  In a series of papers, G. Harmer and D. Abbott study the behavior of random
walks associated with games introduced in 1997 by J. M. R. Parrondo. These
games illustrate an apparent paradox that random and deterministic mixtures of
losing games may produce winning games. In this paper, classical cyclic random
walks on the additive group of integers modulo $m$, a given integer, are used
in a straightforward way to derive the strong law limits of a general class of
games that contains the Parrondo games. We then consider the question of when
random mixtures of fair games related to these walks may result in winning
games. Although the context for these problems is elementary, there remain open
questions. An extension of the structure of these walks to a class of shift
diffusions is also presented, leading to the fact that a random mixture of two
fair shift diffusions may be transient to $+\infty$.
