耶鲁大学博弈论公开课 - 第14集 落后的感应-承诺，间谍，和先行者优势

视频全文转录（中英双语）

以下为完整中英双语转录，已加标点。英文在前，中文在后，逐段对照。

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

[段 1]

All right, so today I want to do something a little bit more mundane than we did on Monday. I want to go back and talk about quantity competition. so in the first half of the course we talked about price competition we talked about quantity competition we talked about competition with differentiated products I want to go back and revisit essentially the Cournot model so this was the Cournot model two firms are producing are choosing their quantities simultaneously firm 1 is choosing Q1 and firm 2 is choosing Q2 and all of this is just reviews so this is all stuff that’s in your notes already already, this is the demand curve. It tells us that prices depend on the total quantity being produced. So this is Q1 plus Q2, and this is prices. Then the demand curve is a straight line of slope B. All right that what this tells us It has slope minus B All right And we know that payoffs are just profits which are price times quantity revenues minus cost times quantity costs. We’ll get assuming constant marginal costs. And we did this model out in full in maybe the third week of class, and we figured out what the best response diagram looked like.

[译文 1]

好的，今天我想做一件比周一稍微平淡一些的事情。我想回顾一下数量竞争。在课程的前半部分，我们讨论了价格竞争、数量竞争以及差异化产品竞争。现在我想回头重新审视一下古诺模型。这就是古诺模型，两个企业同时选择产量，企业1选择Q1，企业2选择Q2。这都是复习内容，这些都在你们的笔记里了。这是需求曲线。它告诉我们价格取决于生产的总量。所以这是Q1加Q2，这是价格。需求曲线是一条斜率为-B的直线。好的，这告诉我们它的斜率是-B。我们知道收益就是利润，即价格乘以数量，收益减去成本乘以数量成本。我们假设边际成本不变。我们在课堂的第三周左右完整地推导了这个模型，我们找出了最佳反应图是什么样子的。

[段 2]

And if you remember correctly, this was the best response for firm 1, taking firm 2’s output as given and this is the best response for firm 2 taking firm 1’s output as given and there were a few other details in here this was the monopoly quantity this was the competitive quantity and so on but this is enough for today and actually we’d done a bit more than that we’d actually worked out in class what the equations were for these best responses here they are and I’m not going to re-derive these today but they’re somewhere in your notes we kind of crunched through some calculus and figured out what firm 1’s best response looks like algebraically. Here it is. So this is the equation of this line. And similarly for firm 2, so this is the equation of this line. All right? And finally we figured out what the Nash equilibrium was and there was no prizes here The Nash equilibrium in Cournot was where these best responses crossed and this is the equation for the Nash equilibrium Have I made a mistake The best, oh thank you, thank you. Best response, thank you. Best response for firm 1 is a function of Q2, exactly. Thanks Jake. Alright, so this is all stuff we did before, I want to go back to this model now to revisit it in the context of thinking about sequential or dynamic games.

[译文 2]

如果你记得正确的话，这是企业1的最佳反应，假设企业2的产量给定；这是企业2的最佳反应，假设企业1的产量给定。这里还有一些其他细节，这是垄断产量，这是竞争产量等等，但今天这些就够了。实际上我们做得比这更多，我们实际上在课堂上已经算出了这些最佳反应的方程在这里，我今天不会再重新推导，但它们在你们的笔记里。我们用一些微积分计算了一下，得出了企业1的最佳反应的代数形式。就这样。这是这条线的方程。企业2的类似，这是这条线的方程。好的？最后我们找出了纳什均衡在哪里，这里没有奖品。古诺模型中的纳什均衡就是这些最佳反应的交点，这是纳什均衡的方程。我是否出错了？最佳反应，谢谢，谢谢。企业1的最佳反应是Q2的函数，完全正确。谢谢Jake。好的，这些都是我们之前做过的，现在我想回到这个模型，在考虑序贯或动态博弈的背景下重新审视它。

[段 3]

So what we’re going to do is, we’re going to imagine that rather than having these firms choose their quantities simultaneously, one firm gets to move first, and the other firm moves after. Let’s be clear, we’re going to assume that one firm is moving first, and the other firm, we’ll assume firm 1 is going to move first, the other firm, firm 2, is going to get to observe what firm 1 has chosen, and then get to make her choice. So we’re just going to see what difference it makes when we go from this classic simultaneous move game into a sequential move game. And this model is fully famous, and I’m almost certainly spelling this wrong, but it’s due to a guy called Stackelberg. So what we’re looking at now is the Stackelberg model. Alright so how do we want to think about this So a natural question to bear in mind is assuming we in this world of quantity competition is it an advantage to get to move first to set one’s quantity first? Or is it an advantage to be able to wait, see what the other firm has done, and then respond? Is there an advantage in going first?

[译文 3]

我们要做的是，想象一下这些企业不是同时选择产量，而是一个企业先行动，另一个企业后行动。让我们明确一下，我们假设一个企业先行动，另一个企业，我们假设企业1先行动，另一个企业企业2将能够观察企业1的选择，然后做出她的选择。所以我们只是想看看当我们从经典的同步行动博弈转变为序贯行动博弈时会产生什么差异。这个模型非常著名，我几乎肯定拼写错误，但这归功于一个叫Stackelberg的人。所以我们现在要看的是Stackelberg模型。好的，我们想怎么思考这个问题？所以要记住的一个自然问题是，在数量竞争的世界里，先行动先设定数量是一个优势吗？还是能够等待，观察其他企业做了什么，然后再回应是一种优势？先行动有优势吗？

[段 4]

Or is there an advantage in knowing a bit more about the other firm and being able to move second? that’s going to be the question at the back of our mind for most of today alright so how are we going to think of this, how are we going to figure this out how are we going to figure this out there shouldn’t be a silence in the room there should be an instant answer how are we going to figure this out we’re going to use backward induction this is going to be an exercise in backward induction we won’t be able to draw a tree here because the game is too complicated because there’s a continuum of actions But nevertheless we are going to use backward induction. So what is using backward induction mean here? Using backward induction means starting at the ends and the end is what the end here is firm to Right firm one is going to move first from two is going to observe that choice and then move So the end of the game is firm two so we’re going to solve out firm twos problem first And we’re actually going to do this entire analysis Firm 2 is going to observe that choice and then move. So the end of the game is Firm 2. So we’re going to solve out Firm 2’s problem first.

[译文 4]

或者在更多地了解其他企业并能够后行动有优势？这将是我们今天大部分时间在脑海中的问题。好的，我们怎么思考这个问题？我们怎么找出答案？我们怎么找出答案？房间里不应该有沉默，应该有一个即时答案。我们怎么找出答案？我们将使用逆向归纳法。这将是一次逆向归纳法的练习。我们不能在这里画树，因为游戏太复杂了，因为有连续统的行动。但我们仍然要使用逆向归纳法。那么使用逆向归纳法在这里意味着什么？使用逆向归纳法意味着从末端开始，这里的末端是企业2。对，企业1将先行动，企业2将观察那个选择然后行动。所以博弈的末端是企业2。所以我们要先解出企业2的问题。

[段 5]

And we’re actually going to do this entire analysis twice. We’re first of all going to do this analysis a bit intuitively, looking at pictures, and then I want to go back and crunch it out in the math. All right, you want to get some used to seeing that we can actually do it crunchly. All right, so this board is just review, so I’m going to get rid of it, I think. never mind. We’re not going to be using this board. This is just what we did do in the simultaneous move game, so we’ll get rid of it. All right, so in the sequential move game, we’re going to start by analyzing the move of Firm 2. So imagine yourself as a manager of Firm 2, you’re coming along to make your output decision. The output of Firm 1 is already set. All right, to analyze Firm 2 first. Firm 2 sees Q1, right, and now let’s choose Q2. So what is Firm 2 going to do? What’s firm 2 going to do? So I claim we already know this. We’ve already solved this problem. When do we solve this problem? we solve this problem, the problem of what Firm 2 is going to do? Well, we already solved it about a month ago when we looked at the simultaneous move game, because what we worked out then was what is Firm 2’s best response for any particular choice that Firm 1 makes?

[译文 5]

实际上我们要做两遍这个分析。首先我们要稍微直观地做这个分析，看图，然后我想回过头来用数学算出来。好的，你想习惯看到我们实际上可以用数学算出来。这块板只是复习，所以我要把它擦掉，我认为。没关系。我们不会用这块板。这只是我们在同步行动博弈中所做的，所以我们会把它擦掉。好的，所以在序贯行动博弈中，我们要从分析企业2的行动开始。想象你自己是企业2的经理，你来做出你的产出决策。企业1的产出已经确定了。好的，先分析企业2。企业2看到Q1，对，现在选择Q2。那么企业2要做什么？企业2要做什么？我声称我们已经知道了。我们已经解决了这个问题。我们什么时候解决这个问题的？我们解决企业2要做什么的问题？嗯，我们大约一个月前在研究同步行动博弈时已经解决了，因为我们当时所做的是，对于企业1的任何特定选择，企业2的最佳反应是什么？

[段 6]

All right? We already solved out that problem. It was kind of a, it took us a while to solve out, but basically it was to maximize Firm 2’s payoff, taking as given Firm 1, we already know what the equation looks like. and it’s just to remind ourselves what the picture looked like. Just to repeat of the picture we had before. We said for any particular choice of Firm 1, Firm 2’s best response can be drawn on a best response diagram and looks like this. It’s exactly the picture we have up there. All right? So this is the best response for Firm 2 taking as given the choice of Firm 1. All right? And we even know the equation of it. I won’t bother to rewrite it there, but we already already know the equation. All right? So in some sense, firm 2’s problem is a problem we’ve already seen. We already worked out months ago what Firm 2 should do taking a Firm 1’s output is given, and that’s exactly the problem Firm 2 finds herself in. She wakes up one morning, Q1 has been said already, and now she must choose Q2 to maximize her profits, so she’s going to choose her best response. And just to remind you how we read this picture for any particular choice of Q1, If we go up to the line and look across, this tells us what Q2 will respond.

[译文 6]

好的？我们已经解决了这个问题。这花了我们一段时间来解决，但基本上就是最大化企业2的收益，假设企业1给定，我们已经知道方程是什么样的。这只是为了提醒我们自己图片是什么样子的。只是重复我们之前的图片。我们说对于企业1的任何特定选择，企业2的最佳反应可以在最佳反应图上画出来，看起来像这样。它正是我们上面有的那张图片。好的？这是企业2的最佳反应，假设企业1的选择给定。好的？我们甚至知道它的方程。我不会费心在那里重写，但我们已经知道方程了。好的？所以从某种意义上说，企业2的问题是一个我们已经见过的问题。我们几个月前已经算出了企业2应该做什么，假设企业1的产出给定，这正是企业2发现自己面临的问题。她有一天早上醒来，Q1已经被设定了，现在她必须选择Q2来最大化她的利润，所以她将选择她的最佳反应。只是提醒你我们怎么看这张图片，对于Q1的任何特定选择，如果我们上到这条线并看过去，这告诉我们Q2将如何回应。

[段 7]

So if Q1 chooses this amount, then Q2 will choose this amount. If Q1 chooses this amount, then Q2 will choose this amount, and so on. All right, so there’s no mystery here. We already know what Firm 2 is going to do. All right? So by definition, by definition, The best response of 2 to Q1 tells us the profit maximizing output of output of of fund 2, taking Q1 as given. All right, so we’ve done the second. of this already. We already know what Firm 2 is going to do. Of course, the additional step here now is that Firm 1 knows that Firm 2 is going to do this. Firm 1’s going to move first, and Firm 1 knows that after she sets her quantity Q1, Firm 2 will respond by choosing her corresponding quantity, which is the best response to it. So if Firm 1 knows that if Firm 1 were to choose this quantity, then Firm 2 will respond by choosing this quantity. And Firm 1 knows that if she chose this smaller quantity, then Firm 2 will respond by choosing this larger quantity. Is that right? Is that right? So Firm 1 can anticipate how Firm 2 is going to respond to each of these choices. All right? So let’s just make that clear. So in particular, if Firm 1 was to choose Q1 hat, I’m not suggesting it should, but if Firm 1 was to choose this Q1 hat, then Firm 1 knows that Firm 2 will produce this quantity, which is the best response to Q1 hat, and if Firm 1 were to choose Q1 double hat, then Firm 2 will respond by choosing the best response to Q1 double hat.

[译文 7]

如果Q1选择这个数量，那么Q2将选择这个数量。如果Q1选择这个数量，那么Q2将选择这个数量，以此类推。好的，这里没有神秘之处。我们已经知道Firm 2要做什么。好的？所以根据定义，根据定义，2对Q1的最佳反应告诉我们Firm 2在给定Q1的情况下的利润最大化产出。好的，所以我们已经完成了这第二个（步骤）。我们已经知道Firm 2要做什么。当然，这里的额外步骤是Firm 1知道Firm 2会这样做。Firm 1将先行动，而且Firm 1知道在她设定她的数量Q1之后，Firm 2将通过选择其相应的数量来回应，这是对该Q1的最佳反应。所以如果Firm 1知道如果Firm 1选择这个数量，那么Firm 2将选择这个数量来回应。而且Firm 1知道如果她选择这个较小的数量，那么Firm 2将选择这个较大的数量来回应。对吗？是这样吗？所以Firm 1可以预见Firm 2将如何回应这些选择中的每一个。好的？让我们把这一点说清楚。所以特别地，如果Firm 1选择这个Q1 hat，我不是在建议它应该这样做，但如果Firm 1选择这个Q1 hat，那么Firm 1知道Firm 2将生产这个数量，即对Q1 hat的最佳反应，而且如果Firm 1选择Q1 double hat，那么Firm 2将通过选择对Q1 double hat的最佳反应来回应。

[段 8]

All right, so this is pretty straightforward so far, but what we’re able to see now is the problem facing Firm 1. which is the interesting problem. The problem facing Firm 1 is, what quantity should Firm 1 choose, knowing that this is how Firm 2 is going to respond? And before we saw this out mathematically, I just want to think it through a little bit. So the first way I want to think this through is is to make the following observation. From Firm 1’s point of view, Firm 1 knows that any Q1 she chooses leads to a response on this line by Firm 2. That’s what Firm 1 knows. So Firm 1 is effectively choosing points on this line. Firm 1 is effectively choosing points on this line. Let me say it again. So what’s actually happening is Firm 1 is choosing Q1 and Firm 2 is responding by choosing a Q2 that puts them on this line. But in effect, that means Firm 1 is choosing points on this line. All right. So you could think of Firm 1’s problem as, choose the joint output. output level on this line that maximizes firm one’s profits. You think of firm one’s problem as choose the combination of outputs on this line, by choosing Q1 and Q2 respond, choose the combination on this line that maximizes profits for firm one.

[译文 8]

好的，到目前为止这很直接，但我们现在能够看到的是 Firm 1 面临的问题，这才是有趣的问题。Firm 1 面临的问题是，在知道 Firm 2 会这样回应的前提下，Firm 1 应该选择什么产量？在数学推导之前，我只想先稍微想一下。第一种思考方式是做一个观察。从 Firm 1 的角度来看，Firm 1 知道，她选择的任何 Q1 都会导致 Firm 2 在这条线上做出回应。这就是 Firm 1 知道的。所以 Firm 1 实际上是在这条线上选择点。Firm 1 实际上是在这条线上选择点。让我再说一遍。所以实际上发生的事情是 Firm 1 选择 Q1，而 Firm 2 通过选择 Q2 来回应，使他们落在这条线上。但实际上，这意味着 Firm 1 在这条线上选择点。好的。所以你可以把 Firm 1 的问题理解为，选择这条线上的 joint output（联合产出）水平，使 Firm 1 的利润最大化。你可以把 Firm 1 的问题理解为，通过选择 Q1 而 Firm 2 回应，选择这条线上使 Firm 1 利润最大化的产出组合。

[段 9]

So I’m belaboring this a little bit because it’s a more general mathematical idea here. How many of you are in Econ 150 right now? So for those you in Econ 150, this should be a very familiar kind of thing. This is a constrained optimization problem, and you’ve been having constrained optimization problems rammed into you for the last month or so, right? So this is an example of a constrained optimization problem. You have to choose a point, but you can’t choose a point freely. You have to choose a point on the line. All right. All right. Okay, so let’s talk about it a bit more before we do the math. All right, let’s actually redraw it again. Since I made a mess of this picture. So one thing you might want to ask is, in making this choice for Firm 1, should Firm 1 choose more or less or the same as it used to choose when the problem was simultaneous? All right? So let’s put in again what it used to choose when the problem was simultaneous. I’ll put it in just faintly. So here’s our old Cornell picture. Looked like this. All right. And this was the quantity that Firm 1 shows in the Kono game. So let me call that Q1C. So certainly one possibility is that Firm 1 could choose her corno quantity.

[译文 9]

所以我在这个问题上多说了一些，因为这涉及一个更一般的数学思想。你们中有多少人现在在上Econ 150？所以对于那些在上Econ 150的人来说，这应该是一种非常熟悉的东西。这是一个约束优化问题，而在过去一个月左右的时间里，你们一直在被灌输约束优化问题，对吧？所以这是一个约束优化问题的例子。你必须选择一个点，但你不能自由地选择这个点。你必须选择线上的一个点。好的。好的。好吧，让我们在做数学之前再多讨论一下。好的，让我们实际上重新画一遍。既然我把这张图弄乱了。所以你可能想问的一个问题是，在为Firm 1做出这个选择时，Firm 1应该比过去问题同时进行时选择更多、更少还是相同的数量？好的？让我们再把问题同时进行时它过去选择的东西放进去。我把它淡淡地画出来。这是我们旧的Cornell图。看起来像这样。好的。这是Firm 1在Kono博弈中显示的数量。所以让我把它称为Q1C。所以Firm 1当然可以选择她的Cornell数量。

[段 10]

Right? She can certainly do that, and she knows that if she does that, firm 2 will respond by choosing the best response of Firm 2 to Firm 1 choosing the corner quantity. But we know what that is. what’s Firm 2’s best response to Firm 1’s corno quantity? It’s Firm 2’s corno quantity, right? So if Firm 2 does that, if Firm 1 chooses the corno quantity, then Firm 2 will also choose the corno quantity. So one thing that Firm 1 could do is effectively choose the old equilibrium. All right, that’s certainly something that’s available to Firm 1. But Firm 1 could also do other things. Firm 1 could produce less than that, or Firm 1. Firm 1 could do more than that. So who thinks Firm 1 should choose the old equilibrium quantity? Who thinks Firm 1 should choose more than that? Who thinks Femm 1 should choose less than that? We’re not getting much of a majority. Let’s just try it with the camera on you, all right? So once again, how many people think that Firm 1 should choose the old equilibrium quantity? A dribbling of hands. And how about less than that? few hands and then they went down again and how about more than that? There’s a majority of a more. Turns out more is correct. That’s good news. It’s good news.

[译文 10]

对吧？她当然可以这么做，而且她知道如果她这么做，Firm 2 会做出反应，选择 Firm 2 对 Firm 1 选择 corner quantity 的最佳反应。但我们知道那是什么。Firm 2 对 Firm 1 的 corner quantity 的最佳反应是什么？就是 Firm 2 的 corner quantity，对吧？所以如果 Firm 2 这么做，如果 Firm 1 选择 corner quantity，那么 Firm 2 也会选择 corner quantity。所以 Firm 1 可以做的一件事就是有效地选择旧的均衡。好的，这当然是 Firm 1 可以做到的事情。但 Firm 1 还可以做其他事情。Firm 1 可以生产比那更少的量，或者 Firm 1……Firm 1 也可以做比那更多的量。那么谁认为 Firm 1 应该选择旧的均衡产量？谁认为 Firm 1 应该选择比那更多的量？谁认为 Firm 1 应该选择比那更少的量？大家意见不太一致。让我们把摄像头对准你来试试，好吗？再来一次，有多少人认为 Firm 1 应该选择旧的均衡产量？零星几只手。比那更少呢？几只手，然后又放下了。比那更多呢？多数人举手了。原来更多是正确的。这是好消息。这是好消息。

[段 11]

Why? Why do we think Firm 1 should produce more than it used to produce before? Let me take us on this. Well, let’s think about it. As Firm 1 produces more, or if Firm 1 were to produce more, then Firm 2, my voice is going, then Firm B, two would produce produces more, or if Firm 1 were to produce more, then Firm 2, then Firm 2 would produce what? Less. As Firm 1 produces more than her corner quantity, Firm 2’s response is to produce less. Do you want to remember the jargon for this? What do we call games where the more I do of my strategy, the less you do of yours? Strategic substitutes. Good, this is a game of strategic substitutes. It’s a game of strategic substitutes. And what that means is that as Q1 goes up, Q2, the best response, affirmed 2 to Q1 goes down. So what? So what? We can look at that, just looking at the picture. Well, the so what is, now we’re in a sequential game, firm one produces more than her corno quantity, she induces firm two to produce less, that’s what, that’s good for firm one, right? My producing more, inducing you to produce less is good for me. It’s going to keep prices higher in the market. Is that right? So let’s just think it through again.

[译文 11]

为什么？为什么我们认为Firm 1应该比之前生产更多？让我带大家一起思考。好，让我们来想想。当Firm 1生产更多，或者说如果Firm 1要生产更多的话，那么Firm 2，我的意思是，那么Firm 2会生产更多，或者说如果Firm 1要生产更多，那么Firm 2会——Firm 2会生产什么？更少。当Firm 1比她的角点产量生产更多时，Firm 2的反应是生产更少。你想记住这个术语吗？我们把这种游戏叫什么？即我的策略做得越多，你的策略就做得越少？策略替代品。很好，这是一个策略替代品的游戏。这是一个策略替代品的游戏。这意味着当Q1上升时，Q2，即Firm 2对Q1的最佳反应，会下降。那又怎样？那又怎样？我们可以看图来说明这一点。好，那又怎样的意思是，现在我们处于一个序贯博弈中，Firm 1比她的角点产量生产更多，她诱导Firm 2生产更少，这对我有利，对吧？我生产更多，诱导你生产更少对我是有利的。这会使市场价格保持更高。是这样吗？让我们再仔细想一想。

[段 12]

In the corneau equilibrium, the choice of firm one was the best choice for firm one, taking the choice of firm two as given. And that was the quite old corner quantity. But now we’re in this Stackelberg setting, this sequential setting, there’s an additional feature. Firm one doesn’t have to take firm two’s output as given. There’s an additional reason for producing, all right, at the margin, which is at the margin if I produce some more units of output, that leads you to produce less, which is good for me. All right? So that suggests that I’m going to produce more than I used to produce under the old assumption. All right. So this suggests that Firm 1 should set Q1 bigger than Q1C to induce Q2 to be less than Q2C. So the first thing we’ve learned, and we’ll see this in the math later, is that Firm 1 will in fact produce more than they used to under KORNO, and that will result in Firm 2 producing less than Korn O. All right? Now we’re already got a lot on the board now that we can actually solve out intuitively the problem. Do we think that Firm 1’s profits by this procedure are the same as they were under Korn O? Are they less than they were under Corn O? Or are they more than they were under Kornow?

[译文 12]

在Cournot均衡中， Firm one的选择是 Firm one的最佳选择，同时把 Firm two的选择视为既定。这就是相当古老的 Cournot产量博弈。但现在我们处于 Stackelberg环境中，这是一种序贯博弈结构，这里有一个额外的特征。 Firm one不必把 Firm two的产出视为既定。在边际上有额外的生产动机，也就是说，在边际上如果我多生产一些产出，那会导致你少生产，这对我是有利的，对吧？这意味着我将会比在旧假设下生产得更多。所以这表明 Firm 1 应该把 Q1 设置得大于 Q1C，以诱导 Q2 小于 Q2C。我们学到的第一件事就是——我们后面会在数学中看到—— Firm 1 实际上将会比在 KORNO 下的产量更多，而这将导致 Firm 2 的产量低于 Korn O，对吧？现在我们在黑板上已经写了很多内容，我们可以直观地解决这个问题。我们认为， Firm 1 通过这种策略获得的利润与在 Korn O 下相同吗？是比在 Corn O 下更少？还是比在 Kornow 下更多？

[段 13]

So let me say again, Firm 1 is going to first now. We’ve argued that Firm 1 is going to produce more. Do we think that Firm 1’s profit in the end of the day are going to be the same as they were under Kornow, higher than they were under Kornow, or lower than they were under Kornow? So let’s have a poll again. Let me get the camera on you guys. So who thinks their profits are going to be the same as they were under Kornow? Who thinks the profits are going to have gone up? Who thinks the profits are going to have gone up? Who thinks profits have gone down? Now we’re in good shape here because indeed the profits have gone up. There’s a very simple argument, why the profits have to have gone up. How do we know the profits must have gone up? Let me actually, is this simple enough, let me grab a mic on this. How do we know the profits just must have gone up? There was a hand at the back. Was a hand of the back? Yes, okay, way at the back. How do we know profits must have gone up here? Way at the back? Yeah. If Firm 1 was going to lower their profits, they wouldn’t have chosen to produce more. All right, good, exactly, exactly.

[译文 13]

让我再说一遍，Firm 1 现在要先行动了。我们已经论证了 Firm 1 会生产更多。我们认为 Firm 1 的利润最终会与 Cornow 时期相同、更高还是更低？让我们再做个民调。让镜头对准你们。谁认为他们的利润会与 Cornow 时期相同？谁认为利润会增加？谁认为利润会增加？谁认为利润下降了？现在我们情况不错，因为利润确实增加了。有一个非常简单的论证，为什么利润一定会增加。我们怎么知道利润一定增加了？让我看看，这个够简单吗，让我拿个麦克风。我们怎么知道利润一定增加了？后面有人举手。后面那位吗？好的，很后面。怎么知道这里的利润一定增加了？很后面那位？是的。如果 Firm 1 想要降低他们的利润，他们就不会选择生产更多了。好的，对，正是，正是。

[段 14]

The fact that Firm 1 has changed their output, in particular producing more, tells you this. they must be able to increase their profits by this maneuver, right? Let’s just think that through. One option that was available to firm one before was to set output at the corno level. If firm one had set output at the corno level, that would have led firm two to set output at the corner level, and in that case, profits would have been exactly the same as before. The fact that firm one has moved away from that must mean there are higher profits available. So in another way, Firm 1 could have had exactly what it had before, so it must be doing at least as well as it was doing before, and the fact that it’s changed means it must be doing better than it was doing before. All right? So indeed, Firm 1’s profits have gone up. We don’t even need any math to prove that. It just must be the case logically. What must have happened to Firm 2’s profits? What do things happen to firm 2’s profits? What do things happen to firm two’s profits. So that’s not, that’s not so immediately obvious, right? All right, what must have happened to Firm 2’s profits? What do things happen to Firm 2’s profits? So that’s not so immediately obvious, right?

[译文 14]

Firm 1 改变了产出，特别是生产更多，这个事实告诉了你这一点。他们一定能够通过这个策略增加利润，对吧？让我们仔细想想。在 Cornow 之前，Firm 1 可选的一个方案是将产出设在 Cornow 水平。如果 Firm 1 将产出设在 Cornow 水平，那会导致 Firm 2 将产出设在 Corner 水平，那样的话利润就会和之前完全一样。Firm 1 偏离了这一点这个事实一定意味着有更高的利润可得。换句话说，Firm 1 本来可以得到和之前完全一样的结果，所以他们一定至少做得和以前一样好，而他们改变了这一点这一事实意味着他们一定做得比之前更好。明白吗？所以确实，Firm 1 的利润增加了。我们甚至不需要任何数学来证明这一点。逻辑上一定如此。Firm 2 的利润一定会发生什么？一定会发生什么？那么这就不是那么显而易见，对吧？一定会发生什么？

[段 15]

It’s obvious, I think, that Firm 1’s profits have gone up here because Firm 1 could have had the same old profits and has chosen something else. But it’s not immediately obvious what happened to Firm 2’s profits, is that right? And before we get to what’s happened to Firm 2’s profits, let’s go through an intermediate step. Let’s try and ask what must have happened to total output in the market, in this example, this nice simple example. Right? That’s not immediately obvious either. Why? We’ve argued that Firm 1’s output went up, but Firm 2’s output went down relative to Cornell. So it’s not immediately obvious whether the sum of those two, Q1 plus Q2, went up or down. We’d like to know what happens to Q1 plus Q2, total output in the market. And by the way, one particular reason we might care about this is, of course, consumers would like it to have gone up. Because if total output’s gone up, prices have gone down, and that’s good for consumers. Right? So if you’re the regulator, if you’re designing this industry, if you’re working for the Justice Department, if you’re working for the European Commission, you’re going to want to know the answer when we switch from a simultaneous, a simultaneous setting to an asymmetric setting where there’s a leader firm and a follower firm, is that going to be good for consumers or bad for consumers?

[译文 15]

很明显，我认为 Firm 1 的利润在这里增加了，因为 Firm 1 本来可以得到和以前一样的利润但选择了其他。但 Firm 2 的利润发生了什么就不那么显而易见了，对吧？在我们讨论 Firm 2 的利润发生了什么之前，让我们先看一个中间步骤。让我们试着问，在这个例子中，这个简单美好的例子中，市场总产出一定会发生什么？那么这也不是那么显而易见。为什么？我们论证了 Firm 1 的产出增加了，但 Firm 2 的产出相对于 Cornow 减少了。所以这两个的总和 Q1 加上 Q2 是增加了还是减少了就不是那么显而易见了。我们想知道 Q1 加上 Q2 会怎样，也就是市场总产出。另外，我们可能关心这一点的一个特别原因是，当然，消费者会希望它增加。因为如果总产出增加了，价格就下降了，这对消费者有利。对吧？所以如果你是一个监管者，如果你是在设计这个产业，如果你是在为司法部工作，如果你是在为欧盟委员会工作，当我们要从同时设定转到一个非对称设定，有一个领导者 Firm 和一个跟随者 Firm 时，你会想知道答案：这对消费者是有利的还是有害的？

[段 16]

All right, well, let’s have a look. Well, we know that firm 1’s output went up, and we know that firm 2’s output went down, but can anyone tell me what happened to total output and why? Let’s have a poll again. Who thinks total output went down? Who thinks total output stayed the same? Who thinks total output stayed the same? Who thinks total output went up? There’s lots of abstentions. Let’s try that again, because too many extensions. Who thinks total output went down? Who thinks total output stayed the same? Who thinks total output went up? That’s pretty split. So I think total out, I know actually, that total output went up. And I claim I can see it on the picture. I claim if you stare at that picture, you can actually see that total output must have gone up. Who’s good at looking at a pitch? Let me get the mic in here. Right, the pitch is there. Whoops. Okay. Let me try this first. So your name is… Andy. Andy, go ahead. Judging by the slope of the line, you know that as it moves, the amount that Q1 change will be greater than the amount that Q2 goes down. Good, good. So what Andy said, I put it in here, everyone, what Andy is saying is look at the slope of the line along which we slid.

[译文 16]

好的，让我们来看看。我们知道 Firm 1 的产出增加了，我们知道 Firm 2 的产出减少了，但有人能告诉我总产出发生了什么以及为什么吗？让我们再做个民调。谁认为总产出下降了？谁认为总产出保持不变？谁认为总产出保持不变？谁认为总产出增加了？有很多弃权的。让我们再试一次，因为扩展太多了。谁认为总产出下降了？谁认为总产出保持不变？谁认为总产出增加了？这相当分散。所以我认为总产出实际上增加了。我声称我能在图上看到这一点。我声称如果你盯着那张图看，你实际上能看到总产出一定增加了。谁擅长看图？让我把麦克风递过去。好的，图在那里。哎呀。让我先试试这个。你叫什么名字？Andy。Andy，请说。根据这条线的斜率，你知道当它移动时，Q1 变化的量会大于 Q2 下降的量。好的，好的。Andy 说的是，请注意，我把它标在这里，大家，Andy 说的是看我们滑动的这条线的斜率。

[段 17]

When we went from Kornow to the Stackelberg Equilibrium, we started here and we slid in this direction down this line. Right, Q1 went up and Q2 went down. And what Andy is pointing out is we can see from the slope of this line that Q1 goes up more than one unit for every unit reduction of Q2. And we said again, for every unit of increase of Q1, there’s less than a proportion, sorry, say again, for every unit of increase of Q1, there’s less than a unit of decrease of Q2. All right. And all they’re saying it is, the slope of this line is less than one. Everyone see that? All right? So we know, by the slope of the line, we know that Q1 had to go up more than Q2 went down, which means total output went up, which means that prices went, what happened to prices? total output went up? That shouldn’t be hard for ever. Everyone’s taking 1.15 here, right? So when total output went up, prices went down, good. Right? Demand curve sloping down is not as important as back conduction, but it’s still quite important. All right, so prices went down. So therefore, we now are ready to say what happened to firm two’s profits. Firm 2 is producing less than before. Firm 2’s cost are the same, and prices have gone down.

[译文 17]

当我们从 Cornow 到 Stackelberg 均衡时，我们从这里开始，沿着这条线向这个方向滑动。对，Q1 增加了，Q2 减少了。Andy 指出的是，我们从这条线的斜率可以看出，Q1 每增加一单位，Q2 就会减少不到一单位。每增加一单位 Q1，Q2 就会减少少于一个单位。大家明白吗？而他们所说的就是，这条线的斜率小于一。大家明白吗？所以我们知道，根据这条线的斜率，我们知道 Q1 一定增加得比 Q2 减少得多，这意味着总产出增加了，这意味着价格……价格发生了什么？总产出增加了？这不应该对任何人来说很难。大家都在这里上 1.15，对吧？所以当总产出增加时，价格下降了，好的。需求曲线向下倾斜虽然不如反垄断传导重要，但仍然相当重要。好的，所以价格下降了。因此，我们现在准备说 Firm 2 的利润发生了什么。Firm 2 生产的比以前少了。Firm 2 的成本是一样的，而价格下降了。

[段 18]

So what’s happened to firm 2’s profits? they must have gone down as well. So Furn 2’s profit has gone down and we know that consumer surplus, so it’s what’s C.S. Consumer surplus has gone up. For those people remember, that 1.15, prices have gone down, quantity’s gone up, so consumer surplus here has gone up. So we’ve analysed everything qualitatively I can think of in this game without reference to any math at all. Is that right? We haven’t done any math there. And we talk about slope of the line, I guess that’s math. I guess that’s math. thing qualitatively I can think of in this game without reference to any math at all. Is that right? We really haven’t done any math there. And we talk about slope of the line. I guess that’s math. I guess that’s math in junior high or something. But we haven’t really done any math there, right? Is that fair? Right? And we already have a pretty good intuition for what’s going to go on in this market. We think Q1’s going to go up. We think Q2 is going down. We think that Firm 1’s profit are going up. We think firm 2’s profits are going down. We think total quantity is going up. All right. Now let’s see if we’re right. Let’s go back and do the math. All right.

[译文 18]

那么 Firm 2 的利润发生了什么？他们一定也下降了。所以 Firm 2 的利润下降了，而我们知道消费者剩余，也就是 C.S.，消费者剩余增加了。对于那些还记得 1.15 的人来说，价格下降了，产量增加了，所以这里的消费者剩余增加了。所以我们已经对这个博弈中的所有事情做了定性分析，我想不出任何数学参考。对吗？我们在那里没有做任何数学。我们讨论了这条线的斜率，我想那是数学。我想那是数学，大概是初中数学吧。但我们真的没有在那里做任何数学，对吧？这样公平吗？我们已经对这个市场将要发生什么有了相当好的直觉。我们认为 Q1 会增加。我们认为 Q2 在下降。我们认为 Firm 1 的利润会增加。我们认为 Firm 2 的利润会下降。我们认为总产量会增加。好的。现在让我们看看我们是否正确。让我们回去做数学。好的。

[段 19]

So I want to spend a bit of time grinding this out. So I don’t claim that doing the math is fun, but I want to prove that we can do it because otherwise everyone’s going to either think that this was all kind of just blah, blah, blah, and or people are going to be scared to do the math when it arrives on a homework assignment. All right. So for now, we’re going to more or less few minutes, we’re going to forget what are economists, and we’re going to turn into nerds. That isn’t a huge transition, but we’ll do it anyway. All right, I’ve got the demand curve up there, but let me, all right, so the demand curve is still there. It’s p equals A minus B, Q1 plus Q2, and I’ve got profits written up there. But let’s put them somewhere more convenient anyway. So p equals A minus B. Q1 plus Q2, and profit is equal to P, profit for firm I is equal to PQI is equal to PQI minus CQI. All right? And what we’re told to do in backward induction is what, first of all, solve things out for firm two, taking firm one as given, and then go back and solve for firm one. So exactly the discussion we’ve just had. formally, we’re now going to do more formally.

[译文 19]

所以我想花点时间来仔细推导这些计算。我并不是说做数学题很有趣，但我想证明我们能够做到，否则大家要么会认为这一切都是废话，要么大家会在作业中出现数学题时害怕去做。好吧。所以现在，大概几分钟的时间，我们要忘记什么是经济学家，我们要变成书呆子。这不是一个很大的转变，但我们还是会这样做。好吧，我这里有需求曲线，让我，好吧，需求曲线还在那里。它是 p 等于 A 减 B，Q1 加 Q2，我这里有利润写着。但让我们把它们放在更方便的地方。所以 p 等于 A 减 B，Q1 加 Q2，利润等于 P，企业 I 的利润等于 PQI 等于 PQI 减 CQI。好吧？我们在逆向归纳法中被告知要做的是，首先，解决企业二的问题，把企业一视为既定，然后再回来解决企业一的问题。这正是我们刚才的讨论。正式地，我们现在要更正式地来做。

[段 20]

All right? So back when induction tells us, solve for firm two first, taking Q1 as given. And what is that’s a sort of boring math problem, it says, it says maximize by choosing Q22, the profit of firm one. So that’s going to be A minus B Q1 minus B Q2. That’s the price times the quantity Q2 minus C Q2. All right. All right. So this bit here, this bit here is the price. these two terms together are revenues, and this term is cost. Alright? Now I could do that. I could grind that bit out, but we already ground that bit out three or four weeks ago, right? So I’m not going to grind it out again. We know how to do that. But by the way, this just remind ourselves what we did. We differentiated with respect to Q2, we set the thing we found, the derivative equal to zero. That was our first order condition, and then we solved for Q2. Is that right? So when we did that, when we did that, we went through the first order. a condition and then we solved it, we know what we actually got, right? So what we actually got was that Q2 star, if you like, or Q2, this is called it, is equal to a minus C over 2b minus Q1 over 2. And in fact, I’ve already given to it up there.

[译文 20]

好吧？所以逆向归纳告诉我们，先解企业二，把 Q1 视为既定。这是什么？这是一个有点无聊的数学问题，它说，通过选择 Q2 来最大化企业一的利润。那就是 A 减 B Q1 减 B Q2。那是价格乘以数量 Q2 减 C Q2。好吧。好吧。所以这部分，这部分是价格。这两个项合在一起是收入，这个项是成本。好吧？现在我可以做那个。我可以仔细推导出那部分，但我们已经在三四周前推导出那部分了，对吧？所以我不打算再推导一遍。我们知道怎么做。但是，顺便说一下，这只是提醒我们自己我们做了什么。我们对 Q2 求导，我们把我们发现的导数设为零。那是我们的一阶条件，然后我们解出了 Q2。对吧？当我们做那个，当我们做那个的时候，我们通过了一阶条件然后解出了它，我们知道我们实际上得到了什么，对吧？所以我们实际上得到的是 Q2 星，或者如果你喜欢的话，Q2，叫做它，等于 a 减 C 除以 2b 减 Q1 除以 2。事实上，我已经把它写在上面的黑板上了。

[段 21]

It’s up on that top, top board. It’s the best response for firm two. So again, I could do this again, but since we did it a few weeks ago, I don’t want to redo it. All right. All right, now the more interesting part. Not thrilling, but a little bit more interesting. All right, now let’s solve for firm one. All right, so what is Firm 1 doing? Firm 1 is also trying to maximize profit. maximize profit. Firm 1 is choosing Q1, and firm 1, at least initially, looks like the same problem. It’s A minus B, Q1, minus B, Q2 times Q1, minus C, Q1. This is the same line we had before, but now, whereas firm 2 was taking Q1 as given, firm 1, Firm 1 knows that Q All right, this is the same line we had before, but now, whereas firm 2 was taking Q1 as given, firm 1 knows that Q2 is given by this formula here. All right, so what we’re going to do is we’re going to plug this Q2 into there. All right? And again, for those of you 115, this isn’t the only way to do it. So, 150, this isn’t the only way we could do it. We could also set up a Lagrangian equation. But for those people who don’t know what that is, don’t worry, we’re going to plug in today.

[译文 21]

它在那个上面的黑板上。这是企业二的最优反应。所以再说一次，我可以再做一次，但既然我们几周前做过，我不想重做。好吧。好吧，现在更有趣的部分。不令人兴奋，但有点更有趣。好吧，现在让我们解企业一。好吧，那么企业一在做什么？企业一也在试图最大化利润。最大化利润。企业一在选择 Q1，而企业一，至少在一开始，看起来是同样的问题。它是 A 减 B，Q1，减 B，Q2 乘以 Q1，减 C，Q1。这和我们之前有的同一行，但现在，企业二是把 Q1 视为既定的，而企业一知道 Q2 是由这个公式给出的。好吧，所以我们要做的是把这个 Q2 代入那里。好吧？再说一次，对于你们 115 的人来说，这不是唯一的方法。所以，150，这不是我们唯一能做的。我们也可以建立一个拉格朗日方程。但对于那些不知道那是什么的人，别担心，我们今天要用代入法。

[段 22]

All right. So we’re going to plug this in, and when we plug it in, we get a right old mess, but let’s do it anyway. So we’re going to get max Q1, A minus B Q1, minus B, big bracket, A minus C over 2B, minus Q1, minus Q1. over 2, why don’t I put this C inside the bracket as well while I’m here, okay? So minus C, close bracket, Q1. All right, so just want to add it, when I put the C inside the bracket. Everyone okay with that? I’m doing algebra on the board, which is not fun, but it’s useful to do occasionally. All right. So eventually, what are we going to do? Eventually, we’re going to differentiate this thing, set it equal to zero, look at our first order condition, and so on. just as we normally would. Eventually, we’re going to use basically 112 level calculus to solve this thing. All right? Everyone remember how to solve a maximization problem? Yeah? But before we do that, let’s tidy up the algebra a bit. All right? So this thing is just tidied it up. So this is equal to max respect to Q1. And notice I’ve got an A minus C here. an A minus C here. And once I take this B inside the bracket, I’ve got a minus A minus C over two here.

[译文 22]

好吧。所以我们要代入这个，当我们代入的时候，我们会得到一团乱，但让我们还是做吧。所以我们要得到 max Q1，A 减 B Q1，减 B，大括号，A 减 C 除以 2B，减 Q1，减 Q1。除以 2，为什么我不也把这个 C 放进括号里呢，既然我在做这个，好吧？所以减 C，合上括号，Q1。好吧，所以当把 C 放进括号里时，只是想加上它。大家都明白吗？我在黑板上做代数，这不好玩，但偶尔做一下是有用的。好吧。那么最终，我们要做什么？最终，我们要对这个东西求导，设为零，看看我们的一阶条件，等等。就像我们通常做的那样。最终，我们要用基本上 112 水平的微积分来解这个东西。好吧？大家都记得怎么解最大化问题吗？是的？但在我们做那个之前，让我们先把代数整理一下。好吧？所以这个东西刚刚被整理好了。所以这等于关于 Q1 的最大值。注意这里我有一个 A 减 C。这里有一个 A 减 C。一旦我把这个 B 放进括号里，我这里有 a 减 A 减 C 除以二。

[段 23]

All right? So I’ve got an A minus C minus C over two. So that’s going to give me an A minus C over two. And I’m really going to pray that the TAs are watching me carefully. I’m going to catch my errors here. Okay? So I think I’m okay so far, but please catch me. All right. What else have I got? I’ve got a minus BQ1 here. And for From in this bracket, I’ve got a minus minus, that’s a plus BQ1 over two. So I have a minus B, I have a minus BQ1 plus a BQ1 over two. So that’s a minus BQ1 over two. So far, so good. And that whole thing is multiplied by Q1. Okay, so far. Let’s multiply out the bracket because otherwise I’ll make. a mistake. So this is the same as saying, A minus C over 2 times Q1 minus B, Q1 squared over 2. So far so good? All right, now we’re at the level where even my very rusty memory of calculus will get us through. Okay, so let’s try and do it. So what we’re going to do is we’re going to differentiate this thing with respect to Q1. All right, it’s differentiate with respect to Q1. Okay, so differentiating with respect to Q1, let’s do it up here. Differentiate with respect to Q1, we get A minus C over 2 from this term, all right, and from the the minus bq squared over 2, we’re going to get, the two is going to cancel rather pleasantly, so we’re going to get B.

[译文 23]

好吧？所以我有一个 A 减 C 减 C 除以二。所以这会给我一个 A 减 C 除以二。而我真的要祈祷助教们在仔细看着我。我会在这里抓住我的错误。好吧？所以我想到目前为止我还可以，但请抓住我。好吧。我还有什么？我这里有一个减 BQ1。而从在这个括号里，我有一个减减，那是一个加 BQ1 除以二。所以我有一个减 B，我有一个减 BQ1 加上一个 BQ1 除以二。所以那是减 BQ1 除以二。到目前为止，还不错。而整个东西乘以 Q1。好吧，到目前为止。让我们把括号展开，否则我会犯一个错误。所以这和说，A 减 C 除以 2 乘以 Q1 减 B，Q1 平方除以 2 是一样的。到目前为止还好？好吧，现在我们到了即使我非常生锈的微积分记忆也能让我们通过的水平。好吧，所以让我们试试做吧。所以我们要做什么？我们要求这个东西关于 Q1 的导数。好吧，是对 Q1 求导。好吧，所以关于 Q1 求导，让我们在这里做。关于 Q1 求导，我们从这项得到 A 减 C 除以 2，好吧，而从那个减 bq 平方除以 2，我们要得到，二会相当愉快地消去，所以我们要得到 B。

[段 24]

Q1 from that term. All right? Everyone with me so far? I’m going through this in slow steps. I agree it’s not exciting, but I want to make sure I don’t make a mistake. All right. So to turn this into a first order condition, what must be true about this derivative? At the maximum, what must be true about this derivative? Said it equal to zero, good. So it equal to zero. All right. And we should just check the second order condition. All right. How do I check the second order condition? I differentiate again and check it’s negative. But if I differentiate again, I’m just going to get minus B. So minus B is certainly negative, all right? So second order condition is okay. So let’s solve it out. Solving this for Q1, I get Q1 is equal to A minus C over 2B. Some people are leaving in sheer boredom. It has to be done occasionally. Q1 is A minus C over 2B. and we’re not done yet. Some people are leaving in sheer boredom, it has to be done occasionally. Q1 is A minus C over 2B. Alright, and we’re not done yet. Now we want to go back and solve that algebraically for Q2. I know what Q1 is now. Q1 is A minus C over 2B. How do I find Q2? Somebody? Shout it out, how do I find Q2?

[译文 24]

Q1 从那一项。好吧？到目前为止大家都跟上了吗？我在慢条斯理地讲解。我同意这不令人兴奋，但我想确保我不犯错误。好吧。那么要把这个变成一阶条件，关于这个导数必须满足什么？在最大值处，关于这个导数必须满足什么？说把它设为零，好的。所以它等于零。好吧。我们应该检查二阶条件。好吧。我怎么检查二阶条件？我再求一次导并检查它是负的。但如果我再求一次导，我只会得到负 B。所以负 B 当然是负的，好吧？所以二阶条件没问题。让我们解出来。解这个求 Q1，我得到 Q1 等于 A 减 C 除以 2B。有些人因为无聊而离开了。这偶尔必须做。Q1 是 A 减 C 除以 2B。而我们还没完成。有些人因为纯粹的无聊而离开，这偶尔必须做。Q1 是 A 减 C 除以 2B。好吧，而我们还没完成。现在我们想回去代数地解 Q2。我现在知道 Q1 是什么了。Q1 是 A 减 C 除以 2B。我怎么找到 Q2？谁来？喊出来，我怎么找到 Q2？

[段 25]

Yeah, I’ve got to plug it in, right? I’m going to go back and plug this Q1 back into this expression here. So I plug it back in, I’ll get Q2 is equal to A minus C over 2B minus a half of A minus C over 2B for a total of A minus C C over 4B. Is that right? That’s what I have in my notes, this looks good. All right. So I’m now done. I’ve now found the equilibrium. I found that in this leader follow ship game, this Stackleberg version of quantity competition, Q1 is given by A minus C over 2B, and Q2 is given by A minus C over 4B. Let’s see how it matches. up with the intuition we developed before without using any boring math. All right? So first of all, we’re comparing Q1 and Q2 with what they used to produce, and what they used to produce is on the top board. What they used to produce, I can use this to guide the camera as well, what they used to produce is A minus C over 3B. Is that right? So our claim was that we think the new Q Q1 is bigger than the old corno quantity C. So now firm 1 is producing A minus C over 2b. Previously it was producing A minus C over 3B, so that is indeed bigger.

[译文 25]

是的，我得把它代回去，对吧？我要回到这里把这个 Q1 代回这个表达式里。代回去之后，我会得到 Q2 等于 A 减 C 除以 2B 减去二分之一乘以 A 减 C 除以 2B，总共是 A 减 C 的平方除以 4B。对吗？这就是我在笔记里写的，看起来没问题。好的，那么我现在完成了。我找到了均衡。我们发现在这个领导者-追随者博弈中，这个 Stackleberg 版本的产量竞争里，Q1 是 A 减 C 除以 2B，Q2 是 A 减 C 除以 4B。让我们看看这与之前不用任何无聊数学得出的直觉是否吻合。好吗？所以首先，我们把 Q1 和 Q2 与它们以前生产的数量进行比较，它们以前生产的数量在上面的白板上。它们以前生产的数量，我也可以用这个来引导镜头，它们以前生产的是 A 减 C 除以 3B。对吗？我们的论点是新的 Q1 大于旧的角落数量。所以现在企业 1 生产的是 A 减 C 除以 2B，之前生产的是 A 减 C 除以 3B，所以确实更大。

[段 26]

That’s good news, right? So this is indeed bigger than Q. 0. And our claim was that firm 2 will produce less than the old corno quantity. So firm 2 used to produce A minus C over 3B, 3B and now it’s producing A minus C over 4B, and that is indeed less than the old corner quantity. All right, so far so good. What about total output? How do I solve for total output? Add the two outputs together. That’s not too hard. Alright, so Q1 plus Q2 is equal to A minus C over 2B which is in fact 3A C over 3A over 4B is that right so it’s going to be 3a minus C over 4b so I just actually for the first time today I skipped a step but is that okay a half plus a quarter is three quarters okay right so so total output is 3A C over 4b what was total output before? what was total output before? I mean it used to to be the corno total output, let’s put it here somewhere, this is bigger than 2A minus C over 3B, which is equal to the corno quantity Q1c plus Q2C. All right, so everything we predicted just when looking at the picture and thinking about the economics, works out in the math. That’s a good thing, we should feel a little bit relieved.

[译文 26]

这是好消息，对吧？所以这确实大于 Q0。我们的论点是企业 2 生产的数量会小于旧的角落数量。企业 2 以前生产 A 减 C 除以 3B，现在生产 A 减 C 除以 4B，确实小于旧的角落数量。好的，到目前为止一切顺利。总产出呢？我怎么计算总产出？把两个产出相加。这不太难。好的，所以 Q1 加 Q2 等于 A 减 C 除以 2B，实际上是 3A 减 C 除以 4B，对吗？所以是 3A 减 C 除以 4B。我刚才实际上今天第一次跳了一步，但这样可以吗？二分之一加四分之一等于四分之三，对。所以总产出是 3A 减 C 除以 4b。之前的总产出是多少？之前的总产出是多少？我是说那曾经是角落总产出，让我们把它放在这里某个地方，这大于 2A 减 C 除以 3B，等于角落数量 Q1c 加 Q2C。好的，所以我们仅通过看图和经济思考所做的所有预测，都在数学中得到了验证。这是一件好事，我们应该感到有点放心。

[段 27]

I’m feeling a little bit relieved. All right? Everything we thought out intuitively, just using the economics, the logic of the situation, when we grind out the algebra, we get the right answers. We get confirming answers. All right? Everyone occasion that there isn’t a particularly fun exercise per se, but I want to do it just to show that you can use backward induction to solve out problems exactly. Backward induction and a little bit of what you look learned in high school and or freshman calculus can get you the answer. Alright. All right. So now I want to leave aside the math and go back to the economics again. So we started off with the question. Who would you rather be? Firm 1 or firm 2? And we know the answer now. Who would you rather be? Firm 1 or firm 2? Firm 1, right? Because firm 1’s profits went up and firm 2’s profits went down. Let’s just talk about this a little bit about what’s going on here. on here. So, Firm 1, previously firm 1 and and firm two’s profits went down. Let’s just talk about this a little bit about what’s going on here. So, firm one, previously, firm one and two were just setting quantities simultaneously. We know now there’s an advantage in going first. So suppose we change the game from the simultaneous move game to a game in which firm one and firm two can make announcements.

[译文 27]

我感觉有点放心了。好吗？我们仅用经济学和情况的逻辑直观思考的所有东西，当我们进行代数运算时，我们得到了正确的答案。我们得到了验证的答案。好吗？每个人都认为这本身并不是一个特别有趣练习，但我之所以想做，只是为了表明你可以用逆向归纳法精确地求解问题。逆向归纳法加上你在高中或大一微积分中学到的一点知识就能让你得到答案。好的。那么现在我想把数学放在一边，回到经济学上来。我们开始时提出了一个问题。你更愿意成为哪一个？企业 1 还是企业 2？我们现在知道答案了。你更愿意成为哪一个？企业 1 还是企业 2？企业 1，对吧？因为企业 1 的利润增加了，企业 2 的利润减少了。让我们稍微谈谈这里发生了什么。企业 1 以前，企业 1 和 2 的利润都下降了。让我们稍微谈谈这里发生了什么。所以，企业 1 以前，企业 1 和 2 只是同时设定产量。我们现在知道先行动是有优势的。所以假设我们把游戏从同时行动游戏改变为一个企业 1 和企业 2 可以发表公告的游戏。

[段 28]

All right? They can announce how much they’re going to produce. So firm one comes in one day and says, I’m going to produce this much. And firm two comes in and can see firm one’s announcement. So I’ve changed it into a sequential game. Firm 1 has announced how much it’s going to produce. Firm 2 is going to go afterwards. Is that really a sequential game? Is that going to make a difference? Why is that? I claim that’s not really enough. Let me say again, we start from the simultaneous move game. If we just change it by simply allowing Firm 1 to announce what they’re going to produce, not actually produce it, but just announce what they’re going to produce, you might think that’s a sequential move game. And you might think that Firm 1 now has an advantage. But I’m claiming that’s not enough really to give Firm 1 an advantage. Why? Why is that not enough? Let’s try and get some ideas here. Come around this side. Yes. So, Patrick, why is that not enough? There’s no credible commitment that you’re going to produce at that level. Good. Good. So imagine these two firms are, let’s say they’re their newspaper-producing firms and one is owned by NBC’s parent company and one is owned by Rupert Murdoch, right? And they’re moving into this new market and the market is a town that both are going to issue newspapers in this market that currently doesn’t have any newspapers and Murdoch simply says, I’m going to produce lots of newspapers.

[译文 28]

好的？它们可以宣布它们将生产多少。所以企业 1 某天进来说，我要生产这么多。企业 2 进来可以看到企业 1 的公告。所以我把它变成了一个序贯博弈。企业 1 已经宣布了它将生产多少。企业 2 将在之后行动。这真的是序贯博弈吗？这会有区别吗？为什么会这样？我声称这实际上还不够。让我再说一遍，我们从同时行动博弈开始。如果我们只是简单地允许企业 1 宣布它们将生产多少，不是实际生产，而只是宣布它们将生产多少，你可能会认为这是序贯行动博弈。你可能会认为企业 1 现在有优势。但我声称这实际上不足以给企业 1 优势。为什么？为什么这不够？让我们在这里得到一些想法。绕到这边来。是的。所以，Patrick，为什么这不够？因为没有可信的承诺你会在那个水平生产。说得好。所以想象这两个企业，它们是，比如说，它们是报纸生产企业，一个由 NBC 的母公司拥有，一个由 Rupert Murdoch 拥有，对吧？它们正在进入这个新市场，这个市场是一个目前没有任何报纸的城镇，Murdoch 只是说，我要生产很多报纸。

[段 29]

There’s no reason for NBC to believe that. So moving first here, it isn’t enough simply to say it going to move first. It isn’t enough even to make a decision that’s reversible. Even if Q1 moves but that decision could be undone, that isn’t enough. What we need, and Patrick gave us the key word, what we need is commitment, a word that came up last week. All right? So for this for moving first to help you here, there really has to be commitment. Let’s just get some of this down. there really needs to be commitment for this to work. So going back to the example of Murdoch and his competitor, Murdoch actually has to build the plant. There actually has to be a factory that he’s built in, and that factory can’t just be sold for scrap. So what creates the commitment in the case when you’ve built the plant is what? Because having built it, it’s a sunk cost. It’s there. So sunk costs can help here. Sunk costs can help. It can help making you committed. Once that money is gone, you can’t get it back, so you’re really committed to that scale. Does everyone know what I mean by sunk cost, by the way? Yep. Okay? So here’s a case where a strategic move entering first and sinking some investments can help you in the marketplace.

[译文 29]

NBC 没有理由相信这一点。所以在这里先行动，光是说先行动是不够的。即使做出一个可撤销的决定也不够。即使 Q1 先行动，但那个决定可以被撤销，那也不够。我们需要的是，而且 Patrick 给出了关键词，我们需要的是承诺，一个上周出现的词。好吧？所以为了让先行动帮助你，在这里真的必须有承诺。让我们把这些记下来。为了让它起作用，真的需要有承诺。所以回到 Murdoch 和他的竞争对手的例子，Murdoch 必须真正建造工厂。必须有一个他已经建造的工厂，而且那个工厂不能只是被当作废铁卖掉。那么在建造了工厂的情况下，是什么创造了承诺？因为一旦建好了，它就是沉没成本。它就在那里。所以沉没成本可以在这里提供帮助。沉没成本可以提供帮助。它可以帮助你做出承诺。一旦那笔钱花出去了，你就拿不回来了，所以你真正投入到那个规模上。大家都理解我说的沉没成本的意思吗？是的。好吧？所以这里有一个例子，战略性地先进入并沉没一些投资可以帮助你在市场中获益。

[段 30]

All right. Let’s also look at this another way. Let’s again go back to the simultaneous move game that we had before where Murdoch and his competitor, the NBC parent corporation, are in fact going to move simultaneously. So both of them are in the business of discussing how big a newspaper plant to put in this new town which hasn’t got a newspaper yet. Somewhere in Alabama or something, I don’t know. All right. And suppose that there are two boardrooms, both of which are avidly trying to discuss how big a newspaper plant to build. All right? So suppose that one of the boardrooms, one of the boardrooms are, these four people on this row, this is, this is the NBC parent company boardroom, and they’re trying to decide how big a plant to bills. All right. And, and And over on the other side of the room is our Murdoch group which is in fact let’s take the row parallel. So these guys over there are our Murdoch News International group. All right. And it’s a simultaneous move back game. So basically we’re in Cornell. Now suppose that Murdoch just to pick a name out of the hat might not be the most moral gentleman in the world, who knows, and suppose that he in fact has hired one of the people in the NBC. a name out of the hat, it might not be the most moral gentleman in the world, who knows, and suppose that he in fact has hired one of the people in the NBC boardroom.

[译文 30]

好的。让我们也用另一种方式来看这个问题。让我们再次回到我们之前的同时行动博弈，其中 Murdoch 和他的竞争对手，NBC 母公司，实际上将同时行动。所以他们两个都在讨论要在这个还没有报纸的新城镇建造多大的报纸工厂。可能在阿拉巴马州的某个地方，我不知道。好的。假设有两个董事会，两个董事会都在热烈讨论要建造多大的报纸工厂。好的？假设其中一个董事会，这排的四个人，这是 NBC 母公司的董事会，他们正在决定要建造多大的工厂。好的。然后，房间另一边的那个董事会，在这一排的另一边。所以那边那些人是我们 Murdoch 新闻国际集团的。好吧。这是一个同时行动的回溯博弈。所以基本上我们在 Cornell。假设 Murdoch 随便挑一个名字，可能不是世界上最道德的绅士，谁知道呢，假设他实际上已经雇用了 NBC 董事会中的一个人。随便挑一个名字，可能不是世界上最道德的绅士，谁知道呢，假设他实际上已经雇用了 NBC 董事会中的一个人。

[段 31]

In fact, this guy, what’s his name? Ryan. He’s hired Ryan to be a spy. So Murdoch has a spy in the NBC boardroom. So Murdoch has a little advantage here information-wise. Why? Because the NBC boardroom doesn’t know what’s going on in the Murdoch boardroom, but the Murdoch boardroom is going to know what’s going on in the NBC boardroom. But to make the problem more interesting, suppose that somebody tells NBC’s parent company, then in fact there is a spy in their boardroom. So these guys know they have a spy. They don’t know who it is. If they knew it was, they’d beat him up or fire him or something. Or maybe not. But they know that someone’s there. Maybe they even suspect it’s Ryan. So what should NBC do here? What decision should NBC make? One thing they could do is they could know it’s Ryan, but Ryan has a sort of Murdoch-like face, so they might just fire him on the, you know, he might be a spy. All right? Or what should they do here? Any takers? you’re in the NBC boardroom, you know that Murdoch has some spy in the camp. What should you do? You can come up with a fake plan and see if it goes back to Murdoch? All right. So your name is? Chris. So Chris is suggesting, come up with a fake plan to feed it back to Murdoch.

[译文 31]

事实上，这家伙，叫什么来着？Ryan。他雇了Ryan当间谍。所以Murdoch在NBC董事会有了间谍。从信息角度来说Murdoch有一点优势。为什么？因为NBC董事会不知道Murdoch董事会里在发生什么，但Murdoch董事会会知道NBC董事会在发生什么。但为了使问题更有趣，假设有人告诉NBC的母公司，实际上他们董事会有间谍。这些人知道他们有间谍，但不知道是谁。如果他们知道是谁，就会揍他一顿或者开除他，或者别的什么。也许吧。但他们知道有人在里面。也许他们甚至怀疑是Ryan。那么NBC应该怎么办？NBC应该做什么决定？他们可以做的事情之一是，他们知道是Ryan，但Ryan有一张像Murdoch一样的脸，所以他们可能就把他开除了，嗯，他可能是间谍。行吧？或者他们应该怎么做？有人要回答吗？如果你们在NBC董事会，你们知道Murdoch在他们的阵营里安插了间谍。你们应该怎么做？你们可以制定一个假计划，然后看它是否会反馈给Murdoch。好的，你叫什么名字？Chris。所以Chris建议，制定一个假计划来反馈给Murdoch。

[段 32]

So Chris has been reading spy novels, right? Right? So this was a John LaCarrie novel. That’s certainly what you do. You’d create a whole bunch of fake information to feed back to the Russians, right? Through this spy who in fact you’ve discovered. All right. And that isn’t bad idea that might that might be a good thing to do it’s pretty hard to do right because ultimately these these actual decisions have to be made in the boardroom contracts have to be signed and so on but I think Chris is on to the right idea here so Chris’s idea is create a fake plan to feedback to Murdoch to give Murdoch some misinformation but there’s another thing you could do I mean it gets one hasn’t contributed yet anyone else yeah here what’s your name um so Osman what would you do um these guys get the first move now effectively because they can just decide they know Say what you just said, but shout out so everyone can hear you. NBC now gets the first move because they can decide and they know the other people are going to respond to it. All right, good, good. So what Osman is suggesting is maybe you don’t feed Murdoch a fake plan. You fake, you feed Murdoch the true plan. Effectively what’s going to happen now is if NBC decide to build a large plant, this information will be fed back to Murdoch and Murdoch is now in the position of being the second over.

[译文 32]

Chris一直在读间谍小说，对吧？对吧？这是一本John LaCarrie的小说。这正是你们通常会做的事情。你们会创造一堆假信息来反馈给俄国人，对吧？通过这个实际上已经被你们发现的间谍。行吧。这不是坏主意，可能是个好主意，但做起来相当困难，因为最终这些实际的决定必须在董事会做出，合同必须签署，等等。但我认为Chris在这里提出了正确的想法，所以Chris的想法是制定一个假计划反馈给Murdoch，给Murdoch一些错误信息。但还有另一件事你们可以做，我是说还没有人发言，其他人有吗，对了，你叫什么名字，嗯，Osman，你会怎么做，嗯，这些人现在有效地获得了第一步，因为他们可以决定他们知道的东西，说出你刚才说的，但要大声说出来让每个人都能听到。NBC现在获得了第一步，因为他们可以做出决定，而且他们知道其他人会对它做出反应。好的，好的。所以Osman建议的是，也许你不要给Murdoch喂假计划。你假装，你给Murdoch喂真实的计划。现在实际上会发生的是，如果NBC决定建设一个大型工厂，这个信息会被反馈给Murdoch，而Murdoch现在处于第二步的位置。

[段 33]

When Murdoch moves, he or she knows what NBC is doing. And NBC knows that Murdoch is going to choose the best response to that so it’s as if NBC has been moved, has been put into the position of firm one and Murdoch has been put into the position of firm two. Even with the correct plan. So the correct thing to do here for NBC is not necessarily to mislead Murdoch, but just go ahead and build a big plant. plant. Have that information be fed to Murdoch and let Murdoch respond to it. So notice, here’s a slightly paradoxical thing. You might think that having a spy in the camp of the other team, having a spy, you might think having a spy would help you. But here, having a spy, or having more information if you like, having more information can actually end up hurting you. All right, everyone see that? Paradoxically, Murdoch ends up losing by the fact that he was able to predict what NBC were going to do. Now there’s a key to this, of course, it was crucial to the arguments, what was crucial to the argument? It was crucial to the argument that NBC knew that Murdoch had a spy. Right? The key here is that the other side, the other players, the other players knew you had, or going to have more information.

[译文 33]

当Murdoch行动时，他或她知道NBC在做什么。而NBC知道Murdoch会选择最佳应对方案，所以这就好像NBC已经被移动了，被放到了公司一的位置，而Murdoch被放到了公司二的位置。即使是使用正确的计划。所以对NBC来说这里正确的事情不一定是误导Murdoch，而是直接建设一个大型工厂。让那个信息被反馈给Murdoch，让Murdoch来应对它。所以注意，这里有一个稍微矛盾的事情。你们可能会认为在对方阵营安插间谍，有间谍，你们可能会认为有间谍会帮助你们。但在这里，有间谍，或者换句话说拥有更多信息，拥有更多信息实际上最终可能伤害你。好的，大家都明白了吗？矛盾的是，Murdoch最终因为能够预测NBC将要做什么而吃亏了。当然，这里有一个关键点，这对论点至关重要，什么对论点至关重要？至关重要的是NBC知道Murdoch有一个间谍。对吧？这里的关键是另一方，其他玩家，其他玩家知道你有，或者即将有更多信息。

[段 34]

So what’s the bigger idea here? There are two bigger ideas. Bigger idea number one is, games being simultaneous or sequential is not really about timing. per se. It’s about information. It’s about who knows what, and who knows that who’s going to know what. In a situation where firm one, our board room over here, knows that Murdoch is going to have this information before Murdoch moves, that’s actually a sequential game. sequential gain. Right? The timing is somewhat relevant. That’s the first observation. And the second observation is already on the board. What we’ve learnt is sometimes, in strategic settings, sometimes, not always, more information can hurt you. Sometimes more information can hurt. We’ll be careful here, because that’s not always true, but sometimes it’s true. And the reason that’s true is, the reason that’s true is, it can lead other players to take actions, in this case to create a large plant, that hurts you. If you put Monday’s lecture together with today’s lecture we’ve seen something, two very similar things arose. On Monday’s lecture, having fewer options burning your boats ended up helping you or having lower payoffs by putting collateral down ended up helping you. And that might seem like a paradox, but it isn’t really a paradox because what happened was, provided the other side knows you have fewer options, they know you’ve burnt your boat, or they know you’ll suffer a future fault on the loan, they know you’ve posted collateral, it will lead them to take behaviour that helps you.

[译文 34]

那么更大的想法是什么？有两个更大的想法。更大的想法一是，游戏是同时进行还是顺序进行，实际上不是关于时间本身的问题。是关于信息。是关于谁知道什么，以及谁知道谁将要知道什么。在一个情况下，公司一，我们这边的董事会，知道Murdoch在Murdoch行动之前将拥有这个信息，这实际上是一个顺序博弈。顺序优势。对吧？时间有些相关。这是第一个观察。第二个观察已经写在黑板上了。我们学到的是，在战略环境中，有时候，不总是，但有时候更多信息可能伤害你。有时候更多信息可能伤害你。我们要小心，因为这不是总是真的，但有时候是真的。之所以这是真的原因是，它可能导致其他玩家采取行动，在这种情况下导致创建大型工厂，伤害你。如果你把周一的讲座和今天的讲座放在一起，我们看到了一些东西，出现了两个非常相似的事情。在周一的讲座中，拥有更少的选择、烧毁你的船最终帮助了你，或者通过放下抵押物获得更低的收益最终帮助了你。这看起来可能像是一个悖论，但实际上并不是一个悖论，因为所发生的是，只要另一方知道你拥有更少的选择，他们知道你烧毁了你的船，或者他们知道你将在贷款上遭受未来的违约，他们知道你放下了抵押物，这将导致他们采取对你有帮助的行为。

[段 35]

In the case of the loan, it led to the lender giving you a business to the lender giving you a big a loan. In the case of the Saxon army, you at least hope it leads to the Saxon army running away. Right? Today, we see that more information can hurt you. And once again, it isn’t really a paradox. It’s the same kind of argument. The fact that the other side know you’re going to have this extra information leads them to take actions that end up hurting you. All right. So in games, unlike in standard single person decision problems, more information can hurt and more options can hurt. All right, and here’s the reason. Alright. Now one other thing to say about this. The game we just looked at, the Stackleberg game we just looked at, is an example of something pretty famous. It’s an example of first mover advantage. It’s an example of a game with a first mover advantage. All right. Now, how many of you’ve heard, how many in the room have heard the term first mover advantage before? As few as that, seriously. The rest of you, how many of you have not heard the term first mover advantage before? I want to see. All right. So I’m always very wary of first move of advantage as a term. If you ask the students in the business school how many of have heard the term, they’ve all heard it, it’s a very popular business school term.

[译文 35]

在贷款的情况下，它导致贷款人给你一笔生意，导致贷款人给你一大笔贷款。在撒克逊军队的情况下，你至少希望它导致撒克逊军队逃跑。对吧？今天，我们看到更多信息可能伤害你。再一次，这实际上并不是一个悖论。这是同一类型的论点。另一方知道你将拥有这些额外信息这一事实，导致他们采取最终伤害你的行动。好的。所以在游戏中，不像标准的单人决策问题，更多信息可能伤害你，更多选择可能伤害你。好的，这就是原因。好了，关于这个还有一件事要说的是。我们刚刚看的这个游戏，我们刚刚看的Stackleberg游戏，是一个相当著名事物的例子。这是先动优势的典型例子。这是一个具有先动优势的游戏例子。好的。在座有多少人之前听说过先动优势这个术语？这么少，真的吗？你们其他人，有多少人之前没有听说过先动优势这个术语？我想看看。好的。所以我总是非常警惕先动优势这个术语。如果你们问商学院的学生有多少人听说过这个术语，他们都听说过，这是一个非常流行的商学院术语。

[段 36]

It’s a very popular term that you see in bad books. So let me just warn you a little bit about this. So if you go to the airport, let’s say to Hartford Airport, and the flights you want to get on is late, which is usually the case, and hence you end up in the bookstore, and you find yourself on the economics and business shelves, you start looking at strategy books. And typically, the kind of book you find at the airport on business or strategy or economics is a pretty bad book. All right, so it has some embossed cover and it says, strategy for dummies, or my boring life by a famous CEO. Right? And I worry about these books, because you end up buying these books, you’ve got time on your hands, and you read them, and they give you absolutely terrible advice. All right? So not always, but almost always. And the kind of things they’ll say is, it’s always a good idea to move first, because that way you’ll have a first mover advantage. All right? This sounds right. I’m always worried about things that sound right because they may mean they’re right, but the problem is if they’re wrong, they’re tempting. They sort of lure you in. So it’s good to move first because that way you’ll have a first move advantage.

[译文 36]

这是一个你在糟糕的书中经常看到的非常流行的术语。所以让我在这里给你们一点警告。所以如果你去机场，比如说去Hartford机场，你想乘坐的航班延误了，这是通常的情况，因此你最终到了书店，你在经济学和商业书架前，你开始看战略书籍。通常，你在机场找到的关于商业或战略或经济学的书是相当糟糕的书。好的，所以它有一个压纹封面，上面写着傻瓜策略，或者某个著名CEO的无聊人生。对吧？我担心这些书，因为你们最终买了这些书，你们手上有时间，你们读它们，它们给你们绝对糟糕的建议。好的？不总是，但几乎总是。它们会说的一类事情是，先行动总是个好主意，因为这样你就会有先动优势。好的？这听起来是对的。我总是担心听起来正确的事情，因为它们可能意味着它们是正确的，但问题是如果它们是错误的，它们很有吸引力。它们会引诱你。所以先行动是好的，因为这样你就会有先动优势。

[段 37]

Sounds right, but it’s nonsense. There are situations of which this is one, there are situations where it’s good to move first. In quantity competition, it’s good to set your quantity to be committed, and that will lead the other side. side to producing a smaller quantity which helps you. So sure, there are situations that In quantity competition, it’s good to set your quantity to be committed, and that will lead the other side to producing a smaller quantity, which helps you. So sure, there are situations, there are games in which you want to move first, but there are also games in which you’d rather move second. Let me give you an example. This is an example we’ve seen in this class before. Rock paper, scissors. If anybody is going to read those books and believe, them, you know, so if anyone’s going to read, you know, how to discover your inner Bill Gates and base their life on it and therefore, you know, think there’s a first move advantage, I want to play rock, paper, scissors, with that person. All right? Everyone happy that you’d rather go second in rock paper scissors? Do I need to prove that, or is that obvious? Okay, okay, good. All right. Let me just expand on it a bit more, a bit more, into a more real world setting. There are plenty of settings in the real world where the advantage of moving second isn’t because, as in rock, paper, scissors, you just get to crush the other guy.

[译文 37]

听起来是对的，但其实是胡说。有些情况确实如此，有些情况下先行动是有好处的。在数量竞争中，设定你的产量并承诺它是有利的，这会促使对方生产更少的产量，从而对你有帮助。是的，有些情形、有些游戏你想先走，但也有些游戏你更愿意后走。让我举个例子。这是我在这门课上已经见过的例子——石头、剪刀、布。如果有人要读那些书并相信它们，想想如何发现自己的内在比尔·盖茨，并以此来安排人生，因而认为先手有优势，我想和这种人玩石头、剪刀、布。好吧？大家都认同在石头、剪刀、布中更愿意后手吗？我需要证明这一点吗，还是这显而易见？好，好，挺好。让我再把它扩展一点，进入更现实的场景。现实中有很多情形，先手的优势并不是像石头、剪刀、布那样仅仅把对手压垮。

[段 38]

It’s simply that you learn from their mistakes. So, for example, in the game of buying new equipment for the office or home, it’s great to move second. The other guy goes out and samples some new piece of equipment. I wait to see if it works and then buy it if it does. All right? Or if I’m sitting up a firm in a new expanding market, let’s say a new part of the former Soviet Union, I’m quite happy to let some other firms go in there first and then watch what they did and try and learn from their mistakes. If I’m sitting a new company, curriculum for a university. I’m quite happy to let other universities, Duke and Cornell and so on, move first, and then I can go in as Yale, see what they did, and for sure they’ll have made mistakes, and I’ll learn from that. All right? So there’s plenty of obvious situations where moving second helps you for the obvious reason that information is often very useful. Right? We argued here, information can hurt you. But there’s plenty of other perfectly natural situations where information is to your advantage. All right? So there are games with first mover advantages, but there are also games with second mover advantages. And let me give you one example of a game that neither has a first mover advantage or a second mover advantage, just to convince you that can happen as well.

[译文 38]

这仅仅是因为你可以从他们的错误中学习。例如，在购买办公室或家庭新设备的游戏中，后手非常好。对方先去尝试某种新设备，我等着看它是否好用，好用再买。或者，如果我要在新的扩张市场——比如说前苏联的新地区——创建一家公司，我很乐意让其他公司先去，然后观察他们的做法，尽量从他们的错误中学习。如果我要为某所大学设计新课程，我很乐意让其他大学——比如杜克、康奈尔等——先走，然后我作为耶鲁进去，看看他们做了什么，当然他们会犯错，我就能从中学习。好吧？所以有很多明显的情形，先手帮助你是因为信息往往非常有用。我们在这里争论过，信息有时会伤害你。但也有很多完全自然的情形，信息对你是有利的。所以有些游戏有先手优势，有些游戏有后手优势。让我给你一个既没有先手优势也没有后手优势的游戏例子，只是为了让你相信这种情况也会发生。

[段 39]

So when you were a child, you probably occasionally had to divide a cake and or candy bar with your sibling. Anyone be in this situation? There was some candy bar or cake, and you had to divide this thing between you and your brother and or sister. Is that right? And there’s a way in which there’s a way in which there. There’s a typical way in which we divide things among siblings in that setting. There’s a game we play to divide it. What’s the game we play? Anyone? I’ll cut and you choose. Or vice versa, right? So I’ll cut and you choose. Neither has a first mover advantage or a second mover advantage. Assuming you can cut accurately. Neither has a first move advantage or a second mover advantage, which is precisely why it’s a good way to divide the candy bar. All right. Now to rub this point home and because we’ve had a very dry lecture up to now. Let’s play a game. All right. So everyone going to wake up now. Math is over. I want to play a game. And the game I want to play to play for the rest of today is called NIM. And it’s not going to surprise you that one of the things we’re going to learn in this game is that sometimes games have a first mover advantage and sometimes games have a second mover advantage.

[译文 39]

当你还是个孩子时，你可能偶尔需要和你的兄弟姐妹分一块蛋糕或糖果。有人遇到过这种情况吗？有一块糖果或蛋糕，你必须和你兄弟或姐妹分这份东西。对吧？在这种情况下，我们通常有一种把东西分给兄弟姐妹的方式。我们玩一个游戏来分配它。那是什么游戏？有人知道吗？我来切，你来选。或者反过来，对吧？所以我来切，你来选。两者都没有先手优势或后手优势。假设你能准确切割。没有先手优势也没有后手优势，这正是它成为分糖果的好方法的原因。好。现在为了把这一点强调到底，也因为我们之前上了一节很枯燥的课，让我们玩一个游戏吧。大家现在都醒醒吧。数学结束了。我想玩一个游戏。今天剩下的时间我想玩的游戏叫做NIM。你们会在这局游戏里学到，有时候游戏有先手优势，有时候有后手优势。

[段 40]

So I’m giving away the punchline. How many of you’ve played NIM before? Okay, if you’ve played it before, you can’t play now, right? So don’t shout out. All right, so this is the game. There are two players, and there are two piles of stones. We’ll make the piles of stones into just chalk lines on the board. The players are going to move sequentially. In each turn, the player whose turn it is to move picks one of the two piles and removes some of the stones, which will just be lines on the board. So they decide how many of those lines to delete. Each time it’s your turn, you get to move again. You can choose the other pile this time. You move sequentially, and the only rule that matters is the person who gets the last stone wins. So for example, here’s a case where there’s one, two, three, four, five, six, seven stones on this pile and one, two, three, four stones on this pile. All right? So I’m going to get some voluntary. to come on the stage and do this, but I was told I should choose a volunteer. So is Li Xing Chang here? Is Leishing Chang here? So Leishing Chang’s going to volunteer for this. I’m volunteering Leishing Chang because it’s his birthday. All right, so can you get a round of applause for Leishing Chang?

[译文 40]

我先把结论透露给你们。你们中有多少人玩过NIM？好，如果你以前玩过，你现在就不能玩了，对吧？所以不要大声喊。好，游戏是这样的。两名玩家，两堆石头。我们把石头堆画成黑板上的粉笔线。玩家轮流行动。每回合，轮到行动的玩家选择两堆中的一堆，移除若干石头——也就是划掉若干粉笔线。于是他们决定划掉多少线。每轮轮到你时，你可以再次行动，这次可以选择另一堆。你依次行动，唯一重要的规则是抢到最后一个石头的玩家获胜。比如，这里有一堆有1、2、3、4、5、6、7块石头，另一堆有1、2、3、4块石头。好吧？我需要找一位自愿者上台来演示，主持人告诉我要选一位志愿者。李兴昌在这里吗？雷兴昌在这里吗？所以雷兴昌要当志愿者。我让雷兴昌来当志愿者，因为今天是他的生日。好，请大家为雷兴昌鼓掌。

[段 41]

He come on and stage. Who wants to play against Leishing Chang? Let’s see. Let’s see. Let’s see. Let’s see it. Let’s see it. Who wants to play against Leishing Chang? Let’s see, let’s see it. Anyone else want to play who hasn’t played a game yet? How about the guy with a Yale hat on back there? Yeah, the white Yale hat. I’m staring right at the guy, it’s not, yeah, you, this is. You want to come up? All right? What’s your name? Evan. Evan. We’ve got Evan and Leasing, is it Leaching? Leashing and Evan. Come up on stage. There’s a step up here. Come on, come on, come on. All right. All right, so we’ll let Lee Sching go first, and we’ll let Evan go second. Have you played this game before? I didn’t hear it. Okay, I didn’t hear what the game is. We’ll explain the rules again for those people are sleeping, all right? All right, if we’re sleeping, then the game is back in induction. But, okay, so, all right, so the rules of the game with this, they’re going to take turns. in each turn, they’re going to pick one of these two piles. This is pile A, and this is pile B, and they’re going to tell me how many of these chalk lines to delete. Yeah, that’s a good idea.

[译文 41]

他上台了。谁想和雷兴昌对决？让我看看，谁想和雷兴昌对决？让我看看，还有谁没玩过这个游戏想上场？后面戴耶鲁帽子的那位怎么样？对，白色的耶鲁帽子。我正看着那个人，就是你，你想要上来吗？好吧？你叫什么名字？埃文。埃文。我们有埃文和雷兴。上台来吧。这儿有台阶。上来，上来，上来。好吧，好吧，我们让李兴先走，让埃文后走。你以前玩过这个游戏吗？我没听清。好，我没听清游戏是什么。我们再给还在睡觉的人解释一下规则，好吧？好，如果我们还在睡觉，那么游戏回到了归纳。好吧，所以游戏的规则是，他们轮流行动，每回合选一堆——这堆是A，那堆是B——然后告诉我划掉多少条粉笔线。是的，这是个好主意。

[段 42]

Thank you. They’re going to tell me how many of these chalk lines to delete, thanks. All right, and I’m going to delete those lines. We’re going to go on playing until somebody gets the last line. The person who gets the last line wins. Okay, person who gets the last line win. You can take, you can get the last line tenor to go. It doesn’t have to be that there’s only one on the board when you get the last line, but the person who gets the last line wins. All right? So you have to pick a pile each go and tell me how many lines to remove. So, Leishing, why don’t you go first? You can choose Pile A or Pile B and tell me how many lines to remove. Any advice from the audience? No advice. You’re on your own here. Can you remove three from Pile A? Three from Pile A, okay? So three from Pile A. Evan, your sub. Let you stand near the board so we can get you in the board so we can get you in it so have a and you remove two from Pile A. Two from Pile A, okay, so Pile A is the Pau-A. Liegeon? Two from Pile B. Two from Pile B. Suspense is building here. One from Pile B. Should go fast at the stage, go on.

[译文 42]

谢谢。他们会告诉我划掉多少条粉笔线，谢谢。好，我把这些线划掉。我们继续玩，直到有人划掉最后一条线。抢到最后一条线的人获胜。好，抢到最后一条线的人获胜。你可以抢到最后一条线后再走。不必在只剩一条线时才抢到最后一条线，但抢到最后一条线的人获胜。好吧？每回合你必须选一堆并告诉我划掉多少条线。所以，雷兴，你先走。你可以选择堆A或堆B，告诉我划掉多少条线。观众有什么建议吗？没有建议。你只能靠自己了。你能划掉堆A的三条吗？堆A的三条，好，堆A三条。埃文，轮到你了。让你站在黑板旁边，这样我们能把你拍进画面——你可以划掉堆A的两条。堆A两条，好，堆A现在只剩下——？雷兴？堆B两条。堆B两条。悬念在升级。堆B一条。快点上台，继续。

[段 43]

Two from Pile A. Last Stone wins. Last own wins. Last own wins. Let’s have to be careful here. It’s his birthday. Come on. Last stone wins. The person who gets the last stone wins. So one from A. All right. One from A. Can’t have stains. Can’t have stains. So. B. One from B. And Lisa is the winner here. All right. All right. All right. All right. Very good. Very good. Let’s get two more volunteers. All right. Everyone understand the game now. Thank you very much, gentlemen. All right. Two more volunteers. Yeah. Have you played before? Come out. I want to get some female volunteers. It can’t be a male only class. There we go. Thank you. This may not be as important economics than doing the Stacklebone model, but it’s probably a little bit more fun. All right. And your names are? John. Christine. John and Christine. Okay. All right. So, okay, so let’s make it a bit more complicated this time. So 1, 2, 3, 4, 5. 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13. And 1, 2, 3, 4, 5, 6. four, five, six, seven. So we’ll pick two prime numbers, if that’s important. Let’s see what happens. What don’t we say, ladies first, I think? So, Christine, your choice? That one has… This one started off at 13, I believe.

[译文 43]

A堆两个。最后一个拿石头的赢。最后一个拿的赢。最后一个拿的赢。这里要小心。是他生日。来吧。最后一个拿石头的赢。谁拿到最后一个石头谁赢。所以A堆一个。好。A堆一个。不会有污渍。不会有污渍。所以B堆。B堆一个。Lisa是赢家。好。好好好。非常棒。非常棒。我们再找两个志愿者。大家现在都懂这个游戏了吧。非常感谢，绅士们。好。再来两个志愿者。你们之前玩过吗？出来吧。我想找一些女性志愿者。不能只有男性。好了。谢谢。这个可能没有Stacklebone模型重要，但可能更有趣。好。你们叫什么名字？John。Christine。John和Christine。好。那么这次我们把它弄得更复杂一点。1、2、3、4、5。1、2、3、4、5、6、7、9、10、11、12、13。还有1、2、3、4、5、6。四、五、六、七。所以我们选两个质数，如果有意义的话。看看会发生什么。女士优先怎么样？Christine，你选哪个？那堆有…这堆是从13开始的，我想。

[段 44]

And this one started off at 7. Assuming I counted correctly. Let me get out of the way. Six from Pile. Six from Pile. Six from by. Okay, one, two, three, four, five. four or five. Um, six from pile A. Six, six from by, okay, one, two, three, four, five. Is that right? Is that right? One, two, three, okay. Three from pile B. Three from pile B. Three from pile A. Three from pile A. One from pile B. One from pile B, all right. One from Pile A. All right, so I think we’re on to this, right? One from Pile B. One from Pile A. All right. One from Pile B. All right. One from Pile A. All right. One from Pile A. Ah, okay, I’ll switch. One from Pile B. All right, so Christine is the winner. Okay, okay. So a round of applause for these two. All right. So I think everyone’s figured out now how to play. play this game, is that right? Has everyone figured it out? All right. Let me take one of these mics and go down and make sure that people have figured this out. All right? So what is the rule for how to play this game? What’s the rule for how to play this game? People haven’t figured out. We should play again. So let’s try over here.

[译文 44]

这堆是从7开始的。如果我数得没错的话。让我让开。堆里六个。堆里六个。六个由…好，一、二、三、四、五。四或五个。嗯，A堆六个。六个由…好，一、二、三、四、五。对吗？对吗？一、二、三，好。B堆三个。B堆三个。A堆三个。A堆三个。B堆一个。B堆一个，好。A堆一个。好，那么我想我们接着这个，对吧？B堆一个。A堆一个。好。B堆一个。好。A堆一个。好。A堆一个。啊，好，我换一下。B堆一个。好，Christine是赢家。好。那么为这两位鼓掌。大家现在都弄清楚怎么玩这个游戏了，对吧？大家都弄清楚了吗？好。让我拿一个麦克风下去确认大家都弄清楚了这个游戏。好吗？这个游戏的规则是什么？这个游戏的规则是什么？人们还没弄清楚。我们应该再玩一次。那么这边来试试。

[段 45]

Someone has an answer. How should you play this game? Yeah. It depends on whether the piles have the same number in them. If they have the same number, you want to be second player, and if they have different numbers, you want to be first player. All right. And what should you do? You want to… Does it assume they have different numbers? If they have different numbers, you want to make them equal. You want to make them equal. All right, all right. So the trick to playing this game is, thank you, very good. The trick to playing this game is, if the piles are uneven, then you want to make them even. You want to make them equal. Everyone to see that? So if you start off with the piles. being unequal, as we did both times, for example, three and two, then you want to be player one. There’s a first mover advantage, and the correct tactic is to equalize the piles. All right? What you’ll notice now is that player two can’t do anything. If they take two from here, you’ll win by taking both of those two. If you take one from here, you’ll equalize the piles again. And if they take then, one from here, you’ve won. All right? So the way to play this game is to equalize the piles. What does that mean?

[译文 45]

有人有答案。这个游戏应该怎么玩？对。取决于堆里的数量是否相同。如果数量相同，你想当第二个玩家；如果数量不同，你想当第一个玩家。好。那你应该怎么做？你想要…这假设它们数量不同？如果数量不同，你想让它们相等。你想让它们相等。好，好。那么这个游戏的诀窍是，谢谢，非常好。这个游戏的诀窍是，如果堆不平衡，那么你想让它们平衡。你想让它们相等。大家看到这一点了吗？所以如果你从堆开始就不相等，正如我们两次所做的，例如，三和二，那么你想当第一个玩家。有先手优势，正确的策略是让堆相等。对吧？你会注意到，现在第二个玩家什么都做不了。如果他们从这里拿两个，你拿这两个就赢了。如果你从这里拿一个，你就再次让堆相等了。如果他们然后从这里拿一个，你就赢了。对吧？所以这个游戏的玩法是让堆相等。这意味着什么？

[段 46]

It means if you, if the initial position has unequal piles, uneven piles, then you would rather be player one. It has a first mover advantage. But if the initial position has an even number of, uh, has even numbers in the piles, then you’d rather be player two. Is that right? Right? Because if you start off with an even number in each pile, the person moving first is going to make them unequal, and thereafter, the next person’s in a winning position. So I want to notice two things from this game. First, notice in this game that from any initial position, we can very quickly tell who is going to win and who’s going to lose, assuming they play well. Three things to know, actually. Third, second, sorry, second, we didn’t actually use backward induction here, but it’s pretty obvious you do want to use backward induction here. You want to figure out what the end game’s going to look like. Is that right? It’s very easy to see this game if you look at the end game. All right? And the third lesson is what we just said in this game, sometimes there’s a first mover advantage, sometimes there’s a second mover advantage. The first lesson I do, have I say called? So sometimes there’s a first lesson. there’s a first mover advantage in this game, sometimes there’s a second mover advantage in this game.

[译文 46]

这意味着如果你最初的堆不平衡、不均匀，那么你宁愿当第一个玩家。它有先手优势。但如果最初的堆有偶数，呃，堆里有偶数，那么你宁愿当第二个玩家。对吧？对吧？因为如果你从每个堆都有偶数开始，先走的人会让它们变得不相等，此后，下一个人就处于必胜位置。所以我想从这个游戏中注意两件事。首先，注意在这个游戏中，从任何初始位置，我们都可以很快判断谁会赢谁会输，假设他们玩得好的话。实际上有三件事要知道。第二，抱歉，第二，我们实际上没有在这里使用逆向归纳法，但很明显你确实想在这里使用逆向归纳法。你想弄清楚终局会是什么样子。对吧？如果你看这个游戏的终局，就很容易理解。好吧？第三个教训是我们刚才在这个游戏中说的，有时有先手优势，有时有后手优势。第一个教训我说了叫什么？所以有时有第一个教训。在这个游戏中，有时有先手优势，有时有后手优势。

[段 47]

All right? I’m just illustrating the point we made earlier that it’s not always the case that you want to move first. Sometimes you want to move second. Now, today, I set up the piles with unequal piles, so there was a first mover advantage, but that was just to give an advantage to the guy whose birthday it was. All right? I could have set things up with equal lines in each pile and made the other guy, the other guy who I, the other guy who go first. guy to go first.

[译文 47]

对吧？我只是在说明我们之前提出的观点，不总是你都想先走。有时你想后走。今天，我把堆设置成不相等的，所以有先手优势，但那只是为了给过生日的人一个优势。对吧？我本可以用相等的线来设置每堆，让另一个人，先走的那个人先走。