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

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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.

Highlights

► A multi-agent spatial Parrondo game model is designed. ► Double actions between individual and its neighbors are discussed. ► The weak and strong paradox conditions are established by theoretical analysis. ► Research results demonstrate some new biological points. ► Competition is an adaptive behavior on the population level too.

Introduction

Parrondo’s paradox is a paradox in game theory and is named after its creator, Parrondo, a Spanish physicist [1]. Parrondo’s paradox claims that two losing games, under random or periodic alternation of their dynamics, can result in a winning game. The seminal papers concerning Parrondo’s Paradox were published by Abbott and Harmer [2], [3] in 1999. Already, Parrondo’s paradox has been confirmed by means of computer simulation, the Brownian ratchet and discrete time Markov chain theory. Moreover, Parrondo’s paradox has been developed into many different versions [4]. The original version of Parrondo’s games involves two games, A and B, each based on tossing biased coins: (1) Game A is a game of tossing biased coin 1 with the probability of winning p1. (2) Game B is a little more complex. If the present capital is a multiple of some integer M, a biased coin 2 is tossed with the probability of winning p2. If not, another biased coin 3 is tossed, with the probability of winning p3.Winning a game earns 1 unit and losing surrenders 1 unit. Playing game A or B is always a losing game, but when these two losing games are played under random or periodic alternation, the combination of the two games is, paradoxically, a winning game via an effective set of probability p1, p2, p3 and modulus M, for instance, p1 = 0.5  ε, p2 = 0.1  ε, p3 = 0.75  ε, M = 3, ε has a small positive value and 0.005 can be chosen, for example. Observing the original version further, we can find that dependence on the capital limits its application in practice. Therefore, Parrondo [5] modified game B in the original version and presented a new version which was related to the history of the games instead of the capital. This new history-dependent structure has enlarged the parameter space of Parrondo’s paradox. Kay [6] further studied the Parrondo’s paradox effect where both game A and game B were history-dependent. Arena [7] devised a new version which was constituted of three games. Game A and game B were the same as the original version while the rule of game C depended on the recent game history of winning or losing. By analyzing the above-mentioned game versions, we find that Parrondo’s paradox requires some form of dependence, such as the dependence on capital and game history. Toral [8] proposed a space-dependent “cooperative Parrondo’s paradox” version. A remarkable difference was that there were i(1, 2,  , N) players instead of only one player involved in the game. On each round, one player ‘i’ was randomly chosen from N persons to play game A or B according to some rules. Game A remained unchanged as was defined in the original Parrondo’s games. Game B depended on the states of winning or losing of two neighbor players, i-1 and i+1. Mihailovic carried out theoretical analysis on cooperative Parrondo’s paradox and provided cooperative game model of discrete-time Markov chain. In addition, he also gave the calculation method of transition probability, and carried out the analysis and computer simulation on one dimension [9] and two dimensions [10] respectively. Since the previous versions have focused on how to modify game B, Toral [11] proposed a modification of game A. There were N players involved in this version as well. Game A was devised that a player i paid for a unit to a player j that was also chosen randomly. Game B remained unchanged as was defined in the original Parrondo’s games or history-dependent version. Ethier [12] has just considered a collective version of Parrondo’s games with probabilities parameterized by ρ  (0, 1] of an infinite number of players collectively choose and individually play at each turn the game that yielded the maximum average profit at that turn. Allison [13] proposed a new form of Parrondo’s paradox, namely the Allison mixture.

Abbott pointed out that [4], Parrondo’s paradox now has connections in physics, biology and economics and other disciplines. The earliest example in physics was the Brownian ratchet. Related ratchet phenomena include the Brazil nut paradox, Longshore drift and the Buy-low sell-high process in stock-market trading. Parrondo effects have also inspired work in the study of negative mobility phenomena [14], reliability theory [15], [16], [17], noise induced synchronization [18], spatial patterns via switching [19], and in controlling chaos [20], [21], [22]. In the area of biology, it has been proposed that Parrondo’s paradox may relate to the dynamics of gene transcription in GCN4 protein and the dynamics of transcription errors in DNA [23]. Parrondo’s paradox has been studied in various interesting scenarios involving population genetics [24], [25], [26], [27]. In terms of the stock market, Boman [28], [29] has used a Parrondian game framework as a toy model for studying the dynamics of insider information. Parrondo’s paradox not only can be used to explain a large number of nonlinear phenomena [7], but also presents its own rich non-linear characteristics. Recent work [30] has shown that Parrondo’s games exhibit fractal patterns in their state space.

Considering both game B’s structure in [8] and game A in [11] proposed by Toral, we combine the two games and establish a multi-agent spatial Parrondo’s paradox version, which reflects the game relationships among individuals in biological and social systems. By using an analytical approach based on discrete-time Markov chain, we analyze game A, game B and random combination of game A+B, and establish the conditions of the paradox. The results being compared with computer simulation ones, the validity of our theoretical method is verified.

Section snippets

Model

Based on the game version provided by Toral, the paper designs a spatial Parrondo game model which is shown in Fig. 1. Considering a population made up of N individuals, each individual occupies a certain space. For any one individual i, all its spatial neighbors compose its survival’s social microhabitat. There are double roles of the relationships between individual i and its neighbors: (1) Competitive interaction between individual i and its neighbors, which is set to game A. When game A is

Discrete-time Markov chain

Transition probability of Markov chain: {Sm, m  0} is assumed to be a homogeneous Markov chain. Its transition probability pab is the conditional probability p{St+1 = bSt = a}(a, b   E) and has nothing to do with m, where E is the state set of the system.

Transition probability Matrix:[P]=[pab]a,bE=p00p01p02p10p11p12p20p21p22where: (1)pab  0, a, b  E; (2) bEpab=1,aE.

The stationary distribution: If non-negative column{πb }satisfies bEπb=1 and πb=aπapab,bE. {πb} is assumed to be the

Results

We use the knowledge mentioned in the third part to analyze the games, and carry out the computer simulation in order to compare with analytical solution. For computer simulation, we define the multi-agent average fitness g (The same as mathematical expectation E in theoretical analysis) asg=WTwhere W=i=1N[Ci(T)-C0] is the multi-agent general profit. Ci(T) is the capital at time T, C0 is the initial capital and T is the game time.

The computation parameters are as follows: the population size

Conclusion

  • (1)

    Based on discrete-time Markov chain, the analytical approach is proposed by the paper with regard to the game model of Parrondo’s paradox in N individuals’ space. By comparing with the results of the computer simulation, we verify the validity of the analytical approach. Then we discuss the case of N = 4 in details. Obviously, the analytical approach applies to any N condition. However, with the increase of N, the transition probability matrix will increase at 2N, which makes the theoretical

Acknowledgments:

The Project was supported by the New Century Excellent Talents in University (Grant No.070003), the Natural Science Foundation of Anhui Province of China (Grant No.11040606M119), and Program for Innovative Research Teams in Anhui University of Technology.

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