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Evolution of cooperation in the spatial public goods game with adaptive reputation assortment
Physics Letters A 380 (2016) 40–47
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Physics Letters A www.elsevier.com/locate/pla
Evolution of cooperation in the spatial public goods game with adaptive reputation assortment Mei-huan Chen a,b , Li Wang a,b , Shi-wen Sun a,b , Juan Wang c,∗ , Cheng-yi Xia a,b,∗∗ a b c
Tianjin Key Laboratory of Intelligence Computing and Novel Software Technology, Tianjin University of Technology, Tianjin 300384, PR China Key Laboratory of Computer Vision and System (Ministry of Education), Tianjin University of Technology, Tianjin 300384, PR China School of Electrical Engineering, Tianjin University of Technology, Tianjin 300384, PR China
a r t i c l e
i n f o
Article history: Received 4 July 2015 Received in revised form 5 September 2015 Accepted 27 September 2015 Available online 1 October 2015 Communicated by C.R. Doering Keywords: Public goods game Reputation assortment Individual diversity Promotion of cooperation Evolutionary game theory
a b s t r a c t We present a new spatial public goods game model, which takes the individual reputation and behavior diversity into account at the same time, to investigate the evolution of cooperation. Initially, each player x will be endowed with an integer R x between 1 and R max to characterize his reputation value, which will be adaptively varied according to the strategy action at each time step. Then, the agents play the game and the system proceeds in accordance with a Fermi-like rule, in which a multiplicative factor (w y ) to denote the individual difference to perform the strategy transfer will be placed before the traditional Fermi probability. For influential participants, w y is set to be 1.0, but be a smaller value w (0 < w < 1) for non-influential ones. Large quantities of simulations demonstrate that the cooperation behavior will be obviously influenced by the reputation threshold (R C ), and the greater the threshold, the higher the fraction of cooperators. The origin of promotion of cooperation will be attributed to the fact that the larger reputation threshold renders the higher heterogeneity in the fraction of two types of players and strategy spreading capability. Our work is conducive to a better understanding of the emergence of cooperation within many real-world systems. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Although the cooperation is not a dominant strategy from the theoretical viewpoints or individual perspectives [1], but it is still a widespread phenomenon within real-world systems ranging from cellular organisms, vertebrates, even to human beings and societies [2]. Thus, understanding the universality of cooperation has become an inter-disciplinary topic which attracts a lot of concerns in the scientific communities [3], including life science, engineering and technology, natural and social sciences. Among them, evolutionary game theory provides a fruitful platform to address the origin of cooperation, and even the so-called social dilemmas [4,5]. In this field, over the past decades, a couple of mechanisms have been proposed to shed light on the persistence and emergence of cooperation inside the population, and five main schemes to promote the collective cooperation, as suggested in Ref. [6], including kin selection [7], direct [8] and indirect reciprocity [9,10], group
* Corresponding author. ** Corresponding author at: Tianjin Key Laboratory of Intelligence Computing and Novel Software Technology, Tianjin University of Technology, Tianjin 300384, PR China. E-mail addresses: [email protected] (J. Wang), [email protected] (C.-y. Xia). http://dx.doi.org/10.1016/j.physleta.2015.09.047 0375-9601/© 2015 Elsevier B.V. All rights reserved.
selection [11] and spatial reciprocity [12], have been identified. In particular, more concerned is the network reciprocity in the recent years [13], a mechanism that has attracted a huge realm of interest inspired by the growing relevance in the field of network science [14,15], and which fosters the survival of cooperators by limiting the number of interactions of playing the game and multiplexing the individual payoff [16–25]. Along this research line, a great myriad of works [26–40] explored the impact of game mechanisms or interaction topology on the cooperative behaviors and uncovered the evolving patterns of cooperation hidden behind the real population, to some extent (see Refs. [41–44] for comprehensive reviews). As evidenced by some researches, modern society is built on the individual or collective credit system, which is accumulated and guaranteed by past acts or behaviors [45]. Any individual lacking the credit or holding the lower reputation will incur the cost of losing the cooperation or coordination since most of people are not willing to cooperate with such an agent. Thus, it is of utmost importance to deeply investigate the role of individual reputation in the evolution of cooperation. In a previous work regarding indirect reciprocity [9], the choice of strategy will be correlated with the individual image score during the game playing, which can be generally considered as a type of the individual reputation to sustain the level of cooperation. Additionally, Milinski et al. [46] utilized
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the human experiments to demonstrate that, through alternating game rounds of public goods or indirect reciprocity, the collective cooperation regarding the public goods game can be kept at a surprisingly high level. Cuesta et al. [47] declared that the reputation really fosters the cooperation through experiments conducted within groups of humans playing an iterated prisoner’s dilemma on a dynamical network. However, in the above-mentioned works, the individuals playing the game are often looked upon as the game peers whose behavioral properties are considered to be identical, while the diversity in the individual behavior or activity may also play a vital role in the collective cooperation. As an example, it is proven that the social diversity of individual behavior will substantially improve the level of cooperation [48], and the more diverse the individual behavior, the more obvious the elevation of cooperation. Meanwhile, the individual diversity can also be characterized by the difference of strategy transfer capability, in which all players will be divided into influential and non-influential ones according to the strategy spreading behavior, and the results have shown that the cooperation can be greatly promoted in the prisoner’s dilemma game [49] or the public goods game [50]. Altogether, provided that we can combine the individual reputation and diversity to discuss the evolution of cooperation, it will be helpful to investigate the cooperation behavior under the more realistic scenarios. In the present work, we propose a model of public goods game (PGG) with adaptive reputation assortment simultaneously taking the individual reputation and strategy transfer difference into account, and extensive simulations demonstrate that the cooperative behaviors will be largely improved after the reputation assortment is introduced. The remainder of this Letter is organized as follows. In Section 2, the game model with adaptive reputation assortment is firstly described in detail. Then, a wealth of simulation results and discussions are depicted in Section 3. At last, some concluding remarks and conclusions are summarized in Section 4. 2. Spatial PGG model with adaptive reputation assortment The model is built on an archetypal spatial public goods game, in which each intersection node of regular lattice with the periodic boundary condition will be occupied by a game player. Initially, each player will be designated as a binary strategy with equal probability: cooperation (C) or defection (D), and endowed with a random integer denoting an individual reputation value R x which lies between 1 and R max . R max signifies the maximally potential reputation and, without loss of generality, is set to be 100 in our model. After that, the system proceeds according to the following elementary time steps. First, each player will participate in G = k + 1 game groups (here, k is the number of nearest neighbors), where one group is centered upon the focal agent (himself) and other k groups are resolved around his k nearest neighbors, respectively. Inside every PGG group, all players will make an independent decision about whether to put a specific contribution into the public resource pool at the same time, that is, to cooperate (contribute) or defect (not contribute). If a player cooperates, he will put a unit contribution into the pool; Otherwise, he will not invest any contribution for the public resource. Then, we will count the number of cooperators (nC ) within the group as the total contribution, which will be multiplied by a factor r ≥ 1 and then evenly distributed among the players who are involved in this PGG group. Therefore, after a game interaction, the payoff that a cooperator or a defector will obtain can be written as follows,
⎧ ⎨ ⎩
PD =
r ∗ nC
=
G PC = P D − 1
r∗
x∈ g s x
G
(1)
41
where r is the synergy factor greater than 1.0 during the game, P D and P C denote the payoff of a defector and a cooperator, respectively; s x represents the strategy value of a player x belonging to group g, and s x = 1 for a cooperator and s x = 0 for a defector. It is evident, from an individual viewpoint, that a defector wins over a cooperator by a unit benefit since the cooperator needs to pay a cost for the group interest, hence the dominant strategy or Nash equilibrium is the defection in the PGG model. However, from the perspective of whole population, the collective benefits are the highest and then individual payoff will also become higher provided that all players take the cooperative strategy. It creates a dilemma for a selfish player, and there is a free-rider problem: why cooperate if one may obtain a greater personal benefit by cheating? Meanwhile, why not cooperate if all players can produce the greatest benefit by coordinating collectively? Consequently, additional mechanisms need to be devised to stimulate the collective cooperation, such as reward or punishment [51,52], reputation effect [53], etc. Next, each player x will accumulate his payoff x by summing the payoffs from each PGG group in which he participates, namely,
x =
g
Px
(2)
g g
where P x denotes the payoff collected by player x when he participates in the PGG group g, and G = 5 since we here consider the von-Neumann neighborhood (i.e., k = 4). Then, player x will try to update his current strategy from a randomly selected neighbor y who also calculated the payoff same as player x with the following Fermi-like rule,
Prob(sx ← s y ) = w y
1 1+e
(x − y )
(3)
K
where K represents the amplitude of noise or its reverse denotes the so-called strength of strategy selection, w y is a multiplicative factor that depends on the reputation of player y. Meanwhile, during the strategy update, his reputation value will increase or decrease 1 if he adopts the cooperation or defection strategy. We will set a reputation threshold R C , which will divide the population into two types of individuals: A-type (influential and highreputation agents) if an individual reputation R x is larger than R C , otherwise B-type (non-influential and low-reputation ones). For these two types of players, we can set his multiplicative factor w y to be different so that the individual heterogeneity can be characterized. Accordingly, based on the reputation value of player y, w y in Eq. (3) can be set as follows,
wy =
1, w,
if t y = A if t y = B , 0 < w < 1
(4)
where t y stands for the individual type determined by whether his reputation value R y is over R C , and t y is equal to A if R y ≥ R C , otherwise t y = B. Since the individual reputation will vary during the time evolution, the individual type may continually change as the reputation evolves. Therefore, R C becomes an important quantity dominating the type classification of population, and then influences the evolution of cooperation within the population. After the above-mentioned basic steps are completed, a full Monte Carlo Simulation (MCS) step is finished. Starting from a randomly chosen player, within one MCS step, each player has a chance on average to adopt the strategy of one of his nearest neighbors. After some transient steps (tr = 45,000), the system evolves into the stationary state in which the average fraction of cooperators (ρC ) almost arrives at a constant value. Among them, ρC is the quantity of most interest and determined by averaging the fraction of cooperators within another ta = 5000 time steps.
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Fig. 1. (Color online.) Fraction of cooperators (ρC ) at the stationary state as a function of synergy factor r. As R C increases, the role of individual reputation assortment becomes much more obvious. The non-influential (i.e., B-type) players hold the weaker strategy transfer capability w = 0.001, other parameters are set to be MCS = 5 × 104 , L = 200, R max = 100 and K = 0.5.
Meanwhile, in order to remove the impact of randomness, the following results are often averaged over 50 independent runs, and lattice size is usually kept to be L = 200. Furthermore, we consider two other different neighborhood setup k = 8 (Moore neighborhood) and k = 24 neighbors, and larger lattice size (e.g., L = 400) is also explored, numerical simulations indicate that qualitatively same evolutionary behaviors can be observed (which will not be presented here). 3. Numerical simulation results In Fig. 1, we plot the fraction of cooperators (ρC ) at the stationary state as a function of synergy factor (r) under different reputation threshold (R C ). The black square symbols represent the results of traditional PGG model without any reputation effect, and other symbols stand for those in our model for various R C . As it is shown, the introduction of reputation assortment can greatly improve the evolution of cooperation. In our model, all players will be divided into two types based on the reputation threshold at each MCS step, one class is the high reputation individuals with R ≥ R C who own the greater spreading capability (w = 1) and become the influential ones; the other class is the low reputation individuals with R < R C who have the smaller transfer factor (w = 0.001). As indicated in Fig. 1, the larger the reputation threshold (R C ), the more obvious the promotion of cooperation. On one hand, the lower threshold of synergy factor leading to the emergence of cooperators (as shown by r C 1 in Fig. 1) becomes smaller and smaller as the R C increases, the similar scenario also takes place for the upper threshold leading to the full cooperation (as pictured by r C 2 in Fig. 1); on the other hand, for a fixed synergy factor (r), once r transcends over the corresponding lower threshold, the cooperator’s fraction will be largely elevated as R C augments. As an example, when r = 3.0, r is less than r C 1 for the traditional case, R C = 60, and R C = 80, thus creating the extinction of cooperation, that is, ρC = 0; However, for R C = 90, 95, 98, r is greater than corresponding the lower threshold r C 1 , hence rendering that ρC > 0, where ρC is equal to 0.326, 0.707 and 0.999, respectively. The possible mechanism to promote the cooperation can be understood as follows. As R C increases, the percentage of high reputation individuals (R > R C ) becomes lower, hence the fraction of A-type ones also becomes smaller. At the same time, larger R C also leads to the situation in which the low reputation individuals hold the higher percentage within the population. Accordingly, under this case, the influential players can own enough space or chance to persuade
Fig. 2. (Color online.) Fraction of cooperators (ρC ) as a function of MCS step under a fixed synergy factor r = 3.3 for different reputation threshold R C . Likewise, as R C grows, the role of individual reputation assortment becomes much more prominent. The non-influential (i.e., B-type) players hold the weaker strategy transfer capability w = 0.001, other parameters are set to be r = 3.3, L = 200, R max = 100 and K = 0.5.
the non-influential neighbors to adopt their own strategies, which will gain a much richer surrounding for the cooperative clusters centering around an influential cooperator to form. As a result, the role of individuals with higher strategy spreading factor will become much more prominent as R C increases, which means that the difference between two classes of individuals become much more evident and further promote the evolution of cooperation. In order to further scrutinize the role of adaptive reputation assortment in the spatial PGG model, the time evolution of cooperator’s fraction at each MCS step is depicted in Fig. 2 for a constant synergy factor (r = 3.3). For the traditional PGG, the spatial reciprocity is not enough to sustain the survival of cooperators, and the fraction of cooperators will monotonically drop down to the zero percentage of cooperators, which means that the defectors fully exploit the cooperators till the cooperation tends to be extinct. Nevertheless, the introduction of reputation effect, the destiny of cooperator begins to turn around, and even the large and cooperative clusters can emerge within the population. When we consider the cases of reputation assortment, the individuals may still not resist the temptation to defect, and the defectors can invade the cooperator’s population which leads to the fall of fraction of cooperators at the initial steps. However, before the stationary state is arrived at, the reciprocal competition between cooperators and defectors may exist, in particular for intermediate R C (R C = 60 or R C = 80). After the reputation assortment is introduced, some players may become the influential ones ( A-type players) and obtain the higher probability to persuade the nearest neighbors to adopt their own strategy. Once the influential cooperators survive, they will become the compact core clusters which help to attract more agents to join the cooperative ones even if the fraction of cooperators is relatively low for R C = 60. While for larger reputation threshold (for example, R C = 95 or R C = 98), a very short enduring period appears and then cooperators win over the superiority of competition with defectors, and the whole population is quickly organized into a giant cooperative cluster which creates a very high cooperation level. As a consequence, the reputation assortment can help the cooperators to avoid the full exploitation from the defectors by creating the cooperative clusters, and ρC tends to become higher and higher, as shown in Fig. 2, when the reputation threshold increases from around 60. These simulation results can again highlight the impact of reputation assortment on the promotion of cooperation. Next, by investigating the interacting patterns between cooperators and defectors, we can further explore the impact of adaptive
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Fig. 3. (Color online.) Characteristic pattern of cooperators and defectors at the initial MCS step (a, MCS = 0) and stationary state step (b–f, MCS = 5 × 104 ). In panel (a), cooperators and defectors are randomly allocated on lattices, and each player also stochastically takes a reputation value from the interval [1, 100]. From panel (b) to (f), stationary distribution of cooperators and defectors are depicted; Among them, red (R ≥ R C ) and pink (R < R C ) dots represent the defectors, while blue (R ≥ R C ) and cyan (R < R C ) dots denote the cooperators. The non-influential (i.e., B-type) players hold the weaker strategy transfer capability w = 0.001, other parameters are set to be r = 3.3, L = 200, R max = 100 and K = 0.5.
reputation assortment on the evolution of cooperation. In Fig. 3, panel (a) presents the distribution of cooperators (blue) and defectors (red) at the initial time step (MCS = 0), in which each cooperator or defector is randomly placed on the lattice and there does not exist any pattern of clustering. Meanwhile, the initial setup in panel (a) will be adopted in the following 5 reputation assortment cases. From panel (b) to (f), we depict the stationary distribution of cooperators (blue or cyan) and defectors (red or pink) for different reputation thresholds, which are set to be 60, 80, 90, 95 and
98, respectively. As displayed in Fig. 3, the cooperative clusters can be effectively organized, and the fraction of cooperators can grow little by little as the reputation threshold increases. It is particulary worth mentioning that many cooperators with high reputation value are often formed into giant clusters, then surrounded by a few low reputation cooperators so that they can defend the survival of cooperators by creating the hierarchical structure from cooperators to defectors; while for defectors, the low reputation agents can dominate the population and develop into a very giant
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Fig. 4. (Color online.) Characteristic pattern of cooperators and defectors at the initial MCS step (a, MCS = 0) and stationary state step (b–f, MCS = 5 × 104 ). In panel (a), cooperators are allocated on the intermediate regions and defectors are allocated on the side regions of lattices, and each player is also assigned a reputation value from the interval [1, 100]. From panel (b) to (f), stationary distribution of cooperators and defectors are depicted; Among them, red (R ≥ R C ) and pink (R < R C ) dots represent the defectors, while blue (R ≥ R C ) and cyan (R < R C ) dots denote the cooperators. The non-influential (i.e., B-type) players hold the weaker strategy transfer capability w = 0.001, other parameters are set to be r = 3.3, L = 200, R max = 100 and K = 0.5.
cluster, but cannot devour all cooperators although some defective agent has penetrated into the cooperative clusters. As R C increases, the size of cooperative clusters will also become larger and larger, and even fully dominate the whole population when R C is equal to 98. Furthermore, we also demonstrate the organizing modes of collective cooperation by fixing all cooperators into the middle of lattice at the initial step, as indicated in Fig. 4, and the similar evolutionary patterns as those in Fig. 3 are also observed. The current clustering patterns again validate the fact that the game micro-
scopic dynamics dominate the evolutionary patterns of evolution, that is, the reputation assortment will help to create the compact and cooperative clusters regardless of the initial choice of strategy of players. For the sake of probing into the role of reputation assortment in depth, we illustrate the distribution of reputation values within the whole population at the steady state in Fig. 5. It is explicitly revealed that the reputation values are uniformly distributed within the interval [1, 100] at the initial time. After we introduce
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Fig. 5. (Color online.) Distribution of reputation value within the whole population at the initial step (a, MCS = 0) and stationary state step (b–f, MCS = 5 × 104 ). In panel (a), reputation values are almost uniformly distributed within the interval [1, 100], whether for cooperators or for defectors. From panel (b) to (f), distribution of reputation at the stationary state is depicted; Among them, red and blue bars represent the relative fractions of defectors and cooperators with various reputation values, respectively. The non-influential (i.e., B-type) players hold the weaker strategy transfer capability w = 0.001, other parameters are set to be r = 3.3, L = 200, R max = 100 and K = 0.5.
the reputation assortment, the cooperators can resist the invasion of defectors so as not to be fully overwhelmed by the defecting players. For example, when the reputation threshold R C is equal to 60, we can observe that nearly 90% of agents are defectors whose reputation values lie between 1 and 10, but around 10% of agents are cooperators whose reputation values are very high and distributed within [91, 100]. As the reputation threshold increases from 80 [panel (c)] to 95 [panel (e)], the fraction of players with low reputation decreases, meanwhile the fraction of players with high reputation increases. Even when R C is up to 98, the whole system is totally composed of cooperators with high reputation values. Thus, as shown in Fig. 5, the population usually evolves into a state in which the system mainly consists of two classes of individuals: one class is the low reputation defector, and the other one is the high reputation cooperator. The competition between cooperators and defectors will drive the cooperators to win the advantage over the defectors during the evolution of collective interactions, and then the cooperative clusters will become larger and larger as the reputation effect works. In an ideal circumstance, the cooperators will dominate the whole population when the reputation threshold is high enough (i.e., R C = 98). In Fig. 6, we plot the evolution of ρC for the specified space r–K under the adaptive reputation assortment, from which we can clearly discern the whole phase transition process from full D to the coexistence of cooperators and defectors (C + D), and then to full C . In panel (a) the reputation threshold R C is equal to 70, but the threshold R C is set to be 95 in panel (b). When the reputa-
tion threshold is lower (R C = 70), the transition from D to C + D is difficult, and the synergy factor for a constant noise strength (K ) transcends the full defection also becomes higher, and similar process happens for the process from C + D to D. Additionally, the region to allow the coexistent cooperators and defectors is also much wider in our phase space. On the contrary, the larger reputation threshold (R C ) can lead to the easier and quicker phase transition within the specified region when compared to the case of R C = 70. It is worth mentioning that the transition points between two spaces becomes lower, and the extraordinarily extended regions to arrive at the full cooperation (C ) renders that the coexistence for cooperators and defectors also displays the narrower room although the D phase is contracted. Therefore, the adaptive reputation assortment can provide a stronger support for the survival of cooperators. Finally, the impact of strategy spreading factor (w) on the cooperation for a fixed reputation threshold (R C = 95) is considered in Fig. 7. It can be clearly observed that the cooperation level, similar to Fig. 1, will be improved as the strategy spreading factor (w) decreases. Specifically, the lower threshold (r C 1 ) of synergy factor denoting the full defection to the appearance of cooperation becomes smaller, and the upper threshold (r C 2 ) representing the coexistence between cooperators and defectors to the full cooperation also has the same tendency. Furthermore, under a fixed synergy factor, as an example, r = 3.5 is less than r C 1 for the traditional version and more than those of other cases for various strategy factors w < 1. Thus, the results again indicate that the co-
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tion and low reputation ones) enlarges, and the influential players can hold a surprisingly high capability to persuade other players to adopt their strategies. Above all, the reputation assortment will help the cooperators to build the cooperative advantages to obtain the higher reputation values, which may further render the cooperators to become influential ones so that they can own a higher transfer ability to easily create the cooperator’s cluster. The greater the difference between two classes of players, the more obvious the enhancement of cooperation level, current results are also compatible with the general conclusions which have claimed that the heterogeneity can hugely favor the evolution of cooperation [13–15]. 4. Conclusions and discussions
Fig. 6. (Color online.) Evolution of fraction of cooperators (ρC ) for the specified r–K parameter space. In panel (a), R C = 70; while in panel (b) R C = 95, and the reputation effect powerfully enhances the cooperation level. The non-influential (i.e., B-type) players hold the weaker strategy transfer capability w = 0.001, other parameters are set to be MCS = 5 × 104 , L = 200, R max = 100.
Fig. 7. (Color online.) Fraction of cooperators (ρC ) at the stationary state as a function of synergy factor (r) for various strategy spreading factor (w). As w decreases, the role of individual reputation assortment becomes much more obvious. The reputation threshold is fixed to be R C = 95, other parameters are set to be MCS = 5 × 104 , L = 200, R max = 100 and K = 0.5.
operation level will be largely enhanced as the strategy spreading factor varies from 0.1 to 0.001. The origin of promotion of cooperation can be attributed to the fact that the difference of strategy spreading factor between two types of players (i.e., high reputa-
To summarize, we propose a spatial public goods game, in which the players are adaptively classified into two types of ones based on a reputation effect, to study the evolution of collective cooperation. Firstly, a reputation value uniformly distributed within the interval [1, 100] is randomly assigned to each player, meanwhile a binary strategy (cooperation or defection) is adopted by an agent with the equal probability. Then, the system proceeds by playing PGG according to the Ferm-like rule, where a multiplicative factor (w y ) will be placed before the common Fermi probability. At the same time, an individual reputation value will be minus or plus one unit if the player decides to take a cooperating or defecting action. Here, w y is dependent on the type of one randomly chosen neighbor (say, y) of a focal player, and w y will be set to be 1.0 if player y is an influential one or a constant w (0 < w < 1) if player y is a non-influential player. Therefore, an agent will be divided into influential or non-influential one based on his own reputation value, in which a player will be influential if a predefined reputation threshold R C is surpassed (i.e., R y ≥ R C ), non-influential otherwise (i.e., R y < R C ). A wide spectrum of numerical simulations are performed on regular lattices to validate the system behavior, where the stationary state will be arrived at or evaluated after enough time steps are discarded. The simulation results demonstrate that the evolution of cooperation can be greatly enhanced by the introduction of adaptive reputation assortment. In particular, the larger the reputation threshold, the fewer the influential players, and the possible origin of cooperation promotion lies that the influential players rendered by the individual assortment can hold enough opportunity and space to persuade the neighboring nodes to adopt his own strategy, in the meantime, the reputation effect helps the cooperators to build accumulative advantages. The evolutionary pattern and statistics of reputation value distribution also uncovers that the reputation assortment is beneficial to the influential cooperators who will finally dominate the collective cooperation. However, the real population structure is far beyond the regularity, and often exhibits the so-called small-world or scale-free properties, hence exploring the evolutionary behavior within the population with heterogeneous topology is also of utmost relevance. Here, we take advantage of regular lattices to simply study the role of reputation or diversity in the promotion of cooperation, and need not involve the impact of heterogeneous topology on the cooperation behavior so that the model is kept to be as simplified as possible. Regarding the strategy update rule, imitation-MAX [9] or replicator dynamics [42] can also be applied, but we adopt the Fermi-like rule so as to keep the consistence and comparison with previous works [48–50]. Thus, it may be deserved the further probings as far as the different interaction topologies or update rules are concerned. Moreover, in the present work, the individual reputation value needs to be recorded at each time step and can be referred by his neighbors. In the future, it will become much more important and necessary if individual reputation cannot be fully
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known by other peers, which means a kind of reputation inference, as shown in Ref. [53]. Anyway, current results can be highly conducive to expounding the possible mechanisms to facilitate the evolution of cooperation within many real-world systems. Acknowledgements This project is in part funded by the National Natural Science Foundation of China (NSFC) under grant Nos. 61203138, 61374169, 61403280 and 71401122. References [1] K. Sigmund, The Calculus of Selfishness, Princeton University Press, Princeton, NJ, 2010. [2] R. Axelrod, The Evolution of Cooperation, Basic Books, New York, 2006. [3] M.A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life, Harvard University Press, Cambridge, MA, 2006. [4] M.J. Osborne, A. Rubinstein, A Course in Game Theory, The MIT Press, Cambridge, MA, 1994. [5] H. Gintis, Game Theory Evolving, Princeton University Press, Princeton, NJ, 2000. [6] M.A. Nowak, Science 314 (2006) 1560–1563. [7] W.D. Hamilton, J. Theor. Biol. 7 (1964) 1–52. [8] R.L. Trivers, Q. Rev. Biol. 46 (1971) 35–37. [9] M.A. Nowak, K. Sigmund, Nature 393 (1998) 573–577. [10] K. Panchanathan, R. Boyd, Nature 432 (2004) 499–502. [11] A. Traulsen, M.A. Nowak, Proc. Natl. Acad. Sci. USA 103 (2006) 10952–10955. [12] M.A. Nowak, R.M. May, Nature 359 (1992) 826–829. [13] F.C. Santos, J.M. Pacheco, Phys. Rev. Lett. 95 (2005) 098104. [14] S. Boccaletti, V. Latora, Y. Moreno, M. Chavezf, D.U. Hwang, Phys. Rep. 424 (2006) 175–308. ¨ M. Romance, [15] S. Boccaletti, G. Bianconi, R. Criado, C.I. Genio, J. Gómez-Gardens, I. Sendiña-Nadal, Z. Wang, M. Zanin, Phys. Rep. 544 (2014) 1–122. [16] Z. Wang, A. Szolnoki, M. Perc, Europhys. Lett. 97 (2012) 48001. [17] L.L. Jiang, M. Perc, Sci. Rep. 3 (2013) 2483. [18] A. Szolnoki, M. Perc, New J. Phys. 15 (2013) 053010. [19] Q. Jin, L. Wang, C.Y. Xia, Z. Wang, Sci. Rep. 4 (2014) 4095. [20] Z. Wang, A. Szolnoki, M. Perc, Sci. Rep. 3 (2013) 1183. [21] X.K. Meng, C.Y. Xia, Z.K. Gao, L. Wang, S.W. Sun, Phys. Lett. A 379 (2015) 767–773.
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[22] Z. Wang, L. Wang, M. Perc, Phys. Rev. E 89 (2014) 052813. [23] C.Y. Xia, Q. Miao, J. Wang, S. Ding, Appl. Comput. Math. 246 (2014) 389–398. [24] Z. Wang, A. Szolnoki, M. Perc, Sci. Rep. 3 (2013) 2470. [25] C.Y. Xia, X.K. Meng, Z. Wang, PLoS ONE 10 (2015) 12954. [26] C.Y. Xia, J. Zhao, J. Wang, Y.L. Wang, H. Zhang, Phys. Scr. 84 (2011) 025802. [27] J.J. Zhang, H.Y. Ning, Z.Y. Yin, S.W. Sun, L. Wang, J.Q. Sun, C.Y. Xia, Front. Phys. 7 (2012) 366–372. [28] X.J. Chen, A. Szolnoki, M. Perc, L. Wang, Phys. Rev. E 85 (2012) 066133. [29] F. Fu, L.H. Liu, L. Wang, Eur. Phys. J. B 56 (2007) 367–372. [30] W.B. Du, Z.X. Wu, K.Q. Cai, Physica A 392 (2013) 3505–3512. [31] X. Ni, W.X. Wang, Y.C. Lai, C. Grebogi, Phys. Rev. E 82 (2010) 066211. [32] Z.H. Rong, Z.X. Wu, G.R. Chen, Europhys. Lett. 102 (6) (2013) 68005. [33] Z.X. Wu, X.J. Xu, Y. Chen, Y.H. Wang, Phys. Rev. E 71 (2005) 037103. [34] H.F. Zhang, R.R. Liu, Z. Wang, H.X. Yang, B.H. Wang, Europhys. Lett. 94 (2012) 18006. [35] C. Zhang, J. Zhang, G. Xie, L. Wang, Eur. Phys. J. B 80 (2011) 217–222. [36] Z. Wang, S. Kokubo, J. Tanimoto, E. Fukuda, K. Shigaki, Phys. Rev. E 88 (2013) 042145. [37] J. Grujíc, C. Fosco, L. Araujo, J.A. Cuesta, A. Sánchez, PLoS ONE 5 (2010) e13749. [38] C. Gracia-Lázaro, A. Ferrer, G. Ruiz, A. Tarancón, J.A. Cuesta, A. Sánchez, Y. Moreno, Proc. Natl. Acad. Sci. USA 109 (2012) 12922–12926. [39] Z. Wang, A. Szolnoki, M. Perc, Sci. Rep. 2 (2012) 369. [40] Z.K. Gao, P.C. Fang, M.S. Ding, D. Yang, N.D. Jin, Phys. Lett. A 379 (2014) 790–797. [41] Z. Wang, L. Wang, A. Szolnoki, M. Perc, Eur. Biophys. J. 88 (2015) 124. [42] C.P. Roca, J.A. Cuesta, A. Sánchez, Phys. Life Rev. 6 (2009) 208–249. ¨ A. Szolnoki, L.M. Floría, Y. Moreno, J. R. Soc. Inter[43] M. Perc, J. Gómez-Gardenes, face 10 (2013) 20120997. [44] Z. Wang, S. Kokubo, M. Jusup, J. Tanimoto, Phys. Life Rev. 14 (2015) 1–30. [45] M. Kosfeld, Rev. Network Econom. 3 (2004) 20–41. [46] M. Milinski, D. Semmann, H.J. Krambeck, Nature 415 (2002) 424–426. [47] J.A. Cuesta, C. Gracia-Lázaro, A. Ferrer, Y. Moreno, A. Sánchez, Sci. Rep. 5 (2015) 7843. [48] M. Perc, A. Szolnoki, Phys. Rev. E 77 (2008) 011904. [49] G. Szabó, A. Szolnoki, Phys. Rev. E 79 (2009) 016106. [50] C.J. Zhu, S.W. Sun, L. Wang, S. Ding, J. Wang, C.Y. Xia, Physica A 406 (2014) 145–154. [51] R. Jiménez, H. Lugo, J.A. Cuesta, A. Sánchez, J. Theor. Biol. 250 (2008) 475–483. [52] Z. Wang, C.Y. Xia, S. Meloni, C.S. Zhou, Y. Moreno, Sci. Rep. 3 (2013) 3095. [53] Z. Wang, Z. Yin, C.Y. Xia, L. Wang, PLoS ONE 7 (2012) e40218.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Evolution of cooperation in the spatial public goods game with adaptive reputation assortment Mei-huan Chen a,b , Li Wang a,b , Shi-wen Sun a,b , Juan Wang c,∗ , Cheng-yi Xia a,b,∗∗ a b c
Tianjin Key Laboratory of Intelligence Computing and Novel Software Technology, Tianjin University of Technology, Tianjin 300384, PR China Key Laboratory of Computer Vision and System (Ministry of Education), Tianjin University of Technology, Tianjin 300384, PR China School of Electrical Engineering, Tianjin University of Technology, Tianjin 300384, PR China
a r t i c l e
i n f o
Article history: Received 4 July 2015 Received in revised form 5 September 2015 Accepted 27 September 2015 Available online 1 October 2015 Communicated by C.R. Doering Keywords: Public goods game Reputation assortment Individual diversity Promotion of cooperation Evolutionary game theory
a b s t r a c t We present a new spatial public goods game model, which takes the individual reputation and behavior diversity into account at the same time, to investigate the evolution of cooperation. Initially, each player x will be endowed with an integer R x between 1 and R max to characterize his reputation value, which will be adaptively varied according to the strategy action at each time step. Then, the agents play the game and the system proceeds in accordance with a Fermi-like rule, in which a multiplicative factor (w y ) to denote the individual difference to perform the strategy transfer will be placed before the traditional Fermi probability. For influential participants, w y is set to be 1.0, but be a smaller value w (0 < w < 1) for non-influential ones. Large quantities of simulations demonstrate that the cooperation behavior will be obviously influenced by the reputation threshold (R C ), and the greater the threshold, the higher the fraction of cooperators. The origin of promotion of cooperation will be attributed to the fact that the larger reputation threshold renders the higher heterogeneity in the fraction of two types of players and strategy spreading capability. Our work is conducive to a better understanding of the emergence of cooperation within many real-world systems. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Although the cooperation is not a dominant strategy from the theoretical viewpoints or individual perspectives [1], but it is still a widespread phenomenon within real-world systems ranging from cellular organisms, vertebrates, even to human beings and societies [2]. Thus, understanding the universality of cooperation has become an inter-disciplinary topic which attracts a lot of concerns in the scientific communities [3], including life science, engineering and technology, natural and social sciences. Among them, evolutionary game theory provides a fruitful platform to address the origin of cooperation, and even the so-called social dilemmas [4,5]. In this field, over the past decades, a couple of mechanisms have been proposed to shed light on the persistence and emergence of cooperation inside the population, and five main schemes to promote the collective cooperation, as suggested in Ref. [6], including kin selection [7], direct [8] and indirect reciprocity [9,10], group
* Corresponding author. ** Corresponding author at: Tianjin Key Laboratory of Intelligence Computing and Novel Software Technology, Tianjin University of Technology, Tianjin 300384, PR China. E-mail addresses: [email protected] (J. Wang), [email protected] (C.-y. Xia). http://dx.doi.org/10.1016/j.physleta.2015.09.047 0375-9601/© 2015 Elsevier B.V. All rights reserved.
selection [11] and spatial reciprocity [12], have been identified. In particular, more concerned is the network reciprocity in the recent years [13], a mechanism that has attracted a huge realm of interest inspired by the growing relevance in the field of network science [14,15], and which fosters the survival of cooperators by limiting the number of interactions of playing the game and multiplexing the individual payoff [16–25]. Along this research line, a great myriad of works [26–40] explored the impact of game mechanisms or interaction topology on the cooperative behaviors and uncovered the evolving patterns of cooperation hidden behind the real population, to some extent (see Refs. [41–44] for comprehensive reviews). As evidenced by some researches, modern society is built on the individual or collective credit system, which is accumulated and guaranteed by past acts or behaviors [45]. Any individual lacking the credit or holding the lower reputation will incur the cost of losing the cooperation or coordination since most of people are not willing to cooperate with such an agent. Thus, it is of utmost importance to deeply investigate the role of individual reputation in the evolution of cooperation. In a previous work regarding indirect reciprocity [9], the choice of strategy will be correlated with the individual image score during the game playing, which can be generally considered as a type of the individual reputation to sustain the level of cooperation. Additionally, Milinski et al. [46] utilized
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the human experiments to demonstrate that, through alternating game rounds of public goods or indirect reciprocity, the collective cooperation regarding the public goods game can be kept at a surprisingly high level. Cuesta et al. [47] declared that the reputation really fosters the cooperation through experiments conducted within groups of humans playing an iterated prisoner’s dilemma on a dynamical network. However, in the above-mentioned works, the individuals playing the game are often looked upon as the game peers whose behavioral properties are considered to be identical, while the diversity in the individual behavior or activity may also play a vital role in the collective cooperation. As an example, it is proven that the social diversity of individual behavior will substantially improve the level of cooperation [48], and the more diverse the individual behavior, the more obvious the elevation of cooperation. Meanwhile, the individual diversity can also be characterized by the difference of strategy transfer capability, in which all players will be divided into influential and non-influential ones according to the strategy spreading behavior, and the results have shown that the cooperation can be greatly promoted in the prisoner’s dilemma game [49] or the public goods game [50]. Altogether, provided that we can combine the individual reputation and diversity to discuss the evolution of cooperation, it will be helpful to investigate the cooperation behavior under the more realistic scenarios. In the present work, we propose a model of public goods game (PGG) with adaptive reputation assortment simultaneously taking the individual reputation and strategy transfer difference into account, and extensive simulations demonstrate that the cooperative behaviors will be largely improved after the reputation assortment is introduced. The remainder of this Letter is organized as follows. In Section 2, the game model with adaptive reputation assortment is firstly described in detail. Then, a wealth of simulation results and discussions are depicted in Section 3. At last, some concluding remarks and conclusions are summarized in Section 4. 2. Spatial PGG model with adaptive reputation assortment The model is built on an archetypal spatial public goods game, in which each intersection node of regular lattice with the periodic boundary condition will be occupied by a game player. Initially, each player will be designated as a binary strategy with equal probability: cooperation (C) or defection (D), and endowed with a random integer denoting an individual reputation value R x which lies between 1 and R max . R max signifies the maximally potential reputation and, without loss of generality, is set to be 100 in our model. After that, the system proceeds according to the following elementary time steps. First, each player will participate in G = k + 1 game groups (here, k is the number of nearest neighbors), where one group is centered upon the focal agent (himself) and other k groups are resolved around his k nearest neighbors, respectively. Inside every PGG group, all players will make an independent decision about whether to put a specific contribution into the public resource pool at the same time, that is, to cooperate (contribute) or defect (not contribute). If a player cooperates, he will put a unit contribution into the pool; Otherwise, he will not invest any contribution for the public resource. Then, we will count the number of cooperators (nC ) within the group as the total contribution, which will be multiplied by a factor r ≥ 1 and then evenly distributed among the players who are involved in this PGG group. Therefore, after a game interaction, the payoff that a cooperator or a defector will obtain can be written as follows,
⎧ ⎨ ⎩
PD =
r ∗ nC
=
G PC = P D − 1
r∗
x∈ g s x
G
(1)
41
where r is the synergy factor greater than 1.0 during the game, P D and P C denote the payoff of a defector and a cooperator, respectively; s x represents the strategy value of a player x belonging to group g, and s x = 1 for a cooperator and s x = 0 for a defector. It is evident, from an individual viewpoint, that a defector wins over a cooperator by a unit benefit since the cooperator needs to pay a cost for the group interest, hence the dominant strategy or Nash equilibrium is the defection in the PGG model. However, from the perspective of whole population, the collective benefits are the highest and then individual payoff will also become higher provided that all players take the cooperative strategy. It creates a dilemma for a selfish player, and there is a free-rider problem: why cooperate if one may obtain a greater personal benefit by cheating? Meanwhile, why not cooperate if all players can produce the greatest benefit by coordinating collectively? Consequently, additional mechanisms need to be devised to stimulate the collective cooperation, such as reward or punishment [51,52], reputation effect [53], etc. Next, each player x will accumulate his payoff x by summing the payoffs from each PGG group in which he participates, namely,
x =
g
Px
(2)
g g
where P x denotes the payoff collected by player x when he participates in the PGG group g, and G = 5 since we here consider the von-Neumann neighborhood (i.e., k = 4). Then, player x will try to update his current strategy from a randomly selected neighbor y who also calculated the payoff same as player x with the following Fermi-like rule,
Prob(sx ← s y ) = w y
1 1+e
(x − y )
(3)
K
where K represents the amplitude of noise or its reverse denotes the so-called strength of strategy selection, w y is a multiplicative factor that depends on the reputation of player y. Meanwhile, during the strategy update, his reputation value will increase or decrease 1 if he adopts the cooperation or defection strategy. We will set a reputation threshold R C , which will divide the population into two types of individuals: A-type (influential and highreputation agents) if an individual reputation R x is larger than R C , otherwise B-type (non-influential and low-reputation ones). For these two types of players, we can set his multiplicative factor w y to be different so that the individual heterogeneity can be characterized. Accordingly, based on the reputation value of player y, w y in Eq. (3) can be set as follows,
wy =
1, w,
if t y = A if t y = B , 0 < w < 1
(4)
where t y stands for the individual type determined by whether his reputation value R y is over R C , and t y is equal to A if R y ≥ R C , otherwise t y = B. Since the individual reputation will vary during the time evolution, the individual type may continually change as the reputation evolves. Therefore, R C becomes an important quantity dominating the type classification of population, and then influences the evolution of cooperation within the population. After the above-mentioned basic steps are completed, a full Monte Carlo Simulation (MCS) step is finished. Starting from a randomly chosen player, within one MCS step, each player has a chance on average to adopt the strategy of one of his nearest neighbors. After some transient steps (tr = 45,000), the system evolves into the stationary state in which the average fraction of cooperators (ρC ) almost arrives at a constant value. Among them, ρC is the quantity of most interest and determined by averaging the fraction of cooperators within another ta = 5000 time steps.
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Fig. 1. (Color online.) Fraction of cooperators (ρC ) at the stationary state as a function of synergy factor r. As R C increases, the role of individual reputation assortment becomes much more obvious. The non-influential (i.e., B-type) players hold the weaker strategy transfer capability w = 0.001, other parameters are set to be MCS = 5 × 104 , L = 200, R max = 100 and K = 0.5.
Meanwhile, in order to remove the impact of randomness, the following results are often averaged over 50 independent runs, and lattice size is usually kept to be L = 200. Furthermore, we consider two other different neighborhood setup k = 8 (Moore neighborhood) and k = 24 neighbors, and larger lattice size (e.g., L = 400) is also explored, numerical simulations indicate that qualitatively same evolutionary behaviors can be observed (which will not be presented here). 3. Numerical simulation results In Fig. 1, we plot the fraction of cooperators (ρC ) at the stationary state as a function of synergy factor (r) under different reputation threshold (R C ). The black square symbols represent the results of traditional PGG model without any reputation effect, and other symbols stand for those in our model for various R C . As it is shown, the introduction of reputation assortment can greatly improve the evolution of cooperation. In our model, all players will be divided into two types based on the reputation threshold at each MCS step, one class is the high reputation individuals with R ≥ R C who own the greater spreading capability (w = 1) and become the influential ones; the other class is the low reputation individuals with R < R C who have the smaller transfer factor (w = 0.001). As indicated in Fig. 1, the larger the reputation threshold (R C ), the more obvious the promotion of cooperation. On one hand, the lower threshold of synergy factor leading to the emergence of cooperators (as shown by r C 1 in Fig. 1) becomes smaller and smaller as the R C increases, the similar scenario also takes place for the upper threshold leading to the full cooperation (as pictured by r C 2 in Fig. 1); on the other hand, for a fixed synergy factor (r), once r transcends over the corresponding lower threshold, the cooperator’s fraction will be largely elevated as R C augments. As an example, when r = 3.0, r is less than r C 1 for the traditional case, R C = 60, and R C = 80, thus creating the extinction of cooperation, that is, ρC = 0; However, for R C = 90, 95, 98, r is greater than corresponding the lower threshold r C 1 , hence rendering that ρC > 0, where ρC is equal to 0.326, 0.707 and 0.999, respectively. The possible mechanism to promote the cooperation can be understood as follows. As R C increases, the percentage of high reputation individuals (R > R C ) becomes lower, hence the fraction of A-type ones also becomes smaller. At the same time, larger R C also leads to the situation in which the low reputation individuals hold the higher percentage within the population. Accordingly, under this case, the influential players can own enough space or chance to persuade
Fig. 2. (Color online.) Fraction of cooperators (ρC ) as a function of MCS step under a fixed synergy factor r = 3.3 for different reputation threshold R C . Likewise, as R C grows, the role of individual reputation assortment becomes much more prominent. The non-influential (i.e., B-type) players hold the weaker strategy transfer capability w = 0.001, other parameters are set to be r = 3.3, L = 200, R max = 100 and K = 0.5.
the non-influential neighbors to adopt their own strategies, which will gain a much richer surrounding for the cooperative clusters centering around an influential cooperator to form. As a result, the role of individuals with higher strategy spreading factor will become much more prominent as R C increases, which means that the difference between two classes of individuals become much more evident and further promote the evolution of cooperation. In order to further scrutinize the role of adaptive reputation assortment in the spatial PGG model, the time evolution of cooperator’s fraction at each MCS step is depicted in Fig. 2 for a constant synergy factor (r = 3.3). For the traditional PGG, the spatial reciprocity is not enough to sustain the survival of cooperators, and the fraction of cooperators will monotonically drop down to the zero percentage of cooperators, which means that the defectors fully exploit the cooperators till the cooperation tends to be extinct. Nevertheless, the introduction of reputation effect, the destiny of cooperator begins to turn around, and even the large and cooperative clusters can emerge within the population. When we consider the cases of reputation assortment, the individuals may still not resist the temptation to defect, and the defectors can invade the cooperator’s population which leads to the fall of fraction of cooperators at the initial steps. However, before the stationary state is arrived at, the reciprocal competition between cooperators and defectors may exist, in particular for intermediate R C (R C = 60 or R C = 80). After the reputation assortment is introduced, some players may become the influential ones ( A-type players) and obtain the higher probability to persuade the nearest neighbors to adopt their own strategy. Once the influential cooperators survive, they will become the compact core clusters which help to attract more agents to join the cooperative ones even if the fraction of cooperators is relatively low for R C = 60. While for larger reputation threshold (for example, R C = 95 or R C = 98), a very short enduring period appears and then cooperators win over the superiority of competition with defectors, and the whole population is quickly organized into a giant cooperative cluster which creates a very high cooperation level. As a consequence, the reputation assortment can help the cooperators to avoid the full exploitation from the defectors by creating the cooperative clusters, and ρC tends to become higher and higher, as shown in Fig. 2, when the reputation threshold increases from around 60. These simulation results can again highlight the impact of reputation assortment on the promotion of cooperation. Next, by investigating the interacting patterns between cooperators and defectors, we can further explore the impact of adaptive
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Fig. 3. (Color online.) Characteristic pattern of cooperators and defectors at the initial MCS step (a, MCS = 0) and stationary state step (b–f, MCS = 5 × 104 ). In panel (a), cooperators and defectors are randomly allocated on lattices, and each player also stochastically takes a reputation value from the interval [1, 100]. From panel (b) to (f), stationary distribution of cooperators and defectors are depicted; Among them, red (R ≥ R C ) and pink (R < R C ) dots represent the defectors, while blue (R ≥ R C ) and cyan (R < R C ) dots denote the cooperators. The non-influential (i.e., B-type) players hold the weaker strategy transfer capability w = 0.001, other parameters are set to be r = 3.3, L = 200, R max = 100 and K = 0.5.
reputation assortment on the evolution of cooperation. In Fig. 3, panel (a) presents the distribution of cooperators (blue) and defectors (red) at the initial time step (MCS = 0), in which each cooperator or defector is randomly placed on the lattice and there does not exist any pattern of clustering. Meanwhile, the initial setup in panel (a) will be adopted in the following 5 reputation assortment cases. From panel (b) to (f), we depict the stationary distribution of cooperators (blue or cyan) and defectors (red or pink) for different reputation thresholds, which are set to be 60, 80, 90, 95 and
98, respectively. As displayed in Fig. 3, the cooperative clusters can be effectively organized, and the fraction of cooperators can grow little by little as the reputation threshold increases. It is particulary worth mentioning that many cooperators with high reputation value are often formed into giant clusters, then surrounded by a few low reputation cooperators so that they can defend the survival of cooperators by creating the hierarchical structure from cooperators to defectors; while for defectors, the low reputation agents can dominate the population and develop into a very giant
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Fig. 4. (Color online.) Characteristic pattern of cooperators and defectors at the initial MCS step (a, MCS = 0) and stationary state step (b–f, MCS = 5 × 104 ). In panel (a), cooperators are allocated on the intermediate regions and defectors are allocated on the side regions of lattices, and each player is also assigned a reputation value from the interval [1, 100]. From panel (b) to (f), stationary distribution of cooperators and defectors are depicted; Among them, red (R ≥ R C ) and pink (R < R C ) dots represent the defectors, while blue (R ≥ R C ) and cyan (R < R C ) dots denote the cooperators. The non-influential (i.e., B-type) players hold the weaker strategy transfer capability w = 0.001, other parameters are set to be r = 3.3, L = 200, R max = 100 and K = 0.5.
cluster, but cannot devour all cooperators although some defective agent has penetrated into the cooperative clusters. As R C increases, the size of cooperative clusters will also become larger and larger, and even fully dominate the whole population when R C is equal to 98. Furthermore, we also demonstrate the organizing modes of collective cooperation by fixing all cooperators into the middle of lattice at the initial step, as indicated in Fig. 4, and the similar evolutionary patterns as those in Fig. 3 are also observed. The current clustering patterns again validate the fact that the game micro-
scopic dynamics dominate the evolutionary patterns of evolution, that is, the reputation assortment will help to create the compact and cooperative clusters regardless of the initial choice of strategy of players. For the sake of probing into the role of reputation assortment in depth, we illustrate the distribution of reputation values within the whole population at the steady state in Fig. 5. It is explicitly revealed that the reputation values are uniformly distributed within the interval [1, 100] at the initial time. After we introduce
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Fig. 5. (Color online.) Distribution of reputation value within the whole population at the initial step (a, MCS = 0) and stationary state step (b–f, MCS = 5 × 104 ). In panel (a), reputation values are almost uniformly distributed within the interval [1, 100], whether for cooperators or for defectors. From panel (b) to (f), distribution of reputation at the stationary state is depicted; Among them, red and blue bars represent the relative fractions of defectors and cooperators with various reputation values, respectively. The non-influential (i.e., B-type) players hold the weaker strategy transfer capability w = 0.001, other parameters are set to be r = 3.3, L = 200, R max = 100 and K = 0.5.
the reputation assortment, the cooperators can resist the invasion of defectors so as not to be fully overwhelmed by the defecting players. For example, when the reputation threshold R C is equal to 60, we can observe that nearly 90% of agents are defectors whose reputation values lie between 1 and 10, but around 10% of agents are cooperators whose reputation values are very high and distributed within [91, 100]. As the reputation threshold increases from 80 [panel (c)] to 95 [panel (e)], the fraction of players with low reputation decreases, meanwhile the fraction of players with high reputation increases. Even when R C is up to 98, the whole system is totally composed of cooperators with high reputation values. Thus, as shown in Fig. 5, the population usually evolves into a state in which the system mainly consists of two classes of individuals: one class is the low reputation defector, and the other one is the high reputation cooperator. The competition between cooperators and defectors will drive the cooperators to win the advantage over the defectors during the evolution of collective interactions, and then the cooperative clusters will become larger and larger as the reputation effect works. In an ideal circumstance, the cooperators will dominate the whole population when the reputation threshold is high enough (i.e., R C = 98). In Fig. 6, we plot the evolution of ρC for the specified space r–K under the adaptive reputation assortment, from which we can clearly discern the whole phase transition process from full D to the coexistence of cooperators and defectors (C + D), and then to full C . In panel (a) the reputation threshold R C is equal to 70, but the threshold R C is set to be 95 in panel (b). When the reputa-
tion threshold is lower (R C = 70), the transition from D to C + D is difficult, and the synergy factor for a constant noise strength (K ) transcends the full defection also becomes higher, and similar process happens for the process from C + D to D. Additionally, the region to allow the coexistent cooperators and defectors is also much wider in our phase space. On the contrary, the larger reputation threshold (R C ) can lead to the easier and quicker phase transition within the specified region when compared to the case of R C = 70. It is worth mentioning that the transition points between two spaces becomes lower, and the extraordinarily extended regions to arrive at the full cooperation (C ) renders that the coexistence for cooperators and defectors also displays the narrower room although the D phase is contracted. Therefore, the adaptive reputation assortment can provide a stronger support for the survival of cooperators. Finally, the impact of strategy spreading factor (w) on the cooperation for a fixed reputation threshold (R C = 95) is considered in Fig. 7. It can be clearly observed that the cooperation level, similar to Fig. 1, will be improved as the strategy spreading factor (w) decreases. Specifically, the lower threshold (r C 1 ) of synergy factor denoting the full defection to the appearance of cooperation becomes smaller, and the upper threshold (r C 2 ) representing the coexistence between cooperators and defectors to the full cooperation also has the same tendency. Furthermore, under a fixed synergy factor, as an example, r = 3.5 is less than r C 1 for the traditional version and more than those of other cases for various strategy factors w < 1. Thus, the results again indicate that the co-
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tion and low reputation ones) enlarges, and the influential players can hold a surprisingly high capability to persuade other players to adopt their strategies. Above all, the reputation assortment will help the cooperators to build the cooperative advantages to obtain the higher reputation values, which may further render the cooperators to become influential ones so that they can own a higher transfer ability to easily create the cooperator’s cluster. The greater the difference between two classes of players, the more obvious the enhancement of cooperation level, current results are also compatible with the general conclusions which have claimed that the heterogeneity can hugely favor the evolution of cooperation [13–15]. 4. Conclusions and discussions
Fig. 6. (Color online.) Evolution of fraction of cooperators (ρC ) for the specified r–K parameter space. In panel (a), R C = 70; while in panel (b) R C = 95, and the reputation effect powerfully enhances the cooperation level. The non-influential (i.e., B-type) players hold the weaker strategy transfer capability w = 0.001, other parameters are set to be MCS = 5 × 104 , L = 200, R max = 100.
Fig. 7. (Color online.) Fraction of cooperators (ρC ) at the stationary state as a function of synergy factor (r) for various strategy spreading factor (w). As w decreases, the role of individual reputation assortment becomes much more obvious. The reputation threshold is fixed to be R C = 95, other parameters are set to be MCS = 5 × 104 , L = 200, R max = 100 and K = 0.5.
operation level will be largely enhanced as the strategy spreading factor varies from 0.1 to 0.001. The origin of promotion of cooperation can be attributed to the fact that the difference of strategy spreading factor between two types of players (i.e., high reputa-
To summarize, we propose a spatial public goods game, in which the players are adaptively classified into two types of ones based on a reputation effect, to study the evolution of collective cooperation. Firstly, a reputation value uniformly distributed within the interval [1, 100] is randomly assigned to each player, meanwhile a binary strategy (cooperation or defection) is adopted by an agent with the equal probability. Then, the system proceeds by playing PGG according to the Ferm-like rule, where a multiplicative factor (w y ) will be placed before the common Fermi probability. At the same time, an individual reputation value will be minus or plus one unit if the player decides to take a cooperating or defecting action. Here, w y is dependent on the type of one randomly chosen neighbor (say, y) of a focal player, and w y will be set to be 1.0 if player y is an influential one or a constant w (0 < w < 1) if player y is a non-influential player. Therefore, an agent will be divided into influential or non-influential one based on his own reputation value, in which a player will be influential if a predefined reputation threshold R C is surpassed (i.e., R y ≥ R C ), non-influential otherwise (i.e., R y < R C ). A wide spectrum of numerical simulations are performed on regular lattices to validate the system behavior, where the stationary state will be arrived at or evaluated after enough time steps are discarded. The simulation results demonstrate that the evolution of cooperation can be greatly enhanced by the introduction of adaptive reputation assortment. In particular, the larger the reputation threshold, the fewer the influential players, and the possible origin of cooperation promotion lies that the influential players rendered by the individual assortment can hold enough opportunity and space to persuade the neighboring nodes to adopt his own strategy, in the meantime, the reputation effect helps the cooperators to build accumulative advantages. The evolutionary pattern and statistics of reputation value distribution also uncovers that the reputation assortment is beneficial to the influential cooperators who will finally dominate the collective cooperation. However, the real population structure is far beyond the regularity, and often exhibits the so-called small-world or scale-free properties, hence exploring the evolutionary behavior within the population with heterogeneous topology is also of utmost relevance. Here, we take advantage of regular lattices to simply study the role of reputation or diversity in the promotion of cooperation, and need not involve the impact of heterogeneous topology on the cooperation behavior so that the model is kept to be as simplified as possible. Regarding the strategy update rule, imitation-MAX [9] or replicator dynamics [42] can also be applied, but we adopt the Fermi-like rule so as to keep the consistence and comparison with previous works [48–50]. Thus, it may be deserved the further probings as far as the different interaction topologies or update rules are concerned. Moreover, in the present work, the individual reputation value needs to be recorded at each time step and can be referred by his neighbors. In the future, it will become much more important and necessary if individual reputation cannot be fully
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known by other peers, which means a kind of reputation inference, as shown in Ref. [53]. Anyway, current results can be highly conducive to expounding the possible mechanisms to facilitate the evolution of cooperation within many real-world systems. Acknowledgements This project is in part funded by the National Natural Science Foundation of China (NSFC) under grant Nos. 61203138, 61374169, 61403280 and 71401122. References [1] K. Sigmund, The Calculus of Selfishness, Princeton University Press, Princeton, NJ, 2010. [2] R. Axelrod, The Evolution of Cooperation, Basic Books, New York, 2006. [3] M.A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life, Harvard University Press, Cambridge, MA, 2006. [4] M.J. Osborne, A. Rubinstein, A Course in Game Theory, The MIT Press, Cambridge, MA, 1994. [5] H. Gintis, Game Theory Evolving, Princeton University Press, Princeton, NJ, 2000. [6] M.A. Nowak, Science 314 (2006) 1560–1563. [7] W.D. Hamilton, J. Theor. Biol. 7 (1964) 1–52. [8] R.L. Trivers, Q. Rev. Biol. 46 (1971) 35–37. [9] M.A. Nowak, K. Sigmund, Nature 393 (1998) 573–577. [10] K. Panchanathan, R. Boyd, Nature 432 (2004) 499–502. [11] A. Traulsen, M.A. Nowak, Proc. Natl. Acad. Sci. USA 103 (2006) 10952–10955. [12] M.A. Nowak, R.M. May, Nature 359 (1992) 826–829. [13] F.C. Santos, J.M. Pacheco, Phys. Rev. Lett. 95 (2005) 098104. [14] S. Boccaletti, V. Latora, Y. Moreno, M. Chavezf, D.U. Hwang, Phys. Rep. 424 (2006) 175–308. ¨ M. Romance, [15] S. Boccaletti, G. Bianconi, R. Criado, C.I. Genio, J. Gómez-Gardens, I. Sendiña-Nadal, Z. Wang, M. Zanin, Phys. Rep. 544 (2014) 1–122. [16] Z. Wang, A. Szolnoki, M. Perc, Europhys. Lett. 97 (2012) 48001. [17] L.L. Jiang, M. Perc, Sci. Rep. 3 (2013) 2483. [18] A. Szolnoki, M. Perc, New J. Phys. 15 (2013) 053010. [19] Q. Jin, L. Wang, C.Y. Xia, Z. Wang, Sci. Rep. 4 (2014) 4095. [20] Z. Wang, A. Szolnoki, M. Perc, Sci. Rep. 3 (2013) 1183. [21] X.K. Meng, C.Y. Xia, Z.K. Gao, L. Wang, S.W. Sun, Phys. Lett. A 379 (2015) 767–773.
47
[22] Z. Wang, L. Wang, M. Perc, Phys. Rev. E 89 (2014) 052813. [23] C.Y. Xia, Q. Miao, J. Wang, S. Ding, Appl. Comput. Math. 246 (2014) 389–398. [24] Z. Wang, A. Szolnoki, M. Perc, Sci. Rep. 3 (2013) 2470. [25] C.Y. Xia, X.K. Meng, Z. Wang, PLoS ONE 10 (2015) 12954. [26] C.Y. Xia, J. Zhao, J. Wang, Y.L. Wang, H. Zhang, Phys. Scr. 84 (2011) 025802. [27] J.J. Zhang, H.Y. Ning, Z.Y. Yin, S.W. Sun, L. Wang, J.Q. Sun, C.Y. Xia, Front. Phys. 7 (2012) 366–372. [28] X.J. Chen, A. Szolnoki, M. Perc, L. Wang, Phys. Rev. E 85 (2012) 066133. [29] F. Fu, L.H. Liu, L. Wang, Eur. Phys. J. B 56 (2007) 367–372. [30] W.B. Du, Z.X. Wu, K.Q. Cai, Physica A 392 (2013) 3505–3512. [31] X. Ni, W.X. Wang, Y.C. Lai, C. Grebogi, Phys. Rev. E 82 (2010) 066211. [32] Z.H. Rong, Z.X. Wu, G.R. Chen, Europhys. Lett. 102 (6) (2013) 68005. [33] Z.X. Wu, X.J. Xu, Y. Chen, Y.H. Wang, Phys. Rev. E 71 (2005) 037103. [34] H.F. Zhang, R.R. Liu, Z. Wang, H.X. Yang, B.H. Wang, Europhys. Lett. 94 (2012) 18006. [35] C. Zhang, J. Zhang, G. Xie, L. Wang, Eur. Phys. J. B 80 (2011) 217–222. [36] Z. Wang, S. Kokubo, J. Tanimoto, E. Fukuda, K. Shigaki, Phys. Rev. E 88 (2013) 042145. [37] J. Grujíc, C. Fosco, L. Araujo, J.A. Cuesta, A. Sánchez, PLoS ONE 5 (2010) e13749. [38] C. Gracia-Lázaro, A. Ferrer, G. Ruiz, A. Tarancón, J.A. Cuesta, A. Sánchez, Y. Moreno, Proc. Natl. Acad. Sci. USA 109 (2012) 12922–12926. [39] Z. Wang, A. Szolnoki, M. Perc, Sci. Rep. 2 (2012) 369. [40] Z.K. Gao, P.C. Fang, M.S. Ding, D. Yang, N.D. Jin, Phys. Lett. A 379 (2014) 790–797. [41] Z. Wang, L. Wang, A. Szolnoki, M. Perc, Eur. Biophys. J. 88 (2015) 124. [42] C.P. Roca, J.A. Cuesta, A. Sánchez, Phys. Life Rev. 6 (2009) 208–249. ¨ A. Szolnoki, L.M. Floría, Y. Moreno, J. R. Soc. Inter[43] M. Perc, J. Gómez-Gardenes, face 10 (2013) 20120997. [44] Z. Wang, S. Kokubo, M. Jusup, J. Tanimoto, Phys. Life Rev. 14 (2015) 1–30. [45] M. Kosfeld, Rev. Network Econom. 3 (2004) 20–41. [46] M. Milinski, D. Semmann, H.J. Krambeck, Nature 415 (2002) 424–426. [47] J.A. Cuesta, C. Gracia-Lázaro, A. Ferrer, Y. Moreno, A. Sánchez, Sci. Rep. 5 (2015) 7843. [48] M. Perc, A. Szolnoki, Phys. Rev. E 77 (2008) 011904. [49] G. Szabó, A. Szolnoki, Phys. Rev. E 79 (2009) 016106. [50] C.J. Zhu, S.W. Sun, L. Wang, S. Ding, J. Wang, C.Y. Xia, Physica A 406 (2014) 145–154. [51] R. Jiménez, H. Lugo, J.A. Cuesta, A. Sánchez, J. Theor. Biol. 250 (2008) 475–483. [52] Z. Wang, C.Y. Xia, S. Meloni, C.S. Zhou, Y. Moreno, Sci. Rep. 3 (2013) 3095. [53] Z. Wang, Z. Yin, C.Y. Xia, L. Wang, PLoS ONE 7 (2012) e40218.