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Incorporating the information from direct and indirect neighbors into fitness evaluation enhances the cooperation in the social dilemmas
Chaos, Solitons and Fractals 77 (2015) 47–52
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Incorporating the information from direct and indirect neighbors into fitness evaluation enhances the cooperation in the social dilemmas Menglong Hu a,b, Juan Wang a,b,∗, Lingcong Kong a,b, Kang An a,b, Tao Bi a,b, Baohong Guo c, Enzeng Dong a,b a
Tianjin Key Laboratory for Control Theory and Complicated Industry Systems, Tianjin University of Technology, Tianjin 300384, PR China School of Electrical Engineering, Tianjin University of Technology, Tianjin 300384, PR China c School of Computer and Communication Engineering, Tianjin University of Technology, Tianjin 300384, PR China b
a r t i c l e
i n f o
Article history: Received 11 March 2015 Accepted 23 April 2015 Available online 16 May 2015
a b s t r a c t We propose an improved fitness evaluation method to investigate the evolution of cooperation in the spatial social dilemmas. In our model, a focal player’s fitness is calculated as the linear combination of his own payoff, the average payoffs of direct and indirect neighbors in which two independent selection parameters (α and β ) are used to control the proportion of various payoff contribution to the current fitness. Then, the fitness-based strategy update rule is still Fermi-like, and asynchronous update is adopted here. A large plethora of numerical simulations are performed to validate the behaviors of the current model, and the results unambiguously demonstrate that the cooperation level is greatly enhanced by introducing the payoffs from the surrounding players. In particular, the influence of direct neighbors become more evident when compared with indirect neighbors since the correlation between focal players and their direct neighbors is much closer. Current outcomes are significant for us to further illustrate the origin and emergence of cooperation within a wide variety of natural and man-made systems. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Although the Darwinian evolutionary theory maintains that the mutual competition dominates an individual benefit and even its final fate [1], extensive cooperation phenomena are still emerging within many natural, biological and social systems, which does not coincide with the evolutionary selection theory [2]. Therefore, exploring the origin of cooperation among selfish agents becomes a challenging and multi-disciplinary topic [3], and the evolutionary game
∗
Corresponding author. Tel.: +8618002042280. E-mail addresses: [email protected] (J. Wang), [email protected] (E. Dong).
http://dx.doi.org/10.1016/j.chaos.2015.04.014 0960-0779/© 2015 Elsevier Ltd. All rights reserved.
theory provides a powerful framework to illustrate the evolution of cooperation [4]. Starting from this, several mechanisms to facilitate the cooperation, which include kin selection [5], direct or indirect reciprocity [6–8], group interaction [9], spatial and networking reciprocity [10,11] and so on, have been proposed to account for the emergency of cooperation. In particular, Nowak & May [10] seminally investigated the cooperative behaviors among selfish agents located on the spatial square lattice, and they found that neighboring individuals taking the cooperative strategy can be organized into compact clusters to avoid the employment of defective ones. After this seminal work, spatial reciprocity can be often used as a viable mean to promote the evolution of cooperation by combing with other elements [12,13]. For several intriguing examples, adding the reputation, reward or
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M. Hu et al. / Chaos, Solitons and Fractals 77 (2015) 47–52
punishment into the spatial game models can effectively enhance the collective cooperation level [14–18]; aspiring the highest fitness can powerfully benefit the evolution of cooperation [19,20]; integrating the weight mechanism into the fitness evaluation can strongly facilitate the cooperative behaviors [21,22], to name but a few. Furthermore, different update rules may lead to some distinct evolutionary results [23–25]. Meanwhile, complex networks also build solid topological structures to support the collective cooperation, such as scale free properties and local world effect [26–33]. In the very recent years, interdependent or multi-layer networks provide a novel and potent support for the understanding of cooperative behaviors within the real-world systems [34–43]. In addition, the co-evolutionary game model, which considers the interplay between the individual state or behaviors and the topological structure, also captures the extensive attention in the field of biology and physics [44,45]. However, most works focus on modifying the individual fitness expression or evaluation so that the collective cooperation is widely promoted, and the environmental information is hardly considered. In fact, the environment may remarkably play an important role from the perspective of individual benefit and decision analysis. As an example, the external conditions in one region, which include traffic infrastructure, tax system, laboring cost and correspondingly complementary industry level and so on, often influence whether other large or multi-national corporations will decide to make an investment at that region. Thus, integrating the neighboring information into the game model is of great significance to understand the widespread cooperation phenomenon. Among them, Wang et al. [46] devised an improved game model which modifies the definition of individual fitness, combing the individual payoff with the average value of payoffs from 4 nearest neighbors through a selection parameter, and the results indicate that the cooperation can be largely elevated. Based on this work, Xia et al. [47] further integrate the maximum payoff among 4 nearest neighbors into the focal player’s fitness computation and find that the introduction of neighboring payoff can help to promote the cooperation level within the population. In the meantime, Ref. [48] take the environmental information in the public goods game model (PGG) into account, and the results suggest that the cooperation level cannot be greatly influenced by the environmental information. Nevertheless, these studies can only discuss the impact of direct neighbors, and the role of indirect neighbors needs to be further explored. Along this research line, in the present work, we investigate the effect of neighboring payoffs by two selection parameters, where one parameter is for the direct neighbors and the other one is for the indirect neighbors. Extensive numerical simulation results infer that the cooperation can be greatly promoted by introducing the environmental information. The rest of this paper is structured as follows. At first, the modified prisoner’s dilemma game model considering two kinds of environmental information is clearly illustrated in Section 2. After that, Section 3 presents large quantities of simulation results which persuasively prove the enhancement of promotion, and qualitatively similar behaviors are also demonstrated by extending the present fitness definition into the snowdrift game. At last, we end the paper with some concluding remarks in Section 4.
2. An improved spatial prisoner’s dilemma game model As we know, the prisoner’s dilemma game (PDG) is one category of game in which the cooperative players are the most difficult to survive in the classical scenarios since the defection is always superior to the cooperation regarding the individual game payoff, that is, the defection is the Nash equilibrium [4]. In this game model, players must simultaneously adopt one of two pure strategies: to cooperate (C) or to defect (D), and their actual payoffs are dependent on their strategy choices. When two players choose the same strategy, they will get the maximally total payoff as the reward (R) for their cooperation, but both of them will obtain the punishment (P) for their defection. However, if one player chooses the cooperation strategy and the other one takes the defective action, the defective player will reap the temptation (T) payoff and cooperative one will gain the sucker’s (S) payoff. Typically, the payoff matrix can be used to describe the above-mentioned individual payoff as follows,
C D
C R T
D S P
(1)
where T > R > P > S and 2R > S + T must be satisfied for the standard PDG model, it is obvious that to defect is the best strategy for a player, regardless of his opponent’s choice. Without lacking the generality, here we consider a kind of weak PDG parameter setup: 1 < T = b < 2, R = 1 and P = S = 0, which is firstly adopted by Nowak & May [10] and nearly captures all features of the standard PDG model. In the model, all individuals are initially placed on the square lattice with periodic boundary conditions, and they are randomly assigned as a cooperator or defector, that is, they will take the cooperation (Si = C) or defection (Si = D) strategy with the equal probability. At each time step, the focal player (say, i) firstly calculates his payoff (Pi ) by playing the games with his 4 nearest neighbors according to Eq. (1), and his direct or indirect neighbors can also compute their corresponding payoff with the same way as Pi . After that, we consider the role of environmental information in the definition of individual fitness, in which we commonly consider the effect of direct and indirect neighbors with the following formula,
fi = (1 − α − β)Pi + α P¯ + β P˜
(2)
where we set 0 ࣘ α , β ࣘ 0.5 so that 0 ࣘ (1 − α − β ) ࣘ 1, P¯ denotes the average value of 4 direct neighbors and P˜ characterizes the average value of 4 indirect neighbors as illustrated in Fig. 1, and P¯ and P˜ are calculated as follows,
P¯ =
j∈Ni
4
Pj
, P˜ =
j ∈Ni
4
Pj
(3)
where Ni and Ni represent the set of direct and indirect neighbors of focal player i. Here, P¯ and P˜ characterize the influence of environmental information in the evaluation of individual fitness, α and β are two independent tunable parameters which are used to quantify the proportion of external information and control the contribution of direct or indirect
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3. Results and discussions
Fig. 1. The neighborhood setup used in our PDG model. The red star node is the focal player in the game, blue circle nodes stand for direct or nearest neighbors and green triangle nodes represent indirect neighbors. The fitness of a focal player is evaluated as the linear combinations of his own payoff and the average payoffs of direct or indirect neighbors according to Eq. (2). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
neighbors. Especially, if we set α = β = 0, the current model is reduced to the traditional PDG model which regards the current payoff as the present fitness of a focal agent; if we only consider the role of direct neighbors, i.e., β = 0, our model is equivalent to the model in Ref.[46]. Then, each player will asynchronously update his current strategy choice according to the classical Fermi rule [49], that is, a focal player i adopts the strategy from one of his nearest neighbors, who is randomly selected (say, j) with the following Fermi probability
Prob(Si ← Sj ) =
1 1+e
−(fj −fi )
As ρ C characterizes the collective cooperation level and becomes the quantity of most interest in the field of evolutionary game theory, at first we illustrate the relationship between ρ C and defection parameter b under a specific setup of α and β in Fig. 2. Here, we consider three different cases for α and β : (i) 0 ࣘ α < 0.5 and β = 0; (ii) α = 0 and 0 ࣘ β < 0.5; (iii) 0 ࣘ α , β < 0.5. In panel (a), we can merely consider the effect of direct neighbors (0 ࣘ α < 0.5) on the cooperation and neglect the role of indirect neighbors (β = 0), and it is explicitly observed that if the direct neighbors’ payoffs are integrated into the fitness evaluation of focal player, the cooperation can be largely enhanced when compared to the traditional case, in which a focal player’s payoff can be regarded as his own fitness. Next, in panel (b), we set α = 0 and allows β > 0, which means that we just add the indirect neighbor’s payoffs into the fitness evaluation of focal player. The results also demonstrate that the cooperation can be slightly promoted when compared with the traditional version, but the augmentation level of cooperation is lower than the previous case. By comparing the results under these two simulations, the role of direct neighbors will become much more salient since their payoffs are also correlated with focal player, and in turn, these payoffs are calculated as a part of the current fitness of focal one. Meanwhile, the strategy update can only take place within the direct neighborhood, which further make the direct neighbors hold an important position in the evolution of cooperation. In the third case, the impact of two kinds of
(4)
K
where fi and fj denote the fitness of player i and j, Si and Sj represent their current strategy states, and K characterizes the magnitude of noise or irrationality during decision-making process. It is obvious that, in our model, the above-mentioned two basic steps comprise a full Monte Carlo simulation (MCS) step, within which each player has a chance on average to update his own strategy based on a pairwise comparison process, i.e., asynchronous update. The steady state will be reached after some transient MCS steps (ts ) have been discarded, and the stationary density of cooperators (ρ C ) will be counted and recorded within some steady and last MCS steps (ta ). In the following section, all numerical results are performed on 100 × 100 (i.e., L = 100) regular lattices, larger lattice sizes (e.g., L = 200 or 400) are also tested and results are not qualitatively varied. In all simulations, each independent run will last 100000 MCS steps and ρ C will be obtained by averaging the last 5000 MCS steps, that is, ts = 95000 and ta = 5000. In addition, all numerical results are averaged over 20 independent runs, and the magnitude of noise K is always set to be 0.1 if not stated.
Fig. 2. Fraction of cooperators (ρ C ) as a function of defection parameter b for different selection parameter setup in the weak PDG. In panel (a), β = 0 is fixed and we can change α so that the effect of direct neighbors is simply considered; In panel (b), we can set α = 0 so as to merely take the role of indirect neighbors into account; In panel (c), the impact of direct and indirect neighbors is commonly considered. The other parameter setup is K = 0.1 and L = 100.
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Fig. 3. Impact of tunable selection parameter (α and β ) on the fraction of cooperators (ρ C ) under a fixed defection parameter. In the top panel (a), b is set to be 1.2, while b is fixed to be 1.4 in the bottom panel (b). The results potentially imply that the role of direct neighbors seems to be more striking under the current model. The other parameter setup is K = 0.1 and L = 100.
environmental information has been simultaneously taken into account and payoffs from direct and indirect neighbors are both used to assess the fitness of focal agent. It is clearly shown in panel (c) that the cooperation will be surprisingly enhanced and even the cooperators fully dominate the whole population, ρ C will be much higher than those in the abovementioned two cases under the same temptation to defect (b), that is, the combination of direct and indirect neighbors will further facilitate the cooperative agents to defend the invasion of defectors, superposition effect of direct and indi-
rect neighbors’ role creating the emergency of cooperation is expressly displayed here. For a fixed defection parameter b, in Fig. 3 we scrutinize the influence of direct (α ) and indirect neighbors (β ) inside all possible parameter ranges, we can show that the integration of environmental information including direct and indirect neighborhood will increasingly modify the cooperative behaviors in the population. In panel (a) of Fig. 3, b is set to be 1.2, α varies from 0 to 0.5 and β changes between 0 and 0.5, and K is always set to be 0.1. It is clearly indicated that the cooperation tends to be extinct when α and β are smaller, but the cooperator’s density becomes higher and higher as α and β enlarge. For a fixed β , ρ C will increase from 0 to 100% when α aggrandizes from 0 to 0.5, that is, the system evolution proceeds the transition from the complete defection to the full cooperation among the whole population. But for a fixed α , the process from all-defection to all-cooperation will not be observed although ρ C continuously augments when β increases from 0 to 0.5. Once again, we can conclude that the impact of direct neighbors on the cooperation becomes much more prominent when compared with the indirect neighbors, and considering the superposition between these two kinds of neighbors will further lead to the emergence of cooperation when we assess the fitness of a focal player. We further examine the role of direct or indirect neighbors in the promotion of cooperation by presenting the characteristic snapshots between cooperators and defectors at the stationary state in Fig. 4. In the top four panels, we set the defection parameter b = 1.05 and tunable factors α and β are set to be different values so as to explore the role of direct and indirect neighbors in detail. In panel (a), we suppose that α = β = 0, the model is reduced to the classic one with
Fig. 4. Characteristic snapshots of cooperators and defectors at the stationary state for two different defection parameters, in which gray (light) dots represent defectors and red (dark) dots stand for cooperators. From panel (a) to (d), the defection parameter is set to be b = 1.05; while for panel (e) to (h), b is fixed to be 1.1. In panels (a) and (d), α = β = 0, and α = β = 0.25 for panels (d) and (h); but for panels (b) and (f), α is 0 and β is 0.25, and the opposite parameter setup is assumed to be α = 0.25 and β = 0 in panels (c) and (g). It is clearly clarified that the role of direct neighbors become much more highlighted. In all simulations K = 0.1 and the lattice size is L = 100. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
M. Hu et al. / Chaos, Solitons and Fractals 77 (2015) 47–52
Fermi update rule, and the cooperators cannot resist the invasion of defectors so that the cooperation becomes extinct; but in panel (b), β is set to be 0.25 and α is still set to be 0, we can observe that only considering the impact of indirect neighbors can help to create some sporadic clusters of cooperators, the moderate cooperation level can be maintained. Nevertheless, if we can just take the direct neighbors into account, which here means α = 0.25 and β = 0, the cooperators will comprise strong and compact clusters to get around the exploitation from the defectors, and it is clearly manifested that the cooperation will be largely elevated, yet the survival space of defectors is kept so that the full cooperation cannot be arrived at. Provided that we simultaneously integrate the impact of direct and indirect neighbors into the fitness evaluation of focal players, the promotion of cooperation will become much more prominent and even the full cooperation has been reached as shown in panel (d). Similar phenomena can also be observed for the larger defection parameter (b = 1.1) from panel (e) to (h), only that merely combining the behavior of indirect neighbors is not enough to form the cooperative clusters and the defectors still dominate the population in panel (f); meanwhile, the full cooperating population cannot emerge even if we take the direct and indirect neighbors into account at the same time. In a word, incorporating the external information including direct and indirect neighbors will greatly facilitate the formation of cooperative clusters and elevate the collective cooperation level, and particularly the role of direct neighbors will become more obvious as shown in Ref.[46]. In order to highlight the role of environmental information including the direct and indirect neighbors in the evolution of cooperation, we extend the current model into snowdrift game (SDG) which represents a mixed optimal strategy for a focal agent. In the SDG, two Nash equilibrium exist and the optimal strategy depends on his opponent’s strategy. As a typical example, there are two people trapped by a snowdrift at the two ends of a road, for every one of them, the best option is not to shovel snow off to free the road and let the other person do it; however, if the other person does not shovel, then the best option is to shovel oneself. Hence, it is possible to create a dilemmatic situation, and the current action will be dependent on the other player’s choice. In the simulation, to simplify the payoff matrix, we set T = 1 + r, S = 1 − r, R = 1 and P = 0 so that T > R > S > P is satisfied. In Fig. 5, we can depict the relationship between ρ C and r and find that the cooperation can be remarkably enhanced by the introduction of environmental information. Altogether, integrating the neighboring resources into the fitness assessment becomes much more beneficial to form the cooperative clusters in the population, whether for the PDG or SDG model. 4. Conclusions To summarize, in this paper, we have proposed an improved fitness evaluation approach in the spatial prisoner’s dilemma, which integrates the payoff from direct and indirect neighbors. Two tunable parameters (i.e., α and β ) are utilized to control the integration intensity of neighbors’ payoff, and α = β = 0 makes the model be reduced to the standard model with Fermi rule, but the model with α > 0 and β =
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Fig. 5. Fraction of cooperators (ρ C ) as a function of ratio of cost to benefit (r) in the SDG when payoffs from two kinds of neighbors are both integrated into the fitness assessment of a focal player. The SDG payoff matrix elements are chosen as follows: T = 1 + r, S = 1 − r, R = 1 and P = 0 in order to keep the ranking order T > R > S > P. It is evidently proved that the cooperation can be highly promoted when the environmental information is introduced. In all simulations, we still keep K = 0.1 and the lattice size is also L = 100.
0 is equivalent to the model in Ref.[46]. Extensive numerical simulations indicate that the cooperation will be dramatically enhanced when we compare the current model with the standard scenario. It is also indicated that the role of direct neighbors will become relatively crucial in the evolution of cooperation, which may be attributed to two main factors: one is that the calculation of payoffs of direct neighbors is also correlated with the focal player; the other one is that the strategy imitation can only allowed within the direct neighborhood. At the same time, the evolutionary pattern and the fraction of cooperators within all possible intervals at the stationary state are also applied to validate the abovementioned picture. Furthermore, the method of integrating the external payoffs from other neighbors into the fitness evaluation is also generalized into another social dilemmasnowdrift game (SDG), and qualitatively same behaviors are also exhibited and the evolution of cooperation is still much more facilitated by this new kind of fitness evaluation mechanism. Anyway, current findings will help to provide an insight into the widespread cooperation phenomena within many natural, biological, economical and social systems.
Acknowledgments This project is partially supported by the National Natural Science Foundation of China (NSFC) under grant no. 71401122, the Development Foundation For Science and Technology of Higher Education in Tianjin under grant no. 20130821, and the National Creative Experiment Project for Undergraduates under grant no. X2013035. References [1] Darwin C. The origin of species. Cambrige, MA: Harward University Press; 1859. (Reprinted, 1964). [2] Gintis H. Game theory evolving. Princeton, NJ: Princeton University Press; 2000. [3] Axelrod R. The evolution of cooperation. New York: Basic Books; 2006. [4] Nowak MA. Evolutionary dynamics: exploring the equations of life. Cambrige, MA: Harvard University Press; 2006. [5] Hamilton WD. The genetical evolution of social behavior: I–II. J Theor Bio 1964;7:1–52. [6] Trivers RL. The evolution of reciprocal althruism. Q Rev Biol 1971;46:35–7. [7] Nowak MA, Sigmund K. Evolution of indirect reciprocity by image scoring. Nature 1998;393:573–7.
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Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Incorporating the information from direct and indirect neighbors into fitness evaluation enhances the cooperation in the social dilemmas Menglong Hu a,b, Juan Wang a,b,∗, Lingcong Kong a,b, Kang An a,b, Tao Bi a,b, Baohong Guo c, Enzeng Dong a,b a
Tianjin Key Laboratory for Control Theory and Complicated Industry Systems, Tianjin University of Technology, Tianjin 300384, PR China School of Electrical Engineering, Tianjin University of Technology, Tianjin 300384, PR China c School of Computer and Communication Engineering, Tianjin University of Technology, Tianjin 300384, PR China b
a r t i c l e
i n f o
Article history: Received 11 March 2015 Accepted 23 April 2015 Available online 16 May 2015
a b s t r a c t We propose an improved fitness evaluation method to investigate the evolution of cooperation in the spatial social dilemmas. In our model, a focal player’s fitness is calculated as the linear combination of his own payoff, the average payoffs of direct and indirect neighbors in which two independent selection parameters (α and β ) are used to control the proportion of various payoff contribution to the current fitness. Then, the fitness-based strategy update rule is still Fermi-like, and asynchronous update is adopted here. A large plethora of numerical simulations are performed to validate the behaviors of the current model, and the results unambiguously demonstrate that the cooperation level is greatly enhanced by introducing the payoffs from the surrounding players. In particular, the influence of direct neighbors become more evident when compared with indirect neighbors since the correlation between focal players and their direct neighbors is much closer. Current outcomes are significant for us to further illustrate the origin and emergence of cooperation within a wide variety of natural and man-made systems. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Although the Darwinian evolutionary theory maintains that the mutual competition dominates an individual benefit and even its final fate [1], extensive cooperation phenomena are still emerging within many natural, biological and social systems, which does not coincide with the evolutionary selection theory [2]. Therefore, exploring the origin of cooperation among selfish agents becomes a challenging and multi-disciplinary topic [3], and the evolutionary game
∗
Corresponding author. Tel.: +8618002042280. E-mail addresses: [email protected] (J. Wang), [email protected] (E. Dong).
http://dx.doi.org/10.1016/j.chaos.2015.04.014 0960-0779/© 2015 Elsevier Ltd. All rights reserved.
theory provides a powerful framework to illustrate the evolution of cooperation [4]. Starting from this, several mechanisms to facilitate the cooperation, which include kin selection [5], direct or indirect reciprocity [6–8], group interaction [9], spatial and networking reciprocity [10,11] and so on, have been proposed to account for the emergency of cooperation. In particular, Nowak & May [10] seminally investigated the cooperative behaviors among selfish agents located on the spatial square lattice, and they found that neighboring individuals taking the cooperative strategy can be organized into compact clusters to avoid the employment of defective ones. After this seminal work, spatial reciprocity can be often used as a viable mean to promote the evolution of cooperation by combing with other elements [12,13]. For several intriguing examples, adding the reputation, reward or
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punishment into the spatial game models can effectively enhance the collective cooperation level [14–18]; aspiring the highest fitness can powerfully benefit the evolution of cooperation [19,20]; integrating the weight mechanism into the fitness evaluation can strongly facilitate the cooperative behaviors [21,22], to name but a few. Furthermore, different update rules may lead to some distinct evolutionary results [23–25]. Meanwhile, complex networks also build solid topological structures to support the collective cooperation, such as scale free properties and local world effect [26–33]. In the very recent years, interdependent or multi-layer networks provide a novel and potent support for the understanding of cooperative behaviors within the real-world systems [34–43]. In addition, the co-evolutionary game model, which considers the interplay between the individual state or behaviors and the topological structure, also captures the extensive attention in the field of biology and physics [44,45]. However, most works focus on modifying the individual fitness expression or evaluation so that the collective cooperation is widely promoted, and the environmental information is hardly considered. In fact, the environment may remarkably play an important role from the perspective of individual benefit and decision analysis. As an example, the external conditions in one region, which include traffic infrastructure, tax system, laboring cost and correspondingly complementary industry level and so on, often influence whether other large or multi-national corporations will decide to make an investment at that region. Thus, integrating the neighboring information into the game model is of great significance to understand the widespread cooperation phenomenon. Among them, Wang et al. [46] devised an improved game model which modifies the definition of individual fitness, combing the individual payoff with the average value of payoffs from 4 nearest neighbors through a selection parameter, and the results indicate that the cooperation can be largely elevated. Based on this work, Xia et al. [47] further integrate the maximum payoff among 4 nearest neighbors into the focal player’s fitness computation and find that the introduction of neighboring payoff can help to promote the cooperation level within the population. In the meantime, Ref. [48] take the environmental information in the public goods game model (PGG) into account, and the results suggest that the cooperation level cannot be greatly influenced by the environmental information. Nevertheless, these studies can only discuss the impact of direct neighbors, and the role of indirect neighbors needs to be further explored. Along this research line, in the present work, we investigate the effect of neighboring payoffs by two selection parameters, where one parameter is for the direct neighbors and the other one is for the indirect neighbors. Extensive numerical simulation results infer that the cooperation can be greatly promoted by introducing the environmental information. The rest of this paper is structured as follows. At first, the modified prisoner’s dilemma game model considering two kinds of environmental information is clearly illustrated in Section 2. After that, Section 3 presents large quantities of simulation results which persuasively prove the enhancement of promotion, and qualitatively similar behaviors are also demonstrated by extending the present fitness definition into the snowdrift game. At last, we end the paper with some concluding remarks in Section 4.
2. An improved spatial prisoner’s dilemma game model As we know, the prisoner’s dilemma game (PDG) is one category of game in which the cooperative players are the most difficult to survive in the classical scenarios since the defection is always superior to the cooperation regarding the individual game payoff, that is, the defection is the Nash equilibrium [4]. In this game model, players must simultaneously adopt one of two pure strategies: to cooperate (C) or to defect (D), and their actual payoffs are dependent on their strategy choices. When two players choose the same strategy, they will get the maximally total payoff as the reward (R) for their cooperation, but both of them will obtain the punishment (P) for their defection. However, if one player chooses the cooperation strategy and the other one takes the defective action, the defective player will reap the temptation (T) payoff and cooperative one will gain the sucker’s (S) payoff. Typically, the payoff matrix can be used to describe the above-mentioned individual payoff as follows,
C D
C R T
D S P
(1)
where T > R > P > S and 2R > S + T must be satisfied for the standard PDG model, it is obvious that to defect is the best strategy for a player, regardless of his opponent’s choice. Without lacking the generality, here we consider a kind of weak PDG parameter setup: 1 < T = b < 2, R = 1 and P = S = 0, which is firstly adopted by Nowak & May [10] and nearly captures all features of the standard PDG model. In the model, all individuals are initially placed on the square lattice with periodic boundary conditions, and they are randomly assigned as a cooperator or defector, that is, they will take the cooperation (Si = C) or defection (Si = D) strategy with the equal probability. At each time step, the focal player (say, i) firstly calculates his payoff (Pi ) by playing the games with his 4 nearest neighbors according to Eq. (1), and his direct or indirect neighbors can also compute their corresponding payoff with the same way as Pi . After that, we consider the role of environmental information in the definition of individual fitness, in which we commonly consider the effect of direct and indirect neighbors with the following formula,
fi = (1 − α − β)Pi + α P¯ + β P˜
(2)
where we set 0 ࣘ α , β ࣘ 0.5 so that 0 ࣘ (1 − α − β ) ࣘ 1, P¯ denotes the average value of 4 direct neighbors and P˜ characterizes the average value of 4 indirect neighbors as illustrated in Fig. 1, and P¯ and P˜ are calculated as follows,
P¯ =
j∈Ni
4
Pj
, P˜ =
j ∈Ni
4
Pj
(3)
where Ni and Ni represent the set of direct and indirect neighbors of focal player i. Here, P¯ and P˜ characterize the influence of environmental information in the evaluation of individual fitness, α and β are two independent tunable parameters which are used to quantify the proportion of external information and control the contribution of direct or indirect
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3. Results and discussions
Fig. 1. The neighborhood setup used in our PDG model. The red star node is the focal player in the game, blue circle nodes stand for direct or nearest neighbors and green triangle nodes represent indirect neighbors. The fitness of a focal player is evaluated as the linear combinations of his own payoff and the average payoffs of direct or indirect neighbors according to Eq. (2). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
neighbors. Especially, if we set α = β = 0, the current model is reduced to the traditional PDG model which regards the current payoff as the present fitness of a focal agent; if we only consider the role of direct neighbors, i.e., β = 0, our model is equivalent to the model in Ref.[46]. Then, each player will asynchronously update his current strategy choice according to the classical Fermi rule [49], that is, a focal player i adopts the strategy from one of his nearest neighbors, who is randomly selected (say, j) with the following Fermi probability
Prob(Si ← Sj ) =
1 1+e
−(fj −fi )
As ρ C characterizes the collective cooperation level and becomes the quantity of most interest in the field of evolutionary game theory, at first we illustrate the relationship between ρ C and defection parameter b under a specific setup of α and β in Fig. 2. Here, we consider three different cases for α and β : (i) 0 ࣘ α < 0.5 and β = 0; (ii) α = 0 and 0 ࣘ β < 0.5; (iii) 0 ࣘ α , β < 0.5. In panel (a), we can merely consider the effect of direct neighbors (0 ࣘ α < 0.5) on the cooperation and neglect the role of indirect neighbors (β = 0), and it is explicitly observed that if the direct neighbors’ payoffs are integrated into the fitness evaluation of focal player, the cooperation can be largely enhanced when compared to the traditional case, in which a focal player’s payoff can be regarded as his own fitness. Next, in panel (b), we set α = 0 and allows β > 0, which means that we just add the indirect neighbor’s payoffs into the fitness evaluation of focal player. The results also demonstrate that the cooperation can be slightly promoted when compared with the traditional version, but the augmentation level of cooperation is lower than the previous case. By comparing the results under these two simulations, the role of direct neighbors will become much more salient since their payoffs are also correlated with focal player, and in turn, these payoffs are calculated as a part of the current fitness of focal one. Meanwhile, the strategy update can only take place within the direct neighborhood, which further make the direct neighbors hold an important position in the evolution of cooperation. In the third case, the impact of two kinds of
(4)
K
where fi and fj denote the fitness of player i and j, Si and Sj represent their current strategy states, and K characterizes the magnitude of noise or irrationality during decision-making process. It is obvious that, in our model, the above-mentioned two basic steps comprise a full Monte Carlo simulation (MCS) step, within which each player has a chance on average to update his own strategy based on a pairwise comparison process, i.e., asynchronous update. The steady state will be reached after some transient MCS steps (ts ) have been discarded, and the stationary density of cooperators (ρ C ) will be counted and recorded within some steady and last MCS steps (ta ). In the following section, all numerical results are performed on 100 × 100 (i.e., L = 100) regular lattices, larger lattice sizes (e.g., L = 200 or 400) are also tested and results are not qualitatively varied. In all simulations, each independent run will last 100000 MCS steps and ρ C will be obtained by averaging the last 5000 MCS steps, that is, ts = 95000 and ta = 5000. In addition, all numerical results are averaged over 20 independent runs, and the magnitude of noise K is always set to be 0.1 if not stated.
Fig. 2. Fraction of cooperators (ρ C ) as a function of defection parameter b for different selection parameter setup in the weak PDG. In panel (a), β = 0 is fixed and we can change α so that the effect of direct neighbors is simply considered; In panel (b), we can set α = 0 so as to merely take the role of indirect neighbors into account; In panel (c), the impact of direct and indirect neighbors is commonly considered. The other parameter setup is K = 0.1 and L = 100.
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Fig. 3. Impact of tunable selection parameter (α and β ) on the fraction of cooperators (ρ C ) under a fixed defection parameter. In the top panel (a), b is set to be 1.2, while b is fixed to be 1.4 in the bottom panel (b). The results potentially imply that the role of direct neighbors seems to be more striking under the current model. The other parameter setup is K = 0.1 and L = 100.
environmental information has been simultaneously taken into account and payoffs from direct and indirect neighbors are both used to assess the fitness of focal agent. It is clearly shown in panel (c) that the cooperation will be surprisingly enhanced and even the cooperators fully dominate the whole population, ρ C will be much higher than those in the abovementioned two cases under the same temptation to defect (b), that is, the combination of direct and indirect neighbors will further facilitate the cooperative agents to defend the invasion of defectors, superposition effect of direct and indi-
rect neighbors’ role creating the emergency of cooperation is expressly displayed here. For a fixed defection parameter b, in Fig. 3 we scrutinize the influence of direct (α ) and indirect neighbors (β ) inside all possible parameter ranges, we can show that the integration of environmental information including direct and indirect neighborhood will increasingly modify the cooperative behaviors in the population. In panel (a) of Fig. 3, b is set to be 1.2, α varies from 0 to 0.5 and β changes between 0 and 0.5, and K is always set to be 0.1. It is clearly indicated that the cooperation tends to be extinct when α and β are smaller, but the cooperator’s density becomes higher and higher as α and β enlarge. For a fixed β , ρ C will increase from 0 to 100% when α aggrandizes from 0 to 0.5, that is, the system evolution proceeds the transition from the complete defection to the full cooperation among the whole population. But for a fixed α , the process from all-defection to all-cooperation will not be observed although ρ C continuously augments when β increases from 0 to 0.5. Once again, we can conclude that the impact of direct neighbors on the cooperation becomes much more prominent when compared with the indirect neighbors, and considering the superposition between these two kinds of neighbors will further lead to the emergence of cooperation when we assess the fitness of a focal player. We further examine the role of direct or indirect neighbors in the promotion of cooperation by presenting the characteristic snapshots between cooperators and defectors at the stationary state in Fig. 4. In the top four panels, we set the defection parameter b = 1.05 and tunable factors α and β are set to be different values so as to explore the role of direct and indirect neighbors in detail. In panel (a), we suppose that α = β = 0, the model is reduced to the classic one with
Fig. 4. Characteristic snapshots of cooperators and defectors at the stationary state for two different defection parameters, in which gray (light) dots represent defectors and red (dark) dots stand for cooperators. From panel (a) to (d), the defection parameter is set to be b = 1.05; while for panel (e) to (h), b is fixed to be 1.1. In panels (a) and (d), α = β = 0, and α = β = 0.25 for panels (d) and (h); but for panels (b) and (f), α is 0 and β is 0.25, and the opposite parameter setup is assumed to be α = 0.25 and β = 0 in panels (c) and (g). It is clearly clarified that the role of direct neighbors become much more highlighted. In all simulations K = 0.1 and the lattice size is L = 100. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
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Fermi update rule, and the cooperators cannot resist the invasion of defectors so that the cooperation becomes extinct; but in panel (b), β is set to be 0.25 and α is still set to be 0, we can observe that only considering the impact of indirect neighbors can help to create some sporadic clusters of cooperators, the moderate cooperation level can be maintained. Nevertheless, if we can just take the direct neighbors into account, which here means α = 0.25 and β = 0, the cooperators will comprise strong and compact clusters to get around the exploitation from the defectors, and it is clearly manifested that the cooperation will be largely elevated, yet the survival space of defectors is kept so that the full cooperation cannot be arrived at. Provided that we simultaneously integrate the impact of direct and indirect neighbors into the fitness evaluation of focal players, the promotion of cooperation will become much more prominent and even the full cooperation has been reached as shown in panel (d). Similar phenomena can also be observed for the larger defection parameter (b = 1.1) from panel (e) to (h), only that merely combining the behavior of indirect neighbors is not enough to form the cooperative clusters and the defectors still dominate the population in panel (f); meanwhile, the full cooperating population cannot emerge even if we take the direct and indirect neighbors into account at the same time. In a word, incorporating the external information including direct and indirect neighbors will greatly facilitate the formation of cooperative clusters and elevate the collective cooperation level, and particularly the role of direct neighbors will become more obvious as shown in Ref.[46]. In order to highlight the role of environmental information including the direct and indirect neighbors in the evolution of cooperation, we extend the current model into snowdrift game (SDG) which represents a mixed optimal strategy for a focal agent. In the SDG, two Nash equilibrium exist and the optimal strategy depends on his opponent’s strategy. As a typical example, there are two people trapped by a snowdrift at the two ends of a road, for every one of them, the best option is not to shovel snow off to free the road and let the other person do it; however, if the other person does not shovel, then the best option is to shovel oneself. Hence, it is possible to create a dilemmatic situation, and the current action will be dependent on the other player’s choice. In the simulation, to simplify the payoff matrix, we set T = 1 + r, S = 1 − r, R = 1 and P = 0 so that T > R > S > P is satisfied. In Fig. 5, we can depict the relationship between ρ C and r and find that the cooperation can be remarkably enhanced by the introduction of environmental information. Altogether, integrating the neighboring resources into the fitness assessment becomes much more beneficial to form the cooperative clusters in the population, whether for the PDG or SDG model. 4. Conclusions To summarize, in this paper, we have proposed an improved fitness evaluation approach in the spatial prisoner’s dilemma, which integrates the payoff from direct and indirect neighbors. Two tunable parameters (i.e., α and β ) are utilized to control the integration intensity of neighbors’ payoff, and α = β = 0 makes the model be reduced to the standard model with Fermi rule, but the model with α > 0 and β =
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Fig. 5. Fraction of cooperators (ρ C ) as a function of ratio of cost to benefit (r) in the SDG when payoffs from two kinds of neighbors are both integrated into the fitness assessment of a focal player. The SDG payoff matrix elements are chosen as follows: T = 1 + r, S = 1 − r, R = 1 and P = 0 in order to keep the ranking order T > R > S > P. It is evidently proved that the cooperation can be highly promoted when the environmental information is introduced. In all simulations, we still keep K = 0.1 and the lattice size is also L = 100.
0 is equivalent to the model in Ref.[46]. Extensive numerical simulations indicate that the cooperation will be dramatically enhanced when we compare the current model with the standard scenario. It is also indicated that the role of direct neighbors will become relatively crucial in the evolution of cooperation, which may be attributed to two main factors: one is that the calculation of payoffs of direct neighbors is also correlated with the focal player; the other one is that the strategy imitation can only allowed within the direct neighborhood. At the same time, the evolutionary pattern and the fraction of cooperators within all possible intervals at the stationary state are also applied to validate the abovementioned picture. Furthermore, the method of integrating the external payoffs from other neighbors into the fitness evaluation is also generalized into another social dilemmasnowdrift game (SDG), and qualitatively same behaviors are also exhibited and the evolution of cooperation is still much more facilitated by this new kind of fitness evaluation mechanism. Anyway, current findings will help to provide an insight into the widespread cooperation phenomena within many natural, biological, economical and social systems.
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