耶鲁大学博弈论公开课 - 第10集 混合战略棒球，约会和支付您的税

视频全文转录（中英双语）

以下为完整中英双语转录，已加标点。英文在前，中文在后，逐段对照。

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

[段 1]

All right, so last time we did something I think substantially harder than anything we’ve done in the class so far. We looked at mixed strategies and in particular we looked at mixed strategy equilibrium. All right? And there was a big idea last time. The big idea was if a player is playing a mixed strategy in equilibrium, then every pure strategy in the mix, that’s to say every pure strategy on which they place some positive weight, must also be a best response to what the other side is doing. And then we used that trick, we used it in this game here, to help us find Nash equilibria. And the way it allowed us to find Nash equilibria is we knew that if, in this case, Venus Williams is mixing between left and right, it must be this case that her payoff is equal to that of right And we used that to find Serena mix And conversely since we knew that Serena is mixing again between little l and little r we knew she must be indifferent between little l and little r and we used that to find Venus’s mix. So I want to go back to this example just for a few moments, just to make one more point, and then we’ll move on. But we’ll still be talking about mix strategies throughout today.

[译文 1]

好的，上一次我们做的东西，我认为比到目前为止我们课堂上做过的任何事情都要难得多。我们研究了混合策略，特别是混合策略均衡。好的？上一次有一个重要的思想。重要的思想是，如果一个参与者在均衡中采用混合策略，那么混合策略中的每一个纯策略，也就是说每一个被赋予正权重的纯策略，也必须是对对方所采取策略的最佳反应。然后我们用了这个技巧，我们在这个博弈中用了它，来帮助我们找出纳什均衡。它允许我们找出纳什均衡的方式是，我们知道如果在这个例子中，Venus Williams在左边和右边之间混合，那么她的收益必须等于右边的收益，我们用这个来找到Serena的混合策略。反过来，由于我们知道Serena又在小l和小r之间混合，我们知道她必定对小l和小r无差异，我们用这个来找到Venus的混合策略。所以我想回到这个例子，花一点时间，再做一个说明，然后我们再继续。但今天我们仍然会一直讨论混合策略。

[段 2]

So this was the mix that we found before we changed the payoffs. we found that Venus’ equilibrium mix was 0.7, 0.3 and Serena’s equilibrium mix was 0.6, 0.4 and a reasonable question at this point would be how do we know that’s really an equilibrium we kind of found it but we didn’t go back and check so what I want to do now is actually do that do that missing step, we rushed a bit last time because we want to get through all the material let’s actually check that in fact P star is a best response to Q star. So what I want to do is, I want to check that Venus’ mix P star is a best response for Venus against Serena’s mix Q star. And the way I going to do that is I going to look at payoffs that Venus gets now we know she now we know she playing against Q star Alright so let look at Venus payoffs I’m going to figure out her payoffs for L, her payoffs for R, and also her payoff for what she’s actually doing, P star.

[译文 2]

这是我们在改变收益之前的混合策略，我们发现Venus的均衡混合是0.7，0.3，Serena的均衡混合是0.6，0.4，这时候一个合理的问题是，我们怎么知道这真的是一个均衡，我们有点找到了，但我们没有回去检验，所以我现在要做的是做那个缺失的步骤，我们上次有点仓促，因为我们想讲完所有内容，让我们实际上检验一下P star确实是对Q star的最佳反应。所以我想做的是，我想检验Venus的混合策略P star对于Venus来说，在Serena的混合策略Q star下是最佳反应。我要做的方法是，我来看Venus现在的收益，我们知道她现在在和Q star对抗。好的，让我们看Venus的收益，我要计算出她选L的收益，她选R的收益，以及她实际上做的事情P star的收益。

[段 3]

Alright, so Venus’s payoffs, if she chooses L against Q star then she gets very similar to what we had on the board last week but now I’m going to put in what Q star is explicitly she gets 50 times 0.6 this is Q star and this is 1 minus Q star so she gets 50 times 0.6 and 80 times 1 minus 0.6 which is 0.4 80 times 0.4 alright and we can work this out and I worked it out at home but if somebody has a calculator that can please check me I think this comes to 0.62 alright somebody should just check that alright and if Venus chose right, remember right here means shooting to Serena right to Serena forehand If she chose right then her payoffs are 90 Q star so 90 plus 20 1 minus Q star so 20 So 90.6 plus 20.4. And again, I work that out at home, and fortunately, that also comes out at 0.62. All right. So what’s Venus’ payoff for P star? We’ve got her payoff for both her pure strategies. So her payoff from actually choosing P star is what? Well, P star is 0.7. So 0.7 of the time, she will actually be playing left. And when she plays left, she’ll get a payoff of 0.62. and 0.3 of the time, 0.3 of the time, she’ll be playing right, and once again, she’ll be getting a payoff of 0.62, and do I have a calculator?

[译文 3]

好的，Venus的收益，如果她选L对抗Q star，那么她得到的和我们上周在黑板上的非常相似，但现在我要明确写出Q star是什么，她得到50乘以0.6，这是Q star，这是1减去Q star，所以她得到50乘以0.6和80乘以1减去0.6也就是0.4，80乘以0.4，好的，我们可以算出来，我在家算过了，但如果有人有计算器请帮我核实一下，我认为结果是0.62，好的应该有人检查一下。如果Venus选右边，记得这里的右边是指向Serena的反手击球，如果她选右边，那么她的收益是90乘以Q star，也就是90，加上20乘以1减去Q star，也就是20，所以90乘以0.6加上20乘以0.4。我在家也算过了，幸运的是，这也得到0.62。好的。那么Venus对P star的收益是多少？我们已经得到了她两个纯策略的收益。那么她实际选择P star的收益是多少？P star是0.7，所以70%的时间，她实际上会出左边。当她出左边时，她会得到0.62的收益。30%的时间，30%的时间她会出右边，再一次，她会得到0.62的收益，我有计算器吗？

[段 4]

Just as a clarification, when you say 0.62, She’ll be playing right, and once again, she’ll be getting a payoff of 0.62. And do I have a calculator? Just as a clarification, when you say p star is 0.7 and 0.3, you mean p star is 0.7. Yes, sorry, thank you. So p star is 0.7. Yes, you’re absolutely right. So this is p star and 1 minus p star. All right, so let’s make that clearer. I was showing what the equilibrium is, but p star itself is 0.7. Thank you. All right. so when Venus plays left with probability 0.7 at 0.7 of the time she’ll get an expected payoff of 0.62 and 0.3 of the time she’ll get an expected payoff again at 0.62 and that’s the kind of math I don’t have to do at home that’s going to come out at 0.62 again assuming my math is correct so what I’ve really done here is confirm what we did already last time we knew that we in fact chose Serena’s mixed Q to make Venus indifferent between left and right. And that’s exactly what we found here. Going left gets 0.62, going right gets 0.62, and hence P star gets 0.62. But I claim we can now see something a little bit else We can now ask the question is P star in fact a best response Well for it not to be a best response for this not to be an equilibrium there would have to be some deviation that Venus could make that would make her strictly better off.

[译文 4]

澄清一下，当你说0.62的时候，她会出右边，再一次，她会得到0.62的收益，我有计算器吗？澄清一下，当你说p star是0.7和0.3，你是说p star是0.7。是的，抱歉，谢谢你。所以p star是0.7。你完全正确。所以这是p star和1减去p star。好的，让我们把它说清楚。我展示的是均衡是什么，但p star本身是0.7。谢谢。好的，所以当Venus以0.7的概率出左边时，70%的时间她会得到0.62的期望收益，30%的时间她会再次得到0.62的期望收益，这是我不需要在家算的那种数学，假设我的计算正确，结果又是0.62。所以我在这里真正做的是确认我们上周已经做过的事情，我们知道我们实际上选择了Serena的混合策略Q来使Venus在左边和右边之间无差异。这正是我们在这里发现的。出左边得到0.62，出右边得到0.62，因此P star得到0.62。但我声称我们现在可以看到一些别的东西。我们现在可以问这个问题，P star实际上是一个最佳反应吗？如果这不是一个均衡的话，那么Venus必定能够进行某种偏离使她严格地更好。

[段 5]

Let me repeat that. If this were not an equilibrium, there would have to be some deviation for Venus that would make her strictly better off. By playing P star, she’s getting a return of 0.62. So one thing she could deviate to is playing left all the time. If she deviates to playing left all the time, her payoff is still 0.62. So she’s not strictly better off. That’s not a strictly profitable deviation. Right? Another thing she could deviate to is she could deviate to playing right. If she deviates to playing right, her payoff will be 0.62. Once Once again, she’s not strictly better off. She’s the same as she was before. So that’s not a strictly profitable deviation. So what have I shown so far? I’ve shown that P star is as good as playing left, and P star is as good as playing right.

[译文 5]

让我重复一下。如果这不是一个均衡，那么必定存在Venus的某种偏离使她严格地更好。通过玩P star，她得到0.62的回报。所以她可以偏离的一件事是一直玩左边。如果她偏离到一直玩左边，她的收益仍然是0.62。所以她并没有严格地更好。那不是一个严格盈利的偏离。对吧？她可以偏离的另一件事是她可以偏离到一直玩右边。如果她偏离到一直玩右边，她的收益将是0.62。再一次，她并没有严格地更好。她和之前一样。所以那也不是一个严格盈利的偏离。那么我到目前为止展示了什么？我展示了P star和玩左边一样好，P star和玩右边一样好。

[段 6]

In fact that how we constructed it So deviating to left is not a strictly profitable deviation and deviating to right is not a strictly profitable deviation But at this point somebody might ask and say OK, you’ve shown me that there’s no way to deviate to a pure strategy in a strictly profitable way, but how about deviating to another mixed strategy? alright so far we’ve shown we’ve shown just up here we can see that Venus has no strictly profitable pure strategy deviation pure strategy deviation she has no strict pure strategy deviation because each of her pure strategies yield the same payoff as did her putative mixed strategy yield the same as P star but how do we know that she doesn’t have a mixed strategy that would be strictly better how do we know that anybody how do we know that No No hands going up Oh is there a hand up Good good Any mix between left and right will still yield 0 Good, good. So any mix, any mix that Venus deviates to will be a mix between left and right, and any mix between left and right will be a mix between 0.62 and 0.62, and hence will yield 0.62. All right, so we’re going to use again this fact we developed last week. The fact we developed last week was that any mixed strategy yields a payoff that is a weighted average of the pure strategy payoffs, the payoff to the pure strategies in the mix.

[译文 6]

实际上这就是我们构建它的方式。所以偏离到左边不是一个严格盈利的偏离，偏离到右边也不是一个严格盈利的偏离。但此时可能有人会问，好，你已经向我展示了没有办法以严格盈利的方式偏离到纯策略，但偏离到另一个混合策略怎么样？好的，到目前为止我们已经展示的，我们刚刚在这里展示的是Venus没有严格盈利的纯策略偏离，因为她每个纯策略都得到和假设的混合策略相同的收益，和P star相同的收益。但我们怎么知道她没有一个会严格更好的混合策略？我们怎么知道？我们怎么知道？没有人举手，哦有人举手吗？好的，好的。任何在左边和右边之间的混合都将仍然得到0.62。好的，好的。所以任何混合，任何Venus偏离到的混合都将是在左边和右边之间的混合，而在左边和右边之间的任何混合都将是在0.62和0.62之间的混合，因此将得到0.62。好的，所以我们再次使用上周我们发展的这个事实。上周我们发展的这个事实是，任何混合策略产生的收益是混合中纯策略收益的加权平均，混合中纯策略的收益。

The user wants me to translate the English speech from segment 7 to segment 12 into Chinese. I need to follow these rules:

1. Only output translations for [段 7] to [段 12]
2. Keep all segment numbers continuous, don’t change or add segments
3. Natural Chinese translation, keep names/brands/numbers/institutions in English
4. No explanations, prefixes, or markdown code blocks

Let me translate each segment:

[段 7]

Any mixed strategy yields a payoff that is a weighted average of the payoff to the pure strategy in the mix. That was our key fact last week. So here, if we’ve shown that there’s no pure strategy deviation that’s strictly profitable, then there can’t be any mixed strategy deviation that’s strictly profitable. Why? Because the mixed strategy deviations must yield payoffs that lie among the pure strategy deviations. All right? So this is a great fact for us. What’s the Because the mixed strategy deviations must yield payoffs that lie among the pure strategy deviations. So this is a great fact for us. What’s the lesson here? the lesson is we only ever have to check for strictly profitable pure strategy deviations. That’s a good job. Why? Because if we had to check for mixed strategy deviations, one by one, we’d be here all night, because there’s an infinite number of possible mixed strategy deviations. All right? But there aren’t so many pure strategy deviations we have to check. Let’s just repeat the idea. Suppose there isn’t any pure strategy deviation that’s profitable. Then there can be any mixed strategy deviation that profitable because the highest the The highest expected return you could ever get from a mixed strategy is one of the pure strategies in the mix and you already checked that none of those are profitable All right So this simple idea the simple idea we developed last time not only helps us to find Nash equilibria, but also to check for Nash equilibria.

[译文 7]

Any mixed strategy yields a payoff that is a weighted average of the payoff to the pure strategy in the mix. That was our key fact last week. So here, if we’ve shown that there’s no pure strategy deviation that’s strictly profitable, then there can’t be any mixed strategy deviation that’s strictly profitable. Why? Because the mixed strategy deviations must yield payoffs that lie among the pure strategy deviations. All right? So this is a great fact for us. What’s the Because the mixed strategy deviations must yield payoffs that lie among the pure strategy deviations. So this is a great fact for us. What’s the lesson here? the lesson is we only ever have to check for strictly profitable pure strategy deviations. That’s a good job. Why? Because if we had to check for mixed strategy deviations, one by one, we’d be here all night, because there’s an infinite number of possible mixed strategy deviations. All right? But there aren’t so many pure strategy deviations we have to check. Let’s just repeat the idea. Suppose there isn’t any pure strategy deviation that’s profitable. Then there can be any mixed strategy deviation that profitable because the highest the The highest expected return you could ever get from a mixed strategy is one of the pure strategies in the mix and you already checked that none of those are profitable All right So this simple idea the simple idea we developed last time not only helps us to find Nash equilibria, but also to check for Nash equilibria.

任何混合策略产生的收益都是该混合中各个纯策略收益的加权平均值。这是我们上周的关键事实。所以在这里，如果我们已经证明没有任何纯策略偏离是严格有利可图的，那么就不可能有任何混合策略偏离是严格有利可图的。为什么？因为混合策略偏离产生的收益必然落在纯策略偏离的收益范围内。明白吗？这对我们来说是一个很好的事实。为什么？因为混合策略偏离产生的收益必然落在纯策略偏离的收益范围内。这对我们来说是一个很好的事实。这里的教训是什么？教训是我们只需要检查严格有利可图的纯策略偏离。这是个好工作。为什么？因为如果我们必须逐一检查混合策略偏离，我们今晚就会一直待在这里，因为有无限多种可能的混合策略偏离。但我们需要检查的纯策略偏离并没有那么多。让我们再重复一下这个想法。假设没有任何纯策略偏离是有利可图的。那么就不可能有混合策略偏离是有利可图的，因为你能从混合策略中获得的最高预期回报就是该混合中某个纯策略的回报，而你已经检查过这些纯策略中没有哪个是有利可图的。所以我们上次提出的这个简单想法不仅帮助我们找到纳什均衡，也帮助我们检验纳什均衡。

[段 8]

All right. Now a lot of people I gathered from feedback from sections were left pretty confused last time. It’s a hard idea. Actually, I looked at the tape over the weekend, and I could see where it could be confusing, but it’s actually, I think what’s really confusing here wasn’t so much, I think this time, wasn’t so much that I could have been clearer, though I’m sure I could have been, it’s that this is really a hard idea, this idea of mixed strategies. So we’re going to work on it again today, but I think one of the ideas that gets people confused is the following idea. They say, look, we found Venus’s equilibrium mix by choosing a p and a 1 minus p to make Serena indifferent. And we found Serena’s equilibrium mix by finding a q and a 1 minus q to make Venus indifferent. And a natural question you hear people ask then is, why is Venus, quote, trying to make Serena indifferent? And why is Serena quote trying to make Venus indifferent And that not really the point here It isn that Venus is trying to make Serena indifferent it that in equilibrium she is going to make Serena indifferent It isn’t her goal in life to make Serena indifferent between left and right, and it isn’t Serena’s goal in life to make Venus indifferent between left and right.

[译文 8]

All right. Now a lot of people I gathered from feedback from sections were left pretty confused last time. It’s a hard idea. Actually, I looked at the tape over the weekend, and I could see where it could be confusing, but it’s actually, I think what’s really confusing here wasn’t so much, I think this time, wasn’t so much that I could have been clearer, though I’m sure I could have been, it’s that this is really a hard idea, this idea of mixed strategies. So we’re going to work on it again today, but I think one of the ideas that gets people confused is the following idea. They say, look, we found Venus’s equilibrium mix by choosing a p and a 1 minus p to make Serena indifferent. And we found Serena’s equilibrium mix by finding a q and a 1 minus q to make Venus indifferent. And a natural question you hear people ask then is, why is Venus, quote, trying to make Serena indifferent? And why is Serena quote trying to make Venus indifferent And that not really the point here It isn that Venus is trying to make Serena indifferent it that in equilibrium she is going to make Serena indifferent It isn’t her goal in life to make Serena indifferent between left and right, and it isn’t Serena’s goal in life to make Venus indifferent between left and right.

好的。从各section的反馈来看，很多人在上次课之后感到相当困惑。这是一个很难的概念。实际上，我在周末看了录像，我能看出哪里可能会让人困惑，但我觉得真正令人困惑的并不是，我这次并不是说，我本可以讲得更清楚，虽然我肯定可以讲得更清楚，而是这个混合策略的概念确实是一个很难的概念。所以我们今天会再讨论它，但我觉得让人们困惑的一个想法是以下这个想法。他们说，看，我们通过选择p和1-p使得Serena无差异来找到Venus的均衡混合。我们通过找到q和1-q使得Venus无差异来找到Serena的均衡混合。然后你听到人们问的一个自然问题是，为什么Venus"试图"让Serena无差异？为什么Serena"试图"让Venus无差异？但这里并不是这个重点。这不是说Venus试图让Serena无差异，而是说在均衡中她会使得Serena无差异。让Serena在左和右之间无差异并不是她的人生目标，让Venus在左和右之间无差异也不是Serena的人生目标。

[段 9]

But in equilibrium, it ends up that they make each other indifferent. alright and the way that we can see that is that if Venus puts something said last time it’s repeat it if Venus puts too much weight more than point seven on left on left then Serena Serena just cheats to left all the time and that can’t possibly be an equilibrium and if Venus puts too much weight on right then Serena cheats to the right all the time and that can’t be an equilibrium so it has to be that what Venus is doing is gonna make Serena exactly indifferent and vice versa. All right. Now let’s see that idea in some other applications. Let’s talk about this a bit before we move on. All right. So it turns out that some very natural applications for mixed strategy equilibria arise in games, in sport. All right. So let talk about a few now Can anybody suggest some other places where we see randomization or at least mixed strategy in equilibria in sporting events Let me actually grab the mic myself Anybody here play football, for example? We’re talking American football now, gridiron game, not the civilized type. So some of you play football, right? So where is the mixing involved in playing football? Where in equilibrium would we expect to see mixed strategies? Yeah, there’s somebody down there.

[译文 9]

But in equilibrium, it ends up that they make each other indifferent. alright and the way that we can see that is that if Venus puts something said last time it’s repeat it if Venus puts too much weight more than point seven on left on left then Serena Serena just cheats to left all the time and that can’t possibly be an equilibrium and if Venus puts too much weight on right then Serena cheats to the right all the time and that can’t be an equilibrium so it has to be that what Venus is doing is gonna make Serena exactly indifferent and vice versa. All right. Now let’s see that idea in some other applications. Let’s talk about this a bit before we move on. All right. So it turns out that some very natural applications for mixed strategy equilibria arise in games, in sport. All right. So let talk about a few now Can anybody suggest some other places where we see randomization or at least mixed strategy in equilibria in sporting events Let me actually grab the mic myself Anybody here play football, for example? We’re talking American football now, gridiron game, not the civilized type. So some of you play football, right? So where is the mixing involved in playing football? Where in equilibrium would we expect to see mixed strategies? Yeah, there’s somebody down there.

但在均衡中，最终结果是他们使得彼此无差异。好的，我们能看到这一点的方式是，如果Venus放太多权重在左边的选择上，超过0.7，那么Serena就会一直作弊选择左边，这不可能是均衡；如果Venus放太多权重在右边，那么Serena就会一直作弊选择右边，这也不可能是均衡。所以必然是Venus所做的会让Serena恰好无差异，反之亦然。好的。现在让我们在其他的应用中看看这个想法。在我们继续之前，让我们先讨论一下这个。好的。事实证明，混合策略均衡的一些非常自然的应用出现在游戏和体育中。好的。那么现在让我们谈谈几个例子。有谁能建议一些其他我们看到随机化或者至少是均衡中的混合策略的体育赛事中的地方？让我自己拿一下麦克风。这里有人踢足球吗？比如美式足球，橄榄球，不是那种文明的游戏。你们中有人踢足球，对吧？那么在踢足球中混合策略涉及哪里？在均衡中我们期望在哪里看到混合策略？是的，下面有人。

[段 10]

Let me go and get them. All right. So shout it out. Running game and passing game. All right, the running game and the passing game, right? So a very simple idea. Whether to run or whether to pass when you have the ball is likely to end up as a mixed strategy equilibrium. The defense is also randomizing between, for example, rushing the passer or playing a run defense. Is that right? This is not exactly a game I know a lot about, but I’m hoping I’m getting close enough here. It couldn’t possibly be a pure strategy equilibrium, other than in very extreme parts of the game, like at the end of the game perhaps. But for most of the game, it’s very unlikely to end up as a pure strategy equilibrium. Much more likely that the offense is mixing between passing and running, and for that matter, between going to the left, going to the right, and going to the center. And the defense is also mixing over its types of defense. So we see that, for those people who were watching yesterday, we see that in football games. Why do we see it else in sport? Some other sports? I can’t have a room full of non-sports fans. How many of you ever watch any sport? Let’s just raise some hands here. All right, some of you do, some of you do.

[译文 10]

Let me go and get them. All right. So shout it out. Running game and passing game. All right, the running game and the passing game, right? So a very simple idea. Whether to run or whether to pass when you have the ball is likely to end up as a mixed strategy equilibrium. The defense is also randomizing between, for example, rushing the passer or playing a run defense. Is that right? This is not exactly a game I know a lot about, but I’m hoping I’m getting close enough here. It couldn’t possibly be a pure strategy equilibrium, other than in very extreme parts of the game, like at the end of the game perhaps. But for most of the game, it’s very unlikely to end up as a pure strategy equilibrium. Much more likely that the offense is mixing between passing and running, and for that matter, between going to the left, going to the right, and going to the center. And the defense is also mixing over its types of defense. So we see that, for those people who were watching yesterday, we see that in football games. Why do we see it else in sport? Some other sports? I can’t have a room full of non-sports fans. How many of you ever watch any sport? Let’s just raise some hands here. All right, some of you do, some of you do.

让我去找他们。好的，大声说出来。跑动进攻和传球进攻。好的，跑动进攻和传球进攻，对吧？所以这是一个非常简单的想法。当你持球时是选择跑动还是传球，这很可能最终成为一个混合策略均衡。防守方也在随机选择，例如，是冲传还是打防跑。对吧？这不是一个我很熟悉的比赛，但我希望我说的足够接近了。这不可能是一个纯策略均衡，除了在比赛的某些非常极端的时刻，比如比赛结束的时候。但对于大部分比赛时间，这非常不可能是一个纯策略均衡。更可能的是进攻方在传球和跑动之间混合，而且同样地，在向左、向右和向中间进攻之间混合。防守方也在其防守类型之间混合。所以我们看到，对于那些昨天看过比赛的人，我们在足球比赛中看到了这一点。为什么我们在其他体育运动中也看到它？其他体育运动？我不相信这里满屋子都是不看体育的人。你们有多少人看过任何体育运动？让我们举手。有一些人有。

[段 11]

All right, so this is baseball playoff season, right? How many of you have been watching the baseball playoffs? Raise your hands if you’ve been watching the baseball playoffs. I’ll let you off. I know you should have been doing my homework. How many of you have been watching the playoffs instead? How many watched the Yankees game last night? All right, quite a few of you. All right, All right, so it hasn’t been very exciting yet, but we’re hoping it’s going to get more exciting, right? So when you’re watching baseball, what kind of things do you see where you just know that there must be mixed strategies involved? There must be randomization involved Now I got a few more hands up Good so you sir Choosing how to pitch the ball Choosing how to pitch the ball So enlarge a little bit more Say a little bit more Fastball versus slider versus change all sorts of different things All right. So, there’s different ways of throwing the ball, all right, and there’s going to be randomization from the pitcher, or at least it’s going to look like there’s randomization by the pitcher, over whether to throw a fastball or a curveball or whatever. How is the hitter randomizing that? How’s the hitter randomizing? Is the hitter randomizing at all? What’s the hitter doing while this is going on?

[译文 11]

All right, so this is baseball playoff season, right? How many of you have been watching the baseball playoffs? Raise your hands if you’ve been watching the baseball playoffs. I’ll let you off. I know you should have been doing my homework. How many of you have been watching the playoffs instead? How many watched the Yankees game last night? All right, quite a few of you. All right, All right, so it hasn’t been very exciting yet, but we’re hoping it’s going to get more exciting, right? So when you’re watching baseball, what kind of things do you see where you just know that there must be mixed strategies involved? There must be randomization involved Now I got a few more hands up Good so you sir Choosing how to pitch the ball Choosing how to pitch the ball So enlarge a little bit more Say a little bit more Fastball versus slider versus change all sorts of different things All right. So, there’s different ways of throwing the ball, all right, and there’s going to be randomization from the pitcher, or at least it’s going to look like there’s randomization by the pitcher, over whether to throw a fastball or a curveball or whatever. How is the hitter randomizing that? How’s the hitter randomizing? Is the hitter randomizing at all? What’s the hitter doing while this is going on?

好的，现在是棒球季后赛季节，对吧？你们有多少人一直在看棒球季后赛？如果你们一直在看棒球季后赛请举手。我放你们一马。我知道你们应该在做我的作业。你们有多少人在看季后赛？有多少人看了昨晚的洋基队比赛？好的，你们中有不少。好的，好吧，所以到目前为止还不太精彩，但我们希望它会变得越来越精彩，对吧？所以当你们在看棒球时，你们看到什么样的事情让你们知道必然涉及混合策略？必然涉及随机化。现在我看到更多手举起来了。好的，你，选择怎么投球。选择怎么投球。所以再展开一点说更多一点。快速球对滑球对变速球，各种不同的东西。好的。所以，有不同的投球方式，而且投手会进行随机化，或者至少看起来投手在进行随机化，决定是投快速球还是曲线球或者其他什么。击球手如何进行随机化？击球手如何随机化？击球手有没有在随机化？在这一切进行的时候击球手在做什么？

[段 12]

Anybody? Yeah? He’s choosing whether to swing or not to swing. Okay, he’s choosing whether to swing or not to swing, although presumably he can do that just after the ball’s thrown. So you sometimes hear the commentator say that that hitter was looking for a fastball, is that right? Or looking for a curveball. The hitter’s trying to anticipate the pitch, is that right? This is not a game I’ve played a lot of either, I’ve played a little bit. You’re trying to anticipate where the ball’s going to be thrown. So the type of ball you throw in baseball and the way in which the pitch being anticipated by the hitter is likely to be a mixed strategy. What else is likely to be a mixed strategy in baseball? What else? Anybody here on the Yale baseball team Ah OK OK I got one volunteer here alright Alright so what else I mean stand up a second let have our Yale baseball team member, alright. Alright, what’s your name? Chris. And where do you play? I’m a pitcher. You’re a pitcher, okay. He’s not going to get on base notes, he’s not going to be able to answer this. Suppose you did get on base, pitchers don’t often get on base, I don’t know. Let’s assume that happens, right? What might you randomize? there you are, you’re standing on base, what might you randomize about?

[译文 12]

Anybody? Yeah? He’s choosing whether to swing or not to swing. Okay, he’s choosing whether to swing or not to swing, although presumably he can do that just after the ball’s thrown. So you sometimes hear the commentator say that that hitter was looking for a fastball, is that right? Or looking for a curveball. The hitter’s trying to anticipate the pitch, is that right? This is not a game I’ve played a lot of either, I’ve played a little bit. You’re trying to anticipate where the ball’s going to be thrown. So the type of ball you throw in baseball and the way in which the pitch being anticipated by the hitter is likely to be a mixed strategy. What else is likely to be a mixed strategy in baseball? What else? Anybody here on the Yale baseball team Ah OK OK I got one volunteer here alright Alright so what else I mean stand up a second let have our Yale baseball team member, alright. Alright, what’s your name? Chris. And where do you play? I’m a pitcher. You’re a pitcher, okay. He’s not going to get on base notes, he’s not going to be able to answer this. Suppose you did get on base, pitchers don’t often get on base, I don’t know. Let’s assume that happens, right? What might you randomize? there you are, you’re standing on base, what might you randomize about?

有人吗？是的？他在选择是否挥棒。好的，他在选择是否挥棒，尽管他大概只能在球投出之后才能做这个决定。所以你有时会听到解说员说那个击球手在找快速球，对吧？或者在找曲线球。击球手试图预判投手会投什么球，对吧？这也不是一个我打过很多次的运动，我打过一点点。你在试图预判球会被投到哪里。所以你在棒球中投的球类型以及击球手预判投手的方式很可能是一个混合策略。在棒球中还有什么很可能是混合策略？还有什么？这里有耶鲁棒球队的人吗？啊好的好的，我这里有一个志愿者。好的，还有什么，我意思是站起来一下，让我们有请我们的耶鲁棒球队成员，好的。好的，你叫什么名字？Chris。你打什么位置？我是一个投手。你是一个投手，好的。他不会上垒的，他没办法回答这个。假设你确实上垒了，投手不经常上垒，我不知道。让我们假设那发生了，对吧？你可能会随机化什么？当你站在那里，你站在垒上，你可能会随机化什么？

[段 13]

Whether to steal second or not? Right, whether to steal or not, whether to try to steal or not. Stay up a second, stay up a second. Alright, so the decision whether to try and steal or not is likely to end up being random, right? And if you’re the pitcher, if you’re the pitcher, what can you do in response to, here’s a pitcher, right, what can you do in response to that? You can either choose to try to pick him off or not. Alright, what else can you, that’s the one that you can try and pick him off, what else? You can be quicker to the plate. You can be quicker to the plate, what else? You can pitch out. You can pitch out, what else? At least those three things, right? At least those three things, okay? Okay, thank you, thank you. All right, I have an expert here, I’m glad I have an expert. All right, so in this case, in this case we can see this randomization going on from the runner whether he attempts to steal the base or not and by the pitcher on whether he throws the pitch out or whether he tries to throw to get to the plate faster All right Okay So we see this in sport We don’t see it well anticipated by sports commentators. Let me give this a bit down a second.

[译文 13]

关于是否盗垒？是的，是否盗垒，是否尝试盗垒。等一下，等一下。好的，所以是否尝试盗垒的决定最终可能是随机的，对吧？如果你是投手，如果你是投手，你该如何应对？投手，你该怎么做？你可以选择是否尝试牵制盗垒。好的，你还能做什么，你能尝试牵制盗垒，还能做什么？你可以更快地投球到本垒。你可以更快地投球到本垒，还能做什么？你可以投外角球。你可以投外角球，还能做什么？至少这三个选择，对吧？至少这三个选择，好吗？好的，谢谢，谢谢。我这里有一位专家，我很庆幸有位专家。好的，在这个例子中，在这个例子中，我们可以看到跑者是否尝试盗垒以及投手是否投外角球或是否试图更快投球到本垒的随机化过程。好的，我们可以在体育运动中看到这一点，但我们没有看到体育评论员很好地预见到这一点。让我再讲一下。

[段 14]

So in baseball, for example, you’ll sometimes see quite sophisticated statistical analyses of baseball in which somebody will have looked at base stealers across the major leagues, and they’ll look at all the instances in which a player was on first base in a position where you think they might steal and they’ll look at what happened on every attempt to steal whether they were in fact caught stealing or not. And they’ll try and measure the value of these things and the conclusion they’ll come to is something like this. They’ll conclude that whether the guy stole or not, whether the guy attempted to steal or not, sorry, or whether they just sat on first base, doesn’t seem to make much difference, they’ll say. I’ll say the payoff for even great base feelers of attempting to steal or not, all right, when you take into account the pick-offs versus just staying put, turns out the payoff in terms of the impact on the game is roughly equal. And then they’ll draw that these analysts will then draw the following conclusion. They’ll say, oh look, speed or the ability to steal bases is their fault. impact on the game is roughly equal. And then they’ll draw that these analysts will then draw the following conclusion. They’ll say, oh look, speed, or the ability to steal bases, is therefore overrated in baseball.

[译文 14]

例如在棒球中，你有时会看到相当复杂的棒球统计分析，其中有人会研究整个大联盟的盗垒者，他们会观察所有球员占据一垒且你认为他们可能会盗垒的情况，他们会观察每次盗垒尝试的结果，无论他们实际上是否被捕盗。他们会尝试衡量这些行为的价值，他们的结论大致是这样的。他们会得出结论，无论这家伙是否盗垒，无论这家伙是否尝试盗垒，抱歉，或者他们是否就停在一垒，似乎没有太大区别，他们会这么说。我会说，即使是优秀的盗垒者，尝试盗垒与否的回报，当你把牵制与留在原地都考虑进去时，事实证明，从对比赛的影响来看，回报大致相等。然后这些分析师就会得出以下结论。他们会说，哦，看啊，速度或者说盗垒能力，因此被高估了。

[段 15]

How have they made a mistake? What’s the mistake they made there? So the premise was, let’s give them the premise, the premise was that when a base stealer is attempting to steal or not, the expected return in terms of outcome of the game is roughly equal whether they attempt to steal or don’t attempt to steal. The conclusion is, therefore stealing doesn’t seem such a big deal. What’s the mistake they’ve made? Yeah, let me borrow it again, sorry. Yeah? The pitcher has to react differently in pitching when he knows that there’s a fast guy on his… Good, good. So our pitcher has to react differently. Let’s talk to our pitcher again, right? So one thing our pitcher said was he wants to get to the plate faster. Right? Right? What does that mean, getting to the plate faster? Let’s just… It means just… Shout out to people who can hear you. Getting the ball to the catcher as fast as possible so that he has the best chance to throw out the runner. Right. Right. So you’re going to pitch from what’s… You not going to do that funny wound thing right You going to Thank you You going to pitch from the stretch I knew there was a term there somewhere I learning American by being here right And you more likely to throw a fastball right There some advantage in throwing a fastball rather than a curveball Both actions of which, both having to move more towards fastballs and pitching from the stretch, are actually costly for the pitcher.

[译文 15]

他们犯了什么错误？他们在这里犯了什么错误？让我们先给出他们的前提，前提是当盗垒者尝试盗垒与否时，从比赛结果来看，预期回报大致相等，无论他们是否尝试盗垒。结论是，因此盗垒似乎不是什么大事。他们犯了什么错误？是的，让我再借用一下，抱歉。是的？当知道有一个速度快的球员在一垒时，投手在投球时必须做出不同的反应。很好，很好。所以我们的投手必须做出不同的反应。让我们再和我们的投手谈谈，对吧？所以我们的投手说的一件事是他想更快地投球到本垒。对吧？对吧？更快地投球到本垒是什么意思？让我们…这意味着就是…大声一点让能听到的人都能听到。尽快把球传给捕手，这样他才有最好的机会把跑者杀出局。对吧？对吧。所以你不做那个滑稽的蓄力动作，你要…谢谢，你要从伸展姿势投球，我知道那里有个术语，我在这里学美国人的说法，从伸展姿势投球，你就会更可能投出快速球。投快速球比投曲球有一些优势，这两种做法，即更多地投快速球和从伸展姿势投球，对投手来说实际上是有代价的。

[段 16]

We’ll get there in a second. Let’s just back up a second. That was good, that’s right, but let’s just back up a second. The premise of these commentators was what? it was that the return to stealing, attempting to steal, seems to be roughly a wash. It seems to be that the expected return when this great base runner attempts to steal the base is roughly the same as the return when they don’t attempt to steal the base. But I claim we knew that was going to be the case. We didn’t have to go and look at the data. Why did we know that was going to be the case? How did we know that you were bound to find a return in that analysis that finds those things roughly equal. Wow. Yeah. If we use randomizing, that means that the returns will be equal. If they weren’t equal, we would just do one or the other all the time. Good. Good. Excellent. Excellent. Since we’re in a mixed strategy equilibrium, since she’s randomizing, it must be the case that the returns are equal All right That makes sense That the big idea here right That the thing we learned last time If this player and these are professional baseball players doing this they been very well trained a lot of money has been spent on getting the tactics right all right?

[译文 16]

我们一会儿会讲到。让我们先退后一下。让我们先退后一下。这很好，这是对的，但让我们先退后一下。这些评论员的前提是什么？前提是盗垒的回报，尝试盗垒，似乎大致持平。似乎是，当这位优秀的盗垒者尝试盗垒时的预期回报，与当他没有尝试盗垒时的回报大致相同。但我声称我们早就知道会是这种情况。我们不需要去看数据。我们怎么知道会是这种情况？我们怎么知道在那个分析中你必然会得出这些事物大致相等的回报？哇，是的。如果我们使用随机化，那就意味着回报会是相等的。如果它们不相等，我们就会一直做其中某一个。很好，很好。非常好，非常好。因为我们处于混合策略均衡中，因为她在随机化，所以必然的情况是回报是相等的。好的，这有道理。这是我们上次学到的大概念。如果这位球员和这些职业棒球运动员这样做，他们已经接受了非常好的训练，花了很多钱来确保战术正确，对吧？

[段 17]

There’s people sitting there who are paid to get the tactics right, right? If it was the case that the return to base stealing wasn’t roughly equal when you attempt to steal or didn’t attempt to steal, right, then you shouldn’t be randomizing. Since you are randomizing, it must be the case that the returns are roughly equal. All right? So that’s the first thing to observe. And the second thing to observe is what we just pointed out. In fact, the value of having a fast base spieler on the team doesn’t show up in the expected return on the occasions on which he attempts to steal or which he does not attempt to steal. It shows up where? It shows up in the fact that the pitching team changes their behavior to make it harder for this guy to steal by going faster to the plate or throwing more fastballs? And where will that show up in the statistics? If you’re just a statistician like me, you just look at the data, where will that show up? I mean, suppose I can’t keep track of every single pitch. I can’t actually observe all these fastballs. Where will I see the effect of all these extra fastballs and pitching from the stretch in the data Somebody It going to show up in the batting average of the guy who hitting behind the base stealer All right?

[译文 17]

有人被付钱来确保战术正确，对吧？如果盗垒的回报在你尝试盗垒与否时大致相等这件事不是真的，对吧，那么你不应该随机化。因为你在随机化，所以必然的情况是回报大致相等。好的吧？这是第一件要观察到的事情。第二件要观察到的事情是我们刚刚指出的。事实上，拥有一个快速的盗垒者在球队中的价值，不会显示在他尝试盗垒或没有尝试盗垒的场合的预期回报中。它显示在哪里？它显示在这样一个事实中：投球方会改变他们的行为，通过更快地投球到本垒或投出更多的快速球，使这个人更难盗垒，而这会显示在统计数据中的什么地方？如果你们只是像我这样的统计学家，你们只看数据，它会显示在哪里？我的意思是，假设我无法追踪每一个投球。我实际上无法观察到所有这些快速球。所有这些额外的快速球和从伸展姿势投球的效果，会显示在数据中的什么地方？有人会说…它会显示在盗垒者后面那位击球手的打击率上。

[段 18]

The guy who’s hitting behind the base stealer is going to have a higher batting average because he’s going to get more pitches, which are fastballs to hit, and more pitches out of the stretch. All right? So if you ignore that effect, you’re going to be in trouble. But we know, we analyzed this properly using gain theory, we know we’re in a mixed-structure equilibrium, him, we know, in fact, the pitching team must be reacting to it, we know there must be a cost in doing that, and the cost turns up in the hitter behind. So when you’re watching the playoffs in the last, now I’m giving you permission to watch a bit of TV at night, after you’ve done my homework assignment, but before anyone else’s homework assignment, you can have a look at these baseball games and have a go at being a little bit better than the commentators who are working on them. all right so one application for mixed strategies is in sport but not the only application let’s just talk about another application a slightly more scary application so after 9-11 there was a lot of talk in the US about the placements of baggage checking machines at airports all right actually still going on at all but there’s a lot of talk then about the placement of machines just to search the luggage of baggage-checking machines at airports.

[译文 18]

盗垒者后面那位击球手会有更高的打击率，因为他会得到更多的球来打，那些是快速球，以及更多的从伸展姿势投出的球。对吧？所以如果你忽略那个效应，你就会陷入麻烦。但我们知道，我们用博弈论正确地分析了这一点，我们知道我们处于混合策略均衡中，他，我们知道，事实上，投球方一定在对此做出反应，我们知道这样做一定有代价，而代价会落在后面的击球手身上。所以当你们在看季后赛的时候，最后，我现在给你们许可让你们在晚上看一会儿电视，在你们做完我的作业之后，但在其他人交作业之前，你们可以看看这些棒球比赛，试着比那些从事这项工作的评论员做得更好一点，好吗？所以混合策略的一个应用是在体育运动中，但不是唯一的应用，让我们谈谈另一个应用，一个稍微更可怕的应用，所以在911之后，美国有很多关于在机场放置行李检查机器的讨论，对吧，实际上现在仍在进行中，但当时有很多关于放置机器来检查行李的讨论。

[段 19]

Actually, there’s still quite a lot of talk, but there was a lot of talk then about the placement of machines to search the luggage that goes on board. The hand luggage was being searched anyway, but to search luggage going into the cabins. And it was pointed out at the time, this has changed since, there weren’t actually enough machines in the US on the day after 9-11 to search every single bag that went into the hold. And you’d hear discussions of the following type. you’d hear these experts on Nightline or whatever, and they’d say, look, there’s no point trying to do this, because if we put all our baggage-searching machines at Logan Airport in Boston, for example, then the terrorists will simply move their attack to O’Hare. And if we put them at O’Hare, then they’ll move their attack to Logan. If we have enough to do both Logan and O’Hare, then they’ll move their attack to some third airport. So there was a sense of doom in the air. it was kind of a depressing time anyway, there was then some doom in the air saying that when you, if you put your baggage searching machines somewhere, all you do is cause the attempted terrorists terrorists attempting to blow up the planes to go elsewhere And you hear the same things today about searching individuals as they go on the plane For example you hear the discussion that says if we only search men travelling alone let say then you’ll quickly end up with all the people carrying bombs being couples or women.

[译文 19]

实际上，关于行李安检机的部署问题，当时仍有大量讨论。手提行李已经在检查了，但托运行李的检查是个问题。当时有人指出，情况已经发生了变化，在911之后的那个日子里，美国实际上没有足够的机器来检查每一个进入货舱的行李。当时你会听到类似这样的讨论。你会听到Nightline或其他节目上的专家们说：看，这样做没有意义，因为如果我们把所有行李安检机都放在波士顿的Logan机场，那么恐怖分子就会把袭击地点改到O’Hare。如果我们把它们放在O’Hare，他们就会改到Logan。如果我们有能力同时覆盖Logan和O’Hare，他们就会改到第三个机场。所以当时空气中弥漫着一种宿命感。那段时期本来就令人沮丧，当时有些悲观论调说，当你把行李安检机放在某个地方时，你所做的只是迫使企图炸毁飞机的恐怖分子转移到别处。如今关于登机人员安检，你也能听到同样的观点。例如有人说，如果我们只检查单独旅行的男性，那么很快所有携带炸弹的人都会变成情侣或女性。

[段 20]

And again, there’s this sense of doom, this sense that says, it’s hopeless. Whatever we do, we’re just going to force the terrorists to do something else, but we won’t have gained anything. So once again, that’s wrong. What’s wrong about that one? What should we be doing in that sense? What should they have done, in fact they did do, with those luggage-baggage searching machines when they were in short supply after 9-11? What do they do with searching people as they get on planes? What do they do? Well, here’s what they didn’t do. They didn’t just put them at certain airports and announce they’re just at these airports. Right? That would have been a crazy thing to do. That would have been hopeless. Entirely hopeless, but not, you know, not wise. What should they have done? What did they do? Anybody want to guess Yeah In name they randomize who they were checking Right right So when they checking passengers they’re going to randomly check passengers, right? When they’re checking, when they think about the baggage machines, a sensible thing to do is to put a big metal box at every single airport and say, we’re not going to tell you which of these boxes actually have baggage checking machines, which effectively is randomizing. from the point of view of the terrorists, they’re not going to know where the baggage checks are going on.

[译文 20]

同样地，又有一种宿命感，有一种声音说这是无望的。无论我们做什么，我们只是迫使恐怖分子做别的事情，但我们什么都得不到。所以再说一遍，那是错误的。那种说法错在哪里？我们应该做什么？在行李安检机紧缺的时候，911之后他们应该做什么——实际上他们做了什么——他们如何处理登机人员的安检？他们应该怎么做？好吧，他们没有做的是，他们没有只在某些机场放置安检机然后宣布它们就在这些机场。对吧？那将是一件疯狂的事情。那将是完全无望的。彻底无望，但不明智。他们应该做什么？他们做了什么？有人想猜吗？他们在名称上随机化了检查对象，对吧？所以当他们检查乘客时，他们会随机检查乘客，对吧？当他们考虑行李安检机时，一个明智的做法是在每个机场都放置一个大的金属箱子，然后说我们不会告诉你这些箱子中哪些实际上有安检设备，这实际上就是随机化。从恐怖分子的角度来看，他们不会知道行李检查会在哪里进行。

[段 21]

That’s worth doing. It isn’t going to perfectly eliminate, well, unfortunately, it isn’t probably going to perfectly eliminate all terrorist attacks, but it does make it harder for the terrorists. So randomization there, whether it’s literally randomizing over who is checked, or whether it’s, as it were, randomizing by concealing where, in fact, you have placed those machines, can be very effective. The hard thing, both in sports and in these military examples, is really mimicking randomization. It’s very hard for us as humans to do it. And there’s a famous story about a military commander, actually an English military commander during an insurgent war in I think it was Malaysia I think it was Malaysia after World War II where again he had to worry about randomizing which convoys to protect and the way in which he figured out that randomizing was the right thing to do, to try and protect these convoys as well as he could with small numbers of troops, and the way in which he randomized was he literally randomized. Every morning he put a bit of paper in his hand and he had somebody, had one of his sergeants pick which hand the bit of paper was there. So we do actually see these random strategies used. The reason we have to literally randomize is because it’s very difficult to do so unless you’re a professional sports player.

[译文 21]

这值得去做。它不会完美地消除——不幸的是，它可能不会完美地消除所有恐怖袭击，但它确实让恐怖分子更难行动。所以在那个情境下，随机化——无论是对检查对象进行真正的随机化，还是通过隐藏机器的实际位置来进行随机化——可能非常有效。困难的地方，无论是体育还是这些军事例子，都是真正模仿随机化。我们作为人类很难做到这一点。有一个关于军事指挥官的著名故事，实际上是一位英国军事指挥官，在二战后的叛乱战争中——我想是在马来西亚——他再次不得不考虑随机化保护哪些车队，以及他如何想出随机化是正确做法，用少量的部队尽可能保护这些车队，而他随机化的方式是他真的做到了随机化。每天早上他在一只手里放一张纸，然后让他的一个中士选择纸在哪只手里。所以我们确实看到这些随机策略被使用。我们必须真正随机化的原因是，除非你是一个职业运动员，否则很难做到这一点。

[段 22]

Okay. But it turns out that mixed strategy equilibria and mixed strategies in general are relevant beyond just these contexts in which you think of people literally randomizing. And I want to look at a different context now. All right, so I want to go back to a game we started a few weeks ago. This isn’t the same game, it’s a sequel, it’s a follow up in our exciting adventure of our dating couple in the classroom. Who were our dating couple? Do we still have them here? That was the guy. Who was, who was the? Yeah, there they are. They’re even sitting close though. classroom. Who were our dating couple? Do we still have them here? That was the guy. Who was the… Yeah, there they are. They’re even sitting closer. What a success here. All right? All right? Can we get the camera on them a second? Stand up a second. Yeah, stand up a second. Thank you. And your name was… Shout out, I’ll just repeat it. David. David, yeah, can you actually look? And look at this. Is this romantic or what? David, and your name is? Nina. Nina and David, okay? And I think we pretty much figured out last time that Nina’s player one and David’s player two. Is that right? Is that right? And as we remember last time, I’ll pick on you in a second.

[译文 22]

好吧。但事实证明，混合策略均衡和一般意义上的混合策略，其相关性超出了你认为是人们真正进行随机化的那些情境。现在我想看一个不同的情境。好吧，我想回到几周前我们开始的一个游戏。这不是同一个游戏，这是一个续集，是关于我们课堂上约会情侣的激动人心冒险的续篇。我们的约会情侣是谁？我们这里还有他们吗？那是那个男的。谁是……是的，他们在那里。他们甚至坐得很近。我们的约会情侣。我们这里还有他们吗？那是那个男的。谁是……是的，他们在那里。他们甚至坐得更近了。这里真是一个成功。好吧？好吧？我们能让他们在镜头前露个面吗？站起来一下。是的，站起来一下。谢谢。你叫什么名字……喊出来，我重复一下。David。大卫，是的，你能不能实际看看？看看这个。这是浪漫还是什么？David，你叫什么名字？Nina。Nina和David，好吗？我想我们上次基本上搞清楚了Nina是一号玩家，David是二号玩家。对吗？是这样吗？正如我们上次记得的，我一会儿会点到你们。

[段 23]

You can sit down a second. So we figured out last time that they were trying to go on a date and they had arranged to go to the movies. they picked out two, in fact three movies, but two that remained viable, and the problem was being typical economics majors who are you both economics majors I didn’t think figure that out they are look at that so being typical economics majors just hopeless at dating they’ve forgotten to tell each other which movie they going to so that you know that work I don know if that worked out well or not but now then life has moved on they going to try it again but this time taking advantage of fall in New England rather than go to a movie they’ve decided on some new activities. So they might either go apple picking, or they might go to the Yale Rep. Alright, whoops, Rep or Rep, let’s see a play. alright and so apple picking is has its advantages in the fall weather its local flavor it’s you know it has certain undertones about the Garden of Eden or something it has I don’t know if we can use the term flavor local otherwise we’re talking about American apples but anyway that’s right and the Yale Rep you know Yale Rep is a good thing to do in New Haven go to a play I think it’s Richard The second is showing now, is that it?

[译文 23]

你可以先坐下。所以我们上次搞清楚了他们试图去约会，他们已经安排好了去看电影。他们选了两部，实际上三部，但两部仍然可行，问题在于他们是典型的经济学专业学生——你们俩都是经济学专业学生对吧——典型的经济学专业学生在约会方面真是无可救药，他们忘记告诉对方要看哪部电影了。所以那件事我也不知道结果好不好。但现在生活继续了，他们要再试一次，这次利用新英格兰的秋天，他们决定进行一些新的活动。所以他们可能去摘苹果，或者去Yale Rep。好的，哎呀，Rep或者Rep，看一出戏剧。好的，摘苹果在秋天的天气里有其优势，当地的flavor——你们知道它有伊甸园或什么的某些意味——它有，我不知道我们能不能用flavor这个词，否则我们在说美国苹果但反正就是这样，Yale Rep你们知道是在New Haven做的一件好事，看一出戏剧。我想现在上演的是《理查德二世》，是这样吗？

[段 24]

Probably not a great date play, but, you know, economists are trying to show they have culture. So, you know, there it goes. And let’s assume the payoffs are like this, as much as they were before, whereby we mean that Nina wants to meet David, but she would, you know, given the choice, she’d rather meet David in the apple fields all right And David who a dark personality likes the sort of darker side of Shakespeare and he also wants to meet Nina but he would rather meet at the Yale Rat, all right? If that’s backwards, I apologize to their preferences, all right? But once again, because they’re still in competent economics pages, they’ve again forgotten to tell each other where they’re going. All right? They’ve forgotten to tell each other where they’re going. So let’s analyze this game again. And we figured out this was a coordination game last time, several weeks ago, and we know in this game, we know what the pure strategy Nash equilibria are. All right? So no prizes to be able to spot them. One of the Nash equilibria in pure strategies, let’s put this in pure strategies. So one of the pure strategy Nash equilibria is for them both to go apple picking and meet up in Bishop’s Orchard or whatever. And another pure strategy equilibrium is for them both to choose the rep.

[译文 24]

可能不是一个很好的约会剧，但你知道，经济学家试图展示他们有文化。所以，就这样吧。让我们假设收益像之前一样，也就是说Nina想要见到David，但给她一个选择的话，她宁愿在苹果园见到David。而David，一个有着阴暗性格的人，喜欢莎士比亚较阴暗的一面，他也想见到Nina，但他宁愿在Yale Rep见面，对吧？如果这反过来了，我向他们的偏好道歉，好吗？但再次，因为他们仍然是能力堪忧的经济学人，他们又忘记告诉对方他们要去哪里了。好吧？他们忘记告诉对方他们要去哪里了。所以让我们再分析这个游戏。我们几周前搞清楚了这是一个协调游戏，我们知道在这个游戏中，我们知道纯策略Nash均衡是什么。好吧？所以能够找出它们是没有奖励的。其中一个纯策略Nash均衡，让我们把它放在纯策略中。其中一个纯策略Nash均衡是他们都去摘苹果，在Bishop’s Orchard或什么地方碰面。另一个纯策略均衡是他们都选择Rep。

[段 25]

All right, we’d figured out that if they were able to communicate, there really a pretty good chance of them managing to coordinate at one of these equilibria but we suspect I think that this is not all that going on here It looks quite likely that come, you know, next Saturday afternoon, when we send these guys out on their dates, they’re going to fail to meet. It’s at least plausible, right? And to test the plausibility of that, let’s ask them, have you managed to meet on a date yet? See, no, no, haven’t managed to meet on a date. See? So I’m proving the point, I’m proving the point that, in fact they haven’t managed to get this coordinated equilibrium yet. All right? All right? So it seems at least plausible that they’re going to fail to coordinate. It seems plausible they’re going to fail to coordinate. And we’d like to sort of capture that idea, and the way we’re going to capture that idea is let’s see if there’s another equilibrium in this game. Well, there certainly isn’t another pure strategy equilibrium in this game, as we know that. So if there’s another equilibrium it better be mixed so let’s try and find a mixed Nash equilibrium in this game and remember this game is called battle of the sexes it’s a famous game this is battle the sexes revisited all right so how are we It’s a famous game.

[译文 25]

好吧，我们已经弄清楚，如果他们能够沟通，他们确实有很大的机会在这些均衡中的一个实现协调，但我们认为这并不是这里发生的全部情况。看起来很有可能，下周六下午，当我们把这些家伙派出去约会时，他们会错过彼此。这至少是合理的，对吧？为了检验这个可能性，让我们问问他们，你们在约会时见过面了吗？看，没有，没有，还没能在约会时见面。看到了吗？所以我在证明这个观点，我在证明他们实际上还没有达到协调均衡。对吧？好的，所以至少看起来他们可能会协调失败。看起来他们可能会协调失败。我们想要捕捉这个想法，而我们要捕捉这个想法的方式是看看这个游戏是否还有另一个均衡。好吧，我们知道，这个游戏肯定没有另一个纯策略均衡了。所以如果有另一个均衡，它应该是混合的，所以让我们试着在这个游戏中找到一个混合纳什均衡，记住这个游戏叫做性别之战，这是一个著名的游戏，这是性别之战回顾，好的，我们怎么做呢？这是一个著名的游戏。

[段 26]

This is Battle of the Sections Revisited. All right. So how are we going to go about finding this mixed Nash equilibrium in the game? We’ll interpret it later, but let’s just work on finding it. So in particular, I’m going to postulate the idea that Nina is going to mix P1 minus P, and David is going to mix Q 1 minus Q. So how do we go about finding David’s equilibrium mix Q 1 minus Q? What’s our trick from last week? Should have got a cold call at this point, but let’s not have to. How am I going to find Q, the equilibrium Q? Somebody? Ah, thank you, thank you. Use Venus’s payoffs. Good, good, good. So to find, it isn’t Venus’ payoffs, it’s Nina’s payoffs, okay? But fair enough. So to find, sorry, to find the Nash equilibrium Q, to find the mix that David using we use Nina payoffs We use Nina payoffs All right let do that All right, so in particular, for Nina, if she goes apple picking, then her payoff is 2 with probability Q if she meets David. and 0 otherwise. And if she goes to the rep, then her payoff is 1 if she meets David. Sorry, I’ve got to be careful. It’s not again. If she goes to the rep, her payoff is 0 if David goes apple picking with probability Q, and her payoff is 1 if she meets David at the rep, which happens with probability 1 minus q.

[译文 26]

这是Sections之战回顾。好吧，那么我们如何在这个游戏中找到这个混合纳什均衡呢？我们稍后会解释，但现在让我们先找出它。特别地，我将假设Nina将以P1减P的方式混合，而David将以Q1减Q的方式混合。那么我们如何找到David的均衡混合Q1减Q呢？我们上周的技巧是什么？这时应该有人被点名回答，但我们先不这样。我要怎么找到Q，均衡Q？有人知道吗？啊，谢谢，谢谢。用Venus的收益。好的，好的，好的。所以要找到，这不是Venus的收益，是Nina的收益，好吧？但也合理。所以要找到，抱歉，要找到纳什均衡Q，要找到David使用的混合，我们使用Nina的收益。我们使用Nina的收益。好的，让我们这样做。好的，特别是对于Nina，如果她去摘苹果，那么如果她遇到David，她的收益是2，概率为Q，否则是0。如果她去rep，那么她的收益是0如果David去摘苹果，概率为Q，她的收益是1如果她在rep遇到David，这发生的概率是1减q。

[段 27]

Is that correct? All right. So this is her payoff from apple picking, and this is her payoff from seeing Richard II. And what do we know if Nina is indeed mixing? What do we know about these two payoffs They must be equal right If Nina is in fact mixing then these two things must be equal and that means what we’re saying is 2q equals 1, 1 minus q, or q equals 2 thirds, I guess it is. No, it’s one-third. Sorry. One-third. One-third. Sorry. Is that right? Here’s one-third. Okay. So our guess is that if there’s a mixed strategy equilibrium, it must be the case that David is assigning a probability a third to going apple-picking, which means he’s assigning probability two-thirds to his more favoured activity, which is going to see Richard II. All right? What about… I’m going to pull these both down. Okay How do we find Nina mix All right so to find to find the Nash equilibrium P to find Nina’s mix what do we do? What’s the trick? Somebody? Use David’s payoffs. Use David’s payoffs. So David’s payoffs if he goes apple picking then he gets a payoff of 1 if he meets Nina there and 0 otherwise and if he goes to the rep he gets a payoff of 0 if Nina’s gone apple picking and he gets a payoff of 2 if he meets Nina at the rep.

[译文 27]

对吗？好的。所以这是她从摘苹果的收益，这是她看Richard II的收益。如果Nina确实在混合，我们知道什么？关于这两个收益我们知道什么？它们必须相等，对吧？如果Nina实际上在混合，那么这两件事必须相等，这意味着我们说的是2q等于1，1减q，或者q等于2/3，我想是。不，是三分之一。抱歉。三分之一。三分之一。抱歉。是对的吗？这是三分之一。好的。所以我们的猜测是，如果有一个混合策略均衡，那么David必须分配三分之一的概率去摘苹果，这意味着他分配了三分之二的概率去他更喜欢的活动，也就是去看Richard II。对吧？…我要把这两个都拉下来。好的，如何找到Nina的混合？好的，为了找到纳什均衡P，为了找到Nina的混合，我们做什么？技巧是什么？有人？用David的收益。用David的收益。所以David的收益，如果他去摘苹果，那么如果他在那里遇到Nina，他得到收益1，否则是0，如果他去rep，如果Nina去摘苹果了，他得到收益0，如果他在rep遇到Nina，他得到收益2。

[段 28]

Alright? And once again, if David is indifferent, it must be these are equal. So if David is in fact mixing between apple picking and going to the rep, it must be that these two are equal. And if we set this out carefully, we’ll get, let’s just see, we’ll get 1p equals 2, 1 minus p equals, If we set this out carefully, we’ll get, let’s just see, we’ll get 1p equals 2, 1 minus p, which is p equals 2 thirds, and 1 minus p equals 1 third. So here we have Nina assigning 2 thirds to going apple picking, which in fact is her more favored thing, and 1 third to going to the rev. All right? Okay, so we just used the same trick as last time. Let’s check that this is, in fact, an equilibrium. All right? So in particular, let’s check that it is, in fact, an equilibrium for Nina to choose 2 thirds, 1 third. All right? Let’s check. So check that P equals two-thirds is in fact the best response for Nina. All right, let’s go back to Nina’s payoffs For Nina if she chose to go apple picking her payoff now is 2 times q but q is equal to a third plus 0 1 minus q And if she chooses to go to the rep, then her payoff is 0 with probability a third, and 1 with probability now 2 thirds.

[译文 28]

对吧？同样，如果David是无所谓的，这两个必须相等。所以如果David实际上在摘苹果和去rep之间混合，这两个必须相等。如果我们仔细列出这个，我们会得到，让我们看看，我们会得到1p等于2，1减p，即p等于2/3，1减p等于1/3。所以这里Nina分配2/3的概率去摘苹果，这实际上是她更喜欢的事，1/3的概率去rev。好的？好的，所以我们用了和上次一样的技巧。让我们检查这确实是一个均衡。好的？特别地，让我们检查Nina选择2/3，1/3确实是一个均衡。好的？让我们检查。检查P等于2/3确实是Nina的最佳反应。好的，让我们回到Nina的收益。对于Nina，如果她选择去摘苹果，她的收益现在是2乘以q，但q等于1/3加上0，1减q，如果她选择去rep，那么她的收益是0，概率为1/3，1，概率为现在2/3。

[段 29]

All I’ve done is I’ve taken the lines I had before and substituted in now what we know must be the correct Q and 1 minus Q. And this gives her a payoff of 2 thirds in either case. If she chooses P, her payoff to P, will be two-thirds of the time she’ll get the payoff from apple picking, which is two-thirds, and one-third of the time she’ll get the payoff from going to the rep, which is two-thirds, for a total of two-thirds. All right? So Nina payoff from either of her pure strategies is two Her payoff from our claimed equilibrium mixed strategy is two so neither of her possible pure strategy deviations were profitable They didn’t lose or anything either, but they weren’t profitable. And by the lesson we started the class with, that means there cannot be any strictly profitable mixed deviation either. So indeed for Nina, P is a best response to Q. And we could do the same for David, but let’s not bother. It’s symmetric. So in this game, we found another equilibrium. The other equilibrium, the new equilibrium is Nina mixed two-thirds, one-third, and David mixed one-third, two-thirds. And we also know the payoff from this equilibrium. The equilibrium for this payoff for both players was two-thirds. There are three equilibria in this game.

[译文 29]

我做的只是把之前的行拿过来，代入现在我们知道必须是正确的Q和1减Q。这给了她在任何情况下2/3的收益。如果她选择P，她对P的收益，在2/3的时间里她会从摘苹果得到收益，是2/3，在1/3的时间里她会从去rep得到收益，是2/3，总共是2/3。好的？所以Nina从她的任何纯策略得到的收益是2。她从我们声称的均衡混合策略得到的收益是2，所以她任何可能的纯策略偏离都不是有利可图的。它们没有损失或什么的，但也不是有利可图的。根据我们上课开始时的教训，这意味着不可能有任何严格有利可图的混合偏离。所以对于Nina来说，P确实是对Q的最佳反应。我们可以对David做同样的事，但让我们不要麻烦。它是对称的。所以在这个游戏中，我们找到了另一个均衡。另一个均衡，新的均衡是Nina混合2/3，1/3，David混合1/3，2/3。我们也知道这个均衡的收益。这个收益对两个玩家来说都是2/3。这个游戏有三个均衡。

[段 30]

They manage to meet at apple picking in which case the payoffs are two and one They manage to meet at the rep In fact that the pure that the second pure strategy equilibrium in which case the payoffs are two and one they manage to meet at the rep that the purest that the second pure strategy equilibrium in which case the payoffs are one and two or they mix both the mix in this way and their payoffs are two-thirds two-thirds why is the payoff so bad in this mixed strategy equilibrium everyone agree this is a pretty lousy payoff like the other equilibrium payoff the worst you got was one, and you’ve sometimes got two. But now you are playing a different equilibrium, and in this different equilibrium, you’re only getting two-thirds. Why are you only getting… What happened? Why has the payoffs got pushed down so far? What’s happening to our poor hapless couple? Or not hapless, I don’t know. What’s happening to our couple? Yeah, here. Ash? Okay. Sometimes they don’t meet. Yeah, they’re failing to meet. Right? The reason… What’s forcing these payoffs down is they’re not meeting very often. How often are they actually meeting? How often are they meeting? Let’s have a look. Let’s go back to the previous board. Here it is. board. Here it is. All right. So they meet when they end up in this box or this box.

[译文 30]

他们在摘苹果时相遇，在这种情况下收益是2和1。他们在rep相遇。事实上第二个纯策略均衡，在这种情况下收益是2和1，他们在rep相遇，那个最纯粹的，第二个纯策略均衡，在这种情况下收益是1和2，或者他们两个都混合以这种方式混合，他们的收益是2/3，2/3。为什么这个混合策略均衡的收益这么差？大家同意这是一个相当糟糕的收益，像其他均衡收益最差的是1，有时得到2。但现在你在玩一个不同的均衡，在这个不同的均衡中，你只得到2/3。为什么你只得到…发生了什么？为什么收益被推得这么低？我们的可怜的不幸情侣发生了什么？或者不是不幸的，我不知道。我们的情侣发生了什么？是的，在这里。Ash？好的。有时他们不能相遇。是的，他们没能相遇。对吧？原因是…是什么导致这些收益下降是他们不经常相遇。他们实际上多久相遇一次？多久相遇一次？让我们看看。让我们回到之前的黑板。这是。黑板。这是。好的。所以当他们最终在这个格子或这个格子里时，他们相遇。

[段 31]

Is that right? All right. So what’s the probability of them ending up in those boxes? Well, ending up in this box is probability two-thirds, one-third. And ending up in this box is probability one-third, two-thirds, is that right? You end up meeting apple picking the two-thirds of the time when Nina goes there times the one-third of the time when David goes there. And you end up meeting at the rep the one-third of the time Nina goes there times the two-thirds of the time that David goes there. Alright? So this is the total probability of meeting. This is the probability of meeting. And it’s equal to I can write Sorry Four Four Is that right Four All right So four of the time they meeting but five of the time and more than half the time more than half the time, they’re screwing up and failing to meet. This is why I call them hopeless dating couple. All right? All right? All right? So this is a very bad equilibrium. But it captures something which is true about the game. What is surely true about this game is that if they just played this game, they wouldn’t meet all the time. In fact, what we’re arguing here is they’d meet less than half of the time. All right, but certainly this idea that we’re given from the pure strategy equilibria that they would magically always manage to meet seems very unlikely, so this does seem to add a little bit of realism to this analysis of the game.

[译文 31]

是这样吗？好吧。那么他们最终出现在这些盒子里的概率是多少？嗯，出现在这个盒子里的概率是三分之二、三分之一。而出现在这个盒子里的概率是三分之一、三分之二，对吧？当Nina去那里时，你们相遇的概率是三分之二，乘以David去那里时的三分之一。当你们在聚会相遇时，Nina去那里的概率是三分之一，乘以David去那里时的三分之二。对吧？所以这就是相遇的总概率。这是相遇的概率。它等于我可以写……抱歉，四……四……是这样吗？好吧，所以四分之一的时间他们相遇，但五分之一的时间……超过一半的时间，超过一半的时间，他们在搞砸并且没能相遇。这就是为什么我称他们为 hopeless 的约会情侣。懂吗？好吧。那么这是一个非常糟糕的均衡。但它捕捉到了这个游戏中的一些真实情况。这个游戏中确实真实的是，如果他们只是玩这个游戏，他们不会每次都相遇。事实上，我们在这里论证的是他们相遇的时间不到一半。好吧，但我们从纯策略均衡中得到的他们会神奇地总是设法相遇的想法似乎非常不可能，所以这确实给这个游戏的分析增加了一点现实主义。

[段 32]

However, it leads to a bit of an interpretation problem. You might ask the question, why on earth are they randomizing in this way? Why are they doing this? It’s bad for everybody. Why are they doing this? All right. And this leads us to think about a second interpretation for what we think mixed strategy equilibria are. Rather than thinking of them literally as randomizing it probably better in this case to think about the following idea We need to think about David mixture as being a statement about what Nina believes David going to do David may not be literally randomizing, but his mixture, Q1 minus Q, we could think of as Nina’s belief about what David’s going to do. and conversely Nina may not literally be randomizing but her p1 minus p we could think of as David’s belief about what Nina’s gonna do and what we’ve done is we’ve found the beliefs we found the beliefs such that these players are exactly indifferent over what they do we found the beliefs for David over what Nina’s going to do such that David doesn’t really quite know what to do. And we found the beliefs that Nina holds about what David’s going to do such that Nina doesn’t quite know what to do. That make sense? All right. So it’s probably better here to think about this not as people literally randomizing, but these mixed strategies being a statement about what people believe in equilibrium All right We come back and look at this game some more later on so our couple, I’m afraid, are not quite out of the woods yet.

[译文 32]

然而，这导致了一个解读问题。你可能会问这个问题，他们到底为什么要以这种方式随机化？为什么他们要这样做？这对每个人都不利。他们为什么要这样做？好吧。这让我们想到对混合策略均衡的第二个解读。与其字面上认为他们在随机化，不如在这种情况下这样想更好：我们需要把David的混合策略看作关于Nina相信David会做什么的陈述。David可能不是真的在随机化，但他的混合策略，Q1减Q，我们可以认为是Nina关于David会做什么的信念。反过来，Nina可能不是真的在随机化，但她的p1减p我们可以认为是David关于Nina会做什么的信念。我们找到了这些信念——我们找到了使得这些玩家对他们的行动完全无差异的信念。我们找到了David关于Nina会做什么的信念，使得David不太确定该怎么做。我们找到了Nina关于David会做什么的信念，使得Nina不太确定该怎么做。明白吗？好吧。所以在这里最好这样想，不是人们真的在随机化，而是这些混合策略是关于人们在均衡中的信念的陈述。我们稍后会回来更多地研究这个游戏，所以我们的情侣，我担心，还没有完全脱离困境。

[段 33]

But I want to spend the rest of the day looking at yet another interpretation of mixed strategy equilibrium. So far we have two. We have people are literally randomizing. We have thinking of these as expressions about what people believe in equilibrium rather than what they’re literally doing. And now I’m going to give you a third interpretation. So for now we can get rid of the Venus and Serena game. so to motivate this third idea I want to think about tax audits alright so none of you here have ever probably ever had to fill out a tax form except for the fact that there seems to be a lot of parents in the room today is it parents weekend? alright so where are the parents from? wave your arms in the air if you’re a parent here ok so at least these guys fact that there seems to be a lot of parents in the room today. Is it parents weekend? Is that what’s going on? All right, so where are the parents from? Wave your arms in the air if you’re a parent here. Yeah, so okay, so at least these guys have probably at some point in their life filled out a tax form. All right? All right, so come tax day, the parents in the room face a choice, and the choice is, are they going to honestly fill out their taxes, or are they going to cheat?

[译文 33]

但我想花今天剩下的时间来讨论混合策略均衡的另一种解释。到目前为止我们有两种。一种是人们真的在随机化。另一种是认为这些是关于人们在均衡中的信念的表述，而不是他们字面上在做什么。现在我要给你们第三种解释。所以现在我们可以摆脱Venus和Serena的游戏了。为了激发这第三个想法，我想谈谈税务审计，好吧。你们这里可能从来没有人填过税表，但似乎今天房间里有不少家长。是家长周末吗？好吧，那这些家长是从哪里来的？如果你是这里的家长，挥挥手。好的，至少这些人可能在他们生命的某个时刻填过税表。懂吗？好吧，到了报税季，在座的家长们面临一个选择，问题是他们是诚实填写税表，还是作弊？

[段 34]

I’m not going to ask them what they did. Well, maybe I will, but for now I won’t ask them what they did. and so they can choose one of two things they can choose to pay their taxes honestly, we’ll call that H or to cheat this is the taxpayer, the parent and at the same time the audit office the auditor has to make a choice and the auditor’s choice is whether to audit you or not. This is not literally true, because literally the auditor can wait until your tax return comes in and then decide whether to audit you. But for now, let’s think of these choices being made simultaneously. We see why that makes it more interesting So let me put down some payoffs here and then I explain them So 2, 0, 4, minus 10, 4, 0, and 0, 4. All right. So how do we interpret this? Let’s look at the auditor’s payoffs first of all. So the auditor is very happy not having to audit your parents and having your parents pay taxes. So we’ll give that a payoff of 4. And it’ll turn out in this game, we’ve decided in the payoffs, that the auditor is equally happy if she actually audits your parents in the year that they cheated. All right? We’ll say that makes the auditor equally happy. All right?

[译文 34]

我不会问他们做了什么。嗯，也许我会，但现在我不会问他们做了什么。所以他们可以选择两件事中的一件。他们可以选择诚实缴税，我们称之为H，或者作弊。这是纳税人，也就是家长，与此同时审计办公室——审计员必须做出选择，审计员的选择是是否审计你。这不是字面上的真实情况，因为字面上审计员可以等到你的税表提交后再决定是否审计你。但现在，让我们认为这些选择是同时做出的。我们看到为什么这让它更有趣。所以让我在这里放一些收益，然后我解释它们。所以是2, 0；4, 减10；4, 0；和0, 4。好的。我们怎么解释这个？让我们先看审计员的收益。所以审计员非常高兴不用审计你们的父母，而且你们的父母缴税了。所以我们给那个收益4。在这个游戏中，我们已经决定收益，审计员如果在她审计你们父母的年份里他们确实作弊了，她同样高兴。好吧？我们说这让审计员同样高兴。

[段 35]

Now the auditor is not so happy if she audits your parents when they’re honest because audits are costly. All right? And the auditor is really unhappy if she fails to audit when the parents cheat it. All right? Let’s look at the, I keep calling them parents, I should stop calling them parents, let’s call them taxpayers. All right So the for the taxpayers what are their payoffs Well we normalize things so that if they honest we give them a payoff of zero All zero That means they correctly fill in their tax form and pay what they supposed to pay But if they conceal some of their income, they pretend to have a third child, then they might be in trouble if they’re audited. If they’re audited, they’re going to have to pay a big fine. Maybe even go to jail, so that’s minus ten. Of course, if they’re not audited, they get to keep a chunk of money, so we’ll call that four. All right, everyone understand the basic idea of this game? All right, in reality, we could add more complications, we could think of different ways to cheat in your taxes, but I don’t want to give tutorials on how to cheat in your taxes here. All right, so it’s not going to take long, staring at this game, to figure out that there are no pure strategy equilibria in this game.

[译文 35]

现在审计员不太高兴如果她审计你们父母而他们是诚实的，因为审计是有成本的。懂吗？而且如果她在父母作弊时没有审计，她真的不高兴。懂吗？让我们看看……我一直在叫他们父母，我应该停止叫他们父母，让我们叫他们纳税人。好吧。那么对于纳税人来说，他们的收益是什么？嗯，我们归一化处理，如果他们是诚实的，我们给他们的收益为零，零。这意味着他们正确填写了税表并支付了他们应该支付的。但如果他们隐瞒了一些收入，假装有了第三个孩子，那么如果被审计他们可能就有麻烦了。如果被审计，他们将不得不支付巨额罚款。甚至可能入狱，所以那是减10。当然，如果他们没被审计，他们就能留下一笔钱，所以我们称之为4。大家理解这个游戏的基本概念吗？好吧，在现实中，我们可以添加更多的复杂性，我们可以考虑不同的方式来逃税，但我不希望在这里提供如何逃税的教程。懂吧。所以盯着这个游戏看不久，就能找出这个游戏中没有纯策略均衡。

[段 36]

Let’s just do that. So from the taxpayer’s point of view, if they’re going to be audited, then they’d rather pay their taxes than not. And if they’re not going to be audited, then according to these payoffs, they’d rather cheat. All right? And from the auditor’s point of view, if they knew everyone was going to pay taxes, then they wouldn bother auditing And if they knew everyone was going to cheat then they of course audit All right So you can quickly see that there no box in which the best responses coincide. There’s no pure strategy of national equilibria. All right? For those people who think this is seeming otherworldly, you will have to pay taxes in a couple of years, and trust me, your parents are paying taxes now. All right? so what we want to do here is we’re going to solve out and find a mixed strategy equilibrium but we’re going to give it a different interpretation to the equilibrium we’ve found so far but the basic initial exercise is what? we’re going to try and find the equilibrium here to find the Nash equilibrium here we know it’s going to be mixed so to find the probability with which taxpayers pay their taxes, and let me already start getting ahead of myself and to say to find the proportion of taxpayers who are going to pay their taxes, what do we do?

[译文 36]

让我们来验证一下。所以从纳税人的角度来看，如果他们要被审计，那么他们宁愿缴税而不是不缴。如果他们不会被审计，那么根据这些收益，他们宁愿作弊。懂吧？从审计员的角度来看，如果他们知道每个人都会缴税，那么他们不会费力去审计。如果他们知道每个人都会作弊，那么他们当然会审计。所以你可以很快看到没有一个格子里最佳反应是一致的。没有纯策略的Nash均衡。懂吧？对于那些觉得这看起来不切实际的人，你几年后就得交税了，相信我，你们的父母现在就在交税。懂吧？所以我们在这里要做的是，我们要解出并找到一个混合策略均衡，但我们给它一个不同于我们迄今为止找到的均衡的解释。但基本的初始练习是什么？我们要去尝试找到这里的均衡，找到这里的Nash均衡，我们知道它将是混合的，所以要找到纳税人缴税的概率，让我已经提前说，找到缴税的纳税人的比例，我们要做什么？

[段 37]

What must be true of that equilibrium proportion Q of taxpayers who pay their taxes? How am I going to find that Q? Shout it out somebody? Yeah, look at the auditor’s payoffs. So from the auditor’s point of view, if the auditor audits, their payoff is 2Q plus 4, 1 minus Q. And if they don’t audit, their payoff is 4Q plus 0, 1 minus Q. All right, everyone see I did this? This is 2Q plus 4, 1 minus Q. This is 4Q plus 0, 1 minus Q. And if indeed the auditor is mixing, then these must be equal. This must be equal. And if they’re equal, let’s just do a little bit of algebra here, and we’ll find that 2q equals 4, 1 minus q. That right So q equals 2 thirds Is that right Four over six yes two Q equals two All right. So our claim is to make the auditor exactly indifferent between whether to audit or not, it must be the case that two-thirds of the parents of the kids in the room are going to be paying their taxes honestly. Which is a third answer, which is kind of worrying. Never mind. All right? Let’s have a look at the taxpayer. To find… Sorry. We found the taxpayer. We found the proportion of taxpayers who are paying their taxes. Now I want to find out the probability of being audited.

[译文 37]

那么这个均衡比例Q，即缴纳税务的纳税人，必须满足什么条件？我该如何找到这个Q？有人知道吗？好的，我们来看审计师的收益。从审计师的角度看，如果审计师进行审计，他们的收益是2Q加上4乘以1减Q。如果他们不审计，收益是4Q加上0乘以1减Q。好的，大家都看到我的计算了吗？这是2Q加4、1减Q。这是4Q加0、1减Q。如果审计师确实在混合策略，那么这两者必须相等。这必须相等。如果它们相等，我们来做一点代数运算，我们会发现2q等于4乘以1减q。对吧，所以q等于三分之二，对吗？四除以六是的，2Q等于2。好的，所以我们的结论是，要使审计师在审计与否之间恰好无差异，教室里孩子的父母中必须有三分之二会诚实地缴税。这是第三个答案，这有点令人担忧。没关系。好的，让我们来看看纳税人。我们找到了缴税的纳税人比例。现在我想找出被审计的概率。

[段 38]

How do I figure out the equilibrium probability of being audited in this model? How do I work out the equilibrium probability of being audited? Shout it out. So the equilibrium probability of being audited, I’m going to use p and 1 minus p, so p is going to be the probability of being audited. How do I find p? Yeah I going to look at the taxpayer payoffs So from the taxpayer point of view if the taxpayer pays their taxes their payoff is just zero and if they cheat their payoff is minus 10p plus 4 1 minus p alright and if indeed the taxpayers are mixing or in other words saying that is not all taxpayers are cheating and not all taxpayers are honestly paying their taxes then these must be equal so if these are equal I’m going to get 4p equals 14 no it doesn’t, I’m going to get 4 equals 14p, let’s try again, 4 equals 14p, that was a bit worrying, 4 equals 14p, which is the same as saying p equals 2 sevenths. Someone can just check my algebra and guess right, hope that’s right. All right so my claim is that the equilibrium here is for two of the taxpayers to pay their taxes and for the auditor to audit two of the time Now we could go back in here and we could check I could do it earlier before, I could plug the P’s and Q’s in here and check that in fact this is an equilibrium, but trust me that I’ve done that.

[译文 38]

我如何计算出这个模型中被审计的均衡概率？我如何计算出被审计的均衡概率？说出来。所以被审计的均衡概率，我用p和1减p来表示，p就是被审计的概率。我如何找到p？是的，我要看纳税人的收益。从纳税人的角度看，如果纳税人缴税，他们的收益就是零，如果他们逃税，收益是负10p加4乘以1减p。好的，如果纳税人确实在混合策略，换句话说，不是所有纳税人都在逃税，也不是所有纳税人都在诚实缴税，那么这两个必须相等。如果这两个相等，我会得到4p等于14，不对，不是这样的，我会得到4等于14p，让我们再试一次，4等于14p，这有点令人担忧，4等于14p，这相当于说p等于七分之二。有人可以检查一下我的代数运算，应该没错。好的，所以我的结论是，这里的均衡是三分之二的纳税人缴税，审计师有七分之二的时间进行审计。现在我们可以回到这里检查，我之前可以做到的，我可以把P和Q代进去检查这确实是一个均衡，但请相信我已经做过了。

[段 39]

Trust me that it’s okay. So here we have an equilibrium, let’s just write down what it is. From the auditor’s point of view, it is that they audit two-sevenths of the time, or two-sevenths of the population. And from the taxpayer’s point of view, it’s that they pay their taxes honestly two-thirds of the time and not otherwise. All right? Now, without focusing too much on these exact numbers for a second, I want So I’m going to focus for a minute on how do we interpret this mixed strategy equilibrium. Right? So from the point of view of the auditor, we’re really back where we were before with the base stealer or the person who’s searching baggage on the air, at the airport. We can think of the auditor literally as randomizing. Right? In fact, there’s some truth to that, it actually is the case, by law, that the We can think of the auditor literally as randomizing. In fact, there’s some truth to that. It actually is the case, by law, that the auditors literally have to randomize. So this 5 sevenths, this has the same interpretations we had before. This is really a randomization.

[译文 39]

请相信这是对的。所以这里我们有一个均衡，让我们把它写下来。从审计师的角度看，是他们有七分之二的时间进行审计，或者说七分二的总体。从纳税人的角度看，是他们有三分之二的时间诚实缴税，否则就不缴。好的？现在，不要太关注这些确切的数字，我想花一点时间谈谈我们如何解释这个混合策略均衡。从审计师的角度看，我们真的回到了之前讨论基础窃贼或在机场搜查行李的人的情况。我们可以把审计师真正看作是随机行动。实际上，这有一定道理，事实上，按照法律，审计师确实必须随机行动。所以这个七分之五，和我们之前有相同的解释。这真的是一种随机化。

[段 40]

But this 2 thirds, 1 third, has a different interpretation and a potentially exciting interpretation. it isn’t that we think that your parents get to tax day work out what their taxes would be and then toss a coin right, well they may be doing that I’m looking at the parents here, I don’t think that’s what they’re doing right, the interpretation here is that the parents some parents are paying their taxes and some parents aren’t paying their taxes, there’s a lot of parents out there, a lot of potential taxpayers and in the population in equilibrium, if these numbers were true two thirds of parents would be paying their taxes and one-third would be cheating. So this is a, a randomization by a player and this is a mixture in the population. Right? The new interpretation here is we could think of the mixed strategy not as players randomizing but as a mix in a large population in which some people are doing one thing and the other group are doing the other. It’s a proportion of people paying taxes. So I don’t know if this two-thirds, one-third is an accurate number for the US. It’s probably not very far off, actually. For Italy, I’m ashamed to say, that number of people who pay taxes is more like 40%, maybe even lower now. And there are countries, I think, where it gets as high as 90%.

[译文 40]

但是这个三分之二和三分之一有不同的解释，而且可能是一个令人兴奋的解释。这并不是说我们认为你们的父母在纳税日算出他们的税后然后抛硬币，对吧，他们可能确实在那样做，我看着这里的父母们，我认为那不是他们正在做的，这里的解释是，有些父母在缴税，有些父母不缴税，有很多父母在外面，有很多潜在的纳税人，在均衡状态的总体中，如果这些数字是真实的，三分之二的父母会缴税，三分之一会逃税。所以这是一个，球员的随机化，这是总体中的一个混合。对吧？这里的新解释是，我们可以把混合策略不是看作球员在随机行动，而是看作一个大群体中的混合，其中一些人在做一件事，另一组人在做另一件事。这是缴税人群的比例。我不知道这个三分之二和三分之一对美国来说是否是准确的数字。实际上它可能不会太离谱。对于意大利，我很惭愧地说，缴税人数的比例更像是40%，现在可能甚至更低。有些国家，我认为，这个比例高达90%。

[段 41]

I think the U.S. rate when they end up auditing is a little higher than this, but not much. All right? Okay. So again, we’re going to think of this not as randomization, but as a prediction of the proportion of American taxpayers who are going to pay their taxes. All right? now I want to use this example in the time we have left to actually think about a policy experiment let’s put this up somewhere we can see it and let’s think about a new tax policy so suppose that congress gets fed up with all these newspaper reports about how two of Americans don pay their taxes or whatever whatever the true proportion is I think it actually a little higher than that but never mind They get fed up with all these reports and they say this isn’t fair we should make people pay their taxes so we’re going to change the law and instead of paying, instead of being in jail for, instead of being in jail for ten years, or the equivalent of a fine of minus 10 if you’re caught cheating, we’re going to raise the fine, or the time in jail, so that it’s now minus 20. So the policy experiment is, let’s raise the fine, the fine for cheating, to minus 20. And the aim of this policy is to try to deter cheating, right?

[译文 41]

我认为美国在被审计时的比例比这个略高，但不会高太多。好的？所以我们再次要把这不是看作随机化，而是作为对美国纳税人缴税比例的预测。好的？现在我想用剩下的时间来实际思考一个政策实验，让我们把它放在某个我们能看到的地方，让我们思考一项新的税收政策，所以假设国会对这些报纸报道感到厌倦，说有多少美国人没有缴税或者不管真实比例是多少，我认为实际上比那略高，但没关系。他们对这些报道感到厌倦，他们说这不公平，我们应该让人们缴税，所以我们将改变法律，不再是，假设你逃税被抓，要坐十年牢，或者相当于负10的罚款，我们要提高罚款，或者坐牢时间，所以现在是负20。所以政策实验是，让我们提高罚款，逃税的罚款，提高到负20。这个政策的目的是试图阻止逃税，对吧？

[段 42]

Seems a plausible thing for government to want to do. All right, let’s redraw the matrix. So here’s the game. 2, 0, 4, minus 20, 4 0 0 4 audit not audit And pay honestly or cheat Alright. So here’s our new payoffs. And let’s ask the question, with this new fine in place, that we’ve raised the fine to being caught not paying your taxes, in the long run, once things have worked their way back into equilibrium again, after a few years, do we expect American tax paying compliance to go up or to go down or what do we expect? What do we think is going to happen? So who thinks it’s going to go up? Who thinks it’s going to go down? Who thinks it’s going to stay the same? Who’s abstaining here? I know the parents are abstaining not really meant to abstain, you have to vote here. Well how are we going to figure this out? How are we going to figure out what’s going to happen to compliance? What happens to tax compliance? Tax compliance, that was our P. No, it wasn’t, sorry, it was our Q. Well, the only way we’re going to figure this out is to work it out. So let’s work out the new Q in equilibrium. All right, let’s do this. So to find out the new Q in equilibrium, once again, we’re going to have to look at the auditor’s payoffs. and the auditor’s payoffs, if they audit, they’re going to get 2Q plus 4, 1 minus Q and if they don’t audit, they’re going to get 4Q plus 0, 1 minus Q.

[译文 42]

这似乎是政府想要做的合理的事情。好的，让我们重画收益矩阵。所以这是游戏。2，0，4，负20，4，0，0，4，审计或不审计，诚实缴税或逃税。好的。这是我们新的收益。让我们问一下，有了新的罚款到位，我们把罚款提高到被抓到不缴税，在长期，一旦事情重新回到均衡，在几年之后，我们预期美国纳税遵从度会上升还是下降，或者我们预期什么？我们认为会发生什么？谁认为会上升？谁认为会下降？谁认为会保持不变？谁弃权了？我知道父母们弃权了，但他们真的不应该弃权，你们必须在这里投票。我们要怎么弄清楚？我们怎么弄清楚遵从度会发生什么变化？税收遵从度发生了什么？那就是我们的P。不，抱歉，那是我们的Q。好的，我们唯一能弄清楚的方法就是算出来。让我们算出新的Q在均衡中。好的，让我们来做这个。要找出新的均衡Q，我们再次必须看审计师的收益。审计师的收益，如果他们审计，他们会得到2Q加4乘以1减Q，如果不审计，他们会得到4Q加0乘以1减Q。

[段 43]

4Q plus 0, 1 minus Q. 2Q plus 4, 1 minus Q. And if the auditor is indifferent, if they’re mixing, it must still be the case that these are equal. And I want to ask you a question. Where have you seen that equation before? Yeah, it’s still there, right? I didn’t delete it. It’s the same equation that sits up there. Is that right? From the auditor’s point of view, given the payoffs to the auditors, nothing has changed So the tax compliance rate that makes the auditor exactly indifferent between auditing your parents and not auditing your parents is still exactly the same as it was before at two In equilibrium, tax compliance hasn’t changed at all. I’m going to say it again, right? The policy was we’re going to double the fines for being caught cheating, And in equilibrium, it made absolutely no difference whatsoever to the equilibrium tax compliance rate. Now why did it make no difference? Why did it make no difference? Well, let’s have a techie answer and then a more intuitive answer. The techie answer is this. What determines the equilibrium tax compliance rate, What determines the equilibrium mix for the column player is what? Is the rows payoffs. Right? What determines the equilibrium mix for the column player are the rows payoffs, row players payoffs. We didn change the row players payoffs so we not going to change the equilibrium mix for the column player All right say again We changed one of the payoffs for the column player but the column players’ equilibrium mix depends on the row players’ payoffs, and we haven’t changed the row players’ payoffs, so we won’t change the equilibrium compliance rate, the equilibrium mixed by the column player.

[译文 43]

4Q加0，1减Q。2Q加4，1减Q。如果审计员是无差异的，如果他们在混合，那么这些仍然必须相等。我想问你们一个问题。你们在哪里见过这个等式？是的，它还在那儿，对吧？我没有删除它。它和上面那个等式是一样的。从审计员的角度来看，考虑到给审计员的收益，没有任何改变。所以使审计员在审计你的父母和不审计你的父母之间完全无差异的税率，在之前仍然是2，与之前完全一样。在均衡中，税务合规完全没有改变。我要再说一遍，好吗？政策是，我们要将被抓到作弊的罚款翻倍，而在均衡中，它对均衡税率完全没有影响。现在为什么没有影响？为什么没有影响？好吧，让我们先给出一个技术性的答案，然后再给出一个更直观的答案。技术性的答案是这样的。决定均衡税率的是什么？决定列玩家均衡混合的是什么？是行玩家的收益。对吧？决定列玩家均衡混合的是行玩家的收益，行玩家的收益。我们没有改变行玩家的收益，所以我们不会改变列玩家的均衡混合。再说一遍，我们改变了列玩家的一个收益，但列玩家的均衡混合取决于行玩家的收益，而我们没有改变行玩家的收益，所以我们不会改变均衡合规率，列玩家的均衡混合。

[段 44]

What will have changed here? What will have changed in the new equilibrium? So we’ve pretty much established that people are cheating as much as they were before in equilibrium. Raoul, can I get Henry here? Wait for the mic. Say again? The probability of audit will have changed. The probability of audit will have changed. What’s going to change is not the Q but the P. The probability with which you’re audited is going to change in this model. All right, let’s just check it. To find the new p, I need to look at the taxpayers’ payoffs, and the taxpayers’ payoffs are now zero if they, sorry, if they pay their taxes honestly, then they get zero. And if they cheat, they get minus 20 with probability p and 4 with probability 1 minus p If they mixing if some of them are paying and some of them are not, this must be the same. And being more careful than I was last time, I hope this gives me 24p is equal to 4 or p equals a sick. alright so the audit rate has gone down from 2 sevenths to 1 sixth I’m guessing that probably wasn’t the goal of the policy although it isn’t necessarily a bad thing right, there is some benefit for society here because audits are costly both to do for the auditor and they’re unpleasant to be audited so the fact we’ve managed to lower the audit rate from 2 sevenths to a sixth is a good thing all right?

[译文 44]

在这里会发生什么变化？在新均衡中会发生什么变化？我们已经基本确定，人们在均衡中的作弊程度和以前一样。劳尔，我能请亨利过来吗？等一下麦克风。再说一遍？审计概率会改变。审计概率会改变。会发生变化的不是Q而是P。你被审计的概率在这个模型中会发生变化。好的，让我们检查一下。为了找到新的p，我需要看纳税人的收益，纳税人的收益现在是零，如果他们诚实纳税，那么收益是零。如果他们作弊，被抓到概率是p，没被抓到概率是1减p，收益分别是负20和正4。如果他们在混合，如果他们中的一些人在纳税而一些人在不纳税，这两个收益必须相等。比上一次更仔细地计算，希望这给我得出24p等于4，或者p等于1/6。好吧，审计率从2/7下降到1/6，我猜这不是政策的初衷，尽管这未必是坏事，对吧，这里有一些对社会的好处，因为审计成本很高，对审计员来说做审计成本很高，而且被审计也令人不愉快，所以事实上我们成功地把审计率从2/7降低到1/6是一件好事，对吧？

[段 45]

But we didn’t manage to raise the compliance rate, all right? So I don’t want to take this model too literally, because it’s just a toy model, but nevertheless, let’s try and draw out some lessons from this model, all right? So here what we did was we changed the payoff to cheating. We made it But nevertheless, let’s try and draw out some lessons from this model. So here what we did was we changed the payoff to cheating. We made it worse. But a different kind of change is we could have changed… Sorry, we changed the payoff negatively to being caught cheating. But a different change we could have done is we could have left the minus 10 in place and we could have raised the payoff to cheating and not getting caught. We could have left this 10 in place and changed this 4, let’s say, to a 6 or an 8. We could increase the benefits to cheating if you’re not caught. What would that have done in equilibrium? What would that have done in equilibrium? So I claim, once again, that would have done nothing in equilibrium to the probability of people paying their taxes, but that would have done what to the audit rate? The audit rate would have gone up. the equilibrium audit rate would have gone up. Alright, let’s tell that story a second.

[译文 45]

但我们没有成功提高合规率，对吧？所以我不想把这个模型看得太字面化，因为它只是一个玩具模型，但尽管如此，让我们试着从这个模型中得出一些教训，对吧？所以这里我们做的是我们改变了作弊的收益。我们让它变差了。但另一种改变是，我们可以保留这个负10，我们可以提高作弊不被抓到的收益。我们可以保留这个10，改变这个4，比如改成6或8。我们可以增加作弊不被抓到时的收益。这在均衡中会怎样？这在均衡中会怎样？我声称，再一次，这在均衡中不会对人们纳税的概率产生任何影响，但这会对审计率产生什么影响？审计率会上升，均衡审计率会上升。好的，让我们再讲一遍这个故事。

[段 46]

Right, so rich people, people who are well paid, have a little bit more to gain from cheating on their taxes if they not caught There more money at stake Alright so my colleagues who are finance professors in the business school have more money on their tax returns than I do so in principle, they gain more if they cheat. Alright, does that mean that they cheat more than me in equilibrium? No, it doesn’t mean that they cheat more than me in equilibrium. What does it mean? It means they get audited more often. alright in equilibrium richer people aren’t necessarily going to cheat more but they are going to get audited more and that’s true right the federal audit rates are designed so they audit the rich more than they audit the poor again it’s not because they think the rich are inherently less honest or the poor are inherently more honest or anything like that it’s simply that the gains to cheating and not getting caught are bigger if you’re rich so you need to audit more to push back into equilibrium alright now suppose we did in fact want to use the policy of raising fines to push up the compliance rate to push down cheating how would we change the law suppose we want to raise the fines for cheating we don like people cheating so we raise the fines but we worried about this result We worried about this result that didn push up compliance rates How can we change the law or change the incentives in the game so that it actually would change compliance rates What could we do?

[译文 46]

好的，所以有钱的人，收入高的人，如果不被抓到的话，在逃税方面有更多的收益。有更多的钱处于危险之中。好的，所以我在商学院教金融的同事们在我的纳税申报表上比我更多的钱，所以原则上，如果他们作弊，他们获得的收益更多。好的，这是否意味着他们在均衡中比我作弊更多？不，这并不意味着他们在均衡中比我作弊更多。这是什么意思？这意味着他们被审计的频率更高。好的，在均衡中有钱的人不一定会作弊更多，但他们会被审计更多，这是真的，对吧，联邦审计率的设计是让它们审计有钱人比审计穷人更多。同样，这不是因为他们认为有钱人天生不诚实或者穷人在本质上更诚实或者任何类似的事情，这仅仅是因为如果你有钱的话，不被抓到的作弊收益更大，所以你需要更多审计来把他们推回均衡。好的，现在假设我们实际上想用提高罚款的政策来提高合规率、降低作弊，我们如何改变法律？假设我们想提高作弊的罚款，我们不喜欢人们作弊，所以我们提高了罚款，但我们担心这个结果，我们担心这个没有提高合规率的结果。我们如何改变法律或改变游戏中的激励，以便它实际上会改变合规率？我们可以做什么？

[段 47]

Yeah. If we change the payoffs of auditing to four, from two to four. Good, good, good. If we want to change the compliance rates, we should change the payoff to the auditor. The problem with the way the auditor is paid here is that the auditor is paid more if they manage to catch people. But audits are costly. And the problem with that is when you raise the fine on the other side, all that happens is the auditors audit less often in equilibrium. So if you want to get a higher compliance rate, one thing you could do is change the payoff to the auditor to make auditing less costly for them or making catching people nicer for them, give them a reward. Or you could simply take it out of Game 3 altogether. You could enforce, you could have a congressional law that sets the audit rates outside of equilibrium. And that been much discussed in Congress over the last five years simply setting audit rates as it were exogenously by Congress Why might that not be a great idea leaving aside economic theory leaving aside game theory Why might it not be a great idea to have Congress set the audit rates rather than some office? Yeah. Most members in Congress have a lot of money, so they’re going to lower the audit rates. Right, so the lady in the front is saying that a lot of congressmen are rather rich, so maybe they have particular incentives here.

[译文 47]

是的。如果我们把审计的收益从2改成4。好的，好的，好的。如果我们要改变合规率，我们应该改变审计员的收益。这里审计员报酬方式的问题是，审计员在抓到人时获得的报酬更多。但审计成本高昂。这样做的问题是，当你在另一边提高罚款时，所有发生的事情就是审计员在均衡中审计的频率降低了。所以如果你想要获得更高的合规率，一件你可以做的事情是改变审计员的收益，使审计对他们来说成本更低，或者使抓到人给他们带来更好的感觉，给他们奖励。或者你完全可以把它从博弈3中完全移除。你可以执行，你可以有一个国会法律，在均衡之外设定审计率。这在过去五年里在国会得到了广泛讨论，简单地由国会外生地设定审计率。为什么这可能不是一个好主意，撇开经济学理论不谈，撇开博弈论不谈。为什么由国会设定审计率而不是某个办公室可能不是一个好主意？是的。大多数国会议员有很多钱，所以他们会降低审计率。好的，所以前面那位女士说的是很多国会议员相当富有，所以他们可能在这里有特殊的激励。

[段 48]

I don’t want to take a particular political stance here, but it could be whatever side of the political spectrum you guys sit on, it could be that you might not trust Congress to get this right. You might think there are going to be political considerations going on in Congress other than just having an efficient tax system. So what do I want to draw out as lessons here? The big lessons from this class are there are three different ways to think about randomization and equilibrium, or out of equilibrium. One is it’s genuinely randomization, another is it could be something about people’s beliefs, and a third way, and a very important way, is it could be telling us something about the proportion of people who are doing something in society. In this case, the proportion of people who are paying tax. All right? And a second important who are doing something in society. In this case, the proportion of people who are paying tax. And a second important lesson I want to draw out here, beyond just finding equilibria, two other things we drew out today. One lesson was when you’re checking equilibria, checking mixed strategy equilibria, you only have to check for pure strategy deviations. be careful you have to check for all possible pure strategy deviations not just the pure strategies that were involved in the mix if the guy has seven strategies and is only mixing on two you have to remember to check the other five and the third lesson I wanted to route today is because of the way equilibria work with mixed strategy equilibria work if I change the column players payoffs it changes the row players equilibrium mix and if I change the row players payoffs, it changes the column players equilibrium mix.

[译文 48]

我不想在这里采取某种特定的政治立场，但无论你们属于政治光谱的哪一方，你们可能不信任国会能把这事做好。你们可能认为国会里会有除了高效税收体系之外的政治考量。那么我在这里想提炼出什么教训呢？这门课的重要教训是，关于随机化和均衡，或者非均衡，有三种不同的思考方式。一种是真正的随机化，另一种可能与人们的信念有关，第三种方式，一种非常重要的方式，是它可能告诉我们社会中做某件事的人的比例。在这个问题中，是纳税人的比例。明白吗？我想在这里提炼的第二个重要教训是，除了找到均衡之外，我们今天还提炼出另外两点。一个教训是，当你在检验均衡、检验混合策略均衡时，你只需要检验纯策略的偏离。要小心，你必须检验所有可能的纯策略偏离，而不仅仅是混合中所涉及的那些纯策略——如果这个人有七种策略但只混合两种，你得记得检验另外五种。第三个我今天想提炼的教训是，由于混合策略均衡的运作方式，如果我改变列玩家的收益，它会改变行玩家的均衡混合，如果我改变行玩家的收益，它会改变列玩家的均衡混合。

[段 49]

Next time we’re going to pick up this idea that mixed strategies can be about proportions of people playing things and take it to a totally different setting, namely evolution. So on Wednesday we’ll start talking about evolution.

[译文 49]

下次我们将延续这个观点——混合策略可以是关于玩某事的人的比例——并把它带到一个完全不同的情境，即进化。所以周三我们将开始讨论进化。