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Discrete–time ratchets, the FokkerPlanck equation and Parrondo’s paradox
 Accepted in Proc. R. Soc. London A. Proc. of SPIE
, 2004
"... Parrondo’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 approaches, based on the FokkerPlanck equation, that rigorously establish the connection between Parro ..."
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Cited by 4 (4 self)
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Parrondo’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 approaches, based on the FokkerPlanck equation, that rigorously establish the connection between Parrondo’s games and a physical model known as the flashing Brownian ratchet. This gives rise to a new set of Parrondo’s games, of which the original games are a special case. For the first time, we perform a complete analysis of the new games via a discretetime Markov chain (DTMC) analysis, producing winning rate equations and an exploration of the parameter space where the paradoxical behaviour occurs. Keywords: Parrondo’s paradox; FokkerPlanck equation; Brownian ratchet. 1.
Exact ratchet description of Parrondo’s games with selftransitions
"... We extend a recently developed relation between the master equation describing the Parrondo’s games and the formalism of the Fokker–Planck equation to the case in which the games are modified with the introduction of “self–transition probabilities”. This accounts for the possibility that the capital ..."
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Cited by 1 (1 self)
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We extend a recently developed relation between the master equation describing the Parrondo’s games and the formalism of the Fokker–Planck equation to the case in which the games are modified with the introduction of “self–transition probabilities”. This accounts for the possibility that the capital can neither increase nor decrease during a game. Using this exact relation, we obtain expressions for the stationary probability and current (games gain) in terms of an effective potential. We also demonstrate that the expressions obtained are nothing but a discretised version of the equivalent expressions in terms of the solution of the Fokker–Planck equation with multiplicative noise. Keywords: Master and Fokker–Planck equations, Parrondo’s games, multiplicative noise 1.
Parrondo’s paradox via redistribution of wealth ∗
"... In Toral’s games, at each turn one member of an ensemble of N ≥ 2 players is selected at random to play. He plays either game A ′ , which involves transferring one unit of capital to a second randomly chosen player, or game B, which is an asymmetric game of chance whose rules depend on the player’s ..."
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In Toral’s games, at each turn one member of an ensemble of N ≥ 2 players is selected at random to play. He plays either game A ′ , which involves transferring one unit of capital to a second randomly chosen player, or game B, which is an asymmetric game of chance whose rules depend on the player’s current capital, and which is fair or losing. Game A ′ is fair (with respect to the ensemble’s total profit), so the Parrondo effect is said to be present if the random mixture γA ′ +(1−γ)B (i.e., play game A ′ with probability γ and play game B otherwise) is winning. Toral demonstrated the Parrondo effect for γ = 1/2 using computer simulation. We prove it, establishing a strong law of large numbers and a central limit theorem for the sequence of profits of the ensemble of players for each γ ∈ (0, 1). We do the same for the nonrandom pattern of games (A ′ ) r B s for all integers r, s ≥ 1. An unexpected relationship between the randommixture case and the nonrandompattern case occurs in the limit as N → ∞. Keywords: Parrondo’s capitaldependent games; Markov chain; stationary distribution; fundamental matrix; strong law of large numbers; central limit theorem.
Reversals of chance in paradoxical games
, 2006
"... We present two collective games with new paradoxical features when they are combined. Besides reproducing the so–called Parrondo effect, where a winning game is obtained from the alternation of two fair games, a new effect appears, i.e., there exists a current inversion when varying the mixing proba ..."
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We present two collective games with new paradoxical features when they are combined. Besides reproducing the so–called Parrondo effect, where a winning game is obtained from the alternation of two fair games, a new effect appears, i.e., there exists a current inversion when varying the mixing probability between the games. We present a detailed study by means of a discrete–time Markov chain analysis, obtaining analytical expressions for the stationary probabilities for a finite number of players. We also provide some qualitatively insight into this new current inversion effect.