Extended Parrondo's game and Brownian ratchets: Strong and weak Parrondo effect

Abstract

Inspired by the flashing ratchet, Parrondo's game presents an apparently paradoxical situation. Parrondo's game consists of two individual games, game A and game B. Game A is a slightly losing coin-tossing game. Game B has two coins, with an integer parameter M. If the current cumulative capital (in discrete unit) is a multiple of M, an unfavorable coin p(b) is used, otherwise a favorable p(g) coin is used. Paradoxically, a combination of game A and game B could lead to a winning game, which is the Parrondo effect. We extend the original Parrondo's game to include the possibility of M being either M(1) or M(2). Also, we distinguish between strong Parrondo effect, i.e., two losing games combine to form a winning game, and weak Parrondo effect, i.e., two games combine to form a better-performing game. We find that when M(2) is not a multiple of M(1), the combination of B(M(1)) and B(M(2)) has strong and weak Parrondo effect for some subsets in the parameter space (p(b),p(g)), while there is neither strong nor weak effect when M(2) is a multiple of M(1). Furthermore, when M(2) is not a multiple of M(1), a stochastic mixture of game A may cancel the strong and weak Parrondo effect. Following a discretization scheme in the literature of Parrondo's game, we establish a link between our extended Parrondo's game with the analysis of discrete Brownian ratchet. We find a relation between the Parrondo effect of our extended model to the macroscopic bias in a discrete ratchet. The slope of a ratchet potential can be mapped to the fair game condition in the extended model, so that under some conditions, the macroscopic bias in a discrete ratchet can provide a good predictor for the game performance of the extended model. On the other hand, our extended model suggests a design of a ratchet in which the potential is a mixture of two periodic potentials.

Winning in sequential Parrondo games by players with short-term memory

“…The above analysis for the fair game boundary in figure 4 allows one to conclude that there exists region in the parameter space (p = 0.49, p b , p g ) where a CD game player can win using α = 0.5, but he can lose using some other values of α. This observation corresponds to the definition of strong Parrondo effect [14] where the combination of two losing game leads to a winning one. The fair game boundary is defined by the condition that the region at the right hand side of the fair game boundary corresponds to the winning region of parameter space (p = 0.49, p g , p b ) when the CD game is played with the associated switching parameter α, while the game is losing in the long run for parameters in the region left of the boundary.…”

“…Chaotic game sequences have also been investigated by Tang et al [11]. Apart from the original Parrondo game, other types of Parrondo games have been constructed such as the Quantum Parrondo Game [12], history-dependent Parrondo Game [13] and Extended Parrondo Game that incorporate different B games without the diffusive A game [14]. Here we address the benefits for player with finite memory in various switching strategies when playing the original Parrondo Game.…”

Paradoxical Survival: Examining the Parrondo Effect across Biology

“…For this reason, the model was termed doubly anomalous. The weak Parrondo effect can also occur in borderline cases, where individual pure games are fair or winning, but mixed games yield enhanced outcomes nevertheless. Exploring the balance between individualistic and social behavior indicates, surprisingly, that fast‐growing inequality can almost be completely eliminated with appropriate tuning, albeit at the cost of shrinking paradoxical growth regimes in the explored parameter space (Figure c).…”

Weak stochastic ratchets and dynamic localization in measurement-induced quantum trajectories

“…Brownian ratchets are deeply connected to so called Parrondo games, when two or more lossy games are combined to give a winning one [30,34,35]. Very recently, the notion of weak Parrondo games and weak Brownian ratchets were introduced in [30] to describe the situation when two or more lossy games are played together to give just less lossy (but not winning) one. We remark that, although both Parrondo games and Brownian ratchets were considered in context of quantum systems [28,[36][37][38][39], this was up to now done via some direct action on the system itself.…”