# Theoretical analysis and numerical simulation of Parrondo’s paradox game in space

Published on Jun 1, 2011in Chaos Solitons & Fractals3.764
· DOI :10.1016/J.CHAOS.2011.01.014
Neng-gang Xie10
Estimated H-index: 10
(Anhui University of Technology),
Yun Chen2
Estimated H-index: 2
(Anhui University of Technology)
+ 3 AuthorsChao Wang7
Estimated H-index: 7
(Anhui University of Technology)
Sources
Abstract
Abstract A multi-agent spatial Parrondo game model is designed according to the cooperative Parrondo’s paradox proposed by Toral. The model is composed of game A and game B. Game A is a zero-sum game between individuals, reflecting competitive interaction between an individual and its neighbors. The winning or losing probability of one individual in game B depends on its neighbors’ winning or losing states, reflecting the dependence that individuals has on microhabitat and the overall constraints that the microhabitat has on individuals. By using the analytical approach based on discrete-time Markov chain, we analyze game A, game B and the random combination of game A+B, and obtain corresponding stationary distribution probability and mathematical expectations. We have established conditions of the weak and strong forms of the Parrondo effect, and compared the computer simulation results with the analytical results so as to verify their validity. The analytical results reflect that competition results in the ratchet effect of game B, which generates Parrondo’s Paradox that the combination of the losing games can produce a winning result.
📖 Papers frequently viewed together
216 Citations
52 Citations
28 Citations
References29
#1Stewart N. EthierH-Index: 19
#2Jiyeon LeeH-Index: 8
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...
1 Citations
#1Derek Abbott (University of Adelaide)H-Index: 83
In 1996, Parrondo's games were first constructed using a simple coin tossing scenario, demonstrating the paradoxical situation where individually losing games combine to win. Parrondo's principle has become paradigmatic for situations where losing strategies or deleterious effects can combine to win. Intriguingly, there are deep connections between the Parrondo effect and a range of physical phenomena, as it turns out that Parrondo's original games are a discrete-time and discrete-space version ...
74 CitationsSource
#1Floyd A. Reed (UMD: University of Maryland, College Park)H-Index: 23
An example is provided where, with antagonistic selection and epistatic interaction of alleles at two loci, an autosomal allele can rise in frequency, persist in the population, and even continue to fixation, despite having an apparently lower average fitness than the alternative allele, in a process similar to Parrondo's paradox.
48 CitationsSource
#1Andrew Allison (University of Adelaide)H-Index: 12
#2Charles E. M. Pearce (University of Adelaide)H-Index: 26
Last. Derek Abbott (University of Adelaide)H-Index: 83
view all 3 authors...
The amount of text stored on the Internet, and in our libraries, continues to expand at an exponential rate. There is a great practical need to locate relevant content. This requires quick automated methods for classifying textual information, according to subject. We propose a quick statistical approach, which can distinguish between 'keywords' and 'noisewords', like 'the' and 'a', without the need to parse the text into its parts of speech. Our classification is based on an F-statistic, which ...
9 CitationsSource
#1Antonio Di Crescenzo (UNISA: University of Salerno)H-Index: 20
Dipartimento di Matematica e InformaticaUniversita di Salerno84084 Fisciano (SA), ItalyE-mail: adicrescenzo@unisa.itAbstractParrondo’s paradox arises in sequences of games in which a winning expectation may beobtained by playing the games in a random order, even though each game in the sequencemay be lost when played individually. We present a suitable version of Parrondo’s paradoxin reliability theory involving two systems in series, the units of the ﬁrst system being lessreliable than those o...
30 Citations
Cooperative Parrondo's games on a regular two-dimensional lattice are analyzed based on computer simulations and on the discrete-time Markov chain model with exact transition probabilities. The paradox appears in the vicinity of the probabilites characteristic of the “voter model”, suggesting practical applications. As in the one-dimensional case, winning and the occurrence of the paradox depend on the number of players.
35 CitationsSource
#1Abraham Boyarsky (Concordia University)H-Index: 15
#2Paweł Góra (Concordia University)H-Index: 15
Last. Md. Shafiqul Islam (Concordia University)H-Index: 2
view all 3 authors...
Abstract Parrondo’s paradox [J.M.R. Parrondo, G.P. Harmer, D. Abbott, New paradoxical games based on Brownian ratchets, Phys. Rev. Lett. 85 (2000), 5226–5229] (see also [O.E. Percus, J.K. Percus, Can two wrongs make a right? Coin-tossing games and Parrondo’s paradox, Math. Intelligencer 24 (3) (2002) 68–72]) states that two losing gambling games when combined one after the other (either deterministically or randomly) can result in a winning game: that is, a losing game followed by a losing game ...
27 CitationsSource
#1Denise M. Wolf (LBNL: Lawrence Berkeley National Laboratory)H-Index: 31
#2Vijay V. Vazirani (Georgia Institute of Technology)H-Index: 61
Last. Adam P. Arkin (LBNL: Lawrence Berkeley National Laboratory)H-Index: 100
view all 3 authors...
Population diversification strategies are ubiquitous among microbes, encompassing random phase-variation (RPV) of pathogenic bacteria, viral latency as observed in some bacteriophage and HIV, and the non-genetic diversity of bacterial stress responses. Precise conditions under which these diversification strategies confer an advantage have not been well defined. We develop a model of population growth conditioned on dynamical environmental and cellular states. Transitions among cellular states, ...
244 CitationsSource
#1Andrew Allison (University of Adelaide)H-Index: 12
#2Derek Abbott (University of Adelaide)H-Index: 83
Last. Charles E. M. Pearce (University of Adelaide)H-Index: 26
view all 3 authors...
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.
8 CitationsSource
#1J. Almeida (Complutense University of Madrid)H-Index: 2
#2Daniel Peralta-Salas (Complutense University of Madrid)H-Index: 15
Last. Miguel Romera (CSIC: Spanish National Research Council)H-Index: 20
view all 3 authors...
Abstract The recently discovered Parrondo's paradox claims that two losing games can result, under random or periodic alternation of their dynamics, in a winning game: “losing + losing = winning”. In this paper we follow Parrondo's philosophy of combining different dynamics and we apply it to the case of one-dimensional quadratic maps. We prove that the periodic mixing of two chaotic dynamics originates an ordered dynamics in certain cases. This provides an explicit example (theoretically and nu...
80 CitationsSource
Cited By13
#1Sung Chan Choi (UofU: University of Utah)H-Index: 1
Parrondo games with spatial dependence were introduced by Toral (2001) and have been studied extensively. In Toral's model, Nplayers are arranged in a circle. The players play either game Aor 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 xsay, wins or loses one unit according to the toss of a p_mcoin, where m\in\{0,1,2\}is ...
1 Citations
#1Joel Weijia Lai (SUTD: Singapore University of Technology and Design)H-Index: 7
#2Kang Hao Cheong (SUTD: Singapore University of Technology and Design)H-Index: 17
How do group dynamics affect individuals within the group? How do individuals, in turn, affect group dynamics? As society comes together, individuals affect the group dynamics and vice versa. Social dynamics look at group dynamics, its effect on individuals, conformity, leadership, networks, and more. In the past two decades, the game theoretic Parrondo’s paradox has been used to model and explain the different aspects of social dynamics. Two losing games can be combined in a certain manner to g...
13 CitationsSource
#1Stewart N. Ethier (UofU: University of Utah)H-Index: 19
#2Jiyeon Lee (Yeungnam University)H-Index: 8
Parrondo games with one-dimensional (1D) spatial dependence were introduced by Toral and extended to the two-dimensional (2D) setting by Mihailovic and Rajkovic. MN players are arranged in an M × 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 non-random pattern of games A and B. Of interest is μB (or μC), the mean profit per turn at equilibrium to the set of MN players playing game B (or game C). Ga...
7 CitationsSource
#1Ye Ye (Anhui University of Technology)H-Index: 9
#2Kang Hao Cheong (Singapore Institute of Technology)H-Index: 17
Last. Neng-gang Xie (Anhui University of Technology)H-Index: 10
view all 4 authors...
A multi-agent Parrondo’s model based on complex networks is used in the current study. For Parrondo’s game A, the individual interaction can be categorized into five types of behavioral patterns: the Matthew effect, harmony, cooperation, poor-competition-rich-cooperation and a random mode. The parameter space of Parrondo’s paradox pertaining to each behavioral pattern, and the gradual change of the parameter space from a two-dimensional lattice to a random network and from a random network to a ...
23 CitationsSource
An ansemble of N players arranged in a circle play a spatially dependent Parrondo game B. One player is randomly selected to play game B, which is based on the toss of a biased coin, with the amount of the bias depending on states of the selected players two nearest neighbors. The player wins one unit with heads and loses one unit with tails. In game A the randomly chosen player transfers one unit of capital to another player who is randomly chosen among N - 1 players. Game A is fair with res...
Source
#1Liu Yang (Anhui University of Technology)H-Index: 1
#2Zheng Kaixuan (Anhui University of Technology)H-Index: 1
Last. Lu Wang (Anhui University of Technology)H-Index: 8
view all 5 authors...
For a multi-agent spatial Parrondo's model that it is composed of games A and B, we use the discrete time Markov chains to derive the probability transition matrix. Then, we respectively deduce the stationary distribution for games A and B played individually and the randomized combination of game A + B. We notice that under a specific set of parameters, two absorbing states instead of a fixed stationary distribution exist in game B. However, the randomized game A + B can jump out of the absorbi...
1 CitationsSource
#1Yin-feng Li (Anhui University of Technology)H-Index: 1
#2Shun-qiang Ye (Anhui University of Technology)H-Index: 5
Last. Lu Wang (Anhui University of Technology)H-Index: 8
view all 6 authors...
For the multi-agent spatial Parrondo’s games, the available theoretical analysis methods based on the discrete-time Markov chain were assumed that the losing and winning states of an ensemble of N players were represented to be the system states. The number of system states was 2N types. However, the theoretical calculations could not be carried out when N became much larger. In this paper, a new theoretical analysis method based on the discrete-time Markov chain is proposed. The characteristic ...
5 CitationsSource
#1Ye Ye (Anhui University of Technology)H-Index: 9
#2Neng-gang Xie (Anhui University of Technology)H-Index: 10
Last. Yuwan Cen (Anhui University of Technology)H-Index: 5
view all 4 authors...
A multi-agent Parrondo’s model is proposed in the paper. The model includes link A based on the rewiring mechanism (the network evolution) + game B (dependent on the spatial neighbors). Moreover, to produce the paradoxical effect and analyze the “agitating” effect of the network evolution, the dynamic processes of the network evolution + game B are studied. The simulation results and the theoretical analysis both show that the network evolution can make game B which is losing produce the winning...
8 CitationsSource
#1Stewart N. Ethier (UofU: University of Utah)H-Index: 19
#2Jiyeon Lee (Yeungnam University)H-Index: 8
Let game B be Toral's cooperative Parrondo game with (one-dimensional) spatial dependence, parameterized by N ≥ 3 and p0, p1, p2, p3 ∈ [0, 1], and let game A be the special case p0 = p1 = p2 = p3 = 1/2. In previous work we investigated μB and μ(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 3 ≤ N ≤ 19, at least, and appeared to converge as N → ∞, suggesting that the P...
12 CitationsSource
#1Stewart N. Ethier (UofU: University of Utah)H-Index: 19
#2Jiyeon Lee (Yeungnam University)H-Index: 8
In Toral's games, at each turn one member of an ensemble of N\ge2players is selected at random to play. He plays either game A' which involves transferring one unit of capital to a second randomly chosen player, or game B which is an asymmetric game of chance whose rules depend on the player's current capital, and which is fair or losing. Game A'is fair (with respect to the ensemble's total profit), so the \textit{Parrondo effect} is said to be present if the random mixture \$\gamma A...
10 CitationsSource