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Asymmetric evaluation of fitness enhances spatial reciprocity in social dilemmas
Chaos, Solitons & Fractals 54 (2013) 76–81
Contents lists available at SciVerse ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Asymmetric evaluation of fitness enhances spatial reciprocity in social dilemmas Yi-Ling Wang ⇑ School of Life Science, Shanxi Normal University, LinFen, Shanxi 041000, China
a r t i c l e
i n f o
Article history: Received 1 February 2013 Accepted 17 May 2013 Available online 20 June 2013
a b s t r a c t Evolutionary game theory has been a powerful tool to understand the ubiquity of cooperation in many real-world systems which ranges from biological to economical and social sciences. In this paper, we propose a novel game model, which considers different fitness evaluation means for focal agent and his randomly chosen neighbor, to characterize the asymmetry of information during the game playing. When playing the game, the focal agent obtains its own fitness which includes its own payoff and the average payoff of his neighbors. However, due to the limited information for the chosen neighbor, his fitness can not be evaluated as the focal agent but just the traditional way, namely his fitness equals to his payoff. Large-scale numerical simulations indicate that this kind of asymmetric fitness computation can have substantial effects on the cooperative behaviors on the regular lattice. Interestingly, the larger the asymmetric factor a, the higher the cooperation level qC. Meanwhile, introducing the asymmetric fitness evaluation can induce the cooperative clusters to become larger and larger as a increases. The phase space between b and a can be enlarged, and thereby the ranges for cooperators to survive in the sea of defectors can be widely extended. Our findings can greatly be conducive to interpret the emergence of cooperation within the population. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Recently, understanding the universal sustainability and persistence of cooperation among egoistic agents has become an active topic in the scientific communities, especially in biology, economics and social science, and the evolutionary game theory provides a powerful framework to clarify many problems related to the emergency of cooperation within the population [1–5]. Huge progresses have been made over the past few decades [6–9], and typical mechanisms to promote the cooperation include the kin section [10], direct and indirect reciprocity [11,12], group selection [13,14], spatial and network reciprocity [15,16]. In particular, the seminal works from Nowak and May [17] proved that the cooperators can form some clusters to prevent the invasion of defectors on the regular ⇑ Tel.: +86 3572051630. E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.chaos.2013.05.015
lattice, and the spatial reciprocity can largely promote the emergency of cooperation between individuals. Thus, the lattice configuration has become the focus of many game theory related works [18–20], and a great number of microscopic mechanisms, such as reward and punishment [21–26], different update rules [27,28], voluntary participation [29], reputation [30], fitness weighting [31,32], payoff aspiration [33–36], neighborhood size [37,38], individual mobility [39–42], age structure [43,44], improved fitness evaluation [45–53], diluted environment [54,55], etc., have been integrated into the regular lattices to explore the effective means to promote the cooperation. Meanwhile, the interaction pattern among individuals in many real-world systems often exhibits complex topological properties in which the nodes characterize the individual units and links represent the interplay between them, including the small-world and scale-free phenomenon, and statistical analysis of and dynamical behaviors on complex networks have been given much more attention
Y.-L. Wang / Chaos, Solitons & Fractals 54 (2013) 76–81
[56–60] over the past few years. Among them, extensive researches have demonstrated that the network reciprocity could become another possible origin of cooperation [61–66]. As mentioned in Ref.[62], Santos et al found that the direct links between hubs can provide sufficient conditions to resist the invasion of defectors and sustain the cooperation within the large-size population, and thereby the complex networks provide a unifying framework for us to understand the emergency of cooperation. Nevertheless, many previous works often assume that the interaction between individuals or fitness computation is symmetric. In some scenarios, the role or information an individual holds is asymmetric when two individuals interact, and can lead to some unexpected results during the game evolution. For example, in the financial or stock markets, the substantial shareholder often hold much more information than the middle or small investors, and so they take over the favorable position when they play the game with others. The resulting asymmetry in the individual role or position may remarkably influence the cooperation behaviors among agents. Thus, it is necessary to investigate the effect of asymmetric interaction between the players on the evolution of cooperation, and helps us to recognize the role of asymmetric information spread in the game dynamics. In this paper, we present a spatial prisoner’s dilemma and snowdrift game model with asymmetric fitness evaluation based on the regular graph, and find that the addition of external information source can create the asymmetric fitness calculation and promote the evolution of cooperation among players. The rest of this paper is structured as follows. In Section 2, our game model with asymmetric fitness is described in detail. Section 3 presents large scale numerical simulation results. At last, some concluding remarks are given in Section 4.
2. Model We consider an evolutionary prisoner’s dilemma game (PDG) with players located on the sites (x) of a L L square lattice with periodic boundary conditions. We can denote the strategy of a player at site x with sx, initially designated as a cooperator (sx = C) or defector (sx = D) with equal probability, and is given a fitness coefficient alpha as well. Following the notation suggested in previous literatures [17,46], we use the rescaled payoff matrix: the temptation to defect T = b (the highest payoff received by a defector if playing against a cooperator), reward for mutual cooperation R = 1, and both the punishment for mutual defection P and the sucker’s payoff S (the lowest payoff received by a cooperator if playing against a defector) equaling to 0, whereby 1 6 b 6 2 ensures a proper payoff ranking. At the same time, this rescaled matrix can only have one parameter b and capture nearly all the features of prisoner’s dilemma game. The game is iterated in accordance with the Monte Carlo (MC) simulation procedure comprising the following elementary steps. First, player x acquires its payoff Px by playing the game with its four nearest neighbors. Then, we evaluate in the same way the payoffs of all the neigh-
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bors of player x and randomly choose one neighbor y for strategy updating. Importantly, when a focal agent updates its strategy, he can only acquire the payoff information for neighbor y. Based on this fact, we can get the fitness of focal agent x as,
F x ¼ ð1 aÞ Px þ a < Px >;
ð1Þ
where denotes the average payoff for all neighbors of player x. When a=0, it returns to the traditional version; and a=1, the fitness is totally determined by the average payoff of its all neighbors. Without loss of generality, we assume that a lies between 0 and 1, we can term a as the asymmetric factor in the fitness evaluation. While for the fitness of neighbor y, since the focal agent can not get more information for his neighbor, there is no any additional factor, namely, Fy = Py. Last, player x adopts the strategy sy from the selected player y in accordance with the probability
Wðsy ! sx Þ ¼
1 ; 1 þ exp½ðF x F y Þ=K
ð2Þ
where K denotes the amplitude of noise or its inverse (1/K), the so-called intensity of selection [67]. During one full Monte Carlo step (MCS) each player has a chance once on average to adopt the strategy from a randomly selected neighbor as described above. Results of Monte Carlo simulations presented below were obtained on populations comprising 100 100 to 400 400 individuals (typically we use L = 200), whereby the fraction of cooperators qC was determined within last 104 full steps of overall 2 105 MCS. In order to overcome the influence of large K values and different initial conditions, longer simulation time was used. Moreover, the final results were averaged over 20 to 40 independent runs for each set of parameter values in order to assure suitable accuracy.
3. Simulation Results and Discussion In the evolutionary game theory, the fraction of cooperators qC during the steady state is the key quantity to measure the collective cooperation level. In Fig. 1, qC is described as a function of the temptation to defect b on the square lattice, and we can observe that qC will generally decrease as b increases, but the cooperation level will be remarkably enhanced when a is greater than 0. At the same time, the critical threshold of defection parameter b in which cooperators completely disappear is also largely augmented as a grows. In particular, the cooperators will still exist even if the defection parameter b arrives at 2.0 where the temptation to defect is exceptionally high. Additionally, we also perform the large-scale simulations on the triangle lattice, and the results, which are shown in Fig. 2, are qualitatively identical with those on the square lattice. In the simulations including the following ones, unless stated deterministically, the intensity of selection K is often set to be 0.1. All these results indicate that the players on the lattice can employ enough information about their neighbors to choose a strategy which will be beneficial to
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Y.-L. Wang / Chaos, Solitons & Fractals 54 (2013) 76–81
Fig. 1. Frequency of cooperator qC in dependence on the parameter b on square lattice. Depicted results were obtained for L = 200, K = 0.1.
the collective cooperation when a is introduced into the fitness evaluation. To further explore the promotion of cooperation in the most wide range, we give the whole phase diagram to identify the full cooperation and full defection between the asymmetric factor and the defection parameter in Fig. 3. It is obvious that the two critical thresholds, which characterize the transition from the full cooperation to coexistence of cooperators and defectors bc1, and from the coexistence state to the full defection bc2, respectively, continuously enlarge a increases on the square lattice. These results indicate that the cooperators can utilize the payoff information of neighbors to resist the invasion of defectors, and sustain the cooperation between the individuals. In addition, the difference between bc1 and bc2 will also amplify as a increases from 0 to 1, which means that the parameter region for the coexistence between cooperators and defectors will become wider and wider. Altogether, the cooperators will hold the more advantageous position when the individuals can refer to other player’s information to make the decision, and thus lead to the higher cooperation level as compared to the traditional prisoner’s dilemma game. Fig. 4 depicts the characteristic snapshots about the distribution of cooperators and defectors on the square lattice
Fig. 2. Frequency of cooperator qC in dependence on the parameter b on triangle lattice. Depicted results were obtained for L = 200, K = 0.1.
Fig. 3. Full b a phase diagram on square lattice. Obviously, with the enhancement of a, C and C+D phases quickly expand. Depicted results were obtained for L = 200.
during the steady state. From the subfigure (A) to (F), the asymmetric factor a is changed from 0, 0.2, 0.4, 0.6, 0.8 to 1.0. When a = 0, the scenario stands for the traditional PDG in which the spatial reciprocity is the only mechanism to sustain the cooperation, and the cooperators can’t form the clusters to defend themselves not to be employed by the defectors. But for a > 0, we can observe that the cooperators can obtain the beneficial conditions to create the cooperative clusters, and size of clusters of cooperators will become larger and larger as a is varied. That is to say, only the spatial structure is not enough to support such high cooperation level, and the asymmetric evaluation of neighbor’s fitness may become the origin of promotion of cooperation in our new game model. Furthermore, for a given defection parameter b = 1.1, we discuss the dynamical evolution of the fraction of cooperators qC(t) as a function of Monte Carlo simulation steps in Fig. 5. Likely, the symmetric payoff calculation can’t provide the sufficient conditions to sustain the collective cooperation in the traditional model (i.e., a = 0), and qC(t) monotonically and continuously decreases until all cooperators are extinct in which qC(t) = 0. While for a > 0, qC(t) usually tends to descend at the initial time step, but quickly ascend after several time steps and then maintains or fluctuates around a specific value which is larger than 0. To be interesting, all players arrive at the consensus and take the cooperation strategy when the fitness of a random neighbor is totally determined by the average payoff of its nearest neighbors (i.e., a = 1). The more the asymmetric factor a, the higher the fraction of cooperators qC at the stationary state. That is to say, the novel asymmetric fitness evaluation is mainly responsible for the enhancement of cooperation. Although the spatial prisoner’s dilemma game has drawn a great deal of attention in the scientific communities, the PDG has also encountered some drawbacks in exploring the emergence of cooperation since it is often difficult to access the proper payoffs. Thus, snowdrift game (SDG), which is also named as hawk-dove game (HDG), has become another good candidate in the studies of cooperative behaviors and social dilemmas. The SDG is similar to PDG except for the order of payoff, and the ranking in the
Y.-L. Wang / Chaos, Solitons & Fractals 54 (2013) 76–81
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Fig. 4. Characteristic snapshots of cooperators (red) and defectors (green) for different values of the parameter a. From A to F, the values of a are 0, 0.2, 0.4, 0.6, 0.8 and 1.0, respectively. C is denoted by red, D is denoted by green. Depicted results were obtained for L = 200,b = 1.11 and K = 0.1.
Fig. 5. Time courses depicting the evolution of cooperation for different values of a. Depicted results were obtained for L = 200,b = 1.11 and K = 0.1, from bottom to top, a is set to be 0, 0.2, 0.4, 0.6, 0.8, and 1.0, respectively.
SDG is T > R > S > P while the ranking is T > R > P > S in the PDG. Without loss of generality, as described in Ref.[18], the normalized payoff configuration is listed as follows: T = 1 + r, R = 1,S = 1 r and P = 0 in which r denotes the ratio of benefit to cost, and r generally lies between 0 and 1. Does the asymmetric fitness evaluation promote the cooperation in the SDG model? We answer this question in Fig. 6. To be exciting, the asymmetric fitness computation will also enhance the collective cooperation level on the square lattice, when compared to the traditional SDG model which is equivalent to a = 0. Likely, the larger the asymmetric factor a, the higher the fraction of cooperators qC, and the novel asymmetric fitness evaluation has again become the reason of the enhancement of cooperation on the spatial lattices. Since the information acquirement is ubiquitous in the real-world systems, our model can to a greater extent explain the emergence of cooperation in many biological, economical and social systems.
Fig. 6. Frequency of cooperator qC in dependence on the parameter r for snowdrift game on square lattice. Depicted results were obtained for L = 200, K = 0.1, from bottom to top, a is set to be 0, 0.2, 0.4, 0.6, 0.8 , and 1.0, respectively.
4. Conclusion In summary, we present a novel asymmetric fitness evaluation method to be applied into the spatial PDG and SDG models on complex networks. In this model, focal individual x can obtain its fitness combining its own payoff and the performance of his neighbors, namely, player x integrates its own payoff and the average payoff of its all nearest neighbors to obtain its fitness: Fx = (1 a) Px + a < Px > , where a is termed as the asymmetric factor. While for the potential opponent, due to the limitation of evaluation information, the evaluation of his fitness still uses the traditional way: fitness equals to his payoff. Large scale simulations are performed on the square and triangle lattices, and the results indicate that the introduction of asymmetric fitness evaluation can greatly promote the fraction of cooperators in the whole population at stationary state. The larger the asymmetric factor a, the higher the
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fraction of cooperators qC, and this phenomenon will be seen in both PDG and SDG models. Meanwhile, by the characteristic snapshots at the steady state, we can observe that larger a will be helpful for cooperators to form giant clusters to defend themselves and resist the invasion of defectors. Thus, we can conclude that our novel asymmetric fitness evaluation creates the persistence and emergence of cooperation among players on the spatial lattices, and current findings are very conducive to help us to deeply understand why the cooperation is a wide-spread behaviors in many real-world systems ranging from biological to economical and social sciences.
Acknowledgements This project is supported by the National Natural Science Foundation of China (Grant No. 11005047), Natural Science Foundation of Shanxi Province(No. 20110110312) and the High-Tech Industrialization Project of Shanxi Province (No. 20110014).
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Contents lists available at SciVerse ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Asymmetric evaluation of fitness enhances spatial reciprocity in social dilemmas Yi-Ling Wang ⇑ School of Life Science, Shanxi Normal University, LinFen, Shanxi 041000, China
a r t i c l e
i n f o
Article history: Received 1 February 2013 Accepted 17 May 2013 Available online 20 June 2013
a b s t r a c t Evolutionary game theory has been a powerful tool to understand the ubiquity of cooperation in many real-world systems which ranges from biological to economical and social sciences. In this paper, we propose a novel game model, which considers different fitness evaluation means for focal agent and his randomly chosen neighbor, to characterize the asymmetry of information during the game playing. When playing the game, the focal agent obtains its own fitness which includes its own payoff and the average payoff of his neighbors. However, due to the limited information for the chosen neighbor, his fitness can not be evaluated as the focal agent but just the traditional way, namely his fitness equals to his payoff. Large-scale numerical simulations indicate that this kind of asymmetric fitness computation can have substantial effects on the cooperative behaviors on the regular lattice. Interestingly, the larger the asymmetric factor a, the higher the cooperation level qC. Meanwhile, introducing the asymmetric fitness evaluation can induce the cooperative clusters to become larger and larger as a increases. The phase space between b and a can be enlarged, and thereby the ranges for cooperators to survive in the sea of defectors can be widely extended. Our findings can greatly be conducive to interpret the emergence of cooperation within the population. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Recently, understanding the universal sustainability and persistence of cooperation among egoistic agents has become an active topic in the scientific communities, especially in biology, economics and social science, and the evolutionary game theory provides a powerful framework to clarify many problems related to the emergency of cooperation within the population [1–5]. Huge progresses have been made over the past few decades [6–9], and typical mechanisms to promote the cooperation include the kin section [10], direct and indirect reciprocity [11,12], group selection [13,14], spatial and network reciprocity [15,16]. In particular, the seminal works from Nowak and May [17] proved that the cooperators can form some clusters to prevent the invasion of defectors on the regular ⇑ Tel.: +86 3572051630. E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.chaos.2013.05.015
lattice, and the spatial reciprocity can largely promote the emergency of cooperation between individuals. Thus, the lattice configuration has become the focus of many game theory related works [18–20], and a great number of microscopic mechanisms, such as reward and punishment [21–26], different update rules [27,28], voluntary participation [29], reputation [30], fitness weighting [31,32], payoff aspiration [33–36], neighborhood size [37,38], individual mobility [39–42], age structure [43,44], improved fitness evaluation [45–53], diluted environment [54,55], etc., have been integrated into the regular lattices to explore the effective means to promote the cooperation. Meanwhile, the interaction pattern among individuals in many real-world systems often exhibits complex topological properties in which the nodes characterize the individual units and links represent the interplay between them, including the small-world and scale-free phenomenon, and statistical analysis of and dynamical behaviors on complex networks have been given much more attention
Y.-L. Wang / Chaos, Solitons & Fractals 54 (2013) 76–81
[56–60] over the past few years. Among them, extensive researches have demonstrated that the network reciprocity could become another possible origin of cooperation [61–66]. As mentioned in Ref.[62], Santos et al found that the direct links between hubs can provide sufficient conditions to resist the invasion of defectors and sustain the cooperation within the large-size population, and thereby the complex networks provide a unifying framework for us to understand the emergency of cooperation. Nevertheless, many previous works often assume that the interaction between individuals or fitness computation is symmetric. In some scenarios, the role or information an individual holds is asymmetric when two individuals interact, and can lead to some unexpected results during the game evolution. For example, in the financial or stock markets, the substantial shareholder often hold much more information than the middle or small investors, and so they take over the favorable position when they play the game with others. The resulting asymmetry in the individual role or position may remarkably influence the cooperation behaviors among agents. Thus, it is necessary to investigate the effect of asymmetric interaction between the players on the evolution of cooperation, and helps us to recognize the role of asymmetric information spread in the game dynamics. In this paper, we present a spatial prisoner’s dilemma and snowdrift game model with asymmetric fitness evaluation based on the regular graph, and find that the addition of external information source can create the asymmetric fitness calculation and promote the evolution of cooperation among players. The rest of this paper is structured as follows. In Section 2, our game model with asymmetric fitness is described in detail. Section 3 presents large scale numerical simulation results. At last, some concluding remarks are given in Section 4.
2. Model We consider an evolutionary prisoner’s dilemma game (PDG) with players located on the sites (x) of a L L square lattice with periodic boundary conditions. We can denote the strategy of a player at site x with sx, initially designated as a cooperator (sx = C) or defector (sx = D) with equal probability, and is given a fitness coefficient alpha as well. Following the notation suggested in previous literatures [17,46], we use the rescaled payoff matrix: the temptation to defect T = b (the highest payoff received by a defector if playing against a cooperator), reward for mutual cooperation R = 1, and both the punishment for mutual defection P and the sucker’s payoff S (the lowest payoff received by a cooperator if playing against a defector) equaling to 0, whereby 1 6 b 6 2 ensures a proper payoff ranking. At the same time, this rescaled matrix can only have one parameter b and capture nearly all the features of prisoner’s dilemma game. The game is iterated in accordance with the Monte Carlo (MC) simulation procedure comprising the following elementary steps. First, player x acquires its payoff Px by playing the game with its four nearest neighbors. Then, we evaluate in the same way the payoffs of all the neigh-
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bors of player x and randomly choose one neighbor y for strategy updating. Importantly, when a focal agent updates its strategy, he can only acquire the payoff information for neighbor y. Based on this fact, we can get the fitness of focal agent x as,
F x ¼ ð1 aÞ Px þ a < Px >;
ð1Þ
where
Wðsy ! sx Þ ¼
1 ; 1 þ exp½ðF x F y Þ=K
ð2Þ
where K denotes the amplitude of noise or its inverse (1/K), the so-called intensity of selection [67]. During one full Monte Carlo step (MCS) each player has a chance once on average to adopt the strategy from a randomly selected neighbor as described above. Results of Monte Carlo simulations presented below were obtained on populations comprising 100 100 to 400 400 individuals (typically we use L = 200), whereby the fraction of cooperators qC was determined within last 104 full steps of overall 2 105 MCS. In order to overcome the influence of large K values and different initial conditions, longer simulation time was used. Moreover, the final results were averaged over 20 to 40 independent runs for each set of parameter values in order to assure suitable accuracy.
3. Simulation Results and Discussion In the evolutionary game theory, the fraction of cooperators qC during the steady state is the key quantity to measure the collective cooperation level. In Fig. 1, qC is described as a function of the temptation to defect b on the square lattice, and we can observe that qC will generally decrease as b increases, but the cooperation level will be remarkably enhanced when a is greater than 0. At the same time, the critical threshold of defection parameter b in which cooperators completely disappear is also largely augmented as a grows. In particular, the cooperators will still exist even if the defection parameter b arrives at 2.0 where the temptation to defect is exceptionally high. Additionally, we also perform the large-scale simulations on the triangle lattice, and the results, which are shown in Fig. 2, are qualitatively identical with those on the square lattice. In the simulations including the following ones, unless stated deterministically, the intensity of selection K is often set to be 0.1. All these results indicate that the players on the lattice can employ enough information about their neighbors to choose a strategy which will be beneficial to
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Fig. 1. Frequency of cooperator qC in dependence on the parameter b on square lattice. Depicted results were obtained for L = 200, K = 0.1.
the collective cooperation when a is introduced into the fitness evaluation. To further explore the promotion of cooperation in the most wide range, we give the whole phase diagram to identify the full cooperation and full defection between the asymmetric factor and the defection parameter in Fig. 3. It is obvious that the two critical thresholds, which characterize the transition from the full cooperation to coexistence of cooperators and defectors bc1, and from the coexistence state to the full defection bc2, respectively, continuously enlarge a increases on the square lattice. These results indicate that the cooperators can utilize the payoff information of neighbors to resist the invasion of defectors, and sustain the cooperation between the individuals. In addition, the difference between bc1 and bc2 will also amplify as a increases from 0 to 1, which means that the parameter region for the coexistence between cooperators and defectors will become wider and wider. Altogether, the cooperators will hold the more advantageous position when the individuals can refer to other player’s information to make the decision, and thus lead to the higher cooperation level as compared to the traditional prisoner’s dilemma game. Fig. 4 depicts the characteristic snapshots about the distribution of cooperators and defectors on the square lattice
Fig. 2. Frequency of cooperator qC in dependence on the parameter b on triangle lattice. Depicted results were obtained for L = 200, K = 0.1.
Fig. 3. Full b a phase diagram on square lattice. Obviously, with the enhancement of a, C and C+D phases quickly expand. Depicted results were obtained for L = 200.
during the steady state. From the subfigure (A) to (F), the asymmetric factor a is changed from 0, 0.2, 0.4, 0.6, 0.8 to 1.0. When a = 0, the scenario stands for the traditional PDG in which the spatial reciprocity is the only mechanism to sustain the cooperation, and the cooperators can’t form the clusters to defend themselves not to be employed by the defectors. But for a > 0, we can observe that the cooperators can obtain the beneficial conditions to create the cooperative clusters, and size of clusters of cooperators will become larger and larger as a is varied. That is to say, only the spatial structure is not enough to support such high cooperation level, and the asymmetric evaluation of neighbor’s fitness may become the origin of promotion of cooperation in our new game model. Furthermore, for a given defection parameter b = 1.1, we discuss the dynamical evolution of the fraction of cooperators qC(t) as a function of Monte Carlo simulation steps in Fig. 5. Likely, the symmetric payoff calculation can’t provide the sufficient conditions to sustain the collective cooperation in the traditional model (i.e., a = 0), and qC(t) monotonically and continuously decreases until all cooperators are extinct in which qC(t) = 0. While for a > 0, qC(t) usually tends to descend at the initial time step, but quickly ascend after several time steps and then maintains or fluctuates around a specific value which is larger than 0. To be interesting, all players arrive at the consensus and take the cooperation strategy when the fitness of a random neighbor is totally determined by the average payoff of its nearest neighbors (i.e., a = 1). The more the asymmetric factor a, the higher the fraction of cooperators qC at the stationary state. That is to say, the novel asymmetric fitness evaluation is mainly responsible for the enhancement of cooperation. Although the spatial prisoner’s dilemma game has drawn a great deal of attention in the scientific communities, the PDG has also encountered some drawbacks in exploring the emergence of cooperation since it is often difficult to access the proper payoffs. Thus, snowdrift game (SDG), which is also named as hawk-dove game (HDG), has become another good candidate in the studies of cooperative behaviors and social dilemmas. The SDG is similar to PDG except for the order of payoff, and the ranking in the
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Fig. 4. Characteristic snapshots of cooperators (red) and defectors (green) for different values of the parameter a. From A to F, the values of a are 0, 0.2, 0.4, 0.6, 0.8 and 1.0, respectively. C is denoted by red, D is denoted by green. Depicted results were obtained for L = 200,b = 1.11 and K = 0.1.
Fig. 5. Time courses depicting the evolution of cooperation for different values of a. Depicted results were obtained for L = 200,b = 1.11 and K = 0.1, from bottom to top, a is set to be 0, 0.2, 0.4, 0.6, 0.8, and 1.0, respectively.
SDG is T > R > S > P while the ranking is T > R > P > S in the PDG. Without loss of generality, as described in Ref.[18], the normalized payoff configuration is listed as follows: T = 1 + r, R = 1,S = 1 r and P = 0 in which r denotes the ratio of benefit to cost, and r generally lies between 0 and 1. Does the asymmetric fitness evaluation promote the cooperation in the SDG model? We answer this question in Fig. 6. To be exciting, the asymmetric fitness computation will also enhance the collective cooperation level on the square lattice, when compared to the traditional SDG model which is equivalent to a = 0. Likely, the larger the asymmetric factor a, the higher the fraction of cooperators qC, and the novel asymmetric fitness evaluation has again become the reason of the enhancement of cooperation on the spatial lattices. Since the information acquirement is ubiquitous in the real-world systems, our model can to a greater extent explain the emergence of cooperation in many biological, economical and social systems.
Fig. 6. Frequency of cooperator qC in dependence on the parameter r for snowdrift game on square lattice. Depicted results were obtained for L = 200, K = 0.1, from bottom to top, a is set to be 0, 0.2, 0.4, 0.6, 0.8 , and 1.0, respectively.
4. Conclusion In summary, we present a novel asymmetric fitness evaluation method to be applied into the spatial PDG and SDG models on complex networks. In this model, focal individual x can obtain its fitness combining its own payoff and the performance of his neighbors, namely, player x integrates its own payoff and the average payoff of its all nearest neighbors to obtain its fitness: Fx = (1 a) Px + a < Px > , where a is termed as the asymmetric factor. While for the potential opponent, due to the limitation of evaluation information, the evaluation of his fitness still uses the traditional way: fitness equals to his payoff. Large scale simulations are performed on the square and triangle lattices, and the results indicate that the introduction of asymmetric fitness evaluation can greatly promote the fraction of cooperators in the whole population at stationary state. The larger the asymmetric factor a, the higher the
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fraction of cooperators qC, and this phenomenon will be seen in both PDG and SDG models. Meanwhile, by the characteristic snapshots at the steady state, we can observe that larger a will be helpful for cooperators to form giant clusters to defend themselves and resist the invasion of defectors. Thus, we can conclude that our novel asymmetric fitness evaluation creates the persistence and emergence of cooperation among players on the spatial lattices, and current findings are very conducive to help us to deeply understand why the cooperation is a wide-spread behaviors in many real-world systems ranging from biological to economical and social sciences.
Acknowledgements This project is supported by the National Natural Science Foundation of China (Grant No. 11005047), Natural Science Foundation of Shanxi Province(No. 20110110312) and the High-Tech Industrialization Project of Shanxi Province (No. 20110014).
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