耶鲁大学博弈论公开课 - 第17集 足球比赛与商业合作之最佳对策

视频全文转录（中英双语）

以下为完整中英双语转录，已加标点。英文在前，中文在后，逐段对照。

由 Groq Whisper 转录 → M2.7 B 方案整文标点 + 分段 → M2.7 段号保留翻译 → 逐段对照。

[段 1]

So last time we introduced a new idea and the new idea was that of best response. And 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. And you could think of this belief as the belief that rationalizes that choice. So if you have a boss, and he or she is going to ask you why you chose the action you did, if you took the best response to some belief, you can say, I took this action because I believed other people were going to do this, and since that was the best you could do under that belief, that will hopefully keep your job. Alright, I promised that today we would look at the most important game in the world, and as announced last time, the most important game is the penalty kick game. 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. 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.

[译文 1]

上次我们介绍了一个新概念，这个新概念就是最佳反应。核心思想是考虑一个策略，这个策略是在你关于其他人、你的对手、其他玩家会做什么的信念给定的情况下，你能做到的最佳选择。你可以把这个信念看作是使那个选择合理化的信念。所以如果你有个老板，他或她会问你为什么选择那个行动，如果你对某种信念采取了最佳反应，你可以说，我采取这个行动是因为我相信其他人会这样做，而既然在那个信念下这是你能做到的最佳选择，这应该能保住你的工作。好吧，我承诺过今天我们要来看世界上最重要的游戏，正如上次宣布的那样，最重要的游戏就是点球游戏。这是一个在足球中出现的游戏，对于那些不幸不是足球迷的人来说，上届世界杯就是由点球决定的。就英格兰而言，英格兰每次世界杯和每次欧洲杯都会被淘汰，因为它在点球大战中输了，通常是输给了德国，不得不这么说。

[段 2]

And more immediately, this weekend, as all of you are thinking, the most important 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 favourite team, Portsmouth, were playing Kai, the head CA’s favourite team, Liverpool and about a third of the way through that game there was a penalty and I’ll let you know what happened later. So keep in the back of your minds that the real world example that matters here is Portsmouth versus Liverpool this weekend. Kai 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. All right. So what we going to do is we going to look at some numbers that approximately are approximately the probability of scoring when you kick the penalty kick in different directions But just make sure everyone do I need to explain what going on here I mean is everyone familiar with the situation? There’s one guy who’s going to run up and kick the ball, the goalkeeper is sending in the goal, and the aim is to kick it into the goal. Is that, is that, that’s really enough? You’ve all seen this, right? If you haven’t seen this, go see it. Come on, this is, all right?

[译文 2]

更直接地说，这个周末，正如你们都在想的，世界上最重要的事件是国会关于伊拉克发生的事情，实际上世界上最重要的事件发生在英格兰，我最喜欢的球队Portsmouth正在比赛，首席CA最喜欢的球队Liverpool，在这场比赛进行了大约三分之一时出现了一个点球，我之后会告诉你们发生了什么。所以在你们脑海中记住，这里重要的现实世界例子是这个周末Portsmouth对Liverpool。 Kai是首席TA，他是斯堪的纳维亚人，所以我完全不知道为什么他支持Liverpool，不过我觉得也许他是这样拼写的。好吧。那么我们要做的是我们要看一些数字，这些数字大约是当你朝不同方向踢点球时进球的概率。但是首先要确保每个人都明白我需要解释这里发生了什么吗？我的意思是每个人都熟悉这种情况吗？有一个人要跑过来踢球，守门员在球门前防守，目标是把球踢进进球。你们都见过这个，对吧？如果你没见过，去看看吧。来吧，这，这就是，够了吗？

[段 3]

So things you should do in life, you know, read Shakespeare and see a soccer game. Okay, so the rough numbers for this are as follows. Actually, later on in the class, I’ll give you some more accurate numbers, but this will do for now. There are three ways the attacker could kick the ball. He could kick the ball to the left, the middle, or the right, 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. And the goalie can dive to the left or the right. In principle, the goalie could stay in the middle. We’ll come back and talk about that later. All right, so this is the guy who is shooting. Let’s call him a shooter. And this is the goalie. And these are roughly… Well let me put out the payoffs for this game and then I explain them So you notice that I just going to put in numbers here and then the negative of the number And the numbers are roughly like this. 4 minus 4. So the numbers are 4 minus 4, 9 minus 9, 6 minus 6, 6 minus 6, 9 minus 9, and 4 minus 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 goalkeeper dives to the left.

[译文 3]

所以你一生中应该做的事情，你知道，读莎士比亚和看足球比赛。好吧，所以这些粗略的数字如下。实际上，在课程的后面，我会给你们一些更准确的数字，但现在这些就够了。进攻者有三种方式可以踢球。他可以把球踢向左边、中间或右边，当然我不应该说"他"。我的意思是这是他或她，但如果我在接下来的过程中说错了，请原谅我。守门员可以扑向左边或右边。原则上，守门员可以待在中间。我们之后会回来说这个。好吧，所以这是射门的人。我们叫他射门者。这是守门员。这些大致是……好吧让我先把这个游戏的收益写出来然后再解释。你们注意到我这里只放数字，然后是数字的负值。数字大致是这样的。4减4。所以数字是4减4，9减9，6减6，6减6，9减9，和4减4。这里的想法是数字4代表如果你把球踢向球门左侧而守门员扑向左侧，你有40%的进球概率。

[段 4]

So the payoff here is something like u1 of left if the goalkeeper dives to the left is equal to 4, by which I mean there’s a 40% chance of scoring. So the payoff for the shooter is his probability of scoring and the payoff for the goalkeeper is just the negative of that. So keep things simple. Okay? And as I said before, for now, we’ll ignore the possibility that the goalkeeper could stay put. All right? So how should we start analysing this important game? Well, we start with the ideas we learnt already several weeks ago now, or more than a week ago, which is the idea of dominant strategies. How should we start analyzing this important game? 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. Does either player here have a dominated strategy? Does either player have a dominated strategy? No, it’s kind of clear that they don’t. Let’s just look at the shooter, for example. So you might think that maybe middle dominates left, but notice that middle has a higher payoff against left for shooting to the left than shooting to the left it has a lower payoff if the goalie dives to the right so not surprisingly in this game it turns out that if the goalie dives 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 and if the goalie dives to the left 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 shooting to the right.

[译文 4]

所以这里的收益是这样的：如果守门员扑向左侧，左侧的u1等于4，我的意思是这代表有40%的进球概率。所以射门者的收益是他的进球概率，而守门员的收益就是这个的负值。保持简单，好吗？正如我之前说的，现在，我们会忽略守门员可能待在原地的可能性。好吧？那么我们应该怎么开始分析这个重要的游戏呢？嗯，我们从几周前——或者说一个多星期前——学到的概念开始，也就是优势策略的概念。我们应该怎么开始分析这个重要的游戏呢？嗯，我们从几周前——或者说一个多星期前——学到的概念开始，也就是优势策略的概念。这里的任意一个玩家有被支配的策略吗？任意一个玩家有被支配的策略吗？没有，这很明显他们都没有。让我们只看射门者，例如。你可能认为中间可能支配左边，但注意到中间在面对左侧时有更高的收益——对于踢向左侧来说——比踢向左侧有更低的收益如果守门员扑向右侧，所以毫不奇怪，在这个游戏中，如果守门员扑向左侧，你最好射向右侧，其次射向中间，最差射向左侧；如果守门员扑向左侧，如果守门员扑向右侧，你最好射向左侧，其次射向中间，最差射向右侧。

[段 5]

And that’s kind of common sense. Okay? Okay, so if we had stopped the class after the first week, where all we learned to do was to delete dominated strategies we be stuck We 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 But luckily, we can do a little bit better than that. Before I do that, let’s just take a poll of the class. 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, 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? And how many of you would shoot to the middle? And how many of you would shoot to the right? We’ve got, you know, kind of an even split there, pretty much an even split. Alright, and we’ll assume, again, 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. Alright, so how should we go about thinking about this? 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.

[译文 5]

这算是常识。好吧？好吧，所以如果我们第一周之后就结束课程，我们当时学到的只是删除被支配的策略，我们会陷入困境。我们对这场比赛无话可说。而正如我之前说的，这是最重要的游戏。所以这对博弈论来说是坏消息。但幸运的是，我们能做得比这更好一点。在我做这个之前，让我们先做个课堂投票。你们中有多少人，如果你们在比赛，我的意思是这将是America，这是一件悲哀的事情开始，别介意，你们为America比赛，你们要踢这个点球，这是世界杯的最后一罚，你们中有多少人，举手，你们有多少人会射向左侧？你们有多少人会射向中间？你们有多少人会射向右侧？我们有，你知道，相当均匀的分配，几乎是均匀的分配。好吧，我们会再次假设，我们会假设这些是正确的数字，我们会看看那个均匀分配是否真的一个好主意。好吧，那么我们应该怎么思考这个？我建议我们做我们上次做的事情，我们开始画一幅图来找出我的预期收益是多少，取决于我相信守门员会做什么。

[段 6]

So this is the same kind of picture we drew last time. So on the horizontal axis is my belief and my belief is essentially the probability that the goalie dives to the right Now, as I did last time, let me put in two axes to make the picture a little easier to draw. So this is 0 and this is 1, and you probably have lines in your notes, but I don’t, so let me just help myself a bit. 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. This will be the basis of my picture. alright, so let’s start with the possibility of shooting to the left, alright, let’s do this in red alright, so if I shoot to the left and the goalie dives to the left, my payoff is what? right it’s four, alright 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 If I shoot to the left and the goalie dives to the right then I score 90% of the time, so my payoff is 0.9. By the way, why is it 90% of the time and not 100% of the time?

[译文 6]

所以这是我们上次画的同一类图。所以在横轴上是我的信念，我的信念本质上是守门员扑向右侧的概率。现在，就像我上次做的，让我画两条轴来让图更容易画。这是0，这是1，你们可能在笔记里有线，但我没有，所以让我帮自己一下。这是2，4，6，8，10。所以这将是2，4，6，8，和10。还有这边，2，4，6，8，和10。2，4，6，8，和10。这将成为我图的基础。好吧，所以让我们从射向左侧的可能性开始，好吧，让我们用红色来做这个，如果我射向左侧而守门员扑向左侧，我的收益是什么？对是4，如果我射向左侧而守门员没有可能扑向右侧，这意味着他们扑向左侧，那么我的收益是4意味着我40%的时间进球。如果我射向左侧而守门员扑向右侧，那么我90%的时间进球，所以我的收益是0.9。顺便问一下，为什么是90%的时间而不是100%的时间？

[段 7]

I could miss. I could miss. I could miss. That happens rather often, it turns out. Well, 10% of the time. alright, so we know this is going to be a straight line in between so let’s put this line in alright, so watch this it’s the expected payoff to player one of shooting to the left as it depends on the probability that the goalkeeper dies to the right alright, and conversely we can put in, well let’s do this in an order So middle so if I shoot to the middle and the goalkeeper dives to the left then my payoff is 0.6 or is it six or I score 0.6 at the time and if I Shoot to the middle and the goalie does to the right. It’s still I still score 60% of the time so Hang on to get it’s a straight straight line in between so this line represents the expected I still score 60% of the time, so, hang on, once again, it’s a straight line in between, so this line represents the expected payoff of shooting to the middle as a function of the probability that the goalkeeper dies to the right. And finally, let’s do it in green, let’s look at the payoffs, expected payoffs, if I shoot to the right. so if I shoot to the right and the goalie dives to the left then I score with probability 0.9 or my payoff is 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 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.

[译文 7]

我可能会踢偏。我可能会踢偏。我可能会踢偏。事实证明这种情况相当常见。好吧，10%的情况下会这样。好吧，我们知道这会是一条介于两者之间的直线，所以让我们画这条线。好吧，看着这个，这是球员一射门到左边时的期望收益，取决于守门员扑向右边的概率。好吧，反过来我们也可以画，好吧让我们按顺序来做。所以中间，如果我射门到中间而守门员扑向左边，那么我的收益是0.6或者是6，或者我当时得分概率是0.6。如果我射门到中间而守门员扑向右边，我仍然有60%的概率得分。所以等一下，介于两者之间的是一条直线，所以这条线代表射门到中间的期望收益，作为守门员扑向右边概率的函数。最后，让我们用绿色来做，让我们看看射门到右边的收益、期望收益。所以如果我射门到右边而守门员扑向左边，那么我有0.9的概率得分，或者我的收益是9。相反，如果我射门到右边而他或她扑向右边，那么我只有40%的概率得分，所以这是我的收益0.4，这是我的绿色线，代表我作为射门者射门到右边时的期望收益，作为守门员扑向右边概率的函数。

[段 8]

Alright, does everyone understand how I constructed this picture? It’s actually an easier picture than the one we constructed last time. Alright. So what does anyone notice from this picture? What the first thing that jumps out at you from this picture Assuming these numbers are true what jumps out at you in this picture Can we get some mics up here Ali, can we get this guy to stand up first, the guy in red? What’s your name? Don’t hold the mic, just shout. Stephen, and there’s no point at which the six, at which shooting at the middle gives a higher expected payoff? Exactly, exactly. So the thing that I hope jumps out of you with this picture is, no great guesses for figuring out this is a half, so if the probability that the goal is going to jump to the right is less than a half, then the best you can do is represented by this green line, which is shoot to the right.

[译文 8]

好吧，大家都理解我是怎么画这张图的吗？实际上这比我们上次画的那张图要简单。好吧，那么有人从这张图上注意到什么了吗？第一眼从这张图上跳入你眼帘的是什么？假设这些数字是真实的，你在图上注意到什么？我们能把麦克风递过来吗阿里，我们能让这位先生先站起来吗，穿红色衣服的那位？你叫什么名字？不要握着麦克风，喊出来就行。斯蒂芬，没有任何一点使得射门到中间给出更高的期望收益，对吧？正是，正是。所以我希望你们从这张图上注意到的关键是，没有很好的猜测能确定这是50%，所以如果球门将要扑向右边的概率小于50%，那么你能做的最好的选择由这条绿色线表示，也就是射门到右边。

[段 9]

So if the goal is going to shoot to the right with the probability less than a half, so if it’s going to dive to the right with the probability less than a half, you should shoot to the right. and conversely, if you think the goalie is going to shoot to the right with probably more than a half then the best you can do is represented by the pink line and that’s shooting to the left so if you think the goalie is going to dive to the right with probably more than a half the best you can do, your best response is to shoot to the left and there is no belief there is no belief you could possibly hold given these numbers in this game that could ever rationalise shooting the ball to the middle is that right Is that right So no so another way middle is not a best response to any belief I can hold about the goalkeeper, but any belief. Alright, so there’s a lesson here, and just to make the lesson resonate again, And 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. And what’s the lesson here? I’m hoping it was going to be obvious.

[译文 9]

所以如果守门员扑向右边的概率小于50%，你应该射门到右边。相反，如果你认为守门员扑向右边的概率大于50%，那么你能做的最好的选择由这条粉色线表示，也就是射门到左边。所以如果你认为守门员扑向右边概率大于50%，你能做的最好的选择是射门到左边。给定这个游戏中的这些数字，没有任何一种你可能持有的信念能够合理化射门到中间，是这样吗？是这样吗？所以不，换句话说，中间不是对我可以持有的关于守门员的任何信念的最佳反应，任何信念。好吧，这里有一个教训，为了让这个教训再次引起共鸣，想象你正在世界杯上为英格兰效力，你不仅要向你的队友、你的教练和你的老板解释你的行为，还要向大约6000万相当愤怒的球迷解释。这个教训是什么？我希望这是显而易见的。

[段 10]

What’s the lesson here? the lesson is alright lesson do not shoot to the middle alright, do not shoot to the middle let me qualify that lesson slightly unless you German alright Germans can do whatever they like Alright Alright? Now it turns out that about a third of the game between my team, Portsmouth, and Kai’s team, Liverpool, this weekend, there was a penalty. 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. Alright? So just confirming these actions, not only did that spoil my weekend but it also spoiled my opportunity to make fun of Kyle so it was really a big deal so this weekend a penalty was missed exactly by somebody ignoring this rule 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 the more general lesson do not choose a strategy do not choose a strategy that is never a best response to any belief. And notice here, just to underline something which came up at the end last time, that doesn’t mean beliefs of the form, the goalie’s going to dive left or the goalie’s going to dive right. It means all probabilities in between.

[译文 10]

这里的教训是什么？教训是好吧教训是不要射门到中间，好吧，不要射门到中间。让我稍微补充一下这个教训，除非你是德国人，好吧德国人可以随便怎么做。好吧，好吧？结果证明，这周末我的球队朴茨茅斯和凯的球队利物浦之间大约三分之一的比赛时间里，有一个点球。朴茨茅斯获得了一个点球。那个要罚点球的人上来罚点球，他把球踢向中间，然后被扑住了。好吧？所以只是确认这些行为，这不仅毁了我的周末，也毁了我取笑凯的机会，所以这真的是件大事。所以这周末有个点球被罚丢了，正是有人忽视了这个规则。所以这里有一个更普遍的教训，更普遍的教训是当然不要选择一个永远不可能是你可以相信的任何东西的最佳反应的策略。更普遍的教训是不要选择一个策略，不要选择一个永远不是任何信念的最佳反应的策略。注意这里，只是为了强调上次结尾时出现的观点，那并不意味着像"守门员将扑向左边"或"守门员将扑向右边"这种形式的信念，而是意味着介于两者之间的所有概率。

[段 11]

We’re allowing you, for example, to hold the belief that it’s equally likely that the goalie dives left or dives right. But if there’s no belief that could possibly justify it, don’t do it. and to underline 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. So when we look at domination and deleted dominated strategies, we got nowhere here here at least we got somewhere, we got rid of the idea of shooting to the middle, now if you can just persuade the English and Portsmouth soccer players of this lesson, I’d be very happy. Alright? Now 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. And let’s just try, any of you on the Yale soccer team? No? Any of you play soccer for your college One or two play soccer for college How many of you have ever played soccer Okay good I was getting worried there for a second Alright, 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?

[译文 11]

我们允许你，例如，持有这样的信念：守门员扑向左边或扑向右边的可能性是相等的。但如果没有任何信念可以证明它是合理的，就不要做。而且为了强调这个游戏中出现的情况，注意在这个游戏中我们能够消除其中一个策略，在这种情况下是射门到中间的策略，尽管没有任何东西是被支配的。所以当我们看支配和删除被支配的策略时，我们在这里没有进展至少我们有所收获，我们去掉了射门到中间的想法，现在如果你能说服英格兰和朴茨茅斯的足球运动员接受这个教训，我会非常高兴。好吧？在我们结束之前，我一直在课堂上强调要时不时回到现实。所以这是足球比赛的一个非常简单模型。实际上让我们试试，你们中有谁在耶鲁足球队吗？没有？你们中有谁为大学踢球？一二个人为大学踢球？你们中有多少人踢过足球？好的，我刚才有点担心。好吧，所以我们上次说的一件事是，当我们提出一个模型并试图从中吸取教训时，我们应该退后一步说，这里缺了什么？

[段 12]

What’s missing here? So let’s try and get some kind of… I’ll come off the stage to make it easier for Jude. What’s missing here? What’s missing in this model of this piece of soccer, this game within a game? What’s missing here? Why is this not necessarily a 100% accurate model? I’ll need some mics up here. You have to really shout because you’re miles from the mic there. You might be better at kicking to the left or to the right, depending on whether you’re right-handed or left-handed. Good, 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 goal is right. All right? So right-footed players find it easier to shoot to the left as facing them, shoot across the goal. All right? Is that true? Everyone confirm that’s true? Yeah? Everyone ever tried to do this? It a little easier to hit the ball hard to the opposite side from the side which is your foot And that the same principle in baseball It a little bit easier to pull the ball hard than it is to hit the ball to the opposite field All right Yes Players don’t make their decision before and then just stick with it, necessarily. All right. So players are making the decision as they’re running up.

[译文 12]

这里缺了什么？让我们试着……我会走下舞台让裘德更容易一些。这里缺了什么？这个足球片段的模型中，这个比赛中的比赛缺了什么？这里缺了什么？为什么这不一定是一个100%准确的模型？我需要这里有一些麦克风。你必须大声喊因为你离麦克风很远。你可能更擅长踢向左边或右边，这取决于你是右脚还是左脚。好，好。所以这里明显缺失的一件事是，我忽略了事实上，右脚球员发现射门到他们的左边更容易，而这实际上是右边球门。好吧？所以右脚球员更容易射门到左边，也就是面对他们射门穿过球门。好吧？这是真的吗？大家都确认这是真的吗？是的？大家都试过这样做吗？用力把球踢向与你的脚相反的一边会更容易，这在棒球中也是同样的原理。用力拉球比把球打到相反的方向稍微容易一些。好吧，是的。球员们不一定是在做出决定后就坚持它。好吧。所以球员们是在他们跑动的过程中做出决定的。

[段 13]

I think that’s okay here, right? We can think of this as the decision happening at the instant to which you kicked. All right? 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 ultimately what matters are we getting that off there? okay let’s hope that goes away are we sure it’s not off my mic? just in case I move my mic a bit lower alright so I’m going to shout louder because my mic is now lower it doesn’t really matter exactly when the decision is made 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, so it’s as if that decision is made instantaneously as the shooter is running up. What else? Can we get this guy here Stand up Shout out The goalie might stay in the middle The goalie might stay in the middle That a good point and of course I abstracted from that here and in fact we come back, I think, I’ll try and put that onto our problem set, but I think you’re right, it is, it is an issue here, is an issue here. Anything else? Well let me put up some real numbers and We’ll see how much it corresponds to what we’ve got here.

[译文 13]

我觉得这里可以这样理解，对吧？我们可以认为决策是在你踢球的那一刻做出的。好吧？所以你说得对，你本可以在更衣室里做决定，或者在中场休息时做决定，但归根结底，重要的是我们能不能把它解决掉？好吧，希望那个能消失。我们确定它不在我的麦克风上吗？以防万一，我把麦克风稍微移低一点。好的，我要大声喊，因为我的麦克风现在低了。决策具体在什么时候做出其实并不重要。说到底，守门员不知道射手的决定，射手也不知道守门员的决定，所以这就像是当射手跑上来的时候，那个决定是瞬间做出的。还有别的吗？能让这位先生站起来吗？致意。守门员可能会守在中间。守门员可能会守在中间。说到点上了，当然我在这里抽象掉了这一点，实际上我们回过头来看，我想，我会试着把它放到我们的问题集中，但我觉得你说得对，这确实是个问题，这确实是个问题。还有别的吗？好吧，让我列一些真实的数字，看看它与我们这里得到的结果有多大程度的对应。

[段 14]

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, 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 Kiyopori and some co-authors, and for everyone at Yale, I’ll make that paper available to you through JSTOR, through the Yale library, so you can go look at it if you’d like. And what they worked out was the following table, and again, we’ll be a little bit careful here. So I’m going to put the left and right in inverted commas here, because actually what they did was because they corrected for people’s natural direction here. So I’m going to put the left and right in inverted commas here because actually what they did was they corrected for people’s natural direction and not natural direction. 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. Of course if you’re left-footed it goes the other way, but they’ve corrected for that. All right, and it turns out that the probabilities of scoring here are as follows 63.6 94.4 89.3 and 43.7 all right so that things are not and 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.

[译文 14]

所以我给你们一些数字，这些数字其实是很久以前编的，但自从我在课堂上用这个游戏以来，有人出去核实了，结果表明先忽略中间的情况，先忽略中间。这些是真实的数字，这些数字来自AER期刊上Kiyopori和一些合著者的一篇论文，对于在耶鲁的各位，我会通过JSTOR、通过耶鲁图书馆把这篇论文提供给你们，这样如果你们愿意的话可以去查阅。他们的工作是得出了下面这个表格，同样，我们会稍微谨慎一点。我要把左边和右边加上引号，因为实际上他们做的是因为他们纠正了人们自然的方向。我要把左边和右边加上引号，因为实际上他们做的是他们纠正了自然方向和非自然方向。所以这里的想法是，如果你是右脚球员，向左射门就是自然方向。所以这里的左意味着自然方向。当然如果你是左脚球员，情况就反过来了，但他们已经对此做了纠正。好吧，结果表明得分的概率如下：63.6、94.4、89.3和43.7。好吧，所以情况并不是，我没有给你们中路的数字，但你们可以看到，那个说稍微好一点的人，你在自然一侧踢球得分的概率稍微更高，这完全正确。

[段 15]

Things still aren’t dominated, and we could sort on 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. I forget who it was who said that, but that does turn out to be true Certainly the goalie staying put is an issue as I said we deal with that in the problem set but there another issue here Let me just raise one more issue One more issue is you have another decision when you run up to hit the penalty other than just left and right Someone who’s played the game. What’s the other decision you really face? Can you get the woman here? What’s the other decision you face? Can you kick up to the top corner or the lower corner? Okay, you can kick up and down, that’s true. Okay, that’s true actually, that’s true, but I meant something else. That’s right, but I meant something else. What else is there? Let’s try this guy here. Spin. Well, that’s getting subtle here. That’s getting subtle. It’s a much more basic thing. What’s a more basic thing? What’s a more basic thing here? Can you get that right in front of you? Speed. Speed, right? 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?

[译文 15]

情况仍然没有被主导，我们可以用完全相同的分析来排序，实际上你们可以看到我编的数字并没有差得很远。但情况并不是完全对称的。我忘了是谁说的，但这确实是真的。当然，守门员保持不动是个问题，就像我说的，我们会在问题集中处理，但这里还有另一个问题。让我再提一个issue。另一个issue是，当你跑上去罚点球时，你还有一个除了左右之外的决定。谁玩过这个游戏。除了左右之外，你面临的另一个决定是什么？能让这位女士过来吗？你面临的另一个决定是什么？你能踢向上角还是向下角吗？好吧，你可以上下踢，这是对的。好吧，这是对的，但这确实是，不过我指的是别的东西。是的，不过我指的是别的东西。还有什么？让我们试试这位先生。旋转。好吧，这变得有点微妙了。这变得有点微妙了。这是一个更基本的东西。什么是更基本的东西？这里什么是更基本的东西？能把你面前那位请过来吗？速度。速度，对吧？所以你面临的另一个决定是，你只是尽量把球踢得尽可能用力，还是尽量把球控制好？

[段 16]

That’s probably as important a decision as deciding depending on which direction to hit, and it turns out to matter. 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 If you can kick the ball very hard but not very accurately then if you try and shoot to the left or right you slightly more likely to miss On the other hand as you shoot to the middle since you kicking the ball hard you’re slightly more likely to score. 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. Alright, 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 goalkeeper to stop it alright and here it goes in the middle, if you look carefully there, didn’t really make it clear enough, you can see suddenly a strategy that looks crazy, shooting to the middle, has suddenly started to seem okay.

[译文 16]

这可能和决定往哪个方向踢一样重要，结果表明这确实有关系。例如，如果你是一种这样的人，我不得不说，我以前就是这种人，如果你是一种能把球踢得相当用力但不太准的人，那么这实际上可能会改变这些数字。如果你能把球踢得很有力但不太准确，那么如果你试着往左或右边射门，你稍微更可能踢偏。另一方面，如果你往中路射门，因为你把球踢得很有力，你稍微更可能得分。现在如果这一切看起来像是什么神秘而不相关的细节，让我们看看为什么这在这个图景中很重要。然后我们就会离开足球，至少今天先这样。好吧，所以如果你是一种能把球踢得有力但不准确的人，那么当你往右边踢时，你得分的概率会降低，因为你可能会踢偏；当你往左边踢时，你得分的概率也会降低，因为你可能会踢偏；而当你往中路踢时，你得分的概率实际上可能会提高，因为你把球踢得很有力，守门员真的很难扑住它。好吧，在这里进了中路，如果你仔细看那里，我之前没有把它说清楚，你们可以看到，突然之间，一个看起来疯狂往中路射门的策略，突然开始看起来还行了。

[段 17]

All right? It turns out, if you look at those dotted lines, there’s an area in the middle, which is the area between here and here this little area here you actually might be just fine shooting to the middle All right 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. And if you’re interested in that, and I realize at this point I’ve probably lost the interest of most Americans in the room, but for the non-Americans of 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. All right, and I should just say that the data I just gave you is real data But it’s actually a mixed ability data This data comes half from the Italian league which is pretty good and half from the French league which sucks so So who knows how much we should trust it all right, okay? So that was our example for the day and our first brash with reality for the day Let’s clean the board and do some work, do some more formal stuff here. So here we have an example, but I want to go back to the generality and to a bit of formalism.

[译文 17]

好吗？结果表明，如果你看那些虚线，中间有一个区域，这是这里的区域和那里的区域之间的这个小区域，你实际上往中路射门可能完全没有问题。好吧。所以在现实中我们需要考虑更多一点，特别是我们需要考虑球员准确击球和/或大力击球的能力。如果你对此感兴趣，而且我意识到在这一点上，我可能已经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，对于对现实世界感兴趣的人，就像我之前说的，我会把那篇文章放到网上，那篇文章会详细介绍所有的细节。好吧，我应该说一下，我刚才给你们的那些数据是真实的数据。但这实际上是混合能力的数据。这些数据一半来自意大利联赛，那里水平相当高，一半来自法国联赛，那里水平不怎么样。所以谁知道我们到底应该有多信任它。好吧，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。让我们把黑板擦干净，做一些工作，做一些更正式的东西。这里我们有一个例子，但我想回到一般性，回到一些形式化。

[段 18]

Oh, by the way, I should tell you that the game ended 0-0, or 0-0 in that game. It’s a moral victory for me, I think. All right. It’s a moral victory for me, I think. Alright. So, I want to be formal about these things I’ve been mentioning informally, and in particular, I want to be formal about the definition of best response. And 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 which is going to correspond to the more general idea of a best response to a belief. All right? And it will allow us to use our notation and just be a little bit more nerdy. So player I’s strategy… player I’s strategy, S-I-hat, I’m going to get a hat to single it out, is a best response, always abbreviated B-R, to the strategy S minus I of the other players if and here our real excuse to use our notation if the payoff from player i from choosing si hat against s minus i is weakly bigger than her payoff from choosing some other strategy, si prime, against s minus i. And this better hold for all si prime. available to player i. Alright, so in previous definitions, we’ve seen the qualifier for all be on the other player’s strategy.

[译文 18]

哦，顺便说一句，我应该告诉你们，那场比赛的比分是0比0。那场比赛是0比0。对我来说是一场精神上的胜利，我想是的。对我来说是一场精神上的胜利，我想是的。好的。所以，我想把我一直非正式提到的这些东西正式化，特别是，我想把最佳反应的定义正式化。我要给出两个不同的最佳反应定义，其中一个对应于对某人采取特定策略如左边和右边的最佳反应，另一个将对应于更一般的最佳反应的概念，即对一种信念的最佳反应。好吗？这将允许我们使用我们的符号，稍微变得更 geek 一点。所以玩家 I 的策略……玩家 I 的策略，S-I-hat，我要加个帽子来特别标出它，是一个最佳反应，总是缩写为 B-R，对其他玩家的策略 S 减 I 成立，如果这里我们真正使用符号的借口是，如果玩家 i 选择 si hat 对抗 s 减 i 的收益，弱大于她选择其他某个策略 si prime 对抗 s 减 i 的收益。而这个条件要对玩家 i 所有可用的 si prime 都成立。好吧，所以在之前的定义中，我们看到对所有条件的限定是在其他玩家的策略上。

[段 19]

Here the qualifier for all is on my strategy. So strategy si hat is the best response to the strategy s minus i of the other players if my payoff from choosing si hat against s minus i is weakly bigger than that from choosing si prime against s minus i. And this better hold for all possible other strategies I could choose. And there’s another way of writing that that’s kind of useful, or equivalently, S i hat solves the following. It maximizes my payoff against S minus i So you all used to that I hoping everyone is used to seeing the term max S I solves the maximization problem, how do I maximize my payoff given that other people are choosing S minus I. And again, for the mathophobics in the Roman, 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. Let’s generalize this definition a little bit. since we want it to allow for more general beliefs so just rewriting player I’s strategy same thing F I hat is a best response but now let’s be careful best response to the belief P about the other player’s choices if, and it’s going to look remarkably similar, except now I’m going to have an expectation. If the expected payoff to player i from choosing s i hat given that she holds this belief p is bigger than her expected payoff from choosing any other strategy, given she holds this belief p.

[译文 19]

这里所有人的限定条件都在我的策略上。所以策略ŝi是对其他玩家策略ŝ⁻i的最佳回应，当我对ŝ⁻i选择ŝi时的收益 weakly 大于我对ŝ⁻i选择s′i时的收益。而且这应该对所有我能选择的其他策略都成立。还有另一种写法也很实用，或者说，ŝi解决了以下问题：它在给定其他人选择ŝ⁻i的情况下最大化我的收益。你们都习惯了，我希望每个人都习惯看到 max sᵢ 解决最大化问题的表述，即在其他人选择ŝ⁻i的情况下，我如何最大化自己的收益。再说一遍，对于罗马礼堂里害怕数学的人，别慌，这只是把我们今天已经见过几次的东西用文字写下来。今天和上次。让我把这个定义稍微推广一下，因为我们想让它适用于更一般的信念。重新写一下玩家i的策略，同样的东西，f̂i是最佳回应，但现在请注意，这是对关于其他玩家选择的信念P的最佳回应，而且看起来会非常相似，只是现在我要引入一个期望。如果玩家i从选择ŝi且她持有这个信念P时的期望收益大于她从选择任何其他策略且持有信念P时的期望收益，那么这就是最佳回应。

[段 20]

And this better hold for all si prime that she could choose. All right, it’s a very similar idea, And 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. This is the expectation given this belief. All right. And once again, we can write it the other way. Or SI hat solves max, and I choose SI, to maximize my expected payoff this time. from choosing S i against S minus i. And what do I mean by expected payoff? Just in our example, just to make clear what that expectation means, so for example, the expected payoff So just to make clear what that expectation means, so for example, the expected payoff to player 1 in the game above from choosing left, given she holds the belief P is equal to the probability that the goalkeeper dives to the left times player 1’s payoff from choosing left against left plus the probability that the goalkeeper dives to the right times player 1’s expected payoff from choosing left against right. So expectation with respect to P just means exactly what you expected alright so this is a little bit of math a little bit of formality is everyone ok with that? I haven’t done anything here all I’ve done is write down slightly boringly and nerdily exactly what we already saw in a couple of occasions thank you thank you thank you Thank you Thanks Dave Thank you Thank you thank you thank you Thanks Jake Thank you All right.

[译文 20]

而且这应该对她能选择的所有s′i都成立。好的，这是一个非常相似的想法，唯一的事情是我稍微滥用了符号，说我的收益取决于我的策略和信念，但我真正意思是期望收益。这是给定这个信念后的期望。同样，我们可以写成另一种形式。或者ŝi解决 max，我选择sᵢ来最大化我的期望收益。这次是从选择sᵢ对ŝ⁻i的期望收益。什么是期望收益？我举个例子来说明这个期望意味着什么。比如，玩家1在上述游戏中从选择左边时的期望收益，给定她持有信念P，等于守门员扑向左边的概率乘以玩家1选择左边对左边的收益，加上守门员扑向右边的概率乘以玩家1选择左边对右边的期望收益。对P求期望正是你理解的那个意思，所以这是一些数学一些形式化，大家都明白了吗？我这里什么都没做，只是稍微无聊地、一本正经地写下了我们在几个场合已经见过的东西。谢谢谢谢谢谢。谢谢Dave。谢谢。谢谢。谢谢Jake。好的。

[段 21]

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 ideas, not the notation, and let’s spend the next half hour applying these ideas to an application. This application is not as important as soccer, but it’s a bit more economics-y, so I can justify it under the economics title of the class. All right, so clearing off my soccer game. so imagine what we’re going to look at is a game involving a partnership it’s a partnership game 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 The idea is this there are two individuals who are going to supply an input to a joint project So that could be a firm it could be a law firm for example and they going to share equally in the profits So one example would be a firm that they both own, and another example would be two of you working as a study group on my homework assignment. So you’re going to share equally in the profits of this firm, or this joint project, but you’re going to supply efforts individually. All right, so let’s just be a bit more formal. So the players are going to be the two agents, the two agents, and they own this firm, let’s call it.

[译文 21]

所以现在这看起来会突然冒出一堆符号，让我们提醒自己，我们真正关心的是想法，而不是符号。让我们花接下来的半小时把这些想法应用到一个案例上。这个案例没有足球那么重要，但更经济学化，所以我可以用这节课的经济学名义来证明它是合理的。好的，清空我的足球游戏。让我们想象我们要研究的是一个涉及合作关系的游戏，这是一个合作博弈，我相信这个博弈在Watson的教材中有详细讨论，或者非常接近的内容，如果你有困难的话。想法是这样的：有两个个体将为一个联合项目提供投入。这可能是一个企业，比如一家律师事务所，他们将平均分享利润。一个例子是他们共同拥有的企业，另一个例子是你们中的两个人在我的作业上作为学习小组一起工作。你们将平均分享这个企业或这个联合项目的利润，但你们将各自提供努力。好的，让我们更正式一点。玩家将是两个代理人，这两个代理人，他们共同拥有这个企业，我们叫它。

[段 22]

They own this firm jointly, and they split the profits. So they share 50% of the profits each. it’s a profit sharing partnership and each agent each agent is going to choose her effort level her effort level to put into this firm to put into this firm So it could be that you deciding as a lawyer how many hours you’re going to spend on the job, so for most of you these decisions will be questionable of whether you spend 20 hours a day at the firm or 21 hours a day at the firm, something like that. For most of you on your homework assignment I’m hoping it’s a little less than that, but not much less than that. Alright, so the strategy choice is 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. And just to mention as we go past it, a novelty here, every game we’ve seen in the class so far has had a discrete number of strategies. Even the game when you chose numbers, you chose numbers 1, 2, 3, 4, 5, up to 100, there were 100 strategies. Here there’s a continuum of strategies. You could choose any real number in the interval 0, 4. So you have a continuum of possible choices.

[译文 22]

他们共同拥有这个企业，并分享利润。他们每人分享50%的利润。这是一个利润分享的合作关系，每个代理人都将选择她的努力水平，她投入这个企业的努力水平。这可能是你作为律师决定在这份工作上花费多少小时。对于你们大多数人来说，这些决定可能值得质疑你是否每天在企业工作20小时还是21小时，差不多就是这样。对于你们大多数人的作业，我希望它比那少一点，但也不会少太多。好的，策略选择是，我们不用小时来衡量，让我们标准化，认为这些选择落在0到4之间。你可以在这段时间内选择任意小时数。顺便提一下，随着我们深入，这里有一个新奇之处：我们在这节课上看到的所有博弈都有离散数量的策略。即使是你选择数字的游戏，你选择1到100的数字，有100个策略。这里有连续统的策略。你可以在区间0到4内选择任何实数。所以你有连续统的可能选择。

[段 23]

That’s not going to bother us, but let’s just point out it’s there. So there’s a continuum of strategies. In principle, you could bill your clients for fractions of a second or fractions of a minute. And let’s wonder what the firm’s profit is given by. So this partnership, this law firm, its profit is given by the following expression. 4 times the effort of player 1 plus the effort of player 2 plus a parameter I’ll call b times the product of their efforts. This is their profit. 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. We’ll assume that b lives between 0 and a quarter. And it’s known. I just want to be able to vary it later. So what the idea here The idea is player 1 directly contributes profits to the firm by working as does player 2 but they also contribute through this interaction term How do we think of that interaction term How do we think of that term B S1 S2 When you working on your homework assignments if your product, the thing you hand in, was just S1 plus S2, then you might think what? You might think there’s no point working in a study group at all.

[译文 23]

这不会困扰我们，但让我们指出它的存在。所以有连续统的策略。原则上，你可以按秒或分的零头来给客户计费。让我们想想这个企业的利润是怎么给出的。这个合作关系，这家律师事务所，其利润由以下表达式给出。4乘以玩家1的努力加上玩家2的努力，再加上一个参数b乘以他们努力的乘积。这就是他们的利润。我现在不会告诉你b是什么，但你的意思是，我不会确切告诉你它是什么，但我会解释它。我们假设b在0和四分之一之间。这是已知的。我只是想在以后能够改变它。这个想法是，玩家1通过工作直接为企业的利润做出贡献，玩家2也是如此，但他们也通过这个交互项做出贡献。我们怎么理解这个交互项？我们怎么理解这个项bS1S2？当你们做作业时，如果你们的产品、你上交的东西只是S1加S2，那么你可能会想什么？你可能会想一起在小组学习根本没有意义。

[段 24]

If the product is just the sum, or a multiple of a sum of the inputs, there wouldn’t be much point working in a team at all it’s the fact you’re getting this extra benefit from working with someone else that makes it worthwhile working as a team to start with, is that right? so we can think of this term as to do with complementarity complementarity or synergy a very unpopular word these days but still synergy so we’re going to assume that when you work together there are some synergies, some of you are good at some parts of the homework Some of them are good at other parts of the homework. And similarly, in this law firm, one of these guys is an expert on intellectual property, and the other one on fraud or something. So I’ve got the agents. I’ve got the strategy set. I know something about the profit of the firm. I need to tell you about their payoffs So the payoffs for player one is going to depend of course on her choice and on the choice of her partner.

[译文 24]

如果产品只是投入的和，或者是投入之和的倍数，那么在团队中工作就没有多大意义。恰恰是因为你从与他人合作中获得了额外的收益，才使得团队合作值得开始，对吗？所以我们可以把这个项看作是关于互补性、互补性或协同作用的，这些天这个词很不受欢迎，但仍然是协同作用。所以我们要假设当你们一起工作时，存在一些协同作用，你们中有些人在作业的某些部分很擅长，有些人在其他部分很擅长。同样，在这家律师事务所，其中一个人是知识产权方面的专家，另一个人是欺诈方面的专家之类的。所以我有了这些代理人。我有了策略集。我对企业利润有所了解。我需要告诉你们他们的收益。所以玩家一的收益当然取决于她的选择和她的搭档的选择。

[段 25]

And it’s going to equal a half, because they’re splitting the profits, so a half of the profit, so a half of 4 times S1 plus S2 plus BS1S2, she gets half of those profits but it also costs her it costs her S1 squared so S1 squared is her effort costs it’s her input costs this is the effort cost and similarly player 2 everything is symmetric here player 2’s payoff is the same thing so this term is the same except we’re going to subtract off player 2’s effort squared, S2 squared so you get the profits of the firm minus the disutility of having missed all that sleep there a guy in about the fifth row there who missed too much sleep so somebody just nudge him that it that it good We won put him on camera just nudge him That it good There you go Next time we’ll use the camera for that, okay? Alright. Whoops, what’s the line on the screen? Alright, so now we have everything we want to analyse this firm and to analyse how things are going to work, either when you’re working on a home work assignments or in the law partnership. And 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.

[译文 25]

它会等于二分之一，因为他们在分配利润，所以利润的一半，即二分之一乘以4乘以S1加S2加B乘S1乘S2，她得到一半的利润，但同时也要付出代价，她要付出S1的平方，所以S1的平方是她的努力成本，是她的投入成本，这是努力成本，类似地，2号玩家一切都是对称的，2号玩家的收益是一样的，所以这个项是相同的，只不过我们要减去2号玩家的努力平方，S2的平方，所以你会得到公司利润减去所有失眠带来的负效用——大约第五排有个家伙睡眠不足，所以有人推他一下，告诉他这很好，我们赢了我们把他放到镜头里推他一下告诉他这很好。就这样，下次我们用摄像头来提醒他，好吗？好了。哎呀，屏幕上的线是什么？好了，现在我们有了分析这个公司的一切，分析事情会如何运作的一切，无论你是在做家庭作业还是在律师事务所工作。再说一次，为了让你们觉得这件事与你们相关，我的意思是这当然是高度简化的，但外面有大量企业是合伙制，而且确实有这种利润分享规则，确实有协同效应。

[段 26]

So this is a relevant issue in a lot of businesses. Alright? Now we’re going to analyze this, no secret here, we’re going to analyze this using the idea of best response. Right? That’s not a surprise to any of you since that’s where we started today. Alright? So in particular I want to figure out what is player 1’s best response to each possible choice of player 2 I want to figure out what’s player 1’s best response to each possible choice S2 of player 2 how should I go about doing that how should I do that What’s player one’s best response for East possible choice, S2, of player two? How should I go about doing that? How should I do that? So here, what we did before was we drew these graphs with probabilities, at least with beliefs of player one, and the problem here is, previously we had a nice simple graph to draw because there were just two strategies for player two. Player two was a goalie, he could dive to the left or the right. The problem is here that player two 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. It’s going to be too hard. We need some other technique. How are we going to find out player one’s best response?

[译文 26]

所以这是一个与许多企业相关的问题。好了？现在我们要分析这个，没什么秘密，我们要用最佳反应的概念来分析。对吧？这对你们任何人都不是意外，因为我们今天就是从这里开始的。好了？具体来说，我想弄清楚2号玩家的每个可能选择下，1号玩家的最佳反应是什么。我想弄清楚对于2号玩家的每个可能选择S2，1号玩家的最佳反应是什么。我应该怎么做？我应该怎么做？在这里，我们之前做的是画出这些图，用概率，至少用1号玩家的信念，问题在于，之前我们有一个很好画的简单图，因为2号玩家只有两种策略。2号玩家是守门员，他可以扑向左或扑向右。问题在于这里2号玩家有连续统的策略，试图在板上画出所有无限多对象的概率分布，超过了我的绘图能力。太难了。我们需要一些其他技术。我们怎么找出1号玩家的最佳反应？

[段 27]

Somebody? Wave your hands in the air. Way, way, way, way back in the corner. Can we… Let’s get the mic… Stand up and wait for the mic to come to you. How are we going to do it? How are we going to figure out what player one’s best response is? Shout loudly. You can find player one payoffs as a function of player two of what player two effort is Good okay okay so we that certainly the first step We got that right Actually we got that So here here player one payoff as a function of what player two chooses and what player one chooses So we have that already, we have player one’s payoff as a function of the two efforts, and now I want to find out what is player one’s best effort, given a particular choice of S2. Uh, can I, uh, yeah, yeah. Take the derivative with a response. Sorry, derivative of S1. Good. Take a derivative and? S1 equals 0. Okay, okay, good, good. So we’re going to use calculus. We’re going to use calculus of one variable. We’re just changing one variable, S1. How many of you, we won’t let the camera say, how many of you have not, no, I’m not going to get your hands at all. 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, alright?

[译文 27]

有人吗？挥挥手。在后面的角落里。能不能……让我们把麦克风拿过来……站起来等麦克风过来。我们怎么做到？我们怎么弄清楚1号玩家的最佳反应是什么？大声喊出来。你可以找出1号玩家的收益作为2号玩家——作为2号玩家努力程度的函数。好，好的，好的，所以我们确实这是第一步我们做对了实际上我们做到了所以这里这里1号玩家的收益作为2号玩家选择什么的函数以及1号玩家选择什么的函数我们已经有了这个，我们有1号玩家的收益作为两个努力的函数，现在我想找出给定S2的特定选择时，1号玩家的最佳努力是什么。呃，我能，呃，是的，是的。对S1求导。抱歉。对S1求导。好的。求导然后呢？S1等于0。好的，好的，好的。所以我们要用微积分。我们要使用单变量微积分。我们只改变一个变量，S1。你们中有多少人，我们不会让摄像头看到，你们中有多少人没有见过我要在黑板上使用的微积分，或者更可能的是，如果你们从高中以来已经忘记了，不要慌，好吗？

[段 28]

There is a chapter in the back of the book, I think it’s chapter 25, that goes over this and refreshes your memory of such calculus, and if you’ve never seen it before, if you haven’t taken for example the equivalent of Math 112, come and see us, we probably try and line up a quick calculus lesson a special section for those people Alright So if what I going to do now is scary come and see us and we deal with it Alright, what I’m going to do, we want to take a derivative of this thing. What we’re going to do is, we’re asking the question, what is the maximum, choosing S1, of this profit? can I multiply the half by the 4 to save myself some time so the profit is 2 S1 plus S2 plus B S1 S2 minus S1 squared we’re asking the question taking S2 as given what S1 maximizes as an 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 zero. So I’m almost bound to get this wrong on the board. So can you all watch me like a hawk a second? All right, so if I differentiate this object, I differentiate, I’m going to find a first order condition in a second. All right so differentiate I going to have two still And then this S1 is going to become a 1 all right and this s1 here is going to become a plus bs2 Everyone happy with that And this s1 squared is going to become a minus 2s1.

[译文 28]

书后面有一章，我想是第25章，回顾并刷新你对这类微积分的记忆，如果你从来没有见过它，如果你没有学过例如相当于数学112的内容，来找我们，我们可能会尝试安排一个快速的微积分课，为那些人开设一个专题。好吧，所以如果我现在要做的事情很吓人，来找我们，我们会处理。好吧，我要做的是，我们想对这个东西求导。我们要做的核心问题是，在选择S1时，这个利润的最大值是什么？我可以把二分之一乘以4来节省一点时间，所以利润是2S1加S2加B乘S1乘S2减去S1的平方。我们要问的问题是，把S2视为给定，什么S1使这个表达式最大化，正如后排那位先生说的，我要对它求导，然后我要令它等于零。所以我在黑板上几乎肯定会把这个搞错。你们能像老鹰一样盯着我吗？好了，所以如果我对这个东西求导，我求导，我将在一秒钟内找到一个一阶条件。好了，求导我将有2仍然。然后这个S1将变成1好吧，这个这里的s1将变成加bS2每个人都对这个满意吗，而这个s1平方将变成减2s1。

[段 29]

All right, that was just differentiating. Everyone happy with the way I differentiated? Is this coming back from high school, the cogs are spinning now. Yeah? And to make this a first order condition, I’m going to say at the best response, put a hat over the one, at the best response, this is equal to zero. Alright? Yeah, can you get the guy again? Shouldn’t that be two? Oh, sorry, never mind. Okay, okay, okay. It’s quite, you’re right to shout out, I mean, doing it on the board, I’m very likely to make mistakes, but okay, so I differentiated this object, this is my first derivative, and I set it equal to zero. 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?

[译文 29]

好了，那只是求导。每个人对我求导的方式满意吗？这从高中回来，大脑齿轮现在转起来了。是吗？要让它成为一个一阶条件，我要说在最佳反应处，在1上加个帽子，在最佳反应处，这等于零。好了？是吗，能再叫一下那个人吗？不应该等于2吗？哦，抱歉，没关系。好的，好的，好的。你喊出来是对的，我的意思是，在黑板上做这个，我很可能犯错误，但是，好的，所以我对这个东西求导了，这是我一阶导数，我令它等于零。现在我一秒钟后将用它来工作，但我想确保我找到的是最大值而不是最小值。我怎么确保我找到的是最大值而不是最小值？

[段 30]

I look at the second derivative, alright, I look at the second derivative, which is the second order condition, so I’m going to differentiate this object again with respect to S1, pretend the hat isn’t there a second, second derivative, all right, look at the second derivative, which is the second order condition, so I’m going to differentiate this object again with respect to S1, pretend the hat isn’t there a second, and none of this has an S1 in it, so that all goes away, and I’m going to get minus 2, right, minus 2 which came from here, minus 2, and that is in fact negative, which is what I wanted to know, right, to find a maximum, I want the second derivative to be negative, all right, So here it is, I’ve got my first order condition, it tells me that the best response to S2 is the S1 hat that solves this equation, that solves this first order condition. Alright? And we can just rewrite that. If I divide through by 2 and rearrange, it’s going to tell me that S1 hat, or if you like, S1 hat is equal to 1 plus B S2. 1 plus B S2. So this thing is equal to player 1’s best response, given S2. All right Now I could go through again and do exactly the same thing for player 2 but I not going to do that because everything symmetric Okay, so everyone happy with that?

[译文 30]

我看二阶导数，好了，我看二阶导数，这就是二阶条件，所以我要对这个东西再对S1求导，假装帽子不在那里一秒钟，二阶导数，好了，看二阶导数，这就是二阶条件，所以我要对这个东西再对S1求导，假装帽子不在那里一秒钟，而且这里面没有任何东西含有S1，所以那些都消失了，我将得到负2，对了，负2是从这里来的，负2，而这实际上是否定的，这正是我想知道的，对吧，为了找到最大值，我希望二阶导数是负的，好了，所以在这里，我有了我的一阶条件，它告诉我对S2的最佳反应是S1帽子，它满足这个方程，满足这个一阶条件。好了？我们可以直接改写一下。如果我两边除以2并重新整理，它将告诉我S1帽子，或者说如果你愿意的话，S1帽子等于1加B乘S2。1加B乘S2。所以这个东西等于1号玩家的最佳反应，给定S2。好了现在我可以再走一遍，对2号玩家做完全相同的事情，但我不打算那么做，因为一切都是对称的。好了，每个人对这个满意吗？

[段 31]

So I could do the same kind of analysis, but we know I’ll get the same answer. So similarly, I would find that S2 hat equals 1 plus B S1, and this is the best response of player 2 as it depends on player 1’s choice of effort, S1. Okay, now I’ve found out what player 1’s best response is to player 2 and what player 2’s best response is to player 1 for each possible choice of player 2 up here and for each possible choice of player 1 down there. Alright, now let’s see if we can get a bit further and to get a bit further, let’s draw a picture. All right, what I’m going to do is I want to draw the two functions we just found and see what they look like. All right so this is all in our notes already I can get rid of it What I can do with it is some more chalk There we go Excuse me. Alright, 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. And there are different choices here. 1, 2, 3, and 4 for player 1. And here’s the 45 degree line. I’m careful I should get this right. 1, 2, 3, and 4 are the possible choices for player 1.

[译文 31]

所以我可以做同样的分析，但我们知道我会得到相同的答案。同样地，我会发现S2 hat等于1加上B S1，这是玩家2的最佳反应，因为它取决于玩家1的努力选择S1。好的，现在我已经找出了玩家1对玩家2的最佳反应，以及玩家2对玩家1的最佳反应，对于上面玩家2的每个可能选择和下面玩家1的每个可能选择都是如此。好的，现在让我们看看能否更进一步，为了更进一步，让我们画个图。好的，我打算做的是画出我们刚刚发现的两个函数，看看它们是什么样的。好的，这些都在我们的笔记里了，我可以把它擦掉。我可以用粉笔来做更多。好了，来了。抱歉。好的，我打算做的是画一个图，横轴是S1，纵轴是S2。这里有不同的选择。1、2、3和4是玩家1的选择。这是一条45度线。我小心点应该画对的。1、2、3和4是玩家1的可能选择。

[段 32]

Okay? Now before I draw it, I better decide what B is going to be. So I’m going to draw for the case, I’m going to draw the best response of player 1, and I’m going to draw the best response of player 2 in a minute, for the case B equals a quarter So we said B was somewhere between 0 and a quarter Let draw the case for b equals a quarter so the expression I want to draw first of all is the best response of player 1 as a function of S2 and we agreed that that was given by 1 plus a quarter now 1 plus a quarter S2 so for each possible choice of S2 I’m going to draw player 1’s best response and we’ll do it in red so if player 2 chooses 0 if player 2 chooses 0 what is player 1’s best response? somebody shout out? 1, ok, so 1 plus a quarter of 0 is 1 so if player 2 chooses 0 player 1’s best response is to choose 1 What if player 2 chooses 4? Player 2 chooses 4, what will be player 1’s best response? All right, so it’ll be 1 plus a quarter times 4. A quarter times 4 is 1, so 1 plus 1 is 2. So player 2’s best response in that case will be 2.

[译文 32]

好的？在画之前，我最好先决定B的值。我要画的是这种情况，我要画玩家1的最佳反应，我一会儿要画玩家2的最佳反应，假设B等于四分之一。我们说B在0到四分之一之间。让我画B等于四分之一的情况。我首先想画的表达式是玩家1作为S2函数的最佳反应，我们一致认为它由1加上四分之一的S2给出，所以对于S2的每个可能选择，我要画出玩家1的最佳反应，我们用红色来做。如果玩家2选择0，如果玩家2选择0，玩家1的最佳反应是什么？有人喊出来吗？1，好的，所以1加上四分之一的0是1，所以如果玩家2选择0，玩家1的最佳反应是选择1。如果玩家2选择4呢？玩家2选择4，玩家1的最佳反应会是什么？好的，那将是1加上四分之一乘以4。四分之一乘以4是1，所以1加1是2。所以在这种情况下玩家2的最佳反应是2。

[段 33]

So if player 1 chooses 4, player 2 should choose, sorry, if player 2 chooses 4, player 1 should choose 2. Alright, and this is a straight line in between. so the line I’ve just drawn is the best response for player 1 as it depends on player 2’s choice everyone happy with how I drew that? I mean I’m assuming you’re taking on faith that it’s a straight line in between but it is alright and 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 1 ok and we can do the the same for player two. We can draw player two’s best response as it depends on the choices of player one. But rather than go through any math, I already know what that line’s going to look like. What does that line look like? Somebody raise your hand, somebody What will player two best response look like as a function of player one choice On the same picture Someone we haven had before We had all these guys before Someone else someone else Yeah there a guy in the middle Can we get into him? Oh, yeah, yeah, maybe it’s easier from that side. That’s probably right. Shout out loud to the mic, can I hear you?

[译文 33]

所以如果玩家1选择4，玩家2应该选择，抱歉，如果玩家2选择4，玩家1应该选择2。好的，这是一条介于其间的直线。我刚画的这条线是玩家1的最佳反应，取决于玩家2的选择。大家对我画的满意吗？我的意思是，我假设你们相信它是一条介于其间的直线，但实际上就是这样。我们读这张图的方式是，你给我一个S2，我读到粉色线，然后往下落，那告诉我玩家1的最佳反应。好的，我们也可以对玩家2做同样的事。我们可以画出玩家2的最佳反应，取决于玩家1的选择。但是不用经过任何数学计算，我已经知道那条线看起来像什么了。那条线看起来像什么？有人举手吗，有人。玩家2的最佳反应作为玩家1选择的函数看起来像什么？在同一张图上。我们之前没见过这个。我们之前都见过这些人。另一个人，另一个人。是的，中间有个人。我们能让他参与吗？哦，好的，好的，也许从那边来更容易。那可能是对的。对着麦克风大声喊，我能听到你吗？

[段 34]

It should be a reflection across the 45 degree line. Right, exactly. So if I drew the equivalent line for player two, which is player two’s best response for each choice of player one, we’re simply flipping the identities of the players which means we’re reflecting everything in that 45 degree line. So it’ll go from 1 here to 2 here, and it’ll look like this. So this is the best response for player 2 for every possible choice of player 1, 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 1, I read up to the blue line and go across, and that tells me player 2’s best response. Okay, now we’re making some progress. What do we notice? Remember we said in one of the lessons of today’s class, the second lesson, the first lesson was don’t shoot towards the middle of the goal, and the second more general lesson was what? It was don ever play a strategy that is not a best response to everything I mean I cheating a little bit here because I ignoring beliefs but trust me that okay in this game All right So are the strategies here that are never a best response to anything? The strategies here… I’ll put it another way. What strategies of player ones are ever a best response?

[译文 34]

它应该是关于45度线的反射。对的，完全正确。所以如果我为玩家2画出等价的线，也就是玩家2对玩家1每个选择的最优反应，我们只是交换了玩家的身份，这意味着我们把一切都关于那条45度线反射。所以它从这里1到那里2，看起来会像这样。所以这是玩家2对玩家1每个可能选择的最优反应，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1的努力水平，我读到蓝线然后横过去，那告诉我玩家2的最优反应。好的，现在我们取得了一些进展。我们注意到什么？记住我们今天课上说过的一些经验教训，第二条经验教训，第一条经验教训是不要朝球门中间射门，第二条更一般的经验教训是什么？是永远不要玩一个不是对所有情况的最优反应。我是说我在作弊一点点，因为我忽略了信念，但相信我在这游戏中这是对的。好的，所以这里的策略中有哪些是永远不是对任何东西的最优反应？这里的策略……我换一种说法。玩家1的哪些策略曾经是最优反应？

[段 35]

Anybody? Why don’t you have a look? All right? If player two chooses zero, then player 1’s best response is 1, and that’s as low as he ever goes. It’s as low as he ever goes. So these strategies down here, these strategies down here, less than 1, are never a best response for player 1. And if player 2 chooses 4, then the synergy leads player 1 to raise his best response all the way up to 2, but these strategies up here, strategies up here, above two, are never a best response for player one. Is that right? So the strategies below one and above two are never a best response for player one. And similarly for player two the lowest player one could ever do is choose zero in which case player two would want to choose 1 so the strategies below 1 are never a best response for player 2 and the strategies above 2 were never a best response for player 2 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. So all of these strategies for player 1 are gone, and all of these strategies for player 1 are gone. You might want to not scribble quite so much on your own notebook, but still.

[译文 35]

有人吗？为什么不看看呢？好的？如果玩家2选择0，那么玩家1的最优反应是1，那是他能达到的最低点。那是他能达到的最低点。所以这里的这些策略，这些下面的策略，小于1的，永远不是玩家1的最优反应。如果玩家2选择4，那么协同作用导致玩家1把他的最优反应提高到2，但这里的这些策略，这些上面的策略，高于2的，永远不是玩家1的最优反应。对吗？所以低于1和高于2的策略永远不是玩家1的最优反应。同样对于玩家2，玩家1能做的最低选择是0，这种情况下玩家2会选择1，所以低于1的策略永远不是玩家2的最优反应，高于2的策略也永远不是玩家2的最优反应。所以让我们实际上，你可能想在自己的笔记本上稍微温柔一点，但在我的黑板上，让我们去掉所有这些永远不是最优反应的策略。所以玩家1的所有这些策略都没了，玩家1的所有这些策略都没了。你可能不想在自己的笔记本上涂写太多，但还是。

[段 36]

And all these strategies for player 2 are gone, and all these strategies for player 2 are gone. And what’s left? A lot of scribble is left. What’s left? so I claim if you look carefully there’s a little box in here a little box in here that’s still alive I’ve deleted all the strategies that are never best response for player 1 and all the strategies that are never best response for player 2 and what I’ve got left is that little box but I can’t see that little box so what I’m going to do is I’m going to redraw that little box And all the strategies that are never responsible are 2, and what I’ve got left is that little box. But I can’t see that little box, so what I’m going to do is, I’m going to redraw that little box. So let’s redraw it. So it goes from 1 to 2 this time. I’m going to blow up that box. so this now is 1, 1 and up here is 2, 2 and let’s put in numbers of quarters so this will be 5 quarters 6 quarters and 7 quarters and over here it will be 5 quarters and 6 quarters and 7 quarters. And let’s just draw how that pink and blue line look in that box. This is just a picture of that little box.

[译文 36]

还有玩家2的所有这些策略也没了，玩家2的所有这些策略也没了。剩下什么？剩下一堆涂鸦。剩下什么？所以我声称如果你仔细看，这里有一个小方块，这里有一个小方块还活着。我删除了所有永远不是玩家1最优反应的策略，以及所有永远不是玩家2最优反应的策略，我剩下的就是那个小方块，但我看不到那个小方块，所以我要做的是我要重新画那个小方块。还有所有永远不是最优反应的策略是2，我剩下的就是那个小方块。但我看不到那个小方块，所以我要做的是我要重新画那个小方块。让我们重新画。所以这次它从1到2。我要把那个方块放大。所以现在是1，1，上面这里是2，2，让我们标上四分之一的数字，所以这将是四分之五、四分之六和四分之七，这边也是四分之五、四分之六和四分之七。让我们就在那个方块里画出粉色和蓝色线的样子。这只是那个小方块的一张图。

[段 37]

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 We can work it out at home and check carefully but this isn that correct All right. All right. What I’ve done is I’ve redrawn the picture we just had and blown it up. And have any of you seen that picture before? Anyone here seen that picture before? That’s the picture we just had, right? Except I’ve changed the numbers a bit. All right. 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 looks exactly the same as it did before, albeit blown up and the numbers changed. So what have we done so far? We said players should never play a strategy that’s never a best response to anything. So we threw those away. Now what’s left? What do we do now? 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. Is that right This should be something familiar from when we were leading dominated strategies The strategies I about to throw away now they not it isn that they not best responses they are best responses to something 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 So what strategies do I have in mind?

[译文 37]

所以结果会是粉线从这里到这里，蓝线从这里到这里。我们可以在家里算一下仔细核对，但这对吗？好的。好的。我刚才做的是把我们刚才的那张图重新画了一遍并放大了。你们有人见过那张图吗？这里有人见过那张图吗？那就是我们刚才的那张图，对吧？只是我稍微改了一下数字。好的。一旦我删除了所有从未是最佳反应的策略，只关注存活下来的那小部分策略，图看起来和之前完全一样，只是放大了，数字变了。到目前为止我们都做了什么？我们说玩家永远不应该玩一个对任何情况都不是最佳反应的策略。所以我们把它们扔掉了。现在剩下什么？我们接下来做什么？所以有些我们没有扔掉的策略是对某些东西的最佳反应，但它们作为最佳反应的那些东西现在已经被扔掉了。对吗？这应该是我们在讲被支配策略时熟悉的东西。我现在要扔掉的策略，它们并不是从未成为最佳反应，它们是对某些东西的最佳反应，但它们作为最佳反应的那些东西，我们知道不会被玩到，因为它们本身对任何东西都不是最佳反应。所以我想到的是什么策略？

[段 38]

What strategies am I about to throw away? well for example for player 1 we know now that player 2 is never going to choose any strategy below 1 so the lowest player 2 will ever choose is 1 and it turns out that the lowest player 2 would ever do in response to anything 1 and above never leads player 1 to choose a strategy less than 5 quarters and the highest player 2 ever chooses is 2 and the highest response that player 1 ever makes to any strategy 2 or less is 6 quarters so all these things bigger than 6 quarters can go let’s be careful here, 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 so we throwing away all of the strategies less than 5 quarters for player 1 and bigger than 6 quarters for player 1 which is 1 and a half for player 1 and similarly for player 2 And if I did this, and again, don’t scribble too much in your notes, but if we just make clear what’s going on here, and I actually delete these strategies, since they’re never going to be played, I end up with a little box again alright alright so everyone see what I did?

[译文 38]

我正要扔掉哪些策略？比如对于玩家1，我们现在知道玩家2永远不会选择1以下的策略，所以玩家2最低会选择1，而事实证明玩家2对任何1及以上策略的最低反应永远不会让玩家1选择低于四分之五的策略，而玩家2最高选择的是2，玩家1对任何2或以下的策略的最高反应是四分之六，所以所有大于四分之六的都可以去掉。让我在这里仔细说明一下，这些我即将删除的策略，并不是它们从未是最佳反应，它们是对某些东西的最佳反应，但它们作为最佳反应的那些东西是永远不会被玩到的，所以它们无关紧要。所以我们扔掉了玩家1所有低于四分之五的策略和大于四分之六的策略，也就是玩家1的四分之一点五，同样地对于玩家2也是这样。如果我这样做，而且同样地，不要在笔记上涂写太多，但如果我们把这里发生的事情弄清楚，然后我实际上删除这些策略，因为它们永远不会被玩到，我就又得到一个小方框了。好的，大家都看到我做了什么？

[段 39]

I started with the game I found out what player 1’s best response was for every possible choice of player 2 and I found out what player 2’s best response was for every possible strategy of player 1 I threw away all strategies that were never a best response then I looked at the strategies that were left I said those strategies that were best response to things that have now been thrown away, but not best response otherwise, I can throw those away too, and when I threw those away, I was left once again with this little box. I could do it again and again and again. I could do it again and again. If I go on doing this exercise again and again and again, what am I going to end up with? So out of that, what am I going to end up with? The intersection, right? If I keep on constructing these boxes with What am I going to end up with? Sounded out what I’m going to end up with? The intersection, right? If I keep on constructing these boxes within boxes, so that next box would be a little box in here, I can’t even draw it, but it’s something like this. If I keep on doing boxes within boxes, I’m going to converge in on that intersection. So if we know people are not going to play something which is never a best response, and we know if people are not going to play something which is never a best response, something which is never a best response, and we know people are not going to play, which is not a best response, etc, etc, etc.

[译文 39]

我从这个博弈开始，弄清了对于玩家2的每一个可能选择，玩家1的最佳反应是什么，我也弄清了对于玩家1的每一个可能策略，玩家2的最佳反应是什么。我扔掉了所有从未是最佳反应的策略。然后我看着剩下的那些策略，我说那些是对现在已经被扔掉的东西的最佳反应，但除此之外不是最佳反应的策略，我也可以把它们扔掉，而当我扔掉这些之后，我又剩下这个小方框了。我可以一次又一次地重复这个过程。如果我不断重复这个练习，最终我会得到什么？所以从中我会得到什么？交集，对吧？如果我继续构造这些盒中盒，让下一个盒子是这里面更小的一个盒子，我甚至画不出来，但大概是这个样子。如果我不断做盒中盒，我会收敛到那个交点。所以如果我们知道人们不会玩一个从未是最佳反应的东西，而且我们知道如果人们不会玩一个从未是最佳反应的东西，一个从未是最佳反应的东西，而且我们知道人们不会玩，不是最佳反应的东西，等等，等等，等等。

[段 40]

Alright? We’re going to converge in in this game to just one strategy for each player, which is where they intersect. Which is where they intersect. Alright? So what we’re going to converge in onto is that S1 star, let’s call it in this case, is equal to 1 plus b s2 star, and that s2 star is equal to 1 plus b s1 star. All right? Actually, we can do it a little better than that since we know the game is symmetric. We know that s1 star is actually equal to s2 star All right All right so taking advantage of the fact that we know S1 equal to S2 because we relying on the 45 degree line I can simplify things by making S1 star equal to S2 star. Alright? So now I’ve got, actually that looks like three equations, it’s really just two equations, because one of them implies the other, alright? 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 star is equal to 1 or S1 star equals S2 star is equal to 1 over 1 minus B. And again, any time I’m doing algebra on the board, someone should check me at home. So just have a quick look at that.

[译文 40]

对吧？在这个博弈中我们会收敛到每个玩家只有一个策略，也就是它们相交的地方。也就是它们相交的地方。对吧？所以我们收敛到的是S1星，让我们把它叫做，在这个例子中，等于1加上b乘以S2星，而S2星等于1加上b乘以S1星。好的？实际上我们可以做得更好一些，因为我们知道这个博弈是对称的。我们知道S1星实际上等于S2星。好的好的，利用我们知道S1等于S2这一事实，因为我们依赖的是45度线，我可以简化，方法是让S1星等于S2星。好的？所以现在我有了，实际上看起来像三个方程，但其实是两个方程，因为其中一个蕴含另一个，对吧？我可以解它们，而如果我解出来，我会得到类似的东西，让我仔细一点，我会得到类似1减去B乘以S1星等于1，或者S1星等于S2星等于1除以1减去B。同样地，任何时候我在黑板上做代数，都应该有人在家里检查我。让我们快速看一下。

[段 41]

Is that right? I think that’s right. My algebra, which is often wrong, suggests that the solution is S1 star equals S2 star equals 1 over 1 minus B. Alright, this is just math, there’s nothing interesting going on. I’m just trying to solve out the equation at this point. Okay, so what do we learn here? We learned that in this game deleting strategies that are never best response and then deleting strategies that are never best response anything that a best response and so on and so forth yielded a single strategy for each player Just one strategy for each player. And that strategy was given by this equation. So if we were a management consultant, 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 question is this amount of work a good amount of work or a bad amount of work you’re working for McKinsey you’ve been hired by Joe Smith and Ann Bloggs to figure out their strategy in working in a team on working on my homework assignments, you’ve figured out how much work they’re going to contribute, is this a good amount of work? Are they contributing too much? Too little?

[译文 41]

对吗？我想那是对的。我的代数，虽然经常出错，表明解是S1星等于S2星等于1除以1减去B。好的，这只是数学，没什么有趣的。我现在只是在试图解出这个方程。所以我们从这里学到什么？我们学到，在这个博弈中，删除从未是最佳反应的策略，然后删除从未是对任何东西的最佳反应的策略的最佳反应，以此类推，最终得到每个玩家只有一个策略。每个玩家只有一个策略。而那个策略由这个方程给出。所以如果我们是管理咨询师，在McKinsey或什么的公司工作，我们被请进来给你建议家庭作业的事，或者给这个法律合伙企业提供关于工作实践的建议，我们会得出一个预测，这就是你们要做多少工作。问题是这个工作量是好的还是坏的？你们为McKinsey工作，你们被Joe Smith和Ann Bloggs雇佣来帮他们想出在团队合作做家庭作业时的策略，你们已经算出了他们会贡献多少工作量，这是一个好的工作量吗？他们是不是贡献太多了？还是太少了？

[段 42]

Of course the answer to that is compared to what? Compared to what? So let me rephrase. 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 Let’s have a poll. Who thinks more? Who thinks more? Let’s turn the camera around for the audience. Let’s have a look. Who thinks more? Who thinks they’re working just right? Who thinks less? A lot of abstentions here. I think they’re working too little here. I think they’re working too little here compared to what’s efficient I’ll get you to solve it out on a homework assignment, so you can actually prove that You can prove that in fact if you if you’re running a contract if there’s a social planner You’d work more, but let’s try and get to grips why? Why is it that when we see these law partners? Or medical partners or whatever happens to be your students working 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 I’m conceding the answer, I’m telling you they’re going to work too little why do they end up not working hard enough can you take us can we get a mic in here yeah working hard enough.

[译文 42]

当然答案取决于和什么比？和什么比？让我换一种说法。这对合伙人或者两个做家庭作业的学生，他们工作的量比有效率的水平是多还是少？让我们做个投票。谁觉得多了？谁觉得多了？让我们把镜头转向观众。让我们看看。谁觉得多了？谁觉得刚刚好？谁觉得少了？这里弃权的好多。我觉得他们工作太少了。我觉得和有效率的水平比他们工作太少了。我会让你们在一个家庭作业里解出来，这样你们可以实际证明这一点。你可以证明，事实上如果你在经营一份合同，如果有社会规划者，你会工作得更多，但让我们试着理解为什么？为什么当我们看到这些法律合伙人？或者医疗合伙人，或者不管你的学生是什么，在一起做家庭作业时？为什么我们往往会得到效率不足的少努力，当我们开始分析策略并求解这个博弈的时候。我承认答案，我告诉你们他们会工作得太少。为什么他们最终不够努力？我们能让麦克风过来吗？是的，不够努力。

[段 43]

Can you take us? Can we get a mic in here? I guess because if they work any harder than that, then the other person is just going to slack off instead. Alright, so there’s something about that. On the other hand, this isn’t really a… I mean, the intuition you’re giving me is kind of a prisoner’s dilemma intuition. Right? Saying, I’m going to let the other guy work and I’ll shirk. but there’s something in that but there’s a little bit more going on here what more is going on I think that’s a good first step there is something of that if there are two people working together there’s about half as much work for each person to do that’s true but that would suggest it doesn’t matter if they slack off what’s going on here so go back to your economics 115 or 150 if you took either of those courses what’s the problem here What’s underlying the problem? Let’s get this guy in pink down here. They only capture 50% of their marginal effort. That’s the point. The benefit of their marginal effort. Good, good. Well, your name is? So Patrick is giving I think it the correct answer here The problem here isn really about the amount of work It isn even by the way about the synergy You might think it because of this synergy that they don take into account correctly That isn’t the problem here.

[译文 43]

你能带我们去吗？我们能在这里放个麦克风吗？我想是因为如果他们再努力工作，另一个人就会偷懒。好吧，这有一定道理。另一方面，这其实不是……我是说，你给我的直觉有点像囚徒困境的直觉。对吧？就是说，我打算让另一个人工作，我就偷懒。但这有一定道理，但这里还有更多的东西在发生。还有什么在发生？我想这是很好的第一步，有一定道理。如果有两个人一起工作，每个人大约只需做一半的工作。这没错，但这意味着他们是否偷懒其实无所谓。这是怎么回事？所以回到你的经济学115或150，如果你修过其中任何一门课程，这里有什么问题？问题的根本是什么？让穿粉红色衣服的这位过来。他们只能获得自己边际努力的50%。这就是关键。他们边际努力的好处。好，好。你叫什么名字？Patrick给出了一个我认为正确的答案。这里真正的问题不是关于工作量。甚至也不是关于协同效应——你可能会这样想，因为这种协同效应，他们没有正确地考虑进去。但这也不是这里的问题。

[段 44]

It turns out even without the synergy, this problem would be there. The problem is what Patrick said. 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 right, 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 of the induced profits of the firm because of profit share and that leads all of us to put in too little effort what’s the general term that captures all such situations in economics it’s an externality there’s an externality here there an externality when I figuring out how much effort to contribute to this firm I don take into account that other half of profits that goes to you So this isn’t to do with the synergy, it isn’t to do with something complicated, it’s something you knew back in 115. If you have profit sharing in a firm, or profit sharing in homework assignment, or any giant project, you have to worry about too little effort being contributed because there’s an externality. My effort benefits you, not just me.

[译文 44]

事实证明，即使没有协同效应，这个问题依然存在。问题在于 Patrick 所说的。问题在于，在边际层面上，我作为这家公司的一名员工，无论是律师事务所还是作业解答小组，我在边际上投入的每一单位额外努力，都由我承担全部成本，但我只能获得一半的收益。在边际上，我为我贡献的额外努力承担成本，但我只能获得公司因利润分享而产生的利润的一半。正是这一点导致我们所有人都投入了过少的努力。在经济学中，捕捉所有这类情况的通用术语是什么？那就是外部性。这里存在一种外部性。当我决定向这家公司贡献多少努力时，我没有把你获得的那另一半利润考虑进去。因此，这与协同效应无关，与什么复杂的事情也无关，而是你在 115 课程中学过的内容。如果你在一家公司实行利润分享，或者在作业分配中实行利润分享，或者在任何大型项目中实行利润分享，你就必须担心贡献的努力过少的问题，因为存在外部性。我的努力对你有益，而不仅仅对我有益。

[段 45]

Now while we’ve got this on the board, let’s just think a little bit more. What would happen if we changed the degree of the synergy? What would happen if we lowered B? So B is the degree to which the synergy across these workers. If we lowered B, what would happen to our picture? Let me redraw our picture unscribbled. We had a picture that looked something like this. This was S1 and this was S2. If we lowered the degree of synergy, what would happen to the effort level that we find by this method What would happen What would happen to the picture And again this is a really 115 kind of exercise We going to be moving lines around Yeah, Henry, isn’t it? So can I come to Henry? The lines will get shallower and eventually become horizontal and vertical respectively. Alright, good. So the pink line is actually going to get steeper, but I know what you mean. 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. It’s down in this direction. Alright? 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.

[译文 45]

现在，趁我们还在黑板上写着这个，让我们再深入思考一下。如果我们改变协同的程度，会发生什么？如果我们降低B，会发生什么？所以B是跨这些工人的协同程度。如果我们降低B，我们的图形会发生什么变化？让我重新画一下我们的图形，不那么潦草。我们之前有一个大概像这样的图形。这是S1，这是S2。如果我们降低协同程度，通过这个方法得出的努力水平会发生什么变化，会发生什么，图会发生什么变化？再说一次，这真的是一个115类型的练习，我们要在线之间移动。是的，Henry，不是吗？所以我能问一下Henry吗？这些线会变得更平缓，最终分别变成水平线和垂直线。好的，很好。所以粉色这条线实际上会变得更陡，但我知道你的意思。粉色这条线会向垂直方向移动，而蓝色这条线会向水平方向移动，注意我们产生的努力量会急剧下降。往这个方向下降。明白吗？所以如果我们降低这里的协同程度，不仅我贡献的努力减少了，而且你知道我贡献的努力减少了，因此你贡献的努力也减少了，以此类推。

[段 46]

So we get this scissors effect of taking effort this way. Alright. we could draw other lessons from this but let me try and move on a little bit 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 point where the pink and the blue line crossed. What do we call that point? What do we call the point where the pink and the blue line cross? that’s an important idea for this class that’s going to turn out to be what’s called a Nash equilibrium we know what it’s called how many of you have heard the term Nash equilibrium before how many of you saw the movie about Nash we’ll come back and talk about that a bit next time so this is a Nash equilibrium but 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? But what does it mean here Why is it what going on at that line What does it tell us What does it tell us that the pink and the blue line cross What makes that point special?

[译文 46]

所以我们得到了这种剪刀效应，把精力往这边拉。好吧。我们可以从中得到其他经验教训，但让我试着继续往下讲，在这个博弈中我们决定通过查看最优反应来解决问题，删除那些从来不是最优反应的东西，再查看，再删除那些从来不是最优反应的东西，如此反复，幸运的是在这个特定的博弈中它收敛了，而且收敛到了粉线和蓝线交叉的那一点。我们称那一点为什么？粉线和蓝线交叉的那一点我们称之为什么？这是这堂课的一个重要概念，它最终会被称为纳什均衡。我们知道它叫什么，你们中有多少人之前听说过纳什均衡这个词？你们中有多少人看过关于纳什的电影？我们下次会回来讨论一下这个。所以这是一个纳什均衡，但我们知道它的行话叫什么，我们也知道我们事先就觉得那会是一个重要的点，因为你们大多数人都上过经济学课程，你们知道每当经济学中出现线条交叉时，那就是重要的，对吧？但这里是什么意思？为什么那是正在发生的事？那告诉我们什么？粉线和蓝线交叉告诉我们什么？是什么让那一点变得特殊？

[段 47]

What does it mean to say the pink and the blue line cross? Can I get that way, three rows behind you in purple? Shout out again. It means that neither player has an incentive to deviate from that point. All right, well that’s correct. That’s correct. So let’s try and read that through. So, I don’t know your name. Your name is? Alan. So Alan is saying if player 1 is choosing this strategy and player 2 is choosing her corresponding strategy here, neither player has an incentive to deviate. Or another way of saying it is, neither player wants to move away. So if player 1 chooses S1 star, player 2 will want to choose S2 star since that’s her best response. And if player 2 is playing S2 star, player 1 will want to play S1 star, since that’s his best response. Neither has any incentive to move away So more succinctly player 1 and player 2 at this point with the lion cross player 1 and player 2 are playing a best response to each other Right? The players are playing a best response to each other. best response to each other alright so clearly in this game it’s where the lines cross let’s go back to the game we played with the numbers everyone had to choose a number and the winner was going to be the person who That was closest to two-thirds of the average.

[译文 47]

说粉线和蓝线交叉是什么意思？我能这样吗，在你后面三排穿着紫色？再喊一遍。这意味着两个玩家都没有动机偏离那个点。好的，嗯这是正确的。这是正确的。所以让我们试着理解一下。所以，我不知道你的名字。你的名字是？Alan。所以 Alan 在说如果 player 1 选择这个策略和 player 2 选择她对应的策略，两者都没有动机偏离。或者换句话说，两个玩家都不想离开。所以如果 player 1 选择 S1*，player 2 会想选择 S2* 因为那是她的最优反应。如果 player 2 玩 S2*，player 1 会想玩 S1*，因为那是他的最优反应。两者都没有动机离开。所以更简洁地说，player 1 和 player 2 在这个点上，狮线交叉，player 1 和 player 2 在玩对彼此的最优反应。对吗？玩家们在玩对彼此的最优反应。对彼此的最优反应。好的，所以显然在这个游戏中这就是线交叉的地方。让我们回到我们玩过的数字游戏，每个人都必须选一个数字，赢家将是那个最接近平均值三分之二的人。

[段 48]

By the way, the winners never picked up their winnings for that, so they still can. So in that game, what’s the Nash equilibrium in that game? Yeah, everyone choosing one. Everyone choosing one. How do we know that’s the Nash equilibrium in that game? How do we know that everyone choosing one is the Nash equilibrium in the game where you all chose numbers Well let just use the definition If everybody chose a one right the average in the class would be one Two-thirds of that would be two-thirds. You can’t go down below one. So everyone’s best response would be to choose one. I’ll say it again. So if everyone chose one, then everyone’s best response would be to choose one. Right? So that would be a Nash equilibrium. Right? Did people play Nash equilibrium when we played that game? No, they didn’t. Not initially at any rate. But notice as we played the game repeatedly, what happened? As we played the game repeatedly, we noticed that play seemed to converge down towards one. Is that right? And in this game, when we analyzed the game repeatedly, it seemed like our analysis converged towards the equilibrium. And that’s not always going to happen, but it’s kind of a nice feature about Nash equilibrium. Sometimes, sometimes play tends to converge there. Nash equilibrium is going to be a huge idea from now to the midterm exam and we’re going to pick it up and see more examples on Wednesday.

[译文 48]

顺便说一下，获胜者们从来没有领取他们的奖金，所以他们仍然可以领取。所以在那个游戏中，那个游戏的纳什均衡是什么？是的，每个人都选择1。每个人都选择1。我们怎么知道那是那个游戏的纳什均衡？我们怎么知道每个人都选择1是在你们都选择数字的游戏中的纳什均衡？好吧，我们就用定义。如果每个人都选择1，班级的平均数就是1。三分之二就是三分之二。你不能低于1。所以每个人的最佳反应就是选择1。我再说一遍。所以如果每个人都选择1，那么每个人的最佳反应就是选择1。对吧？所以这将是一个纳什均衡。对吧？当我们在玩那个游戏时，人们有没有按照纳什均衡来玩？不，他们没有。至少一开始没有。但注意当我们反复玩这个游戏时，发生了什么？当我们反复玩这个游戏时，我们注意到游戏似乎收敛到1。是这样吗？在这个游戏中，当我们重复分析这个游戏时，我们的分析似乎收敛到均衡点。这并不总是会发生，但这是纳什均衡的一个很好的特性。有时，有时游戏往往会收敛到那里。纳什均衡将从现在到期中考试都是一个重要概念，我们会在周三继续讨论并看更多例子。