November 10th, 2019 at 7:31:40 PM
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I recently saw this Vsauce2 video about Parrondo's Paradox, and while I normally really like Vsauce's content, I found the explanation in the video really lacking. It has over a million views, so I figured I might not be the only one who on the forum encountered it, and thought a thread might help illuminate what's really going on.

The "paradox" is that you can define two games, A and B, that are both negative EV on their own; playing either continuously will eventually result in a loss. However, you end up winning in the long-run by simply switching back and forth between the games.

That might sound unremarkable to anyone who has Wonged into a positive count and then switched to vulturing a multi-state slot; blackjack and slots are both -EV overall, but you can make money by judiciously choosing when to play each one. The difference here is that you end up winning even if you randomly choose which game to play each turn, or use a simple pattern like AABBAABB that doesn't take into account what state the game is in (what cards have been dealt, what bonuses are in play, etc.). In fact, Parrondo's original example (which he beautifully explains on this website) doesn't even have "states" — you can set up both games using only even-money bets on a total of three different biased coins. The diagram below maps it out (notation — X(t) is your bankroll at any given time, which will always be an integer since all wins and losses are 1 unit. The green coin favors the player, while the red coins favor the house):

And below is a plot of your capital growth over time using various strategies (notation — the line labeled [3, 2] is your profit from playing game A for 3 rounds, then game B for 2 rounds, then A for 3 again, B for 2 again, etc.):

There's a lot more to say about this, including its connection to thermodynamics and the Brownian Ratchet, as well as explaining why Parrondo's Game B is actually -EV on its own (it might look +EV at first glance). But I'd much rather this be a discussion than a monologue, so I'll leave it there for now. Would love to hear any thoughts folks have on this.

The "paradox" is that you can define two games, A and B, that are both negative EV on their own; playing either continuously will eventually result in a loss. However, you end up winning in the long-run by simply switching back and forth between the games.

That might sound unremarkable to anyone who has Wonged into a positive count and then switched to vulturing a multi-state slot; blackjack and slots are both -EV overall, but you can make money by judiciously choosing when to play each one. The difference here is that you end up winning even if you randomly choose which game to play each turn, or use a simple pattern like AABBAABB that doesn't take into account what state the game is in (what cards have been dealt, what bonuses are in play, etc.). In fact, Parrondo's original example (which he beautifully explains on this website) doesn't even have "states" — you can set up both games using only even-money bets on a total of three different biased coins. The diagram below maps it out (notation — X(t) is your bankroll at any given time, which will always be an integer since all wins and losses are 1 unit. The green coin favors the player, while the red coins favor the house):

And below is a plot of your capital growth over time using various strategies (notation — the line labeled [3, 2] is your profit from playing game A for 3 rounds, then game B for 2 rounds, then A for 3 again, B for 2 again, etc.):

There's a lot more to say about this, including its connection to thermodynamics and the Brownian Ratchet, as well as explaining why Parrondo's Game B is actually -EV on its own (it might look +EV at first glance). But I'd much rather this be a discussion than a monologue, so I'll leave it there for now. Would love to hear any thoughts folks have on this.

"Not only [does] God play dice... he sometimes confuses us by throwing them where they can't be seen." ~ Stephen Hawking

November 11th, 2019 at 3:34:32 AM
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I never found Parrondo's Paradox very interesting. It is actually three games, one of which is positive EV. It gets played often enough to make the overall average positive.

It's not whether you win or lose; it's whether or not you had a good bet.

November 11th, 2019 at 4:37:04 AM
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a fair coin has a slight bias towards its initially upwards face.

a die can be milled to a biased dimension but is almost impossible to gain financially from it.

THREE games????

a die can be milled to a biased dimension but is almost impossible to gain financially from it.

THREE games????

November 11th, 2019 at 6:40:58 AM
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It seems that one of the games must have a periodic oscillatory nature such that it swings from positive EV to negative EV due to some predictable underlying reason. And, the player simply plays this game when it is positive EV and 'Wongs out" to the first game when the second game becomes wildly negative in its expected value.

Okay, I understand this. Interesting, but not particularly profound in my opinion.

Okay, I understand this. Interesting, but not particularly profound in my opinion.

So many better men, a few of them friends, were dead. And a thousand thousand slimy things lived on, and so did I.

November 11th, 2019 at 8:38:14 AM
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There was a fiction book written with this as the main plot point. I forget the title, but it's about a guy who finds a way to apply Parrondo's paradox to playing craps, and he becomes a millionaire gambling in casinos.

November 11th, 2019 at 9:41:26 AM
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Fwiw: Ask the Wizard 149 starts with a discussion of parrondo's paradox.

https://wizardofodds.com/ask-the-wizard/149/

https://wizardofodds.com/ask-the-wizard/149/

If the House lost every hand, they wouldn't deal the game.

November 11th, 2019 at 10:41:12 AM
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As you said, FICTION.Quote:TigerWu, and he becomes a millionaire gambling in casinos.

November 11th, 2019 at 10:42:57 AM
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Quote:TigerWuThere was a fiction book written with this as the main plot point. I forget the title, but it's about a guy who finds a way to apply Parrondo's paradox to playing craps, and he becomes a millionaire gambling in casinos.

I read that and am struggling to remember the name. Pretty good read. However, unless we're talking about different books, that wasn't exactly the plot. As I recall I guy buys a #@$% betting system referring to Parrondo somehow, actually wins with it on luck alone, some goons mixed up in organized crime get interested, and that's as far as I'll go without ruining it too much.

It's not whether you win or lose; it's whether or not you had a good bet.

November 11th, 2019 at 10:51:37 AM
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Quote:WizardI read that and am struggling to remember the name. Pretty good read. However, unless we're talking about different books, that wasn't exactly the plot. As I recall I guy buys a #@$% betting system referring to Parrondo somehow, actually wins with it on luck alone, some goons mixed up in organized crime get interested, and that's as far as I'll go without ruining it too much.

We're probably talking about the same book; it's been about 15 years since I've read it. I figured out it's called "God Doesn't Shoot Craps."