The Demonetization Game


Vsauce!
Kevin here with 100 YouTubers.
Some are friends, some are colleagues — all
are about to be demonetized forever.
Unless… each one chooses their own channel
out of these 100 eggs.
I’ll put one YouTuber in each egg and every
player gets only 50 guesses.
If just one of the 100 YouTubers fails to
find their channel in the egg, everyone loses.
Here’s the setup.
All 100 YouTubers are in the same waiting
room, they’re each assigned a number and
then… one at a time, brought into the egg
room to play the game alone.
Once a YouTuber completes their round, they’re
immediately escorted out the back door so
they can’t go back to the waiting room and
say, like “hey Rhett & Link, you’re hidden
in Egg #93!” or whatever.
No.
No.
That’s cheating.
This game is 50 blind guesses for each YouTuber,
and they all have to find their egg or everyone
is demonetized.
After each round, the eggs are all reset and
re-sealed so that the next YouTuber can’t
see which has been chosen.
Every player has to start fresh.
So, how do you win when you can’t see what’s
in the eggs and you can’t communicate with
any other players?
The most obvious option is for everyone to
just choose randomly and hope for the best.
I mean, you have 50 tries — that’s a lot,
right?
50 out of a 100 eggs gives you 50/50 odds
of finding your egg.
Not bad!
But…
The odds of all 100 YouTubers pulling this
off successfully are virtually impossible
— let’s crunch the numbers.
It’s pretty simple: with 100 eggs and 50
guesses, the odds of selecting the right egg
randomly are 50% — or .5.
One hundred successful attempts at odds of
.5 comes out to…
0.0000000000000000000000000000008.
To get the percentage odds, we just shift
the decimal over two places… and uh, yeah,
there are still 28 zeros.
The odds of me getting drafted in the NBA
or winning the lottery are unfathomably higher
than this demonetization egg game working
out for these YouTubers.
These odds are literally the same as correctly
predicting 100 coin flips in a row.
Try it.
Try to successively predict coin flips.
You may get 5 or 6 in a row right, maybe 10!
But you almost certainly won’t get 50 — let
alone the 100 we need to save our YouTubers.
So sorry, 3Blue1Brown.
And sorry, EpicRapBattles.
There’s just no hope of winning this game.
Your channels, like everyone else’s here,
are cancelled.
Unless…
What if there was a mathematical strategy
that our 100 YouTubers could employ to give
their channels a realistic chance of survival?
Let’s actually play this game and try to
figure it out.
I’ve given each of these YouTubers a number
between 1 and 100, and each one will randomly
go in a numbered egg.
Now we’ve got 100 YouTubers hidden in 100
eggs.
And the secret solution to saving their channels
goes like this…
When it’s their turn to play, each YouTuber
just needs to start by choosing the numbered
egg that matches their personally assigned
number.
If they aren’t in that egg, rather than
choosing another egg randomly, the next egg
to be opened is the number of the YouTuber
inside the one that they just opened.
Each player will repeat this strategy until
they find their egg or until they’ve opened
50 eggs unsuccessfully — in which case everyone
loses.
MaxMoeFoe is player #1, so I will play as
MaxMoeFoe.
Here we go.
Okay, Egg #1 is…
Gus Johnson.
Who’s #62, so we’ll just open up Egg #62.
And?
PhysicsGirl.
And she is #12, so now we just open up Egg
#12…
Nerd City.
#77, so let’s open up Egg #77.
Numberphile, #35.
Okay, let’s check out Egg #35.
And here we have Carson, number #69 — this
is very, very important.
Whatever you do, do not forget 69.
We’ll talk about that later.
But for now let’s open up Egg #69.
AspectScience #7.
Y’know?
This might actually take a little while.
So while I find the MaxMoeFoe channel, let’s
montage.
#20?
#20.
Is Max!
Is me!
MaxMoeFoe!
We did it!
We found, me!
Well, we found Max.
And it only took 31 guesses.
Way below our 50 guess threshold.
Now we need to reset our eggs.
Okay, we’re all set for YouTuber #2 — RedLetterMedia.
Let’s do this.
RedLetterMedia!
I found it already!
That didn’t take long at all.
Okay, now let’s reset and go with YouTuber
#3.
Okay, now It is time for Danny Gonzalez.
Here we go.
Danny Gonzalez!
#3!
Did it again!
Another winner!
That is awesome.
Okay, I am going to reset this and then let’s
talk about what just happened.
Okay, our strategy is working — three YouTubers
have found their eggs.
The odds of the first three YouTubers choosing
successfully with random guessing would have
been .5 x .5 x .5 which equals 12.5%.
Four in a row drops down to 6.25%.
Five in a row… you’re looking at 3.125%
and by seven in a row you’re below 1%.
So how much better is our system?
Well, remember random guessing has the odds
of all 100 YouTubers winning at 0.000 28 zeros
and then an 8 percent..
To put it in perspective, our system increases
the odds more powerfully than turning a penny
into all the money in the world.
We can increase the odds of all saving all
100 YouTubers from effectively 0% to 31%.
And I’ll explain how it works but, here’s
a question.
Did I just make this whole thing up?
No.
Usually, I like to examine old, longstanding
problems in math and science, but this one
is actually pretty new.
In 2003, computer scientists Peter Bro Miltersen
and Anna Gál, in a paper titled, The Cell
Probe Complexity of Succinct Data Structures,
created a version of this game in which red
or blue slips were dropped in 100 little drawers.
To win, every single one of the 100 players
had to correctly guess whether their drawer
contained a red or a blue slip.
The game is a little different, but the odds
of guessing red or blue vs. choosing 50 eggs
out of 100 are exactly the same.
Miltersen’s initial thought was that as
the number of players increases, the probability
of winning the game would trend to zero.
But by starting with the egg matching their
own number, each player is guaranteed that
they’re on a path to an egg that contains
them, and it’s just a question of whether
their egg is within a cycle of 50 attempts.
Okay.
Why?
Because each egg contains one unique number
that points to another unique number and that
creates, as Nick Berry of Datagenetics described,
a circular chain.
One number points into the chain and one number
points out.
So if the YouTuber uses their own number to
point into the chain, they will eventually
find their own number.
Rather than blindly guessing, our system is
a way of tapping into that circular chain.
To get math-y about it, by giving each YouTuber
a number, we’ve created a permutation of
the set that’s a one-to-one mapping of all
100 numbers to itself.
Our strategy makes a cycle so each number
returns to itself, and it’s successful when
all 100 YouTubers find a cycle length of 50
or less.
With 100 boxes, there can only, mathematically,
be one cycle longer than 50.
But that’s all it takes for us to lose the
game, anyway.
So what are the chances of a permutation in
which there’s one cycle longer than 50?
Let’s find out.
The solution to our chance of success equals
1 minus the probability of getting a cycle
longer than 50.
Let’s call the cycle length L. That gives
us 100 choose L possible sets, times L!/L
permutations of the cycle within the set,
times (100 – L)! permutations of the remaining
YouTubers… and that comes out to 100!/L,
which is really, really easy to work with.
We know that there are 100! ways to arrange
the tickets, so the probability of chain length
L is just…
1/L.
Our chance of success is 1 minus all those
long-chain failures combined.
Since we also know that only 1 cycle of 51
or longer is possible in each set, we can
calculate the probability of losing by 1 – (1/51
+ 1/52 + 1/53…
+ 1/98 + 1/99 + 1/100), which comes out to
about .69.
I told you to remember 69.
There’s a 69% chance that we encounter a
cycle longer than 50.
So, the probability of all 100 YouTubers having
a cycle of 50 or under — which means they’d
win the demonetization game using this strategy
— is the remaining .31, or 31%.
31% isn’t amazing, but just like The Game
of Googol, around 30% is waaaay better than…
basically zero.
The weird thing is that even with our solution,
each individual YouTuber’s chance of success
is still 50% just like if they’d guessed
randomly.
The strategy changes the group’s chance
of success.
So, that’s that.
Problem solved.
Until… you tweak the problem.
Miltersen’s game continues to evolve.
Like… what happens if we have more than
100 eggs but some of them are empty?
Reaching an empty egg would stop the cycle
and completely ruin our system.
But even this seemingly impossible version
of the game has inspired computer scientists
to develop a strategy that defies our expectations
of failure.
Which is what we do.
Whether we’re pretending the floor is lava,
whether we’re watching superheroes defeat
galaxy-destroying villains, or we’re trying
to cure diseases that seem impossible to overcome.
We test, push, and expand our limits because
challenges are only impossible until they
aren’t.
And as always, thanks for watching.
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