The Game You Win By Losing (Parrondo's Paradox)

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..