Parrondo's paradox via redistribution of wealth

TL;DR

This paper analyzes how redistributing wealth among players in Parrondo's paradox can turn losing or fair games into winning ones, providing rigorous proofs and exploring the relationship between different game patterns as the number of players grows.

Contribution

It establishes strong laws of large numbers and central limit theorems for the profits in wealth redistribution games, extending previous simulation results with rigorous proofs.

Findings

The random mixture of fair and losing games results in winning outcomes.

The nonrandom pattern of games also produces winning results under certain conditions.

An unexpected relationship emerges between the random mixture and nonrandom pattern as the number of players increases.

Abstract

Toral (2002) considered an ensemble of N\geq2 players. In game B a player is randomly selected to play Parrondo's original capital-dependent game. In game A' two players are randomly selected without replacement, and the first transfers one unit of capital to the second. Game A' is fair (with respect to total capital), game B is losing (or fair), and the random mixture {\gamma}A'+(1-{\gamma})B is winning, as was demonstrated by Toral for {\gamma}=1/2 using computer simulation. We prove this, establishing a strong law of large numbers and a central limit theorem for the sequence of profits of the ensemble of players for each {\gamma}\in(0,1). We do the same for the nonrandom pattern of games (A')^r B^s for all integers r,s\geq1. An unexpected relationship between the random-mixture case and the nonrandom-pattern case occurs in the limit as N\rightarrow\infty.