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Heritability promotes cooperation in spatial public goods games
Physica A 389 (2010) 5719–5724
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Heritability promotes cooperation in spatial public goods games Run-Ran Liu a,∗ , Chun-Xiao Jia a , Bing-Hong Wang a,b a
Department of Modern Physics and Nonlinear Science Center, University of Science and Technology of China, Hefei 230026, PR China
b
Research Center of Complex Systems Science, University of Shanghai for Science and Technology, Shanghai 200093, PR China
article
info
Article history: Received 23 April 2010 Received in revised form 4 July 2010 Available online 17 September 2010 Keywords: Heritability Fitness Cooperation Public goods game
abstract Heritability is ubiquitous within most real biological or social systems. A heritable trait is most simply an offspring’s trait that resembles the parent’s corresponding trait, which can be fitness, strategy, or the way of strategy adoption for evolutionary games. Here we study the effects of heritability on the evolution of spatial public goods games. In our model, the fitness of players is determined by the payoffs from the current interactions and their history. Based on extensive simulations, we find that the density of cooperators is enhanced by increasing the heritability of players over a wide range of the multiplication factor. We attribute the enhancement of cooperation to the inherited fitness that stabilizes the fitness of players, and thus prevents the expansion of defectors effectively. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Recent years have witnessed a great development of evolutionary game theory in studying the ubiquity of cooperation in nature, from human society to the animal world. Understanding the maintenance of cooperative behavior among selfish individuals is an interesting and challenging problem [1–3]. The Prisoner’s Dilemma Game (PDG) as a general metaphor for studying cooperative behaviors, has received much attention in theoretical and experimental studies [4,5]. The PDG seizes the characteristics of conflict between selfish individuals and the collective interests of two involved players, which has become the leading paradigm to explain cooperative behaviors. The public goods game (PGG) is proposed as a multi-person game, which can be regarded as a PDG with more than two participants. In the typical PGG, N players independently and simultaneously decide whether to invest into a public pool (cooperate) or not (defect). The collected investment is multiplied by a multiplication factor r with r > 1, and then equally redistributed among all players irrespective of their actual strategies. Obviously, the whole group will get the maximum interests when everyone contributes all of their possessions into the public pool. However, defection is the better choice no matter what the other’s selection when r < N. For the evolutionary PGG in a well-mixed population, the defectors will dominate the whole system when r < N [6]. Since defecting is the dominating strategy for PDG and PGG, cooperative behavior will disappear, which is opposite to observations in the real world. Therefore, a variety of mechanisms have been proposed to understand the maintenance of cooperation [7], one of which was done by Nowak and May, who showed that the PDG on a simple spatial structure can induce the emergence of cooperation [8]. Inspired by the idea of a spatial game, much attention has been given to evolutionary games on several population structures, including regular lattices [9–22], small-world networks [23–28] and scale-free networks [29–37]. More interestingly, some typical features of human sociality have been incorporated into evolutionary games, which have also witnessed the flourishing of cooperation in structured populations. Such investigations involve memory effects [38,39], aspiration effects [40,41], social tolerance [42,43], social diversity [44], adaptive networks with alternative interactions [45–49], different teaching capabilities [50–55], noise in the strategy adoption [56] and payoff noise [57].
∗
Corresponding author. E-mail addresses: [email protected] (R.-R. Liu), [email protected] (C.-X. Jia), [email protected] (B.-H. Wang).
0378-4371/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2010.08.043
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In order to explain the emergence of cooperation in the real world by the PGG model, some works have introduced various mechanisms that facilitate cooperation. Hauert et al. introduced voluntary participation in PGG and found that it results in a substantial and persistent willingness to cooperate [58]. Szabó et al. further studied the voluntary participation mechanism on a square lattice and then found a cyclic dominance of the strategies and transitions between one-, two-, and three-strategy states [11]. Guan et al. found that the diversity of teaching activity can promote the cooperation level [59]. Santos et al. introduced social diversity by means of heterogeneous graphs and found that cooperation is also remarkably promoted [60]. Yang et al. investigated the effect of individual diversity and found that there exists an optimal value of the diversity parameter that induces the highest level of cooperation [61]. Rong et al. comprehensively studied the effect of degree correlation on a networked PGG [62]. Cao et al. investigated the effect of heterogeneous investment on a networked PGG [63]. Szolnoki et al. studied the impact of critical mass on the evolution of cooperation in spatial PGG, and found that the introduction of critical mass can promote cooperation effectively [64]. In this paper, we focus on another intrinsic feature of individuals, heritability, which is ubiquitous in most real biological or social systems [65]. A filial generation can inherit property, physical quality and other resources from their parents. In [66], Wu et al. introduced heritability into the evolutionary spacial PDG, where the players can inherit fitness from the last generation. The conception of heritability is similar with the memory effect in Ref. [39]. In that work, Qin et al. considered an evolutionary spatial PDG with the memory effect on a square lattice, in which the payoffs that were obtained by C strategy and the payoffs that were obtained by D strategy are accumulated separately and players update their strategies by considering the accumulated payoffs. They found that historical payoff plays a key role in the maintenance of cooperation. Fort studied models where the elements of the payoff matrix were inherited in parallel with the strategy adoption [67]. In our study, we wish to elaborate on the prominent role of heritability and its ability to promote cooperation in the PGG. For this purpose, we consider an evolutionary spatial PGG on a square lattice, where each player can inherit fitness from the last generation at each time step. Meanwhile, their fitness should be correlated with their payoffs from the current interactions. We observe that the introduction of inherited fitness can improve the cooperation level of a system significantly. From simulations, we also find that the inherited fitness plays a pivotal role in the fitness stability, which can prevent the expansion of defectors effectively. 2. Model We consider an evolutionary two-strategy public goods game with players located on a four-neighbor square lattice with periodic boundary conditions, where each player and its nearest neighbors form a group with size G = 5. Initially, either a cooperator or a defector, randomly chosen with equal probability, occupies each site. At each time step, cooperators contribute to the group whereas defectors do not. The total payoff of a certain player i depends on its strategy si and the number of cooperators nc in the neighborhood. Thus, the player i’s payoff is r (nc + si )
− si , G where r represents the multiplication factor. If the player i is a cooperator at this step, si equals 1, otherwise si is zero. At the t-th generation, the fitness fi (t ) of the player i is defined as: Pi =
fi (t ) = τ fi (t − 1) + (1 − τ )Pi (t ),
(1)
(2)
where the parameter τ denotes the heritability of players, which sets the balance between the present and past payoff gains−the relative importance of a previous generations or strength of maternal effects decays with a factor τ per time step [66]. According to Eq. (2), we can write the fitness fi (t ) of the player i in another form: fi (t ) = τ t fi (0) +
t −
τ t −n Pi (n)(1 − τ ).
(3)
n =1
If τ = 1, the current payoff is neglected, the fitness of a player will never change in the evolution. Hence, we have τ in the range of [0,1). In our model, each player’s initial fitness fi (0) is set to 1. We have check that the precise value of fi (0) does not affect the final cooperator density of a system. As the contributions of f (0) to all the players are always same at each time step, the impact of initial fitness is eliminated when the fitness-difference update rule is adopted. When updating strategies, all players imitate the strategy of a randomly chosen neighbor with a probability depending on the fitness difference synchronously, i.e., the probability for player i to adopt a randomly chosen neighbor j’s strategy is given by: Wi =
1 1 + exp[( fi − fj )/κ]
,
(4)
where κ characterizes the noise introduced to permit irrational choices [9] (κ → ∞ leads to random imitation, whereas κ → 0 leads to the deterministic imitation of the better player). Here we have κ = 0.1 for all simulations. Simulations are carried out for a population of N = 100 × 100 individuals. We study the key quantity of cooperator density ρc in the steady state. Equilibrium frequencies of cooperators are obtained by averaging over 2000 Monte Carlo time steps from a total of 22 000 steps, and each data point results from an average of over 10 realizations.
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Fig. 1. (Color online) Stationary fraction of cooperators ρc in dependence on the multiplication factor r for different values of τ .
Fig. 2. (Color online) Stationary fraction of cooperators ρc in dependence on the heritability factor τ for different values of r.
3. Simulation and analysis Fig. 1 shows how ρc varies in dependence on the multiplication factor r for different values of τ . It displays that ρc increases monotonically with the increasing of r, regardless what τ is. Moreover, for a fixed r, different values of τ can affect the final cooperation levels dramatically. To quantify the effects of varying τ on cooperation, we plot ρc as a function of the parameter τ for different values of r, as shown in Fig. 2. From Fig. 2, we can find that the cooperator density ρc increases with the increase of τ for a fixed value of r. Fig. 3 shows the typical distributions of cooperators and defectors on a square lattice at different time steps t for τ = 0 and r = 4.5. During the first ten steps, a large number of cooperators are eliminated, and some small cooperative clusters may survive at this time. After that, many small cooperative clusters are eliminated and some large cooperative clusters form. At the end, all the cooperative clusters disappear, and no cooperator survives. Based on this process, we find that the cooperators have a tendency to form clusters in the evolution although they are eliminated finally for τ = 0 and r = 4.5. In order to understand the process more clearly, we plot the mean fitness of cooperators and defectors on the boundary of cooperative clusters (⟨ fcb ⟩ and ⟨ fdb ⟩) in the evolution (In our study, a player belongs to boundary if it has at least one neighbor who holds an opposite strategy). From Fig. 4(a), we can find that the fitness of cooperators is always larger than that of defectors, which indicates that the cooperators are always eliminated although they earn higher payoffs than those of defectors for τ = 0 and r = 4.5. At the interface of a cooperator cluster, the transitions between C players and D players occur continuously. Once a cooperator changes into a defector, its payoff will increase greatly by exploiting its cooperative neighbors, which may further result in some of its cooperative neighbors changing into defectors. Hence, the introduction of historical payoff can restrict the sudden increase in defector’s fitness, and prevent the continuous change of cooperators into defectors. At the same time, the introduction of historical payoff can also maintain the fitness stability of cooperators so that the cooperators can keep their advantages in fitness and prevent the invasion of defectors. Fig. 5(a) shows the mean fitness of cooperators and defectors (⟨ fc ⟩ and ⟨ fd ⟩) in the equilibrium state. We can find that, in the equilibrium state, ⟨ fc ⟩ is no less than ⟨ fd ⟩ for most values of τ . For τ < 0.2, there is no cooperator, both ⟨ fc ⟩ and ⟨ fd ⟩ equal zero. For τ > 0.5, there is no defector, ⟨ fc ⟩ reaches its maximum, and ⟨ fd ⟩ equals zero. Fig. 5(b) shows the average payoffs of cooperators ⟨Pc ⟩ and defectors ⟨Pd ⟩ as a function of τ . Comparing Fig. 5(a) and (b), we can find that the fitness of
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Fig. 3. A series of snapshots of distribution of cooperators (white) and detectors (black) at different time steps: (a) t = 1, (b) t = 10, (c) t = 100, and (d) t = 1000. The parameter τ is set as 0 and r is set as 4.5.
(a)
(b)
Fig. 4. (Color online) The mean fitness of cooperators and defectors on the boundary (⟨ fcb ⟩ and ⟨fdb ⟩) in the evolution for τ = 0 (a) and τ = 0.1 (b) with r = 4.5.
a
b
Fig. 5. (Color online) (a) The mean fitness of cooperators and defectors (⟨ fc ⟩ and ⟨ fd ⟩) in the equilibrium state, as a function of τ ; (b) the mean payoff of cooperators and defectors (⟨Pc ⟩ and ⟨Pd ⟩) in the equilibrium state, as a function of τ . The parameter r is set as 4.
both C players and D players almost equal the payoffs of them. In the steady state, following Eq. (2) we can get
⟨ fc (t )⟩ = τ ⟨ fc (t − 1)⟩ + (1 − τ )⟨Pc (t )⟩
(5)
⟨ fd (t )⟩ = τ ⟨ fd (t − 1)⟩ + (1 − τ )⟨Pd (t )⟩.
(6)
and According to Fig. 4(b), we find that the fitness of cooperators and defectors can reach a steady state, and we can have fc (t ) = fc (t − 1) and fd (t ) = fd (t − 1). Submitting them into Eqs. (4) and (5) respectively, we can get fc (t ) = Pc (t ) and
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fd (t ) = Pd (t ) in the steady state. These results further indicate that cooperators earn higher payoffs than those of defectors by forming clusters, and the heritability can maintain the advantage of cooperators all the while. 4. Conclusion In Ref. [66], Wu et al. introduced heritability into the evolutionary prisoner’s dilemma game, and they focused on the effects of reproduction time scale and its diversity on the evolutionary cooperation. Interestingly, they found that there exists an intermediate selection time scale that maximizes cooperation. Another factor they find to promote cooperation is a diversity of reproduction time scales. In contrast with that, we paid our attention to the effects of heritability on the evolutionary public goods games and found that the introduction of inherited fitness can improve the cooperation level of a system significantly. In another related work [39], Qin et al. explored the effects of memory in a spatial Prisoner’s Dilemma game. Their memory mechanism is similar to the maternal-effect fitness inheritance in our model. The players update their strategies according to the cumulative historical payoff in both models. However, there exist differences in the way of cumulation. In their model, the payoffs obtained by the C strategy and the payoffs obtained by the D strategy are accumulated separately. While in our model, the payoffs are accumulated regardless what the strategy is. In conclusion, we have explored the effects of heritability on spacial public goods games. In our model, the fitness of each player is based on the instantaneous payoff from the interactions and the inherited fitness from the past, where the heritability of players is controlled by a tunable parameters τ . Interestingly, we have shown that if heritability is considered, then cooperation can emerge, maintain, and even thrive. In order to give an intuitive account of the maintenance of cooperation, we provide some typical snapshots of the system at different time steps and compare the mean fitness of defectors and cooperators in the evolution. It is shown that the introduction of historical payoff can maintain the fitness stability of players and prevent the expansion of defectors. In order to check the robustness of this mechanism, we have introduced the historical payoff into another PGG model proposed by Santos et al. [60], where each individual can take part in games centered on it and its neighbors, and obtains payoffs from each game that it has taken part in. All players update their strategies simultaneously by fitness differences. We find that heritability can facilitate cooperation all the same. We have also applied our model by an asynchronous strategy update rule, where the players update their strategies one by one in a random sequence at each time step. We find that the cooperation can be also improved when inheritable fitness is considered. Finally we have checked that our conclusions are robust with respect to using different strategy updating rules, such as ‘‘imitate the neighbor with the highest fitness’’[8] and the finite population analog of the replicator dynamics [29]. However, there exist limitations in our model. For instance, the ‘‘tit for tat’’ strategy has not been introduced to this model. Acknowledgements This work is funded by the National Basic Research Program of China (973 Program No. 2006CB705500), the National Natural Science Foundation of China (Grant Nos. 10975126, 10635040), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20093402110032). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Heritability promotes cooperation in spatial public goods games Run-Ran Liu a,∗ , Chun-Xiao Jia a , Bing-Hong Wang a,b a
Department of Modern Physics and Nonlinear Science Center, University of Science and Technology of China, Hefei 230026, PR China
b
Research Center of Complex Systems Science, University of Shanghai for Science and Technology, Shanghai 200093, PR China
article
info
Article history: Received 23 April 2010 Received in revised form 4 July 2010 Available online 17 September 2010 Keywords: Heritability Fitness Cooperation Public goods game
abstract Heritability is ubiquitous within most real biological or social systems. A heritable trait is most simply an offspring’s trait that resembles the parent’s corresponding trait, which can be fitness, strategy, or the way of strategy adoption for evolutionary games. Here we study the effects of heritability on the evolution of spatial public goods games. In our model, the fitness of players is determined by the payoffs from the current interactions and their history. Based on extensive simulations, we find that the density of cooperators is enhanced by increasing the heritability of players over a wide range of the multiplication factor. We attribute the enhancement of cooperation to the inherited fitness that stabilizes the fitness of players, and thus prevents the expansion of defectors effectively. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Recent years have witnessed a great development of evolutionary game theory in studying the ubiquity of cooperation in nature, from human society to the animal world. Understanding the maintenance of cooperative behavior among selfish individuals is an interesting and challenging problem [1–3]. The Prisoner’s Dilemma Game (PDG) as a general metaphor for studying cooperative behaviors, has received much attention in theoretical and experimental studies [4,5]. The PDG seizes the characteristics of conflict between selfish individuals and the collective interests of two involved players, which has become the leading paradigm to explain cooperative behaviors. The public goods game (PGG) is proposed as a multi-person game, which can be regarded as a PDG with more than two participants. In the typical PGG, N players independently and simultaneously decide whether to invest into a public pool (cooperate) or not (defect). The collected investment is multiplied by a multiplication factor r with r > 1, and then equally redistributed among all players irrespective of their actual strategies. Obviously, the whole group will get the maximum interests when everyone contributes all of their possessions into the public pool. However, defection is the better choice no matter what the other’s selection when r < N. For the evolutionary PGG in a well-mixed population, the defectors will dominate the whole system when r < N [6]. Since defecting is the dominating strategy for PDG and PGG, cooperative behavior will disappear, which is opposite to observations in the real world. Therefore, a variety of mechanisms have been proposed to understand the maintenance of cooperation [7], one of which was done by Nowak and May, who showed that the PDG on a simple spatial structure can induce the emergence of cooperation [8]. Inspired by the idea of a spatial game, much attention has been given to evolutionary games on several population structures, including regular lattices [9–22], small-world networks [23–28] and scale-free networks [29–37]. More interestingly, some typical features of human sociality have been incorporated into evolutionary games, which have also witnessed the flourishing of cooperation in structured populations. Such investigations involve memory effects [38,39], aspiration effects [40,41], social tolerance [42,43], social diversity [44], adaptive networks with alternative interactions [45–49], different teaching capabilities [50–55], noise in the strategy adoption [56] and payoff noise [57].
∗
Corresponding author. E-mail addresses: [email protected] (R.-R. Liu), [email protected] (C.-X. Jia), [email protected] (B.-H. Wang).
0378-4371/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2010.08.043
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R.-R. Liu et al. / Physica A 389 (2010) 5719–5724
In order to explain the emergence of cooperation in the real world by the PGG model, some works have introduced various mechanisms that facilitate cooperation. Hauert et al. introduced voluntary participation in PGG and found that it results in a substantial and persistent willingness to cooperate [58]. Szabó et al. further studied the voluntary participation mechanism on a square lattice and then found a cyclic dominance of the strategies and transitions between one-, two-, and three-strategy states [11]. Guan et al. found that the diversity of teaching activity can promote the cooperation level [59]. Santos et al. introduced social diversity by means of heterogeneous graphs and found that cooperation is also remarkably promoted [60]. Yang et al. investigated the effect of individual diversity and found that there exists an optimal value of the diversity parameter that induces the highest level of cooperation [61]. Rong et al. comprehensively studied the effect of degree correlation on a networked PGG [62]. Cao et al. investigated the effect of heterogeneous investment on a networked PGG [63]. Szolnoki et al. studied the impact of critical mass on the evolution of cooperation in spatial PGG, and found that the introduction of critical mass can promote cooperation effectively [64]. In this paper, we focus on another intrinsic feature of individuals, heritability, which is ubiquitous in most real biological or social systems [65]. A filial generation can inherit property, physical quality and other resources from their parents. In [66], Wu et al. introduced heritability into the evolutionary spacial PDG, where the players can inherit fitness from the last generation. The conception of heritability is similar with the memory effect in Ref. [39]. In that work, Qin et al. considered an evolutionary spatial PDG with the memory effect on a square lattice, in which the payoffs that were obtained by C strategy and the payoffs that were obtained by D strategy are accumulated separately and players update their strategies by considering the accumulated payoffs. They found that historical payoff plays a key role in the maintenance of cooperation. Fort studied models where the elements of the payoff matrix were inherited in parallel with the strategy adoption [67]. In our study, we wish to elaborate on the prominent role of heritability and its ability to promote cooperation in the PGG. For this purpose, we consider an evolutionary spatial PGG on a square lattice, where each player can inherit fitness from the last generation at each time step. Meanwhile, their fitness should be correlated with their payoffs from the current interactions. We observe that the introduction of inherited fitness can improve the cooperation level of a system significantly. From simulations, we also find that the inherited fitness plays a pivotal role in the fitness stability, which can prevent the expansion of defectors effectively. 2. Model We consider an evolutionary two-strategy public goods game with players located on a four-neighbor square lattice with periodic boundary conditions, where each player and its nearest neighbors form a group with size G = 5. Initially, either a cooperator or a defector, randomly chosen with equal probability, occupies each site. At each time step, cooperators contribute to the group whereas defectors do not. The total payoff of a certain player i depends on its strategy si and the number of cooperators nc in the neighborhood. Thus, the player i’s payoff is r (nc + si )
− si , G where r represents the multiplication factor. If the player i is a cooperator at this step, si equals 1, otherwise si is zero. At the t-th generation, the fitness fi (t ) of the player i is defined as: Pi =
fi (t ) = τ fi (t − 1) + (1 − τ )Pi (t ),
(1)
(2)
where the parameter τ denotes the heritability of players, which sets the balance between the present and past payoff gains−the relative importance of a previous generations or strength of maternal effects decays with a factor τ per time step [66]. According to Eq. (2), we can write the fitness fi (t ) of the player i in another form: fi (t ) = τ t fi (0) +
t −
τ t −n Pi (n)(1 − τ ).
(3)
n =1
If τ = 1, the current payoff is neglected, the fitness of a player will never change in the evolution. Hence, we have τ in the range of [0,1). In our model, each player’s initial fitness fi (0) is set to 1. We have check that the precise value of fi (0) does not affect the final cooperator density of a system. As the contributions of f (0) to all the players are always same at each time step, the impact of initial fitness is eliminated when the fitness-difference update rule is adopted. When updating strategies, all players imitate the strategy of a randomly chosen neighbor with a probability depending on the fitness difference synchronously, i.e., the probability for player i to adopt a randomly chosen neighbor j’s strategy is given by: Wi =
1 1 + exp[( fi − fj )/κ]
,
(4)
where κ characterizes the noise introduced to permit irrational choices [9] (κ → ∞ leads to random imitation, whereas κ → 0 leads to the deterministic imitation of the better player). Here we have κ = 0.1 for all simulations. Simulations are carried out for a population of N = 100 × 100 individuals. We study the key quantity of cooperator density ρc in the steady state. Equilibrium frequencies of cooperators are obtained by averaging over 2000 Monte Carlo time steps from a total of 22 000 steps, and each data point results from an average of over 10 realizations.
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Fig. 1. (Color online) Stationary fraction of cooperators ρc in dependence on the multiplication factor r for different values of τ .
Fig. 2. (Color online) Stationary fraction of cooperators ρc in dependence on the heritability factor τ for different values of r.
3. Simulation and analysis Fig. 1 shows how ρc varies in dependence on the multiplication factor r for different values of τ . It displays that ρc increases monotonically with the increasing of r, regardless what τ is. Moreover, for a fixed r, different values of τ can affect the final cooperation levels dramatically. To quantify the effects of varying τ on cooperation, we plot ρc as a function of the parameter τ for different values of r, as shown in Fig. 2. From Fig. 2, we can find that the cooperator density ρc increases with the increase of τ for a fixed value of r. Fig. 3 shows the typical distributions of cooperators and defectors on a square lattice at different time steps t for τ = 0 and r = 4.5. During the first ten steps, a large number of cooperators are eliminated, and some small cooperative clusters may survive at this time. After that, many small cooperative clusters are eliminated and some large cooperative clusters form. At the end, all the cooperative clusters disappear, and no cooperator survives. Based on this process, we find that the cooperators have a tendency to form clusters in the evolution although they are eliminated finally for τ = 0 and r = 4.5. In order to understand the process more clearly, we plot the mean fitness of cooperators and defectors on the boundary of cooperative clusters (⟨ fcb ⟩ and ⟨ fdb ⟩) in the evolution (In our study, a player belongs to boundary if it has at least one neighbor who holds an opposite strategy). From Fig. 4(a), we can find that the fitness of cooperators is always larger than that of defectors, which indicates that the cooperators are always eliminated although they earn higher payoffs than those of defectors for τ = 0 and r = 4.5. At the interface of a cooperator cluster, the transitions between C players and D players occur continuously. Once a cooperator changes into a defector, its payoff will increase greatly by exploiting its cooperative neighbors, which may further result in some of its cooperative neighbors changing into defectors. Hence, the introduction of historical payoff can restrict the sudden increase in defector’s fitness, and prevent the continuous change of cooperators into defectors. At the same time, the introduction of historical payoff can also maintain the fitness stability of cooperators so that the cooperators can keep their advantages in fitness and prevent the invasion of defectors. Fig. 5(a) shows the mean fitness of cooperators and defectors (⟨ fc ⟩ and ⟨ fd ⟩) in the equilibrium state. We can find that, in the equilibrium state, ⟨ fc ⟩ is no less than ⟨ fd ⟩ for most values of τ . For τ < 0.2, there is no cooperator, both ⟨ fc ⟩ and ⟨ fd ⟩ equal zero. For τ > 0.5, there is no defector, ⟨ fc ⟩ reaches its maximum, and ⟨ fd ⟩ equals zero. Fig. 5(b) shows the average payoffs of cooperators ⟨Pc ⟩ and defectors ⟨Pd ⟩ as a function of τ . Comparing Fig. 5(a) and (b), we can find that the fitness of
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Fig. 3. A series of snapshots of distribution of cooperators (white) and detectors (black) at different time steps: (a) t = 1, (b) t = 10, (c) t = 100, and (d) t = 1000. The parameter τ is set as 0 and r is set as 4.5.
(a)
(b)
Fig. 4. (Color online) The mean fitness of cooperators and defectors on the boundary (⟨ fcb ⟩ and ⟨fdb ⟩) in the evolution for τ = 0 (a) and τ = 0.1 (b) with r = 4.5.
a
b
Fig. 5. (Color online) (a) The mean fitness of cooperators and defectors (⟨ fc ⟩ and ⟨ fd ⟩) in the equilibrium state, as a function of τ ; (b) the mean payoff of cooperators and defectors (⟨Pc ⟩ and ⟨Pd ⟩) in the equilibrium state, as a function of τ . The parameter r is set as 4.
both C players and D players almost equal the payoffs of them. In the steady state, following Eq. (2) we can get
⟨ fc (t )⟩ = τ ⟨ fc (t − 1)⟩ + (1 − τ )⟨Pc (t )⟩
(5)
⟨ fd (t )⟩ = τ ⟨ fd (t − 1)⟩ + (1 − τ )⟨Pd (t )⟩.
(6)
and According to Fig. 4(b), we find that the fitness of cooperators and defectors can reach a steady state, and we can have fc (t ) = fc (t − 1) and fd (t ) = fd (t − 1). Submitting them into Eqs. (4) and (5) respectively, we can get fc (t ) = Pc (t ) and
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fd (t ) = Pd (t ) in the steady state. These results further indicate that cooperators earn higher payoffs than those of defectors by forming clusters, and the heritability can maintain the advantage of cooperators all the while. 4. Conclusion In Ref. [66], Wu et al. introduced heritability into the evolutionary prisoner’s dilemma game, and they focused on the effects of reproduction time scale and its diversity on the evolutionary cooperation. Interestingly, they found that there exists an intermediate selection time scale that maximizes cooperation. Another factor they find to promote cooperation is a diversity of reproduction time scales. In contrast with that, we paid our attention to the effects of heritability on the evolutionary public goods games and found that the introduction of inherited fitness can improve the cooperation level of a system significantly. In another related work [39], Qin et al. explored the effects of memory in a spatial Prisoner’s Dilemma game. Their memory mechanism is similar to the maternal-effect fitness inheritance in our model. The players update their strategies according to the cumulative historical payoff in both models. However, there exist differences in the way of cumulation. In their model, the payoffs obtained by the C strategy and the payoffs obtained by the D strategy are accumulated separately. While in our model, the payoffs are accumulated regardless what the strategy is. In conclusion, we have explored the effects of heritability on spacial public goods games. In our model, the fitness of each player is based on the instantaneous payoff from the interactions and the inherited fitness from the past, where the heritability of players is controlled by a tunable parameters τ . Interestingly, we have shown that if heritability is considered, then cooperation can emerge, maintain, and even thrive. In order to give an intuitive account of the maintenance of cooperation, we provide some typical snapshots of the system at different time steps and compare the mean fitness of defectors and cooperators in the evolution. It is shown that the introduction of historical payoff can maintain the fitness stability of players and prevent the expansion of defectors. In order to check the robustness of this mechanism, we have introduced the historical payoff into another PGG model proposed by Santos et al. [60], where each individual can take part in games centered on it and its neighbors, and obtains payoffs from each game that it has taken part in. All players update their strategies simultaneously by fitness differences. We find that heritability can facilitate cooperation all the same. We have also applied our model by an asynchronous strategy update rule, where the players update their strategies one by one in a random sequence at each time step. We find that the cooperation can be also improved when inheritable fitness is considered. Finally we have checked that our conclusions are robust with respect to using different strategy updating rules, such as ‘‘imitate the neighbor with the highest fitness’’[8] and the finite population analog of the replicator dynamics [29]. However, there exist limitations in our model. For instance, the ‘‘tit for tat’’ strategy has not been introduced to this model. Acknowledgements This work is funded by the National Basic Research Program of China (973 Program No. 2006CB705500), the National Natural Science Foundation of China (Grant Nos. 10975126, 10635040), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20093402110032). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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