耶鲁大学博弈论公开课 - 第6集 信息集和子博弈完美

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 today we have a lot of stuff to get through, but it’s all going to be fairly formal. We’re not going to have time to play a game today. Today’s going to be a day where we have to learn some new ideas. So the reason we need to go through some new formal ideas today is we’ve kind of exhausted what we can do with the ideas we’ve gathered so far. So just to bring us up to date where we are, in the first half of the semester, so before the midterm, we looked at simultaneous move games, and one way to think about those simultaneous move games were games where when I make my choice, I don’t know what you’ve done. And when you make your choice, you don’t know what I’ve done. And since the midterm, we’ve been looking at simple examples of sequential move games, sequential move games under perfect information, in which I typically do know what you did when I get to make my choice. And you know I’m going to know what you did when I get to make my choice. And what I want to be able to do moving forward is I want to be able to look at strategic situations that combine those two settings. I want to be able to analyze games which involve both sequential moves and simultaneous move games And in particular I want to see how we can extend the technique we been focusing on for the last few weeks which is backward induction I want us to see how we can extend the notion of backward induction to cope with games where some parts are sequential and some parts are simultaneous.

[译文 1]

今天我们有很多内容要讲，但都会相当正式。我们今天没有时间玩游戏。今天将是我们需要学习一些新概念的一天。我们之所以需要学习一些新的正式概念，是因为我们已经基本用尽了我们目前所学的 ideas 能做的事情。让我们回顾一下我们目前的位置，在上半学期，也就是期中考试之前，我们研究了同时行动博弈，同时行动博弈的一种理解方式是，当我就做出我的选择时，我不知道你做了什么。而当你做出你的选择时，你也不知道我做了什么。期中考试之后，我们一直在研究序贯行动博弈的例子，在完美信息下的序贯行动博弈中，当我做出我的选择时，我通常知道你做了什么。你也知道当我做出我的选择时，我会知道你做了什么。而我接下来希望能够做的是，我希望能够分析同时包含序贯行动和同时行动博弈的战略情境。我特别想看看我们如何能够扩展过去几周一直在关注的逆向归纳技术，我想让我们看看如何能够将逆向归纳的概念扩展到能够处理部分序贯、部分同时的博弈。

[段 2]

So we’re going to look at a lot of examples and we’re going to introduce some new ideas and I’m going to try and walk you through that today. So that’s our goal. Let’s start with an example. So here’s a very simple game in which player one moves first and has three choices. Let’s call them up, middle, and down. And then player two moves, and player two has two choices from each of these nodes. And we’ll call the choices suggestively up and down, up and down, and he will just call them left and right. And the payoff is as follows, 4, 0, 0, 4, 0, 4, 4, 0, 1, 2, 0, 0. So this is just a standard game of perfect information, much like all the games we seen since the midterm In fact it a relatively easy one So we know how to solve this game We solve this game using what Using Bap and induction And that isn so hard here We know that player 2, if player 2 finds herself up here, she will choose 4 rather than 0. If she finds herself here, she’ll choose 4 rather than 0. And if she finds herself here, she’ll choose 2 rather than 1. So player 1 won’t want to go up here because he’ll get 0, and he won’t want to go into the middle because he’ll get 0, and he won’t want to get, but if he goes down, player 1 will choose left, and player 1 will get 1.

[译文 2]

我们将看很多例子，并引入一些新概念，我今天会尽量引导你们理解。这就是我们的目标。让我们从一个例子开始。这是一个非常简单的博弈，玩家一先行动，有三个选择。我们称之为上、中、下。然后玩家二行动，玩家二在每个节点有两个选择。我们姑且称它们为上或下，上或下，而他称之为左或右。收益如下：4，0，0，4，0，4，4，0，1，2，0，0。这只是一个标准的完美信息博弈，就像期中考试以来我们见过的所有博弈一样。事实上这是一个相对简单的博弈。所以我们知道如何求解这个博弈。我们用逆向归纳法来求解这个博弈。这里并不难。我们知道，如果玩家二发现自己在这里，她会选择4而不是0。如果她发现自己在这里，她会选择4而不是0。如果她发现自己在这里，她会选择2而不是1。所以玩家一不会想上去，因为他会得到0，他也不会想走中间，因为他会得到0，他也不会想得到，但如果他走下，玩家一会选择左，玩家一会得到1。

[段 3]

So player 1 will choose down. So backward induction predicts that player 1 chooses down, and player 2 responds by choosing left. And just staring at this a second, notice that the reason in this game, taking a step back from backward induction second, the reason player one did not want to choose either up or middle was because that move was going to be observed by player two and in either case player two was going to crush player one. Right, so if player one went up, player two was playing this strictly competitive game with player one and player two could design could make a choice that gave 2 4 and 1 0 And conversely if player 1 chose middle player 2 could crush player 1 by choosing up which gave, once again, player 2, 4, and player 1, 0. So there was a good reason here to avoid going into the part of the game following up or middle, and the reason was 2 had a huge second-mover advantage in those parts of the game. That clear to everybody? So I now want to consider a similar but importantly different game. So I’m going to draw the game again. But before I draw it, let me see what I’m going to do. So I want to introduce a new idea, and the new idea is going to be that player 2 will not be able to distinguish between up or middle.

[译文 3]

所以玩家一会选择下。逆向归纳预测玩家一选择下，玩家二回应选择左。只要稍微看一下就会发现，在这个博弈中，退一步来看逆向归纳，玩家一不想选择上或中的原因是，那个移动会被玩家二观察到，而在任何一种情况下玩家二都会碾压玩家一。对，如果玩家一上去，玩家二正在和玩家一玩严格竞争的游戏，玩家二可以设计可以让一个选择给出2、4和1、0。相反，如果玩家一选择中，玩家二可以通过选择上来碾压玩家一，同样再次给出玩家二4，玩家一0。所以有很好的理由避免进入上或中之后的游戏部分，原因是在这些部分玩家二有巨大的后发优势。这一点大家都清楚了吗？现在我想考虑一个类似但又明显不同的博弈。我再画一遍这个博弈。但在我画之前，让我看看我要做什么。我想引入一个新概念，这个新概念就是玩家二将无法区分上或中。

[段 4]

So as I said again, if player 1 chooses down, player 2 will observe that, just as we’ve done before in our standard perfect information games. But if player one chooses either up or middle, I want to capture the idea that player two doesn’t know which of those two choices was made. That’s clearly going to change the game a lot. And the first question is, how do we represent that idea in a tree? So let me try and show a good way to represent that in a tree. The game has the same idea. change the game a lot, and the first question is, how do we represent that idea in a tree? So let me try and show a good way to represent that in a tree. The game has the same structure to it. Player 1 is, again, choosing between up, middle, or down. And player 2, once again, is choosing here, up or down, up or down, and here, left or right. and the payoffs haven’t changed. They’re still 4, 0, 0, 4, 4, 0, 0, 4, 1, 2, and 0, 0. So that’s exactly the same as you have in your notes already. But now I want to adapt this tree to show how we indicate that player 2 cannot distinguish, cannot observe whether 1 chose up or middle, but can observe if player 1 has chosen down.

[译文 4]

正如我所说，如果玩家一选择下，玩家二会观察到这一点，就像我们之前在标准完美信息博弈中所做的那样。但如果玩家一选择上或中，我想表达的意思是，玩家二不知道这两个选择中的哪一个被做了。这显然会大大改变博弈。第一个问题是，我们如何在树中表示这个概念？让我试着展示一个在树中表示的好方法。博弈有相同的结构。玩家一再次选择上、中或下。玩家二再次在这里选择上或下，上或下，以及左或右。收益没有改变。它们仍然是4，0，0，4，4，0，0，4，1，2，以及0，0。所以这和你笔记中的完全一样。但现在我想改编这棵树来展示我们如何表明玩家二无法区分、无法观察到玩家一选择了上还是中，但能够观察到玩家一选择了下。

[段 5]

And the way we do that, very simply, we draw a dotted line between the two nodes of player two, between which two cannot distinguish. So this idea here what this dotted line indicates is that these two nodes are set in the same information set So our new idea here is the idea of an information set I did, I did, I did, I did. Okay, 0, 4, 4, 0. Thank you, Dave. All right, the payoff is going to be the same on the left as on the right. So the idea here is that player 2 cannot distinguish these two nodes. Player 2 knows that she’s in this information set. She knows that player 1 must have chosen either up or middle. She knows that player 1 did not choose down. but she doesn’t know whether she’s really here or here. Okay? Now, what happens in this game? This game is a very different game. Why is it a different game? Well, it’s trying to apply that loose intuition we talked about before. We said previously in the old game that if player one shows up, two knew that player one had chosen up and observed that by choosing down player two could crush one And if player one chose middle player two could observe that player one had chosen middle and this time by choosing up, could crush one.

[译文 5]

我们这样做很简单，我们在这两个玩家二的节点之间画一条虚线，在这两个不能区分的节点之间。所以这个虚线表示的意思是，这两个节点位于同一个信息集中。所以我们这里的新概念是信息集的概念。我、我、我、我。好的，0，4，4，0。谢谢你，Dave。好的，左边的收益和右边的一样。所以这里的意思是，玩家二无法区分这两个节点。玩家二知道她在这个信息集中。她知道玩家一一定选择了上或中之一。她知道玩家一没有选择下。但她不知道她是真的在这里还是在这里。好的？现在，这个博弈中会发生什么？这个博弈是一个完全不同的博弈。为什么它是一个不同的博弈？嗯，它试图应用我们之前谈到的那个松散的直觉。我们之前在旧的博弈中说，如果玩家一出现，二知道玩家一选择了上，并通过选择下观察到这一点，玩家二可以碾压一。如果玩家一选择中，玩家二可以观察到玩家一选择了中，而这次通过选择上，可以碾压一。

[段 6]

The problem is that now in this new game, player two doesn’t know whether he’s here, whether she’s here, in which case she would want to choose down, or here, in which case she’d want to choose up. so player 2’s choice is not so obvious anymore that simple back induction argument has disappeared moreover, player 1 knows that player 2 will not be able to reserve between up or middle so it isn’t necessarily the case that player 1 will want to choose down anymore it’s still true that if player 1 did choose down that player 2 would be able to reserve that and will choose left so that part of the argument is the same what do we think is going to happen here? Well, we don’t know, but let me give a suggestion what might happen here. Player 1 might say, hey, I could randomize between up and middle. I could choose half the time up and half the time middle. If I choose half the time up and half the time middle, player 2 isn’t going to know, in general, isn going to know what I done It isn quite clear what player 2 is going to do And since I randomizing between up and middle whatever player 2 is going to do I going to get half the time 4 and half the time 0 for an expected value of 2.

[译文 6]

问题是，现在在这个新博弈中，玩家二不知道他在这里，她在这里，在这种情况下她想选择下，还是在这里，在这种情况下她想选择上。所以玩家二的选择不再那么明显了，简单的逆向归纳论证已经消失了，而且，玩家一知道玩家二将无法在上或中之间保留，所以玩家一不一定会再想选择下仍然是正确的，如果玩家一确实选择下，玩家二将能够保留那个并会选择左，所以那部分论证是一样的。我们认为这里会发生什么？嗯，我们不知道，但让我提一个可能发生什么的建议。玩家一可能会说，嘿，我可以在上和中之间随机。我可以一半时间选上，一半时间选中。如果我一半时间选上，一半时间选中，玩家二总体上不会知道，不会知道我做了什么。玩家二要做什么还不太清楚。由于我在上和中之间随机，无论玩家二要做什么，我一半时间会得到4，一半时间会得到0，期望值为2。

[段 7]

So I said again, so player 1 might decide in this game to randomize 50-50 between up and middle, knowing that half the time, therefore, he will get 4, and half the time he’ll get 0 for an expected value of 2, which, notice, is better than he got by choosing down. All right? So this change in this game, changing the information in this game, not only led to a different game, it led to a very different outcome. All right? So here, one might, for example, they might randomize between up and middle. And over here we know exactly what one does. One chooses down. One chooses down. So we get very different outcomes because of this change in information in the game. And the theme of today is that information is going to matter. The way we’re going to model information is by thinking about these information sets. And as we go through today, I want to start giving you some formal definitions. So this is the idea, now let’s look at the formal definition. There’s going to be a lot of writing today, so I hope you brought a notepad with some room on it. Alright, so the first formal definition of the day comes off that last example. the formal definition is the idea that I want to capture, I want to capture the idea that players down the tree may not know exactly what was done up the tree.

[译文 7]

所以我又说了一遍，玩家1在这个游戏中可能会决定在up和middle之间随机选择50-50，他知道有一半的时间他会得到4，有一半的时间他会得到0，期望值为2，注意，这比他选择down时要好。好的？所以这个游戏中信息的变化不仅导致了一个不同的游戏，还导致了非常不同的结果。好的？所以在这里，例如，他们可能会在up和middle之间随机选择。而在另一边我们确切知道玩家1做什么。他选择down。他选择down。所以由于游戏中信息的这个变化，我们得到了非常不同的结果。今天的主题是信息将很重要。我们建模信息的方式是通过考虑这些信息集。当我们继续今天的内容时，我想开始给你们一些正式的的定义。所以这是这个想法，现在让我们看看正式的定义。今天会有很多内容要写，所以我希望你们带了一个有足够空间的笔记本。好的，今天的第一个正式定义来自上一个例子。正式的定义是，我想捕捉的想法是，沿着树向下的玩家可能不知道在树上做了什么。

[段 8]

And the formal definition is going to go through the idea of an information set. So an information set of player I, in this case above player 2, but more generally of player I, is a collection, or a set if you like, it a collection of player I nodes between which or I guess it could be more than two so let say among which I cannot distinguish It’s going to turn out that by clever use of information sets, we’re going to be able to use our technology, our technology of drawing trees, to capture all sorts of interesting and increasingly complicated information settings. In this particular game, it’s the case that player one knew that player two was not going to be able to distinguish between up or middle in this tree, and player one knew that player two would be able to distinguish in the left-hand tree. We can even use information sets in a more elaborate tree to capture the idea that player one may not know what player two is going to know. But I won’t do that now, I’ll leave that later, and you’ll see some examples of that on the homework. All right, so we have our formal definition. This is going to be the first of our big tools of the day. But let me just put down a few things that we have to be careful about, a couple of rules.

[译文 8]

正式的的定义将阐述信息集的概念。信息集是关于玩家I的，在这种情况下是上面的玩家2，但更一般地是玩家I，是一个集合，或者说如果你喜欢的话，是一个收集，是玩家I节点的集合，在这些节点之间，或者可能不止两个，所以可以说在这些节点之间，玩家I无法区分。通过巧妙地使用信息集，我们将能够使用我们的技术，我们画树的技术，来捕捉各种有趣的且越来越复杂的信息设置。在这个特定的游戏中，情况是玩家1知道玩家2将无法区分up还是middle在这个树中，而玩家1知道玩家2将能够在左手边的树中区分。我们甚至可以在更复杂的树中使用信息集来捕捉玩家1可能不知道玩家2将会知道什么的想法。但我现在不做那个，我把它留到后面，你们会在作业中看到一些例子。好的，所以我们有了正式的定义。这将是我们今天的大工具之一。但让我先放下一些我们需要注意的事情，几个规则。

[段 9]

So these information sets have to obey certain rules And certain things are not allowed Certain things are not allowed So in particular the following is not allowed Here a tree in which player 1 moves first and player 2 does not observe player 1’s move, so these two nodes are player 2’s nodes, they’re in the same information set, which means player 2 is not meant to be able to distinguish between these two nodes. And suppose, however, the tree looked like this. Okay, so I claim that this is crazy. We couldn’t allow this. It wouldn’t make any sense to allow this. Can anyone see why? Why is this not really a sensible tree? Anyone see that? Why is that not a sensible tree? Yeah, do you want to grab somebody? Yeah, Ali, there’s just the guy behind you. Yeah, that’s good. That’s great. Shout out. If player two knows that he has three choices, then he’ll know he’s at the top node. Exactly, exactly. In this tree, you haven’t got the payoffs in, but if player two observes that she has three choices, she knows she must be at the top node. If she observes she has two choices, she must be at the bottom node. So in this tree it was supposed to be the case that two didn know whether she was here or here but merely by observing how many choices she has she could infer whether she was at the top node or the bottom node So that can make any sense So this is not allowed So we’ll put a great cross through that one.

[译文 9]

所以这些信息集必须遵守某些规则，某些事情是不允许的，某些事情是不允许的。特别是，以下是不允许的。这里有一个树，玩家1先移动，玩家2不观察玩家1的移动，所以这两个节点是玩家2的节点，它们在同一个信息集中，这意味着玩家2不应该能够区分这两个节点。然而，假设树看起来像这样。好的，所以我声称这是疯狂的。我们不能允许这个。允许这个没有任何意义。有人能看出为什么吗？为什么这不是一个合理的树？有人看出那个了吗？为什么这不是一个合理的树？是的，你想叫谁回答？是的，Ali，就是你后面那个人。好的，这样很好。大声说出来。如果玩家2知道她有三个选择，那么她就知道她在顶部节点。正是，正是。在这个树中，你们没有把收益放进去，但如果玩家2观察到有三个选择，她就知道自己一定在顶部节点。如果她观察到有两个选择，她一定在底部节点。所以在之前那个树中，本应是玩家2不知道她是在这里还是在这里，但仅仅通过观察她有多少选择，她就可以推断她是在顶部节点还是底部节点所以这根本说不通所以这是不允许的所以我们会在那个上面画一个大叉。

[段 10]

Now the second thing that’s not allowed is a little bit more subtle, and actually is an interesting thing. And this is just kind of bookkeeping. But the second thing is more interesting. So let’s have a look at it. Here’s a more interesting tree. Player 1 moves first. Player 2 observes that move. All right, and player two moves second. And then at the bottom of this, player one may have another chance to move again. So again, I haven’t put payoffs in here. Player one moves first, player two moves second, and if player two chooses down here or up there, then player one gets to move again. Now I claim again that this is not a sensible tree. It’s not a sensible arrangement of information sets. Can anyone see why this isn’t sensible? Why is this not sensible? Yeah, Steven, yeah. Shout it out. Player one knows what node he’s at based on the first choice that he made. Exactly, exactly. So to get to the upper node here for player one, player one must have chosen up before. And to get to the lower node here, player one must have played down before. So provided that player one remembers his or her own move, she knows where she is. Is that right? So provided player one can recall what she herself did earlier on the tree, she should be able to distinguish these things.

[译文 10]

现在第二个不允许的事情稍微更微妙，实际上是一个有趣的事情。这只是某种记录。但第二个事情更有趣。让我们看看它。这里有一个更有趣的树。玩家1先移动。玩家2观察到那个移动。好的，然后玩家2移动第二。然后在这下面，玩家1可能有另一次移动的机会。同样，我没有把收益放进去。玩家1先移动，玩家2移动第二，如果玩家2在这里选择down或在那里选择up，那么玩家1可以再次移动。现在我再次声称这不是一个合理的树。这不是一个合理的信息集安排。有人能看出为什么这不是合理的吗？为什么这不是合理的？是的，Steven，是的。大声说出来。玩家1知道他自己在哪个节点是基于他做的第一个选择。正是，正是。所以为了到达这里的玩家1的上部节点，玩家1一定之前选择了up。为了到达这里的下部节点，玩家1一定之前玩了down。所以只要玩家1记得他或她自己的移动，她就知道自己在哪个位置。对吗？所以只要玩家1能回忆起她自己之前在树上做了什么，她就应该能够区分这些东西。

[段 11]

So we’re going to rule this out, but I just want to make a remark here. There’s an assumption in ruling it out, and the assumption is we’re assuming perfect recall or perfect memory. We’re assuming perfect recall. And people don’t always, in the real world, players don’t always have perfect recall. There are two reasons, and we’re always going to assume this, but let me just make a remark. There are two reasons why people might not have perfect recall. recall. One reason is, like me, they’re getting old. They simply can’t remember what they did yesterday. So when I’m driving home, I know roughly how many traffic lights I have to go through before I turn right but I sometimes forget which traffic light I at and I turn right too early or too late That doesn happen to you guys but it happens to me as I getting a bit senile Alright So old age would rule out perfect recall A more important example perhaps is if players of games are themselves institutions It’s sometimes useful, and we’ve often talked about it in this class, to imagine a player of a game being a firm, or a country, or some kind of institution, in which the actual decisions may be being taken by different actual people within the firm, institution, or country. And this assumption of perfect recall is saying that the players within the institution knew what the other players within that same institution were doing.

[译文 11]

所以我们要排除这个，但我只是想在这里做一个评论。在排除它时有一个假设，假设是，我们假设完美的回忆或完美的记忆。我们假设完美的回忆。人们并不总是，在现实世界中，玩家并不总是有完美的回忆。有两个原因，我们总是要假设这个，但让我做一个评论。有两个原因人们可能没有完美的回忆。回忆。一个原因是，像我一样，他们变老了。他们就是记不住昨天做了什么。所以当我开车回家时，我大致知道我需要经过多少个红绿灯才右转，但我有时忘记我在哪个红绿灯，我会太早或太晚右转。这不会发生在你们身上，但这发生在我身上，因为我变得有点老年痴呆了。好的，所以年老会导致完美回忆不成立。也许一个更重要的例子是，如果游戏的参与者本身是机构有时这是有用的，我们在这门课中经常讨论，假设一个游戏的参与者是一家公司，或一个国家，或某种机构，实际的决策可能是由同一机构、公司或国家内的不同实际人做出的。这个完美回忆的假设是说，机构内的参与者知道该机构内其他参与者在做什么。

[段 12]

So if we’re modeling General Motors as one player, this assumption is assuming that the chief financial officer and the chief executive officer of GM are observing each other’s actions on the same page. The left hand knows what the right hand’s doing. We are typically going to assume that, but just to make the point, it is an assumption. And it’s quite interesting to see what happens if you relax it. All right, so with that in mind, we can move to our next definition. And this is something I’ve referred to early on in the class, but I want to be formal now. Now we can be formal. We talked earlier on this class about the idea of perfect information So for example when we talked about Zamello theorem we talked about games of perfect information And we said informally what this was. The game of perfect information is a game where each player in the game can observe all previous moves. That was our informal definition, but we can now give a formal definition very simply. Perfect information is a setting where all information sets in the tree contain just one node. alright, we’ll be clear here so what we’re saying here is if we have a tree in which every information set is a singleton we basically never bother with any dotted lines, that’s a game of perfect information and that shouldn’t be a surprise to anybody here because that’s exactly how we drew trees since the mid-term is that right Of course the novelty is we now going to be allowed to look at games of imperfect information The reason we doing this is because it would be interesting as an example we just seen to think about games where information is not perfect.

[译文 12]

所以如果我们把通用汽车建模为一个参与者，这个假设是假设通用汽车的首席财务官和首席执行官在同一个页面上观察彼此的行动。左手知道右手在做什么。我们通常会这样假设，但只是为了说明这一点，这是一个假设。看看如果放松它会发生什么会很有趣。好的，所以考虑到这一点，我们可以转向下一个定义。这是我在课上早期提到过的东西，但我想现在正式一些。现在我们可以正式了。我们之前在这门课上讨论过完美信息的概念。例如当我们讨论Zamello定理时，我们讨论了完美信息的游戏。我们非正式地说了这是什么。完美信息的游戏是游戏中每个玩家可以观察到所有先前移动的游戏。那是我们的非正式定义，但我们现在可以给出一个非常简单的正式定义。完美信息是一种树中所有信息集只包含一个节点的设置。好的，我们在这里说清楚，我们说的是如果我们有一个树，其中每个信息集都是一个单例，我们基本上从不费心画任何虚线，那就是一个完美信息的游戏，这对这里的任何人来说都不应该令人惊讶，因为那正是我们自期中考试以来画树的方式，对吧当然，新颖之处在于我们现在将允许看不完美信息的游戏。我们这样做的原因是，作为一个例子会很有趣，我们刚刚看到考虑信息不完美的游戏。

[段 13]

So what is the definition of imperfect information? Imperfect information, formal definition is not perfect information. We’ve defined what perfect information is, imperfect information is the rest. In the real world, there’s a lot of games that turn out to have imperfect information. There’s lots of strategic situations where I’m going to be able to observe some things that you’ve done, but other things I won’t know quite what you’ve done. Okay, let’s go straight to an example. I don’t think we really need to keep that definition very focal, so let’s get rid of that board. Alright, let’s do an example. There are many examples today. Alright, so this example is going to be a tree in which player 1 moves first. Player 2 cannot observe this move, and sometimes rather than labeling both of these nodes with a 2, I’ll just put a 2 on the information set, just to indicate that both of these nodes belong to player 2, so player 2 moves second, and we’ll call player 1’s move up or down, and we’ll call player 2’s move left or right, kind of suggestively, left or right. Okay. So what’s the information set here? The information set is indicating the fact that player 2 cannot observe whether player 1 moved up or down. Player 2 cannot observe whether player 1 chose up or down. Now why does that matter?

[译文 13]

那么，不完美信息的定义是什么？不完美信息的形式定义就是非完美信息。我们已经定义了什么是完美信息，不完美信息就是除此之外的部分。在现实世界中，有许多博弈实际上具有不完美信息。有许多战略情形，其中我能观察到你所做的一些事情，但其他事情我无法确切知道你做了什么。好，让我们直接看一个例子。我认为我们真的不需要让那个定义太突出，所以我们把那块板擦掉。好，我们来看一个例子。今天有很多例子。这个例子将是一棵博弈树，其中玩家1先行动。玩家2无法观察到这一行动，有时我会在信息集上标注一个2，而不是在这两个节点上分别标注2，这只是为了表明这两个节点都属于玩家2，所以玩家2后行动，我们把玩家1的行动称为上或下，把玩家2的行动称为左或右，带有暗示性地称为左或右。好，这里的信息集是什么？这个信息集表明了玩家2无法观察到玩家1是向上还是向下行动。玩家2无法观察到玩家1选择的是上还是下。为什么这很重要？

[段 14]

I haven’t put the payoffs in yet, but I will in a second. it matters because had this game been a game of perfect information, had this information set, had there been two information sets here, this dotted line not been there, then player two could have chosen separately whether to choose left or right at this node or left or right and left or right at this node But since player two doesn know whether she at the upper node or the lower node she doesn know whether Player 1 chooses up or down she really only has one choice to make here She’s either choosing left at both nodes, or she’s choosing right at both nodes. And just to pull it back to our first example in the class, we saw the same feature there. When we move from a game of perfect information to a game of imperfect information, we reduced the choices available for player two. Here, player two could choose separately, up or down, at these two different nodes. But here, player two only makes one choice that has to apply to both nodes, because player two cannot distinguish those two nodes. All right? So let’s have a look and see, once we put some payoffs on, what it does in this particular game. All right? So here’s some payoffs. 2, 2, minus 1, 3, 3 minus 1, and 0, 0.

[译文 14]

我还没有把收益放进去，但马上会放进去。这很重要，因为如果这个博弈是一个完美信息的博弈，如果这个信息集，如果这里有两个信息集，这根虚线不存在的话，那么玩家二可以分别选择在这个节点选择左或右，以及在那个节点选择左或右。但由于玩家二不知道她是在上方的节点还是下方的节点，她不知道玩家1选择的是上还是下，她实际上在这里只有一个选择要做。她要么在两个节点都选择左，要么在两个节点都选择右。回顾一下我们课堂上的第一个例子，我们在那里看到了同样的特征。当我们从一个完美信息的博弈转变为一个不完美信息的博弈时，我们减少了玩家二的选择。这里，玩家二可以分别在这两个不同的节点选择上或下。但在这里，玩家二只做一个选择，而这个选择必须适用于两个节点，因为玩家二无法区分这两个节点。好吧？让我们来看看，一旦我们把一些收益放上去，在这个特定的博弈中会产生什么结果。好，这里是一些收益。2, 2，-1, 3，3, -1，和0, 0。

[段 15]

Once again, player 2 cannot separately choose at the upper node or the lower node She either choosing left or she choosing right But it turns out that this game is a little easier than the game we started with Why is it easier than the game we started with? It’s easier than the game that we started with because from player 2’s point of view, whether she thinks she’s up here or whether she thinks she’s down here, she has the same best choice in either case. if she thinks she’s at the upper node then by choosing left she’ll get 2 and right she’ll get 3 so right seems better if she thinks she’s at the lower node then choosing left gets minus 1 and right gets 0 so once again right is better so in fact in this particular game regardless of whether player 2 thinks that player 1 shows up and hence she’s at the upper node or whether player 2 thinks that player 1 shows down and hence she’s at the lower node player two is going to make the same choice in this game, namely write. So notice that this particular game actually solves out rather like backward induction. Rather like backward induction. We actually know, even though player two’s choice is a little bit more complicated because she doesn know where she is it actually clear what player two will do at this information set Now if we push this forward a little bit harder we can see why Player one in this game has two strategies up or down and player two has two strategies She either chooses left or right.

[译文 15]

再一次，玩家2无法在上方节点或下方节点分别选择。她要么选择左，要么选择右。但结果表明，这个博弈比我们开始的博弈要简单一些。为什么它比我们开始的博弈更简单？它比我们开始的博弈更简单，因为从玩家2的角度来看，无论她认为自己在这里还是在下面，最好的选择都是一样的。如果她认为自己在上方的节点，那么选择左会得到2，选择右会得到3，所以右似乎更好。如果她认为自己在下方的节点，那么选择左会得到-1，选择右会得到0，所以再次右更好。所以实际上在这个特定的博弈中，无论玩家2认为玩家1选择了上因此她在上方节点，还是玩家2认为玩家1选择了下因此她在下方节点，玩家二在这个博弈中都会做出同样的选择，也就是右。注意，这个特定的博弈实际上像逆向归纳法一样求解。像逆向归纳法一样。我们实际上知道，即使玩家二的选择因为她不知道自己在哪里而稍微复杂一些，但她在这个信息集上会做什么是清楚的。现在，如果我们更进一步推进，我们可以看到为什么在这个博弈中玩家1有两个策略——上或下，而玩家二有两个策略——她要么选择左，要么选择右。

[段 16]

Notice she only has two strategies because she has to choose the same thing at these two nodes. She doesn’t know where she is. All right. Okay. So let’s draw up the matrix for this game and see if it looks familiar. All right, so player one is choosing between up or down, and player two is choosing between left or right, and the payoffs are as follows. Up left is 2, 2. Up right is minus 1, 3. Down left is 3, minus 1. And down right is 0, 0. All right, Down right is 0, 0. So what game is this? It wasn’t meant to be a trick question, so somebody wave their arm in the air. What game is this? Shout it out if you like. This is prisoner’s dilemma, right? This is an old friend of ours. This is prisoner’s dilemma. Again, we saw the very first day. But notice, what have we seen here? This is prisoner’s dilemma that we have seen many, many times. That’s almost unbearably familiar to most of you. Right, now here’s Prisoner’s Dilemma as represented the way in which we talked about games before the midterm. But here is the same game. This is also Prisoner’s Dilemma, but now I’ve drawn in a tree. Here I drew it in a matrix, and here I drew it in a tree. Now that we have information sets, we can represent all the games that we studied before the midterm, all the games that were simultaneous move games, we can study using trees by building information sets.

[译文 16]

注意她只有两个策略，因为她必须在这两个节点上选择同样的东西。她不知道自己在哪里。好吧。好，让我们为这个博弈画出矩阵，看看它是否看起来眼熟。好，玩家1在选择上或下之间选择，玩家2在选择左或右之间选择，收益如下。上左是2, 2。上右是-1, 3。下左是3, -1。下右是0, 0。好，下右是0, 0。这是什么博弈？这不是故意设置的陷阱题，所以有人挥挥手在空中。这是什么博弈？如果你愿意就喊出来。这是囚徒困境，对吧？这是我们的老朋友了。这是囚徒困境。我们第一天就看到过。但注意，我们在这里看到了什么？这是我们见过很多很多次的囚徒困境。这对你们大多数人来说几乎是熟悉得不能再熟悉了。好，现在这是按照我们在期中考试前讨论博弈的方式呈现的囚徒困境。但这里有同样的博弈。这也是囚徒困境，但现在我画成了一棵树。这里我把它画成矩阵形式，这里我画成了树形。现在我们有了信息集，我们就可以表示期中考试前学过的所有博弈，所有同时行动的博弈，我们可以通过构建信息集用树来研究它们。

[段 17]

And what’s the key observation here? It doesn’t really matter whether player one moves first or player two moves first. It doesn’t really matter what’s happening temporally in this game. What matters is information. When player one makes her move, she doesn’t know what player two is going to do. She doesn’t know what two is doing. And when player two makes her move she doesn know what one is doing That a simultaneous move game even if time is passing The key is information not time All right. Now, on the way here, I snuck something in. And I should just tell you what I snuck in. I snuck in what a strategy is. I went from a tree or an extensive form game to a normal form game. And we’ve already done that a couple of times before in the class. We did it with the entry game, for example, about a week ago. But there, all we did was we defined what a strategy was in a game of perfect information. And just to remind you, a strategy in a game of perfect information is a complete plan of action. It tells the player in question what they should do at each of their nodes. But now, we have to be a bit more careful.

[译文 17]

那么关键观察是什么？玩家1先行动还是玩家2先行动其实并不重要。这个博弈中时间上发生了什么并不重要。重要的是信息。当玩家1做出她的行动时，她不知道玩家2要做什么。她不知道2在做什么。当玩家2做出她的行动时，她不知道1在做什么。这是一个同时行动博弈，即使时间在流逝，关键在于信息而不是时间。好，在我来这里的路上，我偷偷加入了一些东西。我应该告诉你们我偷偷加入了什么。我偷偷加入了策略的定义。我从一棵树或扩展式博弈走到了标准式博弈。我们之前在课堂上已经这样做过好几次了。我们用进入博弈这样做过，例如，大约一周前。但在那里，我们只是定义了在一个完美信息博弈中什么是策略。只是提醒你们，在完美信息博弈中，策略是一个完整的行动计划。它告诉相关玩家在每个节点上应该做什么。但现在，我们必须更加小心。

[段 18]

We can’t have a strategy. once we move to imperfect information we can’t have a strategy tell you what to do with each of your nodes because you yourself can’t distinguish between those nodes alright, so we need to adapt our notion, our definition of a strategy to make it appropriate for these more complicated games, so let’s just adapt it in the obvious way definition definition I just define pure strategies for now A pure strategy a pure strategy of player i is a complete plan of action So this is the same as before. It’s a complete plan of action. It’s a complete plan of action. but what does it mean to be a complete plan of action? It can’t tell me what to do at every single node. That can’t be the right definition because I can’t distinguish nodes. So all it can be doing is telling me what to do at each information set. So it specifies what player I should do, should perhaps is the wrong word, let’s just say will do, will do at each of I’s information sets. All right so you go back about a week you see almost exactly the same definition of a strategy but the previous definition told I what to do at each node and this one just tells I what to do at each information center. What I’m doing is tidying up the previous definition so we can apply it to the more interesting games we’re going to look at from now on.

[译文 18]

我们不能有策略，一旦我们进入不完美信息，我们不能有一个策略告诉你用你的每个节点要做什么，因为你自己无法区分那些节点，好，所以我们需要调整我们的概念，调整我们的策略定义，使其适合这些更复杂的博弈，所以让我们用显而易见的方式来调整它，定义，定义我先只定义纯策略。纯策略，玩家i的纯策略是一个完整的行动计划。这和之前一样，是一个完整的行动计划。但作为一个完整的行动计划意味着什么？它不能告诉我每个节点要做什么。这不可能是正确的定义，因为我无法区分节点。所以它能做的只是告诉我每个信息集上要做什么。所以它指定了玩家i在每个i的信息集上会做什么，也许"应该"这个词不对，我们就说"会"做。好，你们回想大约一周前，几乎完全相同的策略定义，但之前的定义告诉i在每个节点上要做什么，而这里只是告诉i在每个信息集上要做什么。我在做的是整理之前的定义，这样我们就可以应用到从现在起要研究的更有趣的博弈上。

[段 19]

All right. So now we have the definition of a strategy. We can carry on the idea. we’ve just seen here. So what’s the idea here? Any game you give me in the form of a tree, I can rewrite the game in the form of a matrix. So let’s see some other examples of that idea. There’s a lot of new ideas today, but some of them are just tidying up and bookkeeping, and some of them are more interesting. All right, so let’s start with a tree. Let’s make it a slightly more interesting tree than the one we’ve seen before. Actually, that’s too interesting. Let’s do it. Let’s start with a tree. Let’s make it a slightly more interesting tree than the one we’ve seen before. Actually, that’s too interesting. Let’s go a little bit slower. So let’s have player one have two choices, and player two have three choices. So here’s a simple tree, and let’s put some payoffs in. But let me just put some letters in for payoffs rather than put in numbers. All right, so we’ll call these actions up and down, and we’ll call this action left, middle, and right, and left, middle, and right. And we’ll call the payoffs A1, A2, B1, B2, C1, C2, D1, D2, E1, E2, and F1, F2. All right, so just to keep track of it.

[译文 19]

好的，现在我们有了策略的定义。我们可以继续推进刚才看到的想法。这个想法是什么？任何你给我的以树的形式呈现的博弈，我都可以把它改写成矩阵的形式。让我们来看看这个想法的其他例子。今天有很多新想法，但其中一些只是整理和记录工作，而另一些则更有趣。好的，让我们从一个树开始。让我们把它做成一个比我们之前看到的稍微有趣一点的树。实际上，那太有趣了。让我们慢一点。所以让玩家一有两个选择，玩家二有三个选择。这里有一个简单的树，让我们放入一些收益。但我先用一些字母来表示收益，而不是数字。好吧，我们把这些动作叫做上和下，把这个动作叫做左、中和右，还有左、中和右。我们把这些收益叫做A1、A2、B1、B2、C1、C2、D1、D2、E1、E2和F1、F2。好吧，只是为了记录。

[段 20]

And I want to show you how we take this tree and turn it into a matrix. Alright? So how do we turn it into a matrix? Well we look and say how many strategies has player 1 got and how many strategies has player 2 got. So player 1 here just has 2 strategies up or down and player 2 has 3 strategies either left middle or right Again they can choose separately at these two nodes so they really just have three choices left middle or right Leave a space here in your notebook, leave a space to the right here, and let’s draw the matrix for this tree down here. All right, so here’s my matrix. Player two is choosing left, middle, or right, and player one is choosing up or down. All right? And the payoffs go in the obvious way. So A1, A2, B1, B2, C1, C2, D1, D2, E1, E2, and F1, F2. All right? So everyone understand that was just a simple exercise to show we can go from an extensive form, a tree, to the normal form, the matrix. All right. However, okay, so that was easy, right? However, there’s an interesting thing here. It isn obvious that this is if I just gave you the matrix it isn obvious that this is the tree from which it came Let me draw another tree that I claim corresponds to that same matrix Here’s another tree.

[译文 20]

我想向你们展示我们如何把这棵树转化成矩阵。明白吗？那么我们如何把它转化成矩阵呢？嗯，我们看一下，玩家一有多少策略，玩家二有多少策略。所以玩家一这里只有两个策略：上或下，玩家二有3个策略：左、中或右。同样，他们可以在这两个节点分别选择，所以他们实际上只有三个选择：左、中或右。在笔记本上留一个空间，右边留一个空间，让我们在这里画一下这棵树的矩阵。好吧，这是我的矩阵。玩家二选择左、中或右，玩家一选择上或下。明白吗？收益以明显的方式填入。所以A1、A2、B1、B2、C1、C2、D1、D2、E1、E2和F1、F2。明白吗？大家都明白这只是一个简单的练习，展示我们可以从扩展式即树，转化到标准式即矩阵。好吧。然而，嗯，这很容易，对吧？然而，这里有一个有趣的现象。如果你只给我这个矩阵，并不明显这就是它来源的那棵树。让我画另一棵树，我声称它对应同一个矩阵。这是另一棵树。

[段 21]

So this other tree, instead of having player 1 move first, it’s going to have player 2 move first. Player 2 better have three choices, and we better call them left, middle, and right. and it better be the case that player 1 is in one big information set, and player 1 only has two choices, which we’ll call up and down. That’s what this matrix is telling us. It’s telling us player 2 had three choices and player 1 had two choices. So that’s true in the matrix I’ve drawn. And let’s be a little bit careful where the payoffs are. So left up, that’s easy, that’s going to be A1, A2. Left down is going to be D1, D2. Middle up is going to be B1, B2. Middle down is going to be E1 E2 Right up is going to be C1 C2 And right down is going to be f1 f2 So I have to be a little bit careful where I put the payoffs but I think that’s right, I just did. And notice that what I did here, I started from this tree, it was an easy operation to construct the matrix, so easy that it was kind of boring, and it’s not that hard to see that I can go the other way and construct this other tree from the matrix. This is also a tree in which player 2 has three strategies and player 1, this is all player 1, has two strategies.

[译文 21]

这另一棵树，不是玩家一先行动，而是让玩家二先行动。玩家二最好有三个选择，我们最好把它们叫做左、中和右，而且玩家一必须在一个大的信息集中，玩家一只有两个选择，我们把它们叫做上和下。这就是这个矩阵告诉我们的。它告诉我们玩家二有三个选择，玩家一有两个选择。所以我画的这个矩阵中这是真的。让我们稍微仔细注意收益在哪里。所以左上，那很简单，就是A1、A2。左下是D1、D2。中上是B1、B2。中下是E1、E2。右上 是C1、C2。右下是F1、F2。所以我必须稍微仔细注意把收益放在哪里，但我认为那是对的，我就这么做了。注意我在这里做的，我从这棵树开始，构建矩阵是一个简单的操作，简单到有点无聊，而且不难看出我可以反过来做，从矩阵构建这另一棵树。这也是一棵玩家二有三个策略的树，而玩家一，这是全部玩家一，有两个策略。

[段 22]

All right, so what? Well, what do we learn from this? All right, well, let’s look at this more carefully. This tree is a tree in which player 1 moved first and player 2 didn’t observe player 1’s choice. Is that right? This is a tree in which player two moved first and player one didn’t observe two’s choice. What are we noticing here? They’re really the same game. There’s no difference between these two games. They’re really the same game. It doesn’t matter whether it’s player one moving first and player two who’s unable to observe one’s choice, or whether it’s player two’s moving first and player one who is unable to observe two’s choice. All that matters is that neither player could observe the other person’s choice before they got to move. They both correspond to exactly the same game. So what’s the message here? The message is something we’ve talked about before in the class, but I’m trying to make it a bit more formal about it. The message is that what matters is information. What matters is information, not time. Not time. Clearly time isn’t an irrelevant thing. I couldn’t know something that hasn’t happened yet. So time is going to have an effect on information. But ultimately what matters is information. What do I know and when did I know it? So the key idea that we trying to capture with these information sets just to repeat is what did the player know and when did they know it That famous expression from the Watergate trials Alright.

[译文 22]

好吧，那又怎样？嗯，我们从这里学到什么？好吧，让我们更仔细地看看这棵树。这是一棵玩家一先行动而玩家二没有观察到玩家一选择的树。对吗？这是一棵玩家二先行动而玩家一没有观察到玩家二选择的树。我们在这里注意到什么？它们实际上是同一个游戏。这两个游戏没有区别。它们实际上是同一个游戏。玩家一先行动而玩家二无法观察到玩家一的选择，或者玩家二先行动而玩家一无法观察到玩家二的选择，这两者之间没有区别。所有重要的只是一个玩家在行动之前无法观察到另一个玩家的选择。它们都对应完全相同的游戏。那么这里的信息是什么？这里的信息是我们之前在课堂上讨论过的，但我正在尝试让它更正式一些。这里重要的是信息重要的是信息，而不是时间。不是时间。当然时间不是无关紧要的事。我不可能知道还没发生的事情。所以时间会对信息产生影响。但归根结底重要的是信息。我知道什么，以及我什么时候知道的。所以我们试图用这些信息集来捕捉的关键想法，只是重复一下，就是玩家知道什么，以及他们什么时候知道的。那个水门事件的著名表达。好吧。

[段 23]

Okay, let’s look at a more interesting example and see if we can actually talk about what’s going to happen in these games. By the end of today, I want to have enough machinery so we can actually start analyzing these games and predicting what’s going to happen. All right. So as we go on, we’ll get more complicated. So let’s get a little bit more complicated now. Once again, here’s a game in which player one is going to have two choices. And we’ll call those choices up or down. It’s getting a familiar theme. And once again, player two is going to move next. And now this time, just to keep things simple, we’ll have player two just have two choices, left or right, or left or right. But now to make things more interesting, let’s have player one move again. So if up right happens then player one gets to move again in which case player one is going to choose up or down And I use a little u and a little d to distinguish it from the ones furthest to the left of the tree Alright, this is a very simple tree. Player 1 moves first, player 2 moves second. I forgot to put a 2 in here. And then if up right has occurred, then player 1 gets to move again. Let’s put some payoffs in.

[译文 23]

好的，让我们看一个更有趣的例子，看看我们是否能真正讨论这些游戏中会发生什么。到今天结束时，我希望有足够的工具来分析这些游戏并预测会发生什么。好的。随着我们继续，会变得更复杂。所以现在让我们让它稍微复杂一点。再一次，这里有一个玩家一会有两个选择的游戏。我们把这些选择叫做上或下。这是一个熟悉的主题。再一次，玩家二接下来要行动。而这次，为了保持简单，我们让玩家二只有两个选择，左或右，或左或右。但现在为了让事情更有趣，让我们让玩家一再次行动。所以如果上右发生了，那么玩家一可以再次行动，在这种情况下玩家一会选择上或下。我用小u和小d来区分它们与树最左边的那些。好吧，这是一棵非常简单的树。玩家一先行动，玩家二后行动。我忘记在这里放个2了。然后如果上右发生了，那么玩家一可以再次行动。让我们放入一些收益。

[段 24]

So let’s have this be 4, 2, 0, 0. 1, 4, 0, 0 again, and 2, 4. All right. Let’s just carry on analyzing this game using exactly the methods we’ve been talking about in the class today so far. So the first thing I’m going to do is I want to turn this into a matrix. I want to turn this into a matrix. And the first thing to do on that route is to try and figure out how many strategies does player 1 have, and how many strategies does player 2 have. And before we even do that, let’s try and figure out how many information sets they have. So I claim that player 2 just has the one information set. Is that right Player 2 just has the one information set But player 1 has two information sets This information set at the beginning of the game and then potentially the second information set further down the tree All right? A strategy must tell the player what to do at each of their information sets. So the strategies for player one are what? Well, one strategy is up and then up again. Another strategy is up and then right. Another strategy is down and then up. And a fourth strategy is down and then right. and notice something which we’ve seen already in this class before there’s a little bit of a redundancy here these two down strategies these two down strategies force the game into a part of the tree where this node will not arise put it less grandly if player one chooses down she knows that she won’t have to make a choice of up or down later on Jake? you mean a little bit instead of a little large?

[译文 24]

所以让我们让这个是4、2、0、0。1、4、0、0再次，还有2、4。好的。让我们继续使用我们今天到目前为止在课堂上讨论的方法来分析这个游戏。所以我要做的第一件事是把它转化成一个矩阵。我想把它转化成一个矩阵。在这条路上的第一件事是尝试找出玩家一有多少策略，玩家二有多少策略。在我们甚至做那之前，让我们尝试找出他们有多少信息集。我声称玩家二只有一个信息集。对吗？玩家二只有一个信息集。但玩家一有两个信息集。这个游戏开始时的信息集，然后可能是树中更下面的第二个信息集。好的。策略必须告诉玩家在他们的每个信息集上要做什么。所以玩家一的策略是什么？嗯，一个策略是上然后再次上。另一个策略是上然后右。第三个策略是下然后上。第四个策略是下然后右。注意我们之前在课堂上已经见过的东西，这里有一点冗余，这两种下策略，这两种下策略把游戏推到了树的一部分，在这个部分这个节点不会出现。不用说得那么宏大，如果玩家一选择下，她知道她之后不需要做出上或下的选择。对吗？与其说得宏大一点不如说得小一点？

[段 25]

If player one chooses down, she knows that she won’t have to make a choice of up or down later on. Jake? You mean little d instead of little r? Ah, thank you. Sorry, thank you. Thanks, Jake. All right, let me start again, since I did the wrong notation. All right, so player one’s choices are up and then up, up and then down, down and then up, and down and then down. Thanks, Jake. Sorry. All right. Now why are the four strategies? It’s a bit of a surprise perhaps, because if player one chooses down, then she knows she will never have to make a choice at her second information set. Nevertheless, nevertheless, we write down everywhere, when we write down a strategy, we have to write down an instruction for every single information set, so we include both of those strategies. Strategies for player two here are a little bit easier. Strategies for player two are just left left or right. All right. With that in mind, let’s draw up the matrix So player one here has four strategies and they are up up up down down up and down down And player two has two strategies, and they are left or right. All right, everyone okay so far? We’re just basically transferring things across. And now we have to transfer the payoffs across. so up, up followed by left is going to be 4, 2.

[译文 25]

如果玩家一选择下，她知道她之后就不必在上或下之间做出选择了。杰克？你说的是小d而不是小r？啊，谢谢。抱歉，谢谢。谢谢，杰克。好，让我重新开始，因为我用了错误的记号。好，所以玩家一的选项是上然后上，上然后下，下然后上，和下然后下。谢谢，杰克。抱歉。好，为什么有四个策略？这可能有点令人惊讶，因为如果玩家一选择下，那么她知道她永远不必在她的第二个信息集上做出选择。尽管如此，尽管如此，我们在写策略时，必须为每一个信息集写出指令，所以我们包含这两个策略。玩家二的策略在这里要简单一些。玩家二的策略只是左左或右右。好，考虑到这一点，让我们来列出矩阵。所以玩家一在这里有四个策略，它们是上上、上下、下上、上下和下下。玩家二有两个策略，它们是左或右。好，大家到目前为止都清楚吗？我们基本上是在把东西转移过去。现在我们要把收益转移过去。所以跟上然后左就是4、2。

[段 26]

Up, up followed by right is going to be 0. So up, up, right. It’s very easy to think of it as up, right, up. So up, right, up is 0, 0. Up, down, left is the same as up, left, down. So it’s again 4, 2. 2 Up down right is going to be up right down So it going to be 1 4 Down up left is the same as saying down left So it going to be 0 0 Down up right is going to be 2 4 Down down left is once again going to be 0 0 And down down right is once again going to be 2 4 All right does everyone see how I got the payoffs I just used those strategies to tell me which way I going through the tree If I combine them, it gives me an entire path and gets me to an end node. All right, and you can see this redundancy we talked about. We pointed out that these things are kind of the same thing, And you can see in the matrix that the bottom four squares of the matrix have repetition. This row is the same as that row. Everyone happy with that? Okay, we have a matrix. Let’s analyze it by finding the Nash equilibria in this game. So to find the Nash equilibria in this game, we’re going to find best responses.

[译文 26]

跟上下然后右就是0。所以上上下。把它想成上右下很容易。所以上右下是0、0。上下左与上左右相同。所以又是4、2。上下右就是上右下。所以是1、4。下上左就是说下左。所以是0、0。下上右就是2、4。下下左又是0、0。下下右又是2、4。好，大家都看到我是怎么得到这些收益的吗？我只是用这些策略来告诉我走哪条路穿过这棵树。如果我把它们结合起来，就给了我一条完整的路径，把我带到终端节点。好，你可以看到我们讨论过的这种冗余。我们指出这些东西其实是同一种东西，你可以在矩阵中看到矩阵的底部四个方格有重复。这一行和那一行是相同的。大家都满意吗？好，我们有了一个矩阵。让我们通过找出这个游戏中的纳什均衡来分析它。所以要找出这个游戏中的纳什均衡，我们要找出最佳反应。

[段 27]

So let’s start by asking, what is the best response to left? So if player two chooses left, player one’s best response is either up-up or up-down. If player two chooses right, then player one’s best response is either down-up or down-down. All right, we’re okay so far? All right. If player one chooses up then player two is going to choose left If player one chooses up then player two best response is to choose right If player one chooses down then player two’s best response is to choose right. And if player one chooses down-down, then player two’s best response is to choose right. Alright, so this is kind of slow, and I just want to be careful. I’m going slow for a reason, we’re going to gradually get harder. I want to be a little bit careful, I can see people looking a little sleepy around the room, I know it’s lunchtime. If you see your neighbor getting sleepy give them a good sharp elbow because I think this is this isn’t a good time to fall asleep in some sense I’m worried you’re gonna miss something and it’s then going to get harder and you’re gonna miss things all right so what are the Nash equilibria in this game we know how to do that the Nash equilibria must be up up followed by left try get them all down up followed by right and down down followed by right I want this three Nash equilibrium okay so it wasn’t so much a big deal I got three equilibrium in this game and if I had simply given you this game in the three Nash equilibria.

[译文 27]

所以让我们从问什么是最佳反应开始？如果玩家二选择左，玩家一的最佳反应是上上或上下。如果玩家二选择右，玩家一的最佳反应是下上或下下。好，我们到目前为止都没问题吧？好。如果玩家一选择上，那么玩家二会选择左。如果玩家一选择上，那么玩家二的最佳反应是选择右。如果玩家一选择下，那么玩家二的最佳反应是选择右。如果玩家一选择下下，那么玩家二的最佳反应是选择右。好，这有点慢，我只想小心一点。我放慢速度是有原因的，我们会逐渐变难。我想要稍微小心一点，我看到房间里有些人看起来有点困了，我知道现在是午餐时间。如果看到旁边的同学犯困，给他们一个狠狠的手臂肘击，因为我认为现在不是睡觉的好时机，我担心你们会错过一些东西，然后它会变得更难，你们就会一直错过东西。好，这个游戏中的纳什均衡是什么，我们知道怎么做，纳什均衡必须是上下然后左，试着把它们全部找出来，下上然后右和下下然后右。我想要这三个纳什均衡。好，所以这没什么大不了的，我在这个游戏中得到了三个均衡，如果我只是在学期前半部分给你们这个游戏，我没有给你们看这棵树，你们从未见过这棵树，我只是给你们这个游戏，说找出这个游戏中的纳什均衡，那会是期中考的一道题，我们就会停在这里。我们会说，好，我找到了这些纳什均衡，也许你们会继续找出混合均衡，我不知道，但基本上我们在这个时候就完成了。

[段 28]

Okay, so it wasn’t such a big deal. I got three equilibria in this game, and if I’d simply given you this game in the first half of the semester, I hadn’t shown you the tree, you’ve never seen this tree, I just gave you this game, and said find the Nash equilibria in this game, and that would have been a question on the midterm, we’d have stopped here. We’d have said, okay, I found these Nash equilibria, maybe you’d have gone on and found mixed ones, I don’t know, but essentially we’d be done at this point. Let’s say again, if we’d started as we would have done before the midterm with me giving you a payoff matrix and asking you to find the Nash Equilibria, then at this point we’d be done. We’d have found the three Nash Equilibria, the three pure strategy Nash Equilibria. The problem is, if we go back to the tree, to the dynamic game, the game that has some action going on in it, and actually look at this game, it’s not clear that all of these Nash equilibria are really equally plausible. Can anyone see what might be a bit implausible about some of these Nash equilibria? What’s implausible about them? Any takers on this Well let look at this game again This game is a little bit complicated It not clear what one should do here perhaps And perhaps it’s not clear what player two should do here, because after all, player two doesn’t know where he is, and he doesn’t know whether player one, if player one gets to move again, is going to choose up or down.

[译文 28]

好，所以这不是什么大不了的事。我在这个游戏中得到了三个均衡，如果我在学期前半部分给你们这个游戏，我没有给你们看这棵树，你们从未见过这棵树，我只是给你们这个游戏，说找出这个游戏中的纳什均衡，那会是期中考的一道题，我们就会停在这里。我们会说，好，我找到了这些纳什均衡，也许你们会继续找出混合均衡，我不知道，但基本上我们在这个时候就完成了。再说一次，如果我们是按照期中考之前我们会做的那样开始的，我给你们一个收益矩阵，问你们找出纳什均衡，那么在这个时候我们就完成了。我们已经找到了三个纳什均衡，三个纯策略纳什均衡。问题是，如果我们回到这棵树，回到动态游戏，这个游戏中有一些行动在进行，实际上看看这个游戏，并不清楚所有这些纳什均衡都是同样合理的。有谁能看出这些纳什均衡中哪些可能有点不合理吗？它们有什么不合理的？有人要回答吗？好，让我们再看看这个游戏。这个游戏有点复杂，也许不清楚应该怎么做。也许不清楚玩家二应该怎么做，因为毕竟，玩家二不知道他在哪里，他也不知道玩家一，如果玩家一还有机会再行动，会选择上还是下。

[段 29]

But, watch the but. Can we get a mic on Patrick? So if you look at it backwards, you can cross out player one’s second choice. He’s always going to choose down, so that’s 1, 4 at that node. So then you know player two is always going to choose right, because his payoff is always 4. So then player one knows which to choose then. He’s going to choose down. Good, good. So let’s just walk through what Patrick just said. That’s very good. So if we just analyze this game the way we’ve been taught to analyze trees, essentially using backward induction, we first of all observe that if player 1 gets to move again here, she’ll know where she is, and she’ll know she’s choosing between 1 and 0, she’s going to choose down. Is that right? She’s going to choose down. But knowing this player 2 even though player 2 doesn know where he is player 2 actually has a pretty easy choice to make He knows that if he chooses left he either gets 2 or 0 but if he chooses right, he gets 4. 4 is bigger than 2, 4 is bigger than 0, so Player 2 is actually going to choose right. And given that, given that Player 2 is going to choose right, player one is essentially choosing between one, if she chooses up, which would be followed by right and down, and two, which would be what happens if she chooses down, followed by right.

[译文 29]

但是，注意这个但是。我们能把麦克风给帕特里克吗？所以如果你倒过来看，你可以划掉玩家一的第二次选择。他总是会选择下，所以那个节点的收益是1、4。所以然后你知道玩家二总是会选择右，因为他的收益总是4。所以然后玩家一知道该选择什么了。他会选择下。好，好。所以让我们来走一遍帕特里克刚才说的。那非常好。所以如果我们用我们被教导的分析树的方法来分析这个游戏，本质上使用逆向归纳法，我们首先观察到如果玩家一有机会在这里再行动，她会知道她在哪里，她会知道她在1和0之间选择，她会选择下。对吗？她会选择下。但知道这一点，玩家二即使不知道他在哪里，玩家二实际上有一个相当容易的选择要做。他知道如果他选择左，他要么得到2要么得到0，但如果他选择右，他得到4。4大于2，4大于0，所以玩家二实际上会选择右。既然如此，既然玩家二会选择右，玩家一本质上是在1和2之间选择，如果她选择上，那就是1，会跟着右和下，还有2，那就是如果她选择下，跟着右会发生什么。

[段 30]

So this game we can essentially analyze through backward induction. It’s not quite backward induction, because we had to add in this little piece about two not knowing where she was, but it turned out no matter where she was, she had a dominant strategy, She had a better strategy once she figures out that player one is going to choose down. Is that right? If we go back and look at these Nash equilibria, the prediction that we just got, which is what? Down for player one, right for player two, and then down again for player one. That strategy is this one So one of these Nash equilibria corresponds to our sensible analysis of this tree But the other two do not These two Nash equilibria are inconsistent with backward induction. They’re inconsistent with backward induction. They’re perfectly good Nash equilibria. If we’d given you this matrix at the midterm, you’d have thought they’re just fine, but it turns out both of these Nash equilibria involve player one choosing a strategy up that we know that player one is not going to do if reached. And one of these Nash equilibria involves player two choosing a strategy left that in fact she’s only choosing because she thinks player one is going to choose up, which in fact we’ve just argued player one is not going to do. The people at the back, there’s a little bit too much volume bouncing off the wall there, so just keep it down on the balcony.

[译文 30]

所以这个游戏我们可以用逆向归纳法来分析。这不完全是逆向归纳法，因为我们不得不加入这一小段关于玩家二不知道她在哪里，但结果表明无论她在哪，她都有一个占优策略，一旦她弄清楚玩家一会选择下，她有一个更好的策略。对吗？如果我们回去看看这些纳什均衡，我们刚刚得到的预测是什么？玩家一下，玩家二右，然后玩家一再下。那个策略是这个。所以这些纳什均衡中有一个对应于我们对这棵树的合理分析。但另外两个不是。这两个纳什均衡与逆向归纳法不一致。它们与逆向归纳法不一致。它们是完美的纳什均衡。如果我们把那个矩阵放在期中考上给你们，你们会认为它们很好，但结果表明这两个纳什均衡都涉及玩家一选择一个策略上，我们知道玩家一如果到了那个节点是不会这样做的。其中一个纳什均衡涉及玩家二选择一个策略左，事实上她之所以选择只是因为她认为玩家一会选择上，而事实上我们刚刚论证了玩家一不会这样做。后面的人，音量有点太大了，声音在墙上反射，所以请在阳台上保持小声。

[段 31]

Thank you. All right? So these two Nash equilibria, they’re perfectly good Nash equilibria of the game, but they don’t make any sense. They’re completely inconsistent with the way we’ve learned to talk about games. All right? They’re perfectly good Nash equilibria of the game, but they don’t make any sense. They’re completely inconsistent with the way we’ve learnt to talk about games. Now we’ve seen this before. We saw it on the entry game. This is a much more complicated, much more interesting example. But we saw in the entry game, when there was one entrant entering into a market, that in that game there were actually two Nash equilibria, and one of them, we argued, was incredible. Here it’s a bit more complicated, but nevertheless, these two equilibria seem like bogus equilibria or phony equilibria or equilibria we wouldn’t really believe in and the reason we don’t believe in them is that they don’t correspond to backward induction and our common sense intuitions about backward induction. So we need some new notion. The aim of the class has been what? We want to be able to model games that have both sequential moves and simultaneous moves and we want to be able to look at the games and use our techniques from both halves of the class. We want to be able to use the idea of Nash equilibrium from the first half of the class and we want to be able to use the idea of backward induction from the second half of the class.

[译文 31]

谢谢。好吧？所以这两个纳什均衡，它们是这个博弈的完全有效的纳什均衡，但它们没有任何意义。它们完全与我们学习讨论博弈的方式不一致。好吧？它们是这个博弈的完全有效的纳什均衡，但它们没有任何意义。它们完全与我们学习讨论博弈的方式不一致。我们之前见过这种情况。我们在进入博弈中见过。这是个更复杂、更有趣的例子。但我们在进入博弈中看到，当有一个进入者进入市场时，那个博弈实际上有两个纳什均衡，其中一个，我们认为，是不可信的。这里更复杂一些，但尽管如此，这两个均衡看起来像是虚假的均衡或伪造的均衡或我们不会真正相信的均衡，我们不相信它们的原因是它们不符合逆向归纳和我们对逆向归纳的常识直觉。所以我们需要一个新的概念。这门课的目标是什么？我们想要能够建立同时包含顺序行动和同时行动的博弈模型，我们想要能够分析这些博弈并使用我们两半课程中的技术。我们想要能够使用第一半课程中的纳什均衡概念，我们想要能够使用第二半课程中的逆向归纳概念。

[段 32]

But what we learning here is that Nash equilibrium if we just take the notion of Nash equilibrium and plonk it down on these sequential move games it will produce equilibria that don make any sense So we need a more refined notion of equilibrium, a better notion of equilibrium, than just Nash equilibrium to deal with these settings where we have both simultaneity and sequential moves. We have both some perfect information and some imperfect information. alright that was one example let me give you a second example if that example wasn’t yet convincing let me leave that example up so far we’ve seen that Nash equilibrium gets us into trouble in these games and we’ve seen it got us into trouble because it basically conflicted with our backward induction intuitions. Now I’m going to show you a different game and we’re going to see it again and actually equilibrium is going to get us into trouble. All right, this is going to be a three-player game. We’ll get more complicated as we go along So another example this time with three players So as an example to get harder I need to be more alert to see if you can follow through Alright so this is a more complicated tree Here’s a tree in which player one moves first and chooses between A or B. And if player one chooses A, the game is over.

[译文 32]

但我们在这里学到的是，如果我们只是把纳什均衡的概念直接套在这些顺序行动博弈上，它会产生没有任何意义的均衡。所以我们需要比纳什均衡更精炼的均衡概念、更好的均衡概念，来处理这些同时包含同时行动和顺序行动的情形。我们同时有完美信息和不完美信息。好的，这是一个例子，让我再给你们一个例子，如果那个例子还不够有说服力的话。让我把那个例子保留在屏幕上到目前为止，我们已经看到纳什均衡在这些博弈中给我们带来麻烦，我们已经看到它给我们带来麻烦是因为它基本上与我们的逆向归纳直觉相冲突。现在我要给你们展示一个不同的博弈，我们会再次看到它的问题，实际上均衡会再次给我们带来麻烦。好的，这将是一个三人博弈。我们会越来越复杂。所以另一个例子，这次有三位玩家。所以作为例子增加难度，我需要更加警觉，看你们能否跟上。好吧，这是个更复杂的树。这是一棵树，其中玩家一先行动，在A或B之间选择。如果玩家一选择A，博弈就结束了。

[段 33]

She gets one, and the other two players get nothing. if she chooses B then players 2 and 3 get to play a little game down here in which 2 moves first in this little sub game and 3 moves second and the payoffs in this sub game are as follows again using player 1’s payoff first so there’s 0, 1, 1 0, 0, 2 0, 0, minus 1, and 2, 1, 0. All right? So this is quite a complicated game. It’s got three players for a start, so it’s going to be a little bit hard to draw it up in a matrix. But nevertheless let me try and do that So I claim that we can model this game as follows It a game in which player 1 is choosing which matrix Let’s call this matrix A and matrix B. Player 1 is choosing the matrix. Player 2 is choosing, let’s call them up and down. Player 2 is choosing up or down. And player 3 is choosing left or right. And notice in this game, players 2 and 3 actually can observe the choice of A or B to start with. So let’s try and put in the payoffs in the correct places. It’s not always easy to do, but let’s try. So A is easy. If player 1 chooses A, then the payoffs in this matrix are somewhat trivial.

[译文 33]

她得到1，其他两位玩家什么都得不到。如果她选择B，那么玩家2和玩家3在这里玩一个小博弈，其中2在这个小子博弈中先行动，3后行动。这个子博弈中的收益如下，同样先给出玩家1的收益：有0, 1, 1；0, 0, 2；0, 0, 负1；和2, 1, 0。好吧？这是一个相当复杂的博弈。它有三个玩家，所以用矩阵来画会稍微困难一些。但尽管如此让我试着做一下。我声称我们可以把这个博弈建模如下：这是一个玩家1选择哪个矩阵的博弈。让我们把这个叫做矩阵A和矩阵B。玩家1选择矩阵。玩家2选择，让我们称它们为上和下。玩家2选择上或下。玩家3选择左或右。注意在这个博弈中，玩家2和玩家3实际上可以先观察到A或B的选择。所以让我们试着把收益放在正确的位置上。这并不总是容易的，但让我们试试。A很简单。如果玩家1选择A，那么这个矩阵中的收益有点平凡。

[段 34]

Because if player 1 chooses A, whatever anyone else does, the payoff is 1, 0, 0. All right? All right? the payoff is 1, 0, 0. All right? So somewhat uninteresting matrix over there. But if player 1 chooses B, then life gets more interesting. Then if player 2 chooses up and player 3 chooses left, we end up here. So that’s 0, 1, 1. if player 2 chooses this is 2 and this is 3 this is 2 and this is 3 if player 2 chooses up and player 3 chooses right then we’re at 0, 0, 2 so this is 0, 0, 2 going in here if player 2 chooses down and player 3 chooses left then we’re at 0, 0, minus 1. Everyone okay with that? And if player 2 chooses down and player 3 chooses down, then we’re down here, which is 2, 1, 0. All right Okay so here a little game Player 1 is choosing the matrix Player 2 is choosing the row in the matrix albeit trivially on the left side. And Player 3 is choosing the column in the matrix, again, albeit trivially on the left-hand side.

[译文 34]

因为如果玩家1选择A，不管其他人怎么做，收益是1, 0, 0。好吧？好吧？收益是1, 0, 0。好吧？所以那边这个矩阵有点无趣。但如果玩家1选择B，生活就更有趣了。然后如果玩家2选上，玩家3选左，我们到达这里。所以那是0, 1, 1。如果玩家2选上，玩家3选右，那么我们得到0, 0, 2，所以这是0, 0, 2写在这里。如果玩家2选下，玩家3选左，那么我们得到0, 0, 负1。大家都同意吗？如果玩家2选下，玩家3选下，那么我们到达这里，也就是2, 1, 0。好吧。所以这里是一个小博弈。玩家1选择矩阵。玩家2选择在矩阵中选择行，虽然在左边那边这很平凡。玩家3选择在矩阵中选择列，同样，虽然在左边那边这很平凡。

[段 35]

We don’t really care about this picture very much. okay so now what well once again we could look for Nash equilibrium in this game it turns out there are lots of Nash equilibrium in this game let me just show you one Nash equilibrium and then we’ll talk about it so I claim that there are lots of Nash equilibrium and one of them one of them is the Nash equilibrium a up left a up left so let’s see what that is in the tree first of all so player one chose a player two to top and left but it followed a so we end up here We end up at 1, 0, 0. a up left is this box in the tree Now let just check that that actually is a Nash equilibrium So we all know how to do this from the first half of the class To check that that’s a Nash equilibrium, we better check that no individual player can do better by deviating. So let’s start with player 1. If player 1 deviates, holding player 2 and 3 fixed, then player 1 will be switching the matrix from matrix A to matrix B. Is that correct? So we’ll move from this box in the left-hand matrix to the equivalent box in the right-hand matrix. And player 1’s payoff will go from 1 to 0.

[译文 35]

我们不太关心这张图。好吧，那么现在我们能做什么呢？又一次，我们可以在这个博弈中寻找纳什均衡。事实证明这个博弈有很多纳什均衡。让我给你们看一个纳什均衡，然后我们会讨论它。所以我声称这个博弈有很多纳什均衡，其中一个其中之一是一个纳什均衡a up left。所以让我们先在树中看看那是什么。所以玩家一选了a，玩家二选了top和left，但它遵循了a，所以我们到达这里。我们到达1, 0, 0。a up left是树中这个格子。现在让我们检查一下这实际上是否是一个纳什均衡。我们都知道怎么做，从课程的前半部分。要检查这是一个纳什均衡，我们最好检查没有单个玩家能通过偏离做得更好。所以让我们从玩家1开始。如果玩家1偏离，在玩家2和3保持不变的情况下，那么玩家1会把矩阵从矩阵A切换到矩阵B。对吗？所以我们会从左边矩阵的这个格子移动到右边矩阵中等价的格子。玩家1的收益会从1变成0。

[段 36]

From 1 to 0. So player 1 doesn’t want to deviate. Everyone happy with that? Player 1 doesn’t want to deviate here. How about player 2? If player 2 deviates, holding players 1 and 3 fixed, then player 1 is going to switch rows in this matrix, so we’ll move from this box to this box. Player 2 was making 0 before, she’s still making 0, so she has no incentive to deviate. And the same argument applies for player 3 because she will be choosing the column holding the row and the matrix fixed so once again she gets zero in either case All right so everyone happy with that So that actually is a Nash equilibrium And again if this had been the midterm I could have set this up I could have given you these matrices, or the story behind them, and I could have asked you whether this was a Nash equilibrium, and the answer would have been yes. But I claim that once again, this is just not a believable Nash equilibrium. It is a Nash equilibrium. Formally it’s an absolute equilibrium, but it’s not a plausible prediction for how this game’s going to be played. Why is it not a plausible prediction for how this game’s going to be played? Come on, see. Stare at the tree a bit. So in the information here, the pre-midterm information, it’s fine.

[译文 36]

从1变成0。所以玩家1不想偏离。大家都同意吗？玩家1在这里不想偏离。玩家2怎么样？如果玩家2偏离，在玩家1和3保持不变的情况下，那么玩家1会在这矩阵中切换行，所以我们会从这个格子移动到这个格子。玩家2之前得到0，她现在仍然得到0，所以她没有偏离的动机。对于玩家3同样的论证也适用，因为她会在行和矩阵保持不变的情况下选择列，所以再一次她在任何情况下都得到0。好吧，所以大家都同意吗？所以这实际上是一个纳什均衡。同样，如果这是期中考的话，我可以这样设置。我可以给你们这些矩阵，或者它们背后的故事，我可以问你们这是否是一个纳什均衡，答案会是是的。但我声称，再一次，这只是一个不可信的纳什均衡。它是一个纳什均衡。正式地它是一个绝对均衡，但它不是对这个博弈将如何进行的可信预测。为什么它不是对这个博弈将如何进行的可信预测？来吧，看看。盯着树看一点。所以在这里的信息中，这是期中考前的信息，这没问题。

[段 37]

But knowing about the actual structure of this game, I claim this makes no sense at all. Why does it make no sense? Well, notice that if player 1 were to switch her action from the prescribed action A to action B, then we’d be here. And notice that the tree from here on in looks like a little game. Is that right? Then we’d be here, all right? And notice that the tree from here on in looks like a little game. Is that right? The tree from here on looks like a little game. All right, so this thing here, let’s put it in green, this thing here is a little game within the game. It’s a sub-game, all right? And this sub-game really only involves two players. the two players that it involves are players 2 and player 3. Player 1’s done. Player 1’s put us into this game. But now in this little sub-game, it’s a little sub-game involving just player 2 and player 3. So we can analyze this little sub-game. If we analyze this little sub-game, what will it give us? What will we find? All right? So let’s look at this sub-game. So look at the green, the green sub game. The game that would have happened had player one chosen B. This is a sub game involving just players two and three, so why don’t they just forget player one?

[译文 37]

但了解了这款游戏的实际结构，我声称这完全毫无意义。为什么毫无意义？嗯，注意如果玩家1把她规定的行动从A改为B，那我们就会在这里。注意从这往后的树看起来像一个小游戏。对吗？那么我们就在这里，对吧？注意从这往后的树看起来像一个小游戏。对吗？这里的树看起来像一个小游戏。好吧，所以这个东西，我们把它标成绿色吧，这个东西是游戏中的一个小游戏。它是一个子游戏，对吧？这个子游戏实际上只涉及两个玩家。它涉及的这两个玩家是玩家2和玩家3。玩家1已经完成了。玩家1把我们带入了这个游戏。但现在在这个小子游戏中，这是一个只涉及玩家2和玩家3的小子游戏。所以我们可以分析这个小子游戏。如果我们分析这个小子游戏，它会给我们什么？我们会发现什么？好吧？让我们看看这个子游戏。看这个绿色的，这个绿色的子游戏。如果玩家1选择了B，那么这个游戏就会发生。这是一个只涉及玩家2和玩家3的子游戏，那他们为什么不直接忽略玩家1呢？

[段 38]

Alright we know what I mean player one is part of the game he getting payoffs but player one has made their move they not really involved anymore So let just look at this game as a game involving players 2 and 3 and let look at the matrix for players 2 and 3 So this is the, actually it corresponds to the matrix above. It’s a matrix in which player 2 is choosing up and down. Here it is, up and down. And simultaneously, player 3 is choosing left or right. Here it is, left or right, at this information set. And the payoffs are 1, 1, 0, 2, 0, minus 1, and 1, 0. All right. All right. So this is, I claim, a representation of this little green game. Perhaps we should put this in green as well. Alright, this thing corresponds to that thing. Alright, everyone okay with that? Alright so if player one had chosen B rather than A then we be involved in a little game a game within a game or a sub involving just players 2 or 3 and we can analyze that game That a straightforward game Here it is And what would we do with that game? We’d look for the Nash equilibrium in that game. So let’s look for the Nash equilibrium in this game. So what do we notice about this game?

[译文 38]

好的，我们知道我的意思了，玩家1是这个游戏的一部分，他获得收益，但玩家1已经做出了选择，他不再真正参与了。所以让我们把这个游戏看作一个涉及玩家2和玩家3的游戏，让我们看看玩家2和玩家3的收益矩阵。这是，实际上它对应上面的矩阵。这是一个矩阵，其中玩家2在选择上和下。在这里，是上和下。同时，玩家3在选择左或右。在这里，是左或右，在这个信息集上。收益分别是1、1、0、2、0、负1和1、0。好的。好的。所以这个，我声称，是这个绿色小游戏的表现。也许我们也应该把它标成绿色。好吧，这个东西对应那个东西。大家都明白吗？好吧，所以如果玩家1选择了B而不是A，那么我们就会陷入一个小游戏，一个游戏中的游戏，或者说一个只涉及玩家2或3的子游戏，我们可以分析那个游戏。这是一个直接的游戏。就是这个。那么我们该如何处理这个游戏呢？我们会寻找那个游戏中的纳什均衡。所以让我们寻找这个游戏中的纳什均衡。这个游戏有什么特点呢？

[段 39]

So if player 3 chooses left, then player 2 would rather choose up. If player 3 chooses right, then player 2 should choose down. If player 2 chooses up, then player 3 would rather choose right, because 2 is bigger than 1. And if player 2 were to choose down, then player 3 would choose right again, because 0 is bigger than minus 1. So in fact, in this little sub-game, in this little sub-game, actually, player 3 has a dominant strategy. If it turned out that we got involved in this little sub-game, player 3 has a dominant strategy, which is to play right. And moreover, this sub-game has just one Nash equilibrium. If I given you this sub on its own it clear that the Nash equilibrium of this sub or this game within a game is down right It down right So what’s that telling us? It’s telling us if player two and three ever get called upon to play in this game, and that only happens when player one chooses B, if player two and three ever get called upon to play in this game, we know from when we were young, or at least from before the midterm, we know that they’re going to play Nash equilibrium in that sub game, and the Nash equilibrium in the sub game is going to have player three choosing right, and player two choosing down.

[译文 39]

所以如果玩家3选择左，那么玩家2宁愿选择上。如果玩家3选择右，那么玩家2应该选择下。如果玩家2选择上，那么玩家3宁愿选择右，因为2大于1。如果玩家2选择下，那么玩家3会再次选择右，因为0大于负1。所以实际上，在这个小子游戏中，在这个小子游戏中，实际上，玩家3有一个优势策略。如果最终我们陷入这个小子游戏，玩家3有一个优势策略，就是选择右。而且，这个子游戏只有一个纳什均衡。如果把这个子游戏单独给你，很明显，这个子游戏或这个游戏中的游戏的纳什均衡是下、右。那么这告诉我们什么？它告诉我们，如果玩家2和玩家3被叫来玩这个游戏，而这只在玩家1选择B时才会发生，如果玩家2和玩家3被叫来玩这个游戏，我们从我们年轻时，或者至少在期中考试之前，我们知道他们会在那个子游戏中玩纳什均衡，而子游戏中的纳什均衡将是玩家3选择右，玩家2选择下。

[段 40]

All right? But the equilibrium we talked about, this equilibrium we argued before about, A, U, L, the equilibrium we talked about before, A, U, L, doesn’t involve player two choosing down. In fact, she chose up. And it doesn’t involve player three choosing right. In fact, she chose left. All right? So let’s sum up. We’ve found that the equation Goal player three choosing right. In fact, she chose left. All right, so let’s sum up. We found an equilibrium of this game. This equilibrium of this game was A, U, L. But I claim this is not a plausible equilibrium. It’s not a plausible equilibrium because it predicts that if we actually were to play the game within the game, we wouldn’t play equilibrium. Let me say it again. In the whole game, in the whole game, A up left is an equilibrium. But I claim it’s a silly equilibrium because it involves the prediction that if in fact we ever got into the game within the game, we would no longer play equilibrium. And that doesn’t seem right. If we’re going to believe in equilibrium, we should be consistent and believe in equilibrium throughout. out. So this brings us to a new idea. And the new idea is going to have two parts to it. The first part is kind of on the board already It something we talk about informally It the notion of a sub It the notion of a sub What a sub It a game within a game I’ve been using that informally, but we need to start thinking about more formally what it means.

[译文 40]

好的？但我们谈论的均衡，我们之前讨论的均衡，A、U、L，我们之前谈论的均衡，A、U、L，不涉及玩家2选择下。实际上，她选择了上。它也不涉及玩家3选择右。实际上，她选择了左。好的？让我们总结一下。我们发现玩家3选择右。实际上，她选择了左。好的，所以让我们总结一下。我们发现了这个游戏的一个均衡。这个游戏的这个均衡是A、U、L。但我声称这不是一个合理的均衡。它不是一个合理的均衡，因为它预测如果我们真的去玩游戏中的游戏，我们就不会玩均衡。让我再说一遍。在整个游戏中，在整个游戏中，A上左是一个均衡。但我声称这是一个愚蠢的均衡，因为它涉及一个预测：如果我们真的陷入游戏中的游戏，我们就不再玩均衡了。这似乎不对。如果我们相信均衡，我们应该保持一致，自始至终都相信均衡。所以这把我们带到了一个新的想法。这个新想法将有两部分。第一部分已经在黑板上了，它是我们之前非正式讨论过的东西，它是子游戏的概念，它是子游戏的概念。什么是子游戏？它是游戏中的游戏。我一直在非正式地使用这个概念，但现在我们需要开始更正式地思考它的含义。

[段 41]

So I talked about it informally, I said that green object is the game that would be played were player one to choose B. And we talked about other sub-games in this class. We talked about the sub-game that would happen in the entry game if one of those rival pizza companies moved in in the Miami market or something. It was a game within a game. When we talked about the Tour de France, we talked about there being a game within a game that is about when you break away. But now I want to be formal about this notion of a game within a game and introduce some nomenclature. So the formal definition is this. Definition a sub is a part of a game informally that looks like a game within the tree And it has three properties. It satisfies the following three properties. So one, since it looks like a game itself, the sub-game must start from a particular point. So it starts, the sub-game must start, it starts from a single node. And let’s just look at the example. In the example we just looked at, the sub-game started from this node here. second, second, it comprises, it comprises all successors, successors to that node So in our example here our sub game here our green sub game here’s the node it starts from, here are all the nodes that are successors of that node, these are the children these are the grandchildren, right, you have this grandparent node you have to have all of his children and all of his grandchildren. so it comprises all the successes of that node and finally, and this is important it does not it does not break up it does not break up any information sets It does not break up any information sets.

[译文 41]

所以我非正式地讨论过它，我说那个绿色的东西就是如果玩家1选择B将会发生的游戏。我们在这个课程中讨论过其他子游戏。我们讨论过如果其中一个竞争对手披萨公司进入迈阿密市场，在进入游戏中会发生什么子游戏。这是一个游戏中的游戏。当我们讨论环法自行车赛时，我们讨论过有一个关于何时突围的游戏中的游戏。但现在我想对游戏中的游戏这个概念进行正式的定义，并引入一些术语。所以正式定义是这样的。定义：一个子游戏是游戏树中的一部分，形式上看起来像一个游戏，它满足以下三个属性。第一，因为它本身看起来像一个游戏，子游戏必须从一个特定的点开始。所以它开始，子游戏必须开始，它从一个单独的节点开始。让我们看看例子。在我们刚才看的例子中，子游戏从这里这个节点开始。第二，第二，它包括，它包括该节点的所有后继者。所以我们这里的例子中，我们的子游戏在这里，我们的绿色子游戏这里是它开始的节点，这里是那个节点的所有后继节点，这些是子节点，这些是孙节点，对吧，你有这个祖节点，你必须拥有他所有的孩子和所有的孙子。所以它包括那个节点的所有后继者。最后，也是重要的，它不，它不会打破任何信息集。它不会打破任何信息集。

[段 42]

So a subgame, informally, it’s just a little game within the game. But slightly more formally, I can’t put one node that’s part of an information set into this subgame, unless I’m going to put all the nodes that are part of that information set into the subgame. Let’s have a look at some examples. We’ve got one example. The subgame, you’re going to put all the nodes that are part of that information set into the subgame. Let’s have a look at some examples. We’ve got one example up there. The entry game looks something like this. The entry game looks something like this. So what are the subgames here? No secrets here. This is a subgame. There’s actually another subgame. Do you want to see what the other subgame is? The whole game is a sub-game. The whole game is itself a sub-game. Somewhat trivially. So this particular game, which is the schematic of the entry game, it has actually two sub-games, but only one proper sub-game. Here’s a more complicated example. This is actually going to be quite a complicated example, just to make life interesting. So one is going to move, then two is going to move, and then one is going to move again. This is all one big information set for player one. And one is going to move like this So again without payoffs this is a little tree and the key point here is this is an information set.

[译文 42]

所以一个子游戏，非正式地说，就是游戏中的一个小游戏。但稍微更正式一点，我不能把一个属于某个信息集的节点放进这个子游戏，除非我打算把属于那个信息集的所有节点都放进子游戏。让我们看一些例子。我们有一个例子。子游戏，你要把属于那个信息集的所有节点都放进子游戏。让我们看一些例子。我们在那上面有一个例子。进入游戏看起来像这样。进入游戏看起来像这样。那么这里的子游戏有哪些？这里没有秘密。这是一个子游戏。实际上还有另一个子游戏。你想知道另一个子游戏是什么吗？整个游戏是一个子游戏。整个游戏本身就是一个子游戏。某种程度上是平凡的。所以这个特别的游戏，也就是进入游戏的示意图，它实际上有两个子游戏，但只有一个真子游戏。这里有一个更复杂的例子。这实际上会是一个相当复杂的例子，只是为了让生活更有趣。所以一个要先动，然后两个要动，然后一个又要动。这对玩家1来说是一个很大的信息集。一个要这样动。同样，没有收益，这是一棵小树，关键点在于这是一个信息集。

[段 43]

Let’s stare at this tree a second and figure out what are and are not sub-games. So first of all, this was a sub-game and this was a sub-game. What about this thing here? Is that a sub-game? It’s not a set. What rule does it break? It’s breaking up an information set, right? It’s breaking up an information set. So that’s no good because of rule three. What about this thing? That doesn’t break up an information set. I’ve got the whole information set in there. Is that any good? No, that’s no good because it doesn’t start from a singleton node. That’s no good. It violates one. and if we do this we look at this piece that piece there that’s also no good, why is that no good? again it breaks up an information set alright, so this is no good again because of rule 3 alright so you can practice at home drawing trees and trying to identify what are and what are not sub All right, so with the definition of a sub-game now formal, right, it’s basically just formalizing something we’ve talked about before, which is the idea of a game within the game, I want to introduce our new, what’s going to be our new solution concepts, and this is going to be the solution concept we’re going to use essentially almost until the final.

[译文 43]

让我们仔细看看这棵树，弄清楚哪些是子游戏，哪些不是子游戏。首先，这是一个子游戏，这也是一个子游戏。这里的这个东西是子游戏吗？它不是一个集合。它违反了哪条规则？它破坏了信息集，对吧？它破坏了信息集。所以这不行，因为规则三。这里的东西没有破坏信息集，我把整个信息集放在里面。这行吗？不行，这不行，因为它不是从单点节点开始的。这不行，它违反了规则一。如果我们再看看这块和那块，那也是不行的，为什么不行？同样，它破坏了信息集，所以这也是不行的，因为规则三。好，大家可以在家里练习画树，试着辨别哪些是子游戏，哪些不是子游戏。好，子游戏的定义现在正式了，基本上只是把我们之前讨论的内容形式化，也就是“游戏中的游戏”这个概念。接下来我要介绍我们的新解概念，这将是我们在几乎直到最后几乎都要使用的解概念。

[段 44]

Definition. So just remember what our task is. Our task is to come up with a solution concept that picks up the idea from the first half of the semester, namely Nash equilibrium, but does so in a way that respects what we’ve learned in the second half of the semester, namely that games have sequential elements and people move by backward induction. So in particular, what we want to rule out are those Nash equilibria that instruct players down the tree to play in sub-games according to strategies that are not Nash equilibria. Say it again. We want to rule out those Nash equilibria that instruct people way down the tree to play according to something which is not a Nash equilibria We want our new notion to say wherever you find yourself in a tree play Nash equilibrium And that’s exactly what the definition is going to say. So a Nash equilibrium, n e, s1 star, s2 star, all the way up to sn star, is a sub-game perfect equilibrium. It’s a sub-game perfect equilibrium, so that’s an SPE. It’s a sub-game perfect equilibrium if it induces a Nash equilibrium in every sub-game of the game. So a sub-game perfect equilibrium, it has to itself be a Nash equilibrium, of course, but it also has to instruct players to play a Nash equilibrium in every sub game. Let’s take that immediately back to… be a Nash equilibrium, of course, but it also has to instruct players to play a Nash equilibrium in every sub-game.

[译文 44]

定义。请记住我们的任务是什么。我们的任务是提出一个解概念，它承接本学期前半段的思想，即Nash均衡，但又以一种尊重本学期后半段所学的方式来做，即游戏具有顺序性，人们通过逆向归纳来行动。因此，我们具体想要排除的是那些Nash均衡，它们指示树中玩家在子游戏中按照非Nash均衡的策略行动。再说一遍。我们想要排除那些Nash均衡，它们指示玩家在树的深处按照非Nash均衡的方式行动。我们希望新概念说明，无论你在树的哪个位置，都要玩Nash均衡。而这正是定义要说的内容。因此，一个Nash均衡，记为NE，s1*、s2*、……、sn*，若它在每个子游戏中都诱导出一个Nash均衡，则它是子博弈完美均衡，即SPE。一个子博弈完美均衡首先必须本身是一个Nash均衡，当然，但它还必须指示玩家在每个子游戏中都玩Nash均衡。让我们立刻把它带回……它本身当然必须是一个Nash均衡，但也必须指示玩家在每个子游戏中都玩Nash均衡。

[段 45]

Let’s take that immediately back to our examples. In this example, in this example, we know, let’s bring it down, in this example, we know that this is a sub-game. We know that in this sub-game, there is only one Nash equilibrium, and that Nash equilibrium involves player 2 choosing down and player 3 choosing right. So we know that player 2 is going to choose down according to that equilibrium, and player 3 is going to choose right according to that equilibrium. So if we now have to look for an equilibrium of the whole game, let’s go back to player 1’s choice. player 1 if they choose A will get 1 if they chose B then they know that this Nash equilibrium will be played so they’ll get 2 they prefer 2 to 1 so the sub game perfect equilibrium here is player 1 chooses B player 2 chooses down and player 3 chooses right This is an equilibrium of the game and it induces, here it is, it induces an equilibrium in the sub game So in that example the sub game perfect equilibrium is found by first of all looking in the sub game find the equilibrium in the sub game and then go back and look at the equilibrium in the whole game The equilibrium we end up with, it is a Nash equilibrium in the whole game, but more importantly, it induces a Nash equilibrium in the sub game.

[译文 45]

让我们立刻回到我们的例子。在这个例子中，我们知道这是一个子游戏。我们知道在这个子游戏中只有唯一一个Nash均衡，而这个Nash均衡涉及玩家2选择下、玩家3选择右。因此我们知道，根据该均衡，玩家2会选择下，玩家3会选择右。现在如果我们要在整个游戏中寻找一个均衡，回到玩家1的选择。如果玩家1选择A，他会得到1；如果选择B，那么他知道这个Nash均衡会被实施，所以他会得到2，他更喜欢2而不是1。所以这里的子博弈完美均衡是：玩家1选择B，玩家2选择下，玩家3选择右。这是一个游戏的均衡，并且它在子游戏中诱导出一个均衡。在这个例子中，子博弈完美均衡的求法是：首先在子游戏中寻找均衡，然后在整体游戏中再寻找均衡。我们最终得到的均衡，它是整个游戏中的Nash均衡，但更重要的是，它在子游戏中诱导出一个Nash均衡。

[段 46]

Let’s just go back to our other example, then I’ll stop. So our other example was here. Here was our other example. And we claimed, hang on everybody, we claimed that the good equilibrium here, the one we believed in, was down, down, right. All right? Where are the sub-games in this game? Where are the sub-games in this tree? Anybody? So So I claim there’s only one real subgame here, and that’s this piece. This is a subgame. What’s the Nash equilibrium of this somewhat trivial subgame? The Nash equilibrium of this somewhat trivial subgame is that player 1 must choose down. So for a Nash equilibrium to be a subgame perfect equilibrium, here are our three Nash equilibria, 1, 2, 3. For this Nash equilibrium to be a subgame perfect equilibrium, it’s got to instruct player one to choose down in the trivial sub game. And here it is. This is our sub game perfect equilibrium in this game. Now I know today was a lot of formal stuff, a lot of new ideas. When we come back on Monday, we’ll first of all give you a game that refreshes these ideas, and then we’ll go straight to applications. So trust me, there will be applications. It will be useful. See you on Monday. there’s a homework to come in and there’s another one on the

[译文 46]

让我们回到另一个例子，然后我就停下来。我们的另一个例子在这里。这是我们的另一个例子。我们声称，大家稍等，我们声称这里的好均衡，即我们相信的均衡，是下、下、右。好的？这个游戏中的子游戏在哪里？这棵树中的子游戏在哪里？有人吗？所以我声称这里只有一个真正的子游戏，就是这部分。这是一个子游戏。这个相当平凡的子游戏的Nash均衡是什么？这个相当平凡的子游戏的Nash均衡是玩家1必须选择下。所以对于一个Nash均衡成为子博弈完美均衡，这里有我们的三个Nash均衡，分别是1、2、3。对于这个Nash均衡要成为子博弈完美均衡，它必须指示玩家1在平凡子游戏中选择下。就这样。这就是这个游戏中的子博弈完美均衡。我知道今天讲了很多正式的内容，很多新概念。当我们周一回来时，首先会给你们一个游戏来复习这些概念，然后直接进入应用。所以相信我，会有应用的。会很有用。周一见。还有一份作业要交，还有一份在……