Adventures in the Middle of the Road

Imagine you are in a casino designed by a depraved psychologist. There are no slot machines, no card games. The only tokens are your own dollar bills. There are only two games.

The first, Game A, is simple: You pay the casino $1, and you always lose. It's sort of like playing the lottery.

The second, Game B, at least gives you a chance to win, but is more expensive and is still stacked in the house's favor. You count your money. If the total is an even amount, you win $3, Otherwise you pay $5.

You must play 100 rounds before the casino will let you leave. Which game would you choose?

It's clear that neither game gives you a chance to keep any of your money. In Game A, you lose $1 every round. In Game B, you lose $2 every two rounds.

Round | A | B |

1 | $99 | $103 |

2 | $98 | $98 |

3 | $97 | $101 |

4 | $96 | $96 |

5 | $95 | $99 |

6 | $94 | $94 |

7 | $93 | $97 |

8 | $92 | $92 |

9 | $91 | $95 |

10 | $90 | $90 |

Either way, by round 10 you've lost $10, and by round 100 you will have lost all $100.

But there's a loophole. The psychologist said you had to play 100 rounds, but did not specify 100 rounds of the *same game*. Given the choice between two losing games, can you devise a strategy where you win money?

Suppose you were to alternate, playing Game A then Game B.

Round | Money |

1 - A | $99 |

2 - B | $94 |

3 - A | $93 |

4 - B | $88 |

5 - A | $87 |

6 - B | $82 |

7 - A | $81 |

8 - B | $76 |

9 - A | $75 |

10 - B | $70 |

This is not the answer. After you lose $1 in Game A, you always have an odd amount for Game B. With this strategy, you actually lose money three times as fast. You'll need to bring an additional $200.

But—you're probably already guessed this—everything reverses if you start with Game B.

Round | Money |

1 - B | $103 |

2 - A | $102 |

3 - B | $105 |

4 - A | $104 |

5 - B | $107 |

6 - A | $106 |

7 - B | $109 |

8 - A | $108 |

9 - B | $111 |

10 - A | $110 |

A more complex form of this scenario was put forth in 1996 by Juan M.R. Parrondo. The idea that two losing strategies can be combined to make a winning strategy has come to be known as Parrondo's paradox.

The paradox has real-world applications. Javier Buceta discovered viruses that adopt strategies enabling them to survive in an environment with wide temperature swings. A strategy called volatility pumping enables stock owners to make money on losing stocks (although transaction costs would likely eat up all the profits). And Adam Clare suggests ways players could use Parrondo's paradox to improve their survival chances in an MMO game.

By mixing and matching strategies to fit the situation, it's possible to create a strategy better than either of the two alternatives. But that shouldn't be a surprise.

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