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Promotion of cooperation due to diversity of players in the spatial public goods game with increasing neighborhood size
Physica A 406 (2014) 145–154
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Physica A journal homepage: www.elsevier.com/locate/physa
Promotion of cooperation due to diversity of players in the spatial public goods game with increasing neighborhood size Cheng-jie Zhu a,b , Shi-wen Sun a,b , Li Wang a,b , Shuai Ding c,d , Juan Wang e , Cheng-yi Xia a,b,∗ a
Tianjin Key Laboratory of Intelligence Computing and Novel Software Technology, Tianjin University of Technology, Tianjin 300384, PR China b Key Laboratory of Computer Vision and System (Ministry of Education), Tianjin University of Technology, Tianjin 300384, PR China c
Key Laboratory of Process Optimization and Intelligent Decision-Making (Ministry of Education), Hefei University of Technology, Anhui Hefei 230009, PR China d
School of Management, Hefei University of Technology, Anhui Hefei 230009, PR China
e
School of Electrical Engineering, Tianjin University of Technology, Tianjin 300384, PR China
highlights • • • • •
We propose an evolutionary public goods game model with individual diversity. Individual diversity can be characterized by two types of players with different strategy transfer capability. An intermediate fraction of influential players facilitates the collective cooperation. The promotion of cooperation is also influenced by the increasing size of neighborhood. The competition between the individual diversity and interaction range determines the evolution of cooperation.
article
info
Article history: Received 21 January 2014 Received in revised form 2 March 2014 Available online 19 March 2014 Keywords: Cooperation promotion Public goods game Influential players Increasing neighborhood size Spatial reciprocity Individual diversity
∗
abstract It is well-known that individual diversity is a typical feature within the collective population. To model this kind of characteristics, we propose an evolutionary model of public goods game with two types of players (named as A and B), where players are located on the sites of a square lattice satisfying the periodic boundary conditions. The evolution of the strategy distribution is governed by iterated strategy adoption from a randomly selected neighbor with a probability, which not only depends on the payoff difference between players, but also on the type of the neighbor. For B-type agents, we pose a prefactor (0 < w < 1) to the strategy transfer probability, which implies the lower teaching activity or strategy convincing performance; but w is always set to be 1 for A-type agents, hence it means that A-type players are influential ones who own a larger strategy spreading chance. Furthermore, we also consider the competition between two opposite effects when the number of nearest neighbors (k) is increased from 4 to 24. Within a range of the portion of A-type influential players, the inhomogeneous teaching activity in strategy transfer yields a relevant increase (dependent on w ) in the density of cooperators characterizing the promotion of cooperation. Current findings are of utmost importance for us to understand the evolution of cooperation under many real-world circumstances, such as the natural, biological, economic and even social systems. © 2014 Elsevier B.V. All rights reserved.
Corresponding author. Tel.: +86 2260216865. E-mail addresses: [email protected] (L. Wang), [email protected] (C.-y. Xia).
http://dx.doi.org/10.1016/j.physa.2014.03.035 0378-4371/© 2014 Elsevier B.V. All rights reserved.
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1. Introduction Although the defection is an evolutionarily stable strategy (ESS) which denotes the dominant one from an individual viewpoint, cooperation is often found to be persistent and emergent within many real-world systems (e.g., natural, biological, economic and even social systems) [1,2]. Thus, how to interpret the evolution of cooperation under real situations becomes a challenging topic, which attracts the attention of many researchers from ecology, biology, mathematics, economics, physics, engineering science and so on [3,4]. Among them, the evolutionary game theory provides an effectively theoretical framework which has shed some light upon this long-standing puzzle [5,6]. For instance, various game models are proposed to illustrate the evolution of cooperation behavior in a population of selfish individuals [1–6], and one of the most investigated games is the prisoner’s dilemma game (PDG) [7–22] which is a classical two-player game model. In the PDG, each player can only adopt two pure strategies: to cooperate (C) or to defect (D); Then, the players will obtain their payoff by playing the game in pairs, where they will get the reward (R) and punishment (P) payoff if they take the same action; while a defecting individual will be tempted to acquire a higher payoff (T ) and the cooperating one can only receive the sucker’s payoff (S) provided that they choose the distinct strategies. The PDG becomes valid whenever the condition T > R > P > S is satisfied and unfortunately, the well-mixing population does not facilitate the collective cooperation, and some additional mechanisms need to be added to the PDG so that the cooperation thrives. It deserves to be mentioned that Nowak and May [23] seminally introduced the spatial structure into the PDG to render the emergence of cooperation as cooperative clusters come into being within the system. Starting from this point, a vast majority of works are devoted to exploring the evolution of cooperation in the spatially structured population based on the PDG (referring to the recent reviews [24–26]). Analogously, T > R > S > P creates another typical pair-wise game: snowdrift game (SDG), and the cooperative behaviors of SDG in the structured population also receive a great deal of concern within the scientific communities [27–31]. Recently, large quantities of evidences have demonstrated that the interaction structure of real-world systems is neither well-mixing nor regular, and to a certain extent exhibits a small world effect [32] or scale-free [33] property. Systems of such complex topology are usually named as the so-called complex networks in which nodes characterize the interacting agents and links mimic their interactions, and complex networks can be found within almost all natural, social and manmade engineering systems [34–36]. Meanwhile, the cooperative behaviors are also found to be greatly promoted when players interact on complex networks [25,26,37], especially provided that the network of contacts is scale-free [38–43] or the individual behavior exhibits some heterogeneity [44,45]. Conversely, complex networks can be generated by implementing a specific growth mechanism in which newcomers are linked to old nodes with a probability that is determined not by their degree, but by their benefit or fitness [46–48]. Additionally, the coevolution between the structure and cooperation behavior is studied in many works [49–51], and the evolution of cooperation in interdependent or multiplex networks has also become an active topic in the very recent years [52,53]. However, aforementioned works do not take into account the group effect or interaction among agents [54], which, to a greater extent, characterizes the public cooperation, such as social insurance, global climate change, free trade zone, to name but a few. As a stylized formalization, public goods game (PGG) is often used to elucidate the role of group interaction in the public cooperation [55], in which each individual within a group can independently and voluntarily choose to contribute or not. Likely, we can identify two different strategies: to cooperate (C) if the player contributes, or to defect (D) otherwise. Each cooperator contributes 1 to the group, but the defector contributes nothing. Then, the total contribution will be multiplied by a synergy factor r > 1 which represents the benefit reached by cooperating. Finally, the gained contribution is evenly distributed among all players within this group, irrespective of their contribution. Assuming that there are nc cooperators within a PGG group of n players, the payoff obtained by each cooperator and defector can be written as follows,
πc = r ∗ nc /n − 1 πd = r ∗ nc /n
(1)
where πc and πd denote the corresponding payoffs of a cooperator and a defector, and r is a synergy factor greater than 1. It is clearly shown that the defecting player has always larger payoff than the cooperator in the same PGG group [56]. As such, defection is a strictly dominant strategy, and hence D becomes an evolutionary stable strategy. This finding implies that any body will not contribute to the group, but full cooperation will obviously lead to the better result if r > 1. Therefore, a social dilemma arises since full cooperation is more profitable for any player than full defection, whereas the individual selfish interest prevents the desirable outcome and the tragedy of commons is widespread within the field of public cooperation [57]. Therefore, how to avoid this tragedy poses an urgent task facing the scientific communities [58–60]. So far, a large body of literature focuses on the role of pair-wise or group interactions in the elevation of cooperation level [25,26,54], but there is a rigorous assumption that each player owns the identical strategy enforcement probability during the game playing, that is, the individual diversity is usually ignored and it is far away from the realities under a couple of realworld circumstances. For example, during the epidemic outbreaks for a specific disease (e.g. influenza), the adolescent and old people often have lower immunity and face a higher risk, while adults may contract this disease with the fewer chance as they are often much stronger. To characterize this type of diversity, the inhomogeneous strategy adoption probability is firstly mentioned in the evolutionary PDG model [61,62]. Among them, Szabó et al. [63,64] propose a novel spatial PDG with two types of players located on the sites of a square lattice, and they correlate the strategy adoption probability with the payoff difference and the type of players, in which the teaching ability is distinguished for two types of players. They find that the density of cooperators can be relatively increased by the nonidentical teaching behavior within a range of
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the portion of influential players, meanwhile the promotion of cooperation is also influenced by the size of neighborhood. However, their works can only cover the PDG model with pair-wise interaction on the square lattice and group effect is not taken into account. In the present study, we extend this work into the public goods game model to further investigate the role of group interaction in the public cooperation. Our results indicate that the density of influential players (h) and the neighborhood size (k) can noticeably influence the behavior of public cooperation, and the intermediate h often leads to the emergence of cooperation for a specific neighborhood setup and individual strategy adoption behavior. Meanwhile, our model is also different from the work of Guan et al. [65] which mainly focuses on the effect of in-homogeneous activity and noise parameter on the cooperative behaviors in the spatial public goods game. The remainder of this paper is organized as follows. At first, we in detail introduce the PGG model with two types of players in Section 2. Then large-scale numerical simulations are presented in Section 3. At last, the concluding remarks and discussions are provided in Section 4. 2. The PGG model with two types of players In our public goods game model, two types of players (tx = A or B) are placed on the sites x of an L × L regular lattice with periodic boundary condition, the fraction of A and B-type players are set to be h and 1 − h, which are randomly distributed on the square lattice and remain unchanged during the simulation. Additionally, every player x has k nearest neighbors and is designated as a cooperator (C ) sx = 1 or a defector (D) sx = 0 with equal probability (i.e., 50%) at the initial time step. Each player needs to attend G = k + 1 PGG groups where one group is centered around himself and other k groups focus on his nearest neighbors. In every PGG group, all players can simultaneously decide whether to make a contribution, and we count the number of cooperators so as to calculate the PGG payoff according to Eq. (1). Thus, the total payoff of a player x can be summed over as follows, Px =
Pxg
g ∈Ωx
s j∈g j − sx = r g ∈Ωx
=
G
r ∗ ngc g ∈Ωx
G
− sx
(2)
where Ωx denotes the set of PGG groups in which player x participates and g is one element in Ωx , sx and sj stand for the g strategy of player x and j, nc is the number of cooperators within group g and G is the total number of PGG groups. After that, the system evolution proceeds with a random sequential strategy update (i.e., asynchronous update). Player x picks up a nearest neighbor y (whose payoff can be computed in the same way as player x) stochastically and tries to imitate the strategy sy with the following probability: 1
Prob(sx ← sy ) = wy 1+e
(Px −Py )/k
(3)
K
where (Px − Py )/k characterizes the difference of normalized payoffs between x and y, K denotes the uncertainty affecting the strategy adoption of player x which is analogous to the temperature as introduced in the kinetic Ising model, and the multiplicative factor wy signifies the strategy spreading capability of player y and can be expressed in the following way,
wy =
1,
w,
if ty = A if ty = B, 0 < w < 1.
(4)
Here we can observe that A-type players hold the higher probability to pass their current strategy on to other ones during the strategy evolution, that is, they can easily convince their neighbors to adopt the strategy they are just following. Hence, A-type agents are influential ones in our model. On the contrary, B-type individuals are non-influential ones who have lower possibility to impose their own strategy to other ones. To a certain extent, this feature can characterize the individual diversity often found within many real-world networks, and can be correlated with the personal reputation, power, age and so on. A full Monte Carlo step (MCS) includes the above-depicted basic steps, and each player has a chance once on average to modify the current strategy into that of one of nearest neighbors. After a suitable relaxation time (tr ), the system develops into the stationary state characterized by the average fraction of cooperators (ρ ), which is the quantity of most interest and determined by averaging the fraction of cooperators within another ta time steps. Furthermore, we consider three different neighborhood setups as shown in Fig. 1: k = 4 (Von Neumann neighborhood), k = 8 (Moore neighborhood) and k = 24 neighbors, and further discuss the role of game interaction range during the evolution of cooperation. Larger neighborhood size (e.g., k = 48) is also explored and has the qualitatively same behavior as the case of k = 24 and will not be presented here.
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a
b
c
Fig. 1. Three typical neighborhood setups used in our PGG model. (a) Left panel: Von Neumann neighborhood (k = 4); (b) Middle panel: Moore neighborhood (k = 8); (c) Right panel: k = 24 neighborhood setup.
3. Numerical simulation results Extensive numerical simulations are performed to explore the behavior of our PGG model on L × L square lattices with periodic boundary conditions. Main simulation parameters are set to be L = 200, the relaxation time tr = 5 × 104 MCS and ta = 104 MCS, and other parameters will be provided during the following analysis. At first, in Fig. 2, we will illustrate the relationship between the fraction of cooperators (ρ ) at the stationary state and normalized synergy factor r /G (which is utilized so as to compare the results for different neighborhood setups, while we still use the usual synergy factor r in other figures). As expected, a general trend is observed in Fig. 2 that the cooperation level will be elevated as the synergy factor (r /G) increases under a given fraction (h) of A-type players, but h will also markedly influence the cooperative behavior on the square lattice. When h is small (e.g., h = 0.2), ρ transcends 0 for a very low synergy factor (r /G) (r /G at this point is called the lower critical synergy factor (r /G)cb ) and continuously increases into the full cooperation. In fact, smaller h means that there are fewer influential individuals who have the stronger strategy transfer ability and other individuals are non-influential. Thus, the influential cooperators can have a chance to impose their strategy into other ones, and then help the population to escape from the full defection at a lower synergy factor. After that, fewer influential cooperators need enough stimulation to convince all players to reach a consensus (i.e., full cooperation). On the other hand, the full defection cannot be avoided until higher r /G is reached for a larger h(= 0.8), that is, (r /G)cb is higher, but the cooperation can be sharply increased into the state of full cooperation (ρ = 1.0) once (r /G) is over (r /G)cb . Likely, larger h signifies that there are adequate influential players who compete for the cooperation, and it is difficult for cooperators to win over defectors, but the cooperation will be sharply enhanced into a very high level provided that contributing agents gain the advantage during the evolution of cooperation. At the same time, there are subtle differences for various neighborhood setups since it covers the interaction range in which the players participate. It can be clearly observed that the larger the neighborhood size k, the lower the critical synergy factor (r /G)cb . In fact, the underlying topology will be approaching the well-mixed case which eventually leads to the full defection for any finite value of r as k becomes larger and larger. In addition, according to Fig. 2, the promotion of cooperation will become more pronounced as the neighborhood size is larger. For example, when k is equal to 24, the cooperation level ρ can suddenly increase from 0% to 100% within a narrower interval of r even in the lower density of A-type players (i.e., h = 0.2) after (r /G)cb is surpassed. Whereas for k = 4, the fraction of cooperators (ρ ) will be continuously elevated as (r /G) varies for lower values of h (i.e., h = 0.2, 0.4), but the full cooperation cannot be arrived at even though r /G is increased to 1.2. The situation of k = 8 is rightly in between the above-mentioned two cases. For the spatial lattice with different neighborhood, the only difference between k = 4 and 24 may come from the fact that the clustering effect exists in the case of k = 24, but is absent in k = 4. A large number of evidences have demonstrated that the clustering among players is an important factor to facilitate the cooperators to create the giant clusters to resist the employment of defectors [25,26, 66–68]. Thus, the public cooperation can be dramatically enhanced under the situation which allows two types of players with different imitating abilities to exist within the population, especially when the density of influential players is lower for the larger neighborhood range. In order to explore the role of fraction of A-type influential players (h), we illustrate the effect of h on the stationary density of cooperators ρ for different w values under the fixed synergy coefficient r in Fig. 3. For brevity, here we can only consider the Moore neighborhood (k = 8) in the current simulation setup and results for k = 4 and k = 24 are neglected. The numerical results clearly indicate that, within a range of A-type influential players (i.e., h ∈ [0.15, 0.5]), different strategy transfer capability w usually leads to the optimal promotion of cooperation, especially in the case of w < 1, and it is worth noting that the smaller the multiplicative factor w , the more prominent the optimized elevation of cooperation. In other words, the discrepancy of teaching ability between two types of players supports the emergence of optimal cooperation phenomenon, and the optimal cooperation disappears when two types of players hold the same strategy spreading probability (i.e., w = 1). Furthermore, too few influential players are not enough to attract the players to create the giant clusters as much as possible, but the cooperation will also deteriorate when too many A-type players lie within the whole population and compete with each other. On the contrary, intermediate density of influential players allows enough space to form the
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a
r/G
b
r/G
c
r/G Fig. 2. (Color online) The fraction of cooperators (ρ ) as a function of normalized synergy factor (r /G) under various neighborhood setups. The density of influential players and different neighborhood setups nontrivially influences the collective cooperation level in our PGG model. The larger the neighborhood size, the more pronounced the role of influential players. From panel (a) to panel (c), the neighborhood size is set to be 4, 8 and 24, respectively. Other parameters are set to be L = 200, K = 2.4 and w = 0.001.
cooperative giant clusters so as to arrive at the optimal cooperation in the lattice. In a word, existence of two types of players and resulting discrepancy of individual teaching ability facilitates the evolution of cooperation on the structured population. The characteristic snapshots of players with different types and strategies are shown in Fig. 4 so that we can observe the distribution of cooperators and defectors at the stationary state. Likely, we can only consider the Moore neighborhood,
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Fig. 3. (Color online) Relationship between stationary density of cooperators (ρ ) and fraction of A-type individuals (h) for different multiplicative factors w on regular lattices with Moore neighborhood (k = 8). The existence of influential player or strategy transfer discrepancy yields the optimal cooperation level among the population in which the fraction of influential players often lies between 0.15 and 0.5 as w < 1, but the cooperation will disappear when all players have the identical strategy transfer probability (w = 1). Other parameters are set to be L = 200, K = 2.4 and r = 6.5.
qualitatively similar snapshots are also present for Von Neumann neighborhood and 24-neighborhood. As expected, the cooperation level will be greatly increased as the synergy factor r becomes higher after r crosses over the critical threshold rbc . In particular, A-type cooperators (AC) and B-type cooperators (BC) readily interweave together to create the cooperative clusters to defend not to be invaded by A-type defectors (AD) and B-type defectors (BD), and the smaller the density h, the easier the clusters of cooperators are formed when the synergy factor r is fixed. Meanwhile, ACs are usually surrounded by BCs and BDs often encircle ADs, and this implies again that A-type players are influential ones in our model. With respect to the role of density of A-type h, lower h will facilitate the evolution of cooperation when r is smaller (e.g., r <= 5.5); while for larger r (e.g., r ≥ 5.5), there exists an intermediate density h, which often lies between 0.15 and 0.5, to lead to a circumstance that allows the survival of cooperators as many as possible. Henceforth, ACs act as a determinant role for the emergence of cooperation once again since the ACs eventually suppress the ADs and impose the cooperation strategy into their neighbors regardless of the type of neighbors. Current snapshots well agree with the above-mentioned results in the previous two figures, and dramatically enrich the understanding of spread of cooperative strategies. By examining the dynamical evolution of fraction of cooperators in the whole population, which is described in Fig. 5, we can further understand the role of two types of players in the promotion of cooperation under a specific synergy factor (r = 6.5). In Fig. 5, the multiplicative factor of strategy spreading is fixed to be w = 0.001 which means that the difference between two types of players is very large, and we can modify the density of influential players h to observe the evolution of cooperation. When h is smaller (e.g., h = 0.2 or 0.4), the fraction of cooperators ρ(t ) decreases into a minimum at first and then ρ(t ) is little by little elevated into a higher stationary value; while for higher h (e.g., h = 0.6, 0.8 and 1.0), ρ(t ) is continuously reduced to 0 which means a full defection status. From this phenomenon, it is indicated that the lower h is beneficial for the influential cooperative player to spread their strategy to other neighbors, further to create the cooperative clusters to defend them not to be invaded by the defectors. However, when h is large enough (h > 0.5) to approach the traditional spatial public goods game, in which there is a large quantity of influential players in the population and some of them may act as the barrier of strategy spreading of A-type cooperators so that the cooperators cannot resist the employment of defectors and thus results in the very low cooperation level, which is similar to the results shown in Fig. 4. In fact, from Fig. 3 we can also observe that when h surpasses 0.5, the stationary fraction of cooperators will dramatically descend, and even the full defection state is arrived at as h is larger than 0.54. The simulation results in Fig. 5 again verified this nontrivial influence of two types of players in the evolution of cooperation, and influential players can effectively impose their strategy only when there exist a variety of non-influential players in the whole population. Thus, the existence of players with different strategy transfer capability may be responsible for the promotion of cooperation, especially when influential players can only hold a very small ratio within two types of players. At present the effect of noise strength on the cooperation behavior is ignored where K is often set to be K = 2.4, and it is necessary to discuss the strategy evolution in the wider range of K . Fig. 6 depicts the full phase diagrams between synergy factor r and noise strength K , which exhibits a very rich phenomenon about the strategy transition. In accordance with Fig. 6, when the density of influential players (h) is too small (h = 0 and 0.2) or too large (h = 1.0), there only exist two kinds of phases: full defection (C ) and existence of cooperators and defectors (C + D), while the full cooperation (C ) is absent. In reality, if we carefully observe the cases of h = 0 and h = 1, there is only a type of players (B or A) in the system and the transition from D to C + D is difficult to be implemented, where the larger critical synergy factor (rbc ) between D and C + D is needed and the D-region becomes wider. But when h is increased to 0.2, the D-region tends to become much narrower, and the critical synergy factor rbc is smaller so that the C + D phase can easily reach under this condition. In the meantime,
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Fig. 4. Characteristic snapshots at the stationary state for different model parameters when considering the Moore neighborhood (i.e., k = 8). From top to bottom, at each row the density of A-type players is fixed to be h = 0.2, 0.4, 0.6, 0.8, 1.0, and so on; While for every row, the synergy factor is set to be, from left to right, r = 4.5, 5.5, 6.5, 7.5 and 8.5, respectively, and the only exception is that r takes the value of 8.8 at the most right panel of the bottom row in order to avoid the full defection of players. Among them, the white and green points denote the A-type cooperators (AC) and defectors (AD); red and black ones represent B-type cooperators (BC) and defectors (BD). Other parameters include L = 200, K = 2.4 and w = 0.001. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. (Color online) Dynamical evolution of the fraction of cooperators [ρ(t )] as a function of MCS time step on regular lattices with Moore neighborhood (k = 8), in which each MCS time step can be divided into 1000 sub-steps. It is again demonstrated that cooperators can only survive at the smaller fraction of influential players (h), while the larger h does not support the cooperation. Other parameters are set to be L = 200, K = 2.4, w = 0.001 and r = 6.5.
it is worthy to note that the same K value here means the different noise parameter for different neighborhood size (k) as pointed in Ref. [69]. In addition, when h is smaller, the full C phase may also exist only that the r value is large enough. However, the moderate density of A-type players is set, such that from h = 0.2 to h = 0.8, the full cooperation phase (C ) is possible to emerge in the system except for the D and C + D phases and there are two types of phase transitions
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a
b
c
d
e
f
Fig. 6. Phase diagrams between the synergy factor (r) and noise strength (K ) when two-type of players are staged on the regular lattices with Moore neighborhood (k = 8). In the top two panels, the density of A-type players (h) is 0 (panel (a)) and 0.2 (panel (b)), h is 0.4 (panel (c)) and 0.6 (panel (d)) in the second row, and h is set to be 0.8 (panel (e)) and 1.0 (panel (f)) in the bottom two panels. Different colored regions stand for the collective strategy phases characterizing the full defection (C , Cyan), coexistence of cooperation and defection (C + D, Magenta), and full cooperation (D, Yellow), respectively. Although the full cooperation phase is here absent at small h values, full C phase also exists but larger r value is necessary to reach it. Other parameters are fixed to be L = 200 and w = 0.001, respectively.
including the transition from D to C + D and transition from C + D to C , which are distinguished from the cases of h = 0 or h = 1.0. On one hand, when taking into account the transition from D to C + D, we can observe that the curve composed of critical synergy factor (rbc ) is approximately monotonic except that the curve approaches a kind of weakly bell-shape when h is around 0.4, the rbc curve is almost similar to that of the traditional prisoner’s dilemma game pointed out by Ref. [63]. On the other hand, the transition from the mixing phase to full cooperation one is very different from the phase transition behavior in the classical case where the curve connected by critical value (rtc ) between C + D and C is usually bell-shaped. But in our model, this kind of transition line from C + D to C is inverted bell-shaped, which means that the pessimistic case for the cooperation within the r − K phase space appears and the region allowing for the existence of cooperators and defectors becomes narrower and narrower. Anyway, Fig. 6 indicates that the introduction of moderate influential players into the spatial game model leads to very rich behavior of phase transition, and opens a new route to understand the role of individual diversity in the evolution of collective cooperation.
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4. Conclusions and discussions To sum up, under the framework of public goods game, we investigate the influence of two types of players on the evolution of cooperation in the structured population. In our model, the players located on regular lattices are divided into two classes of ones with different strategy transfer abilities: one class of players (i.e., A-type) is influential where they hold the higher strategy spreading factor wy = 1; while the other class (i.e., B-type) is non-influential in which their strategy transfer coefficient wy = w is a constant between 0 and 1. At the same time, the effect of increasing neighborhood size on the evolution of cooperation is also taken into account. Large quantities of numerical simulations have demonstrated that the cooperative behavior has been greatly promoted by the in-homogeneous teaching capability under a specific ratio of influential players, when compared to the traditional public goods game on the square lattices. As the density of influential players (h) changes, the cooperation behaviors can be dramatically modified. For the Von Neumann (k = 4) and Moore neighborhood (k = 8), there exists the optimal h for the enhancement of cooperation; while for the neighborhood size k = 24, the promotion of cooperation becomes more pronounced as the density of A-type players becomes small. That is, the larger neighborhood with a less fraction of A-type players can be more beneficial for the whole system since the imitation mechanism for the non-influential players favors (disfavors) the cooperation (defection). Meanwhile, the optimal region supporting the cooperation largely depends on the strategy transfer pre-factor w . Furthermore, we plot the full r − K phase diagrams for the Moore neighborhood setup, and observe that the phase transition exhibits a very rich phenomenon which is determined by the density of influential players, and the region of coexistence between cooperators and defectors may be broadened or shrinking as h varies between 0 and 1. In addition, the characteristic snapshots about the distribution of cooperators and defectors and dynamical evolution of fraction of cooperators are also used to illustrate the role of h in the promotion of cooperation. However, the current researches can only cover the constant strategy transfer factor for each type of players, and the adaptive strategy transfer capability deserves to be further explored in the future. In any case, our work is conducive to deeply understanding the evolution of cooperation in the collective population with individual diversity. Acknowledgments This project is partially supported by the National Natural Science Foundation of China under Grant Nos. 61374169, 71201042 and 61203138, the Scientific Research Foundation for the Returned Overseas Chinese Scholars (Ministry of Education) and the Development Foundation for Science and Technology of Higher Education in Tianjin under Grant No. 20130821. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
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Physica A journal homepage: www.elsevier.com/locate/physa
Promotion of cooperation due to diversity of players in the spatial public goods game with increasing neighborhood size Cheng-jie Zhu a,b , Shi-wen Sun a,b , Li Wang a,b , Shuai Ding c,d , Juan Wang e , Cheng-yi Xia a,b,∗ a
Tianjin Key Laboratory of Intelligence Computing and Novel Software Technology, Tianjin University of Technology, Tianjin 300384, PR China b Key Laboratory of Computer Vision and System (Ministry of Education), Tianjin University of Technology, Tianjin 300384, PR China c
Key Laboratory of Process Optimization and Intelligent Decision-Making (Ministry of Education), Hefei University of Technology, Anhui Hefei 230009, PR China d
School of Management, Hefei University of Technology, Anhui Hefei 230009, PR China
e
School of Electrical Engineering, Tianjin University of Technology, Tianjin 300384, PR China
highlights • • • • •
We propose an evolutionary public goods game model with individual diversity. Individual diversity can be characterized by two types of players with different strategy transfer capability. An intermediate fraction of influential players facilitates the collective cooperation. The promotion of cooperation is also influenced by the increasing size of neighborhood. The competition between the individual diversity and interaction range determines the evolution of cooperation.
article
info
Article history: Received 21 January 2014 Received in revised form 2 March 2014 Available online 19 March 2014 Keywords: Cooperation promotion Public goods game Influential players Increasing neighborhood size Spatial reciprocity Individual diversity
∗
abstract It is well-known that individual diversity is a typical feature within the collective population. To model this kind of characteristics, we propose an evolutionary model of public goods game with two types of players (named as A and B), where players are located on the sites of a square lattice satisfying the periodic boundary conditions. The evolution of the strategy distribution is governed by iterated strategy adoption from a randomly selected neighbor with a probability, which not only depends on the payoff difference between players, but also on the type of the neighbor. For B-type agents, we pose a prefactor (0 < w < 1) to the strategy transfer probability, which implies the lower teaching activity or strategy convincing performance; but w is always set to be 1 for A-type agents, hence it means that A-type players are influential ones who own a larger strategy spreading chance. Furthermore, we also consider the competition between two opposite effects when the number of nearest neighbors (k) is increased from 4 to 24. Within a range of the portion of A-type influential players, the inhomogeneous teaching activity in strategy transfer yields a relevant increase (dependent on w ) in the density of cooperators characterizing the promotion of cooperation. Current findings are of utmost importance for us to understand the evolution of cooperation under many real-world circumstances, such as the natural, biological, economic and even social systems. © 2014 Elsevier B.V. All rights reserved.
Corresponding author. Tel.: +86 2260216865. E-mail addresses: [email protected] (L. Wang), [email protected] (C.-y. Xia).
http://dx.doi.org/10.1016/j.physa.2014.03.035 0378-4371/© 2014 Elsevier B.V. All rights reserved.
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1. Introduction Although the defection is an evolutionarily stable strategy (ESS) which denotes the dominant one from an individual viewpoint, cooperation is often found to be persistent and emergent within many real-world systems (e.g., natural, biological, economic and even social systems) [1,2]. Thus, how to interpret the evolution of cooperation under real situations becomes a challenging topic, which attracts the attention of many researchers from ecology, biology, mathematics, economics, physics, engineering science and so on [3,4]. Among them, the evolutionary game theory provides an effectively theoretical framework which has shed some light upon this long-standing puzzle [5,6]. For instance, various game models are proposed to illustrate the evolution of cooperation behavior in a population of selfish individuals [1–6], and one of the most investigated games is the prisoner’s dilemma game (PDG) [7–22] which is a classical two-player game model. In the PDG, each player can only adopt two pure strategies: to cooperate (C) or to defect (D); Then, the players will obtain their payoff by playing the game in pairs, where they will get the reward (R) and punishment (P) payoff if they take the same action; while a defecting individual will be tempted to acquire a higher payoff (T ) and the cooperating one can only receive the sucker’s payoff (S) provided that they choose the distinct strategies. The PDG becomes valid whenever the condition T > R > P > S is satisfied and unfortunately, the well-mixing population does not facilitate the collective cooperation, and some additional mechanisms need to be added to the PDG so that the cooperation thrives. It deserves to be mentioned that Nowak and May [23] seminally introduced the spatial structure into the PDG to render the emergence of cooperation as cooperative clusters come into being within the system. Starting from this point, a vast majority of works are devoted to exploring the evolution of cooperation in the spatially structured population based on the PDG (referring to the recent reviews [24–26]). Analogously, T > R > S > P creates another typical pair-wise game: snowdrift game (SDG), and the cooperative behaviors of SDG in the structured population also receive a great deal of concern within the scientific communities [27–31]. Recently, large quantities of evidences have demonstrated that the interaction structure of real-world systems is neither well-mixing nor regular, and to a certain extent exhibits a small world effect [32] or scale-free [33] property. Systems of such complex topology are usually named as the so-called complex networks in which nodes characterize the interacting agents and links mimic their interactions, and complex networks can be found within almost all natural, social and manmade engineering systems [34–36]. Meanwhile, the cooperative behaviors are also found to be greatly promoted when players interact on complex networks [25,26,37], especially provided that the network of contacts is scale-free [38–43] or the individual behavior exhibits some heterogeneity [44,45]. Conversely, complex networks can be generated by implementing a specific growth mechanism in which newcomers are linked to old nodes with a probability that is determined not by their degree, but by their benefit or fitness [46–48]. Additionally, the coevolution between the structure and cooperation behavior is studied in many works [49–51], and the evolution of cooperation in interdependent or multiplex networks has also become an active topic in the very recent years [52,53]. However, aforementioned works do not take into account the group effect or interaction among agents [54], which, to a greater extent, characterizes the public cooperation, such as social insurance, global climate change, free trade zone, to name but a few. As a stylized formalization, public goods game (PGG) is often used to elucidate the role of group interaction in the public cooperation [55], in which each individual within a group can independently and voluntarily choose to contribute or not. Likely, we can identify two different strategies: to cooperate (C) if the player contributes, or to defect (D) otherwise. Each cooperator contributes 1 to the group, but the defector contributes nothing. Then, the total contribution will be multiplied by a synergy factor r > 1 which represents the benefit reached by cooperating. Finally, the gained contribution is evenly distributed among all players within this group, irrespective of their contribution. Assuming that there are nc cooperators within a PGG group of n players, the payoff obtained by each cooperator and defector can be written as follows,
πc = r ∗ nc /n − 1 πd = r ∗ nc /n
(1)
where πc and πd denote the corresponding payoffs of a cooperator and a defector, and r is a synergy factor greater than 1. It is clearly shown that the defecting player has always larger payoff than the cooperator in the same PGG group [56]. As such, defection is a strictly dominant strategy, and hence D becomes an evolutionary stable strategy. This finding implies that any body will not contribute to the group, but full cooperation will obviously lead to the better result if r > 1. Therefore, a social dilemma arises since full cooperation is more profitable for any player than full defection, whereas the individual selfish interest prevents the desirable outcome and the tragedy of commons is widespread within the field of public cooperation [57]. Therefore, how to avoid this tragedy poses an urgent task facing the scientific communities [58–60]. So far, a large body of literature focuses on the role of pair-wise or group interactions in the elevation of cooperation level [25,26,54], but there is a rigorous assumption that each player owns the identical strategy enforcement probability during the game playing, that is, the individual diversity is usually ignored and it is far away from the realities under a couple of realworld circumstances. For example, during the epidemic outbreaks for a specific disease (e.g. influenza), the adolescent and old people often have lower immunity and face a higher risk, while adults may contract this disease with the fewer chance as they are often much stronger. To characterize this type of diversity, the inhomogeneous strategy adoption probability is firstly mentioned in the evolutionary PDG model [61,62]. Among them, Szabó et al. [63,64] propose a novel spatial PDG with two types of players located on the sites of a square lattice, and they correlate the strategy adoption probability with the payoff difference and the type of players, in which the teaching ability is distinguished for two types of players. They find that the density of cooperators can be relatively increased by the nonidentical teaching behavior within a range of
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the portion of influential players, meanwhile the promotion of cooperation is also influenced by the size of neighborhood. However, their works can only cover the PDG model with pair-wise interaction on the square lattice and group effect is not taken into account. In the present study, we extend this work into the public goods game model to further investigate the role of group interaction in the public cooperation. Our results indicate that the density of influential players (h) and the neighborhood size (k) can noticeably influence the behavior of public cooperation, and the intermediate h often leads to the emergence of cooperation for a specific neighborhood setup and individual strategy adoption behavior. Meanwhile, our model is also different from the work of Guan et al. [65] which mainly focuses on the effect of in-homogeneous activity and noise parameter on the cooperative behaviors in the spatial public goods game. The remainder of this paper is organized as follows. At first, we in detail introduce the PGG model with two types of players in Section 2. Then large-scale numerical simulations are presented in Section 3. At last, the concluding remarks and discussions are provided in Section 4. 2. The PGG model with two types of players In our public goods game model, two types of players (tx = A or B) are placed on the sites x of an L × L regular lattice with periodic boundary condition, the fraction of A and B-type players are set to be h and 1 − h, which are randomly distributed on the square lattice and remain unchanged during the simulation. Additionally, every player x has k nearest neighbors and is designated as a cooperator (C ) sx = 1 or a defector (D) sx = 0 with equal probability (i.e., 50%) at the initial time step. Each player needs to attend G = k + 1 PGG groups where one group is centered around himself and other k groups focus on his nearest neighbors. In every PGG group, all players can simultaneously decide whether to make a contribution, and we count the number of cooperators so as to calculate the PGG payoff according to Eq. (1). Thus, the total payoff of a player x can be summed over as follows, Px =
Pxg
g ∈Ωx
s j∈g j − sx = r g ∈Ωx
=
G
r ∗ ngc g ∈Ωx
G
− sx
(2)
where Ωx denotes the set of PGG groups in which player x participates and g is one element in Ωx , sx and sj stand for the g strategy of player x and j, nc is the number of cooperators within group g and G is the total number of PGG groups. After that, the system evolution proceeds with a random sequential strategy update (i.e., asynchronous update). Player x picks up a nearest neighbor y (whose payoff can be computed in the same way as player x) stochastically and tries to imitate the strategy sy with the following probability: 1
Prob(sx ← sy ) = wy 1+e
(Px −Py )/k
(3)
K
where (Px − Py )/k characterizes the difference of normalized payoffs between x and y, K denotes the uncertainty affecting the strategy adoption of player x which is analogous to the temperature as introduced in the kinetic Ising model, and the multiplicative factor wy signifies the strategy spreading capability of player y and can be expressed in the following way,
wy =
1,
w,
if ty = A if ty = B, 0 < w < 1.
(4)
Here we can observe that A-type players hold the higher probability to pass their current strategy on to other ones during the strategy evolution, that is, they can easily convince their neighbors to adopt the strategy they are just following. Hence, A-type agents are influential ones in our model. On the contrary, B-type individuals are non-influential ones who have lower possibility to impose their own strategy to other ones. To a certain extent, this feature can characterize the individual diversity often found within many real-world networks, and can be correlated with the personal reputation, power, age and so on. A full Monte Carlo step (MCS) includes the above-depicted basic steps, and each player has a chance once on average to modify the current strategy into that of one of nearest neighbors. After a suitable relaxation time (tr ), the system develops into the stationary state characterized by the average fraction of cooperators (ρ ), which is the quantity of most interest and determined by averaging the fraction of cooperators within another ta time steps. Furthermore, we consider three different neighborhood setups as shown in Fig. 1: k = 4 (Von Neumann neighborhood), k = 8 (Moore neighborhood) and k = 24 neighbors, and further discuss the role of game interaction range during the evolution of cooperation. Larger neighborhood size (e.g., k = 48) is also explored and has the qualitatively same behavior as the case of k = 24 and will not be presented here.
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a
b
c
Fig. 1. Three typical neighborhood setups used in our PGG model. (a) Left panel: Von Neumann neighborhood (k = 4); (b) Middle panel: Moore neighborhood (k = 8); (c) Right panel: k = 24 neighborhood setup.
3. Numerical simulation results Extensive numerical simulations are performed to explore the behavior of our PGG model on L × L square lattices with periodic boundary conditions. Main simulation parameters are set to be L = 200, the relaxation time tr = 5 × 104 MCS and ta = 104 MCS, and other parameters will be provided during the following analysis. At first, in Fig. 2, we will illustrate the relationship between the fraction of cooperators (ρ ) at the stationary state and normalized synergy factor r /G (which is utilized so as to compare the results for different neighborhood setups, while we still use the usual synergy factor r in other figures). As expected, a general trend is observed in Fig. 2 that the cooperation level will be elevated as the synergy factor (r /G) increases under a given fraction (h) of A-type players, but h will also markedly influence the cooperative behavior on the square lattice. When h is small (e.g., h = 0.2), ρ transcends 0 for a very low synergy factor (r /G) (r /G at this point is called the lower critical synergy factor (r /G)cb ) and continuously increases into the full cooperation. In fact, smaller h means that there are fewer influential individuals who have the stronger strategy transfer ability and other individuals are non-influential. Thus, the influential cooperators can have a chance to impose their strategy into other ones, and then help the population to escape from the full defection at a lower synergy factor. After that, fewer influential cooperators need enough stimulation to convince all players to reach a consensus (i.e., full cooperation). On the other hand, the full defection cannot be avoided until higher r /G is reached for a larger h(= 0.8), that is, (r /G)cb is higher, but the cooperation can be sharply increased into the state of full cooperation (ρ = 1.0) once (r /G) is over (r /G)cb . Likely, larger h signifies that there are adequate influential players who compete for the cooperation, and it is difficult for cooperators to win over defectors, but the cooperation will be sharply enhanced into a very high level provided that contributing agents gain the advantage during the evolution of cooperation. At the same time, there are subtle differences for various neighborhood setups since it covers the interaction range in which the players participate. It can be clearly observed that the larger the neighborhood size k, the lower the critical synergy factor (r /G)cb . In fact, the underlying topology will be approaching the well-mixed case which eventually leads to the full defection for any finite value of r as k becomes larger and larger. In addition, according to Fig. 2, the promotion of cooperation will become more pronounced as the neighborhood size is larger. For example, when k is equal to 24, the cooperation level ρ can suddenly increase from 0% to 100% within a narrower interval of r even in the lower density of A-type players (i.e., h = 0.2) after (r /G)cb is surpassed. Whereas for k = 4, the fraction of cooperators (ρ ) will be continuously elevated as (r /G) varies for lower values of h (i.e., h = 0.2, 0.4), but the full cooperation cannot be arrived at even though r /G is increased to 1.2. The situation of k = 8 is rightly in between the above-mentioned two cases. For the spatial lattice with different neighborhood, the only difference between k = 4 and 24 may come from the fact that the clustering effect exists in the case of k = 24, but is absent in k = 4. A large number of evidences have demonstrated that the clustering among players is an important factor to facilitate the cooperators to create the giant clusters to resist the employment of defectors [25,26, 66–68]. Thus, the public cooperation can be dramatically enhanced under the situation which allows two types of players with different imitating abilities to exist within the population, especially when the density of influential players is lower for the larger neighborhood range. In order to explore the role of fraction of A-type influential players (h), we illustrate the effect of h on the stationary density of cooperators ρ for different w values under the fixed synergy coefficient r in Fig. 3. For brevity, here we can only consider the Moore neighborhood (k = 8) in the current simulation setup and results for k = 4 and k = 24 are neglected. The numerical results clearly indicate that, within a range of A-type influential players (i.e., h ∈ [0.15, 0.5]), different strategy transfer capability w usually leads to the optimal promotion of cooperation, especially in the case of w < 1, and it is worth noting that the smaller the multiplicative factor w , the more prominent the optimized elevation of cooperation. In other words, the discrepancy of teaching ability between two types of players supports the emergence of optimal cooperation phenomenon, and the optimal cooperation disappears when two types of players hold the same strategy spreading probability (i.e., w = 1). Furthermore, too few influential players are not enough to attract the players to create the giant clusters as much as possible, but the cooperation will also deteriorate when too many A-type players lie within the whole population and compete with each other. On the contrary, intermediate density of influential players allows enough space to form the
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a
r/G
b
r/G
c
r/G Fig. 2. (Color online) The fraction of cooperators (ρ ) as a function of normalized synergy factor (r /G) under various neighborhood setups. The density of influential players and different neighborhood setups nontrivially influences the collective cooperation level in our PGG model. The larger the neighborhood size, the more pronounced the role of influential players. From panel (a) to panel (c), the neighborhood size is set to be 4, 8 and 24, respectively. Other parameters are set to be L = 200, K = 2.4 and w = 0.001.
cooperative giant clusters so as to arrive at the optimal cooperation in the lattice. In a word, existence of two types of players and resulting discrepancy of individual teaching ability facilitates the evolution of cooperation on the structured population. The characteristic snapshots of players with different types and strategies are shown in Fig. 4 so that we can observe the distribution of cooperators and defectors at the stationary state. Likely, we can only consider the Moore neighborhood,
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Fig. 3. (Color online) Relationship between stationary density of cooperators (ρ ) and fraction of A-type individuals (h) for different multiplicative factors w on regular lattices with Moore neighborhood (k = 8). The existence of influential player or strategy transfer discrepancy yields the optimal cooperation level among the population in which the fraction of influential players often lies between 0.15 and 0.5 as w < 1, but the cooperation will disappear when all players have the identical strategy transfer probability (w = 1). Other parameters are set to be L = 200, K = 2.4 and r = 6.5.
qualitatively similar snapshots are also present for Von Neumann neighborhood and 24-neighborhood. As expected, the cooperation level will be greatly increased as the synergy factor r becomes higher after r crosses over the critical threshold rbc . In particular, A-type cooperators (AC) and B-type cooperators (BC) readily interweave together to create the cooperative clusters to defend not to be invaded by A-type defectors (AD) and B-type defectors (BD), and the smaller the density h, the easier the clusters of cooperators are formed when the synergy factor r is fixed. Meanwhile, ACs are usually surrounded by BCs and BDs often encircle ADs, and this implies again that A-type players are influential ones in our model. With respect to the role of density of A-type h, lower h will facilitate the evolution of cooperation when r is smaller (e.g., r <= 5.5); while for larger r (e.g., r ≥ 5.5), there exists an intermediate density h, which often lies between 0.15 and 0.5, to lead to a circumstance that allows the survival of cooperators as many as possible. Henceforth, ACs act as a determinant role for the emergence of cooperation once again since the ACs eventually suppress the ADs and impose the cooperation strategy into their neighbors regardless of the type of neighbors. Current snapshots well agree with the above-mentioned results in the previous two figures, and dramatically enrich the understanding of spread of cooperative strategies. By examining the dynamical evolution of fraction of cooperators in the whole population, which is described in Fig. 5, we can further understand the role of two types of players in the promotion of cooperation under a specific synergy factor (r = 6.5). In Fig. 5, the multiplicative factor of strategy spreading is fixed to be w = 0.001 which means that the difference between two types of players is very large, and we can modify the density of influential players h to observe the evolution of cooperation. When h is smaller (e.g., h = 0.2 or 0.4), the fraction of cooperators ρ(t ) decreases into a minimum at first and then ρ(t ) is little by little elevated into a higher stationary value; while for higher h (e.g., h = 0.6, 0.8 and 1.0), ρ(t ) is continuously reduced to 0 which means a full defection status. From this phenomenon, it is indicated that the lower h is beneficial for the influential cooperative player to spread their strategy to other neighbors, further to create the cooperative clusters to defend them not to be invaded by the defectors. However, when h is large enough (h > 0.5) to approach the traditional spatial public goods game, in which there is a large quantity of influential players in the population and some of them may act as the barrier of strategy spreading of A-type cooperators so that the cooperators cannot resist the employment of defectors and thus results in the very low cooperation level, which is similar to the results shown in Fig. 4. In fact, from Fig. 3 we can also observe that when h surpasses 0.5, the stationary fraction of cooperators will dramatically descend, and even the full defection state is arrived at as h is larger than 0.54. The simulation results in Fig. 5 again verified this nontrivial influence of two types of players in the evolution of cooperation, and influential players can effectively impose their strategy only when there exist a variety of non-influential players in the whole population. Thus, the existence of players with different strategy transfer capability may be responsible for the promotion of cooperation, especially when influential players can only hold a very small ratio within two types of players. At present the effect of noise strength on the cooperation behavior is ignored where K is often set to be K = 2.4, and it is necessary to discuss the strategy evolution in the wider range of K . Fig. 6 depicts the full phase diagrams between synergy factor r and noise strength K , which exhibits a very rich phenomenon about the strategy transition. In accordance with Fig. 6, when the density of influential players (h) is too small (h = 0 and 0.2) or too large (h = 1.0), there only exist two kinds of phases: full defection (C ) and existence of cooperators and defectors (C + D), while the full cooperation (C ) is absent. In reality, if we carefully observe the cases of h = 0 and h = 1, there is only a type of players (B or A) in the system and the transition from D to C + D is difficult to be implemented, where the larger critical synergy factor (rbc ) between D and C + D is needed and the D-region becomes wider. But when h is increased to 0.2, the D-region tends to become much narrower, and the critical synergy factor rbc is smaller so that the C + D phase can easily reach under this condition. In the meantime,
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Fig. 4. Characteristic snapshots at the stationary state for different model parameters when considering the Moore neighborhood (i.e., k = 8). From top to bottom, at each row the density of A-type players is fixed to be h = 0.2, 0.4, 0.6, 0.8, 1.0, and so on; While for every row, the synergy factor is set to be, from left to right, r = 4.5, 5.5, 6.5, 7.5 and 8.5, respectively, and the only exception is that r takes the value of 8.8 at the most right panel of the bottom row in order to avoid the full defection of players. Among them, the white and green points denote the A-type cooperators (AC) and defectors (AD); red and black ones represent B-type cooperators (BC) and defectors (BD). Other parameters include L = 200, K = 2.4 and w = 0.001. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. (Color online) Dynamical evolution of the fraction of cooperators [ρ(t )] as a function of MCS time step on regular lattices with Moore neighborhood (k = 8), in which each MCS time step can be divided into 1000 sub-steps. It is again demonstrated that cooperators can only survive at the smaller fraction of influential players (h), while the larger h does not support the cooperation. Other parameters are set to be L = 200, K = 2.4, w = 0.001 and r = 6.5.
it is worthy to note that the same K value here means the different noise parameter for different neighborhood size (k) as pointed in Ref. [69]. In addition, when h is smaller, the full C phase may also exist only that the r value is large enough. However, the moderate density of A-type players is set, such that from h = 0.2 to h = 0.8, the full cooperation phase (C ) is possible to emerge in the system except for the D and C + D phases and there are two types of phase transitions
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a
b
c
d
e
f
Fig. 6. Phase diagrams between the synergy factor (r) and noise strength (K ) when two-type of players are staged on the regular lattices with Moore neighborhood (k = 8). In the top two panels, the density of A-type players (h) is 0 (panel (a)) and 0.2 (panel (b)), h is 0.4 (panel (c)) and 0.6 (panel (d)) in the second row, and h is set to be 0.8 (panel (e)) and 1.0 (panel (f)) in the bottom two panels. Different colored regions stand for the collective strategy phases characterizing the full defection (C , Cyan), coexistence of cooperation and defection (C + D, Magenta), and full cooperation (D, Yellow), respectively. Although the full cooperation phase is here absent at small h values, full C phase also exists but larger r value is necessary to reach it. Other parameters are fixed to be L = 200 and w = 0.001, respectively.
including the transition from D to C + D and transition from C + D to C , which are distinguished from the cases of h = 0 or h = 1.0. On one hand, when taking into account the transition from D to C + D, we can observe that the curve composed of critical synergy factor (rbc ) is approximately monotonic except that the curve approaches a kind of weakly bell-shape when h is around 0.4, the rbc curve is almost similar to that of the traditional prisoner’s dilemma game pointed out by Ref. [63]. On the other hand, the transition from the mixing phase to full cooperation one is very different from the phase transition behavior in the classical case where the curve connected by critical value (rtc ) between C + D and C is usually bell-shaped. But in our model, this kind of transition line from C + D to C is inverted bell-shaped, which means that the pessimistic case for the cooperation within the r − K phase space appears and the region allowing for the existence of cooperators and defectors becomes narrower and narrower. Anyway, Fig. 6 indicates that the introduction of moderate influential players into the spatial game model leads to very rich behavior of phase transition, and opens a new route to understand the role of individual diversity in the evolution of collective cooperation.
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4. Conclusions and discussions To sum up, under the framework of public goods game, we investigate the influence of two types of players on the evolution of cooperation in the structured population. In our model, the players located on regular lattices are divided into two classes of ones with different strategy transfer abilities: one class of players (i.e., A-type) is influential where they hold the higher strategy spreading factor wy = 1; while the other class (i.e., B-type) is non-influential in which their strategy transfer coefficient wy = w is a constant between 0 and 1. At the same time, the effect of increasing neighborhood size on the evolution of cooperation is also taken into account. Large quantities of numerical simulations have demonstrated that the cooperative behavior has been greatly promoted by the in-homogeneous teaching capability under a specific ratio of influential players, when compared to the traditional public goods game on the square lattices. As the density of influential players (h) changes, the cooperation behaviors can be dramatically modified. For the Von Neumann (k = 4) and Moore neighborhood (k = 8), there exists the optimal h for the enhancement of cooperation; while for the neighborhood size k = 24, the promotion of cooperation becomes more pronounced as the density of A-type players becomes small. That is, the larger neighborhood with a less fraction of A-type players can be more beneficial for the whole system since the imitation mechanism for the non-influential players favors (disfavors) the cooperation (defection). Meanwhile, the optimal region supporting the cooperation largely depends on the strategy transfer pre-factor w . Furthermore, we plot the full r − K phase diagrams for the Moore neighborhood setup, and observe that the phase transition exhibits a very rich phenomenon which is determined by the density of influential players, and the region of coexistence between cooperators and defectors may be broadened or shrinking as h varies between 0 and 1. In addition, the characteristic snapshots about the distribution of cooperators and defectors and dynamical evolution of fraction of cooperators are also used to illustrate the role of h in the promotion of cooperation. However, the current researches can only cover the constant strategy transfer factor for each type of players, and the adaptive strategy transfer capability deserves to be further explored in the future. In any case, our work is conducive to deeply understanding the evolution of cooperation in the collective population with individual diversity. Acknowledgments This project is partially supported by the National Natural Science Foundation of China under Grant Nos. 61374169, 71201042 and 61203138, the Scientific Research Foundation for the Returned Overseas Chinese Scholars (Ministry of Education) and the Development Foundation for Science and Technology of Higher Education in Tianjin under Grant No. 20130821. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
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