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In the original version of Parrondo’s paradox, two losing sequences of games of chance are combined to form a winning sequence. The games in the first sequence depend on a single parameter , while those in the second depend on two parameters and . The paradox is said to occur because there exist choices of , and such that the individual sequences of games are losing but a sequence constructed by choosing randomly between the games at each step is winning.

At first sight, such behavior seems surprising. However, we contend in this paper that it should not be seen as surprising. On the contrary, we showthat such behaviour is typical in situations in which we randomly create a sequence from games whose winning regions can be defined on the same parameter space.

Before we discuss this issue, we investigate in some detail the issue of when sequences of games, such as those proposed by Parrondo, should be considered to be winning, losing or fair.

Parrondo’s games are essentially Markov games. They belong to the same class as Snakes and Ladders. The important distinguishing feature of Parrondo’s games is that the transition probabilities may vary in time. It is as though “snakes,” “ladders” and “dice” were being added and removed while the game was still in progress. Parrondo’s games are not homogeneous in time and do not necessarily settle down to an equilibrium. They model non-equilibrium processes in physics.

We formulate Parrondo’s games as an inhomogeneous sequence of Markov transition operators, with rewards. Parrondo’s “paradox” is shown to be equivalent to saying that the expected value of the reward, from the whole process, is not a linear function of the Markov operators. When we say that a is “winning” or “losing” then we must be careful to include the whole process in our definition of the word “game.” Specifically, we must include the time varying probability vector in our calculations.We give practical rules for calculating the expected value of the return from sequences of Parrondo’s games. We include a worked example and a comparison between the theory and a simulation.

We apply visualization techniques, from physics and engineering, to an inhomogeneous Markov process and show that the limiting set or “attractor” of this process has fractal geometry. This is in contrast to the relevant theory for Markov processes where the stable, equilibrium limiting set is a single point in the state space.We show histograms of simulations and describe methods for calculating the capacity dimension and the moments of the fractal attractors. We indicate how to construct optimal forms of Parrondo’s games and describe a symmetrical family of games which includes the optimal form, as a limiting case.We investigate the fractal geometry of the attractors for this symmetrical family of games. The resulting geometry is very interesting, even beautiful.

It has been shown that it is possible to construct two games that when played individually lose, but alternating randomly or deterministically between them can win. This apparent paradox has been dubbed “Parrondo’s paradox.” The original games are capital-dependent, which means that the winning and losing probabilities depend on how much capital the player currently has. Recently, new games have been devised, that are not capital-dependent, but historydependent. We present some analytical results using discrete-time Markovchain theory, which is accompanied by computer simulations of the games.

In this paper, we provide an introduction to quantum game theory through discussion of ways of converting classical games into the quantum regime. We illustrate how a quantum-based approach can simulate all possible classical game histories in parallel, for the example of Parrondo’s games.

A version of John Conway’s game of Life is presented where the normal binary values of the cells are replaced by oscillators which can represent a superposition of states. The original game of Life is reproduced in the classical limit, but in general additional properties not seen in the original game are present that display some of the effects of a quantum mechanical Life. In particular, interference effects are seen.