Parrondo's Paradox: Winning Two Games You're Guaranteed to Lose

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JM Parrondo is a casino and con artist's worst nightmare. In the 1990s, he invented two games that are sure to lose you everything. They're both mathematically designed to make you go broke, but play them one after another and you are guaranteed to win.

Parrondo's Paradox was dreamed up in the 1990s by physicist Juan Manuel Rodriguez Parrondo. It spawned a whole new approach to games — specifically, a distrustful approach to games by those who were sure the odds were stacked in their favor. The paradox is simple: two games, if played separately, will always result in you losing your shirt. They're played with a biased coin to make sure of it. If you switch off between them, though, you'll win a fortune. Suddenly, your loss turns into a win.

The first game is simple and always the same. You flip a coin, knowing that the two-faced, lying, no-good cheater you're playing against has weighted it so that your chance of winning is not fifty-fifty. Instead your chance of winning is (0.5 - x), with x being whatever the cheater dared weight it with. If you win, you get a dollar. If you lose, you lose a dollar. Since whenever "x" is more than zero you'll lose slightly more than you'll win, you are guaranteed to lose over the long run.


Your second game is played for the same stakes (win or lose a dollar) but two different ways. First, you look at the money you have in dollars and see if it's a multiple of three. If it isn't, out comes another biased coin, and it gives you odds of winning at (0.75 - x). That means your chance of winning is seventy-five percent, minus whatever "x" was in the initial game. Well, that doesn't look too bad. Why is this a losing game?

If the money you have is not a multiple of three, then you play against the really bad coin. This coin is weighted so that your chance of winning is (0.10 - x). That is a less than a one-tenth chance of winning. Since there are two times as many non-multiples of three than there are multiples of three, you will be playing with the coin that gives you an over-ninety percent chance of losing twice as much as the one that gives you a less-than seventy-five percent chance of winning. In other words, you will lose this game, and you will lose badly.


Tests have shown that, played one hundred times, either games results in lost money as long as "x" is bigger than zero. You can't beat those odds.

But you can. Oh yes, you can. Switch between the games, playing the first one twice and then the next one twice, and you will win money. It has been shown that, with x = 0.005, and with other values, depending on the sequence of the game, the winnings stack up. You don't even have stick to an orderly way of switching back and forth between the games. Randomly flip an (unbiased) coin, and you will still win in the long run. (There are many iterations of Parrondo's game. Some of them have different values of "x," and in these the second game is played with different multiples, so you'll play with the first coin only if your winnings are a multiple of four or five or six and so on. There are many ways to win.)


This isn't just a mathematical abstraction that Parrondo came up with. It's based on a physics concept. If people placed water at the bottom of a long, gentle slope, the water would just stay there. If they placed it at the bottom of a spiky slope, like the one simulated by the second game, it would also just sit there. Any water placed at the top of either slope would roll down it. But if the slope flashed back and forth between its smooth and spiky counterpart, the water would actually travel uphill. Parrondo considered this physical finding, and translated it into a game to come up with his paradox. The paradox is now being studied by investors and financial analysts, eager to see how they, too can juggle losing assets to make the "slide uphill."

Via Seneca twice and The Official Parrondo's Paragraph Page. Image: PEB