How often does the Parrondo effect appear?

摘要

It is already known that two simple losing games can derive a winning game when randomly combined. This special behaviour is known as Parrondo's Paradox, or the Parrondo effect. In this paper we estimate the volume of the parameter space that determines the Parrondo effect. In other words, how often two randomly chosen games are losing while their proper combination is leading to a winning game. Results for a typical class of relevant games indicate that the Parrondo effect is very unusual, because it appears with a probability of 0.0306%. By adding two more dimensions to the parameter space, the family of regions that exhibit the Parrondo effect is studied.