Parrondo's paradox via redistribution of wealth

Abstract

In Toral's games, at each turn one member of an ensemble of $N\ge2$ players is selected at random to play.  He plays either game $A'$, which involves transferring one unit of capital to a second randomly chosen player, or game $B$, which is an asymmetric game of chance whose rules depend on the player's current capital, and which is fair or losing.  Game $A'$ is fair (with respect to the ensemble's total profit), so the \textit{Parrondo effect} is said to be present if the random mixture $\gamma A'+(1-\gamma)B$ (i.e., play game $A'$ with probability $\gamma$ and play game $B$ otherwise) is winning.  Toral demonstrated the Parrondo effect for $\gamma=1/2$ using computer simulation.  We prove it, 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 $r,s\ge1$.  An unexpected relationship between the random-mixture case and the nonrandom-pattern case occurs in the limit as $N\to\infty$.