Vsauce! Kevin here with two games. If you play Game A -- you are guaranteed to lose.
If you play Game B -- you are guaranteed to lose.
But.
If you alternate back and forth between playing Game A and playing Game B -- you are guaranteed
to win.
How? How can two losing games combine into one winning one? Is it just a cheap game hack
or something more? And does this mean that grandma was wrong and it’s actually possible
for two wrongs to make a right? Welcome to Parrondo’s Paradox.
Physicist Juan Parrondo who created this paradox demonstrated with flipping a
biased coin. A coin that, instead of having 50/50 odds of landing on heads or tails has
49.5/50.5 odds. So slightly unfair, biased odds. And… that’s not gonna work for us.
Because precision-biased coins that look like normal coins are likely impossible, or as
statisticians Andrew Gelman and Deborah Nolan wrote in a 2002 paper, “You can load a die
but you can’t bias a coin.”
So, instead, I made a series of games using 3 roulette wheels with a total of 114 spaces
and 1 Markov Chain. Because…PENGUINS.
Seriously, this paradox can be visualized using a penguin slide toy. The same mechanism
that moves these penguins to the top allows
us to turn two losing games into our own money-printing
machine. This is… this is really loud. I’ll explain how later but for now let’s actually
talk about our games.
Okay, Game A works like this: it’s a game of chance in which your odds of winning are
slightly lower than your odds of losing. The American roulette wheel has 38 spaces -- 18
red and 18 black, and these two 2 green zeroes. The rules of Game A are that you can only
bet on red or black. Either way, our odds of winning are 18/38, or 47.36% -- so a little
less than 50/50. Our 53.6% chance of losing means the house edge is 5.2%, which we’ll
round down to 5%. Because of that 5% losing edge, theoretically, every time we bet $1
on this game, we’ll lose about 5 cents. If we start with $100 and keep playing this
game, we’ll be totally broke after about 2,000 spins. In the long run, when we play
Game A, we’re guaranteed to lose.
Come onnnn, red!
Since definitely losing sounds terrible, let’s give Game B a shot.
Before we do that you may want to add up all the numbers on a roulette wheel and just tell
me the sum in the comments.
Okay. B is composed of two games of chance, each with different odds. You’ll still bet
$1, but the wheel that you play depends on how much money you have left. We’ll call
these B1 and B2. You’ll play B1 only if your total money is a multiple of M and…
let’s say that M = 3. So if your leftover money is a multiple of 3, like 93 or 81 or
66, then you have to play wheel B1. If your bankroll balance is not a multiple of 3, then
you’ll play wheel B2.
Here’s the catch.
With B1, you’re only allowed to make what’s called a Corner bet -- choosing an intersection
of 4 numbers, so you’ll win $1 if the ball lands on any of those four. So like, a corner
bet of 26, 27, 29, or 30. A corner bet means your odds of winning are just 4 out of 38,
or… about 10%. So when you play B1, you’re going to lose 90% of the time. Losing isn’t
guaranteed, but you’re going to lose a lot more often than not. Game B1 is a pretty bad
game for the player.
Don’t worry, though, B2 is much better. You get to choose a combination of winning
spaces -- red or black AND odd or even. So, if you choose red and evens, you’d win every
time the ball lands on a red space or an even space, even if it’s a black even space.
The two green spaces are also winners for you on B2. This allows B2’s odds to shift
significantly in your favor, with 18 reds and 9 black evens plus the two 2 greens giving
you a total of 29 chances to win out of 38 possible spaces -- and that, my friends, is
76%.
The good news is that since there are more possible money counts that aren’t multiples
of 3 than there are, you’ll be playing the
much-friendlier B2 a lot more than the nearly-impossible
B1. So that means most of the time, with Game B, you’ll be winning, right? No.
Here’s the thing.
B1 is a loss 90% of the time. And since you’ll also lose roughly every 1 out of 4 times playing
Game B2, overall you’re guaranteed to lose if you just play the B Games. Game B is actually
a Markov Chain, a stochastic process that takes a situation like ours -- different states
with varying probabilities -- and generates a loss by B1’s disproportionate effect on
our fortunes.
Let me explain.
The key is that even though there are only three possible states of our money balance
-- a multiple of 3, like 90 or 93, or two possible states in between multiples of 3,
like 91 and 92 -- our probabilities of playing the bad B1 game vs. the better B2 game aren’t
a simple ⅓ and ⅔. What’s surprising is that a Markov Chain analysis, and if you
want to learn more about Markov chains I’ll link you to a course on those in the description,
shows that our probability of playing the bad B1 game is actually closer to 40%. And
the winning edge that the good B2 game gives us isn’t enough to make up for the terrible
B1.
All of that is to say that Game A is a simple guaranteed loser. And Game B is a kinda complex
guaranteed loser. And… this is getting dreadful so let’s bring back our penguins and lighten
the mood. Our flightless bird friends will help us visualize how we leverage two negatives
to create a positive.
In a 2000 paper titled, “Parrondo’s paradoxical games and the discrete Brownian ratchet,”
Derek Abbott, Parrondo and others, described a process known as the “flashing brownian
ratchet” as an analogy for how the paradox works. Basically, directed motion is achieved
by alternating between a sawtooth and a flat potential. I’ll show you. These penguins
clearly reach the top of the slide, even though they themselves aren’t moving, because they’re
being carried back and forth between
two downward-sloped-sawtooth shapes and a flat shape. These shapes slope
downward. And this shape here -- sorry, penguin! -- is flat. Yet the penguins climb to the
top by alternating between them. Ok, that’s loud, sorry penguins. Wheeee! The end. Excuse
me one second, Happy Feet.
Think of Game A as our flat shape because the odds of winning Game A are close to 50/50
with a slight bias toward losing. Think of Game B as our sawtooth shape because the odds
are much steeper in Game B1 and much better in Game B2, creating a distinct asymmetry.
Look at that! But instead of moving plastic penguins uphill, by alternating between playing
Game A and playing Game B, we, the player, are carried upward by our combination of losing
games into winning.
To put it another way, we can get really simple and math-y about this. Back to the North Pole
with you! Say you start with $100, and each time you play Game A, you lose $1. That’s
it -- done. That’s just how Game A works in this simplified scenario. You’re a loser
with no hope of winning. And if you played Game A 100 times, your financial trajectory
goes downward until you’re broke.
There are two strains of Game B: if the amount of money you have left is an even number,
like $82, then you win $3. If it’s an odd number -- say, you’re down to $71 -- then
you lose $5. If you only played Game B, you’d be broke even faster than if you’d just
auto-lost Game A. At least with Game A you’d get to play that game 100 times.
Both Game A and Game B are clearly 100% losing games, but when you switch back and forth,
you can make them profitable. Check it out. Play Game A and you’re down to $99. That’s
an odd number, so you don’t want to play Game B or else you’d lose another $5…
instead you can play Game A again and lose another dollar. Now you’re at $98 -- when
you switch to Game B, you win $3, and suddenly you’re up to $101. Lose Game A to get to
$100, win Game B to get to $103. You can repeat a cycle of A/B switching to amass infinite
wealth despite playing two games that on their own are guaranteed losers.
But is that really a paradox? When we examine the best way to flip coins or spin roulette
wheels, it seems like just a cheap trick to manually hack a couple of games. It’s not.
The paradoxical situation is that you can alternate playing Parrondo’s two losing
games randomly and it will still produce a winning one.
We can use Stan Wagon’s Parrondo Paradox wolfram simulator to prove it. This simulator
uses Parrondo's biased coin flip odds and allows us to set the number of flips and how
many times we want to repeat that experiment. Parrondo's numbers for Games A, B1 and B2
are almost identical to our roulette example. So, we set the number of flips in a game and
how many times we'll repeat that game, and each time we click New Run, the simulator
crunches those numbers... which is a lot faster than flipping an impossible biased coin 2
million times.
Let’s set this to run 1 repetition of 1,000 flips. When we play in a specific pattern
-- BBABA -- we clearly win over time.
But what happens if we choose Game A or Game B randomly? Most of the time we win over 1,000
flips, occasionally we don’t -- that's just variance. But when we run 2,000 simulations
of 1,000 flips, a pattern emerges: we see a clear upward slope. We don't win as much
as when we carefully orchestrate the best approach to the two games, but... we win.
Despite having no plan and no specific hack, alternating randomly between Games A and B
yields a long-term win.
So even if it takes a while, even it’s done randomly, by alternating between these two
losing games, we actually get a winning result. We get to the top of the slide -- like our
penguins.
But here’s the question.
Can you use this strategy to guarantee that you win at a casino? No. Parrondo's Paradox
depends on being able to interact within two games, and you just can't do that in a casino.
Real casino games like slot machine spins or the way roulette is actually played are
based on entirely separate events -- one outcome will never influence the outcome of the next
round or a different game, like a roulette result will never steer you toward an advantageous
round of Pai Gow.
By guaranteeing that games never intersect, casinos avoid the possibility of an exploit
or a weird Parrondo situation. The outcome of every 'bettable' event in a casino depends
on nothing before it or after it. It exists in its own impenetrable bubble that pops as
soon as that round is over and then is blown up again -- and that keeps the games fair
and keeps the casino’s math predictable.
The independence of each game is actually one of the reasons you so rarely see new games
at casinos. It’s just hard to come up with games that are totally independent, give the
house enough of an edge that they win over time but also give the player enough of a
chance that they want to play and have fun playing.
What’s exciting about Parrondo’s Paradox is not how to win money playing two games
that no casino would never allow. It’s figuring out how to apply its surprising property to
other fields of study. Researchers are working on real world applications for it in disciplines
ranging from quantum mechanics to biogenetics.
And for the rest of us, it can be helpful to realize that two losing games can become
one winning one. That, in a way, as weird as it sounds, two wrongs can make a right.
And even what seems, by all accounts, like a totally hopeless situation can, undeniably,
mathematically, be turned into a winning one.
And as always, thanks for watching.