Parrondo’s paradox via redistribution of wealth∗

In Toral’s games, at each turn one member of an ensemble of N ≥ 2 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 Parrondo effect is said to be present if the random mixture γA′+(1−γ)B (i.e., play game A′ with probability γ and play game B otherwise) is winning. Toral demonstrated the Parrondo effect for γ = 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 γ ∈ (0, 1). We do the same for the nonrandom pattern of games (A′)rBs for all integers r, s ≥ 1. An unexpected relationship between the random-mixture case and the nonrandom-pattern case occurs in the limit as N →∞.