Cooperative Parrondo’s Games

Parrondo’s paradox [1–4] shows that the combination of two losing strategies can lead to a winning result. The paradox can be phrased in terms of very simple gambling games in which some unit (say, 1 euro) is won or lost with a given probability. We can imagine that the games consist on tossing different biased coins and that a “capital” C(t) is built. Every time a game is played (a coin is tossed) time increases by one unit, t → t + 1, and the capital increases or decreases by 1, C(t) → C(t) ± 1. Games are classified as winning, losing or fair if the average capital 〈C(t)〉 increases, decreases or remains constan t with time, respectively. The original version of the paradox is based on the two following basic games that can be played at any time t: • Game A: there is a probability p of winning. • Game B: If the capital C(t) is a multiple of 3, the probability of winning is p1, otherwise, the probability of winning is p2. Game A is fair if p = 1/2 and it is easy to prove that game B is fair if the condition (1 − p1)(1 − p2) 2 = p1p 2 2 holds. Choosing, for example, p = 0.5 − ǫ, p1 = 0.1 − ǫ, p2 = 0.75−ǫ, with ǫ a small positive number it turns out that both game A and game B, when played by themselves, are losing games. The surprise arises when game A and B are played alternatively, either in succession such as AABBAABBAABB. . . or randomly by choosing (with probability 1/2) the next game to be played. We will denote, for short, game “A+B” the case in which games A or B are chosen randomly at each time step and game “[2, 2]” corresponds to the sequence AABBAABBAABB. . .