2017, Vol.22 No.6, 529-534 Article ID 1007-1202(2017)06-0529-06 DOI 10.1007/s11859-017-1284-8
Emergence of Group Cooperation in Public Goods Game on Regular Small-World Network □ ZHANG Yingqing, FAN Ruguo†, LUO Ming
0
Introduction
Economics and Management School, Wuhan University, Wuhan 430072, Hubei, China © Wuhan University and Springer-Verlag Berlin Heidelberg 2017
Abstract: The regular small-world network, which contains the properties of small-world network and regular network, has recently received substantial attention and has been applied in researches on 2-person games. However, it is a common phenomenon that cooperation always appears as a group behavior. In order to investigate the mechanism of group cooperation, we propose an evolutionary multi-person game model on a regular small-world network based on public goods game theory. Then, to make a comparison of frequency of cooperation among different networks, we carry out simulations on three kinds of networks with the same configuration of average degree: the square lattice, regular small-world network and random regular network. The results of simulation show that the group cooperation will emerge among these three networks when the enhancement factor r exceeds a threshold. Furthermore, time required for full cooperation on regular small-world network is slightly longer than the other networks, which indicates that the compact interactions and random interactions will promote cooperation, while the longer-range links are the obstacles in the emergence of cooperation. In addition, the cooperation would be promoted further by enhancing the random interactions on regular small-world network. Key words: regular small-world network; public goods game; group cooperation CLC number: N 941 Received date: 2017-06-20 Foundation item: Supported by the National Natural Science Foundation of China (71601148), the National Social Science Foundation of China ( 14ZDA062) and Humanities and Social Science Research Foundation of Ministry of Education (14JDGC012) Biography: ZHANG Yingqing, male, Ph. D. candidate, research direction: complex science management, spatial evolutionary game and complex network. E-mail:
[email protected] † To whom correspondence should be addressed. E-mail:
[email protected]
Cooperation is ubiquitous in the real world, ranging from biological systems to social systems. However, revealing the mechanisms responsible for group cooperation remains a challenge. Up to now, there is a ton of researches revealing that evolutionary game theory is a successful paradigm for properly addressing cooperative behaviors, such as the Prisoners Dilemma Game (PDG) and the Snowdrift Game (SG)[1]. In particular, the Public Goods Game (PGG) provides the theoretical framework for properly addressing group interactions [2]. In the classical PGG model, it is difficult to maintain the cooperation in the end due to the additional cost for cooperation. But the group cooperation widely exists in the real world. This arouses the question of how to explain the emergence and evolution of group cooperation by the public goods game theory. In the last few years, some mechanisms of volunteering [3], reputation [4], rewards or punishment [5] have been proposed and indeed play crucial roles in the emergence of cooperation. Unfortunately, there are still some phenomena that cannot be explained by those mechanisms, such as the second-order free-riders [6] or the anti-punishment [7]. Benefiting from the network science and complexity theory, networks structures have attracted mounting attention. It is worth mentioning that a seminal research introduced by Nowak et al [8] has explored evolutionary games on spatial lattices, and showed that spatial topologies formed by the local interactions among the players can also greatly improve co-
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operation level in absence of mechanisms of the reward and the punishment, which can be called the spatial or network reciprocity[9]. The introduction of network triggers the development of evolutionary game theory, which are collectively called the spatial evolutionary game. So far, the researches of spatial public good game have been studied in depth. In broad strokes, these literatures have focused on two different networks: homogeneous and heterogeneous networks. The square lattice, kagome network, hexagonal lattice and so on belong to homogeneous networks, while small-world network, BA scale-free network, or community network are classified as heterogeneous networks. From these previous studies we can see that homogeneous networks are a class of simple abstract networks that can subtract effect from the heterogeneity originated from individual network position. So the mechanisms, such as punishment, reward or reputation, will be purely investigated on such homogeneous networks. However, these networks are not quite consistent with real world, such as small-world or power-law property. On the other hand, the heterogeneous networks basically coincide with real world, thereby have been paid more attention to. But a major drawback is that it is difficult to eliminate the influence of individual heterogeneity associated with network position. Thus, an underlying network is needed to make trade-off between the homogeneous and heterogeneous properties. Based on this consideration, Szabó et al [10] proposed an algorithm to generate the regular small-world network, which can incorporate small world property into a regular network. In such network, not only has every individual the same degree, but also the network has a high clustering coefficient as well as low average path length[11]. Because of this, many evolutionary game studies have focused on such network. For example, Szabó et al [10] firstly used a regular small-world structure to describe the fixed partnership, then investigated the effects of anneals and quenched randomness based on the voluntary PDG with three strategies, and found that partially random partnerships played an important role in the maintenance of cooperation. Subsequently, Guan et al [12] have studied the PDG with nonlinear attractive effect on regular small-world network, and showed that the emergence and persistence of cooperation could be induced by this simple rules, and also pointed that long-range links can be a double-edged sword for cooperation. In addition, Wu et al [13] have introduced a PDG with heterogeneous influence of different individual on regular small-world networks, and showed that this sim-
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ple preferential selection rule can promote cooperation in the whole population by means of enhancing the random interactions of the underlying network. However, to the best of our knowledge, massive works mainly focused on the 2-person games instead of multi-person games. Inspired by this observation and in accordance with most already works, in this paper, we consider a multiperson game based on the PGG model on regular small-world network to investigate the emergence and the internal mechanism of group cooperation. Our results show that the longer-range links are the obstacles in the process of cooperation, while the local interactions and random interactions will promote cooperation. The paper is organized as follows. Section 1 introduces the regular small-world network briefly. The model is proposed in Section 2. Then the corresponding simulation results are given in Section 3. At last, the conclusions are provided in Section 4.
1 The Regular Small-World Network According to the current literatures, many real networks can be regarded as a small world network, social network with community structure, and BA scale-free network with the property of power law [14,15]. Individual generally are heterogeneous in above-mentioned networks, and a great deal of researches indicate that heterogeneity has a profound effect on cooperation [16]. Besides, what other mechanisms are responsible for emergence of cooperation? In order to answer this question, the factor of heterogeneity of network should be subtracted at first. Therefore, a grid or lattice network often used to be the structure population of evolutionary game. A grid or lattice network is called regular network, in which individuals have the same degree. And a regular small-world network is a particular regular network with the small-world property. A regular small-world network can generally be represented as set G=(V, E). Here, V and E are the sets of all nodes and edges, respectively, on which each node denotes individual or player and the edges denote the relations between them. As shown in Fig.1 and Fig.2, a square lattice and a regular small-world network are represented, respectively. As described in Ref.[10], a regular small-world network starts from a square lattice. Take an example as depicted in Fig.2, firstly, AJ link between individual A and J is chosen randomly and removed. Then, individual
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Fig. 3 Fig. 1
Illustration of average clustering coefficient (Aver-C)
and average path length (Aver-P) with the increment of
Structure of a square lattice
rewiring probability p
2
The Model
2.1
Fig. 2
Structure of a regular small-world network
A is rewired to the randomly chosen individual B. In order to keep fixed degree on every individual, individual B should eliminate one of the previous links (here BC). Similarly, individual C should add a new link (here CD) so as to keep the same degree. This process is repeated. Finally, a regular small-world network is constructed until all individuals with the same degree. By adjusting the rewiring probability p, different regular small-world networks with different properties will be constructed. If p=0 and p=1, the structures reproduce two extreme cases: a square lattice and random regular network, respectively. As shown in Fig.3, once the rewiring probability exists , the average path length will be decreased precipitously, while the average clustering coefficient is changed more gently. As we all known, small-world property is the network structure with low average path length and high average clustering coefficient. Namely, the regular small-world network with significant small-world property can be constructed by lower rewiring probability p.
Social Diversity Social diversity represents that every individual of social network will not participate in the game in one group, but often in multiple groups formed by himself and all his neighbors [17], which is used for depicting the situations for multiple social circles or enterprise cluster organizations. Taking the individual A of a square lattice in Fig.4 as an example [18], player A has 4 neighbors and will totally participate in 5 PGG in groups centered on him and all his neighbors, namely group A is the group centered on player A and the other four groups B,C,D and E are the groups centered on all neighbors of player A, respectively.
Fig. 4
A schematic example network with multiple groups around the individual A
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2.2
The Spatial PGG Model In particular, the Public Goods Game (PGG) provides reasonable explanation for group interactions of multiple players, such as environment problems, public resources and team work. According to the definition of the study [19,20], in the typical PGG model, cooperators contribute a cost (c=1) into the common pool to realize the public goods, while defectors contribute nothing in each of groups. Then the total investments are multiplied by an enhancement factor r(r>1) and equally redistributed among all players regardless of their contributions. Under the theoretical analysis, cooperators will gain less while defectors will get more. As a result, the difficulty of cooperation maintains in the end due to the additional cost for cooperation. This can be called a typical social dilemma. In this research, we use typical PGG model with social diversity to represent the social dilemma among multiple groups. The PGG game is carried out on the three networks (square lattice, regular small-world network and random regular network) with total L2 individuals. Therefore, in each group with G individuals, the payoffs for the cooperator and defector are given as mx , y r c
Nc sx c G
(1)
where sx denotes the strategy of individual x( sx =1 if Cooperator, sx =0 if Defector), y stands for one group of individual x composed by y and his neighbors, y x , x denotes the groups set belonging to individual x, Nc represents the total cooperators in group y, Nc ≤ G , c is the cost, for simplicity, we set c 1 , r (r>1) is the multiplication factor, which can be characterized as the profitability of public goods. Then the total payoff of individual x can be expressed as (2) M x mx , y yx
2.3 The Strategy Updating Rule of Spatial PGG Model Initially, each player chooses rationally a strategy with equal probability, and obtains payoff based on the Equations (1) and (2). Here, in each run, each player has a chance to update his strategy averagely according to the Monte Carlo simulation method. The individual x will update his strategy sx by learning the strategy sy of a randomly selected neighbor individual y with the following Fermi rule [21],
W ( sx s y )
1 1 exp[( M x M y ) / K ]
(3)
Here K characterizes the bounded rationality or uncertainty of individuals during the course of making decision. When K→0 and K→∞ denote the completely deterministic learning and random learning, respectively. In this paper, we set K=0.1 as in Ref. [20]. M x and M y are the total payoff of individual x and y, respectively. According to the Fermi rule, individual x will learn the strategy sy from individual y if M∨y M x .
3 Simulation Results and Discussions We assume that individuals can be seem as nodes of networks, and each node has G=4 neighbors, in other words, everyone will participate in PGG in 5 groups. All simulations are carried out on these three networks (square lattice, regular small-world network and random regular network) with LL=900. The important index for characterizing the cooperative behavior is the frequency of cooperation, which is defined as the ratio of the cooperators in the total number of individuals at the steady state. Initially, the individual is randomly distributed to be a cooperator or a defector with the equal probability, namely the initial ratio of cooperators is 0.5. The results are obtained by averaging from last 50 time steps of total 500 time steps. Inspecting from Fig.5, it is clear to see that the frequency of cooperation monotonically increases as the enhancement factor r increases on the whole, and the rate of increase of square lattice is faster than that of regular small-world network. Our results on the square lattice is analogous to the result of Ref.[19], which shows that the cooperation will be survived when the enhancement factor r is bigger than 3.74. What is the mechanism responsible for the difference between square lattice and regular small-world network in terms of frequency of cooperation? The mechanism responsible for square lattice is analyzed firstly. From the stylized example as shown in Fig.1, all individuals are restricted to play games with his four neighbors and the groups are centered on them, which can be considered as a close relationship. When the enhancement factor r exceeds a threshold (here 3.74), the cooperative clusters can be emerged easily on such compact spatial structure formed by local neighbors, so as to resist the invasion of defectors. While on the regular small-world network, the close relationship between
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local neighbors will be disrupted due to the longer-range links of regular small-world network, so the defectors will invade the cooperative clusters. As a result, the number of cooperators will decrease under the regular small-world network.
Fig.6
Frequency of cooperation as a function of enhancement
factor r for different rewiring probability p on square lattice (SL, if p=0), regular small-world network (RSW, if 0<p<1) and random regular network (RRN, if p=1) Fig.5
Frequency of cooperation as a function of the
enhancement factor r on square lattice (SL) and regular small-world network (RSW), p=0.1
Therefore, our results corroborate the argument that the compact interactions may promote the cooperation [8], and also find that longer-range links of regular smallworld network will inhibit the cooperation, but are different from the results based on the prisoner’s dilemma game which indicates that the level of cooperation is either promoted or inhibited by longer-range links [12]. We attribute this phenomenon to the difference between 2-persons game and multi-persons game. To further investigate the mechanisms of the evolution of cooperation, we carry out simulations on three kinds of networks by changing the rewiring probability p. When p=0, a square lattice will be constructed, in which every individual is connected to 4 nearest neighbors, and at the other extreme p=1, a random regular network is constructed where each individual being connected to 4 randomly chosen neighbors. Between the two extremes (0<p<1), the mixed situation of these two scenarios, namely the regular small-world network, will be constructed. For the sake of simplicity, herein we consider three values of regular small-world network, p=0.02, p=0.1 and p=0.5, respectively. The corresponding results are shown in Fig.6. As shown in Fig.6, it can be clearly seen that the cooperation will emerge among the square lattice, regular small-network and random regular network when the enhancement factor r exceeds a threshold. As compared with the other networks, time required for full cooperation on regular small-world network is slightly longer. As
analysis above, we can also attribute the decrease of cooperation to the longer-range links of the regular smallworld networks. On further analysis, we can see that the cooperation will be promoted when the rewiring probability becomes bigger (p=0.1, 0.5). In other words, the randomness of interactions enhances the cooperation. Our results are corroborated by the works in Refs.[22, 23], and cooperation will be promoted by strengthening the randomness between the geographical interactions, random mobility or adjusting social links randomly. The main reason lies in that the cooperative diffusion rate will be enhanced based on the random interactions. As shown in Fig.3, the average path length of network becomes smaller as the rewiring probability increases, which improves the rate of strategy diffusion and easily forms cooperators cluster.
4
Conclusion
In this paper, we investigate the mechanisms responsible for the emergence of group cooperation based on square lattice, regular small-world networks and random regular network with PGG model. The results of simulation show that the group cooperation will emerge among these three networks when the enhancement factor r exceeds a threshold. As compared with the other networks, time required for full cooperation on regular small-world network is slightly longer, which indicates that the longer-range links are the obstacles for cooperation, while the local interactions and random interactions will promote cooperation by forming compact clusters for cooperators and increasing cooperative diffusion rate.
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We also find that the cooperation would be promoted further by enhancing the random interactions on regular small-world network. In the light of the arguments presented in this paper, we could deepen our knowledge on the cooperation behaviors within strangers and close friends or relatives to some extent. On the one hand, cooperation within strangers could be ascribed to the development of information technology, which can enhance the security and convenience of cooperation. As results in our work, more random interactions means more channels to diffuse heterogeneous strategies. On the other hand, close friends or relatives can be considered as the compact interactions in the square lattice, so they will easily form cooperative clusters. We believe this study can help us gain a thorough understanding about the group cooperation in realistic systems.
[9] Nowak M A. Evolving cooperation[J]. Journal of Theoretical Biology, 2012, 299(4): 1-8. [10] Szabó G, Vukov J. Cooperation for volunteering and partially random partnerships[J]. Physical Review E, 2004, 69(3): 1-7. [11] Zhou Z Y, Mao B H, Hao H M, et al. Regular small-world network[J]. Chinese Physics Letters, 2009, 26(11): 1-3. [12] Guan J Y, Wu Z X, Huang Z G, et al. Prisoner’s dilemma game with nonlinear attractive effect on regular small-world network[J].
Chinese
Physics
Letters,
2006,
23(10):
2874-2877. [13] Wu Z X, Xu Z X, Wang X J. Prisoner’s dilemma game with heterogeneous influential effect on regular small-world networks[J]. Chinese Physics Letters, 2006, 23(3): 531-534. [14] Vukov J, Szabó G, Szolnoki A. Evolutionary prisoner’s dilemma game on Newman-Watts networks[J]. Physical Review E, 2008, 77(2): 1-5. [15] Barabási A L, Albert R. Emergence of scaling in random networks[J]. Science, 1999, 286(5439): 509-512.
References [1] Hofbauer J, Sigmund K. Evolutionary Games and Population Dynamics[M]. Cambridge: Cambridge University Press, 1998. [2] Perc M, Gómez-Gardeñes J, Szolnoki A, et al. Evolutionary dynamics of group interactions on structured populations: a review[J]. Journal of the Royal Society Interface, 2013, 10(80):1-13. [3] Hauert C, De Monte S, Hofbauer J, et al. Volunteering as red queen mechanism for cooperation in public goods games[J]. Science, 2002, 296(5570): 1129-1132. [4] Brandt H, Hauert C, Sigmund K. Punishment and reputation in spatial public goods games[J]. Proceedings of the Royal Society of London B: Biological Sciences, 2003, 270(1519): 1099-1104. [5] Szolnoki A, Perc M. Reward and cooperation in the spatial public goods game[J]. EPL (Europhysics Letters), 2010, 92(3): 1-6. [6] Perc M. Sustainable institutionalized punishment requires elimination of second-order free-riders[J]. Scientific Reports, 2012, 2(344): 1-6. [7] Nikiforakis N. Punishment and counter-punishment in public good games: Can we really govern ourselves?[J]. Journal of Public Economics, 2008, 92(1): 91-112. [8] Nowak M A, May R M. Evolutionary games and spatial chaos[J]. Nature, 1992, 359(6398): 826-829.
[16] Fan R G, Zhang Y Q, Luo M, et al. Promotion of cooperation induced by heterogeneity of both investment and payoff allocation in spatial public goods game[J]. Physica A: Statistical Mechanics and Its Applications, 2017, 465(1): 454-463. [17] Santos F C, Santos M D, Pacheco J M. Social diversity promotes the emergence of cooperation in public goods games[J]. Nature, 2008, 454(7201): 213-216. [18] Perc M. High-performance parallel computing in the classroom using the public goods game as an example[J]. European Journal of Physics, 2017, 38(4): 045801. [19] Szolnoki A, Perc M, Szabó G. Topology-independent impact of noise on cooperation in spatial public goods games[J]. Physical Review E, 2009, 80(5): 1-5. [20] Szabó G, Hauert C. Phase transitions and volunteering in spatial public goods games[J]. Physical Review Letters, 2002, 89(11): 1-4. [21] Axelrod R, Ford G R, Riolo R L, et al. Beyond geography: Cooperation with persistent links in the absence of clustered neighborhoods[J]. Personality and Social Psychology Review, 2002, 6(4): 341-346. [22] Sicardi E A, Fort H, Vainstein M H, et al. Random mobility and spatial structure often enhance cooperation[J]. Journal of Theoretical Biology, 2009, 256(2): 240-246. [23] Wu B, Zhou D, Fu F, et al. Evolution of cooperation on stochastic dynamical networks[J]. PloS One, 2010, 5(6): 1-7.
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