

Parrondo's paradox: A losing strategy that wins - thisisnotmyname
http://en.wikipedia.org/wiki/Parrondo%27s_paradox

======
est
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>

~~~
khafra
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.

------
tansey
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).

------
GFischer
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/> :)

------
danteembermage
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.

~~~
BrandonM
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.

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

~~~
tel
_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.

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

~~~
tel
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.

