Discrete--time ratchets, the Fokker--Planck equation and Parrondo's paradox

TL;DR

This paper reviews recent approaches linking Parrondo's paradoxical games to physical ratchet models using the Fokker-Planck equation, introduces new game variants, and analyzes their behavior through Markov chain methods.

Contribution

It establishes a rigorous connection between Parrondo's games and flashing Brownian ratchets, introduces new game variants, and provides a comprehensive Markov chain analysis of their paradoxical behavior.

Findings

Identification of parameter regions with paradoxical winning behavior

Derivation of winning rate equations for new games

Extension of original Parrondo's games to a broader class

Abstract

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 Fokker-Planck 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 discrete-time Markov chain (DTMC) analysis, producing winning rate equations and an exploration of the parameter space where the paradoxical behaviour occurs.