Parrondo's paradox arises in sequences of games in which a winning expectation may be obtained by playing the games in a random order, even though each game in the sequence may be lost when played individually. We present a suitable version of Parrondo's paradox in reliability theory involving two systems in series, the units of the first system being less reliable than those of the second. If the first system is modified so that the distributions of its new units are mixtures of the previous distributions with equal probabilities, then under suitable conditions the new system is shown to be more reliable than the second in the "usual stochastic order" sense.
We construct games of chance from simpler games of chance. We show that it may happen that the simpler games of chance are fair or unfavourable to a player andyet the new combined game is favourable -- this is a counter-intuitive phenomenoknown as Parrondo's paradox. We observe that all of the games in question are random walks in periodic environments (RWPE) when viewed on the proper time scale. Consequently, we use RWPE techniques to derive conditions under which Parrondo's paradox occurs.
A Brownian ratchet is a one-dimensional diffusion process that drifts toward a minimum of a periodic asymmetric sawtooth potential. A flashing Brownian ratchet is a process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, producing directed motion. These processes have been of interest to physicists and biologists for nearly 25 years. The flashing Brownian ratchet is the process that motivated Parrondo's paradox, in which two fair games of chance, when alternated, produce a winning game. Parrondo's games are relatively simple, being discrete in tim…
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 n…
Toral (2002) considered an ensemble of N\geq2 players. In game B a player is randomly selected to play Parrondo's original capital-dependent game. In game A' two players are randomly selected without replacement, and the first transfers one unit of capital to the second. Game A' is fair (with respect to total capital), game B is losing (or fair), and the random mixture γA'+(1-γ)B is winning, as was demonstrated by Toral for γ=1/2 using computer simulation. We prove this, establishing a strong law of large numbers and a central limit theorem for the sequence of profits of the ensemble of player…
Toral introduced so-called cooperative Parrondo games, in which there are N players (3 or more) arranged in a circle. At each turn one player is randomly chosen to play. He plays either game A or game B. Game A results in a win or loss of one unit based on the toss of a fair coin. Game B results in a win or loss of one unit based on the toss of a biased coin, with the amount of the bias depending on whether none, one, or two of the player's two nearest neighbors have won their most recent games. Game A is fair, so the games are said to exhibit the Parrondo effect if game B is losing or fair an…
The antique Mills Futurity slot machine has two unusual features. First, if a player loses 10 times in a row, the 10 lost coins are returned. Second, the payout distribution varies from coup to coup in a manner that is nonrandom and periodic with period 10. It follows that the machine is driven by a 100-state irreducible period-10 Markov chain. Here, we evaluate the stationary distribution of the Markov chain, and this leads to a strong law of large numbers and a central limit theorem for the sequence of payouts. Following a suggestion of Pyke [In Mathematical Statistics and Applications: Fest…
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…
Parrondo's coin-tossing games comprise two games, $A$ and $B$. The result of game $A$ is determined by the toss of a fair coin. The result of game $B$ is determined by the toss of a $p_0$-coin if capital is a multiple of $r$, and by the toss of a $p_1$-coin otherwise. In either game, the player wins one unit with heads and loses one unit with tails. Game $B$ is fair if $(1-p_0)(1-p_1)^{r-1}=p_0\,p_1^{r-1}$. In a previous paper we showed that, if the parameters of game $B$, namely $r$, $p_0$, and $p_1$, are allowed to be arbitrary, subject to the fairness constraint, and if the two (fair) games…
Toral introduced so-called cooperative Parrondo games, in which there are N players (3 or more) arranged in a circle. At each turn one player is randomly chosen to play. He plays either game A or game B, depending on the strategy. Game A results in a win or loss of one unit based on the toss of a fair coin. Game B results in a win or loss of one unit based on the toss of a biased coin, with the amount of the bias depending on whether none, one, or two of the player's two nearest neighbors have won their most recent games. Game A is fair, so the games are said to exhibit the Parrondo effect if …
Parrondo games with one-dimensional spatial dependence were introduced by Toral and extended to the two-dimensional setting by Mihailović and Rajković. $MN$ players are arranged in an $M\times N$ array. There are three games, the fair, spatially independent game $A$, the spatially dependent game $B$, and game $C$, which is a random mixture or nonrandom pattern of games $A$ and $B$. Of interest is $μ_B$ (or $μ_C$), the mean profit per turn at equilibrium to the set of $MN$ players playing game $B$ (or game $C$). Game $A$ is fair, so if $μ_B\le0$ and $μ_C>0$, then we say the Parrondo effect is p…
If the parameters of the original Parrondo games $A$ and $B$ are allowed to be arbitrary, subject to a fairness constraint, and if the two (fair) games $A$ and $B$ are played in an arbitrary periodic sequence, then the rate of profit can not only be positive, it can be arbitrarily close to 1 (i.e., 100%).
Parrondo games with spatial dependence were introduced by Toral (2001) and have been studied extensively. In Toral's model $N$ players are arranged in a circle. The players play either game $A$ or game $B$. In game $A$, a randomly chosen player wins or loses one unit according to the toss of a fair coin. In game $B$, which depends on parameters $p_0,p_1,p_2\in[0,1]$, a randomly chosen player, player $x$ say, wins or loses one unit according to the toss of a $p_m$-coin, where $m\in\{0,1,2\}$ is the number of nearest neighbors of player $x$ who won their most recent game. In this paper, we repla…
A quantum walker moves on the integers with four extra degrees of freedom, performing a coin-shift operation to alter its internal state and position at discrete units of time. The time evolution is described by a unitary process. We focus on finding the limit probability law for the position of the walker and study it by means of Fourier analysis. The quantum walker exhibits both localization and a ballistic behavior. Our two results are given as limit theorems for a 2-period time-dependent walk and they describe the location of the walker after it has repeated the unitary process a large num…
Let game B be Toral's cooperative Parrondo game with (one-dimensional) spatial dependence, parameterized by N (3 or more) and p_0, p_1, p_2, p_3 in [0,1], and let game A be the special case p_0=p_1=p_2=p_3=1/2. In previous work we investigated mu_B and mu_(1/2,1/2), the mean profits per turn to the ensemble of N players always playing game B and always playing the randomly mixed game (1/2)(A+B). These means were computable for N=3,4,5,...,19, at least, and appeared to converge as N approaches infinity, suggesting that the Parrondo region (i.e., the region in which mu_B is nonpositive and mu_(1…
We study Toral's Parrondo games with $N$ players and one-dimensional spatial dependence as modified by Xie et al. Specifically, we use computer graphics to sketch the Parrondo and anti-Parrondo regions for $3\le N\le 9$. Our work was motivated by a recent paper of Li et al., who applied a state space reduction method to this model, reducing the number of states from $2^N$ to $N+1$. We show that their reduced Markov chains are inconsistent with the model of Xie et al.
Let game B be Toral's cooperative Parrondo game with (one-dimensional) spatial dependence, parameterized by N (3 or more) and p_0,p_1,p_2,p_3 in [0,1], and let game A be the special case p_0=p_1=p_2=p_3=1/2. Let mu_B (resp., mu_(1/2,1/2)) denote the mean profit per turn to the ensemble of N players always playing game B (resp., always playing the randomly mixed game (1/2)(A+B)). In previous work we showed that, under certain conditions, both sequences converge and the limits can be expressed in terms of a parameterized spin system on the one-dimensional integer lattice. Of course one can get s…
Parrondo games with spatial dependence were introduced by Toral (2001) and have been studied extensively. In Toral's model, $N$ players are arranged in a circle. The players play either game $A$ or game $B$. In game $A$, a randomly chosen player wins or loses one unit according to the toss of a fair coin. In game $B$, which depends on parameters $p_0,p_1,p_2\in[0,1]$, a randomly chosen player, player $x$ say, wins or loses one unit according to the toss of a $p_m$-coin, where $m\in\{0,1,2\}$ is the number of nearest neighbors of player $x$ who won their most recent game. In this paper, we repl…
We consider a collective version of Parrondo's games with probabilities parametrized by rho in (0,1) in which a fraction phi in (0,1] of an infinite number of players collectively choose and individually play at each turn the game that yields the maximum average profit at that turn. Dinis and Parrondo (2003) and Van den Broeck and Cleuren (2004) studied the asymptotic behavior of this greedy strategy, which corresponds to a piecewise-linear discrete dynamical system in a subset of the plane, for rho=1/3 and three choices of phi. We study its asymptotic behavior for all (rho,phi) in (0,1)x(0,1]…
In Newcomb's paradox you choose to receive either the contents of a particular closed box, or the contents of both that closed box and another one. Before you choose, a prediction algorithm deduces your choice, and fills the two boxes based on that deduction. Newcomb's paradox is that game theory appears to provide two conflicting recommendations for what choice you should make in this scenario. We analyze Newcomb's paradox using a recent extension of game theory in which the players set conditional probability distributions in a Bayes net. We show that the two game theory recommendations in N…
