The paradox of Parrondo's games

  title={The paradox of Parrondo's games},
  author={Gregory Peter Harmer and Derek Abbott and Peter G. Taylor},
  journal={Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences},
  pages={247 - 259}
  • G. Harmer, D. Abbott, P. Taylor
  • Published 8 February 2000
  • Economics
  • Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
We introduce Parrondo's paradox that involves games of chance. We consider two fair games, A and B, both of which can be made to lose by changing a biasing parameter. An apparently paradoxical situation arises when the two games are played in any alternating order. A winning expectation is produced, even though both games A and B are losing when we play them individually. We develop an explanation of the phenomenon in terms of a Brownian ratchet model, and also develop a mathematical analysis… 

Figures and Tables from this paper

Inspired by the flashing Brownian ratchet, Parrondo's games present an apparently paradoxical situation. The games can be realized as coin tossing events. Game A uses a single biased coin while game
Parrondo's Paradox is Not Paradoxical
Parrondo's paradox concerns two games that are played in an alternating sequence. An analysis of each game in isolation shows them both to be losing games (i.e., to have a negative expectation).
Mathematical background of Parrondo's paradox
  • E. Behrends
  • Economics
    SPIE International Symposium on Fluctuations and Noise
  • 2004
Parrondo's paradox states that there are losing gambling games which, when being combined stochastically or in a suitable deterministic way, give rise to winning games. Here we investigate the
Brownian ratchets and Parrondo's games.
The combination of Parrondo's games can be therefore considered as a discrete-time Brownian ratchet, and the parameter space in which the paradoxical effect occurs is found and the winning rate analysis is carried out.
How strong can the Parrondo effect be?
It is shown that if the parameters of the games are allowed to be arbitrary, subject to a fairness constraint, and if the two games A and B are played in an arbitrary periodic sequence, then the rate of profit can not only be positive, but can also be arbitrarily close to 1 (i.e. 100%).
Two Issues Surrounding Parrondo’s Paradox
In the original version of Parrondo’s paradox, two losing sequences of games of chance are combined to form a winning sequence. The games in the first sequence depend on a single parameter p, while
Developments in Parrondo’s Paradox
Parrondo’s paradox is the well-known counterintuitive situation where individually losing strategies or deleterious effects can combine to win. In 1996, Parrondo’s games were devised illustrating
On Parrondo's paradox: how to construct unfair games by composing fair games
Abstract We construct games of chance from simpler games of chance. We show that it may happen that the simpler games of chance are fair or unfavourable to a player and yet the new combined game is
Discrete–time ratchets, the Fokker–Planck equation and Parrondo's paradox
Parrond's games manifest the apparent paradox where losing strategies can be combined to win and have generated significant multidisciplinary interest in the literature. Here we review two recent


Parrondo's paradox
We introduce Parrondo’s paradox that involves games of chance. We consider two fair gambling games, A and B, both of which can be made to have a losing expectation by changing a biasing parameter « .
The problem of detailed balance for the Feynman-Smoluchowski Engine (FSE) and the Multiple Pawl Paradox
The Feynman-Smoluchowski Engine (FSE) is simply a ratchet and pawl device, connected by a shaft to a vane, that is small enough to rectify the effect of random bombardment of gas molecules on the
Criticism of Feynman’s analysis of the ratchet as an engine
The well‐known discussion on an engine consisting of a ratchet and a pawl in [R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison‐Wesley, Reading, MA, 1963), Vol. 1,
Brownian ratchets in physics and biology
Thirty years ago Feynman et al. presented a paradox in the Lectures on Physics: an imagined device could let Brownian motion do work by allowing it in one direction and blocking it in the opposite
Forced thermal ratchets.
  • Magnasco
  • Physics
    Physical review letters
  • 1993
We consider a Brownian particle in a periodic potential under heavy damping. The second law forbids it from displaying any net drift speed, even if the symmetry of the potential is broken. But if the
Brownian rectifiers: How to convert brownian motion into directed transport
It is generally appreciated that in accordance with the second law of thermodynamics usable work cannot be extracted if only equilibrium fluctuations act. The physical mechanism that prevents to
A First Course on Stochastic Processes
The Basic Limit Theorem of Markov Chains and Applications and Classical Examples of Continuous Time Markov chains are presented.
A motor protein model and how it relates to stochastic resonance, feynman’s ratchet, and maxwell’s demon
A motor protein turns chemical energy into motion, but it differs from an ordinary engine in that random Brownian kicks become important. Below we propose a description where the energy input is used
Fluctuation driven ratchets: Molecular motors.
Even if the net force is always zero, flow is induced by a fluctuation of the energy barrier, but only at flipping times roughly in between the adiabatic adjustment times on the left and right of the barrier.