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#### 你永远不可能解决的问题

A Problem You'll Never Solve

Vsauce!

Kevin here, with a really simple question.

Do you want this box of 1000 candies

and a mystery box which contains either nothing or a million candies or

do you want just the mystery box?

Obviously you’ll take both boxes

because you’re getting the mystery box either way.

Might as well grab some guaranteed candy too,right?

Right.

Wrong.

Maybe.

Honestly，I don’t know.

The thing is…

When it comes taking both boxes or just the mystery box,

almost everyone watching this

video will be absolutely sure that they knowthe right answer.

This is barely a problem, let alone one you’llnever solve.

But here’s what’s interesting…

Half of you will be certain that the obvious answer is to take both boxes,

and the other half of you will be just as sure

the obvious answer is to take only the mystery box.

How is that possible?

And why is there suddenly a Grandayy genieon my table?

Let’s dissect this problem.

Box A has clear value.

It’s literally clear —

you can see that the contents are 1000 candies.

The issue is Mystery Box B.

The contents of Box B are determined in advance

by our omniscient, all-knowing Grandayy genie,

who predicts what you’ll choose with near-perfectaccuracy.

If he predicts you’ll choose both boxes,

he’s put nothing in Box B.

If he predicts that you’ll only choose mystery Box B,

he’s placed… a million candies.

You can’t see inside Box B,

you can’t touch it, and you

don’t know what the genie

has predicted before you actually choose.

Here’s a question.

Who even came up with this?

Did I just make this whole thing up?

No.

Theoretical physicist William Newcomb devisedthis problem in 1960.

philosopher Robert Nozick detailed the deep philosophical fracture that

makes the two equally-obvious choices bothright and both wrong.

Here’s why.

If you decide to take both boxes,

the genie will likely have predicted that and put nothing

in mystery Box B —

maybe genies don’t like greedy players or something.

So if you choose both Box A and

box B, you’ll wind up

with only a few handfuls of candy.

If you decide to take only mystery Box B,

the genie will almost certainly have predicted that,

too, and put a million candies inside… maybe

as a reward for your courageous choice.

Either way, it’s now obviously better

for you to take mystery Box B because a million

is a much better prize than 1000.

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 %

of over 30,000 survey respondents chose to take

only mystery Box B. Here’s what the

other 46.5% thought:

The genie has already either put a million candies

in the mystery box… or not.

He could’ve setup the boxes a day,

a week, a month ago!

The candy isn’t going to suddenly

appear or disappear based on your decision.

If he’s filled Box B with candy and you take both boxes,

you’ll get a million plus

1000 more from Box A,

which you can eat right away to celebrate your amazingly clever rationale.

If he didn’t fill Box B… he just didn’t.

You take both boxes and win your small

prize and this way you don’t walk away empty-handed.

You can’t really lose.

Worst case scenario,

the mystery box is empty but you still get 1000 pieces of candy which

is 1000 more than zero.

So should you take both boxes or just BoxB?

What is actually going on here?

Because it’s pitting two equally valid methodsof reasoning against each other: Expected

Utility and Strategic Dominance.

Let’s recap the two options with a little math…

so we can get serious.

You may not have a sweet tooth, so let’sswitch prizes from candy to money: Box A now

contains \$ 1,000,

and Box B either has \$ 1 million dollars or no dollars. First,

we can see our possible outcomes with a simple payoff matrix. Basically,

we’ll just write out the fourscenarios.

Excuse me, Grandayy.

If the genie predicts you’ll take Box B and you choose Box B,

you’ll get \$ 1,000,000.

If he predicts you’ll take Box B but you choose both boxes,

you’ll win \$ 1,001,000

— the million in Box B and the \$ 1,000 in Box A.

If he predicts you’re greedy and

will take both boxes but you choose just Box B, then

you get zero dollars.

And if the Genie’s prediction is both boxes,

and you choose both boxes, your prize is just

the \$1,000 from box A.

To put it another way,

these are the outcomes when his prediction is right and these are

the outcomes when his prediction is wrong.

Okay.

We mapped out the potential outcomes, nowwhat?

How do we figure out which choice is right? Well,

we can actually calculate how valuable a choice is to you —

that’s Expected Utility.

It’s like the math of making a decision.

You simply take the result of a choice and multiply it

by the probability of the outcome.

That’ll give you a numerical value to helpinform your decision. So,

let’s say the genie

has a 90 % chance of predicting right.

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.

So if we choose both boxes,

there’s a 10 % chance we win two money-filled boxes and a

90% chance that we’re left with just the\$1,000.

We multiply the.1 probability that he’s wrong

by the payoff of \$ 1,001,000 from both

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

that’s.9 multiplied by just the \$1,000Box A payoff.

This equals \$101,000.

If we assume that the genie is right 9 times out of 10,

each time we chose both boxes,

we’d theoretically gain \$101,000.

Now let’s find the Expected Utility

of choosing only Box B so that we can compare the two

values and determine the best choice.

We get a million dollars

if we choose Box B when the genie predicts our choice correctly.

If we stick with his 90 % accuracy rate,

we multiply.9 by the \$ 1,000,000 payoff and then

add.1 times the \$ 0 from the empty box when he’s wrong

for a theoretical gain of \$ 900,000 per game.

By using Expected Utility as a reasoning framework,

the best choice is to take only mystery Box B,

because an average payoff

of \$ 900,000 is clearly better than \$ 101,000.

Obviously!

That’s the right way to solve this problem.

Until it isn’t.

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,

look, the genie has put the money

in the mystery box or he hasn’t, your choice

comes down to taking whatever is in that box,

or taking whatever is in that box plus Box A.

The mystery box has a value of n,

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,

so your choice is between taking n

or taking n + \$ 1,000.

So no matter what’s inside Box B, your decisionis: do you want just something, or do you

want something plus \$1,000 bucks?

You’re gonna get the something either way,

so you might as well grab the extra cash.

That’s the right way to solve this problem.

Until the Expected Utility people come back and prove that it… isn’t.

Newcomb’s Paradox presents a problem with,what mathematician Martin Gardner described as,

two flawless arguments that are contradictory.

Choosing just Box B makes perfect sense.

Choosing both boxes makes perfect sense.

So… are you still certain one is the obviousanswer?

Are we only left with our own personal perception of the proper solution?

I don’t know.

Piet Hein, a puzzlemaker, mathematician,

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

That Life is two locked boxes, eachContaining the other’s key.”

My question is: are you team Both or teamjust B?

Thank you for subscribing to me.

Sorry for rhyming?

And as always — thanks for watching.

Is it nice?

Is it smart? Well,

the smartest thing

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I actually help make this thing you can see me here

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That’s me right there.

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So to make your doorstep smarter and support

everyone’s brains go to CuriosityBox.com.

And click over here to watch more Vsauce2.

Thanks.

Bye.

hjh