Limit theorems for Parrondo's paradox

TL;DR

This paper proves strong law of large numbers and central limit theorems for Parrondo's paradox, analyzing when combining losing games results in a winning strategy, with explicit formulas for mean and variance.

Contribution

It establishes rigorous probabilistic results for Parrondo's paradox in both capital-dependent and history-dependent games, including explicit formulas for key parameters.

Findings

Strong law of large numbers for profits in Parrondo's games

Central limit theorem with explicit mean and variance formulas

Conditions under which the Parrondo effect occurs

Abstract

That there exist two losing games that can be combined, either by random mixture or by nonrandom alternation, to form a winning game is known as Parrondo's paradox. We establish a strong law of large numbers and a central limit theorem for the Parrondo player's sequence of profits, both in a one-parameter family of capital-dependent games and in a two-parameter family of history-dependent games, with the potentially winning game being either a random mixture or a nonrandom pattern of the two losing games. We derive formulas for the mean and variance parameters of the central limit theorem in nearly all such scenarios; formulas for the mean permit an analysis of when the Parrondo effect is present.