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 know the right answer.
This is barely a problem, let alone one you’ll never 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 genie on 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-perfect accuracy.
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 devised this problem in 1960.
And a decade later, philosopher Robert Nozick detailed the deep philosophical fracture that
makes the two equally-obvious choices both right and both wrong.
It’s a contradiction.
It’s an antinomic paradox.
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, 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 Box B?
What is actually going on here?
Why exactly has Newcomb’s Paradox confounded minds for decades?
Because it’s pitting two equally valid methods of 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’s switch 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 four scenarios.
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, now what?
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 help inform your decision.
So, let’s say the genie has a 90% chance of predicting right.
We’d calculate the expected utility of choosing both boxes like this:
A 90% chance he’s right means there’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, which means Box B would be empty -- so
that’s .9 multiplied by just the $1,000 Box 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, “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 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 decision is: 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 obvious answer?
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:
“A bit beyond perception’s reach I sometimes believe I see
That Life is two locked boxes, each Containing the other’s key.”
My question is: are you team Both or team just B?
Thank you for subscribing to me.
Sorry for rhyming?
And as always -- thanks for watching.