这张图片中看到了什么？

Can we get some mikes up here?

我们可以在这里拿一些麦克风吗？

So Ale, can we get this guy?

那么艾尔，我们能找到这个人吗？

Stand up first, the guy in red.

红衣人先站起来。

What's your name?

你叫什么名字？

Don't hold the mike; just shout.

不要握住麦克风；只是喊叫。

Student: There's no point at which the 6, at which it shooting in the middle gets a higher payoff.

学生：在中间射击的 6 没有任何意义会获得更高的回报。

Professor Ben Polak: Exactly, exactly.

本·波拉克教授：完全正确。

So the thing that I hope jumps out at you from this picture is (no great guesses about figuring out this is a ½), so if the probability that the goalie's going to jump to the right is less than a ½, then the best you can do is represented by this green line, which is shoot to the right.

所以我希望你从这张图片中跳出来的事情是（没有很好的猜测来弄清楚这是 1/2），所以如果守门员跳到右边的概率小于 1/2，那么你能做的最好的事情就是用这条绿线表示，即向右射门。

So the goalie is going to shoot to the right with the probability less than a ½, sorry he's going to dive to the right with the probability less than a ½, you should shoot to the right.

所以守门员将以小于 1/2 的概率射向右侧，抱歉，他以小于 1/2 的概率向右侧射门，你应该向右侧射门。

Conversely, if you think the goalie's going to shoot to the right with probability more than a ½, then the best you can do is represented by the pink line, and that's shooting to the left, or if you think the goalie's going to dive to the right with the probability more than a ½, the best you can do, your best response is to shoot to the left.

相反，如果你认为守门员将以超过 1/2 的概率向右侧射门，那么你能做的最好的事情就是用粉红色的线表示，那就是向左射门，或者如果你认为守门员将以超过 1/2 的概率向右侧射门，你可以做的最好的反应是向左射门。

And there is no belief you could possibly hold given these numbers in this game that could ever rationalize shooting the ball to the middle.

考虑到这场比赛中的这些数字，你不可能相信你可以合理地将球投向中间。

Is that right?

是这样吗？

So no: to say it another way, middle is not a best response to any belief I can hold about the goal keeper, to any belief.

所以不：换句话说，中间并不是对我对守门员的任何信念、任何信念的最佳回应。

So there's a lesson here, and it's pretty much (just to make the lesson resonate again): imagine there you are in the World Cup, you're playing for England, you have to justify your actions not only to your teammates and your manager, and your boss, but to about 60 million rather angry fans.

所以这里有一个教训，而且很重要（只是为了让这个教训再次产生共鸣）：想象一下，你在世界杯上，你为英格兰队效力，你不仅必须向你的队友、你的经理和你的老板，而且还要向大约 6000 万愤怒的球迷证明你的行为是正当的。

What's the lesson here?

这里有什么教训？

I'm hoping it was going to be obvious, what's the lesson here?

我希望这会是显而易见的，这里的教训是什么？

The lesson is, do not shoot to the middle.

教训是，不要射到中间。

Let me qualify that lesson slightly, unless you're German.

让我稍微限定一下这个教训，除非你是德国人。

Germans can do whatever they like.

德国人可以为所欲为。

Now, it turns out that about a third of the game between my team Portsmouth and Kaj's team Liverpool, this weekend there was a penalty.

现在，事实证明，本周末我的朴茨茅斯队和卡伊的利物浦队之间大约三分之一的比赛中出现了点球。

Portsmouth had a penalty and the guy who was going to take the penalty came up to kick the penalty and he kicked it to the middle and it was saved.

朴茨茅斯获得了一个点球，原本要主罚点球的家伙上前主罚点球，他把球踢到了中路，球被扑出。

So just confirming these actions, not only did that spoil my weekend but it also spoiled my opportunity to make fun of Kaj all week, so it was really a big deal.

因此，仅仅确认这些行为，不仅破坏了我的周末，而且还破坏了我整个星期取笑 Kaj 的机会，所以这确实是一件大事。

So this weekend a penalty was missed exactly by somebody ignoring this rule.

所以这个周末，有人无视这条规则，错过了一个点球。

There's a more general lesson here, and the more general lesson is, of course: do not choose a strategy that is never a best response to anything you could believe.

这里有一个更普遍的教训，当然，更普遍的教训是：不要选择对你所相信的任何事情都不是最佳反应的策略。

The more general lesson, do not choose a strategy that is never a best response to any belief.

更普遍的教训是，不要选择永远不是对任何信念的最佳回应的策略。

Notice here, just to underline something which came up at the end last time, that doesn't just mean beliefs of the form, the goalie's going to dive left or the goalie's going to dive right.

注意这里，只是为了强调上次最后出现的一些事情，这不仅仅意味着形式的信念，守门员将向左扑救或守门员将向右扑扑。

It means all probabilities in between.

这意味着介于两者之间的所有概率。

So we're allowing you to, for example, to hold the belief that it's equally likely that the goalie dives left or dives right.

例如，我们允许您相信守门员向左扑或向右扑的可能性是相同的。

But if there's no belief that could possibly justify it, don't do it.

但如果没有任何信念可以证明它是合理的，那就不要这样做。

And underlining what arises in this game, notice that in this game we're able to eliminate one of the strategies, in this case the strategy of shooting to the middle, even though nothing was dominated.

强调这场比赛中出现的情况，请注意，在这场比赛中，我们能够消除其中一种策略，在本例中是向中间射击的策略，即使没有任何东西被支配。

So when we looked at domination and deleted dominated strategies, we got nowhere here.

因此，当我们研究支配并删除支配策略时，我们一无所获。

Here, at least, we got somewhere, we got rid of the idea of shooting to the middle.

在这里，至少我们到达了某个地方，我们摆脱了向中间射击的想法。

Now if you can just persuade the English and Portsmouth soccer players of this lesson, I'd be very happy.

现在，如果你能说服英国和朴茨茅斯的足球运动员接受这一课，我会非常高兴。

So before we leave it, I've been making a point in this class of coming back to reality from time to time, so this is a very simple model of the soccer game in reality.

所以在我们离开之前，我在这堂课上一直强调时不时地回到现实，所以这是现实中足球比赛的一个非常简单的模型。

Let's just try, any of you on the Yale Soccer Team?

让我们试试，你们有耶鲁足球队的成员吗？

No?

不？

Have any of you played soccer for your college?

你们有人在大学踢过足球吗？

One or two.

一两个。

Have you ever played soccer?

你踢过足球吗？

How many of you have ever played soccer?

你们中有多少人踢过足球？

Okay, good I was getting worried there for a second.

好吧，好吧，我有一瞬间有点担心。

So, one thing we said last time was when we put up a model and try and draw lessons from it, we should just take a step back and say, what's missing here?

所以，我们上次说过的一件事是，当我们建立一个模型并尝试从中吸取教训时，我们应该退一步说，这里缺少什么？

So let's try and get some kind of ─ I'll come off the stage to make it easier for Jude.

所以，让我们尝试采取某种措施 — 我会下台，让裘德更容易接受。

What's missing here?

这里缺少什么？

What's missing in this model of this piece of soccer, this game within a game?

这个足球模型、这个游戏中的游戏缺少什么？

What's missing here?

这里缺少什么？

Why is this not necessarily a hundred percent accurate model?

为什么这个模型不一定是百分百准确的？

I'll need some mikes up here.

我这里需要一些麦克风。

Can you?

你可以吗？

You have to really shout because you're miles from the mike there.

你必须大声喊叫，因为你距离那里的麦克风有几英里远。

Student: You might be better kicking to the left or to the right depending on whether you're right handed or left handed.

学生：根据你是右手还是左手，你可能向左踢还是向右踢更好。

Professor Ben Polak: Good, so one thing that's clearly missing here is I'm ignoring that in fact right footed players find it easier to shoot to their left, which is actually the goalie's right.

本·波拉克教授：很好，所以这里明显遗漏的一件事是我忽略了事实上右脚球员更容易向左侧射门，这实际上是守门员的右侧。

So right footed players find it easier to shoot to the left as facing, to shoot across the goal.

因此，右脚球员发现更容易向左侧射门，射门越过球门。

Does everyone confirm that's true?

大家都确认这是真的吗？

Yeah, anyone ever tried to this?

是的，有人尝试过这个吗？

It's a little easier to hit the ball hard to the opposite side from the side which is your foot and that's the same principle in baseball.

从你的脚一侧用力将球击向另一侧要容易一些，这与棒球中的原理相同。

It's a little bit easier to pull the ball hard then it is to hit the ball to the opposite field.

用力拉球比把球打到对面场地要容易一些。

Yes?

是的？

Student: Players don't make their decision before, and then stick with it necessarily.

学生：玩家不会先做出决定，然后就一定要坚持下去。

Professor Ben Polak: All right, so players are making decisions as they're running up.

本·波拉克教授：好吧，所以球员们在跑动时做出决定。

I think that's okay here, right?

我觉得这里没问题吧？

We can think of this as the decision happening at the instant at which you kicked.

我们可以将其视为您踢球那一刻发生的决定。

So you're right that you could have made your decision back in the locker room, or you could have made the decision at half time, but ultimately what matters ─ let's hope that goes away.

所以你是对的，你本可以在更衣室里做出决定，或者你可以在中场休息时做出决定，但最终重要的是——让我们希望这种情况能够消失。

We sure that it's not off my mike?

我们确定这不是我的麦克风的问题吗？

Just in case I'll move my mike a bit lower.

以防万一我会把麦克风调低一点。

So I'm going to shout louder because my mike is now lower.

所以我要喊得更大声，因为我的麦克风现在更低了。

It doesn't really matter exactly when the decision is made.

何时做出决定并不重要。

At the end of the day, the goalie doesn't know the decision of the shooter and the shooter doesn't know the decision of the goalie.

归根结底，守门员不知道射手的决定，射手也不知道守门员的决定。

So it's as if that decision is made instantaneously as the shooter is running up.

所以就好像射手在跑动时立即做出了决定一样。

What else?

还有什么？

Yeah, can we get this.

是的，我们能得到这个吗？

Tae can you get this guy here?

Tae 你能把这个人带到这里来吗？

Stand up.

起来。

Shout out.

喊出来。

Professor Ben Polak: The goalie might stay in the middle.

本·波拉克教授：守门员可能会呆在中间。

That's a good point, of course, I've abstracted from that here, and in fact, we'll come back, I think, I'll try and put that onto a problem set, but I think you're right, it is an issue here.

当然，这是一个很好的观点，我已经在这里进行了抽象，事实上，我们会回来，我想，我会尝试将其放入问题集中，但我认为你是对的，这是这里的一个问题。

Anything else?

还要别的吗？

Well let me put up some real numbers and we'll see about how much the correspondence to what we've got here.

好吧，让我提供一些真实的数字，我们将看看与我们这里得到的数据有多少对应。

So I gave you some numbers I made up actually a long time ago, but since I've been using this game in class, somebody went out and checked.

所以我给了你们一些我很久以前编的数字，但由于我一直在课堂上使用这个游戏，所以有人出去检查了。

And it turns out that ignoring middle for a second, ignoring middle ─ so these are real numbers, and these numbers come from a paper in the AER by Chiappori and some co-authors and for everyone at Yale, I'll make that paper available to you through JSTORE or through the Yale Library, so you can go look at it if you'd like.

事实证明，暂时忽略中间，忽略中间 ── 所以这些是实数，这些数字来自 Chiappori 和一些合著者在 AER 上发表的一篇论文，对于耶鲁的每个人，我将通过 JSTORE 或通过耶鲁图书馆将该论文提供给你们，所以如果你们愿意，可以去看看。

What they worked out was the following table.

他们得出的结果是下表。

And again, we need to be a little bit careful here.

再说一遍，我们在这里需要小心一点。

So I'm going to put the left and right in inverted commas because actually what they did was they corrected for people's natural direction and not natural direction.

所以我要把左和右放在引号中，因为实际上他们所做的是他们纠正了人们的自然方向而不是自然方向。

So the idea here is shooting to the left if you're right footed is the natural direction, so left here means the natural direction.

所以这里的想法是，如果你是右脚，则向左射门是自然方向，所以这里的左侧意味着自然方向。

Of course, if you're left footed it goes the other way, but they've corrected for that.

当然，如果你是左脚，那就相反，但他们已经纠正了这一点。

It turns out that the probabilities of scoring here are as follows, 63.6, 94.4, 89.3, and 43.7.

事实证明，这里得分的概率如下，63.6、94.4、89.3、43.7。

So things are not--I haven't given you the numbers for the middle but--So you can see that whoever it was who said, you're slightly better off, you score with slightly higher probabilities when you kick to your natural side is exactly right.

所以事情不是——我没有给你中间的数字，但是——所以你可以看到，无论是谁说的，你的情况稍微好一点，当你踢到你的自然一侧时，你得分的概率稍高，这是完全正确的。

The thing is still not dominated and we could still have done exactly the same analysis, and actually you can see I'm not very far off in the numbers I made up, but things are not perfectly symmetric.

事情仍然没有占主导地位，我们仍然可以做完全相同的分析，实际上你可以看到我所编的数字离我并不遥远，但事情并不完全对称。

I forget who it was who said that, but that does turn out to be true.

我忘了这句话是谁说的，但事实证明这是真的。

Certainly the goalie staying put is an issue, as I said we'll deal with that in the problem set, but there's another issue here.

当然，守门员留在原地是一个问题，正如我所说，我们将在问题集中处理这个问题，但这里还有另一个问题。

Let me just raise one more issue.

让我再提出一个问题。

One more issue is, you have another decision when you run up to hit this, hit the penalty other than just left and right.

还有一个问题是，当你跑过去击中这个球时，你还有另一个决定，击中点球，而不是只向左和向右击球。

Someone whose played the game, what's the other decision you really face?

玩过这款游戏的人，您真正面临的另一个决定是什么？

Can I get the woman here?

我可以把那个女人叫到这里吗？

What's the other decision you face?

您面临的其他决定是什么？

Professor Ben Polak: Okay, you can kick up and down, that's true.

本·波拉克教授：好的，你可以上下踢腿，没错。

Okay, that's true actually, that's true.

好吧，确实如此，确实如此。

But I meant something else, that's right, but I meant something else.

但我的意思是别的，没错，但我的意思是别的。

What else is there?

还有什么？

Try this guy here.

在这里试试这个家伙。

Professor Ben Polak: Well that's getting subtle here.

Ben Polak 教授：嗯，这有点微妙。

It's a much more basic thing, what's a more basic thing?

这是一个更基本的事情，什么是更基本的事情？

What's a more basic thing here?

这里更基本的事情是什么？

Take it, yeah right in front of you.

拿走吧，就在你面前。

Professor Ben Polak: Speed, right.

Ben Polak 教授：速度，对。

So another decision you face is do you just try and kick this ball as hard as you can or do you try and place it?

所以你面临的另一个决定是，你是尽你所能地尝试踢这个球，还是尝试将它放置好？

That's probably as important a decision as placing it, as deciding which direction to hit, and it turns out to matter.

这可能与放置它、决定击中哪个方向一样重要，而且事实证明它很重要。

So, for example, if you're the kind of person, (which is I have to say all I ever was), if you're the kind of person who can kick the ball fairly hard but not very accurately, then it actually might change these numbers.

所以，举例来说，如果你是那种人（这是我不得不说的所有我曾经说过的），如果你是那种可以相当用力但不是很准确地踢球的人，那么它实际上可能会改变这些数字。

If you can kick the ball very hard, but not very accurately, then if you try and shoot to the left or right, you're slightly more likely to miss.

如果你可以非常用力地踢球，但不是很准确，那么如果你尝试向左或向右射门，那么你射偏的可能性会稍大一些。

On the other hand, as you shoot to the middle, since you're kicking the ball hard, you're slightly more likely to score.

另一方面，当你射向中间时，由于你用力踢球，你得分的可能性会稍大一些。

Now, if this all seems like arcane and irrelevant detail, let's just see why this matters in the picture, and then we'll leave soccer, at least for today.

现在，如果这一切看起来像是神秘和不相关的细节，让我们看看为什么这在图片中很重要，然后我们就离开足球，至少今天是这样。

So if you're the kind of person who can kick the ball hard but not accurately, then it's going to lower the probabilities of scoring as you kick towards the right because you might miss, and it's going to lower the probability of scoring as you hit towards the left because you're likely to miss, and it might actually raise the probability of your scoring as you hit towards the middle, because you hit the ball so hard it's really pretty hard for the goal keeper to stop it.

因此，如果你是那种能用力踢球但不准确的人，那么当你向右侧踢球时，它会降低得分的概率，因为你可能会错过；当你向左侧踢球时，它会降低得分的概率，因为你可能会错过；而当你向中间踢球时，它实际上可能会提高你得分的概率，因为你击球的力度太大，守门员很难阻止它。

Here it goes in the middle, and if you look carefully there, I didn't really make it clear enough, you can see, suddenly a strategy that looked crazy shooting to the middle, that suddenly started to seem okay.

这里它在中间，如果你仔细看那里，我并没有真正说得足够清楚，你可以看到，突然一个看起来疯狂的策略向中间射击，突然开始看起来还不错。

It turns out, if you look at those dotted lines, there's an area in the middle, the area between here and here, this little area here, you actually might be just fine shooting to the middle.

事实证明，如果你看那些虚线，中间有一个区域，这里和这里之间的区域，这里的这个小区域，你实际上可能可以很好地拍摄到中间。

So in reality, we need to take into account a little bit more, and in particular, we need to take into account the abilities of players to hit the ball accurately and/or hard.

所以实际上，我们需要考虑更多一点，特别是，我们需要考虑球员准确和/或用力击球的能力。

And if those people, if you're interested in that ─ and I realize at this point I probably lost the interest of most Americans in the room, but for the non-Americans in the room, the people who are interested in the real world ─ as I said before, I'll put that article online and that goes through all the gory detail of this.

如果那些人，如果你对此感兴趣——我意识到，在这一点上，我可能失去了房间里大多数美国人的兴趣，但对于房间里的非美国人，对现实世界感兴趣的人——正如我之前所说，我会把这篇文章放到网上，其中会详细介绍这件事的所有血淋淋的细节。

I should just say that the data I just gave you is real data but it's actually mixed ability data.

我只能说，我刚才给你的数据是真实的数据，但它实际上是混合的能力数据。

This data comes half from the Italian league, which is pretty good and half from the French league, which sucks.

这个数据一半来自意大利联赛，还不错，一半来自法国联赛，很糟糕。

So who knows how much we should trust it.

所以谁知道我们应该信任它多少。

Okay, so that was our example for the day and our first brush with reality for the day.

好的，这就是我们今天的例子，也是我们今天第一次接触现实。

Let's clean the board and do some work.

让我们清洁电路板并做一些工作。

Do a bit more formal stuff here.

在这里做一些更正式的事情。

So here we have an example but I want to go back to the generality and to a bit of formalism.

这里我们有一个例子，但我想回到一般性和一点形式主义。

By the way, I should tell you that the game ended nil-nil or 0-0.

顺便说一句，我应该告诉你，比赛以零比零或0-0结束。

It's a moral victory for me I think.

我认为这对我来说是道德上的胜利。

So I want to be formal about these things I've been mentioning informally.

所以我想正式地谈论我非正式地提到的这些事情。

And in particular, I want to be formal about the definition of best response.

特别是，我想正式地定义最佳响应的定义。

I'm going to put down two different definitions of best response, one of which corresponds to best response to somebody else playing a particular strategy like left and right, and the other is just going to correspond to the more general idea of a best response to a belief.

我将给出最佳反应的两种不同定义，其中一个对应于对其他人采取左和右等特定策略的最佳反应，而另一种则对应于对信念的最佳反应的更一般概念。

It'll allow us to use our notation and just be a little bit more nerdy.

它将允许我们使用我们的符号并且变得更书呆子一点。

So Player i's strategy, Ŝi (there's going to be a hat to single it out) is a best response (always abbreviated BR), to the strategy S- i of the other players if ─ and here's our real excuse to use our notation ─ if the payoff from Player i from choosing Ŝi against S- i is weakly bigger than her payoff from choosing some other strategy, Si', against S- i and this better hold for all Si' available to Player i.

因此，玩家 i 的策略 Ŝi（将有一顶帽子将其挑出来）是对其他玩家的策略 S- i 的最佳响应（总是缩写为 BR），如果 ─ 这是我们使用符号的真正借口 ─ 如果玩家 i 选择 Ŝi 对抗 S- i 的收益略大于她选择其他策略 Si' 对抗 S- i 的收益，并且这对于玩家可用的所有 Si' 更好岛

So in previous definitions, we've seen the qualifier, for all, be on the other player's strategy.

因此，在之前的定义中，我们已经看到预选赛，对于所有人来说，都取决于其他玩家的策略。

Here, the qualifier for all is on my strategy.

在这里，所有人的预选赛都在我的策略上。

So strategy Ŝi is a best response to the strategy S- i of the other players if my payoff from choosing Ŝi against S- i, is weakly bigger than that from choosing Si' against S- i, and this better hold for all possible other strategies i could choose.

因此，如果我选择 Ŝi 对抗 S- i 的收益远大于选择 Si' 对抗 S- i 的收益，则策略 Ŝi 是对其他玩家策略 S- i 的最佳反应，并且这对于我可以选择的所有可能的其他策略来说更好。

There's another way of writing that, that's kind of useful, or equivalently, Ŝi solves the following.

还有另一种写法，这很有用，或者等效地，Ŝi 解决了以下问题。

It maximizes my payoff against S- i.

它使我对 S-i 的收益最大化。

So you're all used to, I'm hoping everyone is used to seeing the term max.

所以你们都习惯了，我希望每个人都习惯看到max这个词。

As I solve the maximization problem, how do I maximize my payoff given that other people are choosing S- i?

当我解决最大化问题时，考虑到其他人都选择 S-i，我如何最大化我的收益？

Again, for the math phobics in the room, don't panic, this is just writing down in words what we've already seen a couple of times already today, well today and last time.

再说一次，对于房间里的数学恐惧症患者，不要惊慌，这只是用文字写下我们今天已经见过几次的东西，今天和上次。

Let's generalize this definition a little bit, since we want it to allow for more general beliefs.

让我们稍微概括一下这个定义，因为我们希望它允许更普遍的信念。

So just rewriting, Player i's strategy, same thing, Ŝi is a best response.

所以只需重写，玩家 i 的策略，同样的事情，Ŝi 是最好的回应。

But now let's be careful, best response to the belief P about the other player's choices, if ─ and it is going to look remarkably similar except now I'm going to have expectation ─ if the expected payoff to Player i from choosing Ŝi, given that she holds this belief P, is bigger than her expected payoff from choosing any other strategy, given she holds this belief P; and this better hold for all Si' that she could choose.

但现在让我们小心一点，对关于其他玩家的选择的信念 P 的最佳回应是，如果 — 并且看起来非常相似，除了现在我将有期望 — 如果玩家 i 从选择 Ŝi 获得的预期收益（考虑到她持有这种信念 P）大于她选择任何其他策略的预期收益（考虑到她持有这种信念 P）；这对她可以选择的所有 Si' 来说都更好。

So very similar idea, but the only thing is, I'm slightly abusing notation here by saying that my payoff depends on my strategy and a belief, but what I really mean is my expected payoff.

非常相似的想法，但唯一的是，我在这里稍微滥用了符号，说我的回报取决于我的策略和信念，但我真正的意思是我的预期回报。

This is the expectation given this belief.

这是给予这种信念的期望。

Once again, we can write it the other way, or Ŝi solves max when I choose Si, to maximize my expected payoff this time from choosing Si against S-i.

再次，我们可以用另一种方式写它，或者当我选择 Si 时 Ŝi 求解 max，以最大化我这次选择 Si 而不是 S-i 的预期收益。

What do I mean by expected payoff?

预期收益是什么意思？

Just in our example, so just to make clear what that expectation means, so for example, the expected payoff to Player i in the game above from choosing left given she holds the belief P is equal to the probability that the goal keeper dived to the left, times Player i's payoff from choosing left against left, plus the probability that the goal keeper dived to the right, times Player i's expected payoff from choosing left against right.

就在我们的例子中，为了明确这个期望的含义，例如，在上面的游戏中，玩家 i 持有信念 P 时选择向左的预期收益等于守门员向左扑救的概率，乘以玩家 i 选择左侧对抗左侧的收益，加上守门员向右扑救的概率，乘以玩家 i 选择左侧对抗右侧的预期收益。

Okay, so expectation with respect to P just means exactly what you expect it to mean.

好的，所以对 P 的期望正是您所期望的意思。

So this is a little bit of math, a little formality, but is everyone okay with that?

所以这有点数学，有点形式，但每个人都同意吗？

I haven't done anything here.

我在这里什么也没做。

All I've done is write down slightly boringly and nerdily, exactly what we already saw in a couple of occasions.

我所做的只是略显无聊和书呆子般地写下来，这正是我们在几次场合所看到的。

Professor Ben Polak: Thank you.

本·波拉克教授：谢谢。

So that right now is going to seem a little bit like a sudden blast of notation, so let's just remind ourselves what we really care about is the idea, it's not the notation, and let's spend the next half hour applying these ideas to an application.

所以现在看起来有点像突然爆发的符号，所以让我们提醒自己我们真正关心的是想法，而不是符号，让我们在接下来的半个小时里将这些想法应用到应用程序中。

So this application is not as important as soccer, but it's a bit more Economicsy, so I can justify it under the Economics title of the class.

所以这个应用程序不像足球那么重要，但它更经济一些，所以我可以在课程的经济学标题下证明它的合理性。

So clearing off my soccer game.

所以清理我的足球比赛。

So imagine--What we're going to look at is a game involving a partnership.

所以想象一下——我们将要看到的是一款涉及合作的游戏。

So Partnership Game.

所以伙伴关系游戏。

And I believe this game is covered in some detail in the Watson textbook, or something very close to it is, if you're having trouble.

我相信 Watson 教科书中对这个游戏有一些详细的介绍，或者非常接近它的内容，如果您遇到麻烦的话。

The idea is this.

想法是这样的。

There are two individuals who are going to supply an input to a joint project.

有两个人将为联合项目提供意见。

So that could be a firm, it could be a law firm, for example, and they're going to share equally in the profits .

例如，这可能是一家公司，也可能是一家律师事务所，他们将平等分享利润。

So one example would be a firm that they both earn, sorry, they both own, and another example would be two of you working as a study group on my homework assignment.

因此，一个例子是他们都赚钱的一家公司，抱歉，他们都是自己的，另一个例子是你们两个作为一个学习小组来完成我的家庭作业。

So they're going to share equally in the profits of this firm, or this joint project, but you're going to supply efforts individually.

因此，他们将平等分享这家公司或这个联合项目的利润，但你将单独付出努力。

So let's just be a bit more formal.

那么我们就正式一点吧。

So the players are going to be the two agents and they own this firm let's call it.

因此，球员将成为两名经纪人，他们拥有这家公司（我们可以这样称呼它）。

They own this firm jointly and they split the profits, so they share 50% of the profits each.

他们共同拥有这家公司，并分割利润，因此他们各分享 50% 的利润。

So it's a profit- sharing partnership.

所以这是一种利润分享的伙伴关系。

Each agent is going to choose her effort level to put into this firm.

每个代理人都会选择她在这家公司投入的努力水平。

So, it could be that you're deciding, as a lawyer, how many hours you're going to spend on the job.

因此，作为一名律师，您可能正在决定要在工作上花费多少时间。

So for most of you these decisions will be a question of whether you spend 20 hours a day at the firm or 21 hours a day at the firm, something like that.

因此，对于你们中的大多数人来说，这些决定将取决于您是每天在公司工作 20 小时还是每天在公司工作 21 小时，诸如此类。

For most of you on your homework assignments, I'm hoping it's a little less than that, but not much less than that.

对于你们大多数人的家庭作业来说，我希望比这个少一点，但也不会比这个少很多。

So the strategy choices, we're not going to do it in hours, let's just normalize and regard these choices as living in 0 to 4, and you can choose any number of hours between 0 and 4.

所以策略选择，我们不会以小时来进行，我们只是将这些选择标准化并视为生活在0到4之间，你可以选择0到4之间的任何小时数。

Just to mention as we go past it, a novelty here.

顺便提一下，当我们经过它时，这里有一个新奇之处。

Every game we've seen in the class so far has had a discrete number of strategies.

到目前为止，我们在课堂上看到的每款游戏都有不同数量的策略。

Even the game, when you chose numbers, you chose numbers 1, 2, 3, 4, 5, up to a 100, there were 100 strategies.

即使是游戏，当你选择数字时，你选择数字1、2、3、4、5，一直到100，有100种策略。

Here there's a continuum of strategies.

这里有一个连续的策略。

You could choose any real number in the interval [0, 4].

您可以选择区间 [0, 4] 中的任何实数。

So you have a continuum of possible choices.

所以你有一系列可能的选择。

That's not going to bother us but let's point out it's there.

这不会打扰我们，但让我们指出它的存在。

So there's a continuum of strategies.

所以有一个连续的策略。

In principle, you could bill your clients for fractions of a second or fractions of a minute.

原则上，您可以按几分之一秒或几分之一分钟向客户收取费用。

Let's wonder what the firm profit is given by.

让我们想知道公司的利润是由什么给出的。

So this partnership, this law firm, its profit is given by the following expression: 4 times the effort of Player I, plus the effort of Player II, plus a parameter I'll call B, times the product of their efforts.

所以这个合伙企业，这个律师事务所，它的利润由下面的表达式给出：4倍玩家I的努力，加上玩家II的努力，加上一个我称之为B的参数，乘以他们努力的乘积。

This is their profit.

这是他们的利润。

And I won't tell you what B is for now, but let's just--I mean I won't tell you exactly what it is--but I'll explain it.

现在我不会告诉你 B 是什么，但我们只是——我的意思是我不会告诉你它到底是什么——但我会解释一下。

We'll assume that B lives between 0 and a 1/4 and it's known, I just want to be able to vary it later.

我们假设 B 位于 0 到 1/4 之间，并且这是已知的，我只是希望稍后能够改变它。

So what's the idea here?

那么这里的想法是什么？

The idea is Player I directly contributes profits to the firm by working, as does Player II.

这个想法是玩家一通过工作直接为公司贡献利润，玩家二也是如此。

But they also contribute through this interaction term.

但他们也通过这个交互项做出贡献。

How do we think of that interaction term?

我们如何看待这个交互术语？

How do we think of that term B S1 S2?

我们如何看待 B S1 S2 这个术语？

When you're working on your homework assignments, if your product, the thing you hand in was just S1 + S2, then you might think what?

当你做作业的时候，如果你的产品，你交的东西只是S1 + S2，那么你可能会想什么呢？

You might think there's no point working in a study group at all.

您可能认为在学习小组中工作根本没有意义。

If the product is just the sum or multiple of a sum of the inputs, there wouldn't be much of a point working in a team at all.

如果产品只是输入总和或总和的倍数，那么在团队中工作根本没有多大意义。

It's the fact you're getting this extra benefit from working with someone else that makes it worth while working as a team to start with.

事实上，你从与其他人合作中获得了额外的好处，这使得作为一个团队工作一开始是值得的。

Is that right?

是这样吗？

So we can think of this term has to do with complementarity, or synergy, a very unpopular word these days but still: synergy.

因此，我们可以认为这个术语与互补性或协同作用有关，这是一个现在非常不流行的词，但仍然是：协同作用。

So we're going to assume that when you work together there are some synergies.

所以我们假设当你们一起工作时会产生一些协同效应。

Some of you are good at some parts of the homework, some of them are good at other parts of the homework.

你们中的一些人擅长作业的某些部分，另一些人擅长作业的其他部分。

And so in this law firm, one of these guys is an expert on intellectual property and the other one on fraud or something.

因此，在这家律师事务所中，其中一位是知识产权专家，另一位是欺诈或其他方面的专家。

So I've got the agents.

所以我有代理。

I've got the strategy set.

我已经制定了策略。

I know something about the profits of the firm.

我对公司的利润有所了解。

I need to tell you about their payoffs.

我需要告诉你他们的回报。

So the payoffs: the payoff for Player I is going to depend, of course, on her choice and on the choice of her partner, and it's going to equal a ½--because they're splitting the profits--so a ½ of the profits.

因此，收益：玩家 I 的收益当然取决于她的选择和她的合作伙伴的选择，这将等于 1/2——因为他们正在分割利润——所以是 1/2 的利润。

So a ½ of 4 times S1 plus S2 plus B S1S2.

所以 4 乘以 S1 加 S2 加 B S1S2 的 1/2。

She gets half of those profits but it also costs her S1 squared.

她获得了一半的利润，但也损失了她的 S1 平方。

So S1 squared is her effort costs, it's her input costs.

所以 S1 的平方就是她的努力成本，也就是她的投入成本。

This is the effort cost.

这就是努力成本。

Similarly, Player II--everything's symmetric here--Player II's payoff is the same thing.

同样，玩家 II——这里一切都是对称的——玩家 II 的回报是相同的。

This term is the same except we're going to subtract off Player II's efforts squared: S2 squared.

这个术语是相同的，只是我们要减去玩家 II 的努力的平方：S2 平方。

So you get the profits of the firm minus the disutility of having missed all that sleep.

这样你就得到了公司的利润减去错过所有睡眠的负效用。

There's a guy in about the fifth row there who's missed too much sleep, so somebody just nudge him.

大约第五排有一个人睡得太久了，所以有人轻轻推了推他。

That's it, good.

就是这样，很好。

We won't put him on camera just nudge him.

我们不会把他放在镜头前，只是轻推他。

That's it, good.

就是这样，很好。

There you go.

就这样吧。

Next time we'll use the camera for that.

下次我们将使用相机。

So now we have everything we want to analyze this firm and to analyze how things are going to work, either when you're working on your homework assignments or in the law partnership.

现在，我们拥有了分析这家公司以及分析事情将如何运作的一切，无论是在您做家庭作业时还是在法律合伙企业中。

Again, just to make this relevant to you, I mean this is very stylized of course, but a huge number of businesses out there are partnerships and do have this kind of profit sharing rule and do have synergies.

再说一遍，只是为了让这与您相关，我的意思是，这当然是非常程式化的，但是大量的企业存在合作伙伴关系，并且确实有这种利润分享规则，并且确实具有协同效应。

So this is a relevant issue in a lot of businesses.

因此，这是许多企业面临的一个相关问题。

Now we're going to analyze this--no secret here--we're going to analyze this using the idea of best response.

现在我们要分析这一点——这里不是秘密——我们将使用最佳响应的想法来分析这一点。

That's not a surprise to any of you since that's where we started the day.

这对你们中的任何人来说都不奇怪，因为这就是我们今天开始的地方。

So, in particular, I want to figure out what is Player I's best response to each possible choice of Player II?

所以，我特别想弄清楚玩家 I 对玩家 II 的每个可能选择的最佳反应是什么？

What is Player I's best response for each possible choice S2 of Player II?

对于玩家 II 的每个可能选择 S2，玩家 I 的最佳反应是什么？

How should I go about doing that?

我应该怎样做呢？

How should I do that?

我该怎么做呢？

So here what we did before was we drew these graphs with probabilities, with beliefs of Player I and the problem here is, previously we had a nice simple graph to draw because there were just two strategies for Player II.

所以我们之前所做的就是用概率和玩家 I 的信念绘制这些图表，这里的问题是，之前我们有一个很好的简单图表可以绘制，因为玩家 II 只有两种策略。

Player II was a goalie, he could dive to the left or the right.

球员二是一名守门员，他可以向左或向右扑救。

Problem is here, that Player II has a continuum of strategies and trying to draw all possible probabilities over an infinite number of objects on the board is more than my drawing can do.

问题在于，玩家 II 有一个连续的策略，并且试图在棋盘上无限数量的对象上绘制所有可能的概率，这超出了我的绘图能力。

Too hard.

太难了。

So we need some other technique.

所以我们需要一些其他的技术。

How are we going to find out Player I's best response?

我们如何找出玩家 I 的最佳反应？

Somebody?

有人吗？

Wave your hands in the air, way back in the corner.

在空中挥动双手，远离角落。

Can I, can we, let's get the mike.

我可以吗，我们可以吗，我们去拿麦克风吧。

Stand up but wait for the mike to come to you.

站起来，等待麦克风向您走来。

How are we going to do it?

我们要怎么做呢？

How are we going to figure out what Player I's best response is?

我们如何找出玩家 I 的最佳反应是什么？

Shout loudly.

大声喊叫。

Professor Ben Polak: Good, okay.

本·波拉克教授：好的，好的。

That's certainly the first step.

这当然是第一步。

We've got that, actually we've got that.

我们已经得到了，实际上我们已经得到了。

So here's Player I's payoff as a function of what Player II chooses and what Player I chooses, so we have that already.

因此，这是玩家 I 的收益作为玩家 II 选择的内容和玩家 I 选择的函数的函数，所以我们已经有了。

We have Player I's payoff as a function of the two efforts and now I want to find out what is Player I's best efforts given a particular choice of S2?

我们将玩家 I 的回报作为两项努力的函数，现在我想找出在给定 S2 的特定选择的情况下，玩家 I 的最佳努力是什么？

Yeah.

是的。

Professor Ben Polak: Okay, good.

本·波拉克教授：好的，很好。

So we're going to use calculus.

所以我们要使用微积分。

We're going to use calculus of one variable.

我们将使用一个变量的微积分。

We're just changing one variable, S1.

我们只是改变一个变量，S1。

How many of you--we won't let the camera see you--how many of you have not--I'm not going to show hands at all.

你们中有多少人——我们不会让摄像机看到你们——你们中有多少人没有——我根本不会举手。

If you have not seen the calculus that I'm about to use on the board, or more likely, if you've forgotten it since high school, don't panic.

如果您还没有看到我将在黑板上使用的微积分，或者更有可能的是，如果您从高中就忘记了它，请不要惊慌。

There is a chapter in the back of the book, I think it's chapter 25, that goes over this, it refreshes your memory of such calculus.

书的后面有一章，我想是第25章，回顾了这一点，它刷新了你对这种微积分的记忆。

And if you haven't, if you've never seen it before, if you haven't taken, for example, the equivalent of Math 112, come and see us.

如果您还没有，如果您以前从未见过，如果您还没有学过，例如，相当于数学 112 的课程，请来见我们。

We'll probably try and line up a quick calculus lesson, a special section for those people.

我们可能会尝试安排一门快速微积分课程，这是专门为这些人准备的部分。

So if what I'm going to do now is scary, come and see us and we'll deal with it.

因此，如果我现在要做的事情令人恐惧，请来见我们，我们会处理它。

All right, what I'm going to do, we want to take a derivative of this thing.

好吧，我要做的就是，我们想要对这个东西进行衍生。

What we're going to do is, we're asking the question, what is the maximum, choosing S1, of this profit.

我们要做的是，问这个利润的最大值是多少（选择 S1）。

Can I multiply the ½ by the 4 just to save myself some time?

我可以将 1/2 乘以 4 来节省一些时间吗？

So the profit is 2 S1 plus S2 plus B S1 S2 minus S1 squared.

所以利润是 2 S1 加 S2 加 B S1 S2 减去 S1 的平方。

We're asking the question, taking S2 as given, what S1maximizes this expression and as the gentleman at the back said, I'm going to differentiate and then I'm going to set the thing equal to 0.

我们问的问题是，给定 S2，S1 最大化这个表达式，正如后面那位先生所说，我将进行微分，然后将其设置为 0。

So I'm almost bound to get this wrong on the board.

所以我几乎肯定会在黑板上犯这个错误。

So can you all watch me like a hawk a second?

那么你们能像鹰一样看着我吗？

So if we differentiate this object, I'm going to find a first order condition in a second.

因此，如果我们对这个对象进行微分，我将在一秒钟内找到一阶条件。

All right, so we differentiate.

好吧，所以我们要区分。

I'm going to have 2 still, and then this S1 is going to become a 1, and this S1 here is going to become a plus B S2, everyone happy with that?

我还会有2，然后这个S1会变成1，这里的S1会变成plus B S2，大家都满意吗？

This S1 squared is going to become a minus 2S1.

这个 S1 的平方将变成负 2S1。

That was just differentiating.

那只是差异化。

Everyone happy with the way I differentiated?

每个人都对我的差异化方式感到满意吗？

Is this coming back from high school?

这是高中回来的吗？

The cogs are spinning now?

齿轮现在在旋转吗？

To make this a first order condition, I'm going to say "at the best response," put a hat over the 1.

为了使其成为一阶条件，我要说“处于最佳响应”，在 1 上加一顶帽子。

At the best response this is equal to 0.

在最佳响应下，该值等于 0。

Yeah Tae, can you get the guy again.

是的，泰，你能再找那个人吗？

Professor Ben Polak: Okay, you're right to shout out because I'm very--I mean doing it on the board I'm very likely to make mistakes, but okay.

Ben Polak 教授：好吧，你大喊大叫是对的，因为我非常——我的意思是在黑板上做这件事我很可能会犯错误，但是好吧。

So I differentiated this object, this is my first derivative and I set it equal to 0.

所以我对这个对象进行了微分，这是我的一阶导数，我将其设置为 0。

Now in a second I'm going to work with that, but I want to make sure I'm going to find a maximum and not a minimum, so how do I make sure I'm finding a maximum and not a minimum?

现在我要处理这个问题，但我想确保我会找到最大值而不是最小值，那么我如何确保找到最大值而不是最小值呢？

I take a look at the second derivative, which is the second order condition.

我看一下二阶导数，这是二阶条件。

So I'm going to differentiate this object again with respect to S1,.

所以我要再次微分这个对象关于S1，。

Pretend the hat isn't there a second.

假装帽子不在那儿。

And none of this has an S1 in it, so that all goes away.

而且这些都没有 S1，所以一切都消失了。

And I'm going to get minus 2, which came from here: minus 2 and that is in fact negative, which is what I wanted to know.

我会得到负2，它来自这里：负2，实际上是负数，这就是我想知道的。

To find a maximum I want the second derivative to be negative.

为了找到最大值，我希望二阶导数为负。

So here it is, I've got my first order condition.

所以这就是我的第一个订单条件。

It tells me that the best response to S2 is the Ŝ1 that solves this equation, that solves this first order condition.

它告诉我，对 S2 的最佳响应是求解该方程的 Ŝ1，即求解该一阶条件的函数。

We can just rewrite that, if I divide through by 2 and rearrange, it's going to tell me that Ŝ1, or if you like, Ŝ1 is equal to 1 plus B S2.

我们可以重写它，如果我除以 2 并重新排列，它会告诉我 Ŝ1，或者如果你愿意，Ŝ1 等于 1 加 B S2。

So this thing is equal to Player I's best response given S2.

所以这个东西等于玩家 I 对 S2 的最佳反应。

Now I could go through again and do exactly the same thing for Player II, but I'm not going to do that because everything's symmetric.

现在我可以再次经历并为玩家 II 做完全相同的事情，但我不会这样做，因为一切都是对称的。

So everyone happy with that?

那么大家对此都满意吗？

So I could at the same--I could do the same kind of analysis but we know I'll get the same answer.

所以我可以同时进行同样的分析，但我们知道我会得到相同的答案。

So similarly, I would find that Ŝ2 equals 1 plus B S1 and this is the best response of Player II, as it depends on Player I's choice of effort S1.

因此，类似地，我会发现 Ŝ2 等于 1 加 B S1，这是玩家 II 的最佳响应，因为它取决于玩家 I 对努力 S1 的选择。

Okay, now I found out what Player I's best response is to Player II, and what Player II's best response is to Player I for each possible choice of Player II up here, and for each possible choice of Player I down there.

好的，现在我发现了玩家 I 对玩家 II 的最佳反应是什么，以及玩家 II 对玩家 I 的最佳反应是什么，对于这里玩家 II 的每个可能的选择，以及下面玩家 I 的每个可能的选择。

Now, let's see if we can get a bit further.

现在，让我们看看是否可以更进一步。

And to get a bit further, let's draw a picture.

为了更进一步，让我们画一张图。

What I'm going to do is I want to draw the two functions we just found and see what they look like.

我要做的就是画出我们刚刚找到的两个函数，看看它们是什么样子的。

This is all in your notes already, so I can get rid of it.

这些都已经在你的笔记中了，所以我可以摆脱它。

What I can do with is some more chalk.

我能做的就是再用一些粉笔。

Excuse me.

打扰一下。

So what I'm going to do is, let's draw a picture that has S1 on the horizontal axis and S2 on the vertical axis.

所以我要做的是，让我们画一张横轴为 S1、纵轴为 S2 的图片。

And there are different choices here 1, 2, 3, and 4 for Player I, and here's the 45º line.

对于玩家 I，这里有不同的选择 1、2、3 和 4，这是 45° 线。

If I'm careful I should get this right 1, 2, 3, and 4 are the possible choices for Player I.

如果我小心的话，我应该能猜对，1、2、3 和 4 是玩家 I 的可能选择。

Now before I draw it I better decide what B is going to be.

现在，在绘制之前，我最好先决定 B 是什么。

Okay, so I'm going to draw for the case--I'm going to draw the best response of Player I and I'm going to draw the best response for Player II in a minute, for the case B equals 1/4.

好的，所以我要画出这种情况——对于情况 B 等于 1/4，我将在一分钟内画出玩家 I 的最佳响应，并且我将画出玩家 II 的最佳响应。

So we said B was somewhere between 0 and 1/4, let's draw the case for B equals 1/4.

所以我们说 B 介于 0 和 1/4 之间，让我们画出 B 等于 1/4 的情况。

So the expression I want to draw first of all is the best response of Player I as a function of S2and we agreed that that was given by 1 plus 1/4 now, 1 plus 1/4 S2.

所以我首先要画的表达式是玩家 I 作为 S2 的函数的最佳响应，我们一致认为这是由 1 加 1/4 现在给出的，1 加 1/4 S2 。

So for each possible choice of S2, I'm going to draw Player I's best response and we'll do it in red.

因此，对于 S2 的每个可能选择，我将绘制玩家 I 的最佳响应，并用红色表示。

So if Player II chooses 0, what is Player I's best response?

那么如果玩家 II 选择 0，那么玩家 I 的最佳反应是什么？

Somebody shout out.

有人喊出来。

Professor Ben Polak: 1, okay.

本·波拉克教授：1，好的。

So 1 plus 1/4 of 0 is 1, so if Player II chooses 0, Player I's best response is to choose 1.

所以 1 加 0 的 1/4 就是 1，所以如果玩家 II 选择 0，玩家 I 的最佳反应是选择 1。

What if Player II chooses 4?

如果玩家 II 选择 4 怎么办？

If Player II chooses 4, what would be Player I's best response?

如果玩家 II 选择 4，玩家 I 的最佳反应是什么？

So it'll be 1 plus 1/4 times 4, 1/4 times 4 is 1, so 1 plus 1 is 2, so Player II's best response in that case will be 2.

所以它将是 1 加 1/4 乘以 4，1/4 乘以 4 是 1，所以 1 加 1 是 2，所以玩家 II 在这种情况下的最佳反应将是 2。

So if Player I chooses 4, Player II should choose, I'm sorry, Player II chooses 4, Player I should choose 2, and this is a straight line in between.

所以如果玩家 I 选择 4，玩家 II 应该选择，对不起，玩家 II 选择 4，玩家 I 应该选择 2，这是一条中间的直线。

So the line I've just drawn is the best response for Player I as it depends on Player II's choice.

所以我刚刚画的线是对玩家 I 的最佳响应，因为它取决于玩家 II 的选择。

Everyone happy with how I drew that?

大家对我的画法满意吗？

I'm assuming you're taking it on faith that it is a straight line in between, but it is.

我假设你相信它是中间的一条直线，但事实确实如此。

The way we read this graph, is you give me an S2, I read across to the pink line and drop down, and that tells me the best response for Player I.

我们阅读这张图的方式是，你给我一个 S2，我读到粉色线并下拉，这告诉我玩家 I 的最佳反应。

Now we can do the same for Player II, we can draw Player II's best response as it depends on the choices of Player I, but rather than go through any math, I already know what that line's going to look like.

现在我们可以对玩家 II 做同样的事情，我们可以画出玩家 II 的最佳响应，因为它取决于玩家 I 的选择，但我不需要进行任何数学计算，我已经知道那条线会是什么样子。

What does that line look like?

那条线是什么样子的？

Somebody raise your hand.

有人举手。

Somebody?

有人吗？

What will Player II's best response look like as a function of Player I's choice in the same picture?

根据同一张图片中玩家 I 的选择，玩家 II 的最佳反应会是什么样子？

Someone we haven't had before, we've had all these guys before, someone else.

我们以前没有过的人，我们以前有过所有这些人，其他人。

Yeah, there's a guy in the middle, can we get in to him?

是的，中间有一个人，我们可以接近他吗？

Yeah, maybe it's easier from that side.

是的，也许从那方面来说更容易。

Shout out loud so the mike can hear you.

大声喊出来，以便麦克风能够听到您的声音。

Professor Ben Polak: Right, exactly.

本·波拉克教授：对，完全正确。

So if I drew the equivalent line for Player II, which is Player II's best response for each choice of Player I, we're simply flipping the identities of the players, which means we'll be reflecting everything in that 45º line.

因此，如果我为玩家 II 画一条等效线，即玩家 II 对玩家 I 的每个选择的最佳反应，我们只是翻转玩家的身份，这意味着我们将反映该 45° 线中的所有内容。

So it'll go from 1 here to 2 here, and it'll look like this.

所以它会从这里的 1 变成这里的 2，它看起来像这样。

So this the best response for Player II for every possible choice of Player I, and just to make sure we understand it, what this blue line tells me is you give me an S1, an effort level of Player I, I read up to the blue line and go across and that tells me Player II's best response.

因此，这是玩家 II 对于玩家 I 的每个可能选择的最佳响应，只是为了确保我们理解它，这条蓝线告诉我的是，你给我一个 S1，即玩家 I 的努力水平，我读到蓝线并越过，它告诉我玩家 II 的最佳响应。

Okay, now we're making some progress.

好的，现在我们正在取得一些进展。

What do we notice?

我们注意到什么？

Remember we said that one of the lessons of today's class, the second lesson.

还记得我们说过今天这堂课的第一课，第二课。

The first lesson was don't shoot towards the middle of the goal and the second more general lesson was what?

第一个教训是不要射向球门中间，第二个更一般的教训是什么？

It was don't ever play a strategy that is not a best response to everything.

永远不要采取不能对所有事情做出最佳反应的策略。

I admit I'm cheating a little bit here because I'm ignoring beliefs, but trust me that's okay in this game.

我承认我在这里有点作弊，因为我忽略了信仰，但相信我，在这个游戏中这没关系。

So are there strategies here that are never a best response to anything?

那么，是否有一些策略永远都不是对任何事情的最佳反应呢？

Put another way, what strategies of Player I's are ever a best response?

换句话说，玩家 I 的哪些策略是最佳回应？

Anybody?

有人吗？

Well, let's have a look.

好吧，让我们来看看。

If Player II chooses 0 then Player I's best response is 1, and that's as low as he ever goes.

如果玩家 II 选择 0，那么玩家 I 的最佳反应就是 1，这是他所能做出的最低反应。

So these strategies down here less than 1 are never a best response for Player I.

因此，对于玩家 I 来说，这些小于 1 的策略从来都不是最佳应对策略。

If Player II chooses 4, then the synergy leads Player I to raise his best response all the way up to 2, but these strategies up here above 2 are never a best response for Player I.

如果玩家 II 选择 4，那么协同作用会导致玩家 I 将他的最佳响应一直提高到 2，但是高于 2 的这些策略对于玩家 I 来说永远不是最佳响应。

Is that right?

是这样吗？

So the strategies below 1 and above 2 are never a best response for Player I.

因此，对于玩家 I 来说，低于 1 和高于 2 的策略从来都不是最佳应对策略。

Similarly, for Player II, the lowest Player I could ever do, is choose 0, in which case Player II would want to choose 1, so the strategies below 1 are never a best response for Player II.

类似地，对于玩家 II，我可以做的最低玩家选择是选择 0，在这种情况下，玩家 II 会想要选择 1，因此低于 1 的策略对于玩家 II 来说永远不是最佳响应。

And the strategies above 2 were never a best response for Player II.

对于玩家 II 来说，上述 2 种策略从来都不是最佳应对策略。

So let's actually--you might want to be a little bit gentle in your own notebook--but on my board let's get rid of all these strategies that are never a best response.

所以，实际上——你可能想在自己的笔记本上温和一点——但在我的董事会上，让我们摆脱所有这些永远不是最佳回应的策略。

So all of these strategies for Player I are gone, and all of these strategies for Player I are gone.

因此，针对玩家 I 的所有这些策略都消失了，针对玩家 I 的所有这些策略也都消失了。

You might want to not scribble quite so much on your own notebook, but still.

您可能不想在自己的笔记本上乱写太多，但仍然如此。

And all these strategies for Player II are gone, and all these strategies for Player II are gone, and what's left?

而玩家二的所有这些策略都消失了，玩家二的所有这些策略都消失了，还剩下什么？

A lot of scribble is left.

留下了很多字迹。

What's left?

还剩下什么？

So I claim if you look carefully there's a little box in here that's still alive.

所以我断言，如果你仔细看的话，这里有一个小盒子还活着。

I've deleted all the strategies that were best--that are never best responses for Player I and all the strategies that are never best responses for Player II, and what I've got left is that little box.

我已经删除了所有最好的策略——对于玩家 I 来说，这些策略永远不是最佳响应，而对于玩家 II 来说，所有策略从来都不是最佳响应，我剩下的就是那个小盒子。

But I can't see that little box, so what I'm going to do is I'm going to redraw that little box.

但我看不到那个小盒子，所以我要做的就是重新绘制那个小盒子。

So let's redraw it.

那么让我们重新绘制它。

So it goes from 1 to 2 this time.

所以这次从1变成2。

I'm just going to blow up that box.

我只想炸毁那个盒子。

So this now is 1, 1 and up here is 2, 2 and let's put in numbers of quarters, so this will be, what will it be?

现在是 1, 1，这里是 2, 2，让我们输入季度数，所以这将是，它将是多少？

It'll be 5/4, 6/4 and 7/4, and over here it'll be 5/4, and 6/4, and 7/4.

它将是 5/4、6/4 和 7/4，这里将是 5/4、6/4 和 7/4。

And let's just draw how those, that pink and blue line look in that box.

让我们画出那些粉色和蓝色的线在那个盒子里的样子。

This is just a picture of that little box, so it's going to turn out it goes from--the pink line goes from here to here and the blue line goes from here to here.

这只是那个小盒子的图片，所以它会从——粉红色的线从这里到这里，蓝色的线从这里到这里。

We can work it out at home and check it carefully, but this isn't that incorrect.

我们可以在家算出并仔细检查，但这并不是那么错误。

So what I've done is I've redrawn the picture we just had and blown it up.

所以我所做的就是重新绘制我们刚才的图片并将其放大。

And have any of you seen that picture before?

你们以前见过那张照片吗？

Anyone here seen that picture before?

这里有人以前见过那张照片吗？

That's the picture we just had, except I've changed the numbers a bit.

这就是我们刚刚的图片，只是我稍微改变了数字。

Once I deleted all the strategies that were never a best response and just focused on that little box of strategies that survived, the picture looked exactly the same as it did before, albeit it blown up and the numbers changed.

一旦我删除了所有从来都不是最佳反应的策略，只关注幸存下来的那一小盒策略，图片看起来就和以前一模一样，尽管它被放大了，数字也发生了变化。

So what have we done so far?

那么到目前为止我们做了什么？

We said players should never play a strategy that's never a best response to anything, so we threw those away.

我们说玩家永远不应该采取对任何事情都不是最佳反应的策略，所以我们放弃了这些策略。

Now what's left?

现在还剩下什么？

What should we do now?

我们现在应该做什么？

So some of the strategies that we didn't throw away were best responses to things, but the things they were best responses to have now been thrown away.

因此，我们没有抛弃的一些策略是对事物的最佳反应，但它们是对事物的最佳反应现在已经被抛弃了。

Is that right?

是这样吗？

This should be something familiar from when we were deleting dominated strategies.

这在我们删除主导策略时应该是熟悉的。

The strategies I'm about to throw away now, they're not--it isn't that they're not best responses, they are best responses to something.

我现在要抛弃的策略不是——并不是说它们不是最佳反应，而是对某些事情的最佳反应。

But the things they were best responses to, we know are not going to be played, because they themselves were not best responses to anything.

但我们知道，他们对某些事情的最佳反应不会被播放，因为它们本身并不是对任何事情的最佳反应。

So what strategies do I have in mind?

那么我有什么策略呢？

What strategies am I about to throw away?

我要放弃哪些策略？

Well, for example, for Player I we know now that Player II is never going to choose any strategy below 1, and so the lowest Player II will ever choose is 1, and it turns out that the lowest Player II would ever do in response to anything is 1 and above, never leads Player I to choose a strategy less than 5/4.

好吧，例如，对于玩家 I，我们现在知道玩家 II 永远不会选择任何低于 1 的策略，因此玩家 II 将选择的最低策略是 1，而事实证明，玩家 II 对 1 及以上的任何事物做出的最低反应将永远不会导致玩家 I 选择小于 5/4 的策略。

The highest Player II ever chooses is 2, and the highest response that Player I ever makes to any strategy 2 or less is 6/4, so all these things bigger than 6/4 can go.

玩家 II 选择的最高值是 2，而玩家 I 对任何策略 2 或更少的策略做出的最高响应是 6/4，因此所有大于 6/4 的事情都可以进行。

Let's be careful here.

让我们小心点。

These strategies I'm about to delete, it isn't that they're never best responses, they were best responses to things, but the things they were best responses to, are things that are never going to be played, so they're irrelevant.

我要删除的这些策略，并不是说它们永远不是最佳反应，它们是对事物的最佳反应，而是它们对事物的最佳反应是永远不会被玩的东西，所以它们是无关紧要的。

So we're throwing away all of the strategies less than 5/4 for Player I and bigger than 6/4 for Player I, (which is 1½ for Player I) and similarly for Player II.

因此，我们将丢弃所有小于玩家 I 5/4 的策略和大于玩家 I 6/4 的策略（对于玩家 I 为 1½），对于玩家 II 也类似。

And if I did this--and again, don't scribble too much in your notes--but if we just make it clear what's going on here, I'm actually going to delete these strategies since they're never going to be played--I end up with a little box again.

如果我这样做了——再说一遍，不要在笔记中写太多——但如果我们只是弄清楚这里发生了什么，我实际上会删除这些策略，因为它们永远不会被玩——我最终会再次得到一个小盒子。

So everyone see what I did?

那么大家都看到我做了什么吗？

I started with a game.

我从一个游戏开始。

I found out what Player I's best response was for every possible choice of Player II, and I found out what Player II's best response was for every possible strategy of Player I.

我发现了玩家 I 对玩家 II 的每种可能选择的最佳反应是什么，并且我发现了玩家 II 对于玩家 I 的每种可能策略的最佳反应是什么。

I threw away all strategies that were never a best response, then I looked at the strategies that were left.

我扔掉了所有从来都不是最佳反应的策略，然后我查看了剩下的策略。

I said those strategies that were a best response to things that have now been thrown away, but not best response otherwise, I can throw those away too.

我说那些策略是对现在已经扔掉的东西的最佳反应，但不是最好的反应，我也可以扔掉它们。

And when I threw those away, I was left once again with a little box, and I could do it again, and again, and again.

当我把它们扔掉时，我又剩下了一个小盒子，我可以一次又一次地这样做。

If I go on doing this exercise again, and again, and again what am I going to end up with?

如果我继续一次又一次地做这个练习，我最终会得到什么？

Shout it out, what am I going to end up with?

大声喊出来，我最后会怎样？

The intersection, right?

十字路口，对吗？

If I keep on constructing these boxes within boxes, so the next box would be a little box in here.

如果我继续在盒子中构建这些盒子，那么下一个盒子将是这里的一个小盒子。

I'm not going to draw it, but it's something like this.

我不会画它，但它是这样的。

But if you keep on doing boxes within boxes, I'm going to converge in on that intersection.

但如果你继续在盒子里做盒子，我就会在那个十字路口汇聚。

So if we know people are not going to play a best response--that's never a best response, and we know if we people are not going to play something which is never a best response, and we know people are not going to play which is not a best response, which is not a best etc., etc., etc. We're going to converge in, in this game, to just one strategy for each player, which is where they intersect.

因此，如果我们知道人们不会做出最好的反应——那永远不是最好的反应，我们知道如果我们人们不会玩一些永远不是最佳反应的东西，我们知道人们不会玩这不是最好的反应，这不是最好的等等。在这个游戏中，我们将集中在每个玩家的一个策略上，这就是他们的交叉点。

So what we're going to converge in on to, is the S1*, let's call it in this case, is equal to 1 plus B S2* and that S2* is equal to 1 plus B S1*.

因此，我们要收敛的是 S1*（在本例中我们称其为 S1*）等于 1 加 B S2*，并且 S2* 等于 1 加 B S1*。

Actually, we can do it a little better than that, since we know the game is symmetric, we know that S1* is actually equal to S2*.

实际上，我们可以做得更好一点，因为我们知道游戏是对称的，所以我们知道 S1* 实际上等于 S2*。

So taking advantage of the fact that we know S1 is equal to S2 (because we're lying on the 45º line), I can simplify things by making S1* equal to S2*.

因此，利用我们知道 S1 等于 S2（因为我们位于 45° 线上）这一事实，我可以通过使 S1* 等于 S2* 来简化事情。

So now I've got--actually, that looks like three equations, it's really just two equations, because one of them implies the other.

所以现在我得到了——实际上，这看起来像三个方程，实际上只是两个方程，因为其中一个蕴含着另一个。

And I can solve them, and if I solve them out I'm going to get something like (let me just be careful) I'm going to get something like: 1 minus B S1* is equal to 1, or S1* equals S2* is equal to 1 over (1--B).

我可以解决它们，如果我解决了它们，我会得到类似的结果（让我小心一点）我会得到类似的结果：1 减去 B S1* 等于 1，或者 S1* 等于 S2* 等于 1 除以 (1--B)。

And again, anytime I'm doing algebra on the board, someone should check me at home, so just have a quick look at that.

再说一次，每当我在黑板上做代数时，应该有人在家检查我，所以快速看一下即可。

Is that right?

是这样吗？

I think that's right.

我认为这是对的。

My algebra, which is often wrong, suggests that the solution is S1* equals S2* equals 1 over (1--B).

我的代数经常是错误的，它表明解决方案是 S1* 等于 S2* 等于 1 除以 (1--B)。

But what I'm doing, it's just math, there's nothing interesting going on.

但我所做的只是数学，没有什么有趣的事情发生。

I'm just trying to solve out for the equation of this point.

我只是想解出这一点的方程。

So what did we learn here?

那么我们在这里学到了什么？

We learned that in this game deleting strategies that are never best response, and then deleting strategies that are never best response to anything that is a best response and so on and so forth, yielded a single strategy for each player.

我们了解到，在这个游戏中，删除永远不是最佳响应的策略，然后删除永远不是最佳响应的策略，等等，为每个玩家产生一个单一的策略。

Just one strategy for each player and that strategy was given by this equation.

每个玩家只有一个策略，该策略是由这个方程给出的。

So if we were management consultants working for Mckinsey or something, and we were brought in to advise you on your homework assignments, or this law partnership on their work practices, we would come down with a prediction that this is how much work you're going to get.

因此，如果我们是为麦肯锡或其他公司工作的管理顾问，并且我们被请来为你的家庭作业提供建议，或者为这个法律合伙人提供他们的工作实践建议，我们会得出一个预测，这就是你将得到的工作量。

Question, is this amount of work a good amount of work or a bad amount of work?

请问，这个工作量是好工作量还是坏工作量？

Here you are, you're working for Mckinsey, you've been hired by Joe Smith and Ann Blogs to figure out their strategy, working on a problem in a team on working on my homework assignments.

在这里，你在麦肯锡工作，你被 Joe Smith 和 Ann Blogs 雇佣来制定他们的策略，在团队中解决我的家庭作业中的问题。

You figured out how much work they're going to contribute.

你算出了他们将贡献多少工作。

Is this a good amount of work?

这是一个很好的工作量吗？

Are they contributing too much, too little?

他们的贡献是太多还是太少？

Because the answer is, depending on, compared to what?

因为答案取决于与什么相比？

So let me rephrase, are these people, are these pair of partners in the firm, or two students working on their homework assignments, are they working more or less than an efficient level?

那么让我换个说法，这些人、公司里的这对合伙人，还是两个正在做家庭作业的学生，​​他们的工作效率是高于还是低于有效水平？

Let's have a poll, who thinks more?

我们来做个投票，看看谁的想法更多？

Turn the camera out into the audience, let's have a look.

把镜头转向观众，我们来看看。

Who thinks more?

谁想得更多？

Who thinks they're working just right?

谁认为他们工作得恰到好处？

Who thinks less?

谁想得少？

A lot of abstentions here.

这里有很多弃权票。

I think they're working too little here compared to what's efficient.

我认为与效率相比，他们在这里的工作太少了。

I'll get you to solve it out on a homework assignment, so you can actually prove that.

我会让你在家庭作业中解决这个问题，这样你就可以真正证明这一点。

You can prove that, in fact, if you were writing a contract, if there was a social planner you'd work more.

事实上，你可以证明，如果你正在写一份合同，如果有一个社会规划师，你会工作得更多。

But let's try and get to grips why.

但让我们试着弄清楚原因。

Why is it that when we see these law partners, or medical partners, or whatever it happens to be, or students together on a homework assignment, why is it we tend to get inefficiently little effort when we start figuring out the strategy and working through the game?

为什么当我们看到这些法律合作伙伴、医疗合作伙伴或其他任何东西，或者学生一起做作业时，为什么当我们开始制定策略并完成游戏时，我们往往会付出很少的努力？

I'm conceding the answer.

我承认这个答案。

I'm telling you they're going to work too little.

我告诉你他们的工作量太少了。

Why do they end up not working hard enough?

为什么他们最终不够努力？

Any takers?

有接受者吗？

Can we get a mike in here, yeah.

我们可以在这里拿个麦克风吗？

Student: Because if they work any harder than that, then the other person is just going to slack off instead.

学生：因为如果他们比这更努力，那么另一个人就会偷懒。

Professor Ben Polak: All right, so there's something about that, there's something.

本·波拉克教授：好吧，所以这是有道理的，有道理的。

On the other hand, this isn't really, I mean the intuition you're giving me is kind of a Prisoner's Dilemma intuition.

另一方面，这并不是真的，我的意思是你给我的直觉是一种囚徒困境的直觉。

Saying I'm going to let the other guy work and I'll shirk.

说我要让另一个人工作，我就会逃避。

But there's something, I think there's something in that, but there's a little bit more going on here, what more is going on?

但有一些东西，我认为其中有一些东西，但是这里发生了更多的事情，还发生了什么？

I think that's a good first step.

我认为这是很好的第一步。

There is something of that.

确实有这样的事情。

Yeah.

是的。

Student: If there are two people working together, there's about half as much work for each person to do.

学生：如果两个人一起工作，每个人要做的工作量大约是一半。

Professor Ben Polak: That's true, but that would suggest it doesn't matter if they slack off.

本·波拉克教授：确实如此，但这表明他们偷懒也没关系。

What's going on here, so go back to your Economics 115 or 150, if you took either of those courses.

这是怎么回事，如果您选修过经济学 115 或 150 课程，请返回这些课程。

What's the problem here?

这里有什么问题？

What's underlying the problem?

问题的根源是什么？

Let's get this guy in the pink down here.

让我们把这个穿粉红色衣服的家伙带到这里来。

Student: They only capture fifty percent of their marginal benefit.

学生：他们只获得了边际收益的百分之五十。

Professor Ben Polak: That's the point.

本·波拉克教授：这就是重点。

Good, well your name is?

很好，那么你叫什么名字？

Professor Ben Polak: So Patrick is giving, I think, it's the correct answer here.

本·波拉克教授：我认为帕特里克给出的答案是正确的。

The problem here isn't really about the amount of work.

这里的问题实际上并不在于工作量。

It isn't even, by the way, about the synergy.

顺便说一句，这甚至与协同作用无关。

You might think it's because of this synergy that they don't take into account correctly.

您可能认为这是因为他们没有正确考虑这种协同作用。

That isn't the problem here.

那不是这里的问题。

It turns out even without the synergy this problem would be there.

事实证明，即使没有协同作用，这个问题也会存在。

The problem is what Patrick said.

问题是帕特里克所说的。

The problem is that at the margin, I, a worker in this firm, be it a law partnership or a homework solving group, I put in, I bear the cost, at the margin, I bear the full cost at the margin for any extra unit of effort I put in, but I only reap half the benefits.

问题是，在边际上，我作为这个公司的员工，无论是法律合伙人还是家庭作业小组，我投入，我承担成本，在边际上，我付出任何额外的努力，我都承担边际上的全部成本，但我只获得一半的收益。

At the margin, I'm reaping, I'm bearing the cost for the extra unit of effort I contribute, but I'm only reaping half of the induced profits of the firm, because of profit sharing.

在边际上，我正在收获，我承担着我贡献的额外单位努力的成本，但由于利润分享，我只收获了公司诱导利润的一半。

That leads all of us to put in too little effort.

这导致我们所有人付出的努力太少。

What's the general term that captures all such situations in Economics?

经济学中涵盖所有此类情况的通用术语是什么？

It's an "externality." It's an externality.

这是一种“外部性”。这是一种外在性。

There's an externality here.

这里有一个外部性。

When I'm figuring out how much effort to contribute to this firm I don't take into account that other half of profits that goes to you.

当我计算为这家公司贡献多少努力时，我没有考虑到你的另一半利润。

So this isn't to do with the synergy.

所以这与协同作用无关。

It isn't to do with something complicated.

这与复杂的事情无关。

It's something you knew back in 115.

这是 115 年你就知道的事情。

If you have profit sharing in a firm or profit sharing in homework assignment, or any joint projects, you have to worry about too little effort being contributed because there's an externality.

如果你在公司中有利润分享，或者在家庭作业或任何联合项目中分享利润，你就必须担心贡献的努力太少，因为存在外部性。

My efforts benefits you not just me.

我的努力不仅对我有利，对你也有利。

While we've got this on the board, let's just think a little bit more.

虽然我们已经把这个想法写在了板上，但让我们再多思考一下。

What would happen if we changed the degree of the synergy?

如果我们改变协同程度会发生什么？

What would happen if we lowered B?

如果我们降低 B 值会发生什么？

So B is the degree to which the synergy across these workers, if we lowered B, what would happen to our picture?

那么B是这些工人之间协同作用的程度，如果我们降低B，我们的画面会发生什么？

Let me redraw a picture unscribbled.

让我重画一张未潦草的图画。

We had a picture that looked something like this.

我们有一张看起来像这样的照片。

This was S1 and this was S2.

这是S1，这是S2。

If we lowered the degree of synergy, what would happen to the effort level that we'd find by this method?

如果我们降低协同程度，我们通过这种方法找到的努力水平会发生什么变化？

What would happen?

会发生什么？

What would happen to the picture?

图片会发生什么情况？

Anybody, again this is a 115 kind of exercise, we're going to be moving lines around.

各位，这又是 115 种练习，我们将四处移动线路。

Yeah, Henry isn't it?

是啊，亨利不是吗？

So let's get a mike in to Henry.

让我们给亨利接个麦克风。

Student: The lines will get shallower and eventually become horizontal and vertical respectively.

学生：线条会变浅，最终分别变成水平和垂直。

Professor Ben Polak: All right, good.

Ben Polak 教授：好的，很好。

So the pink line is actually going to get steeper, but I know what you mean.

所以粉红线实际上会变得更陡，但我知道你的意思。

So the pink line is going to move towards the vertical, and the blue line is going to move towards the horizontal, and notice that the amount of effort that we generate, goes down dramatically, goes down in this direction.

因此，粉红色的线将向垂直方向移动，蓝色的线将向水平方向移动，请注意，我们产生的努力量急剧下降，沿着这个方向下降。

So if we lower the synergy here, not only do I contribute less effort, but you know I contribute less effort, and therefore you contribute less effort and so on.

因此，如果我们降低这里的协同作用，不仅我贡献的努力更少，而且你知道我贡献的努力更少，因此你贡献的努力也更少，等等。

So we get this scissors effect of looking at it this way.

所以我们这样看就得到了剪刀效应。

We could draw other lessons from this, but let me try and move on a little bit.

我们可以从中吸取其他教训，但让我尝试继续前进一点。

We decided in this game to solve it by looking at best responses, deleting things that were never a best response, looking again, deleting things that were never a best response, and so on and so forth, and luckily, in this particular game, things converged and they converged to the points where the pink and the blue line crossed.

我们决定在这个游戏中通过查看最佳响应、删除从来都不是最佳响应的东西、再次查看、删除从来都不是最佳响应的东西等等来解决这个问题，幸运的是，在这个特定的游戏中，事情收敛了，它们收敛到粉红线和蓝线交叉的点。

What do we call that point?

我们称这个点为啥？

What do we call the point where the pink and the blue line cross?

粉色线和蓝色线交叉的点叫什么？

That's an important idea for this class.

这是本课程的一个重要想法。

That's going to turn out to be what's called a Nash Equilibrium .

这将成为所谓的纳什均衡。

So we know what it's called.

所以我们知道它叫什么。

How many of you have heard the term Nash Equilibrium before?

你们中有多少人以前听说过纳什均衡这个词？

How many of you saw the movie about Nash?

你们中有多少人看过有关纳什的电影？

We'll come back and talk about that a bit next time.

我们下次会回来讨论这个问题。

So this is a Nash Equilibrium, but okay we know what it is in jargon, and we know, we kind of knew that was going to be an important point, because most of you have taken Economics courses before and you know that whenever lines cross in Economics it's important, right?

所以这是一个纳什均衡，但是好吧，我们知道它的行话是什么，我们知道，我们知道这将是一个重要的点，因为你们大多数人之前都学过经济学课程，并且您知道每当经济学中的界限交叉时它都很重要，对吗？

But what does it mean here?

但这是什么意思呢？

Why is it--what's going on at that line?

为什么——那条线上发生了什么？

What does it tell us that the pink and the blue line cross?

粉色线和蓝色线交叉告诉我们什么？

What makes that point special?

这一点有何特别之处？

What does it mean to say the pink and the blue line cross?

粉色线和蓝色线交叉是什么意思？

Can I get the guy way back, like three rows behind you in purple?

我可以让那个穿着紫色衣服的家伙往后退，比如你后面三排吗？

Shout out again.

再次大喊一声。

Student: It means that neither player has an incentive to deviate from that point.

学生：这意味着两个玩家都没有动机偏离这一点。

Professor Ben Polak: All right, well that's correct.

Ben Polak 教授：好吧，这是正确的。

So let's try and read that through.

那么让我们试着通读一下。

So I don't know your name, your name is?

所以我不知道你的名字，你的名字是？

Professor Ben Polak: So Allen is saying if Player I is choosing this strategy and Player II is choosing her corresponding strategy here, neither player has an incentive to deviate.

Ben Polak 教授：艾伦的意思是，如果玩家 I 选择这个策略，而玩家 II 在这里选择相应的策略，那么两个玩家都没有动机偏离。

Another way of saying it is: neither player wants to move away.

另一种说法是：两个球员都不想离开。

So if Player I chooses S1*, Player II will want to choose S2* since that's her best response.

因此，如果玩家 I 选择 S1*，玩家 II 将选择 S2*，因为这是她的最佳反应。

If Player II is playing S2*, Player I will want to play S1* since that's his best response.

如果玩家 II 正在玩 S2*，玩家 I 将想要玩 S1*，因为这是他的最佳反应。

Neither has any incentive to move away.

两人都没有任何离开的动力。

So more succinctly, Player I and Player II, at this point where the lines cross, Player I and Player II are playing a best response to each other.

更简洁地说，玩家一和玩家二，在线交叉的这一点上，玩家一和玩家二正在对彼此做出最好的回应。

The players are playing a best response to each other.

球员们正在互相做出最好的回应。

So clearly in this game, it's where the lines cross.

很明显，在这个游戏中，这就是界限的交叉点。

Let's go back to the game we played with the numbers.

让我们回到我们玩的数字游戏。

Everyone had to choose a number and the winner was going to the person who was closest to 2/3 of the average.

每个人都必须选择一个数字，最接近平均值 2/3 的人获胜。

(By the way, the winner's never picked up their winnings for that, so they still can.) So in that game, what's the Nash Equilibrium in that game?

（顺便说一句，获胜者从未因此获得奖金，所以他们仍然可以。）那么在那场比赛中，那场比赛的纳什均衡是什么？

Everyone choosing 1.

大家都选1。

How do we know that's the Nash Equilibrium of that game?

我们怎么知道这就是该游戏的纳什均衡？

How do we know that everyone choosing 1 is the Nash Equilibrium in the game where you all chose numbers?

我们怎么知道每个人都选择 1 就是你们都选择数字的游戏中的纳什均衡？

Well let's just use the definition.

好吧，让我们使用定义。

If everybody chose a 1, the average in the class would be 1, 2/3 of that would be 2/3 and you can't go down below 1, so everyone's best response would be to choose 1.

如果每个人都选择 1，那么班级的平均分就是 1，其中 2/3 就是 2/3，而且你不能低于 1，所以每个人的最佳反应就是选择 1。

Say it again: if everyone chose 1, then everyone's best response would be to choose 1, so that would be a Nash Equilibrium.

再说一遍：如果每个人都选择1，那么每个人的最佳反应都是选择1，那么这就是纳什均衡。

Did people play Nash Equilibrium when we played that game?

当我们玩那个游戏时，人们玩纳什均衡吗？

No, they didn't.

不，他们没有。

Not initially at any rate, but notice as we played the game repeatedly, what happens?

无论如何，一开始不会，但请注意，当我们反复玩游戏时，会发生什么？

As we played the game repeatedly, we noticed that play seemed to converge down towards 1.

当我们反复玩这个游戏时，我们注意到游戏似乎向 1 收敛。

Is that right?

是这样吗？

In this game, when we analyzed the game repeatedly, it seemed like our analysis converged towards the equilibrium.

在这个博弈中，当我们反复分析博弈时，我们的分析似乎趋向于均衡。

Now that's not always going to happen but it's kind of a nice feature about Nash Equilibrium.

虽然这种情况并不总是会发生，但这是纳什均衡的一个很好的功能。

Sometimes play tends to converge there.

有时，游戏往往会集中在那里。

Nash Equilibrium's going to be a huge idea from now to the mid-term exam and we're going to pick it up and see more examples on Wednesday.

从现在到期中考试，纳什均衡将成为一个伟大的想法，我们将在周三学习它并看到更多例子。