Parrondo's paradox via redistribution of wealth

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.