Parrondo's paradox: A losing strategy that wins

There is a famous Chinese story, perhaps it's related so I'd share.<p>The King of Qi[1] held many horse race with his general TianJi, there are three rounds, but every round the King's horse is better than Tianji's. So the general loses everytime.<p>One of general's stratagy advisor[2] came up with an idea: Use general's worst horse to race's King's best horse and lose for round one, then use the best horse to beat King's average horse in round two, next use the average horse to beat King's worst horse in the final round. So at last the general wins.<p>[1]: <a href="http://en.wikipedia.org/wiki/Qi_(state)" rel="nofollow">http://en.wikipedia.org/wiki/Qi_(state)</a><p>[2]: <a href="http://en.wikipedia.org/wiki/Sun_Bin" rel="nofollow">http://en.wikipedia.org/wiki/Sun_Bin</a>

This is a very interesting phenomenon. I think one application of it may be in no-limit poker.<p>You have two categories of hands: rags and monsters. If you just wait for monsters and don't play rags, you won't likely get paid off against strong opponents. Instead, many successful players use a strategy where they play a lot of hands, hoping to hit with rags one hand (negative EV) and catching a monster shortly after to get paid off huge because their opponents don't give them sufficient credit for a strong hand (due to their recent loose play with rags).

I found the linked paradoxes on the Wikipedia article more interesting:<p><a href="http://en.wikipedia.org/wiki/Simpson%27s_paradox" rel="nofollow">http://en.wikipedia.org/wiki/Simpson%27s_paradox</a><p>and<p><a href="http://en.wikipedia.org/wiki/Braess_paradox" rel="nofollow">http://en.wikipedia.org/wiki/Braess_paradox</a><p>And the discussion on whether it can rightfully be called a paradox:<p>"it was debated whether the word 'paradox' is an appropriate description given that the Parrondo effect can be understood in mathematical terms"<p>the funny counterpoint:<p>"Is Parrondo's paradox really a "paradox"? This question is sometimes asked by mathematicians, whereas physicists usually don't worry about such things."<p>This is a case of <a href="http://xkcd.com/214/" rel="nofollow">http://xkcd.com/214/</a> :)

This reminds me of "doubling" in Roulette. If you bet $1 and win you profit $1. If you lose, bet $2. If you win you win 2 - 1, if you lose you bet 4 and perhaps win 4 - 2 - 1 = $1. With an unlimited supply of capital you will earn positive returns for any reasonably positive probability of winning even if significantly less than 50%. Similar to the Parrondo's example the result requires bets that depend on capital which depend on previous iterations of the game and so the different games are not truly independent.

This requires a <i>very</i> unnatural set-up (a game that has an average payoff that depends on your capital modulo some number).