Exact probability distribution functions for Parrondo's games

TL;DR

This paper derives exact probability distribution functions for Parrondo's games using Fourier transforms, revealing oscillations and limiting distributions, with potential applications in portfolio optimization.

Contribution

It provides the first exact solutions for the distributions in both capital and history dependent Parrondo's games, enhancing understanding of their probabilistic behavior.

Findings

Oscillations near the maximum of the distribution

Two limiting distributions for odd and even rounds

Potential application to portfolio optimization

Abstract

We consider discrete time Brownian ratchet models: Parrondo's games. Using the Fourier transform, we calculate the exact probability distribution functions for both the capital dependent and history dependent Parrondo's games. We find that in some cases there are oscillations near the maximum of the probability distribution, and after many rounds there are two limiting distributions, for the odd and even total number of rounds of gambling. We assume that the solution of the aforementioned models can be applied to portfolio optimization.