Parrondo's games as a discrete ratchet

TL;DR

This paper establishes a mathematical connection between Parrondo's games and ratchet models by discretizing the Fokker-Planck equation, revealing how game probabilities relate to potential tilts and fairness.

Contribution

It introduces a master equation framework that links Parrondo's games with ratchet potentials, providing a new perspective on their underlying dynamics.

Findings

Periodic potentials correspond to fair games

Winning games produce a tilted potential

Provides a precise relation between game probabilities and ratchet potential

Abstract

We write the master equation describing the Parrondo's games as a consistent discretization of the Fokker--Planck equation for an overdamped Brownian particle describing a ratchet. Our expressions, besides giving further insight on the relation between ratchets and Parrondo's games, allow us to precisely relate the games probabilities and the ratchet potential such that periodic potentials correspond to fair games and winning games produce a tilted potential.