耶鲁大学博弈论公开课 - 第22集 混合策略定义及其在网球比赛中的应用

视频全文转录（中英双语）

以下为完整中英双语转录，已加标点。英文在前，中文在后，逐段对照。

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

[段 1]

So last time we saw this, we saw an example of a mixed strategy, which was to play a third, a third, a third in our rock, paper, scissors game. All right, and today we’re going to be formal, we’re going to define this. We’re going to define mixed strategies, and we’re going to talk about them, and it’s going to take a while. All right, so let’s start with a formal definition. Definition. A mixed strategy, and I’ll develop notation as I’m going along, so let me call it P-sub-I, I being the person who’s playing it, P-sub-I, is a randomization. It’s a randomization. over I’s pure strategies. All right, so in particular, we’re going to use the notation P-sub-I of S-I to be the probability that player I plays S-I, given that he’s mixing using P-I. All right, so P-I-S-I is the probability that P.I assigns to the pure strategy SI, and to the pure strategy, SI, and to immediately refer that back to our example. So, for example, if I’m playing third, a third, a third, in rock-paper scissors, then P. then P-I is a third, a third, a third, and P-I of rock, that would P-I of R is a third. All right, so without belaboring it, that’s all I’m doing here, is developing some notation. All right? Let’s immediately encounter two things you might have questions about.

[译文 1]

上一次我们看到这个例子时，我们看到了一个混合策略的例子，也就是在我们的石头、布、剪刀游戏中以三分之一、三分之一、三分之一的方式出招。好的，今天我们要把这个形式化，我们要给出定义。我们要定义混合策略，然后讨论它们，这会花一些时间。好的，让我们从一个正式的定义开始。定义。一个混合策略，我会边讲边建立符号体系，让我称之为P下标I，I表示执行它的人，P下标I，是一种随机化。是对I的纯策略的随机化。好的，特别是，我们要用符号P下标I of S下标I表示当I使用P-I进行混合时，I出S下标I策略的概率。好的，所以P-I-S下标I是P.I分配给纯策略SI的概率，对于纯策略SI，立即将其与我们之前的例子联系起来。举个例子，如果我在石头、布、剪刀中以三分之一、三分之一、三分之一出招，那么P.I就是三分之一、三分之一、三分之一，P-I of rock，也就是P-I(R)就是三分之一。好的，不赘述了，这就是我在这里做的，建立一些符号体系。好的？让我们立即遇到两个你可能有问题的地方。

[段 2]

So the first is that in principle, P-I-S-I could be zero. So just because I’m playing a mixed strategy, it doesn’t mean I have to involve all of my strategies. All right, I could be playing a mixed strategy on two of my strategies and leave the other one with zero probability. So, for example, again, in rock, paper, scissors, we could think of the strategy, a half, a half, zero. Right, in this strategy, I assign, I play rock half the time, I play paper half the time, but I never play scissors. All right. understand that. And while we’re here, it took at the other extreme. The probability assigned by mixed strategy to a particular SI could be one, right? It could be that I assign all the probability to a particular strategy. What would we call a mixed strategy that assigns probability one to one of the pure strategies? What’s a good name for that? That’s a pure strategy, right? That’s a pure strategy. All right? So notice that we can think of pure strategies as the special case of a mixed strategy that assigned all the weight to a particular pure strategy. So, for example, if P.I. Rock was one, that’s equivalent to saying that I’m playing the pure strategy rock. All right, i.e., a pure strategy. All right, so there’s nothing here. I’m just being a little bit nerdy about developing notation and making sure that everything is in place.

[译文 2]

第一个是原则上P-I-S下标I可以是零。所以仅仅因为我采用混合策略，并不意味着我必须涉及我所有的策略。好的，我可以在我的两个策略上采用混合策略，而让另一个策略的概率为零。举个例子，还是在石头、布、剪刀中，我们可以考虑这个策略，二分之一、二分之一、零。在这个策略中，我分配一半的时间出石头，一半的时间出布，但我从不出剪刀。好的。希望这能帮助你理解。趁这个机会再看看另一个极端。混合策略分配给某个特定SI的概率可以是一，对吧？我可以把所有概率都分配给某个特定策略。我们怎么称呼给某个纯策略分配概率为一的混合策略呢？什么是一个好的名称？那就是纯策略，对吧？那就是纯策略。好的？所以请注意，我们可以把纯策略看作是混合策略的一个特例，即把所有权重都分配给某个特定的纯策略。举个例子，如果P.I的Rock是一，那就等同于说我正在执行纯策略rock。好的，即一个纯策略。好的，所以这里没有什么特别的。我只是在稍微啰嗦地建立符号体系，确保一切都到位了。

[段 3]

And just to point out again, one consequence of this is we’ve now got our pure strategies embedded in our mixed strategies. Right? When I’ve got a mixed strategy, I really am including in those all of the pure strategies. All right. So let’s proceed. All right. So now I want to think about what are the payoffs that I get from mixed strategies. And again, I’m going to go a little slowly because it’s a little tricky at first, and we’ll get used to this. Don’t panic. We’ll get used to this as we go on and as you see them in homework assignments and in class. All right. So let’s talk about the payoffs from a mixed strategy. Payoffs from mixed strategies. payoffs from mixed strategies and in particular what we’re going to worry about are expected payoffs so the expected payoff of the mixed strategy p let’s be consistent and call it PI, the mixed strategy PI, is what? It’s the weighted average it’s a weighted average or a weighted mixture if you like of the expected payoffs of each of the pure strategies in the mix. All right, so this is a long way of saying something and again I think is a little bit obvious, but let me just say it again, the way in which we figure out the expected payoff of a mixed strategy is we take the appropriately weighted average of the expected payoffs I would get from the pure strategies over which I’m mixing.

[译文 3]

只是再指出一点，这样做的一个后果是我们现在把纯策略嵌入到了混合策略中。对吧？当我有一个混合策略时，我真的在其中包含了所有纯策略。好的。让我们继续。好的。现在我想考虑从混合策略中我得到什么收益。同样，我会稍微慢一点讲，因为一开始有点棘手，我们会习惯的。别慌。随着我们在课堂作业和课堂上的应用，我们会逐渐习惯的。好的。让我们谈谈混合策略的收益。混合策略的收益。混合策略的收益，特别是我们要关注的是期望收益，所以混合策略P的期望收益——让我们保持一致，称它为PI，混合策略PI，是什么？它是加权平均，它是对混合中每个纯策略的期望收益的加权平均或加权混合，如果你愿意这么叫的话。好的，这是一种很长的表述方式，我想这也有点显而易见，但让我再说一遍，我们计算混合策略期望收益的方式是，我对我正在混合的纯策略分别会获得的期望收益取适当的加权平均。

[段 4]

All right, to make that less abstract, let’s immediately look at an example. So here’s an example, we’ll come back to several times, but just once today. And this is a game you’ve seen before. Here is the game Battle of the Sexes, in which player A can choose capital, player one can choose capital A and B, and player two can choose little A and B. All right? And what I want to do is, I want to figure out, I want to figure out the payoff from a particular strategies. So suppose that P is what’s being played by player 1, and P is, let’s say, one-fifth, four-fifths. So what do I mean by that? I mean that player 1 is assigning a fifth to playing A and four-fifths. All right? And suppose that Q, so I’m going to use P in Q because it’s convenient to do so rather than calling P1 and P2, so suppose that Q is the mixture that player two’s choosing, and she’s choosing a half-half. So she’s putting a probability a half on little A and probability a half on little B. And just to notice that I switch notation on you a little bit, for this example, to keep life easy, I’m going to use P to be rose mixtures, and Q to be columns mixtures. And the question I want to answer is, what is the expected payoff in this case of P?

[译文 4]

好的，为了让这不那么抽象，让我们立即看一个例子。这里有一个例子，我们会多次回到它，但今天只讲一次。这是你之前见过的游戏。这是一个性别战游戏，其中玩家A可以选择大写的A和B，玩家二可以选择小写的A和B。好的？我想要做的是，我想计算从特定策略中获得的收益。假设P是玩家1正在执行的，P是，比如，五分之一、五分之四。所以我是什么意思？ 我的意思是玩家1分配五分之一给执行A，五分之四。好的？假设Q，所以我用P和Q是因为这样方便，而不是叫做P1和P2，所以假设Q是玩家2选择的混合，她选择二分之一、二分之一。所以她给小写A分配二分之一概率，给小写B分配二分之一概率。只是注意到我稍微切换了符号，为了这个例子，让生活简单一点，我用P来表示行混合，Q来表示列混合。我想回答的问题是，在这种情况下P的期望收益是多少？

[段 5]

What is P’s expected payoff? All right? And the way I’m going to do that is I’m first of all going to ask what is the expected payoff of each of the pure strategies that P involves, over which P is, the pure strategy is involved in P. Right? So to start off, right? So the first step is ask, what is the expected payoff for player one of playing A against Q? And what is the expected payoff for player one of playing B against Q? And what is the expected payoff for player one of playing B against Q? That’ll be our first question, and we’ll come back and construct the payoff of P. All right. So these are things we can do, I think. So the expected payoff of A against Q is what? Well, half the time, if you play A, you’re going to find your opponent is playing little A, in which case you’ll get two. And half the time when you play A, you’ll find your opponent is playing B, in which case you’ll get zero. All right. So let’s just write that up. So I’m going to get two, with probability a half, plus zero with probability a half. All right, everyone happy with that. And that gives me one. Okay, please correct my math unless it’s very easy at the board to make mistakes, but I think that one is right.

[译文 5]

P的期望收益是多少？好的？我要做的是，首先我要问，P所涉及的每个纯策略的期望收益是多少？P在其中涉及纯策略的那个混合。好的？所以首先第一步，问，对抗Q时执行A对玩家一的期望收益是多少？对抗Q时执行B对玩家一的期望收益是多少？对抗Q时执行B对玩家一的期望收益是多少？那将是我们的第一个问题，然后我们再回来构建P的收益。好的。这些是我们能做的，我想。所以对抗Q时A的期望收益是多少？嗯，一半的时间，如果你出A，你会发现你的对手出小写A，在这种情况下你会得到二。一半的时间当你出A时，你会发现你的对手出B，在这种情况下你会得到零。好的。让我们把它写下来。我会得到二，概率二分之一，加上零，概率二分之一。好的，大家都同意这个吧。这给了我1。除非在黑板上很容易犯错，否则请纠正我的计算，但我认为1是正确的。

[段 6]

Okay? Conversely, what if I played B? What’s the expected payoff for the role player of playing B against Q, where Q is a half-half?

[译文 6]

好吗？反过来，如果我出B呢？对抗Q时（Q是二分之一、二分之一）执行B的角色玩家的期望收益是多少？

[段 7]

So half the time, when I play B, I’ll meet a player two playing little A, and I’ll get zero. it’s q where q is a half half so half the time when i play b i’ll meet a player two playing little a and i’ll get zero and half the time i’ll find player two is playing little b and i’ll get one all right so let’s write that up so i’ll get zero half the time and i’ll get one half the time for an average of a half all right okay that’s the first thing i ask and now to finish the job i now want to figure out what is the expected payoff for player one of using p against q that was the question i really wanted to start off with all right what’s the way to think about this well p is one-fifth of the time according to p one fifth of the time player one is playing a and four-fifths of the time player one is playing b is that right so to work out the expected payoff what we’re going to do is we’re going to take one-fifth of the time in which case he’s playing a and he’ll get the expected payoff he would have got from playing a against q and four-fifth of the time he’s going to be playing b in which case he’ll get the expected payoff from playing b against q all right and now just plugging in some numbers to that from above so we’ve got one-fifth of the time he’s doing the expected payoff from a against q and that’s this number we worked out already all right so this number here can come down here one and four-fifth of the time he’s playing b against q in which case his expected payoff was a half so this a half comes in here a half comes in here all right everyone okay so far i’m how constructed it so far is this podium in the way of you guys are you okay let me push it slightly you’re okay all right and so the total here is what it’s going to be uh one-fifth of one plus four-fifths of a half uh four-fifths of a half is two-fifths so i’ve got a total of three-fifths right so the total here is three-fifths all right everyone understand how i did that okay now while it’s here let’s notice something when i played p some of the time i played a and some of the time i played b all right and when i played when i ended up playing a i got a’s expected payoff and when i played b i got b’s expected payoff so the number i ended up with three-fifths must lie between the payoff i would have got from a which is one and the payoff i would have got from b which is a half is that right is that right so three-fifths lies between a half and one everyone okay with that yeah now that’s a very that’s a simple but very general and very useful idea it turns out right the idea here is that the payoff i’m going to get must lie between the expected payoffs i would have got from the pure strategies let me said again in general when i play a mix strategy the expected payoff i get is a weighted average of the expected payoffs of each of the pure strategies in the mix and weighted averages always lie inside the payoffs that are involved in the mix all right all right so let me try and push that simple idea a little harder suppose i was going to take the average height in the class average height in the class average heighten this class.

[译文 7]

所以有一半的时间，当我选择B时，我会遇到一个选择小a的玩家，那我得0分。这是q，q是二分之一。所以有一半的时间当我选择B时，我会遇到一个选择小b的玩家，那我得1分。好，让我们写下来。所以有一半的时间得0分，一半的时间得1分，平均是二分之一。好，这就是我问的第一个问题。现在要完成这道题，我想算出玩家一使用p策略对抗q策略的期望收益。这才是我真正想要开始的问题。好的，怎么思考这个问题呢？p是五分之一的时间，根据p，玩家一有五分之一的时间在选择a，五分之四的时间在选择b，对吧？所以要计算期望收益，我们取五分之一的时间，在这种情况下他选择a，他会得到选择a对抗q的期望收益，然后五分之四的时间他选择b，他会得到选择b对抗q的期望收益。好，现在把上面算出的数字代入。所以五分之一的时间，他得到a对抗q的期望收益，就是这个我们之前算出的数字。所以这个数字可以写在这里。五分之四的时间他选择b对抗q，他的期望收益是二分之一。所以这个二分之一写在这里。二分之一写在这里。好，大家都跟上了吗？我到目前为止建构的是这样的。讲台挡着你们了吗？你们还好吗？让我稍微推一下。你们还好吗？所以这里的总数是 uh 五分之一乘以1加上五分之四乘以二分之一。五分之四乘以二分之一是五分之二，所以总数是五分之三。所以总数是五分之三。大家都理解我怎么算的吗？好，现在在这里我们注意到，当我使用p时，有时我选择a，有时我选择b。当我最终选择a时，我得到a的期望收益；当我选择b时，我得到b的期望收益。所以我得到的数字五分之三必然介于从a得到的收益（1）和从b得到的收益（二分之一）之间，对吧？五分之三介于二分之一和1之间，大家都同意吗？好，这是一个非常简单但非常普遍且非常有用的想法。确实如此，这里的想法是，我将要获得的收益必然介于我本来能从混合中的每个纯策略获得的期望收益之间。让我再说一遍：一般来说，当我使用混合策略时，我获得的期望收益是混合中每个纯策略期望收益的加权平均值，而加权平均值总是位于混合中所涉及的所有收益之间。好，好，让我试着把这个简单的想法推得更深入一些。假设我要取这个班的平均身高。

[段 8]

So let me just, rather than use the class, let us use some TAs here. So let me get the, let me get these three TAs to stand up a second. All right. And suppose I wanted to figure out the average height of these three TAs, all right. So it’s not close to go, so I used to see what’s going on here. So I think from where I’m standing, I’ve got that Alley is the tallest and Mercer is the smallest, is the smallest, is that right? So I don’t know instantaneously what this average would be, but I claim that any weighted average of their three heights is going to give me a number that’s somewhere between the smallest height and height of the three. which is Myrto’s height, and the tallest height of the three, which is Ale’s height. Is that right? Is that correct? All right, so that’s a pretty general idea. Thanks, thanks guys. I’ll come back in a second. All right. And let’s think about this somewhere else. I think about the batting average of a team. All right, let’s think about the team batting average in baseball, right? Let’s think, let’s use the Yankees, for example. We know that the team batting average, the average batting average of the Yankees, I don’t know what it is instantly. I didn’t look it up this morning, but I know it lies somewhere between the player who has the highest batting average, which I’m guessing is Jeter, I’m guessing, and the lowest, and the person on the team who has the lowest batting average, who’s probably one of the pitchers who played, who batted a few times in one of those inter-league games.

[译文 8]

好，让我用这些TAs而不是这个班。让我请这三位TAs站起来一下。好。假设我想计算这三位TAs的平均身高。好。从我站的地方看，我觉得Alley最高，Mercer最矮，对吧？所以我不知道这个平均数会是多少，但我断言他们三个身高的任何加权平均值都会给出一个介于三个人中最矮的身高（也就是Myrto的身高）和三个人中最高的身高（也就是Ale的身高）之间的数字。对吧？大家都同意吗？好，这是一个相当普遍的想法。谢谢，谢谢你们。我马上回来。好，让我们从另一个角度思考。我想想一支棒球队的打击率。好，让我们想想棒球中球队的平均打击率，对吧？让我们想想，用洋基队为例。我们知道球队的平均打击率，我不知道它是多少。我今天早上没有查，但我知道它必然介于球队中打击率最高的球员（我猜是Jeter）和最低的（可能是某位在联盟间比赛中打过几次的投手）之间。

[段 9]

All right? It would be better if I’d use the Mets, but I’d I feel I should take pity on Mets fans this week and not mention them. All right, right? So this is a very simple idea. It’s deceptively simple. It says averages, weighted averages lie between the highest thing over which you’re averaging and the lowest thing over which you’re averaging. All right? Everyone okay with that idea? All right? Now, this very simple idea is going to have an enormous consequence. And here’s the enormous consequence. Simple idea, big consequence. big consequence. So there’s going to be a lesson that follows from this incredibly simple idea. And this is the lesson. If a mixed strategy, if a mixed strategy is a best response, right? So if a mixed strategy is the best thing you can be doing, then each of the pure strategies in the mix I’m being a little bit loose here but I mean assigned positive probability in the mix for those people who are nerd enough to worry about it each of the pure strategies in the mix must themselves be best responses. So in particular, in particular, each must yield the same expected payoff. All right, so here’s a big conclusion that follows from that incredibly simple idea about average. is lying between the highest one and the lowest one. And let’s draw ourselves from that lesson to this big conclusion.

[译文 9]

好吧？如果我用大都会队会更好，但我认为这周我应该同情一下大都会队的球迷，就不提他们了。好吧？好的？这是一个非常简单的想法。它看似简单实则深奥。它表明加权平均值必然位于所平均的最高值和最低值之间。好吧？大家都理解这个想法吗？好？现在这个非常简单的想法将会有巨大的影响。这就是巨大的影响。简单的想法，巨大的影响。巨大的影响。所以这个 incredibly 简单的想法之后将会有一个 lesson。这就是 lesson。如果一个混合策略，如果一个混合策略是最佳反应，对吧？所以如果一个混合策略是你能做的最好的事情，那么混合中的每个纯策略——我在这里稍微随意一点，但我是说在混合中被赋予正概率的——对于那些足够 geek 以至于担心这个问题的人来说，混合中的每个纯策略本身必须也是最佳反应。所以特别是，特别是，每个都必须产生相同的期望收益。好，所以这里有一个从这个 incredibly 简单的关于平均数介于最高和最低之间的想法推导出的重大结论。让我们从这个 lesson 推导出这个重大结论。

[段 10]

What is the conclusion? The conclusion is, if a mixed strategy is a best response, if it’s the best thing I can do is to play a mixed strategy, then each of the pure strategies which I’m playing in that mix, which I’m assigning positive probability to in that mix, must themselves be best responses. All right. In particular, each of them therefore must yield the same expected payoff. So let’s go back to our example. Sorry, guys, can I steal my three TAs again? And suppose the game, suppose the thing I’m involved, and I should have made this easier before, let me come down a little bit, I’ll stand above here. This is good, it’s good, okay? So suppose the game I’m involved in, the payoff in the game is a game in which I have to choose the tallest group of my TAs. So my payoff is going to be the average height of whichever subgroup of my TAs I pick. All right, and these are my three choices. So if I pick more than one of them, I’m going to get a weighted average, that’s a mixed strategy. My aim here is to maximize the height of whatever subgroup I pick. All right. So in this game, here’s my three choice. Here’s my three pure strategies. My three pure strategies are to pick Merto, Alley, or Jake.

[译文 10]

结论是什么？结论是，如果一个混合策略是最佳反应，如果玩混合策略是我能做的最好的事情，那么在这个混合策略中我赋予正概率的每个纯策略本身必须是最佳反应。好。特别地，因此每个必须产生相同的期望收益。好，让我们回到我们的例子。对不起，各位，我能用一下我的三位TAs吗？假设游戏，假设我参与的游戏，我应该之前把它弄得更容易一些。让我稍微下来一点，我站在这里。这是好的，没问题，好吧？所以假设我参与的游戏，游戏的收益是一个我必须选择我的TAs中最矮组的游戏。所以我的收益将是我选择的我的TAs子组的平均身高。好，这是我的三个选择。所以如果我选择超过一个，我会得到一个加权平均值，那就是混合策略。我的目标在这里是最大化我选择的任何子组的身高。好，在这个游戏中，这是我的三个选择。这是我的三个纯策略。我的三个纯策略是选择Merto、Alley或Jake。

[段 11]

All right, there’s my three pure strategies. And my mixture, I could mix these two, I could mix these two, I could mix all three. But remember, my payoff here is to get the group, the average, as high as I can. So how am I going to get the average as high as I can? How am I get the average as high as I can? He’s cheating, yeah. I get the average as high as I can. I’m going to kick out Mertu for a start, right? Sorry, because Mertu’s just bringing down the average. Is that right? Average height, I should say. All right, there’s nothing, right? And actually, all right? And I think I’m going to kick out Jake as well, I think. Can you people tell them from the front? I think I’m probably going to kick out Jake as well because that way I just have. alley, right? So if it was the case I was picking both of them, it would have to be they were equally tall, right? But the… from the front, I think I’m probably going to kick out Jake as well, because that way I just have Alley. So if it was the case I was picking both of them, it would have to be they were equally tall, right, but since they’re not equally tall, I should just pick the best one.

[译文 11]

好的，这是我的三个纯策略。我的混合策略，我可以混合这两个，我可以混合这两个，我可以混合全部三个。但记住，我的收益在这里是让这个组合的平均身高尽可能高。我怎样才能让平均身高尽可能高？我怎样才能让平均身高尽可能高？他在作弊，是的。我要让平均身高尽可能高。我首先要踢掉Mertu，对不起，因为Mertu只会拉低平均身高。对吧？平均身高，我应该说。好的，没什么？实际上，好吧？从前面看，我认为我也可能会踢掉Jake，因为那样我就只剩下 Alley了。所以如果我要同时选他们两个，他们必须身高相同，对吧？但从前面看，我认为我也可能会踢掉Jake，因为那样我就只剩下 Alley了。所以如果我要同时选他们两个，他们必须身高相同，对吧，但因为他们身高不同，我应该只选最好的那个。

[段 12]

Let’s go back to my Yankees example. If I want to pick a sub-team of the Yankees, I’m allowed to pick any number of people to have the highest batting average in that sub-team. The way to do it is to find the Yankee who has the highest batting average and just pick him. Let’s do one more example. Let me use the front row of students here. So here’s my… Can I get this front row of students to stand up a second? This part of the row. And suppose my aim in life is to construct the highest average GPA. I’m not going to embarrass these guys and ask them what their GPAs are. So my aim in life here is to pick some subgroup of these 1, 2, 3, 4, 5, 6. seven, eight students, such that the average of that, the average GPA of that subgroup is as high as I can make it. All right? So what will I do here? So this being Yale, I’ll just find the people who have the 4.0 GPAs, and just pick them. Is that right? All right, you might think, well, why not include somebody who has a 3.9 GPA? That’s pretty good. So why not? Because if there’s anybody in this group who has a 4.0 GPA, I’d do, better just to pick that person. The 3.9 person would just be pulling down the average.

[译文 12]

让我们回到我的洋基队例子。如果我想从洋基队中挑选一个子团队，我可以选择任意数量的人，使得该子团队中拥有最高打击率的人就是我的选择。方法是找到洋基队中打击率最高的人，然后只选他。让我再举一个例子。让我用这里的前排学生。所以这是我的……我能请这排前排的学生站起来一下吗？这排的这边。假设我的人生目标是构建最高的平均GPA。我不会为难这些家伙问他们GPA是多少。所以我这里的人生目标是，从这1、2、3、4、5、6、七、八个学生中挑选一个子群体，使得该子群体的平均GPA尽可能高。好吧？那我该怎么做？因为这是在耶鲁，我就找那些有4.0 GPA的人，然后只选他们。对吗？好吧，你可能会想，嗯，为什么不把有3.9 GPA的人也算进来？那也已经很好了。那为什么不呢？因为如果这个群体里有人有4.0 GPA，我最好只选那个人。3.9的人只会拉低平均值。

[段 13]

Now, suppose there’s nobody with a 40 GPA, and suppose it’s the case that three of these guys, three of these people, let’s say these three people, have a 3.9 GPA. So these three have 3.9 GPA, imagine that. And these other people, they’ve got horrible grades like B-plus somewhere, all right? All right, right? So these are our future law school students, and these are the people who are, who knows what they can end up doing, being president probably. All right, right, right, right, right. So to construct the group with the highest average GPA, what am I going to do? Well, first I’ll throw out all these guys with low GPA, so they can all sit down. Right, and I’ll look at these last three, and these last three, if they’re all in the group, they better all have the same GPA, right? Why not? If I’m trying to maximize the average of my group, if any of them had a lower GPA, I should kick them out. And if one of them has a higher GPA, I should kick them out. And if one of them has a higher GPA, than the other two, I should kick out both the other two. So if I’m including all three of them them must have the same GPA, which I’m going to assume is 3.9, to assume you can still make it into law school.

[译文 13]

现在，假设没有人有4.0 GPA，假设这三个人，这几个人，比如这三个人，有3.9的GPA。假设这三个人有3.9的GPA。而其他这些人，他们的成绩很差，比如B+左右，好吧？好吧，对吧？所以这些是我们未来的法学院学生，而这些是那些谁知道他们最终能做什么的人，也许能当总统。好吧，对，对，对，对，对。要构建平均GPA最高的群体，我该怎么做？嗯，首先我会淘汰所有GPA低的人，所以他们都可以坐下了。对，我会看最后这三个人，如果这三个人都在群体里，他们最好都有相同的GPA，对吧？为什么？如果我试图最大化我群体的平均值，如果其中任何一个人的GPA较低，我就应该把他们踢出去。如果其中一个人的GPA较高，我就应该把他踢出去。如果其中一个人的GPA高于另外两个人，我就应该把另外两个人都踢出去。所以如果我把这三个人都包括进来，他们必须有相同的GPA，我假设是3.9，足以让他们还能进入法学院。

[段 14]

All right? Ever understand that? Yep. Yep. Okay, thanks, thanks, guys. all right? So that’s the, that’s the way I want to think about this. All right? So the idea here is, if I’m using a mixed strategy as a best response, right, it must be the case that everything on which I’m mixing is itself best. And the reason is, if it wasn’t, kick out the thing that isn’t best, and my average will go up. All right. All right. So that leads us to the next idea, but before I do, just for formality, let me add a definition. The definition is this. A mixed strategy profile. What I’m going to do now, I’m going to define Nash equilibrium again, just we have it on a note somewhere. So a mixed strategy profile, should be a hyphen there, P1 star, P2 star, all the way up to PN star, is a mixed mixed strategy Nash equilibrium if for each player I, so for each player I, over here, for each player I, I, each player I, which player I, that player I, that player’s mixed strategy, P.I. is a best response for player I, to the strategies everyone else is picking P-minus I-star. So I’m exploiting our, by now, well-developed notation for player’s strategies. Let me get some more short. All right. So this definition of Nash equilibrium, it’s exactly the same as the definition of of Nash equilibrium we’ve been using now for several weeks, except everywhere where before we saw a pure strategy, I’ve, which was an A, I have replaced it with a P.

[译文 14]

好吗？明白了吗？明白了。好的，谢谢，谢谢，你们。好吧？所以这就是我想思考这个问题的方式。好吧？所以这里的想法是，如果我使用混合策略作为最佳回应，对，那么我所混合的每一个策略本身都必须是最优的。原因在于，如果不是，我就把不是最优的那个去掉，我的平均值就会上升。好吧。好吧。所以这引导我们到下一个想法，但在继续之前，为了正式性，让我加一个定义。定义是这样的。混合策略配置文件。我现在要做的，是重新定义纳什均衡，就像我们在某个笔记上看到的那样。所以混合策略配置文件，这里应该有个连字符，P1星、P2星，一直到PN星，是一个混合策略纳什均衡，如果对于每个玩家I，对于这里的每个玩家I，每个玩家I，哪个玩家I，那个玩家I，那个玩家的混合策略，P.I.是玩家I相对于其他所有人所选择的策略P减I星的最佳回应。所以我利用了我们现在已经相当完善的玩家策略符号。让我再简短一些。好吧。所以这个纳什均衡的定义，它和我们几周以来一直使用的纳什均衡的定义完全一样，只是原来看到纯策略A的地方，我用P替换了它。

[段 15]

It’s the same definition, except I’m using mixed strategies instead of pure strategies. But an implication… strategy, which was an S, I have replaced it with a P. It’s the same definition, except I’m using mixed strategies instead of pure strategies. But an implication of our lesson is what? It’s that if P.I. Star is part of a Nash equilibrium. So if P.I. Star is a best response to whatever else is doing, P. Minus I Star, then each of the pure strategies involved in P.I. Star, must itself be a best response. All right? So an implication of the lesson is the lesson implies the following. If P.I. Star of a particular strategy is positive. In other words, I’m using this strategy in my mix, then that strategy is also a best response. to what everyone else is doing. All right. Okay. So from a math point of view, this is the big idea of the day. This board. Okay, and if you’re having trouble reading this at the back, trust me, I’ve written that up on the handout that will appear magically on the computer at the end of class. And at the moment you’re staring at this, it’s all a bit new. And as well as being, new, you’re saying, okay, but so what? Why do I care about this seemingly mundane fact? All right. And the reason we’re going to turn out to care about this seemingly mundane fact is that this fact is going to make it remarkably easy to find Nash equilibrium.

[译文 15]

这是相同的定义，只是我使用混合策略而不是纯策略。但我们这节课的一个含义是……策略，也就是S，我用P替换了它。这是相同的定义，只是我使用混合策略而不是纯策略。但我们这节课的一个含义是什么？那就是如果P.I.星是纳什均衡的一部分。如果P.I.星是对其他所有人的行为的最佳回应P减I星，那么P.I.星中所涉及的每个纯策略本身都必须是一个最佳回应。好吧？所以这节课的一个含义是，这节课意味着以下内容。如果某个特定策略的P.I.星为正。换句话说，我在我的混合中使用了这个策略，那么该策略本身也是一个最佳回应。对其他所有人正在做的。好吧。好的。从数学的角度来看，这是今天的大想法。这块黑板。好的，如果你在后面看不清楚，相信我，我已经把它写在讲义上了，下课后会神奇地出现在电脑上。当你们盯着这个看的时候，这一切都有点新鲜。而且除了新鲜之外，你可能会说，好吧，那又怎样？我为什么要关心这个看似平淡无奇的事实？好吧。我们最终会关心这个看似平淡无奇的事实的原因是，这个事实将使找到纳什均衡变得非常容易。

[段 16]

All right. This fact, this lesson, this idea that if I’m playing a pure strategy as part of the mix, it must itself be a best response. That’s going to be the trick we’re going to use in finding mixed strategy and nasty equilibrium. All right. Now the only way I can illustrate that to you is to do it. So I’m going to spend the rest of today just doing that. I’m going to look at a game and we’re going to go through this game. We’ll discuss it a little bit because it’s a fun game and we’re going to find the mixed strategy equilibria of this game. Okay, everyone know where we’re going? I want to make sure before I go on, are people looking very sort of deer in the headlamps? That was a lot of formality to get through in a short period of time. So how, anyone want to ask a question at this point? A UK? Okay to go on? All right? So just remember that the conclusion here comes from this very simple idea. The simple idea is the payoff to a weighted average must lie between the best and worst thing involved in the average, and therefore, if I’m including things, in there as part of a best response, they must all be good. That’s the simple idea. This is the dramatic conclusion.

[译文 16]

好吧。这个事实，这节课，这个想法，如果我作为混合策略的一部分在玩一个纯策略，它本身必须是一个最佳回应。这将成为我们在寻找混合策略和纳什均衡时使用的诀窍。好吧。现在我说明这一点的唯一方法就是做给你们看。所以我今天剩下的时间都会这样做。我将看一个游戏，我们将经历这个游戏。我们会稍微讨论一下，因为它是一个有趣的游戏，我们将找到这个游戏的混合策略均衡。好吧，大家知道我们要做什么吗？在我继续之前，我想确保人们对接下来要讲的内容感到非常迷茫？短时间内经历了这么多形式化的东西。嗯，有谁在这个时候想问一个问题吗？一个英国人？好吧，可以继续吗？好吧？所以记住，这里的结论来自这个非常简单的想法。简单的想法是，加权平均的收益必须介于平均值中所涉及的最佳和最差事物之间，因此，如果我包括了一些东西，作为最佳回应的一部分，它们一定都是好的。这就是简单的想法。这是戏剧性的结论。

[段 17]

All right, so any way to prove this to you, and any way to prove you this is useful, is to go ahead and do it. What I’m going to do is, I’m going to clean these boards, and I’m going to start showing an example. And again, don’t panic.

[译文 17]

好吧，总之，证明这一点给你们看的任何方式，以及证明这有用的任何方式，就是继续去做。我要做的是，擦掉这些黑板，然后开始展示一个例子。再说一次，不要惊慌。

[段 18]

I think a lot of people at this part of the class have a tendency to panic because it’s a new. idea and it seems like a lot of math around none of it’s very hard math it’s all kind of arithmetic it’s just this idea of not panicking all right so the example I want to look at is going to be from tennis and I’m going to consider a game within a game played by two tennis players and let’s call them Venus and Serena Williams all right so Venus and Serena Williams so a couple of years ago we used to use Venus and Serena Williams for this example and then for a while I worried that you wouldn’t even remember who Venus and Serena Williams were until we picked any two random Russians but now we’re back seems like we’re back to picking Venus and Serena all right so the game within the game is this suppose that they’re playing and Serena is at the net right serena is at the net and the ball is on venus’s court and venus has reached the ball and venus has to decide whether to try to hit a passing shot uh past serena on serena’s left or on serena’s right and notice i’m going to exclude the possibility of throwing up a lob for now all right just to make this manageable so basically the choice facing facing venus is should she try to pass serena to serene is left which is up a lob for now, all right, just to make this manageable.

[译文 18]

我认为班上这部分人很容易恐慌，因为这是一个新概念，看起来涉及很多数学知识，但其实都不难，全是算术，关键就是不要恐慌。好的，我想举的例子来自网球，我要考虑两个网球选手之间的一局小游戏，我们姑且称他们为Venus和Serena Williams。好，Venus和Serena Williams，几年前我们常用Venus和Serena Williams这个例子，后来有段时间我担心你们根本不记得Venus和Serena Williams是谁了，直到我们随便挑两个俄罗斯选手，但现在又回来了，似乎我们又回到选择Venus和Serena了。好，这个局中局是这样的，假设她们正在比赛，Serena在网前，Serena在网前，球在Venus的场地这边，Venus已经接到球，Venus必须决定是否尝试打一个穿越球，绕过Serena的左侧或右侧。注意，我暂时不考虑挑高球的可能性，好吧，这样才好处理。所以基本上Venus面临的选择是，她应该尝试把球穿越到Serena的左侧，也就是挑高球，哦不对，我再重复一遍，好吧，这样才好处理。

[段 19]

So basically, the choice facing Venus is, should she try to pass Serena to Serena’s left, which is Serena’s backhand side, or to Serena’s right, which is Serena’s forehand side? All right, everyone, do people are familiar enough with tennis to understand what I’m talking about? Yeah, yeah? So we’re going to assume this is Wimbledon, otherwise no one would be at the net to start with, I guess. This is at Wimbledon, all right, and let’s try and put up some payoffs here. All right, so these are going to be the payoff. So I think that this example is originally due to Dixit, but it doesn’t, it’s not a big deal. I think this example is due to Dixit and Skee. All right, so here’s some numbers, and I’ll explain the numbers in a minute. So this is 50-50, 80-20, 90-10, and 2080. Okay? So what are these numbers? So first of all, let me just explain what the strategies are. So I’m assuming the row player is Venus and the column player is Serena. I’m assuming that if Venus chooses L, that means she attempts to pass Serena to Serena to Serena’s left. All right, well orient things from Serena’s point of view. And if she hits right, that means she’s attempting to play. pass Serena on Serena’s right. And if Serena chooses L, that means she cheats slightly towards her left.

[译文 19]

所以基本上，Venus面临的选择是，她应该尝试把球穿越到Serena的左侧，也就是Serena的反手侧，还是到Serena的右侧，也就是Serena的正手侧？好的，大家对网球足够熟悉，能理解我在说什么吧？是的，是的？那么我们假设这是在温布尔登，否则一开始就没人会在网前，我猜。这是在温布尔登，好吧，我们试着列出一些收益。好的，这些就是收益。我认为这个例子最初出自Dixit，但没什么大不了的，我认为这个例子出自Dixit和Skeen。好的，这里有一些数字，我等会儿解释这些数字。这是50-50，80-20，90-10，还有2080。好吗？这些数字是什么意思？那么首先，让我解释一下策略是什么。我假设行玩家是Venus，列玩家是Serena。我假设如果Venus选择L，意思是她尝试把球穿越到Serena的左侧。好，让我们从Serena的角度来定向。如果她打右侧，意思是她尝试把球穿越到Serena的右侧。如果Serena选择L，意思她稍微向左侧移动。

[段 20]

Not cheats in the sense of breaking the rules, but cheats in terms of where she’s standing or leaning. And if she chooses right, that means she cheats slightly towards her right. So this is cheating towards her back hand, and this is cheating towards her forehand, assuming she’s right-handed, which she, in fact, is. Okay. And what do these numbers mean? So let’s start with the E. easy ones. So if Venus chooses left and Serena chooses right, then Serena has guessed wrong, is that correct? Right? Serena’s guessed wrong. In which case Venus wins the point 80% of the time and Serena wins it 20% of the time. And conversely, if Venus chooses right and Serena chooses left, then again, Serena has guessed wrong. And this time, Venus wins the point 90% of the time, and Serena wins the point 10% of the time. All right, and this should be a familiar idea by now, but why is it the case that these 90s and 80s are not 100%? Why is it the case that if Serena guesses wrong, Venus doesn’t win 100% of the time? Anybody? Sometimes we can show hands, and we get some mics up? Why hasn’t it 100% here? Somebody? Yeah, Patrick? Wait for the mic. Sometimes she hits it out of balance when she serves? Right. This isn’t even a serve. This is a passing shot, but the same is true.

[译文 20]

不是作弊违反规则的意思，而是指她站位或身体倾斜的位置。如果她选择右，意思她稍微向右侧倾斜。所以这是向她的反手侧倾斜，这是向她的正手侧倾斜，假设她是右手选手，事实上她确实是。好的。这些数字是什么意思？让我们先从简单的开始。如果Venus选择左，Serena选择右，那么Serena猜错了，对吗？对？Serena猜错了。这种情况下，Venus赢得这一分80%的时间，Serena赢得20%。反过来，如果Venus选择右，Serena选择左，那么同样Serena猜错了。这种情况下，Venus赢得这一分90%的时间，Serena赢得这一分10%。好的，这应该是一个你们现在熟悉的概念了，但为什么这些90和80不是100%？为什么Serena猜错时，Venus不是100%赢得这一分？有人知道吗？有时候我们可以举手，我们拿几个麦克风上来？为什么这里不是100%？有人？是的，Patrick？等一下麦克风。有时候她击球失去平衡时？对了。这甚至不是发球，这是穿越球，但道理是一样的。

[段 21]

So sometimes you’re just successfully going to hit it past Serena, but the ball’s going to sail out. All right? So that happens 10% of the time here and 20% of the time here. Look at the other two boxes. If Venus hits Serena’s left and Serena guess is left, then we’re going to assume that Serena’s going to reach the ball and make a volley, but her volley only manages to go in, get over the net and go in, half the time. So the payoffs are 50-50. Half the time Venus wins the point and half the time Serena wins the point. And conversely, if Venus hits the ball to Serena’s right and Serena guesses correctly and chooses rights, then we’re in this box. Once again, Serena has guessed correctly and she’s going to successfully reach the volley. she gets it in 80% of the time. So Venus wins the point 20% of the time and Serena wins it 80% of the time. All right. So just to finish up the description of the game here, notice that we’re assuming that Serena is a little better at volleying to her right than she is volleying to her left. Right? So this is her forehand volley, and we’re going to assume that that’s stronger than her backhand volley. And conversely, we’re assuming that Venus’s passing shot is a little better when she shoots it to Serena’s left than when she shoots it to Serena’s right.

[译文 21]

所以有时候你成功地穿越了Serena，但球会出界。好吗？所以这里这种情况发生10%的时间，这里是20%的时间。看看其他两个格子。如果Venus打向Serena的左侧，Serena也猜左侧，那么我们假设Serena会够到球并打出截击，但她的截击只有一半的时间能成功过网并落在界内。所以收益是50-50。一半的时间Venus赢得这一分，一半的时间Serena赢得这一分。反过来，如果Venus把球打向Serena的右侧，Serena猜对了并选择右侧，那么我们就在这个格子里。Serena再次猜对了，她会成功地够到截击。她有80%的时间能把球打回去。所以Venus赢得这一分20%的时间，Serena赢得这一分80%的时间。好的。让我把这个游戏的描述讲完，注意我们假设Serena在向右侧截击方面比向左侧截击稍微强一些。对吧？所以这是她的正手截击，我们假设这比她的反手截击更强。反过来，我们假设Venus的穿越球在打向Serena左侧时比打向右侧时稍微更好一些。

[段 22]

This is her cross-court passing shot, and this is her down-the-line passing shot. So none of that fine detail matters a great deal, but just if you’re interested, that’s why the numbers come from. Okay? I’m not claiming this is true data, by the way. I made up these numbers. Actually, I think Dix had made up these numbers. I forget where I got them from. All right. So, okay, everyone understand the game. All right. So now imagine, either imagine you… Actually, I think Dixson made up these numbers. I forget where I got them from. All right. So, okay, everyone understand the game. All right? So now imagine, either imagine you are Venus or Serena, or imagine, perhaps, more realistically, that you’ve become Venus or Serena’s coach. Do you have any members of the tennis team here? No, no? Well, imagine you’ve become their coach, all right? So you take this class, and then you apply to replace their father as being their coach. All right? That’s a tough assignment, I would think. So an obvious question is, you are coaching Venus before Wimbled and you know this situation’s going to arise and you might want to coach Venus on what should she do here? Should she try and pass Serena down the line or should she try and hit the cross-court volley? cross-court passing shot. And notice that this is a question of, should you, Venus, play to your strength, which is the cross-court passing shot? or should you play to Serena’s weakness?

[译文 22]

这是她的斜线穿越球，这是她的直线穿越球。这些细节并不重要，但如果你感兴趣的话，这就是这些数字的来源。好吗？我并不是说这是真实数据，顺便说一句。这些数字是我编的。实际上，我认为Dixit编过这些数字。我忘了从哪里拿的了。好的。那么，好，大家理解这个游戏了吧？好的。现在想象，你要么想象你是Venus或Serena，要么更现实地想象，你已经成为Venus或Serena的教练。你们这里有网球校队的成员吗？没有？没有？那么想象你成为了她们的教练，好吗？你上了这门课，然后你申请取代她们的父亲成为她们的教练。好吧，我认为这是一个艰巨的任务。那么一个显而易见的问题是，你在温布尔登之前指导Venus，你知道这种情况会出现，你可能想指导Venus在这里应该怎么做？她应该尝试直线穿越Serena还是应该尝试斜线穿越球？斜线穿越球。注意，这是一个问题，你应该，Venus，发挥你的强项，也就是斜线穿越球？还是应该攻击Serena的弱点？

[段 23]

All right? Which is, which would be to hit it to Serena’s backhand. Right? Playing to your strength is to choose right, and playing to Serena’s weakness is to choose left. And conversely for Serena, should you lean towards your strength, which I guess is leaning to the right, or should you lean towards Venus’s weakness, which I guess is leaning right? And when you look at coaching manuals on this stuff, or you listen to the terrible guys who commentate on tennis for ESPN, oh no, I’m getting in trouble again, very nice guys who commentate on tennis for ESPN, they say just incredibly dumb things at this point. Right? They say things like, you should always play to your strengths and don’t worry about the other person’s weakness. Right? I think it won’t take much time today to figure out that’s not great advice. All right. But can people at least see that this is a, This is a difficult problem, all right? It’s not an immediately obvious problem. Is that correct? Is that correct? And one reason it’s not immediately obvious is not only is no strategy dominated here, but there is no pure strategy Nash equilibrium in this game. All right, in this little sub game. There is no pure strategy Nash equilibrium. Notice that I did the qualifier now. Previously I would just have said Nash equilibrium, but now that we have have mixed strategies in the picture, I’m going to talk about pure strategy National Equilibria to be those that are only involving pure strategies.

[译文 23]

对吧？也就是，打向Serena的反手侧。发挥你的强项就是选择右，攻击Serena的弱点就是选择左。反过来对Serena来说，你应该倾向于你的强项，我想应该是向右倾斜，还是应该倾向于Venus的弱点，我想应该是向右倾斜？当你看这方面的教练手册，或者你听那些在ESPN解说的糟糕家伙们，哦不，我又惹麻烦了，那些解说网球非常好的先生们，他们在这一点上会说一些极其愚蠢的话。对吧？比如说，你应该永远发挥你的强项，不要担心对方的弱点。对吧？我想今天花不了多少时间就能弄清楚那不是什么好建议。好的。但大家至少能看到这是一个困难的问题，对吧？这不是一个显而易见的问题。是这样吗？是这样吗？其中一个原因这不是显而易见的问题，不仅是因为没有策略是劣势的，而且在这个游戏中没有纯策略纳什均衡。好的，在这个小子博弈中没有纯策略纳什均衡。注意我现在加了限定词。以前我可能只会说纳什均衡，但现在我们把混合策略纳入考量，我要说的是纯策略纳什均衡，也就是只涉及纯策略的那些。

[段 24]

Okay, so why is the no pure strategy Nash equilibrium? Well, let’s have a look. So if Venus, if Serena thought that Venus was going to choose left, then her best response, not surprisingly, is to lean left. And if Serena thought that Venus was going to choose right, then her best response is to cheat to the right. So 50 is bigger than 20 and 80. is bigger than 10, right? And conversely, if Venus thought that Serena was cheating a bit to the left, then her best response is to hit it to Serena’s right. And if Venus thought Serena was leaning to the right, then Venus’s best response is to hit it to Serena’s left. Okay? So I think that’s not at all surprising, when you think about it, not at a tall surprising you’re going to get this little cycle like this, but we can see a medial, that these best responses never coincide, so there is no pure strategy equilibrium. Let’s try that up. There’s no pure strategy in that equilibrium. All right. So that leaves us a bit stuck, except I guess it’s, I guess you know what the next question is going to be, and I shouldn’t leave it in too much suspense. The next question is, going to be, okay, there’s no pure strategy national equilibrium, but we’ve just introduced a new idea, which was what?

[译文 24]

好的，那么为什么没有纯策略纳什均衡呢？让我们来看看。如果 Venus，如果 Serena 以为 Venus 会选择左边，那么她的最佳反应毫不意外的是向左倾斜。而如果 Serena 以为 Venus 会选择右边，那么她的最佳反应就是向右倾斜一些。所以 50 大于 20，80 大于 10，对吧？反过来，如果 Venus 以为 Serena 稍微向左倾斜了一些，那么她的最佳反应就是向 Serena 的右边击球。而如果 Venus 以为 Serena 向右倾斜了，那么 Venus 的最佳反应就是向 Serena 的左边击球。好吗？所以我觉得这完全不令人意外，当你仔细想想的时候，一点都不令人意外，你会得到这样一个小的循环，但我们可以看到，这些最佳反应永远不会重合，所以没有纯策略均衡。让我们验证一下。没有纯策略均衡。好的。那么这就让我们有点卡住了，不过我想你们知道下一个问题是什么，我不应该留太多悬念。下一个问题是，好吧，没有纯策略纳什均衡，但我们刚刚引入了一个新的概念，那是什么？

[段 25]

It was Nash equilibrium in mixed strategies. Maybe there’s going to be a mixed strategy national equilibrium. In fact, there is. There is going to be one. All right. So our exercise now is, let’s find a mixed strategy strategy. Nash equilibrium. And before we find it, let’s just interpret what it’s going to mean. A mixed strategy Nash equilibrium in this game is going to be a mix for Venus between hitting the ball to Serena’s left and Serena’s right, and a mix for Serena between leaning left and leaning right, such that each person’s mix, each person’s randomization, is a best response to the other person’s randomization. Thank you. and leaning right, such that each person’s mix, each person’s randomization, is a best response to the other person’s randomization. All right? And since these players are sisters and have played each other many, many times, not just in competition, but probably in practice, it seems like a reasonable idea that they might have arrived in playing each other at a mixed strategy, Nash, equilibrium. All right, that’s what we’re going to try and do. how are we going to do that. Okay. So what we’re going to do is we’re going to exploit the trick that we have here, the lesson here. The lesson we have here says, if players are playing a mixed strategy as part of a national equilibrium, each of the pure strategies involved in the mix, each of their pure strategies must itself be a best response.

[译文 25]

那就是混合策略中的纳什均衡。也许会有一个混合策略纳什均衡。事实上，会有的。会有的一个。好的。所以我们现在要做的是，让我们找到一个混合策略纳什均衡。在找到它之前，让我们先解释一下这意味着什么。这个游戏中的混合策略纳什均衡将是 Venus 在向 Serena 的左边和右边击球之间的混合，以及 Serena 在向左倾斜和向右倾斜之间的混合，使得每个人的混合，每个人的随机化，是对另一个人随机化的最佳反应。谢谢。以及向右倾斜，使得每个人的混合，每个人的随机化，是对另一个人随机化的最佳反应。好吗？而且由于这些选手是姐妹，已经互相打过很多很多次了，不仅是在比赛中，可能也是在练习中，这似乎是一个合理的想法，她们可能已经达到了以混合策略纳什均衡来互相比赛的状态。好的，这就是我们要尝试做的。我们要怎么做呢？好的。我们要做的就是利用我们这里的技巧，这里的教训。这里的教训说，如果玩家作为纳什均衡的一部分在玩混合策略，每个涉及在混合中的纯策略，每个她们的纯策略本身都必须是一个最佳反应。

[段 26]

We’re going to use that idea. All right, so let’s try and do that. All right, so I’m hoping that by doing this, I’m going to illustrate you immediately that this idea is actually useful, at least useful if you end up coaching the Williams sisters. All right, so I want to keep this so you can still read it. Let me bring it down a bit. Can people still read it? Okay. So what I want to do is I want to find a mixture for Serena and a mixture for Venus that are equilibrium. All right, and having just put it up there, let me bring it down again. This was not so intelligent of me. I actually want to bring in some notation. So as before, let’s assume that Serena’s mix is, let’s use Q and 1 minus Q to be. Serena’s mix and let’s use P and 1 minus P to be Venus’s mix. Okay, just establish that notation. So here’s the trick. All right? So this is the slightly magic bit of the class. Okay, so pay attention. I’m about to pull a rabbit out of a hat. trick to find what should I do first to find serena’s Nash equilibrium mix so that’s Q1 minus Q what I’m going to do is I’m going to do is I’m going to look at Venus’s payoffs to To find Serena’s Nash equilibrium mix, the trick is to look at Venus’s payoffs.

[译文 26]

我们要用这个想法。好的，让我们来试试看。所以我希望这样做，我能立即向你们说明这个想法实际上是有用的，至少如果你们最终成为威廉姆斯姐妹的教练会很有用。好的，所以我想保留这个让你们还能看到。让我把它放下来一点。大家还能看到吗？好的。所以我想做的是为 Serena 和 Venus 找到一个混合策略作为均衡。好的，我刚放上去，让我再把它放下来。这不太明智。我实际上想引入一些符号。和之前一样，让我们假设 Serena 的混合是，让我们用 Q 和 1 减 Q 来表示。Serena 的混合，让我们用 P 和 1 减 P 来表示 Venus 的混合。好的，就先建立这个符号。好的，这就是技巧。好了。这就是这节课稍微有点神奇的部分。好了，注意看。我要从帽子里变出一只兔子。技巧是找到我应该先做什么来找到 Serena 的纳什均衡混合，也就是 Q、1 减 Q，我要做的是我要做的是我要看 Venus 的收益。

[段 27]

That’s going to be my magic trick. All right. Let’s try and see why. So let’s look at Venus’s payoffs. Venus’s payoffs against Q. So Serena is choosing Q1 minus Q. So what are Venus is payoffs? So if she chooses if she chooses left, then her payoff is 50 with probability Q. And I’m going to use the pointer here and hope that the camera can see this too. she gets 50 with probability Q and she gets 80 with probability 1 minus Q. Okay. If she chooses right then she gets 90 with probability Q. 90 with probability Q, and she gets 20 with probability 1 minus Q. Oh, I meant to point to that. 20 with probability 1 minus Q. So what? So what is this? We’re looking for a mixed strategy Nash equilibrium. so in particular not only Serena is mixing but in this case what we’re claiming is Venus is mixing as well all right so if Venus is mixing as well, that means that Venus is using the strategy left with some probability, p, and using the strategy right with some probability, 1 minus p. since Venus sometimes chooses left and sometimes chooses right as her best response to Q, as her best response to Serena, what must be true of the payoff to left and the payoff to right? Let’s go through again, right?

[译文 27]

这将是我的魔术技巧。好的。让我们看看为什么。好的，让我们看 Venus 的收益。Venus 对 Q 的收益。Serena 正在选择 Q、1 减 Q。所以 Venus 的收益是什么？如果她选择左边，那么她的收益是概率 Q 下获得 50。我要用指示器，希望相机也能看到。她以概率 Q 获得 50，以概率 1 减 Q 获得 80。好的。如果她选择右边，那么她以概率 Q 获得 90。以概率 Q 获得 90，以概率 1 减 Q 获得 20。哦，我本来想指向那个。以概率 1 减 Q 获得 20。所以呢？所以这是什么？我们正在寻找一个混合策略纳什均衡。所以特别是不仅 Serena 在混合，而且我们在这里声称的是 Venus 也在混合，好的，所以如果 Venus 也在混合，这意味着 Venus 以某个概率 P 使用左边策略，以某个概率 1 减 P 使用右边策略。既然 Venus 有时选择左边，有时选择右边作为对 Q 的最佳反应，作为对 Serena 的最佳反应，那么左边的收益和右边的收益必须满足什么条件呢？让我们再过一遍，对吧？

[段 28]

So we’re going to assume that Venus is mixing. So sometimes she chooses left and sometimes she chooses right. And she’s going to be, she’s in an equilibrium, so she’s choosing a best response. So whatever that mixed p1 minus p is, it’s a best response. Right? Since she playing a best response of p and that sometimes involves choosing left and sometimes involves choosing right it must be the case that what It must be the case that both left itself and right itself are both themselves best response. Right? If she’s mixing between them, it must be that both choosing left or choosing right are themselves best responses. If they weren’t, she should just drop them out of the mix. that would raise her average payoff. Right? Just like we dropped out the short TAs to get a high height. And we dropped out the failing Yale students to get a high GPA. Sorry. All right? All right? So if Venus is mixing in this Nash equilibrium, then the payoff to left and to right must be equal. They must both be best both left and right must be a best response so in particular the expected payoffs must be the same Is that right Is that correct Okay So what does that allow me to do It allows me to put an equal sign in here. It allows me to put an equal sign in here.

[译文 28]

所以我们要假设 Venus 在混合。所以有时她选择左边，有时她选择右边。她将处于均衡中，所以她正在选择最佳反应。所以不管那个混合 P、1 减 P 是什么，它都是一个最佳反应。对吧？既然她在玩最佳反应 P，而它有时涉及选择左边，有时涉及选择右边，那么必须满足的是左边和右边本身都必须是最佳反应。对吧？如果她在它们之间混合，那么选择左边或选择右边本身都必须是最佳反应。如果不是，她应该把它们从混合中剔除。那会提高她的平均收益。对吧？就像我们剔除矮的助教来获得高身高一样。我们剔除不及格的耶鲁学生来获得高 GPA。对不起。好的？好的？所以如果 Venus 在这个纳什均衡中混合，那么左边的收益和右边的收益必须相等。它们都必须都是最佳反应，所以特别是期望收益必须相同。对吧？是这样吗？是这样吗？好的。那么这允许我做什么呢？它允许我在这个里面放一个等号。它允许我在这个里面放一个等号。

[段 29]

Right? Since left is a best response and right is a best response, they’re both best responses, they must yield the same expected payoff. here’s their expected payoffs, they must be the same and now I’ve got one equation and one unknown and now I’m down to algebra so let me do the algebra, I claim this expression is equal to that expression so simplifying a bit I’m going to get, and you should just watch make sure I don’t get this wrong I’m going to get 40q so this implies 40q is equal to 61 minus q, so I took this 50 onto this side and this 20 onto that side, so I have 40q equals 61 minus q, and that implies that q is equal to 0.6. So those last two steps were just algebra So what was the trick here The trick was I found Q which is how Serena is mixing by looking at Venus’s payoffs, knowing that Venus is mixing, and hence I can set Venus’s payoffs equal to another. Say that again. I found the way in which Serena is mixing by knowing that if Venus is mixing, her expected payoffs must be equal, and I solved out for Serena’s mix. This is Serena’s mix. Okay? Let’s do it again. Let’s do it again. Here I’m wishing I had another board. I don’t want to lose those numbers entirely, so I’m going to try and squeeze in a bit.

[译文 29]

对吧？既然左边是一个最佳反应，右边也是一个最佳反应，它们都是最佳反应，它们必须产生相同的期望收益。这里是它们的期望收益，它们必须相同，而现在我有了一个方程和一个未知数，现在我进入代数了，让我做代数，我声称这个表达式等于那个表达式，所以简化一下，我会得到，你们应该只是看着我确保我不会搞错这个，我会得到 40q，所以这意味着 40q 等于 61 减 q，所以我把这边这个 50移到那边，把这边这个 20移到那边，所以我有 40q 等于 61 减 q，这意味着 q 等于 0.6。所以最后两步只是代数。这里的技巧是什么？技巧是我通过看 Venus 的收益找到了 Q，也就是 Serena 混合的方式，知道 Venus 在混合，因此我可以把 Venus 的收益设为相等。再说一遍。我找到了 Serena 混合的方式，通过知道如果 Venus 在混合，她的期望收益必须相等，然后我解出了 Serena 的混合。这是 Serena 的混合。好的？让我们再做一次。让我们再做一次。这里我真希望还有另一块黑板。我不想完全失去那些数字，所以我尽量挤进去一点。

[段 30]

I know I can do. Let’s get rid of this one entirely. There we go. That works. Let’s get rid of this one entirely. I can still see my numbers. And let’s do the converse. Let’s do the trick again. This time, what I’m going to do is I’m going to figure out how Venus is mixing. I know how Serena’s mixing now, so now I’m going to work out how Venus is mixing. Now, to figure out how… It’s hidden now. To figure out how Serena was mixing, I used Venus’s payoffs. So to find out how Venus is mixing, what am I going to do? I’m going to use Serena’s payoffs. All right? So to find Venus’s mix, which is P1 minus P, let’s be careful, it’s her Nash equilibrium mix, use Serena’s payoffs. Here we go. So if Serena chooses this is S payoffs if Serena chooses L then her payoffs will be what So, and again, we’ll just watch, make sure I don’t get this wrong, and I’ll point to the things to try and help myself a bit. So with probability P, she’ll get 50. so 50 with probability P and with probability 1 minus P she’ll get 10 alright and if she chooses to lean to the right to lean towards her forehand then with probability P she’ll get 20 and with probability 1 minus p, she’ll get 80.

[译文 30]

我知道我可以做到。把这个完全去掉。好。这样可以。把这个完全去掉。我还能看到我的数字。让我们做相反的。让我们再用这个技巧。这次，我要做的是算出Venus如何混合。我现在知道Serena如何混合了，所以我现在要算出Venus如何混合。现在，要算出如何……现在隐藏了。要算出Serena如何混合，我用了Venus的收益。那么要找出Venus如何混合，我要做什么？我要用Serena的收益。明白吗？为了找到Venus的混合，也就是P1减P，要小心，这是她的纳什均衡混合，使用Serena的收益。开始吧。所以如果Serena选择这个，这是S的收益，如果Serena选择L，那么她的收益是多少？好，我们再看，确保我不会搞错，我会指着这些东西来帮助自己。所以以概率P，她会得到50。所以以概率P得到50，以概率1减P得到10，好，如果她选择向右倾斜，向她的正手方向倾斜，那么以概率P她会得到20，以概率1减P，她会得到80。

[段 31]

Okay. And we know that Serena is mixing. We know that Serena is mixing So since Serena is mixing what must be true of these two payoffs What must be true of the two payoffs the payoff to little l and the payoff to little r What must be true about them since Serena is using a mixture of these two strategies in Nash equilibrium? It must be the case that both little l is a best response and little r is a best response, in which case the payoff must be equal. Thank you. They must be equal. These must be equal. These must be equal. They must be equal since Serena is indifferent between choosing left or right and hence is mixing over them. All right. So again, using the fact that they’re equal reduces this to algebra. And again, I’ll probably get this wrong, but let me try. So I claim, let’s take 20 away from here. I’ve got 30p equals 70, 1 minus p. I hope that’s right. That looks right. Again, this is just algebra at this point. So I took 20 away from here and 10 away from there, and this implies that p equals 0 So I claim I have now found the mixed strategy Nash equilibrium Here it is The Nash equilibrium is as follows Let’s be careful. This is Venus’s mix.

[译文 31]

好的。我们知道Serena是混合的。我们知道Serena是混合的。所以既然Serena是混合的，这两个收益必须满足什么条件？这两个收益必须满足什么条件？小l的收益和小r的收益？既然Serena在纳什均衡中使用这两个策略的混合，它们必须满足什么条件？必须是小l和小r都是最佳对策，在这种情况下收益必须相等。谢谢。它们必须相等。这些必须相等。这些必须相等。它们必须相等，因为Serena在选择左或右之间是无差异的，因此对它们进行混合。好。所以同样，利用它们相等的事实归结为代数。再说一遍，我可能会搞错，但让我试试。所以我主张从这里减去20。我得到30p等于70，1减p。我希望这是对的。看起来对。这现在就是代数了。所以我从这里减去20，从那里减去10，这意味着p等于0。所以我主张我找到了混合策略纳什均衡。在这里。它是这样的。纳什均衡如下。让我们小心。这是Venus的混合。

[段 32]

So Venus is mixing 0.7, 0.3, 0.7 on left and 0.3 on right and Serena is mixing 0.6 0.4 alright so this is Venus’ mix and this is Serena’s mix Venus is shooting to the left of Serena with probability 0.7 and Serena is leaning that way with probability 0.6 alright So we were able to find this Nash equilibrium by using the trick before. Now let’s just reinforce this a little bit by talking about it. So suppose it were the case, suppose it were the case that Serena, instead of leaning to the left 0.6 of the time, leant to the left more than 0.6 of the time. So suppose you’re Venus’s coach, you’re Venus’s coach, and suppose you know that Serena leans to the left more than 0.6 of the time. What would you advise Venus to do? Alright, let me try it again. So suppose you’re Venus’ coach and suppose you’ve observed the fact that Serena leans to the left more than 0.6 of the time. What would you advise Venus to do? Yeah, pass to the right. Shout out, yeah. Pass to the right, exactly. So if Serena cheats to the left more than 0.6 of the time, then Venus’ best response is always to shoot to the right. Always to shoot to the right. That maximizes her chance of winning the point. And conversely if Serena leans to the left less than 0 of the time if Serena leans to the left less than 0 of the time then Venus should do what Shoot to the left all the time.

[译文 32]

所以Venus混合0.7、0.3，左边0.7，右边0.3，Serena混合0.6、0.4。好，所以这是Venus的混合，这是Serena的混合。Venus以0.7的概率向左击球，Serena以0.6的概率向那个方向倾斜。好，所以我们能够通过之前的技巧找到这个纳什均衡。现在让我们通过讨论来稍微加强一下。假设是这样，假设Serena不是以0.6的概率向左倾斜，而是以更高的概率向左倾斜。所以假设你是Venus的教练，你是Venus的教练，并且假设你知道Serena以高于0.6的概率向左倾斜。你会建议Venus做什么？好，让我再试一次。所以假设你是Venus的教练并且假设你观察到Serena以高于0.6的概率向左倾斜这个事实。你会建议Venus做什么？是的，向右传球。喊出来，是的。向右传球，正是如此。所以如果Serena以高于0.6的概率向左倾斜，那么Venus的最佳对策就是总是向右击球。总是向右击球。这最大化她赢得这一分的机会。相反，如果Serena向左倾斜的概率小于0，如果Serena向左倾斜的概率小于0，那么Venus应该做什么？总是向左击球。

[段 33]

All right? So if Serena doesn’t choose exactly this mix, if Serena doesn’t choose exactly this mix, then Venus’s best response is actually a pure strategy. I’ll say it again. If Serena leans to the left too often, more than 0.6, then Venus should just go right. And if Serena leans to the left too little, then Venus should always go left. And we can do exactly the same the other way around. If Venus shoots to the right, so does her cross-hand passing shot more than 0.7 of the time, and you’re Serena’s coach, what should you tell Serena to do? Go that way all the time. Go that way all the time. So if Venus is hitting it to Serena’s left more than 0.7 of the time, Serena should just always go to her left. And if Venus is hitting to the left less than 0.7 of the time, so to the right more than 0.3 of the time, then Serena should always go to the right So that how this kind of comes back into the sort of coaching manuals if you like Okay So how am I doing so far? Have I lost everyone yet, or are people still with me? How many of you play tennis, ever? All right, so all your tennis is going to dramatically improve after today, right? All right. Okay. So now let’s make life more interesting.

[译文 33]

明白吗？所以如果Serena不选择正好这个混合，如果Serena不选择正好这个混合，那么Venus的最佳对策实际上是一种纯策略。我再说一遍。如果Serena向左倾斜过于频繁，高于0.6，那么Venus就应该向右。如果Serena向左倾斜过于少，那么Venus就应该总是向左。我们可以对调来做完全相同的事情。如果Venus向右击球，所以她的反手传球以高于0.7的概率进行，而你是Serena的教练，你应该告诉Serena做什么？总是往那个方向走。总是往那个方向走。所以如果Venus以高于0.7的概率向Serena的左边击球，Serena就应该总是向左走。如果Venus以低于0.7的概率向左击，也就是以高于0.3的概率向右击，那么Serena就应该总是向右。所以这就是如何回到教练手册之类的东西如果你喜欢的话。好，我到目前为止做得怎么样？我已经把大家弄丢了吗，还是大家还跟着我？你们中有多少人打网球？好的，所以今天之后你们所有的网球水平都会显著提高，对吧？好的。好，现在让我们让生活更有趣一些。

[段 34]

Let’s go back to the start, We figured out this isn’t equilibrium. This is how Venus and Serena play. Venus and Serena know each other perfectly well. They know that they mix this way. They’re going to best respond to it. This is where they end up. But in the meantime, Serena hires a new coach. And Serena’s new coach is just very, very good at teaching Serena how to play at the net, and in particular, how to hit the backhand volley. All right? So Serena’s new coach, let’s say it’s Tony Roche or somebody, is just a brilliant coach, and Tony Roche is able to improve Serena’s backhand volley, and that changes these payoffs. So you should rewrite the whole matrix, but I’m going to cheat. All right? So the new game is exactly the same as it was everywhere else except for now when Serena gets to the backhand volley she gets it in 70 of the time So there used to be 50 in that box, and now it’s 30-70. So the game has changed because Serena has got better at hitting backhand volleys. We want to figure out how is this going to affect play at Wimbledon. All right? How is this going to affect play at Wimbledon? Now, it doesn’t take much to check that there is still no pure strategy Nash equilibrium. It’s still the case, in fact, even more so, that Serena’s best response to Venus choosing left is to lean to the left.

[译文 34]

让我们回到开始，我们弄清楚了这不是均衡。这是Venus和Serena的比赛方式。Venus和Serena非常了解对方。她们知道她们这样混合。她们会对此做出最佳对策。这就是她们最终的结果。但与此同时，Serena聘请了一位新教练。Serena的新教练非常非常擅长教Serena如何打网前，特别是如何打反手截击。好吗？所以Serena的新教练，我们姑且说是Tony Roche或什么人，是一位出色的教练，Tony Roche能够提高Serena的反手截击，这改变了这些收益。所以你应该重写整个矩阵，但我会作弊。好吗？所以新游戏在其他所有地方都和以前一样，除了现在当Serena有机会打反手截击时，她有70的概率成功。所以那个格子里原来是50，现在是30-70。所以游戏变了，因为Serena提高了打反手截击的能力。我们想弄清楚这将如何影响温布尔登的比赛。好吗？这将如何影响温布尔登的比赛？现在，不需要太多检查就可以知道仍然没有纯策略纳什均衡。事实上，Serena对Venus选择左边的最佳对策是向左倾斜，这仍然是，甚至更是如此。

[段 35]

All right? So it’s still the case that the best responses do not coincide, there is still no pure strategy equilibrium. All right? And what we’re going to do, of course, is we’re going to find a mixed strategy equilibrium. But before we do so, let’s think about this intuitively. Let’s see if we can intuit an answer. All right? I’m guessing we can’t. But let’s see if we can intuit an answer. All right? So Serena has… Let’s see if we can intuit an answer. I’m guessing we can’t, but let’s see if we can intuit an answer. So Serena has improved her backhand volley, and hence when she reaches it, she gets it in more often. So one effect, you might think, is what we might want to call a direct effect. I think there’s two effects here. there are two effects one of these I’m going to call the direct effect and by effect I mean in particular an effect on how Serena should play the game so since Serena has improved her backhand volley when she reaches that volley she gets it in more often so one might say in that case, you’re Serena’s coach, in that case you should lean to the left more often than you did before, right? Because at least when you get that backhand volley, you’re going to get it in more often So the direct effect says Serena should lean left more In other words Q should go up Is that right So Serena now better at playing this backhand volley so she may as well favour it a bit more and hence Q will go up So that’s the direct effects, but of course there’s a but coming.

[译文 35]

明白吗？所以最佳对策仍然不重合，仍然没有纯策略均衡。好吗？我们将要做的，当然，我们要找到混合策略均衡。但在这样做之前，让我们直观地思考一下。让我们看看我们是否能直观地得到一个答案。好吗？我猜我们不能。但让我们看看我们是否能直观地得到一个答案。好吗？所以Serena……让我们看看我们是否能直观地得到一个答案。我猜我们不能，但让我们看看我们是否能直观地得到一个答案。所以Serena提高了她的反手截击，因此当她有机会打时，她成功打进的次数更多了。所以一个影响，你可能会想到，是我们可能想要称之为直接影响。我这里有两个影响。有两个影响，其中一个我称之为直接影响，而影响我指的是特别是对Serena应该如何玩游戏的影响，所以既然Serena提高了她的反手截击，当她有机会打那个反手截击时，她成功打进的次数更多了，所以人们可能会说，作为Serena的教练，在这种情况下你应该比以前更频繁地向左倾斜，对吧？因为至少当你有机会打那个反手截击时，你会更频繁地成功。所以直接影响说的是Serena应该更多地向左倾斜。换句话说，Q应该上升。对吗？所以Serena现在更擅长打这个反手截击，所以她不妨更多地偏向它，因此Q会上升。所以这是直接影响，但当然有一个但是要来了。

[段 36]

What’s the but? Again, let’s see my tennis players here. Raise your hands if you play tennis. Oh, suddenly nobody plays tennis. Come on, raise your hands. All right, okay. What’s the but here? We think Serena’s backhand’s improved, so she might be tempted to play towards her backhand a bit more often. What’s the but? What’s the but here? So I claim the but is this. You tell me if I’m wrong. the but is that Venus, she’s her sister after all, right? So Venus knows that Serena’s backhand has improved, so Venus is going to hit it to Serena’s left less often than before. Is that right? Right, so since Serena’s backhand has improved, Venus is going to hit it to Serena’s backhand less often than before and that might make Serena less inclined to cheat towards her backhand because the ball is coming that way less often All right So this is an indirect or a strategic effect The strategic effect is Venus hits L less often, so Serena should reduce the number of times that she leans to the left because the ball’s coming that way fewer times. Now notice that these two effects go in opposite directions. Is that right? One of them tends to argue that Q would go up. That’s the direct effect. And the other one is more subtle. it says, when I think about not just how my play has improved but also how the other person is going to respond to knowing that my play has improved, that’s the more subtle effects, and that’s going to push Q down.

[译文 36]

但是呢？让我们看看这里打网球的人。有没有人打网球的？请举手。哦，突然没人打网球了。来吧，举手。好吧，好的。但是呢？但是呢？我们认为塞雷娜的反手有所提高，所以她可能会更倾向于更多地朝自己的反手方向打。但是呢？但是呢？所以我声称这个但是就是这个。告诉我我是否错了。但是就是维纳斯，毕竟她是她的妹妹，对吧？所以维纳斯知道塞雷娜的反手提高了，所以维纳斯会比以前更少地把球打到塞雷娜的左侧。对吗？是的，所以由于塞雷娜的反手提高了，维纳斯会比以前更少地把球打到塞雷娜的反手方向，这可能会让塞雷娜更不倾向于偏向她的反手方向，因为球朝那个方向来的次数变少了。好的，所以这是一个间接的或战略性的效应。这个战略性效应是维纳斯更少地打左侧，所以塞雷娜应该减少她向左倾斜的次数，因为球朝那个方向来的次数变少了。现在注意，这两个效应的方向是相反的。对吗？其中一个倾向于认为Q会上升。这是直接效应。另一个更微妙。它说的是，当我不仅考虑我的打法提高了多少，而且考虑对方在知道我的打法提高后会如何回应时，这就是更微妙的效应，而这会把Q往下推。

[段 37]

That’s going to make it less likely, that’s going to argue against leaning to the left. Alright, so imagine you’re going to be Serena’s coach. Which of these effects do you think is going to let have a poll Which of these effects do you think is going to win The direct effect or the indirect effect The direct effect or the strategic effect Who thinks the direct effect Who would advise Serena to play to her strength a bit more and lean left a bit more? Who thinks the direct effect? Raise your hands, let’s have a poll. And who thinks the indirect effect, the effect of Serena hitting it that way less often is going to win? And who’s abstaining and basically refusing to be a coach? Okay, quite a number of you, all right? All right. Well, we’re going to find out. We’re going to find out. We’re going to find out by resolving for the Nash equilibrium. What we’re going to do is redo the calculation we did before, starting with Serena. So to find Serena’s mix, to find Serena’s new equilibrium mix, what do we have to do? The question is, in equilibrium, is Serena going to lean to the left more, so Q is going to go up, or less, so Q is going to go down? So I need to find out what is Serena’s new equilibrium mix, what’s the new Q?

[译文 37]

这会使它的可能性降低，这会反对向左倾斜。好的，假设你要当塞雷娜的教练。你认为哪个效应会占上风？直接效应还是间接效应？直接效应还是战略性效应？谁认为直接效应？谁会建议塞雷娜更多地发挥她的优势，更向左倾斜一点？谁认为直接效应？举手，让我们做个调查。谁认为间接效应，塞雷娜朝那个方向打得更少这个效应会占上风？谁弃权了基本上拒绝当教练？好的，你们中有不少人是这样。好吧，我们会知道的。我们会知道的。我们会通过求解纳什均衡来知道。我们要做的是重做我们之前做过的计算，从塞雷娜开始。所以要找到塞雷娜的混合，找到塞雷娜新的均衡混合，我们该怎么做？问题是，在均衡中，塞雷娜会更多地倾斜到左侧，所以Q会上升，还是更少，所以Q会下降？所以我需要找出塞雷娜新的均衡混合是什么，新的Q是什么？

[段 38]

How do I go about finding Serena’s equilibrium Q? What’s the trick here? Shout it out. One, two, three. What’s the… Use Venus’s payoffs, all right? To find the new Q for Serena, use Venus’s payoffs. And let’s do that. So from Venus’s point of view, if she chooses left, then her payoffs are now, and again I should use the pointer, 30 with probability q, this is the new q, and 80 with probability 1 minus q. 30 with probability q plus 80 with probability 1 minus q. Again, this is the new q. I should really give it, I should call it q prime or something but I won And if she chooses right then her payoff is what It going to be 90 with probability Q and 20 with probability 1 minus Q 90 with probability Q and 20 with probability 1 minus Q. and what do we know about these two payoffs if Venus is mixing in equilibrium and we know she’s mixing in equilibrium because we saw there was no pure strategy equilibrium so what do we know about these two payoffs since Venus is using both these strategies in equilibrium they must be the same since she’s using both these strategies these strategies must be equally good they must both be best responses so these two payoffs are equal since they’re equal all I have to do is solve out for Q, so let’s do it.

[译文 38]

我如何找出塞雷娜的均衡Q？诀窍是什么？喊出来。一、二、三。什么……用维纳斯的收益，对吧？要找出塞雷娜的新Q，使用维纳斯的收益。让我们来做这个。所以从维纳斯的角度来看，如果她选择左侧，那么她的收益现在是，同样我应该用指示器，当概率为q时是30，这是新的q，当概率为1减q时是80。当概率为q时是30加上当概率为1减q时是80。同样，这是新的q。我真的应该给它，我应该叫它q撇什么的但我不会。如果她选择右侧，那么她的收益是什么？当概率为Q时是90，当概率为1减Q时是20。当概率为Q时是90，当概率为1减Q时是20。我们知道关于这两个收益的什么，如果维纳斯在均衡中混合，我们知道她是在均衡中混合，因为我们看到没有纯策略均衡，那么关于这两个收益我们知道了什么？因为维纳斯在均衡中使用了这两个策略，它们一定是一样的，因为她使用了这两个策略，这些策略一定同样好，它们一定都是最佳反应，所以这两个收益是相等的，因为它们相等，我要做的就是解出Q，让我们来做。

[段 39]

So I’m going to get 90 minus 30 is 60Q is equal to 80 minus 20 which is 61 minus Q so Q equals 0 All right So if I did the algebra too quickly just trust me I think I got it right All right From here on in it was just algebra All right so if I did the algebra too quickly just trust me I think I got it right All right from here on in it was just algebra All right so what have I found out Did Q go up or down? Well, it used to be, Q used to be what? 0.6, and now it’s 0.5. Let me ask what I think is an easy question. Did it go up or down? It went down, it went down, right? It went down, right? The Q went down. The equilibrium Q went down. So which effect turned out to be bigger? The direct effect of playing more to your strength or the indirect effect of taking into account that your opponent is going to play less often to your strength? Which effect turned out to be the bigger effect? The indirect effect, the strategic effect. Of course, I rigged it. I want the strategic effect to be bigger because this is a course about strategy. All right? All right? All right? But the strategic effect actually won here.

[译文 39]

所以我得到90减30是60Q等于80减20是61减Q，所以Q等于0。好吧，如果我做代数太快了就相信我，我认为我做对了。从这里开始就是代数了。好吧，所以如果我做代数太快了就相信我，我认为我做对了。好吧，从这里开始就是代数了。好吧，所以我发现了什么？Q是上升了还是下降了？嗯，它以前是，Q以前是多少？0.6，现在它是0.5。让我问一个我认为简单的问题。它上升了还是下降了？它下降了，它下降了，对吧？它下降了，对吧？Q下降了。均衡Q下降了。那么哪个效应实际上更大？发挥你优势的直接效应，还是考虑到你的对手会更少地朝你的优势方向打的间接效应？哪个效应实际上更大？间接效应，战略性效应。当然，我做了手脚。我希望战略性效应更大，因为这是一门关于策略的课程。好的？好的？好的？但战略性效应在这里实际上赢了。

[段 40]

The strategic effect, the indirect effect, is bigger. All right? And that’s good news for me, because it says the slightly dumb coach who didn’t bother to take game theory would have stopped at the direct effects and they have told Serena to go the wrong way But the smart coach who takes my class and therefore somehow contributes to my salary in an extraordinarily indirect way gets it right All right? All right, the strategic effect is bigger. Now we can also solve out for Venus’s new mix, but I, and we’ll do it in a second, but before I do it, let me just point out that we actually, we really can now intuit Venus’s effects. Maybe not the exact numbers, but we can intuit it. So I claim, if we think this through carefully, we know whether Venus is shooting more to the left than she was before or less to the left than she was before. Notice that in the new equilibrium, Serena is going less often to her left, even though she’s better at hitting the back end. She’s better at hitting the ball when she gets there. So since Serena is leaning left less often, what must be true about Venus in this new equilibrium? It must be the case that Venus is hitting the ball to the left less often. All right, that make sense?

[译文 40]

战略性效应，间接效应，更大。好的？这对我来说是好消息，因为它说的是，稍微笨一点的教练，不会费心学博弈论，就会停在直接效应上，他们会让塞雷娜走错方向。但聪明的教练，上我的课，因此以某种非常间接的方式为我的工资做出贡献的人，会做对。好的？好的，战略性效应更大。现在我们也可以解出维纳斯的新混合，但我，等等，我们会做，但在我做之前，让我指出，我们实际上，现在我们真的可以直观理解维纳斯的效果。也许不是确切的数字，但我们可以直观理解。所以我声称，如果我们仔细思考，我们知道维纳斯是比以前更多地朝左侧打还是更少朝左侧打。注意，在新的均衡中，塞雷娜即使在击球时反手更好，她也会更少地去她的左侧。当她到达那里时她更擅长击球。所以既然塞雷娜向左倾斜的次数更少，在新的均衡中关于维纳斯一定是什么情况？一定是维纳斯更少地把球打到左侧。好的，这有道理吗？

[段 41]

We have enough information already on the board to tell us that. Nevertheless, let’s do the math. Let’s do the math. So, all right, let’s go and retrieve a board. All right. Nevertheless, let’s do the math. Let’s do the math. So, all right. Let’s go and retrieve a board. Let’s do that. Just for completeness, let’s figure out exactly what Venus does do. All right. So to figure out what Venus is going to do, what’s our trick? I want to figure out how Venus is going to mix. I’m going to find out Venus’ new P. How do I find out Venus’ new equilibrium mix? I look at Serena’s payoffs. All right. So if Serena chooses left, her payoff is, and I read it off quickly this time is 70p plus 10 1 minus p and if Serena chooses right her payoff is 20p plus 80 1 p I just praying that the CAs are going to catch me if I make a mistake here And I know these have to be equal, because I know that in fact Venus is mixing, sorry, I know that Serena is mixing, all right? So I know these must be equal. So since they’re equal, I can solve out, and hope that I’ve got this right. so I’ve got 50p equals 71 minus p so p is equal to 7 twelfths so again that’s just algebra I rushed it a bit it’s just algebra same idea just algebra so 7 twelfths is indeed smaller than what it used to be because it used to be 7 tenths All right, so that confirms our result.

[译文 41]

我们板上有足够的信息告诉我们这一点。尽管如此，让我们做数学。让我们做数学。所以，好的，让我们去拿一块白板。好的。尽管如此，让我们做数学。让我们做数学。所以，好的。让我们去拿一块白板。让我们这样做。为了完整性，让我们找出维纳斯实际上做什么。好的。所以要找出维纳斯要做什么，我们的诀窍是什么？我想找出维纳斯如何混合。我要找出维纳斯的新P。我如何找出维纳斯的新均衡混合？我看塞雷娜的收益。好的。所以如果塞雷娜选择左侧，她的收益是，这次我快速读出来，是70p加101减p，如果塞雷娜选择右侧，她的收益是20p加801p我只是祈祷如果我这里犯错CA们会抓住我。我知道这些必须相等，因为我确实知道维纳斯在混合，抱歉，我知道塞雷娜在混合，好的？所以我知道这些必须相等。所以既然它们相等，我可以解出来，希望我得到正确的结果。所以我得到50p等于71减p，所以p等于12分之7。所以再次，这只是代数，我赶了一点，只是代数，同样的想法，只是代数。12分之7确实小于它以前的值，因为它以前是10分之7。好的，所以这证实了我们的结果。

[段 42]

Okay. So the strategic effect dominated. Venus shot to Serena’s backhand less often and as a consequence so much, so much so that Serena actually found it worthwhile going more to the right than she used to before All right Now let just talk this through one more time This was a comparative statics exercise We looked at a game, we found an equilibrium, we changed something fundamental about the game, and we looked again to look at the new equilibrium. That’s called comparative statics. And let’s talk through the intuition. Before we made any changes, Venus was indifferent. right before we made any changes Venus was indifferent she was indifferent between shooting to the left and shooting to the right then we improved Serena’s ability to hit the volley to her left improved her backhand volley if we had not changed the way Serena played if we had not changed the way Serena played then then what would Venus have done? So suppose in fact Serena’s Q had not changed. If Serena’s Q had not changed, remembering that Venus was indifferent before how would Venus have changed her play Somebody So if we started from the old Q and then we improved Serena ability to play the backhand volley and if Q didn change what would Venus have done She never ever have shot to the left anymore, she’d only have shot to the right.

[译文 42]

好的。所以策略效果主导了。大威廉姆斯减少了对小威廉姆斯的反手位击球次数，而且减少得非常多，以至于小威廉姆斯发现值得比之前更多地往右侧移动。好，现在我们再梳理一遍这个过程。这是一个比较静态分析的练习。我们研究了一个博弈，找到了一个均衡，然后我们改变了博弈的某个根本要素，再次观察新的均衡。这就叫比较静态分析。让我们谈谈其中的直觉。在我们做任何改变之前，大威廉姆斯是无差异的。对，在我们做任何改变之前，大威廉姆斯是无差异的，她在向左击球和向右击球之间是无差异的。然后我们提升了小威廉姆斯向左侧击球的能力，提升了她的反手截击技术。如果我们没有改变小威廉姆斯的打法，如果我们没有改变她的打法，那么大威廉姆斯会怎么做？所以假设实际上小威廉姆斯的Q没有改变。如果小威廉姆斯的Q没有改变，记得之前大威廉姆斯是无差异的，大威廉姆斯会如何改变她的打法？所以如果我们从旧的Q开始，然后我们提升了小威廉姆斯反手截击的能力，如果Q没有改变，大威廉姆斯会怎么做？她就再也不会向左击球了，她只会向右击球。

[段 43]

She’d only have shot to the right, which can’t possibly be in equilibrium. So something about Serena’s play has to bring Venus back into equilibrium, it brings Venus back into being indifferent, and what was it? It was Serena moving to the left less often and moving to the right more often. All right, so again, if we didn’t change Q, Venus would only go to the right, so we need to reduce Q, have Serena go to the right, to bring Venus back into equilibrium. And And conversely, if Venus hadn’t changed her behavior, if Venus had gone on shooting exactly the same as she was, p and 1 minus p as before, then Serena would have only gone to the left. And that can’t be in equilibrium, so it must be something about Venus’ play brings Serena back into equilibrium, and what is it? It’s that Venus starts shooting to the right more often. All right, so just two reminders, two, before you leave, two reminders, wait, wait, wait, and what is it? It’s that Venus starts shooting to the right more often. All right, so just two reminders. Before you leave, two reminders. Wait, wait, wait, wait. First, in about five minutes’ time, a handout will magically appear on the website that goes through these arguments again, all of them in two other games. So you can have a look at the handout.

[译文 43]

她只会向右击球，而这不可能是均衡状态。所以小威廉姆斯的打法中必然有某些因素让大威廉姆斯回归均衡，让大威廉姆斯重新变得无差异，那是什么？是小威廉姆斯向左移动的次数减少，向右移动的次数增加。好吧，再重复一遍，如果我们没有改变Q，大威廉姆斯就只会向右，所以我们需要减少Q，让小威廉姆斯向右，来让大威廉姆斯回归均衡。相反，如果大威廉姆斯没有改变她的行为，如果她继续像之前一样击球，p和1减p和之前一样，那么小威廉姆斯就只会向左。这不可能是均衡，所以必然是大威廉姆斯的打法中有什么因素让小威廉姆斯回归均衡，那是什么？是维纳斯开始更多地向右击球。好吧，还有两个提醒。离开之前，有两个提醒。等一下，等一下。是维纳斯开始更多地向右击球。好吧，还有两个提醒。离开之前，有两个提醒。等一下，等一下，等一下。

[段 44]

Second thing, a problem set has already appeared by magic on that website that gives you lots of examples like this to work on. All right, play tennis over the weekend for practice and we’ll see you on Monday.

[译文 44]

首先，大约五分钟后，一份讲义会神奇地出现在网站上，它会再次梳理这些论证，以及另外两个博弈中的所有论证。所以你可以看看这份讲义。第二件事，一个问题集已经神奇地出现在那个网站上了，它给你提供了很多像这样的例子来练习。好，周末去打球练习一下，我们周一见。