Vsauce! Kevin here, with a game that gives you quantifiable, verifiable strength against
your opponents and even the advantage of going first. Simple math and logic conspire to delude
you into believing that the odds are in YOUR favor -- but in the long run I'm gonna win
every time. And even when you understand how this paradox works, you STILL probably won't
be able to rationalize it.
It goes like this: I’ve got four different dice, and I’ve replaced their regular sides
of 1-6 with different combinations of numbers. Here are some maps to make it easier to see.
A has 4’s and 0’s, B has all 3’s, C has 2’s and 6’s, and D has 5’s and 1’s.
You choose one and I’ll choose one. We roll and the highest number wins. And because I’m
so nice -- I’m gonna let YOU choose the best die first.
WHICH IS THE BEST?
The simplest, safest way to choose the best one is to add up the sides and take the highest
total. C has a total of 20. Those 2’s aren’t great, but the 6’s beat anything. C seems
like the best overall choice. B and D each add to 18 and A only has 16.
But if you choose C, I’ll choose B -- and I’m gonna win most of the time. LOOK. Here’s
a table of all the possible rolls both of us can have. Of the 36 possible combinations
of rolls, B comes out on top 24 times to C’s 12. You’ve chosen what looks like the best
die, and I’ll still beat you 2 out of 3 times.
Okay. So. Just adding up the sides isn’t good enough, so maybe you factor in standard
deviation on the sides of the dice… hold on... hold on... I’m gonna give you a game-changing
hint. Here it is:
A beats B. B beats C. and C beats D. These charts account for every single roll in that
progression, and I’m showing you the data to prove that A is at the top of a chain of
dominance.
You know that due to the transitive property, if A beats B and B beats C, then A should
also beat C. And if C beats D, then A and B should beat D, too. It’s like lining people
up by height, if A is taller than B and B is taller than C, then A has to be taller
than C. Makes sense. Right?
Alright. Let’s play again -- and it sounds like you want to choose A this time? Because
it beats everything? Because I literally just showed you that the transitive property makes
A better than B, C and D, RIGHT? WRONG! If you choose A, I’m gonna take the WORST choice
at the very END of the ‘greater than’ chain. I’m choosing D to go against your
A, and I’m going to crush you with my D. How?
Before I explain I must tell you this video is SPONSORED BY ME.
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Okay so here’s why I’m choosing the worst die to beat your best -- wait. These dice
are too easy. So many numbers repeat that it’s elementary -- and you’re a genius.
Let’s use another set of numbers from 0 to 12. This set has the same property as the
easier ones, A is better than B, B is better than C, and C is better than D. If we get
a tie, we’ll just roll again. You choose A, the strongest die.
I want the weakest die because… it’s actually not.
D is at the losing end of our chain, and my D die is still going to win nearly ⅔ of
the time against your strong A. Which seems impossible but let’s test it!
So despite D seeming to be the weakest and A being the strongest, out of 15 tolls, D
won exactly twice as many times as A. D is somehow stronger than A. In this game, no
matter WHAT die you choose, I’m gonna beat you twice as often as you beat me -- even
with the A > B > C > D chain of dominance, I am ALWAYS going to pick a better die.
Because these dice aren’t transitive. These are Efron’s Dice. Here’s the proof: My
voice cracked. Who cares? Math!
These charts reveal the truth about the matchups in our 0-12 dice, and I’ve labeled the squares
that each die wins. Just like I said, A beats B around ⅔ of the time, and it’s the same
for B vs. C and C vs. D.
But let’s look at D vs. A -- despite being worse against C, which is worse than B, which
is worse than A, D is actually STRONGER than A. Which makes NO SENSE... IT MAKES NO SENSE.
But it’s mathematically true.
Bradley Efron is a statistician who has won just about every award in the statistics game
over the last 60 years. And he’s still going -- in 2018 he won the International Prize
in Statistics, which is basically the Nobel Prize of stats. And he invented what we now
call Efron’s Dice -- sets of dice that are maddeningly nontransitive, and that nontransitivity
is cloaked so carefully that it’s almost impossible to grasp even when you know it’s
true.
If you choose D? Fine, I’ll just choose C, which we know beats D most of the time.
If you choose B? I can choose A and win a lot more than I lose.
C? I choose B. A? I choose D. You literally have NO WAY to win this game in the long term,
because I can ALWAYS choose a die that’s better than yours -- even if you choose the
best die first.
And that’s the veridical paradox here -- how can you have the best possible choice that…
isn’t? How can a die prove it’s the strongest and then LOSE to the weakest? How can a die
verifiably lose to everything except when it’s up against the one it SHOULDN’T be
able to win against?
The problem isn’t the dice. It’s us.
We fall into a cognitive trap of thinking in absolute terms when real value is often
relative. In this game, the concept of better or worse is deceptive -- it depends completely
on the situation. Our brains, especially when it comes to numbers, want to sort patterns
and construct algorithms and employ heuristics that identify the best choice. And we LOVE
ranking things, whether it’s music or sports or cereal tier lists.
But… life isn’t always transitive. And if you get stuck thinking in absolutes, you’re
gonna get duped by the gamemaster who knows when the real value lies in the relative -- and
they’re gonna win most of the time.
SO UNSTICK YOUR BRAIN.
And as always -- thanks for watching. I was gonna throw these at the camera. But that
would be bad. So. There.