Physica A 413 (2014) 86–93

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Physica A

journal homepage: www.elsevier.com/locate/physa

Social influence promotes cooperation in the public goodsgameTe Wu a, Feng Fu b, Puxuan Dou a,∗, Long Wang c

a Center for Complex Systems, Xidian University, Xi’an 710071, Chinab Theoretical Biology, Institute of Integrative Biology, ETH Zürich, Zürich, Switzerlandc Center for Systems and Control, State Key Laboratory for Turbulence and Complex Systems, Peking University, Beijing 100871, China

h i g h l i g h t s

• Propose influence-based imitation pattern for strategy updating.• Influence-based imitation effectively enhances cooperation level.• Cooperation-promoting effects of this imitation are robust with variation of population structures.

a r t i c l e i n f o

Article history:Received 19 December 2013Received in revised form 21 May 2014Available online 1 July 2014

Keywords:Public goods gamePreferential selection ruleEvolution of cooperationSocial influence

a b s t r a c t

Previous studies mainly consider the random selection pattern in which individuals ran-domly choose reference models from their neighbors for strategy updating. However, therandom selection pattern is unable to capture all real world circumstances. We institute aspatial model to investigate the effects of influence-based reference selection pattern onthe evolution of cooperation in the context of public goods games. Whenever experiencingstrategy updating, all the individuals each choose one of its neighbors as a reference withthe probability proportional to this neighbor’s influence. Levels of individuals’ influence aredynamical. When an individual is imitated, the level of its influence increases, thus consti-tuting a positive feedback between the frequencies of individuals being imitated and thelikelihood for them to be reference models. We find that the level of collective cooperationcan be enhanced whenever the influence-based reference selection pattern is integratedinto the strategy updating process. Results also show that the evolution of cooperation canbe promoted when the increase in individuals’ frequency of being imitated upholds theirinfluence in large magnitude. Our work may improve the understanding of how influence-based selection patterns promote cooperative behavior.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

Cooperative behavior is omnipresent in almost all realistic systems ranging from biological sphere to economic systemsto human society [1]. Thus understanding the mechanisms for the emergence and maintenance of cooperative behaviorbecomes an important question, which has attracted increasing attention of scientists from many academic communities.Among others, evolutionary game theory has become one of the most prevailing yet decisive approaches to understanding

∗ Corresponding author. Tel.: +86 5318789401741.E-mail addresses:wute@pku.edu.cn (T. Wu), fufeng@gmail.com (F. Fu), puxuandou@163.com (P. Dou), longwang@pku.edu.cn (L. Wang).

http://dx.doi.org/10.1016/j.physa.2014.06.0400378-4371/© 2014 Elsevier B.V. All rights reserved.

T. Wu et al. / Physica A 413 (2014) 86–93 87

how cooperation evolves in various social dilemma situations [2,3]. In the framework of evolutionary game theory, the clas-sical paradigms include the prisoner’s dilemma game [4–8], the snowdrift game [8–11]. As an alternative, the public goodsgame, which is also regarded as a multi-person prisoner’s dilemma game and governed by group interactions, is also widelyemployed as a classical paradigm for studying the evolution of cooperative behavior [12]. In a typical public goods gameplayed by G individuals, all individuals simultaneously decide whether to cooperate or not. Cooperators each contribute anamount of c to the public pool, while defectors do not. The total contribution is multiplied by a factor r , and then dividedequally among all group members regardless of their contributions. Obviously, defective strategy is the best choice for in-dividuals as r < G, while the group can maximize their payoff as a whole if all members do cooperate. The inconsistencyin the best choice for individuals and for the group gives rise to the dilemma. More precisely, replicator equations speakthat defectors dominate the whole population in a well-mixed population for 1 < r < G [13,14]. To overcome the socialdilemma,manymechanisms promoting the evolution of cooperation have been proposed [15–21]. Hauert et al. introduced avoluntary participationmechanism into the public goods game, and found that cooperative behavior is promoted by addingthe third strategy, namely, loner [16]. Szabó et al. further studied the voluntary participationmechanism in the spatial publicgoods game and found that the introduction of loners leads to a cyclic dominance between the three strategies and improvesthe cooperation level remarkably [17]. Santos et al. introduced social diversity through structuring the populations by het-erogeneous graphs, and found that it can also result in a substantial and persistent cooperation [18]. Li et al. studied theeffects of individuals’ memory on the evolution of cooperation under group interactions and found that one step memoryis optimal to promote cooperation [19]. For other mechanisms, please see Refs. [22,23].

It should be emphasized that though evolutionary games on complex networks have been very fruitful [3,24–43],previousworksmainly focused on random selection rule inwhich individuals randomly choose referencemodels from theirneighbors for strategy updating. However, realistic situations do not always go this way. Generally, individuals’ influenceexhibits diversity and is time-changing. In other words, some individuals may influence others more deeply and morefrequently, whichmeans that the selection of references does not agreewith the random selection pattern. These influentialindividuals’ behaviors are more likely to be imitated [44–46], leading to the expansion of their followers, which perforcefurther enhances their influences. Closely related examples include the number of movie stars’ fans, attractiveness of mallsto customers, capitals in banks, popularity of politicians among populace. The herd behaviors in stock market naturally fallsinto this category [47,48], since most of investors belonging to the same group follow the action of the selected prestigiousone who decides to buy, sell or hold the stocks [49,50]. For more literature on Matthew effects and related rich-get richerphenomena, please see Refs. [51–54]. Of interest, Szolnoki et al. have found that incorporating the wisdoms of group intothe strategy updating can greatly promote cooperative behavior [55].

These observations catalyze many studies on how the non-random reference selection pattern affects the evolution ofcooperation. Gao et al. [5] studied an extended spatial prisoner’s dilemma game. In the model, recommended role mecha-nism is introduced where individuals are allowed to recommend the ones, which they have imitated, to their neighbors forstrategy updating. They found that cooperation can be substantially promoted ascribable to this simple recommendedmech-anism. Very recently, Wang et al. introduced an age-related preferential selection mechanism into the prisoner’s dilemmagame [56]. Under this mechanism, players can select a reference for strategy imitation from their neighbors with biasescorrelated to their ages. They found that larger age parameter can markedly promote the formation of large cooperatorclusters. Moreover, Refs. [57,58] have reported that the inhomogeneous activity of teaching can promote cooperation. Howindividuals’ aging affects the evolution of cooperation is further investigated in Ref. [59].

We here aim to investigate the effects of influence-based reference selection pattern on the evolution of cooperationin the public goods game. We resort the square lattice to structure the population and study the effects of the proposedreference selection pattern in the context of group interactions. Next we show how the influence-based reference selec-tion pattern affects the evolution of cooperation in more realistic networks including Barabási–Albert (BA) scale-free net-works [60] and Watts–Strogatz (WS) small-world networks [61]. In the influence-based reference selection pattern, eachplayer is initialized with equal influence, denoted by a positive real number ∈ [0, 1]. During the evolutionary process, eachplayer acquires its payoff by interactingwith all his nearest neighbors. After that, each individual chooses one of its neighborsas a reference with the probability proportional to these neighbors’ influences. If one neighbor is picked up and successfullyimitated, its influence increases by an influence factor α. In a broader sense, a feedback loop forms between the influenceand the frequency of being picked up as model individual [37]. By using agent-based Monte Carlo simulations, we havefound that the influence-based reference selection pattern is substantially beneficial for the emergence and sustenance ofcooperation. Moreover, this positive effect of promoting cooperation is robust against the variation of interaction networks.Last but not least, we would like to emphasize that influence factor and reputation [19,62] are totally different concepts. In-fluence represents how often individuals are imitated. Influence factor determines how fast individuals’ influences expand.Reputation in Refs. [19,62] is directly determined by how often individuals have cooperated.

2. Model

Weconsider the evolutionary public goods game on a square latticewith periodic boundary conditions and vonNeumannneighborhood [3]. Each node on the square lattice is occupied by an individual. Initially, each individual is designated eitheras a cooperator or a defector with equal probability. Every individual participates in G = 5 groups centered on its nearestneighbors and himself, respectively. In each group cooperators each contribute c = 1 to the public pool, while defectors

88 T. Wu et al. / Physica A 413 (2014) 86–93

Fig. 1. Fraction of cooperators as a function of η with different α. Larger α promotes cooperation level more. Each data point is obtained by averaging thefraction of cooperators in a window of 1000 time steps after a transient of 30000 time steps for 100 independent strategy initializations.

contribute nothing. The total contribution is subsequentlymultiplied by the factor r (1 < r < N). The public goods is equallyshared among these G group members. Thus the payoff of player i obtained from group g is Pg

i = r ·

j∈Ω sj · c/G − si · c ,where Ω includes all the members in that group. si takes value of one if individual i is a cooperator and zero otherwise. Theoverall payoff of individuals i, Pi, is summed over all the payoffs of these G groups. In mathematical language, it reads asPi =

Gg=1 P

gi . Each individual i is assigned an equal influence factor θi(0) on the outset. We would like to point out that the

variation of θi(0) does not change the qualitative results. At the end of each round, player i selects a player j from its directneighbors as a reference to learn. The probability that j is chosen is given by

Ai→j =θj(t)θm(t)

, (1)

where the sum runs over all neighbors of i.Then the system enters the strategy updating phase. Following previous studies [3,17,26], player i adopts the strategy of

player j with the probability

W =1

1 + exp[(Pi − Pj)/κ], (2)

where κ denotes the amplitude of noise or its inverse (1/κ), the so-called intensity of selection [63,64]. In accordance withmost previous studies we set κ to be 0.5, implying that players with higher payoff spread with high probability [65–67]. Ifplayer i does adopt the strategy of its neighbor j, the influence of j increases in the following way

θj(t + 1) = θj(t) + α, (3)

where α (α ≥ 0) denotes the influence factor. In the case of α = 0, the traditionally random reference selection pattern isrecovered where player j is randomly chosen out of i’s neighbors. For α > 0, the neighbor’s influence will be increased afterbeing imitated, and the high influential neighbors of player i have large probabilities to be selected as potential references.In present study, we focus on the impact of α on the evolution of cooperation.

3. Results

The simulation results were carried out in a population comprising 50 × 50 individuals who occupy the nodes of aperiodic square lattice. We confirm that except a very narrow width of η, the present results see little qualitative change.The critical phenomenon within this width calls for more sophisticated methods [3,63,68], which falls outside of our mainconcentration here. Individuals synchronously update their strategies. When the evolutionary process reaches equilibrium,the average number of cooperators is computed to characterize the cooperation level ρc . Each data point is obtained byaveraging over 103 generations after 3 × 104 generations as a transient process. The final results are averaged over 100independent runs to overcome the influence of initial conditions.

To better compare the present study with the traditional ones, we normalize the enhancement factor as η = r/G. As iswell-known, cooperators have chance to survive only if η > 0.76 on the classic spatial public goods game [64]. It is worthnoting that even η is as high as the group size, cooperation level is still not one hundred percent for α = 0, which is inline with results reported in Ref. [64]. Whenever η exceeds the threshold 1.04, cooperators pervade the whole population.These results can serve as a reference point for estimating the impact of α on the evolution of cooperation. We first showhow cooperation evolves for different value of α in Fig. 1. Note that α = 0 responds to the traditional version of the publicgoods game. For α = 1, the evolution of cooperation is conspicuously promoted in comparison with the case of α = 0. Asα increases, the cooperation level can be further enhanced. We naturally come to the conclusion that the influence-based

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Fig. 2. Probability distribution of cooperators and defectors in each of the five different influence spans. As α increases from 1 to 7, the advantage ofcooperators in each span over defectors gets more conspicuous. When the evolutionary process arrives at the time step of 2000, we begin to average thenumber of cooperators and defectors located in each of the five spans, until the evolutionary process terminates, where either cooperators or defectorspervade the whole population, or a predetermined time step is exceeded.

Fig. 3. Cooperation level as a function of the parameter combination of α and η. As α continues to increase, the cooperation-promoting effect gets lessremarkable. Each data point is obtained by averaging the fraction of cooperators in a window of 1000 time steps after a transient of 30000 time steps for100 independent strategy initializations.

reference selection pattern is effective to improve cooperation level. And, the more rapidly the influence upswings withindividuals’ frequency of being imitated, the more favorable for raising the cooperation level.

To observe the impact of α on different influence levels further, we plot the probability distribution of the normalizedinfluence θi(t) = θi(t)/θmax(t)with differentα in the equilibrium state, where θmax(t) is themaximal influence in thewholesystem (see Fig. 2). For α = 1, cooperators hold slight dominance over defectors in number. As α increases from 1 to 7,the number difference between influential cooperators and influential defectors gets more remarkable. These observationsbespeak that the increase in α enhances the number of influential cooperators and suppress the existence of influentialdefectors in the whole range of the normalized influence level. These results further corroborate that the increment ofinfluence factor α favors the evolution of cooperation more, with the preferential selection pattern being integrated.

To provide comprehensive view on how α influences cooperators’ evolutionary fate, we illustrate the cooperation levelwith respect to the combined point of parameter α and η in Fig. 3. One can find that for each fixed α, there always exist twothresholds in terms of η. ηs less than the smaller one put cooperators in disadvantageous place, and eventually cooperatorsvanish from the population. In contrast, cooperators wipe out defectors once η exceeds another threshold. Of interest, eithercooperators or defectors are likely to win out the evolutionary competition for η locating in between these two thresholds.

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Fig. 4. Time evolutions of the fraction of cooperators for different α. There is a tradeoff between the equilibrium fraction of cooperators and the time forthe population to arrive at it. Each curve is obtained for a representative evolutionary process. Parameter η = 0.85.

Fig. 5. Distribution of the normalized mean payoffs of individuals Pi , with i denoting cooperators or defectors, for different normalized influence in theequilibrium state. When the evolutionary process arrives at the time step of 2000, we begin to average the payoff of cooperators and defectors located ineach of the five spans, until the evolutionary process terminates, where either cooperators or defectors pervade the whole population, or a predeterminedtime step is exceeded. Here blue denotes cooperators, while red defectors. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

Statistically speaking, they coexist. This illustration clearly shows that the two thresholdsmove towards left asα appreciates,in line with aforementioned analysis.

Fig. 4 shows the change tendency of cooperators’ number as the evolution advances for different value of α. Recallingthe initial settings of individuals, cooperators and defectors approximately own the same fraction. Interestingly, though αvaries, the number of cooperators exhibits much similar tendency. Cooperation level sharply decreases and then increasesto the equilibrium state rapidly. This is readily understood with bearing in mind that the initialization of strategies is bene-ficial for defectors, since they can easily exploit the neighboring cooperators and hence spread quickly [69,70]. Ensuing thecooperation level reverberates to a higher level. Larger α enables to stabilize the population in higher cooperation levels,which is achieved with prolonged evolutionary time. The following explanation is responsible for this time prolongation.The influence-based reference selection pattern can improve not only cooperators’, but also defectors’ influences. The highinfluence of defectors prolongs the attenuation of defectors, thus cooperators need longer time to assimilate defectors.

Let us further explain how cooperative behavior is promoted through such a preferential selection pattern. Concentratedon the normalized mean payoffs Pi of cooperators and defectors, one can see that for fixed α, the normalized mean payoff ofcooperators increases with the rise of the normalized influence level and always exceeds that of defectors during the whole

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Fig. 6. Time evolution of the strategy switching rate for different values of α. Here ρc→d denotes the rate that cooperators switch to defectors, and ρd→cthe rate that defectors switch to cooperators. Each curve is obtained for a representative evolutionary process. The parameter η is set to 0.85.

Fig. 7. Evolution of cooperation. (a) Fraction of cooperators as a function of η on the BA scale-free network with different α. (b) Fraction of cooperators asa function of η on the WS small-world network with different α. Each data point is obtained by averaging the fraction of cooperators in a window of 1000time steps after a transient of 15000 time steps for 30 independent strategy initializations. Each network is independently generated for 30 runs.

interval of the normalized influence level. Moreover, the increase in α can further enlarge the normalized mean payoff gapbetween cooperators and defectors. This indicates that cooperators, especially high influential ones, have a great advantageover defectors in acquiring payoffs under this preferential selection pattern (see Fig. 5).

We next investigate the rate of cooperators switching to defectors, ρc→d, and that of defectors switching to cooperators,ρd→c , during the evolutionary process. Fig. 6 demonstrates that at the outset of the evolution, ρd→c is largely suppressedby ρc→d. With the evolutionary process moving forward, ρd→c exceeds ρc→d, which resonates with time evolution of coop-eration (see Fig. 4). Besides, it is clearly shown that the ρd→c in the last panel (α = 5) arrives at the maximum value. Thismeans that cooperators along the boundary are more frequently selected as references with the increment of α. This obser-vation can be attributed to the feedback trait of the preferential reference selection pattern. The higher mean payoff makescooperators more likely to be imitated successfully, thus increasing their influences, which in turn raises the possibilities ofbeing chosen as references. This positive feedback favors the expansion of cooperators. For defectors, high influential onescan helpmaintain the flourish of defectors initially. However, the formation of defector clusters leads to their self-inhibitionand thus their ultimate demise. These factors together strengthen the stability of cooperator clusters.

To test the robustness of the preferential selection pattern against the variation in population structure, we here inves-tigate how cooperation evolves on the WS small-world network [71] and BA scale-free network [72–74]. We generate bothnetworks with N = 1000 nodes. The average connectivity is z = 4. Equilibrium fractions of cooperators are determinedwithin 103 time generations after 15 × 104 generations as a transient process. For BA scale-free network (see Fig. 7(a)), itsheterogeneity can upgrade the equilibrium fraction of cooperators to higher values [18,72]. In this settings, cooperators can

92 T. Wu et al. / Physica A 413 (2014) 86–93

get a foothold even η is low as 0.52. Whenever taking into account the preferential reference selection pattern, the require-ment on η for cooperators to survive is further relaxed. Likewise, Fig. 7(b) shows that the preferential reference selectionrule can also engender positive effects on the evolution of cooperation on WS small-world network. These qualitativelysimilar results, both for heterogeneous and homogeneous networks, validate the robustness of influences-based referenceselection pattern on promoting cooperation.

4. Discussion and summary

Many previous studies suggest that preferential selection is an effective pattern to promote cooperation in social dilem-mas. In Ref. [75], Wang et al. studied the effects of a preferential selectionmechanism on the evolution of cooperation in thespatial prisoner’s dilemma game. They found that, for positive tunable parameter, the increase of tunable parameter valuecan promote the evolution of cooperationmonotonously, while for negative tunable parameter, the result is completely op-posite. In Ref. [74], Yang et al. proposed a preferential selection pattern for achievingmaximum cooperation in evolutionarygames on scale-free network. They claimed that there exists a moderate parameter value maximizing the cooperation level.In these studies, individuals’ influence is either restricted to the connectivity of the network, which is kept unchanged oncethe network is formed, or directly related to their present payoffs. Experimental studies on twitter pointed out that popularusers who have a large number of followers are not necessarily influential in terms of spawning retweets or replies [76,77],which suggests that one’s influence is not determined by its degree. This lends us the solid ground to introduce a dynamicgrowth of influence into the public goods game to represent realistic behaviors. In other words, we simulate a more realis-tic scenario, wherein not all individuals have constant influence, and not all high-payoff individuals unvaryingly have highinfluence.

In summary, we have studied the evolution of cooperation on a square latticewith an influence-based reference selectionpattern. By considering asymmetric and heterogeneous influential effects inmany social groups, we estimate an individual’sinfluence based on the frequency of being imitated. We find that under the influence-based reference selection patterncooperative behavior can be obviously promoted. Compared with the random selection patterns, our results show that theincrement of influence factor α is able tomonotonously promote the evolution of cooperation. Additionally, we observe thathigh influential cooperators have a great advantage over high influential defectors in acquiring payoffs under the proposedpattern. Besides, we also verify that the cooperation-enhancing effect is robust against the variation of interaction networks.This simple model offers an efficient way to explore the real social behaviors. We hope that our study can inspire furthereffort to invest in the preferential reference selection mechanism.

Acknowledgments

This work was supported by National 973 Program (2012CB821203) and NSFC (10926195 and 10972003).

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