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你永远不可能解决的问题

A Problem You'll Never Solve

欢迎来到维瑟科普
Vsauce!
我是凯文 我想问大家一个非常简单的问题
Kevin here, with a really simple question.
你是想要一个装一千颗糖的盒子
Do you want this box of 1000 candies
还是一个可能装一百万颗糖但也可能什么都没有的神秘盒子
and a mystery box which contains either nothing or a million candies or
或者说你就是想要这个神秘盒子
do you want just the mystery box?
很明显 你会选择同时要两个盒子
Obviously you’ll take both boxes
因为这样一来 你总是能拿到神秘盒子
because you’re getting the mystery box either way.
还能拿到一些确定到手的糖果 对吧
Might as well grab some guaranteed candy too,right?

Right.

Wrong.
也许
Maybe.
老实说 我也不知道
Honestly,I don’t know.
问题在于
The thing is…
当考虑同时拿两个盒子还是只要神秘盒子的时候
When it comes taking both boxes or just the mystery box,
几乎所有观看这个视频的人
almost everyone watching this
都十分确定他们知道正确答案
video will be absolutely sure that they knowthe right answer.
这几乎不成问题 更谈不上让你犯难
This is barely a problem, let alone one you’llnever solve.
不过 有意思的是
But here’s what’s interesting…
可以肯定 你们中的一半人明显是会同时拿两个盒子
Half of you will be certain that the obvious answer is to take both boxes,
另一半人 同样坚定地相信
and the other half of you will be just as sure
最明显的答案是只拿神秘盒子
the obvious answer is to take only the mystery box.
怎么会这样呢
How is that possible?
啊 为什么我的桌子上忽然有了个神秘的精灵
And why is there suddenly a Grandayy genieon my table?
我们一起来仔细剖析这个问题
Let’s dissect this problem.
盒子A有明确的价值
Box A has clear value.
至少表面看来是明确的
It’s literally clear —
你可以看到里面装着一千颗糖
you can see that the contents are 1000 candies.
关键问题在神奇盒子B
The issue is Mystery Box B.
盒子B的内容是由
The contents of Box B are determined in advance
全知全能的精灵所提前决定的
by our omniscient, all-knowing Grandayy genie,
精灵会用他几近完美的准确率来预测你的选择
who predicts what you’ll choose with near-perfectaccuracy.
如果他预测你会同时选择两个盒子
If he predicts you’ll choose both boxes,
那么他不会往盒子B里放任何东西
he’s put nothing in Box B.
如果他预测你只会选择神秘盒子B
If he predicts that you’ll only choose mystery Box B,
他会放一百万颗糖在里面
he’s placed… a million candies.
你无法看到盒子B的内容
You can’t see inside Box B,
也不能去摸
you can’t touch it, and you
你也不知道
don’t know what the genie
在你选择之前精灵预测了什么
has predicted before you actually choose.
那么问题来了
Here’s a question.
是谁想到这个问题的呢
Who even came up with this?
是我编造出来的吗
Did I just make this whole thing up?
当然不是
No.
理论物理学家威廉·纽科姆在1960年提出了这样一个问题
Theoretical physicist William Newcomb devisedthis problem in 1960.
十年后
And a decade later,
哲学家罗伯特·诺齐克将这个哲学上的问题具体化
philosopher Robert Nozick detailed the deep philosophical fracture that
让两个同样明显的选择都对又都错
makes the two equally-obvious choices bothright and both wrong.
这就是一种矛盾
It’s a contradiction.
是一种悖论了
It’s an antinomic paradox.
那我来说明一下原因
Here’s why.
如果你决定同时拿两个盒子
If you decide to take both boxes,
那么精灵很可能会在预测后
the genie will likely have predicted that and put nothing
让盒子B空空如也
in mystery Box B —
或许因为精灵并不喜欢贪婪的人
maybe genies don’t like greedy players or something.
所以如果你同时选择盒子A和盒子B
So if you choose both Box A and
你就结束了
box B, you’ll wind up
只获得你的一小把糖
with only a few handfuls of candy.
假如你只选择神秘盒子B
If you decide to take only mystery Box B,
精灵几乎确定也能预测到
the genie will almost certainly have predicted that,
并且把一百万颗糖放在里面
too, and put a million candies inside… maybe
作为你勇敢选择的奖励
as a reward for your courageous choice.
无论怎样 对你而言
Either way, it’s now obviously better
选择B盒子明显更好
for you to take mystery Box B because a million
因为一百万颗是一个比一千好太多的奖励
is a much better prize than 1000.
这是看待这个这个问题的一种方式
That’s one way to look at this problem,
《卫报》2016年的民意调查显示
and in a 2016 poll from The Guardian, 53.5 %
在超过三万受访对象中 有53.5%的人选择
of over 30,000 survey respondents chose to take
只拿神秘盒子B
only mystery Box B. Here’s what the
剩下的46.5%的人认为
other 46.5% thought:
精灵要么已经在神秘盒子里放入了一百万颗糖
The genie has already either put a million candies
要么就什么都没放
in the mystery box… or not.
他可以在一天前 一星期前或者一个月前
He could’ve setup the boxes a day,
设置这些盒子
a week, a month ago!
但是糖果不会
The candy isn’t going to suddenly
因为你的选择突然出现或消失
appear or disappear based on your decision.
如果他已经在盒子B里放了糖果 而你又同时拿了两个盒子
If he’s filled Box B with candy and you take both boxes,
你将会得到一百万颗糖
you’ll get a million plus
再加上一千颗 多于来自盒子A里的糖
1000 more from Box A,
你可以立马吃了来庆祝自己非一般的聪明才智
which you can eat right away to celebrate your amazingly clever rationale.
但是如果他没有填满盒子B……真的没放东西
If he didn’t fill Box B… he just didn’t.
那么你拿走两个盒子 赢得了小奖励
You take both boxes and win your small
你也就不会空手而归
prize and this way you don’t walk away empty-handed.
你总没有损失
You can’t really lose.
最坏的情况
Worst case scenario,
不过是神秘盒子B是空的 你仍能获得1000颗糖果
the mystery box is empty but you still get 1000 pieces of candy which
聊胜于无
is 1000 more than zero.
所以你到底是应该拿两个盒子还是只拿盒子B呢
So should you take both boxes or just BoxB?
这儿该怎样选呢
What is actually going on here?
到底为什么纽科姆的悖论困惑了人们几十年呢
Why exactly has Newcomb’s Paradox confoundedminds for decades?
因为这个悖论就是个陷进 让两种同样有依据的想法互相争辩
Because it’s pitting two equally valid methodsof reasoning against each other: Expected
这就是预期效用和战略优势
Utility and Strategic Dominance.
让我们通过一点儿数学来重述这两个选择
Let’s recap the two options with a little math…
所以我们现在认真点
so we can get serious.
你可能不喜欢吃甜食 那我们把奖励改成钱
You may not have a sweet tooth, so let’sswitch prizes from candy to money: Box A now
现在盒子A有一千美元
contains $ 1,000,
盒子B要么有一百万美元 要么一分钱也没有
and Box B either has $ 1 million dollars or no dollars. First,
首先 我们来看看简单的支付矩阵带来的可能的结果
we can see our possible outcomes with a simple payoff matrix. Basically,
基本上就是写出四种情况
we’ll just write out the fourscenarios.
下去待会吧 神秘精灵
Excuse me, Grandayy.
如果精灵预测你会选择盒子B 而你恰好也这么做了
If the genie predicts you’ll take Box B and you choose Box B,
你就会获得一百万美元
you’ll get $ 1,000,000.
如果他预测你选择盒子B 而你却选择了两个盒子
If he predicts you’ll take Box B but you choose both boxes,
你会赢得一百万一千美元
you’ll win $ 1,001,000
即盒子B里的一百万加上盒子A里的一千美元
— the million in Box B and the $ 1,000 in Box A.
假设他预测你是个贪婪的人
If he predicts you’re greedy and
会拿走两个盒子 但你却只选择了盒子B
will take both boxes but you choose just Box B, then
那么你就什么都得不到
you get zero dollars.
如果精灵预测你会选择两个盒子
And if the Genie’s prediction is both boxes,
而你也的确选择了两个盒子
and you choose both boxes, your prize is just
那么你将获得盒子A里的一千美元
the $1,000 from box A.
换句话说
To put it another way,
这些是他正确预测的结果
these are the outcomes when his prediction is right and these are
而这里是他错误预测的结果
the outcomes when his prediction is wrong.
好了
Okay.
我们画出了可能的结果 现在要怎么做呢
We mapped out the potential outcomes, nowwhat?
我们要怎么分辨哪种选择是对的呢
How do we figure out which choice is right? Well,
嗯 我们可以计算这个选择对你的实际价值
we can actually calculate how valuable a choice is to you —
这就是预期效用
that’s Expected Utility.
这就像是在数学里面做一个决定
It’s like the math of making a decision.
你可以简单地将其中一个选择的结果
You simply take the result of a choice and multiply it
乘以这个结果的可能性
by the probability of the outcome.
你将会得到一个数值 帮助你做出你的决定
That’ll give you a numerical value to helpinform your decision. So,
让我们假设
let’s say the genie
精灵预测正确的机会是90%
has a 90 % chance of predicting right.
我们所计算出选择两个盒子的预期效用就像这样
We’d calculate the expected utility of choosingboth boxes like this:
90%的正确率意味着
A 90 % chance he’s right means there
10%的错误率
’ s a 10 % chance that he’s wrong.
所以 如果我们选择两个盒子
So if we choose both boxes,
就有10%的机会赢得两个装有钱的盒子
there’s a 10 % chance we win two money-filled boxes and a
和90%的机会只能获得一千美元
90% chance that we’re left with just the$1,000.
我们将10%的错误率
We multiply the.1 probability that he’s wrong
乘以从两个盒子中得到的一百万零一千美元
by the payoff of $ 1,001,000 from both
接着加上
boxes and add that to the 90 % chance he’s right,
90%的意味着盒子B中空空如也的正确率
which means Box B would be empty — so
与盒子A中的一千美元的乘积
that’s.9 multiplied by just the $1,000Box A payoff.
就得到了十万零一千美元的结果
This equals $101,000.
假设精灵十次中有九次都可以选择正确
If we assume that the genie is right 9 times out of 10,
而我们每次都选择两个盒子
each time we chose both boxes,
理论上我们可以拥有十万一千美元
we’d theoretically gain $101,000.
现在让我们看一下
Now let’s find the Expected Utility
只选择盒子B的预期效用 来比较这两种选择的价值
of choosing only Box B so that we can compare the two
然后选择更好的那一种
values and determine the best choice.
我们有一百万美元
We get a million dollars
假设我们在精灵预测正确的情况下选择盒子B
if we choose Box B when the genie predicts our choice correctly.
我们仍然假设这个正确率为90%
If we stick with his 90 % accuracy rate,
我们用90%乘以一百万
we multiply.9 by the $ 1,000,000 payoff and then
然后加上10%的错误率与盒子里一分也没有的几率的乘积
add.1 times the $ 0 from the empty box when he’s wrong
由此理论上我们每一次能得到九十万美元
for a theoretical gain of $ 900,000 per game.
通过预期效用的框架推论
By using Expected Utility as a reasoning framework,
最好的选择是只拿盒子B
the best choice is to take only mystery Box B,
因为这种选择平均九万美元的回报
because an average payoff
很明显比十万零一千要高
of $ 900,000 is clearly better than $ 101,000.
很显然
Obviously!
这是解决这个问题的正确方式
That’s the right way to solve this problem.
到此为止是这样没错
Until it isn’t.
这种想法突然闯进脑中叫嚣着
The Dominance Principle waltzes in and shouts,
“在哪种场景下我可以获利最大呢”
“ In which scenario can I win the most? ” Because,
因为你看 精灵把钱
look, the genie has put the money
放在或者不放在神秘盒子里
in the mystery box or he hasn’t, your choice
你的选择都取决于盒子里的东西
comes down to taking whatever is in that box,
或者是这个盒子加盒子A的东西
or taking whatever is in that box plus Box A.
这个神秘盒子都有未知数n的价值
The mystery box has a value of n,
而精灵已经提前决定好了它的价值
and the genie has determined that value in advance.
n要么是零 要么是一百万美元
n is either $ 0 or $ 1 million dollars,
所以你其实是在选择
so your choice is between taking n
要n还是n加一千美元
or taking n + $ 1,000.
因此 不管盒子B里是什么 你的决定都是
So no matter what’s inside Box B, your decisionis: do you want just something, or do you
你只想要这些东西还是这些东西加一千美元
want something plus $1,000 bucks?
无论如何你都能得到一些东西
You’re gonna get the something either way,
所以不妨抓紧了这额外的钱
so you might as well grab the extra cash.
这才是解决问题的正确方式
That’s the right way to solve this problem.
到人们回过来证明预期效用结果 其实不是这样
Until the Expected Utility people come back and prove that it… isn’t.
纽科姆的悖论提出了一个问题 就像数学家马丁·加纳得所描述的那样
Newcomb’s Paradox presents a problem with,what mathematician Martin Gardner described as,
一对相反的完美论点
two flawless arguments that are contradictory.
只选择盒子B很完美
Choosing just Box B makes perfect sense.
两个盒子都要也很完美
Choosing both boxes makes perfect sense.
所以 你是否依然确信哪一个是很明显的答案
So… are you still certain one is the obviousanswer?
在寻找合适的解决方案时 我们是否只思考了我们的个人看法呢
Are we only left with our own personal perception of the proper solution?
我不知道
I don’t know.
皮埃特 一个谜题制造者 数学家 诗人
Piet Hein, a puzzlemaker, mathematician,
在总结这个困惑时写道
and poet summarized this confusion when he wrote:
“一点超越认知的发现使我有时相信我所看到的那样
“A bit beyond perception’s reachI sometimes believe I see
生活就像两个锁住的盒子 其中一个里面包含了另一个的钥匙.”
That Life is two locked boxes, eachContaining the other’s key.”
我想知道的是 你是选择两个还是只选择B?
My question is: are you team Both or teamjust B?
感谢大家订阅我
Thank you for subscribing to me.
很抱歉押了个韵
Sorry for rhyming?
一如既往地 感谢观看
And as always — thanks for watching.
你们家的门槛是什么样的呢
What’s your doorstep like?
漂亮的?
Is it nice?
还是时髦的?
Is it smart? Well,
当然最时髦的能够邮寄到你家口的事
the smartest thing
就是这个Curiosity Box
that you can get mailed to your doorstep is the Curiosity Box
这个全新的Box11现在已经上市
and the brand new box 11 is now available.
实际上 我在致力于做这件事
I actually help make this thing you can see me here
让你能在全新的杂志上看到我
on the brand new magazine.
我就在这儿
That’s me right there.
这是一个关于科学玩具 迷题
This is a quarterly subscription of science toys, puzzles,
一本书 一件定制的衣服的季刊
a book, a custom t-shirt and
而其中一部分收益将用来支持阿兹海默的研究
a portion of the proceeds goes to Alzheimer’sresearch.
因此快让你的家门口更加时髦
So to make your doorstep smarter and support
每个人的想法都能提交到Curiositybox网站
everyone’s brains go to CuriosityBox.com.
点击这儿观看唯瑟科普第二集
And click over here to watch more Vsauce2.
谢谢大家的收看
Thanks.

Bye.

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视频概述

你是想要一千颗糖还是一百万零一千颗?

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视频来源

https://www.youtube.com/watch?v=ejUixWn_gE0

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