4.5. Parrondo's paradox: Two bad things becoming a good one


Game theory is an important branch of Mathematics strongly linked to gambling. Mathematicians say that a “losing game” is one in which you tend to lose on average. All the games played in casinos are losing games (otherwise it wouldn't be a business for the house). But sometimes the unexpected happens: Two different losing games played back and forth can surprisingly result in an averaged positive outcome!

This counterintuitive result became popular after a paper published in the journal Nature by G.P Harmer and D. Abbott. The authors were inspired by the ideas originally presented by the Spanish physicist J.M. Parrondo in a conference in 1996. His seminal idea was the translation of a well-known problem in Physics —the induction of directed motion through the rectification of stochastic thermal fluctuations, so-called Brownian motors or ratchets (err... what?)— into the language of game theory.

All right, let's define the two losing games, say Game A and Game B.

First, Game A. Consider tossing a coin. Getting 'heads' you win 1 euro, getting 'tails' you lose 1 euro. With a fair coin, heads come up with a probability of 0.5 (that is, 50%), and tails come up with the same probability. Statistically you can play that game with no significant profit or loss of money in the long run. But now, suppose that the coin is designed to fall with slightly greater probability as 'tails': for instance, 0.505. With this biased coin you will lose in average. In fact, the more you play, the more money you lose. You'll finally be penniless. Well, this is our losing Game A.

Now let's define Game B. This time the player gambles on a slightly more complex game involving two coins (Coin '1' and Coin '2'). Coin 1 is biased so it gives you odds of winning at 0.745 (that means your chance of winning is almost seventy-five percent). Well, this is in fact a really good coin for you... but all that glitters is not gold because there is also Coin 2 in Game B, which is an awfully unfair coin with only a probability of 0,095 for you to win (less than a onetenth chance of winning). When do we have to toss the good Coin 1 and when do we have to toss the bad Coin 2? In Parrondo's original game —there are many variants— Game B depends on the capital. If your total money is a multiple of three (you know, 0€, 3€, 6€, 9€, 12€...) you have to flip Coin 1; if it is not, you are forced to toss the bad Coin 2. Not only is Coin 2 really bad (while Coin 1 is only slightly good), but also you would play against bad Coin 2 more times (because there are two times as many non-multiples of three than there are multiples of three). Game B is definitely a losing game!

Note that the apparently weird values for the probabilities given here (0.505, 0.745, 0.095) can be written in terms of a parameter ε=0.005 in a clearer way as 1/2+ε (for Game A), 3/4-ε (for Coin 1 of Game B), and 1/10-ε (for Coin 2 of Game B). Anyway, these particular values can be changed giving similar results, so don't worry about the specific numbers.

Although both Games A and B are losing games, it turns out that playing two rounds of Game A followed by two of Game B (and so on) actually produces a steadily increasing capital. That's Parrondo's paradox. Surprisingly, playing two bad games alternatively results in a good thing! Furthermore, the Parrondo's paradox is also true when the games are switched at random! You would become multimillionaire with that simple strategy.