Parrondo's paradox: A losing strategy that wins | Hacker News Reader

Parrondo's paradox: A losing strategy that wins

en.wikipedia.org

There is a famous Chinese story, perhaps it's related so I'd share.

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.

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.

[1]: http://en.wikipedia.org/wiki/Qi_(state)

[2]: http://en.wikipedia.org/wiki/Sun_Bin

Sounds like nontransitive dice: http://en.wikipedia.org/wiki/Nontransitive_dice

Still a helluva lot easier to understand than Parrondo, here; at least for me.

I'm told that chess teams will sometimes do this. Usually, the nth best player on each team play against each other. The same strategy you describe can be used to increase the team's chance of winning the match. I don't know if it's considered unsportsmanlike.

It's also an excellent parable about defeating a superior enemy in detail.

This is a very interesting phenomenon. I think one application of it may be in no-limit poker.

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:

http://en.wikipedia.org/wiki/Simpson%27s_paradox

and

http://en.wikipedia.org/wiki/Braess_paradox

And the discussion on whether it can rightfully be called a paradox:

"it was debated whether the word 'paradox' is an appropriate description given that the Parrondo effect can be understood in mathematical terms"

the funny counterpoint:

"Is Parrondo's paradox really a "paradox"? This question is sometimes asked by mathematicians, whereas physicists usually don't worry about such things."

This is a case of http://xkcd.com/214/ :)

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.

That's the Martingale System (http://en.wikipedia.org/wiki/Martingale_system), and it fails because no one has an unlimited supply of capital. Lose 20 times in a row and you've lost $2,097,151 since your last win. And you will eventually lose 20 times in a row. Since you're only winning $1 every time you win, you'll have to play many times to win any significant amount of money.

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

With this understanding, the paradox resolves itself: The individual games are losing only under a distribution that differs from that which is actually encountered when playing the compound game.

You don't need a modulo-style set up. You just need a dependence of the two games such that a player can predict when to play each game to always be optimal. The ratchet is just one example where the player always avoids the really bad odds in game B that help to make it a losing game.

It's exactly the same thing as Simpson's Paradox.

Sure, the particular distributions used are a detail; but it still needs a very odd setup, right?

Not really. I'd say the necessary elements are:

  + A choice between two "games", each losing on average.
  + At least one game has periods of local payoff great enough to overwhelm the long-run losses in the other game
  + Some flow of information between the games so that playing one game will help you predict the payoff periods of the other game.

The example that flies to mind is investment. Game A is to lose value of money you hold on to via inflation. Game B is the generally losing game of day trading. The net game can be profitable as long as you spend time in A learning to accurately predict game B's upswings.

Of course, the real information flow from game B to game A is already heavily capitalized making the net game even more difficult.

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