Parrondo's Paradox and Poker

Parrondo’s paradox is the well-known counterintuitive situation where individually losing strategies or deleterious effects can combine to win…. Over the past ten years, a number of authors have pointed to the generality of Parrondian behavior, and many examples ranging from physics to population genetics have been reported. In its most general form, Parrondo’s paradox can occur where there is a nonlinear interaction of random behavior with an asymmetry, and can be mathematically understood in terms of a convex linear combination.

From Developments in Parrondo’s Paradox (Derek Abbott)

One of my new favorite pastimes is identifying real world scenarios that I think are examples of Parrondo’s Paradox (PP). Here are some from the world of poker:

  • Morton’s Theorem describes situations during the play of hands wherein you may employ a strategy that is losing against two individual opponents, but against both simultaneously it’s a winning strategy.
  • There is a related scenario that exists in tournament poker known as implicit collusion, which is a correct strategy when there is a big jump in payouts from one place to the next.
  • There are a host of small theoretical advantages to having a short stack (fewer chips than your opponents) in the modern table-stakes form of poker where players can be “all-in” and not risk more than they have in front of them. The advantage comes from playing against two larger-stacked opponents in a hand where you could be forced to fold a hand if you had more chips to risk, but because you are all-in you get have a free draw to a possibly winning hand.
  • In the rec.gambling newsgroup in the mid 90s it was discovered that you could “lose by winning” (a reverse PP). This relates to an age-old debate about whether money-management was important for advantage gamblers (like good poker players), who in the interest of “maximizing returns” have a tendency to play as big as they can as long as they think they have an edge. Those that understood the issue realized that if you go broke at any point, you take yourself out of the game and can’t realize your (now purely theoretical) advantage. Thus the well-known Kelly criterion for maximizing log(returns) is also the right strategy for maximizing absolute returns in most practical applications. This is due to the facts that: (1) there is a practical minimum bet size (i.e. you can’t bet arbitrarily small amounts and if your bankroll drops below the minimum you are effectively broke); (2) bankrolls are always finite (but if you had an infinite bankroll, the PP would not exist). Note that the whole confusion and debate might have been obviated if it weren’t for the fact that PP wasn’t really studied as such or well known until 1996 (two years after that rec.gambling thread).

So, what are your favorite examples of Parrondo’s Paradox?

Hat tip to Alex Ryan, and thanks to Derek Abbott for discussion and allowing me to post his paper.

  • John Wallace

    Hey Rafe:

    “One of my new favorite pastimes…”.

    Man, good to see you’re still a geek!


  • Are you suggesting that two g33ks make a l33t? Very Parrondian of you!

  • John Wallace

    lol. Miss seeing you, Rafe! Shla sends her love. We’ll be in your neck of the woods during July, but something tells me you won’t be home. :(

    GL at the WSOP this year!

  • John Wallace

    BTW, just finished Phil’s Little Blue Book on poker. Good read! Now if I can ever find you on when I’m around, I can take you down in a HU game.

    I pity the fool!


  • John Wallace

    lol. Miss seeing you, Rafe! Shla sends her love. We'll be in your neck of the woods during July, but something tells me you won't be home. :(

    GL at the WSOP this year!

  • Anonymous

    Does that mean If I play Phil Helmuth and Chris Fergusson individually I will lose, but if I play them together I win, cool.

    • Rafe Furst

      You do have a much better chance, yes. In a cash game you have “parrondian arbitrage” opportunities. In a tournament your supposition still holds, but you need to compare beating both of them in a single freezeout vs playing them back-to-back heads up.