The paradox of Parrondo's games

We introduce Parrondo's paradox that involves games of chance. We consider two fair games, A and B, both of which can be made to lose by changing a bi asing parameter. An apparently paradoxical situation arises when the two ga mes are played in any alternating order. A winning expectation is produced, even though both games A and B are losing when we play them individually. We develop an explanation of the phenomenon in terms of a Brownian ratchet model, and also develop a mathematical analysis using discrete-time Markov chains. From the analysis we investigate the range of parameter values for which Parrondo's paradox exists.