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

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While working on the interplay between thermodynamics and info hypothesis, scientist Juan Parrondo realized that a system of losing games could really produce a winning outcome. And it is not just by playing them in a highly-strategic way; it is even viable for two ensured dorks to generate a victor when played arbitrarily.

At the heart of that counterintuitive determination is a test of our comprehending of probability and relationships between events. From the easiest money-based games and coin flip episodes to theoretical perpetual motion machines like the Brownian ratchet and flashing Brownian ratchet, Parrondo’s Paradox investigates elements of game theory that are motivating research in fields like biogenetics, finance, and evolutionary biology.

*** SOURCES ***

Derek Abbot, Peter Taylor, and Juan Parrando, “Parrondo’s Paradoxical Games and the Discrete Brownian Ratchet”

Stan Wagon’s Parrondo Paradox Demonstration

Abhijit Kar Gupta and Sourabh Banerjee, “Parrondo’s Paradox: New Results and New Ideas”

New York Times, “Paradox In Game Theory: Losing Strategy That Wins”

S. N. Ethier and Jiyeon Lee, “The flashing Brownian ratchet and Parrondo’s paradox”

Andrew Gelman and Deborah Nolan, “You can load a die but you can not prejudice a coin”

*** LINKS ***

Huge Thanks To Paula Lieber

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  1. Quick point — those added green spaces throw off the math a little bit and make game B2 very slightly biased toward the player. With a 38.36% chance of a player in the B1/B2 Markov Chain playing B1, we can calculate (4/38)*(.3836) to find B1’s share of the overall win: .04037. We can do the same for B2, which he plays 61.64% of the time: (29/38)*(.6164) for a share of .47041. When adding those two together, .04037 + .47041, we get a winning probability of 0.51078, or 51%.

    How do we tweak this to get a truly losing Game B? We take away those additional green spaces. By giving a player just 27 spaces to win on B2, the value becomes (27/38)*(.6164) = 0.43796, putting the combination of B1 and B2’s payoffs well below the 50% threshold and turning it into a proper loser.

    tldr; Make the green spaces losers in our B2, and my roulette version of Parrondo’s biased coin flips align properly.

    Also, check the footnote in the video about the North Pole penguins. 😉


  2. This seems horribly explained to me, At no point did he discuss the payout of the other 2 games.
    He started with Red vs Black… that pays out 1:1 ok, great.

    Now he goes to B1 and B2… Corner bet is about the 1/10 of winning, but it pays 8:1. But the doesn’t ever say this… He just says you ‘Win’ 10% of the time…
    The last bet isn’t really possible on a roulette wheel, but if you have a 76% chance to win, you’d expect it pays like 1.25:1 or something.

    But I think he’s saying during this whole thing that everything pays 1:1?

    He then quickly abandons the actual example he gave and switches to a different thing where you have -1, +3 and -5 but now you’re allowed to choose between a guaranteed win or not?

    It was unclear in the first example if you were allowed to choose or not, it sounded forced. If its a multiple of 3 you must play B1, if not B2. But if its a choice…you’d just never play B1?

    I feel there’s a lot more they needed to explicitly state that is extremely counter intuitive. It seems like the twist to make this whole thing work is totally based on the fact that the B2 game is extremely biased towards the player.
    That’s not a thing that happens, so what’s the point or application of this?

    I just don’t think it makes any sense with the example given, especially if you’re not explicit that for some reason everything is a 1:1 payout, every ‘win’ is equal.

    And then the final comments about how Casinos avoid this by not allowing interaction between games etc… doesn’t make sense. It’s not possible because it requires one of your ‘games’ to have a component that isn’t possible, it requires having a game biased in your favour.

    It’s kind of like saying “Three objects are actually lighter than 2 of them!” and then revealing it’s because the 3rd one has anti-gravity. It’s not a paradox, you just made one of the elements in the game a thing that doesn’t exist.

  3. Dejenny Pilipovic

    I had to stop watching…that smokers soar throat voice at certain words kills it for me.

  4. That sum of all the numbers on the roulette wheel, the sum of numbers from 1 to 36, is 666.

  5. Peter El Ghazal

    What is 19* 39

  6. Peter El Ghazal


  7. 666


  9. game “B2” doesn’t exist. FAIL

  10. Two negatives make a positive.

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