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Game theory is an important branch of maths strongly linked to gambling. Mathematicians say that a “losing game” is one in which you tend to loose in average. All the games played in the casino 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 inspired by the ideas originally presented at a conference in 1996 by the Spanish physicist J.M. Parrondo. 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. In a fair coin, heads come up with a probability of 0.5 (that is, a 50%), and tails comes 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 the Game B. This time the player gambles on a slightly 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 in Game B there is also the Coin 2, which is an awful unfair coin with a probability for you to win of only 0.095 (less than a one-tenth 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 the original Parrondo's 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 the Coin 2 is really bad (while the 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). Definitely Game B is 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, 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 the Parrondo's paradox. Surprisingly playing two bad games alternatively results in a good thing! What is more, the Parrondo's paradox is also true when the games are switched at random! You would become multimillionaire with that simply strategy.

 

REFERENCES:

https://en.wikipedia.org/wiki/Parrondo%27s_paradox

Harmer, G. P. & Abbott, D. Game theory: Losing strategies can win by Parrondo's paradox. Nature 402, 864 1999

http://www.nature.com/news/1999/991223/full/news991223-13.html

http://elpais.com/diario/2000/01/05/sociedad/947026815_850215.html

M. Feito. PhD thesis: http://eprints.ucm.es/10680/1/T31799.pdf

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