耶鲁大学博弈论公开课 - 第15集 迭代剔除和中位选民定理

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]

Last time, we talked a lot about the idea, I didn’t mention the name, but here’s some jargon for you, we talked about the idea of deleting dominated strategies, looking at a game, figuring out which strategies are dominated, deleting them, looking at the game again, looking at which strategies are now dominated, deleting those, and so on and so forth. And this process is called the iterative deletion of dominated strategies. This is not a great title for Barry’s book. You’d be better than that to get the $250, right? This is a boring title. Whatever mind, iterative deletion of dominated strategies. The idea, it embodies the idea of putting yourself in someone else’s shoes and trying to figure out what they’re going to do, and then think about them, putting themselves in your shoes, figuring out what you’re going to do, and so on and so forth. And last time, we saw already that this is a very powerful idea, all right, in that game last time, but we also saw it’s a dangerous idea to take too literally, that sometimes this can get you to overthink the problem, and actually, as in that numbers game last time, the optimal, the best choice, the winning choice, might not involve so many rounds. All right, we also saw in that numbers game last time that in some games, but by no means all games, in some games, this process actually converges to a single choice in that numbers game, it converged to one.

[译文 1]

上次我们讨论了很多关于这个想法，我没有提到这个名字，但这里有一些术语给你们，我们讨论了删除被支配策略的想法，看一个博弈，找出哪些策略是被支配的，删除它们，再次看这个博弈，看现在哪些策略是被支配的，删除那些，以此类推。这个过程叫做迭代删除被支配策略。这不是一个好书名，巴里应该比这更有文采，对吧？能得到那250美元？太无聊的标题了。无所谓，迭代删除被支配策略。这个想法体现了换位思考的理念，试图弄清楚对方会怎么做，然后考虑他们，换位思考你的位置，弄清楚你会怎么做，以此类推。上次我们已经看到这个想法非常强大，在上次那个博弈中，但我们也看到太字面地理解这个想法是危险的，有时候这会让你过度思考问题，实际上，就像上次那个数字博弈一样，最优的、最佳的、获胜的选择可能不需要那么多轮。我们在上次那个数字博弈中也看到，在某些博弈中，但绝非所有博弈，这个过程实际上会收敛到单一选择，在那个数字博弈中，它收敛到一个。

[段 2]

All right. So just say once again what this idea is, this idea is, you yourself should not play a dominated strategy. Delete those. Delete those for everyone else. Whereas everyone else is not going to play a dominated strategy, delete those. Look at the game with all those, all those dominated strategies deleted. See if there are any strategies that are now dominated, delete those. Look again, et cetera, et cetera. All right, I could write that all up. We saw it in practice last time. It’s probably just easy to have the idea there. All right. One tip about this. Try to identify all the dominated strategies of all players before you delete. Then delete. Then look again. Try to identify all the dominated strategies of all players again and then delete. That process will prevent you from getting into trouble. All right. So today, I want to be a little bit less abstract, if you like, and I want to look at an application. I want to look at a famous application from politics. All right. So we’re going to look at a model of politics. All right. And the idea is going to be this. We’re going to imagine that there are two candidates. Two candidates. And what these candidates are doing is they’re choosing their political positions for an election. So these are the players and the strategies are going to be, they’re going to going to choose positions on a spectrum, on a political spectrum.

[译文 2]

好的，让我再说一遍这个想法是什么，这个想法是，你自己不应该玩被支配的策略。删除那些。为其他人也删除那些。由于其他人不会玩被支配的策略，删除那些。在所有那些被支配策略都被删除后看这个博弈。看看现在是否有任何被支配的策略，删除那些。再看一遍，以此类推。我可以把这些都写下来。我们上次在实践中看到了。那里有这个想法可能更容易理解。关于这个有一点提示。在删除之前，试着找出所有玩家的所有被支配策略。然后删除。然后再看一遍。再次尝试找出所有玩家的所有被支配策略，然后再删除。这个过程会让你避免陷入麻烦。好的，今天我想让内容不那么抽象，如果你愿意的话，我想看一个应用。我想看一个政治方面的著名应用。好的，我们要看看一个政治模型。好的，这个想法是这样的。我们要想象有两个候选人。两个候选人。他们正在做的是为选举选择他们的政治立场。所以这些是玩家，策略将是，他们将选择在政治光谱上的位置。

[段 3]

And to make it life easy, we’re going to assume that this political spectrum has ten positions. So here are the positions. We’ll call them one, two, three, four, five, six, six, seven, eight, nine and ten. All right, not very difficult. What’s the idea here? We haven’t finished driving the game. Watch the idea. The idea is that these positions are left-wing positions, and these positions are right-wing positions. Can people not see past the podium? Let’s get it out of the way. All right. So you could think of these extreme left-wing positions. These are people who, I guess, they don’t eat anything except for fruit, and they think the trees should have the vote. All right, and these guys out here, these are the extreme right-wing positions, so they think the poor shouldn’t have the vote and they eat immigrants. I’m an immigrant. I do care. The candidates here are going to try and choose positions. We’re going to assume, this is not realistic, we’re going to assume for now, that there are 10% of the voters at each of these positions. All right? So there’s 10% of the voters at each position. So they’re uniformly distributed. All right. And we’re going to assume that voters will eventually vote for the closest candidate. All right. So voters vote for the closest candidate. the candidate whose position is closest to their own, and we’ll need a tie-breaking assumption for the closest candidate, or the candidate whose position is closest to their own, and we’ll need a tie-breaking assumption, and we’ll do the obvious tie break.

[译文 3]

为了让事情简单，我们假设这个政治光谱有十个位置。这是这些位置。我们称它们为一、二、三、四、五、六、七、八、九和十。好的，不难。想法是什么？我们还没完成博弈的构建。看着这个想法。这个想法是这些位置是左翼立场，这些位置是右翼立场。人们看不到讲台后面吗？让我们把话说清楚。好的，所以你可以想到这些极端左翼立场。这些人，我想他们除了水果什么都不吃，他们认为树应该有投票权。好的，而那边这些人，这些是极端右翼立场，所以他们认为穷人不应该有投票权，他们吃移民。我是移民。我很在意。这里的候选人将尝试选择立场。我们将假设，这不是现实的，我们现在假设，有10%的选民在每个这样的位置上。好的？所以每个位置有10%的选民。所以他们是均匀分布的。好的，我们将假设选民最终会投票给最近的候选人。好的。所以选民投票给最近的候选人。立场最接近他们自己立场的候选人，我们需要一个平局打破假设对于最近的候选人，或者立场最接近他们自己立场的候选人，我们需要一个平局打破假设，我们会做明显的平局打破。

[段 4]

If there’s a tie, then the voters split. The voters of that position split evenly. All right, so there’s a tie, half the voters go, that position go for one of the candidates, and half of them go for the other. All right, so here’s a game. I’ve got the players, that’s the candidates, I’ve got the strategies, that’s the political positions. What am I missing? I’m missing payoffs, right? I’m missing payoffs, and it matters in this game a lot how we specify the payoffs, but we’re going to assume that the payoffs are that the candidates aim to maximize their share of the vote. So that’s not the only thing we could have assumed. We could have assumed that all they really care about is winning, and that winning gave them a high payoff, and that losing gave them nothing. I’m going to assume a little bit more, I’m going to assume that they haven’t felt maximized right. Max will do. They want to max their share of the vote. The reason this isn’t a terrible assumption is you could think that getting a higher share of the vote gives you a mandate, or if this is a primary election, you know, a low. larger share of the vote gives you a bigger push for the next primary or whatever. All right, so we’re going to assume that they’re trying to maximize their share of the vote.

[译文 4]

如果出现平局，那么选民会分裂。那个位置的选民平均分配。好的，所以出现平局，一半的选民，那个位置的选民会去其中一个候选人，另一半会去另一个。好的，这里有一个博弈。我有玩家，那是候选人，我有策略，那是政治立场。我缺什么？我缺收益，对吧？我缺收益，在这个博弈中我们如何指定收益很重要，但我们将假设收益是候选人旨在最大化他们的选票份额。所以这不是我们可以假设的唯一情况。我们可以假设他们真正关心的只是获胜，而获胜给他们带来高收益，失败给他们带来零收益。我将假设更多一点，我将假设他们没有感到最大化权利。马克思会做。他们想要最大化他们的选票份额。这个假设不是太糟糕的原因是，你可以想到获得更高的选票份额给你一个授权，或者如果这是初选，你知道，低。更高的选票份额给你在下一次初选或什么的更大的推动力。好的，所以我们将假设他们试图最大化他们的选票份额。

[段 5]

All right, so we want to know what’s going to happen in this game. It seems like a pretty natural, pretty important game. Okay, so given what we’ve learned so far in the class, a natural first question to ask is, are any of the strategies here, there are ten strategies, one, two, three, four, five, six, nine, nine, ten, are any of the strategies dominated? Are any strategies to know me? Let’s get some, let me get my microphones up and ready a little bit. So any strategy is normally? How are this this gentleman in blue? So stand up and shout? Yeah. Okay, um, one and ten are both dominated? All right, so, uh, this gentleman whose name is… Stephen? Stephen, says one and ten. Let’s, we’ll come back to ten, but you’re right, we’ll come back to ten. So he says that position one, strategy one, choosing the most extreme left-wing position, is a dominated strategy. What dominates it, by the way? Stephen, want to shout out? Two. So, for example, so that our conjecture here is, is that two dominates one. Two dominates one. That’s the claim. Right? And let’s just be careful here. What does it mean to say that two dominates one? It means, it means that choosing position two always gives me a higher share of the vote, than choosing position one, no matter where the other candidate positions herself.

[译文 5]

好的，所以我们想知道这个博弈中会发生什么。这似乎是一个相当自然、相当重要的博弈。好的，鉴于我们到目前为止在课堂上学到的，自然要问的第一个问题是，这里的任何策略，有十个策略，一、二、三、四、五、六、七、八、九、十，是否有任何策略是被支配的？是否有策略让我知道吗？让我们得到一些，让我把麦克风准备好一点。任何一个策略通常是？这个蓝衣绅士怎么样？站起来大声说？好的，嗯，一和十都是被支配的？好的，所以，呃，这位先生的名字是…斯蒂芬？斯蒂芬，说一和十。我们回到十，但你对了，我们回到十。所以他说位置一，策略一，选择最极端的左翼立场，是一个被支配的策略。顺便问一下，是什么支配它？斯蒂芬，想大声说出来吗？二。所以，例如，我们的猜想是，二支配一。二支配一。这是主张。对吧？让我们在这里小心一点。说二支配一是什么意思？它的意思是，无论其他候选人把自己放在什么位置，选择位置二总是给我更高的选票份额，而不是选择位置一。

[段 6]

Right? It does not mean two beats one, right? All right. All right. So let’s take Stevens conjecture and see if it’s true. All right. All right. So in particular, let’s start working out what share of the votes you’d get if you chose position one or position two against different positions. The other other guy can choose. All right. Okay. So for example, so what we’re going to do is we’re going to test does to dominate one. And while we’re doing this, we’ll figure out how these payoffs work as well. All right. Well, how about versus one? So suppose the other candidate has chosen position one. All right? So the other candidate has chosen position one. Then we’re going to compare my payoff from choosing one against the other candidate’s payoff from choosing one. We’re going to compare this with my payoff from choosing two against the other candidate choosing one. All right? Let’s figure out what that is. So what’s my, what’s my, what’s my share of the vote if I choose one and the other candidate chooses one? 50%. That was pretty easy, right? Must be a tie for everybody. So I’ll get 50% of the vote. And what’s my share? share of the vote if I choose two and the other candidate chooses one? 90% right? In fact, he will get, or she will get all the voters at one, I’ll get everyone else, so I get 90%.

[译文 6]

对吧？它不意味着二打败一，对吧？好的。好的。让我们拿斯蒂芬斯的猜想，看看它是否正确。好的。好的。所以特别地，让我们开始计算如果你选择位置一或位置二对抗不同位置时你会得到多少选票份额。另一方可以选择的位置。好的。好的。例如，我们要做的就是测试二是否支配一。当我们做这个的时候，我们也会弄清楚这些收益是如何运作的。好的。那么，对抗一怎么样？如果其他候选人选择了位置一。好的？如果其他候选人选择了位置一。那么我们要比较我从选择一得到的收益对抗其他候选人从选择一得到的收益。我们要把这个和我从选择二对抗其他候选人选择一得到的收益进行比较。好的？让我们算出那是什么。如果我选择一而其他候选人选择一，我的选票份额是多少？50%。这很简单，对吧？每个人都一定是平局。所以我将得到50%的选票。如果我选择二而其他候选人选择一，我的选票份额是多少？90%？实际上，他会，或者她将得到位置一上的所有选民，我将得到其他所有人，所以我得到90%。

[段 7]

So in this case, choosing two is better than choosing one. But of course we’re not done yet. We have to consider other possible positions of my opponent. So suppose my opponent chooses two. So now we’re comparing my choosing one, my opponent chooses two, and the I would get if I choose two when my opponent chooses to. Now we’re comparing my choosing one, what my opponent chooses two, and the payoff I would get if I choose two when my opponent chooses two. Okay, so what’s my payoff if I choose one and my opponent chooses two? I get all the voters right on top of me at position one, and she gets everyone else, is that right? So I get 10%, and what about if we both choose two? 50%? Everyone happy with that? All right, so 50% in this case. And once again, two did better than one. Everyone are happy with that? So we showed this choice was better, and this choice was better, and we’re going to keep on going. Against three. So now we’re comparing me choosing one, my choosing one versus three, and my choosing two versus three. If I choose one against three, what share of the vote do I get? I’m going to get all the people at position one and half the people at position two for a total of 15.

[译文 7]

所以在这种情况下，选择二比选择一更好。但我们当然还没有完成。我们必须考虑对手的其他可能位置。假设对手选择二。现在我们比较我选择一、对手选择二的情况，以及当对手选择二时我选择二的情况。现在我们比较我选择一、对手选择二的情况，以及当对手选择二时我选择二的情况。好，如果我选择一而对手选择二，我的收益是多少？我得到所有正好在我位置一的选民，而她得到其他人，对吗？所以我得到10%，那如果我们都选择二呢？50%？大家都满意吗？好的，这种情况是50%。再一次，二比一更好。大家都满意吗？所以我们证明了这种选择更好，那种选择更好，我们将继续下去。对抗三。现在我们比较我选择一对抗三，以及我选择二对抗三。如果我选择一对抗三，我得到多少选票？我会得到所有在一位置的人以及在二位置的一半人，总共15%。

[段 8]

And if I chose two against three, what do I get? I get 20. I get all the people at one, and now I get all the people at two, so I get 20%. So we’re okay again. Let’s do one more, just to see a pattern here. So if I choose one against four in comparing my choosing two against four. In the former case, if I choose one against four, how many votes do I get? I get all the people at one and all the people at two, right? So I get 20% of the votes. She gets everyone else. And here, if I choose two against four, I get all the people at one, all the people at two, and what, half the people at three. All right, so that comes out as 25%. And once again, And so on. Now, I could go on right the way through here, but I’m going to stop, because it gets a bit boring after a while, right? Because I’m guessing you can see a pattern emerging here. What’s the pattern here? What’s the pattern here? Somebody? Somebody raised their hands? Somebody raise their hands? We can get a mic out there? Can we get a mic in this guy in white? Stand up, stand up. Yep. Two is better than one. Stand up and shouts? Two is always better than one.

[译文 8]

如果我选择二对抗三，我得到什么？我得到20。我得到所有在一位置的人，现在我得到所有在二位置的人，所以我得到20%。所以我们再次没问题。让我们再看一个，只是为了看一个模式。所以如果我选择一对抗四，比较我选择二对抗四。在前一种情况下，如果我选择一对抗四，我得到多少票？我得到所有在一位置的人和所有在二位置的人，对吧？所以我得到20%的选票。她得到其他人。而这里，如果我选择二对抗四，我得到所有在一位置的人、所有在二位置的人，还有三位置的一半人。好的，那是25%。再一次，以此类推。我可以一直这样做下去，但我会停下来，因为过一会儿就有点无聊了，对吧？因为我想你们能看到这里出现了一个模式。这里的模式是什么？这里的模式是什么？有人吗？有人举手了？有人举手吗？我们能拿个麦克风出来吗？能让穿白衣服的那个人拿麦克风吗？站起来，站起来。是的。二永远比一好。站起来大声说出来？二永远比一好。

[段 9]

Two is always better than one. But we can see more than that in the pattern. That’s right, but what’s the pattern here? Mike back here? Whatever’s closer to five is better than what’s ever farther away from five? That’s not quite the pattern I’m looking for. Let’s look at the numbers here. We had 15, 20, then this one up to 20, 25, and so on. All right, so what’s the pattern here? Yes, Stephen again. If your opponent doesn’t choose one or two, then you’re always going to be 5% better off if you choose 2 than 1. Exactly. So ignoring these first two positions are a bit weird that choosing two always gave me 5% more votes than choosing one, regardless of what the other person chooses. I mean, if you want, you can fill in the next, whatever it is, the next six positions and see, but you’ll see that very clearly that Stevens right, that choosing two will always get me 5% more of the votes than choosing one from here on down, and in fact they’ll go up in intervals of five. So in fact, it is true, so we can conclude. that two dominates, in fact, we can do a bit better than that, we can say strictly dominates one. All right? Two strictly dominates one. All right. Is anything else dominated here?

[译文 9]

二永远比一好。但我们可以在模式中看到更多。不止于此。是的，但这里的模式是什么？迈克在后面？接近五的比远离五的好？这不是我想要的模式。让我们看看这些数字。我们有15、20，然后这个到20、25，以此类推。好的，这里的模式是什么？是的，斯蒂芬。你对手如果不选择一或二，那么你选择二总是比选择一好5%。完全正确。所以忽略前两个位置有点奇怪，选择二总是给我比选择一多5%的选票，不管对方选择什么。我的意思是，如果你愿意，你可以填入接下来的六个位置，看看，但你会很清楚地看到斯蒂芬是对的，从这里开始选择二总是比选择一多5%的选票，事实上它们会以五为间隔递增。所以事实上，这是真的，所以我们可以得出结论，二支配一，事实上我们可以做得更好，我们可以说二严格支配一。好吗？二严格支配一。还有什么被支配的吗？

[段 10]

Stephen already told us that 10 is dominated. I’m not going to go through that argument. Can everyone see that symmetric? Hang a second. Can everyone see that 10 is going to be dominated by 9 in exactly the same way? Yeah, is that obvious? It’s just really the same argument coming in another direction. All right? So similarly, nine strictly dominates ten. And again, we’re not saying choosing two beats choosing one, right? two wins against one, or nine wins against ten. Right? We’re saying choosing two two always does better than choosing one regardless of what the other person chooses. And choosing nine always is better than choosing 10 regardless of what the other person chooses. All right, there was a question. Are we okay? You want to get the mic in for the question? Yep, stand, stand, stands. Shout out everyone. I think you may have defined already, but I couldn’t find a definition of strictly in the book we have. Ah, okay. Well, you have one on the handout. I think there probably is one in there somewhere, but just in case there’s one that we’ve had two in class, there’s one on the handout. Okay? All right. So, all right, so, all right, we’ve concluded. just in case, there’s one, we’ve had two in class, there’s one on the handout. Okay, good. All right, so, all right, so we’ve concluded that one is dominated by two and ten is dominated by nine.

[译文 10]

斯蒂芬已经告诉我们10被支配了。我不打算重复那个论证。大家都能看到10将以完全相同的方式被9支配吗？是的，这显而易见吗？这实际上只是同一个论证从另一个方向进来。好吗？所以同样，九严格支配十。我们不是说选择二战胜选择一，对吧？二战胜一，或者九战胜十。对吗？我们的意思是，无论对手选择什么，选择二总是比选择一好。无论对手选择什么，选择九总是比选择十好。好吗？有个问题。我们还好吗？你想为那个问题拿麦克风吗？是的，站起来，站起来，大声说出来。我认为你可能已经定义了，但我找不到书中对strictly的定义。啊，好的。好，你有一份在讲义上。我想可能里面有一个，但我们课堂上已经有两次了，讲义上有一个。好吗？好的。好的，所以我们得出结论。只是以防万一，课堂上已经有两次了，讲义上有一个。好的。好的，所以我们得出结论，一被二支配，十被九支配。

[段 11]

Is anything else dominated here? Well, what about, what about thinking about whether two is dominated by three? After all, I’ve forgotten your name, but that what’s your name there? Yes. Scypta had suggested earlier on, you want to be close to five, so three is closer to five than two, is two dominates by three. Anyone else? Any other hands? There’s a hand here. Can I get this lady here? Well, once you delete the dominated strategies, then you kind of go through it again, and then two is dominated by three. All right, good, so we’re getting ahead of us. Good. So we’ll get there. Hold that thought. Hold that thought. But what about right now before we delete anything? So let’s… Let’s check it out. Let’s check it out. So in particular, let’s just try this process. So how about my payoff versus one, for example? So two against one gives me 90% of the votes. Three against one gives me 85% of the votes. Which is actually lower, right? This is lower. So it’s clear that three doesn’t dominate two. Right? Three doesn’t dominate two. In particular, if the other candidate were to choose position one, I would get a higher share of the vote choosing two than I would have done if I had chosen three. All right? So in fact, it’s not the case that two is dominated by three.

[译文 11]

这里还有什么被支配的吗？嗯，二被三支配怎么样？毕竟，我忘了你的名字，但你在那里叫什么名字？是的。Scypta之前建议过，你要接近五，所以三比二更接近五，二是被三支配的。还有其他人吗？还有其他手吗？这里有一只手。我能让这位女士发言吗？一旦你删除了被支配的策略，你就要再过一遍，然后二是被三支配的。好的，我们超前了。好的，我们会到那里的。保留那个想法。保留那个想法。但现在在我们删除任何东西之前呢？让我们……让我们检查一下。让我们检查一下。所以特别是，让我们试试这个过程。比如我的收益相对于一怎么样？二相对于一给我90%的选票。三相对于一给我85%的选票。实际上更低，对吧？这个更低。所以很明显三不支配二。对吧？三不支配二。特别是，如果另一个候选人选择位置一，我选择二会比选择三得到更高的选票份额。好吗？所以事实上，二被三支配的情况并不成立。

[段 12]

But the woman whose name I either didn’t get or forgot. So Christine has pointed something else. So, when do you wait for them, I can point that out again. So we know that. that two is not dominated, in particular not dominated by three, but, what’s the but? When you delete the dominated strategy of two dominating one, or one being dominated, when you delete that and ten, then it is. All right. So what Christine is arguing is it, even though it’s the case that two is not a dominated strategy, if we do the process of iterative deletion of dominated strategies and we delete the dominated strategies, then maybe we should look again and see if it’s dominated now. All right. So let’s be careful here. We’re not saying that we’re going to delete the voters at 1, or delete the voters at 10, though we might wish to, right? We’re just saying we want, all we’re saying is, we know that the candidates aren’t going to position themselves at 1 and aren’t going to position themselves at 10, so in some sense we’re deleting those strategies. The voters are still there. All right, all right, so let’s try that. So if we, there’s a better, there’s a bit, There’s a but here, I guess, but if we delete the strategies 1 and 10, which were dominated, then does 3 dominate 2?

[译文 12]

但那位女士的名字我或者没听到或者忘记了。所以Christine指出了另一点。所以，当你等他们的时候，我可以再指出来。我们知道那个，二不是被支配的，特别是不是被三支配的，但是，但是呢？当你删除被支配策略时，即二支配一，或一被支配，当你删除那个和十的时候，那就是了。好的。所以Christine论证的是，即使二是被支配策略的情况不成立，如果我们进行迭代删除被支配策略的过程，我们删除了被支配策略，那么也许我们应该再看一看，看看现在它是否被支配了。好吗？让我们仔细一点。我们不是说我们要删除一位置的选民，或者删除十位置的选民，虽然我们可能希望这样做，对吧？我们只是说，我们知道候选人不会把自己定位在一和十，所以在某种意义上我们是在删除那些策略。选民还在那里。好的，好的，让我们试试。所以，如果我们在删除被支配的一和十策略之后，三支配二吗？

[段 13]

And once again, we can try it out. So let’s try it out. All right, so the payoff from choosing 1, it’s start against 2. So the payoff from choosing 2 against 2 is 50%. The payoff from choosing 3 against 2 is what? What’s the payoff from choosing 3 against 2? What’s the payoff from choosing 3 against 2? 80%, right, the person at 2 is going to get all the ones at 1, all the ones at 2, and 3 will get everything else, so this is 80% versus 3, the payoff of 2 against 3 is all the people at 1, all the people at 2, so it’s 20%. The payoff from choosing 3 against 3 is, of course, 50%. So so far, so good, so far choosing 3 is better. Against 4, 4, the payoff from choosing 2 against 4 is what? What’s the payoff from choosing 2 against 4? Again, all the people at 1, The payoff from choosing two against four is what? What’s the payoff from choosing two against four? Again, all the people at one, all the people at two, and half the people at three, so that’s 25%. The payoff from choosing three against four is going to be all the people at one, all the people at two, and all the people at three for a total of 30%, which again is bigger. versus five, the payoff from choosing two, I can’t afford to raise this a bit, against five is all the people at one, all the people at two, and all the people at three.

[译文 13]

好的，我们再来验证一下。让我来试试。好的，选择1的收益，它是从对阵2开始的。所以选择2对阵2的收益是50%。选择3对阵2的收益是多少？选择3对阵2的收益是多少？80%，对吧，位置2的人会获得所有在位置1的人，所有在位置2的人，而3会获得其余的所有人，所以对阵3时，选择2的收益是所有在位置1的人，所有在位置2的人，所以是20%。选择3对阵3的收益当然是50%。到目前为止还不错，到目前为止选择3更好。对阵4时，选择2对阵4的收益是多少？选择2对阵4的收益是多少？同样，所有在位置1的人，所有在位置2的人，以及位置3的一半的人，所以是25%。选择3对阵4的收益将是所有在位置1的人，所有在位置2的人，以及所有在位置3的人，总共30%，这又更大一些。对阵5时，选择2的收益，我稍微提高一下，对阵5是所有在位置1的人，所有在位置2的人，以及所有在位置3的人。

[段 14]

30%, whereas the payoff from choosing three against five is all the people at one, all people two, all people at three, and half the people at four. 35%. And again, you can see exactly the same pattern’s going to emerge here. From here on down, if I bothered to do it, we’d find that choosing three always gets me five percent more of the vote than I would have had from choosing two against any of these higher numbers. The same pattern, Stephen pointed out before. So I’ve forgotten your name again, I’m sorry. So Christine is correct in saying that once we delete the strategies one and ten, once we realize that those positions are not going to be chosen by our sophisticated candidates, then we realize that probably choosing two and, of course, nine, isn’t a good idea either. All right? So two and nine are not dominated, but they are dominated, but they are dominated realize one and ten won’t be played, won’t be chosen, all right? Okay, so where are we going here? Where are we going? Where is this, where is this discussion going to end up? Let’s come back to our original game. All right, we’ve argued that one and ten shouldn’t be chosen because they’re dominated, we’d argue that once we realize there aren’t going to be played, that two and nine aren’t going to be played, what’s going to end up being played?

[译文 14]

30%，而选择3对阵5的收益是所有在位置1的人，所有在位置2的人，所有在位置3的人，以及位置4的一半的人。35%。同样，你可以看到完全相同的模式会出现。从这里往下，如果我费心去做，我们会发现选择3总是比从对阵这些更高数字时选择2多获得5%的选票。同样的模式，斯蒂芬之前指出过。我又忘了你的名字了，抱歉。所以克里斯汀说得对，一旦我们删除策略1和10，一旦我们意识到这些位置不会被我们的老练的候选人选择，那么我们就意识到选择2当然还有9也不是好主意。对吧？所以2和9不是被占优的，但它们是，当意识到1和10不会被采用、不会被选择时，对吧？好的，那我们接下来会怎样？这个讨论最终会走向哪里？让我们回到我们最初的游戏。好的，我们已经论证了1和10不应该被选择，因为它们是被占优的，我们会论证一旦我们意识到它们不会被采用，2和9也不会被采用，最终会被采用的是什么？

[段 15]

Well, let’s just go slower, so we were able to dominate 1 and 10 in the first round, so we know this won’t be played, and we know this won’t be played. Remember, the voters are still there, we’re just deleting the strategies. Then we dominated, then we ruled out, I’ll put double crosses for in the second round, two and nine, and then we ruled out three and eight. And then we ruled out, we would have done, four and seven, and that leads just five and six. All right? So if we do the procedure of iteratively deleting dominated strategies, going back again, looking what’s dominated, doing it again, doing it again, all that’s left is five and six. So this procedure leads us to conclude that the candidates will choose positions five and six. Does five dominate six or six or six dominate five or anything like that? No, at that point it’s just a tie, right? So at the end, the procedure, so iterative del, it’s called it, Dell, leads to deleting all except five and six. Five and six. All right. So this was just a simple exercise, I think, I mean, I’m hoping that was a simple exercise. I’m going to, I’m hoping that was a simple exercise. Let me just look out there. Everyone following that, okay, that really wasn’t meant to be hard.

[译文 15]

好吧，让我们慢一点。我们在第一轮能够占优1和10，所以我们知道这不会被采用，我们知道这不会被采用。记住，投票者仍然在那里，我们只是在删除策略。然后我们在第二轮占优了，我会在这里打双叉，2和9，然后我们排除了3和8。然后我们排除了，我们本应该做4和7，这就只剩下5和6了。对吧？如果我们执行迭代删除被占优策略的程序，再回头看什么是被占优的，再做一次，再做一次，剩下的就是5和6。所以这个程序引导我们得出结论：候选人将选择位置5和6。5是否占优6，或者6是否占优5，或者类似的情况？不，在那一点上只是一个平局，对吧？所以最终，程序，迭代删除，叫做它，戴尔，导致删除除了5和6以外的所有策略。5和6。好的。这只是一个简单的练习，我认为，我的意思是，我希望那是一个简单的练习。我希望那是一个简单的练习。让我看看那边。大家都跟上吗，好吧，那真的不难。

[段 16]

All right? But what’s relevant about this exercise is not that the fact it’s hard, in fact it isn’t hard. What’s relevant is that it’s relevant. All right, it’s about the real world. In particular, this is a famous model in political science. Anyone know the name of this model? What’s the prediction here? The prediction here is that the candidates are going to be squeezed towards the middle. They’re going to choose positions very close to each other and very close to the center. What’s that call? Do you want to know? Hold the thought on that. Hold on that. Someone at the back? This is the median voter theorem. Thank you. Thank you. Let’s have a new board. This one will do fine. So the prediction is candidates crowd the centre. candidate crowd the centre and this is called in political science the median voter theorem it’s called the median voter theorem because the voters at the centre in this case five and six actually get to decide not just the election but decide therefore what policies are are in place. How does this do as a prediction of the real world? How does this do as a prediction of the real world? Well, there are examples in American history in which the median voter theorem looks pretty good. They’re pretty old examples. We’re going to be before you guys were born, but let me mention them anyway.

[译文 16]

好吧？但这个练习的相关之处不在于它很难，事实上它并不难。相关之处在于它是有意义的。好的，它关乎真实世界。特别是，这是政治学中一个著名的模型。有人知道这个模型的名字吗？这里的预测是什么？这里的预测是候选人将被挤向中间。他们会选择非常接近彼此且非常接近中心的位置。这叫什么？你想知道吗？先记着这个。等一下。后面那位同学？这是中间选民定理。谢谢。谢谢。让我们用一块新黑板。这个就够了。所以预测是候选人聚集在中心。候选人聚集在中心，这在政治学中被称为中间选民定理，它被称为中间选民定理是因为在这种情况下处于中心位置的投票者，实际上是5和6，决定的不仅仅是选举，还因此决定了什么政策被实施。作为对真实世界的预测，这个预测做得怎么样？作为对真实世界的预测，这个预测做得怎么样？嗯，在美国历史上有一些例子，中间选民定理看起来相当好。它们是相当古老的例子。是在你们出生之前的事了，但让我还是提一下。

[段 17]

So I think a lot of people, if you go back today and you look at tapes of the Kennedy Nixon election in 1960, some American helped me out here, 60, right? Right, I think of 60. So in the 19th, In the 1960 election between Kennedy and Nixon, if you go over and look at those tapes, it’s amazing how conservative Kennedy looks when you think about his reputation today. They really were crowding the center. And eight years later, Nixon seemed to have learned his lesson. So if you look at the 68 election, which Nixon won, Nixon sounds on those tapes because his policies being put forward in that election, he looks extremely liberal for a Republican. So again, they were crowding that center space. Some people argue that that’s exactly what Clinton did in 92. He took the Democratic Party to the right, a. towards the center, in order to pick up those central voters and win. So there are examples in American history of elections that seem to, in which the median voter theorem seems to do well. And we’ll come back and discuss it more in a second, but before I do, let me mention that the same model, exactly the same idea, has an application in economics. Somebody mentioned already over there. The application in economics is to do with product placement. the thing that Jake’s working on, right?

[译文 17]

所以我认为很多人，如果你今天回顾一下，看1960年肯尼迪和尼克松竞选的录像带，美国有人帮我确认一下，60年，对吧？对，我想是60年。所以在19…在1960年肯尼迪和尼克松之间的选举中，如果你去翻看那些录像带，令人惊讶的是，当你想到他今天的声誉时，肯尼迪看起来是多么保守。他们确实聚集在中心。八年后，尼克松似乎吸取了教训。所以如果你看68年的选举，那是尼克松获胜的，尼克松在那些录像带上的声音，因为他那次选举中提出的政策，对于一个共和党人来说，他看起来极其开明。所以再说一次，他们聚集在中心空间。一些人认为这正是克林顿在92年所做的。他把民主党带向右…朝向中心，以便赢得那些中间选民并获胜。所以美国历史上确实有一些例子，选举似乎…其中中间选民定理似乎做得很好。我们稍后会回来更详细地讨论它，但在我这样做之前，让我提一下同样的模型，完全相同的思想，在经济学中有一个应用。那边已经有人提过了。经济学中的应用与产品定位有关。就是杰克正在做的东西，对吧？

[段 18]

In product placement, you might think, if I think you’re placing a gas station, you might think it would be nice if gas stations spread themselves evenly out over the town or out over the road. So whenever you happen to be, when you ran out of gas, there was a gas station close by. That would be convenient. But unfortunately, gas stations, as we all know, they tend to crowd into the same corners. They tend to be on the same road junctions. Why? Because they’re trying to, they’re competing for voters who happen to be close or running out of gas. at those moments, and by crowding together, they avoid being out-competed by each other in terms of position. I said that a bit quickly, but we’ll come back and look at that idea in more detail later on in the class. All right. So in politics, this is about candidates crowding close together towards the centre to try and get as many voters who are close to them, and in economics, this is about firms crowding together to try and get shoppers who are close to them. All right. And the people associated with this are a guy called Anthony Downs, who did this in political science in a book in 1957, and in economics, a guy called Hotelling, who wrote a paper about this in 1929.

[译文 18]

在产品定位中，你可能会想，如果我想到你放置一个加油站，你可能会认为如果加油站均匀地分布在整个城镇或整条道路上会很好。所以无论你什么时候，当你用完汽油时，总有一家加油站就在附近。那会很方便。但不幸的是，加油站，正如我们都知道的，它们往往聚集在同一个角落。它们往往在同一个路口。为什么？因为它们试图…它们正在争夺那些恰好在附近或即将用完汽油的选民。在那些时刻，通过聚集在一起，它们避免了彼此在位置上被对方竞争出局。我说得太快了一点，但我们稍后会回来在课堂上更详细地研究这个想法。好的。所以在政治学中，这是关于候选人聚集在一起靠近中心，试图获得尽可能多的靠近他们的选民，在经济学中，这是关于企业聚集在一起，试图获得尽可能多的靠近它们的购物者。好的。与这个相关的人是安东尼·唐斯，他在1957年的一本书中在政治学中做了这个，还有在经济学中，是一位叫霍特林的人，他在1929年写了一篇关于这个的论文。

[段 19]

And in political science, they call this simultaneous discovery. I think 1929 is a bit earlier, I don’t mind. All right. Now, what I want to do is, do now, and I want to do this periodically in the class, we have a model up there, and we’ve got to analyze it, we’ve used a bit of game theory to analyze it, I want us to talk about whether we think it’s right, what do we think of the strengths and weaknesses of this model? All right, so what I want you to start thinking about is do you believe this model? This is it’s a good model of politics. What’s it missing? All right, let me come off the stage a second and I encourage you to get involved into discussion. How many of your polycy majors? Now raise your hands, your polycymaids? A bit more. By the way, how many of you have polycymedators have seen this model before? Some of you’ve seen it before. All right. So, what’s this model missing? And you all live in America. You all live in a democracy. I mean, you don’t all live in America. A lot of you live in a democracy. So you have some idea about this. So what are we missing here? So can we get the woman here, Ali? Yeah. Voters are not evenly distributed, 10%, 10%, 10%.

[译文 19]

在政治学中，他们称之为同时发现。我认为1929年稍早了一些，我不在意。好吧。现在我想做的是，现在做这件事，我想在课堂上定期做这件事，我们上面有一个模型，我们必须分析它，我们已经用了一点博弈论来分析它，我想让我们讨论一下我们认为它是否正确，我们对这个模型的优点和缺点有什么看法？好的，所以我想让你们开始思考的是你们相信这个模型吗？这是一个不错的政治模型。它缺少什么？好吧，让我暂时离开舞台，我鼓励你们积极参与讨论。你们中有多少人是政策专业的？现在举手，你们的政策专业学生？再高一点。顺便问一下，你们中有多少政策制定者以前见过这个模型？你们中有些人以前见过。好的。那么，这个模型缺少什么？你们都住在美国。你们都生活在民主制度中。我是说，你们不都住在美国。你们中很多人都生活在民主制度中。所以你们对此有一些了解。那么我们在这里缺少什么？能把这位女士请过来吗，Ali？是的。选民不是均匀分布的，10%，10%，10%。

[段 20]

All right. So one thing that seems odd about the way we set up this model is that the voters are not evenly distributed. Let me let me go and put that up and I’ll be a bit, it’s be tiring, but let me go to and fro a bit. All right? So one thing, what’s missing, or wrong if you like, one thing is the voters are not evenly distributed in the real world. Okay? Good. Anything else? Good. We’ll talk about each these in turn.

[译文 20]

好的。那么我们建立这个模型的方式中有一个看起来很奇怪的地方，就是选民不是均匀分布的。让我让我去把它展示出来，我会稍微，有点累，但是让我来回走一下。好吧？所以有一件事，如果我们愿意说的话，就是缺失的，或者说是错误的，一件事是选民在现实世界中不是均匀分布的。好的？很好。还有别的吗？好的。我们会依次讨论这些问题。

[段 21]

Can I get the guy over here? distributed in the real world okay good anything else good we’ll talk about each these in turn let me can get the guy over here sorry I’m not making this easy for the for Jude let me stay down here I’ll collect have a few and then go back to the board the model doesn’t take into account differences between the groups in voter turnout based on the candidates good so so there’s an issue of turnout right we assumed here that everyone votes is that right but in fact not everyone votes there’s always the option of not voting all right remind me to put that one up that’s okay not voting let’s remind me these how are the gentleman here in yellow there’s a difference between the primary and general election good good so we assumed we kind of jumps to just one election and at least in the American system for better or for worse I could voice an opinion on that there are lots of little elections on the way there are primaries in all these states that you never know you probably have heard of but states you probably have heard of on the way all right anything else something else that I can we get the where’s the other mic can we get the the the woman in green here American voters often vote based on character rather than position all right so so there’s more than just positions another way of saying that so to stop me if this is wrong but one way to say that we’ve been assuming that politics is one dimensional left right and among the other things things involved character or even among issues even among political issues it could be more than one dimension it could be that there’s a dimension about the environment and a dimension about foreign policy and maybe another dimension about redistributed politics all right so there’s that’s a that’s an important idea let me get a couple more is we can we get the the model might not apply with more than two candidates good good to another issue here is I just assumed there were two candidates all right we know in some elections there was a third candidate for example there was Nader in the in the election in the well I guess he was there in the other one as well but he was particularly there in 2000 so by the way I don’t mean this as a joke in some sense there’s always a third candidate and that comes from the point we had earlier because not voting is always an option all right so in in principle in that election in 2000 there was at these four candidates as it were four things you could do your vote you could vote as it was in that time gore bush neither or not voting and I’ll leave it to you to decide which of those is equivalent to not voting anyone else there’s one other thing I think was really small things one of really glaring the thing here on the glaring thing I don’t know I haven’t had yet make Ali help me out here so let me get this guy here extrapolating from the fact that there are different things which people take into account not everyone closes to six might actually vote for six they might be current that’s true although that’s that may be a question of dimension I had a different thing in mind I mean let me say it so I think this is important I vote a candidate could say during an election that I’m a moderate candidate I’m at position five but you might not believe him or her right you’ve got the candidates track record unless that candidate can commit to policies at position five you might know full well that the candidate is in fact a position 10 or a position one candidate is that right so unless the candidate can actually commits to a position all right there may be a problem in them simply announcing my position is moderate so again let’s think of an example here couple of examples in in the Bush Gore election we saw President Bush Bush as was saying I’m not a right-wing candidate I’m a compassionate conservative and you guys believed him I don’t know maybe he was I don’t know he’s track record in Texas that probably did actually look a lot like that or today if you look at both primaries going on now you’re seeing various candidates on both the Republican side and the Democratic side seek to position themselves but as you all know those candidates almost all have track records and not everybody believes that Clinton that Hillary Clinton is quite so centrist as she now seems and not everyone believes that the former governor of Massachusetts is quite so conservative as he now seems right so so it’s not clear that you can just choose your position willy-nilly all right let’s put some of these up all right so I’m going to forget remember so you have to help me out a bit so we have we have many candidates we have many candidates more than just two or we have not voting which is related to that actually and we have choosing your position the inability to commit to a position all right so So your position…

[译文 21]

我能请这边这位先生吗？在现实世界中不是均匀分布的，好的，还有别的吗？好的，我们会依次讨论这些问题。让我能请这边这位先生，抱歉我没有让Jude容易一点，让我留在这里，我会收集一些然后回到讲台。这个模型没有考虑到基于候选人群体之间选民投票率的差异，好的，所以这里有一个投票率的问题对吧，我们在这里假设每个人都投票，对吧？但实际上不是每个人都会投票，不投票总是一个选项。好的，提醒我把它加上去，那没问题，不投票。让我们提醒我，这边的这位先生，在初选和大选之间有一个区别。好的，很好，所以我们假设我们跳到只有一场选举，至少在美国制度中，不管好坏，我可以对此发表意见，在到达大选之前有很多小选举，有各州的初选，你们可能听说过但可能没有听说过的那些州，好的，还有别的吗？我能请绿衣服的那位女士吗？美国选民经常基于性格而不是立场投票。好的，所以不仅仅是立场，另一方面这样说，我们一直在假设政治是一维的——左和右，除此之外还有其他事情，比如性格，甚至在政治问题中，可能不止一个维度，可能有一个关于环境的维度，一个关于外交政策的维度，也许还有另一个关于再分配政治的维度。好的，所以这是一个重要的想法。让我再收集几个，能请那位先生吗？这个模型可能不适用于超过两个候选人的情况。好的，很好，这是另一个问题，我只是假设有两个候选人，我们知道在一些选举中有第三方候选人，例如，在那次选举中有Nader，我猜他在另外一次选举中也在，但他特别出现在2000年，所以顺便说一下，我不认为这是一个笑话，在某种意义上，总是有第三位候选人，而这来自于我们之前提到的观点，因为不投票总是一个选项。所以在2000年的那次选举中，实际上有这四位候选人，你可以做四件事，你可以投票，当时是戈尔、布什、两者都不选，或者不投票，我让你们自己决定其中哪个相当于不投票。还有别的吗？还有一件事，我认为是一个非常小的细节，其中一个非常明显的事情在这里，关于明显的事情，我不知道我还没有提到，让Ali帮我一下。让我请这位先生，从人们考虑的不同事物外推出来，不是每个人都会投票给六，他们可能是现在的，那是真的，虽然这可能是维度的问题，我脑子里有不同的事情，我的意思是让我说出来，我认为这很重要，我投票给一位候选人可能在选举期间说我是一个温和的候选人，我在第五的位置，但你们可能不相信他或她，对吧，除非那位候选人有记录，能够承诺第五位置的政策，你们可能完全知道那位候选人实际上是在第十位或第一位，候选人是这样，对吧？除非候选人实际上能承诺一个位置。好的，这可能有一个问题，就是他们简单地宣布我的立场是温和的，所以再说一遍，让我们在这里想一些例子，几个例子，在布什和戈尔选举中，我们看到布什总统当时说，我不是右翼候选人，我是一个有人道主义的保守主义者，你们相信他，我不知道，也许他是，我不知道他在德克萨斯州的记录可能实际上看起来很像那样，或者今天如果你看现在的两场初选，你看到共和党这边和民主党这边的各种候选人都试图给自己定位，但你们都知道那些候选人几乎都有记录，不是每个人都相信希拉里·克林顿像她现在看起来的那样是中间派，不是每个人都相信马萨诸塞州前州长像他现在看起来的那样是保守派，对吧。所以这不清楚你是否可以随意选择你的立场。好的，让我们把其中一些放上去，好的，所以我会忘记，所以你们得帮我一下。所以我们有很多候选人，我们有很多候选人，不只是两个，或者我们有不投票，这与这实际上是相关的，我们有选择你的立场的困难，无法承诺一个立场，好的。所以你的立场…

[段 22]

And we have choosing your position, the inability to commit to a position. All right, so your position may be not believed. And I want to claim that that’s connected to the idea that you can’t commit yourself to your policies. And we had some others, we had primaries. And we had other dimensions. So we had high dimensions. Have I got the main ones now? I think I got most of the main ones. So there’s lots of things, when we look at this model, there’s lots of things that are missing. A lot of things are not seeming to be captured by it. And here’s where I want to get a little bit more religious, if you like. It’s tempting. to say, look, we read up this model, it’s missing all this stuff, so it must be useless. That’s just, you know, we shouldn’t model it. You might even go further. You might say, modeling, you’re always going to miss stuff out, so let’s just not bother. And that, I think, is the wrong conclusion here. The right conclusion, I think, is the reason we write down these models is to try and capture and test our intuitions, in this case the intuition about crowding towards the center to get votes. But of course these models abstract from very important parts of reality. And the next step to do is what?

[译文 22]

我们有选择你的立场，无法承诺一个立场。好的，所以你的立场可能不被相信。我想声称这与你无法承诺你的政策有关。我们还有其他一些，我们有初选。我们有其他维度。所以我们有高维度。我现在得到主要的几点了吗？我想我得到了大部分主要的。所以有很多事情，当我们看这个模型，有很多事情缺失。很多事情似乎没有被它捕捉到。而这里是我想稍微更深入一点的地方，如果你愿意这么说的话。这很诱人。可以说，看，我们写下了这个模型，它缺少所有这些东西，所以它一定没用。这只是，你知道，我们不应该建立模型。你甚至可能更进一步。你可能会说，建立模型，你总是会遗漏一些东西，所以我们干脆别费心了。而我认为，这是这里错误的结论。正确的结论，我认为，是我们写下这些模型的原因是试图捕捉和检验我们的直觉，在这个案例中是关于向中心拥挤以获得选票的直觉。但当然，这些模型从现实的非常重要的部分中抽象出来了。下一步要做的是？

[段 23]

It’s to say, okay, now let’s try to enrich the model, to add more into the model, and see if you get a different results, and if so, why? So you start with your basic model, then you add in, you enrich the model, and you see if the results change, and that’ll help you explain why you’re getting different results in different settings. So the sort of religious part of today is, is models are always abstractions. You want to use them to see what’s missing and then add things back in to see if they make a difference and if so how. And let’s go through these a little bit. So it turns out that the very first one, voters are not evenly distributed, is certainly true. So that will be true, but it turns out that if you make the voters distributed more realistically, let’s say in a kind of bell curve shape, it makes actually no difference to the results at all. So this is true, but it doesn’t actually change the result. So this one were okay, whoops, the result survives this change. All right? What about the one about many candidates? Well, clearly that matters a lot, right? We know that the many candidates in 92 election and the 2000 election, and we know that not voting is important. So this one I’m going to let you see whether it makes a difference.

[译文 23]

所以这么说，好吧，现在让我们尝试丰富模型，在模型中加入更多内容，看看是否能得到不同的结果，如果能，为什么？那么你先从基本模型开始，然后你加入内容，丰富模型，看看结果是否改变，这会帮助你解释为什么在不同设置下会得到不同的结果。今天这一类带有一点宗教性质的内容是，模型始终是抽象的。你想用它们来看看缺了什么，然后加回去看看是否有影响，如果有的话，影响如何。让我们逐一来看这些。首先，投票者并不是均匀分布的，这是确实成立的。所以这一点是对的，但结果表明，如果你让投票者的分布更现实一些，比如说呈钟形曲线，实际上对结果没有任何影响。所以这一点是对的，但它实际上并没有改变结果。所以这一点——好吧，结果经受住了这个变化。好的？那么多候选人呢？显然这很重要，对吧？我们知道1992年选举和2000年选举中有很多候选人，我们知道不投票很重要。所以这一点我要让你们看看它是否会产生影响。

[段 24]

This one’s going to be on the problem set. All right, so I’m actually going to get you to do this. Let’s go back to the model, add an extra candidate, and see what happens. You can learn whether it makes a difference. I’m not going to give it away. The one about committing to policy. This is important. It’s really hard for candidates to convince you that they really are a left or a right or a centrist candidate. Let me give a historical example outside of America, but in the real world, in England, in the in the 97, in the 80s and 90s, the Labor Party in England had lost election after election after election. And in each election, it would come forward and say, we are not. really a left-wing party. We’re a centrist party these days, and people would say, we don’t believe you. All right? So what Tony Blair, the guy who won the 97th election, did with new labor policy, was he managed to commit to a centrist policy by literally committing to it. He took the, in England you have to publish your economic and financial plans of the government five years and ahead. So he took the financial plans of the conservative government that was then in power five years ahead and he said, these are my financial plans. I’m going to announce these today as my financial pounds five years head.

[译文 24]

这个问题集里会有这一题。好的，所以实际上我要让你们来做这个。让我们回到模型，加入一个额外的候选人，看看会发生什么。你可以了解到它是否会产生影响。我不会直接告诉你们。关于承诺政策的这一点，这很重要。候选人很难让你相信他们真的是左派、右派还是中间派候选人。让我举一个美国以外的历史例子，但在现实世界中，在英格兰，在80年代和90年代，英国工党输掉了一场又一场的选举。在每一次选举中，它都会站出来说，我们并不是真的一个左翼政党。我们现在是一个中间派政党，人们会说，我们不相信你。好吧？所以托尼·布莱尔，这个赢得了1997年选举的人，在新工党政策方面所做的，是他成功地通过真正地承诺来承诺一个中间派政策。在英国，你必须提前五年公布政府的经济和金融计划。所以他拿走了当时执政的保守党政府五年后的金融计划，他说，这些就是我的金融计划。我今天要宣布这些，作为我五年后的财政英镑。

[段 25]

They’re exactly the same policies as the conservative government has. So not only was he choosing a position close to the conservatives, he was choosing exactly their position. And it came down to what somebody else said here, it came, that left election on the other dimension, which might have been character or whatever, all right? And that won, that won by a big way. So nevertheless, even having given you that historical example, we’re going to come back and we’ll look later on in the class at a model in which politicians cannot choose their position. but rather, you know their positions ahead of time. So we’ll do this one. We’ll do this one later. And these other two primaries and higher dimensions, they’re both great points. We’re not going to have time to do them in this course, but if you take a class in the political science department on voting and on elections, on modeling elections, you will see that both of these have been modeled and discussed in great detail. There’s even some empirical work on just how high dimensions are really involved in elections. All right. So this is great stuff for this. Eric. and discussed in great detail. There’s even some empirical work on just how high dimensions are really involved in elections. All right, so this is great stuff for this area of political science.

[译文 25]

这些与保守党政府的政策完全一样。所以他不仅选择了接近保守党的立场，而是选择了他们完全相同的立场。然后这归结为这里其他人所说的，它归结为——左翼选举在另一个维度上，可能是性格或其他什么，好的？而这赢了，这以很大的优势赢了。所以尽管已经给你们举了这个历史例子，我们还是要回来，我们将在后面的课程中看一个模型，在这个模型中政治家不能选择他们的立场，而是提前知道他们的立场。所以我们来做这个。我们稍后会做这个。而另外两个初选和高维度，它们都是很好的观点。我们在这门课程中没有时间来做它们，但如果你在政治科学系选修一门关于投票和选举、选举建模的课程，你会看到这两个都已经被建模并详细讨论过了。甚至有一些实证工作关于高维度在选举中究竟涉及多少。好的，所以这对于政治科学的这个领域是非常好的内容。

[段 26]

All right, I want to inspire you to take some more advanced courses, albeit in another department. All right, so have I convinced you that there’s something you can do with interdilition? We used itridelition in a relatively abstract setting, abstract setting, or not an abstract setting, but a play setting last time, in which we were choosing numbers, and then we talked about it in terms of Hannibal invading Rome, and now we’ve seen how it applies here in an election game, and I think that’s probably enough for intradellation. So what I want to do now is I want to switch course and introduce a new tool. This is what we’ll do typical in the class. We’ll have new tools, we’ll see how they apply, we see what we can learn with them, then we’ll move on to a new tool. So I’m going to switch to a new tool. Any questions at this. point. Everyone happy at this point? Well, happy is too strong. People satisfied at this point. All right, okay. All right, so we’re going to move away from this example, and I’m going to introduce a new subject, and I’ll need several boards, I think. So that we’ll collect them. All right, so that little semi-religious speech about modeling, I think economists, those of you who are economics majors, you’re thinking, yeah, what’s the problem?

[译文 26]

好的，我想激励你们去选修一些更高级的课程，尽管是在另一个系。好的，那么我是否说服了你们 interdiction 有什么可以做的？我们在上一次一个相对抽象的设置中使用了 interdiction——不是抽象的设置，而是一个游戏设置，我们在其中选择数字，然后我们用汉尼拔入侵罗马的术语来讨论，现在我们已经看到它如何应用在这里的选举游戏中，我认为这对于 interdiction 来说大概就足够了。那么我现在想做的是切换课程，引入一个新工具。这就是我们通常在课堂上会做的。我们会有新工具，我们会看到它们如何应用，我们能看到用它们能学到什么，然后我们会转向一个新工具。所以我要切换到一个新工具。在这个时候有什么问题吗？大家现在满意吗？嗯，满意太夸张了。大家现在至少满意吧。好的，好的。那么我们要离开这个例子，我要引入一个新主题，我需要几块黑板，我想。那么我们会收集它们。好的，关于建模的那段有点像布道的讲话，我认为经济学家们，你们这些经济学专业的学生，你们在想，是啊，这有什么问题？

[段 27]

For those of you who are history majors, or for the little science majors, that’s more of an issue, right? So you really want to think about why we’re modeling, what do we get for this exercise? All right, I want to move to a different approach now, and the different approach The different approach to analyzing games is going to be about what we’re going to call Vest Response. And to kick this off, let’s look at an example. the game, very simple game with two players, player one is choosing up, middle, or down, and player two is choosing left or right. And the payoffs are like that as follows. They’re nothing very interesting. They’re just some numbers thought up for the purpose. 4-2 and 2-3. So they’re 5102, 134-1, 4-2 and 2-2 and 2-3. All right. So first of all, let’s try and analyze this game. analyze this game using the tools we’ve learned so far. So does either player have a dominated strategy? Let’s take this. So I claim that neither player has a dominated strategy. Let’s just check. So, for example, you might think down is dominated. You might think down is dominated, but if we look carefully, we see that down actually does better than up, than up against right, and down does better than middle against left. And similarly we can see that right is not dominated because it does better against up than does left, but left isn’t dominated because it does better against middle than does right.

[译文 27]

对于你们历史专业的学生，或者是小理科专业的学生，这就更成问题了，对吧？所以你们真的想思考为什么我们要建模，我们通过这个练习得到了什么？好的，我现在要转向一种不同的方法，而分析博弈的不同方法将是关于我们要称之为 Best Response 的东西。为了启动这个，让我们看一个例子。这个博弈，一个非常简单的博弈，有两个参与者，玩家一选择上、中或下，玩家二选择左或右。收益如下所示。它们没什么非常有趣的。它们只是为了这个目的而想出来的一些数字。4-2 和 2-3。所以它们是 5102、134-1、4-2 和 2-2 和 2-3。好的。那么首先，让我们尝试用我们目前学到的工具来分析这个博弈。那么任何一个玩家有一个被支配的策略吗？让我们看看。我声称没有一个玩家有被支配的策略。让我们检查一下。例如，你可能认为下是被支配的。你可能认为下是被支配的，但如果我们仔细看，我们发现下实际上比上做得更好，在面对右的时候，而下在面对左的时候也比中做得更好。同样地，我们可以看到右不是被支配的，因为它面对上的时候比左做得更好，但左也不是被支配的，因为它面对中的时候比右做得更好。

[段 28]

And so and so forth. You can work through this game and you can check pretty quickly that nothing is dominated. So if the only tool I taught you in this class was dominated strategies and the idea of deletion of dominant strategies, we’d be stuck. Right? Wouldn’t better get started on this game. Nevertheless, it’s a pretty simple game, pretty straightforward game. Imagine your player one in this game, your player one in this game. What do you think you’d do? What would you do if you’re player one? Let’s have a show of hand, actually. So let’s just see. Look at it a bit. No extensions. How many of you would choose up? Raise your hand for up. Raise your hand for up, raise your hands up. But you keep your hand up. You’re at a S-M student. You keep your head up. Oh, your math is wrong. Okay, okay, okay. You took accounting last year. No one’s choosing up, no one’s choosing up at all. Okay. How about middle? One person chose middle? How about down? Oh my goodness. Okay, there’s a massive majority for down. All right. So let’s think about this a second. What if you knew that player two was going to choose left.

[译文 28]

如此等等。你可以仔细检查这个博弈，很快就会发现没有什么是被支配的。所以如果我在这个课教你们的唯一工具是被支配策略和剔除被支配策略的想法，我们就会陷入困境。对吧？不能更好地开始分析这个博弈。尽管如此，这是一个相当简单的博弈，相当直接的博弈。假设你是这个博弈中的玩家一，你在博弈中的玩家一。如果你是玩家一，你会怎么做？让我们举手表决，实际上。让我们看看。有多少人会选择上？选择上的举手。选择上的举手。但是你把手举起来。你是一个经济学学生。你把手举起来。哦，你的数学是错的。好的，好的，好的。你去年选了会计。没有人选上，没有人选上。好的。那么中的呢？一个人选了中？下的呢？天哪。好的，有一个压倒性的多数选择下。好的。让我们想一想这一秒。如果你知道玩家二要选择左，会怎样。

[段 29]

What would you choose if you just happen to know that player two was going to choose left up everyone agree because because five is bigger than one and five is bigger than four so up is and to use a technical term we’ll define formula next time up is the best response to left everyone agree with that right so up as I say does best against left and what does best against right middle middle does best against right after all four is bigger than zero and four is bigger than two so middle does best against right okay so here’s an interesting thing all All of you chose down, but up does best against left, and middle does best against right. Let’s just push this a bit harder. Suppose that you work for somebody. Some of you’re going to go out of Yale and be your own boss, but most of you are going to, like me, are going to end up working for somebody. And suppose this game came along in your line of work and you’d chosen up. And then your boss came to you and looks rather fiercely at you, so this is Rick Levin or somebody looking rather fiercely at me, and says, why did you choose up, right? And what can I do? I can say, oh, I chose up because I thought the other guy was going to choose left.

[译文 29]

如果你们恰好知道玩家二会选择左边，你们会选什么？大家都同意，因为5大于1，5也大于4，所以上是对左边的最佳应对，大家都同意吧？所以如我所说，上对左边表现最好，什么对右边表现最好呢？中间对右边表现最好，毕竟4大于0，4也大于2，所以中间对右边表现最好。好的，这里有个有趣的现象，你们所有人都选了下面，但上对左边表现最好，中间对右边表现最好。让我们再深入一点。假设你受雇于某人。你们有些人会离开耶鲁成为自己的老板，但大多数人，像我一样，最终会受雇于某人。假设这个游戏出现在你的工作领域，而你选择了上。然后你的老板来找你，非常严厉地看着你，就是Rick Levin或者某个这样看着我的人，说，你为什么选上面？那么我能说什么呢？我可以说，哦，我选上面是因为我认为对方会选左边。

[段 30]

I have a way of rationalizing choosing up. I believe the other guy is going to choose left, I can rationalize choosing up. Is that right? So it might save me from getting fired at least for now. Similarly, if I chose middle and President Levin comes along to me and say, why did you choose middle, I can say, okay, I chose middle because I believed that the other guy was going to choose right and middle was the best thing for me to do against right. And once again, if I’m lucky, I won’t get fired. It seems like I’ve managed to rationalize choosing middle. But down, all of you chose down. Pretty much all of you chose down. Down is going to be a little trickier. How am I going to rationalize choosing down? Can I rationalize choosing down? You all chose it. I’m your boss now. Why did you choose down? Somebody hasn’t said anything yet. Can I get the mics up and perhaps the person right in front of you? Stand up and shout? It’s the safer answer if you can’t decide whether or not your opponent’s going to choose left or right because you don’t have to worry about having a payoff of zero. All right. So in some sense it’s safer. It avoids the zero, although I mean it’s true to avoid the zero, but it also doesn’t get the five and the four.

[译文 30]

我有一种方式来合理化选择上面。我相信对方会选择左边，我可以合理化选择上面。对吗？所以这至少可能暂时保住我的工作。类似地，如果我选了中间，校长Levin走过来问我，你为什么选中间，我可以说，好的，我选中间是因为我认为对方会选择右边，而中间是我对右边的最佳应对。同样，如果我运气好的话，我就不会被解雇。看来我已经成功合理化选择中间了。但下面呢，你们都选了下面。几乎所有人都选了下面。下面会有点棘手。我要怎么合理化选择下面呢？我能合理化选择下面吗？你们都选了它。我是你们的老板了。你为什么选下面？有人还没说过话。我能把麦克风递上去吗，也许前面那个人？站起来喊？如果你们无法决定对手会选择左边还是右边，这是个更安全的选择，因为你不用担心得到零回报。好吧。所以在某种意义上它更安全。它避免了零，虽然我的意思是避免零确实是真的，但它也得不到5和4。

[段 31]

But I think the key part of your answer was I might not know, I might not be sure whether my opponent is choosing left or right. For example, I might believe that it’s equally likely that they choose left and right. Is that right? So let’s look at something. Let’s look at my payoffs from choosing these three options, up, middle, and down, if I think it’s equally likely that my opponent will choose left and right. Okay? So what we’re going to look at is the expect the expected payoff of up versus, let me call it a half-half. A half-half being equally likely that my opponent chooses left and right. So what’s my expected payoff from choosing up, where I believe the other person’s going to choose left and right equally? It’s what? It’s a half times five. plus a half times zero for a total of two and a half. Is that right? What about my expected payoff from choosing middle against a half-half? So in this case where I think it’s equally likely that my opponent’s going to choose left or right. So in this case, my expected payoff is a half of one. 1 plus a half of 4 for a total of, again, 2 and 1 half. Everyone happy with that? Is my math correct so far? All right. And how about my expected payoff from choosing down versus a half?

[译文 31]

但我认为你回答的关键部分是，我可能不知道，我可能不确定我的对手是选择左边还是右边。例如，我可能相信他们选择左边和右边的概率是相等的。对吗？让我们来看看一些东西。让我们看看我从选择这三个选项——上、中、下——中得到的回报，如果我认为对手选择左边和右边的概率相等的话。好吗？所以我们要看的是，选择上的期望回报与……让我称之为一半一半。一半一半意味着对手选择左边和右边的概率相等。那么，如果我相信对方选择左边和右边的概率相等，我选择上的期望回报是多少？它是……是二分之一乘以5。加上二分之一乘以0，总共是2.5。对吗？那么我选择中间对抗一半一半的期望回报呢？在这个情况下，我认为对手选择左边或右边的概率相等。所以在这个情况下，我的期望回报是二分之一的1。1加上二分之一的4，总共又是2又二分之一。大家对这个满意吗？我的数学到目前为止对吗？好的。那么我选择下面对抗一半的期望回报呢？

[段 32]

A half. This is going to equal a half of 4. plus a half. This is a half, a half. So this is going to equal a half of four plus a half of two, which is a half of six, so that’s three. So it turns out that, as the gentleman out there said, if I thought it was equally likely that my opponent was going to choose left or right, then actually my best choice, my best response, is to choose down. Not only, not only is it a good compromise, not only is it safe, it’s actually the best I can do. It maximizes my expected payoff. All right, all right. Now we’re not quite there yet. We know that up does best against left, we know that middle does best against right, and we know that down does best if I think it was equally likely that the person’s going to choose left and right. But that’s not the only beliefs I could have. For example, I could believe that it’s twice as likely that the person’s going to choose left as right. It could be that I think they’re twice as likely to choose left as right. So equally likely was probabilities of half and half. What probabilities are associated with thinking it’s twice as likely they’re going to choose left than right? What probabilities are associated with that?

[译文 32]

一半。这将等于二分之一的4。加上二分之一。这是一半，一半。所以这将等于二分之四加二分之二，这是二分之一的六，也就是3。所以事实证明，正如那边那位先生说的，如果我认为对手选择左边和右边的概率相等，那么实际上我的最佳选择、我的最佳应对是选择下面。不仅如此，不仅是一个好的折衷，不仅安全，实际上这是我所能做的最好的。它最大化了我的期望回报。好的，好的。现在我们还没到那一步。我们知道上对左边表现最好，我们知道中间对右边表现最好，我们知道如果我认为对方选择左边和右边的概率相等，下面表现最好。但那不是我可以拥有的唯一信念。例如，我可以相信对方选择左边的概率是选择右边的两倍。可能我认为他们选择左边的可能性是选择右边的两倍。一半一半是概率各半。如果我认为他们选择左边的可能性是选择右边的两倍，那相关的概率是多少？那相关的概率是多少？

[段 33]

Two-thirds, one-third, right? If I thought there was two-thirds probability they’d choose left and one-third probability they’d choose right, then I think it’s twice as likely they’re going to produce left. and I could redo this calculation. And in principle, I could redo this calculation for every single possible probability you could think of. Right? But that would get very boring. So rather than do that, let’s draw a picture. What I want to do is, I want to draw a picture here all right, in which, on the horizontal axis, I’m going to put the probability of the other guy choosing right. It’s my belief that the other guy is going to choose right. Think of this as a belief. All right, and on the vertical axis, I’ll put two vertical axes in just for the sake of things. On the vertical axis, I’ll put my expected payoffs. my expected payoffs. All right. And let’s start by considering my payoffs on this picture. Let’s just to help myself a bit. Let me put some points in here. So I’ve got one, two, three, four, five. One, two, three, four, five. You should probably do the same in your notes. You probably have line paper that makes this easier. Okay, let’s start by plotting my expected payoff from choosing the strategy up. So this isn’t so hard. We know that if I choose up and there’s probability zero that the other guy is going to choose right, that’s the same as saying I choose up and the other guy chooses…

[译文 33]

三分之二，三分之一，对吧？如果我认为他们有三分之二的概率选择左边，三分之一的概率选择右边，那么我认为他们选择左边的可能性是两倍。我可以重新做这个计算。原则上，我可以为你们能想到的每一个可能的概率重新做这个计算。对吧？但那样会非常无聊。所以与其那样做，不如让我们画张图。我想在这里画张图，好吧，在这张图上，在横轴上，我要标上对方选择右边的概率。这是我认为对方会选择右边的信念。可以把这当作一种信念。好的，在纵轴上，我要标上两个纵轴，只是为了方便起见。在纵轴上，我要标上我的期望回报。我的期望回报。好的。让我们从在这张图上考虑我的回报开始。让我帮自己一下。让我在这里标一些点。我有1、2、3、4、5。1、2、3、4、5。你们在笔记里也应该这样做。你们可能有横线纸让这更容易。好的，让我们从标出我选择上这个策略的期望回报开始。这并不难。我们知道如果我选择上，对方选择右边的概率为零，这就等于说我选择上，对方选择了……

[段 34]

Let’s try again. Right, if there’s probability zero that the other guy chooses right, that’s the same as saying that the other guy chooses right. That’s the same as saying that the other guy chooses. guy is going to choose left. Okay. So that’s so the probability zero of the other guy choosing left is the same as that they’re probably zero that they’re, not again, probably zero of their choosing right is the same as saying they just they choose left, so that my payoff from from up against that is given by the top box up there, I get five. All right? So this is my payoff from choosing up against left. All right? And conversely, if there was probability one that the other guy is going to choose right and I choose up, then I get zero. So this point here corresponds to my payoff if I choose up and they choose right. And we already know that if I look at probability a half, which is here, that the payoff I get from choosing, the expected, payoff I get from choosing up against a half half is two and a half. All right. So here’s a third point we can use. Okay. And what is this graph going to look like other than these three points? What’s it going to look like other than these three points?

[译文 34]

让我们再试一次。对，如果对方选择右边的概率为零，这就等于说对方选择了右边。这就等于说对方……对方会选择左边。好的。所以对方选择右边的概率为零等于他们选择左边，所以那是我对抗那种情况从上得到的回报，由上面的那个框给出，我得到5。好吧？所以这是我选择上对抗左边的回报。好吧？反过来，如果对方选择右边的概率是1，而我选择上，那么我得到0。所以这个点对应的是我选择上而他们选择右边时的回报。我们已经知道，如果我看概率一半，也就是在这里，从选择上对抗一半一半中得到的回报是2.5。好的。所以这是我们可以用的第三个点。好的。除了这三个点，这个图会是什么样子？除了这三个点，它会是什么样子？

[段 35]

It’s going to be a straight line. Okay, it’s going to be a straight line. Okay, so you can confirm that for yourself at home, but in fact, if I draw a straight line, this will actually be Correct. So what is this? This is my expected payoff from choosing up against the probability of the other person choosing right. And what is it actually equal to, as to, to confirm the equation, it’s one minus the probability that they choose right. In that case, I’ll get payoff of five, plus the probability that they choose right, in which case I get a payoff of zero. So it doesn’t really matter, but that’s actually the equation of that line. All right. What about choosing middle? Well, once again, we can plot that point. So if the other person’s choosing right with probability zero, that’s the same as saying they’re going to choose left. And if I chose middle against left, I get what? I get one. I should use this pointer. I get one. All right, so that’s here. And if I knew they were choosing right for sure, that’s probably one they’re choosing right. And if I choose middle, I get four. That’s here. And we’ve already agreed that if I think it’s equally likely they’re going to choose right and left. So if there’s probably a half of them choosing right, I get two and a half from choosing middle.

[译文 35]

这将是一条直线。好的，这是一条直线。所以你们可以自己在家验证，但实际上，如果我画一条直线，这实际上是正确的。那么这是什么？这是我从选择"上"策略相对于对方选择"右"策略的概率所获得的预期收益。那么它实际上等于什么，为了确认这个等式，它等于一减去他们选择"右"的概率。在这种情况下，我将获得5的收益，加上他们选择"右"的概率，在这种情况下我获得0的收益。所以这实际上并不重要，但这就是那条直线的方程。好的。那么选择"中"策略呢？好，我们再一次可以标出那个点。所以如果对方以零概率选择"右"，这等同于说他们将选择"左"。如果我选择"中"对抗"左"，我得到什么？我得到1。我应该用这个指示笔。我得到1。好的，就是这里。如果我知道他们肯定会选择"右"，这可能是他们选择"右"的概率为一。如果我选择"中"，我得到4。就在这里。我们已经同意，如果我认为他们选择"右"和"左"的概率相等。那么如果有大约一半的概率他们选择"右"，我选择"中"将得到2.5。

[段 36]

So putting that all together, I know that they must go through here again, and once again, I get a straight line. missed slightly, but that’s more or less right. So what is this? This is the payoff to player one of choosing middle against left. This is the payoff to player one of choosing middle against right. And the line in between, this line here, is the expected payoff to player one of choosing middle as a function of the probability that other people choose right. And we could, again, we could write down the equation. It’s going to be one minus the probability they choose right times one plus the probability that they choose right times four. And again, don’t worry too much about the equation. We’re not really going to do any math here. I’m just putting it in for completeness. All right? Everyone following me so far? Okay. So let’s put in now the third of these lines. I really should use a different color already. Let me switch colors. Let’s put in the line that corresponds to choosing down. So that third line, not to belabor the point, is going to go through here, through four and through two. So this is the payoff from choosing down against lefts, and this is the payoff from choosing down against right. And I got those from looking at these payoffs here and here.

[译文 36]

那么综合以上，我知道它们必须再次经过这里，然后我再次得到一条直线。稍微有点偏差，但大致正确。那么这是什么？这是玩家一选择"中"对抗"左"的收益。这是玩家一选择"中"对抗"右"的收益。而中间的那条线，这条线，是玩家一选择"中"的预期收益，作为其他人选择"右"的概率的函数。我们可以再一次写出方程。它等于一减去他们选择"右"的概率乘以一，加上他们选择"右"的概率乘以四。同样，不要太担心这个方程。我们这里不会真正做任何数学运算。我只是把它放在那里以示完整。好的？到目前为止大家都跟上了吗？好的。那么让我们现在画出第三条线。我应该用一个不同的颜色了。让我换个颜色。让我们画出对应于选择"下"的线。那么这第三条线，不多说，将经过这里，经过4，经过2。所以这是从选择"下"对抗"左"获得的收益，这是从选择"下"对抗"右"获得的收益。我是从看这里的收益和这里的收益得到的。

[段 37]

And in between, once again, it’s a straight line. so here’s the straight line and this line is let’s take it out here. This line is the expected payoff to play a one from choosing down as it depends on the probability that the other person chooses right. And once again we can write down the equation. It’s 1 minus PR times 4 plus PR times to right and I claim that this picture really tells me everything I could possibly want to know when my boss comes along when President Levin or Barry Nailbuff is now walks in the back of the room when my boss comes along and asks me why did you do that right? Why do you do that? I can now say so for example let’s put in these crossing points here For example, what I can say is if I think it’s very likely let’s call this point X if I think that the probability that the other person is going to choose right is less than X then the highest payoff I can get corresponds to this line that’s the highest line and that line corresponds to me choosing up.

[译文 37]

而在中间，又一次，是一条直线。所以这是直线，而这条线让我们把它拿出来。这里这条线是玩家一从选择"下"获得的预期收益，取决于对方选择"右"的概率。我们又一次可以写出方程。它是1减去PR乘以4加上PR乘以"右"的收益。我断言这张图真的告诉我所有我需要知道的东西，当我的老板走过来，当Levin校长或者Barry Nailbuff现在走进房间后面，当我的老板走过来问我为什么你那样做了？正确？你为什么那样做？我现在可以说了举个例子让我们在这里标出这些交叉点。举个例子，我能说的是，如果我认为非常可能——让我们称这个点为X——如果我认为对方选择"右"的概率小于X，那么我能获得的最高收益对应于这条线，这是最高的线，而这条线对应于我选择"上"。

[段 38]

So if I think the probability that they’re going to choose right is less than X then my best response is to choose up Conversely if I think it’s more likely than this crossing point why I think it’s very likely they’re going to choose right it’s more likely than why the probability they’re going to choose right is more than why then the highest line of these three is this one which corresponds to my choosing middle so over here my best response is to choose middle all right let’s put that in so over here So over here, my best response middle, so over here my best response is to choose middle, all right, let’s put that in, so over here my best response is middle, and over here my best response is up, and in between it’s going to turn out that the highest line, so if my belief that the person’s going to choose right is more than X and less than Y, then my highest expected payoff comes from that blue line, and in that case, my best response is down. I can rationalize all three of these decisions and hope that Barry won’t fire me. And we can do more than that, if we’re being nerdy, but I’m not going to be this nerdy today, what we could actually do is we could solve out for the X and for the Y.

[译文 38]

所以如果我认为他们选择"右"的概率小于X，那么我的最优反应是选择"上"。相反，如果我认为比这个交叉点更可能——我认为他们非常可能选择"右"，比Y更可能，如果他们选择"右"的概率大于Y，那么这三条线中最高的是这条，它对应于我选择"中"。所以在这边，我的最优反应是选择"中"。好的，让我们把它标出来。所以在这边，我的最优反应是"中"，而在这边我的最优反应是"上"，而在这之间，结果会是最高的那条线，所以如果我认为这个人选择"右"的概率大于X且小于Y，那么我最高的预期收益来自那条蓝线，在这种情况下，我的最优反应是"下"。我可以为这三个决定都给出合理解释，希望Barry不会解雇我。我们可以做得更多，如果我们很书呆子的话，但我今天不会这么书呆子，我们实际上能做的是我们可以求出X和Y的值。

[段 39]

How would we go about, I mean, I don’t want to do it because I’ll probably get it wrong, but if I wanted to solve out for this X and the Y, since this is a QR class, Let’s talk about it a second. How would I do it? Somebody who’s a, somebody who’s not a math major, tell me how I solve out for the, maybe the math majors can’t do it, actually, it’s too simple. Somebody who is a math major, tell me how I’d solve out for the X and the Y. Let me, it’s going to mic on this person here. Is that the intersection points equal, the equations equal to each other and solve? Good, good, to define it. To solve out for X, X is when, X is when this blue line equals, sorry, this blue line equals this red line, right? Right? So what we’re going to do is take the equations for those two lines. So here’s one of those equations, and here’s the other one, right? Set the P in those equations equal to X. I’ve got two equations in one unknown, so I can, sorry, I’ve got one equation and one unknown, I set these two things equal to each other, I’ve got one equation and one unknown, so I can solve out for X. I think I did that at home.

[译文 39]

我们要怎么做，我是说，我不想做因为我可能会出错，但如果我想求出这个X和Y的值，因为这是一个QR课堂，让我们讨论一下。我要怎么做？某个不是数学专业的人，告诉我我怎么求出——也许数学专业的人反而做不了，实际上这太简单了。某个是数学专业的人，告诉我我怎么求出X和Y的值。让我麦克风对着这个人。是让这些交点相等，让这些方程相等然后求解吗？好的，好的，定义它。要求出X，X是当——X是当这条蓝线等于——抱歉，这条蓝线等于这条红线，对吧？对吧？所以我们要做的就是取这两条线的方程。这是其中一个方程，这是另一个，对吧？把那些方程中的P设为X。我有两个方程一个未知数，所以我能——抱歉，我有一个方程和一个未知数，我把这两个东西设为相等，我有一个方程和一个未知数，所以我可以求出X。我觉得我在家算过了。

[段 40]

If you do that at home, I think, but don’t trust me, you’ll find out that X is equal to a third. Just to repeat, again, I know I’ve got some mathphobics in the audience, so let me just slow down a second. All I’m doing here is I’m saying, look at this equation of the pink line, look at this equation of the blue line, X is when they cross, i.e., they are equal to each other. Take these two equations, put an equal sign between them, replace this PR throughout, with X, I’m going to have one equation and one unknown, and that, even the mathphobics in the audience did in high school. Is that right? All right. Now, we’re going to use, this isn’t, I’m going a bit slower here, but I’m going to use this method next time to introduce you to the most important class, sorry, the most important game we’re going to see in the whole course. In fact, it’s the most important game in the world. What is the most important game in the world? Don’t go yet. Don’t go yet. What’s the most important game in the world? We’re going to use this idea of best response. We’re going to think about it. We’re going to learn from it. In the most important game in the world. What is the most important game in the world?

[译文 40]

如果你们在家算这个，我认为，但不要相信我，你们会发现X等于三分之一。我再说一遍，我知道观众中有些人害怕数学，所以让我放慢一点。我在这里所做的就是，我在说，看这条粉色线的方程，看这条蓝线的方程，X就是它们交叉的时候，也就是说，它们彼此相等。取这两个方程，在它们之间放一个等号，把这里的PR全部替换成X，我将有一个方程和一个未知数，而这个，即使观众中害怕数学的人高中时也做过。对吧？好的。现在，我们要用——这不是——我这里说得慢一点，但我下次要用这个方法向你们介绍最重要的课，抱歉，是整个课程中最重要的博弈。事实上，这是世界上最重要的博弈。什么是世界上最重要的博弈？别走。别走。什么是世界上最重要的博弈？我们要用这个最优反应的概念。我们要去思考它。我们要从它那里学习。在世界上最重要的博弈中。什么是世界上最重要的博弈？

[段 41]

Football. Soccer, actually. Soccer is the most important game. All right? So we’re going to show how this technique is going to help us shoot penalty shots in soccer. Before you go, that’s Barry Nailbuff at the back. He has copies of the book. It’s a great book. Go buy it from him.

[译文 41]

足球。其实是英式足球。英式足球是最重要的运动。好的？那么我们要展示这项技术如何帮助我们踢点球。在你们离开之前，后面那位是Barry Nailbuff。他有这本书的 copies。是很棒的一本书。去向他买吧。