Parrondo’s paradox

Parrondo’s paradox, a paradox in game theory, has been described as: A combination of losing strategies becomes a winning strategy. It is named after its creator, Juan Parrondo, who discovered the paradox in 1996. The paradox states that:

“There exist pairs of games, each with a higher probability of losing than winning, for which it is possible to construct a winning strategy by playing the games alternately.”

Parrondo devised the paradox in connection with his analysis of the Brownian ratchet, a thought experiment about a machine that can purportedly extract energy from random heat motions popularized by physicist Richard Feynman. However, the paradox disappears when rigorously analyzed.

There is a discussion about the use of the word paradox as the text by Dereck Aboot:

Is Parrondo’s paradox really a “paradox”? This question is sometimes asked by mathematicians, whereas physicists usually don’t worry about such things. The first thing to point out is that “Parrondo’s paradox” is just a name, just like the “Braess’ paradox” or “Simpson’s paradox.” Secondly, as is the case with most of these named paradoxes they are all really apparent paradoxes. People drop the word “apparent” in these cases as it is a mouthful, and it is obvious anyway. So no one claims these are paradoxes in the strict sense. In the wide sense, a paradox is simply something that is counterintuitive. Parrondo’s games certainly are counterintuitive—at least until you have intensively studied them for a few months. The truth is we still keep finding new surprising things to delight us, as we research these games. I have had one mathematician complain that the games always were obvious to him and hence we should not use the word “paradox.” He is either a genius or never really understood it in the first place. In either case, it is not worth arguing with people like that.

Parrondo’s paradox is used extensively in game theory, and its application in engineering, population dynamics, financial risk, among others. Parrondo’s games are of little practical use such as for investing in stock markets as the original games require the payoff from at least one of the interacting games to depend on the player’s capital. However, the games need not be restricted to their original form and work continues in generalizing the phenomenon. Similarities to volatility pumping and the two-envelope problem have been pointed out. Simple finance textbook models of security returns have been used to prove that individual investments with negative median long-term returns may be easily combined into diversified portfolios with positive median long-term returns. Similarly, a model that is often used to illustrate optimal betting rules has been used to prove that splitting bets between multiple games can turn a negative median long-term return into a positive one.

See also

Game Theory, Portfolio theory, Stock market

Material

• https://scholar.google.com.au/citations?hl=en&user=aeNdbrUAAAAJ
• http://www.eleceng.adelaide.edu.au/Groups/parrondo/
• http://www.cut-the-knot.org/ctk/Parrondo.shtml
• Parrondo’s paradox and epistemology
• Parrondo effect in quantum mechanics
• Financial diversification and Parrondo
• http://mathworld.wolfram.com/ParrondosParadox.html
• Ball, Philip Good news for losers. Nature news

Papers

• Harmer, G. P., & Abbott, D. (1999). Parrondo’s paradox. Statistical Science, 206-213.
• Harmer, G. P., & Abbott, D. (1999). Game theory: Losing strategies can win by Parrondo’s paradox. Nature, 402(6764), 864-864.
• Harmer, G. P., & Abbott, D. (2002). A review of Parrondo’s paradox. Fluctuation and Noise Letters, 2(02), R71-R107.
• Amengual, P., Allison, A., Toral, R., & Abbott, D. (2004, August). Discrete-time ratchets, the Fokker–Planck equation and Parrondo’s paradox. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (Vol. 460, No. 2048, pp. 2269-2284). The Royal Society.
• Abbott, D. (2010). Asymmetry and disorder: A decade of Parrondo’s paradox. Fluctuation and Noise Letters, 9(01), 129-156.
• Shu, J. J., & Wang, Q. W. (2014). Beyond Parrondo’s paradox. Scientific Reports 4 (4244): 1-9
• Toral, R. (2002). Capital redistribution brings wealth by Parrondo’s paradox. Fluctuation and Noise Letters, 2(04), L305-L311.
• Flitney, A. P., Ng, J., & Abbott, D. (2002). Quantum Parrondo’s games. Physica A: Statistical Mechanics and its Applications, 314(1), 35-42.
• Iyengar, R., & Kohli, R. (2003). Why Parrondo’s paradox is irrelevant for utility theory, stock buying, and the emergence of life. Complexity, 9(1), 23-27.
• Thomas K. Philips and Andrew B. Feldman, Parrondo’s Paradox is not Paradoxical, Social Science Research Network (SSRN) Working Papers, August 2004
• Stutzer, M. (2010). The Paradox of Diversification. The Journal of Investing, 19(1), 32-35.

Books

• Paulos, J. A. (2007). A mathematician plays the stock market. Basic Books.