Parrondo paradox

The Parrondo paradox is a paradox of game theory , which says that it is possible in certain cases to make up two games to secure (long term) loss a game with a sure profit, when played alternately.

The Spanish physicist Juan Manuel Rodriguez Parrondo (* 1964) described the paradox in 1996 in the context of Brownian ratchets and Maxwell's demon . It was analyzed by biomedical engineer Derek Abbott in 1999, who named it after Parrondo.

It has since been used to explain a wide variety of phenomena, for example in evolutionary and population biology and in financial investments. For example, it was attempted to explain why living things alternate between single-cell and multicellular formulas or why animal populations alternate between colonies and migration of individuals.

example

The standard example consists of two games A, B of tossing coins. The loss and profit are the same amount (one unit).

Game A: tossing a coin with a probability of winning (with a small ). ${\ displaystyle P = 0 {,} 5- \ epsilon}$ ${\ displaystyle \ epsilon> 0}$
Game B: You toss two coins, depending on the remaining equity. If the capital is a multiple of an integer , you toss a bad coin (probability of winning ), if not a good coin (probability of winning ). ${\ displaystyle M}$ ${\ displaystyle 0 {,} 1- \ epsilon}$ ${\ displaystyle 0 {,} 75- \ epsilon}$

In the long run, every game by itself is certain to result in a loss (with a suitable choice of probabilities and of ). In their original 1999 paper, Harmer and Abbott used and and demonstrated the loss property by simulation. If, on the other hand, you play the two games alternately in the order AABBAABB…, you get a winning strategy (shown by Abbott, Harmer with the theory of the Markov chains ). However, that depends a lot on the order (ABABAB ... is a losing game). The best strategy depends on. Stringing AB together is best for and , ABBAB is best for . ${\ displaystyle M}$ ${\ displaystyle M = 3}$ ${\ displaystyle \ epsilon = 0 {,} 005}$ ${\ displaystyle M}$ ${\ displaystyle M = 2}$ ${\ displaystyle M = 4}$ ${\ displaystyle M = 3}$

Others

According to Abbott, it probably cannot be applied to casino games, since the assumption is convex linear combinations in a nonlinear parameter space, but linear parameter spaces are so far known for casino games. An article in the New York Times quoted Abbot's view that the phenomenon of increased popularity for US President Bill Clinton during the Lewinsky affair after he no longer denied his sexual relationship with intern Monica Lewinsky could be in the context of the Parrondo Read paradoxes.

While searching for practical applications for Parrondo reactions in the analysis of investment strategies, the physicist Sergei Maslow found indications that loss-making stocks could be combined into a profit portfolio through a suitable rattle effect.

literature

Shu, Wang, Beyond Parrondos Paradox, Scientific Reports, Volume 4, 2019, pp. 1-9, Arxiv

Web links

Abbott's website on Parrondo's Paradox , University of Adelaide, web archive
Alex Bogomolny, Parrondo Paradox, Cut the Knot
Parrondo's Paradox at Mathworld

Individual evidence

^ As a critique of Richard Feynman's treatment of this phenomenon

↑ GP Harmer, D. Abbott: Loosing strategies can win by Parrondo's paradox, Nature, Volume 402, 1999, p. 864.

↑ Harmer, Abbott, Parrondo's paradox, Statistical Science, Volume 14, 1999, pp. 206-213. Project Euclid

↑ Hang Hao Cheong, Jin Ming Koh, Michael C. Jones, Multicellular survival as a consequence of Parrondo's paradox, Proc. Nat. Acad. USA, Vol. 115, 2018, pp. E5258-E5259, PMID 29752380

↑ Zong Xuan Tan, Kang Hao Cheong, Nomadic-colonial life strategies enable paradoxical survival and growth despite habitat destruction, eLife. 6, 2017, e21673. PMID 28084993

↑ Christian Hesse : Attention : Think Traps, The Most Amazing Everyday Mistakes and How to See Through Them . Beck, Munich 2011, ISBN 978-3-406-62204-5 . P. 123.

↑ Christian Hesse: Attention : Think Traps, The Most Amazing Everyday Mistakes and How to See Through Them . Beck, Munich 2011, p. 124.

[1] As a critique of Richard Feynman's treatment of this phenomenon

[2] GP Harmer, D. Abbott: Loosing strategies can win by Parrondo's paradox, Nature, Volume 402, 1999, p. 864.

[3] Harmer, Abbott, Parrondo's paradox, Statistical Science, Volume 14, 1999, pp. 206-213. Project Euclid

[4] Hang Hao Cheong, Jin Ming Koh, Michael C. Jones, Multicellular survival as a consequence of Parrondo's paradox, Proc. Nat. Acad. USA, Vol. 115, 2018, pp. E5258-E5259, PMID 29752380

[5] Zong Xuan Tan, Kang Hao Cheong, Nomadic-colonial life strategies enable paradoxical survival and growth despite habitat destruction, eLife. 6, 2017, e21673. PMID 28084993

[6] Christian Hesse : Attention : Think Traps, The Most Amazing Everyday Mistakes and How to See Through Them . Beck, Munich 2011, ISBN 978-3-406-62204-5 . P. 123.

[7] Christian Hesse: Attention : Think Traps, The Most Amazing Everyday Mistakes and How to See Through Them . Beck, Munich 2011, p. 124.