Abstract
That there exist two losing games that can be combined, either by random mixture or by nonrandom alternation, to form a winning game is known as Parrondo's paradox. We establish a strong law of large numbers and a central limit theorem for the Parrondo player's sequence of profits, both in a one-parameter family of capital-dependent games and in a two-parameter family of history-dependent games, with the potentially winning game being either a random mixture or a nonrandom pattern of the two losing games. We derive formulas for the mean and variance parameters of the central limit theorem in nearly all such scenarios; formulas for the mean permit an analysis of when the Parrondo effect is present.
Highlights
The original Parrondo (1996)games can be described as follows: Let p := −ǫ and p0 := 10 − ǫ, p1 := 4 − ǫ, (1)where ǫ > 0 is a small bias parameter
In some formulations of Parrondo’s games, the player’s cumulative profit Sn after n games is described by some type of random walk {Sn}n≥1, and a Markov chain {Xn}n≥0 is defined in terms of {Sn}n≥1; for example, Xn ≡ ξ0 + Sn in the capital-dependent games, where ξ0 denotes initial capital
We claim that the conditions of the stationary, strong mixing central limit theorem apply to {ξn}n≥1
Summary
Where ǫ > 0 is a small bias parameter (less than 1/10, ). In game A, the player tosses a p-coin (i.e., p is the probability of heads). Certain nonrandom patterns, including AAB, ABB, and AABB but excluding AB, are winning as well, again for ǫ sufficiently small
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