Preferential selection based on degree difference in the spatial prisoner's dilemma games

Cooperation among selfish individuals is ubiquitous in the real world ranging from biological to social systems [1]. Spontaneous emergence and maintenance of cooperation among selfish individuals is an interesting problem in recent years, and evolutionary game theory provides a powerful framework to address the issue [2]. As one of the most famous games and a paradigm for studying cooperative behaviors through pairwise interactions [3], the prisoner's dilemma game (PDG) has been widely explored [4]. In the typical PDG, two players simultaneously take either cooperation or defection. The players' payoffs depend on the decisions of both. They both receive the reward R and the punishment P upon mutual cooperation and mutual defection, respectively. However, once a cooperator meets a defector, the defector will exploit the temptation T and the cooperator is left with the sucker's payoff S. These payoffs satisfy the inequalities T > R > P > S and $T+S<2R$. Under these conditions, it is obvious that a selfish (rational) player has no motivation to cooperate because the payoffs for defecting strictly dominate those for cooperating irrespective of the opponent's strategy. In a well-mixed population, cooperators cannot resist invasion of defectors and are doomed to extinction [5].

Since the pioneering work by Nowak and May [6] and with the rapid development of complex networks [7], ample effort has been put into the evolutionary PDGs and other evolutionary games on different structured networks such as square lattices [8,9], regular random [10], small-world [11,12], scale-free [13,14] and adaptive [15–17] networks. Different from the case in a well-mixed population, cooperation can emerge and sustain in structured populations, where spatial topology helps cooperators support each other through spontaneously forming clusters. Apart from network reciprocity, a number of mechanisms aimed at sustaining cooperation in spatial games have also been proposed, including individuals' persistence [18], voluntary participation [19], social diversity [20,21], memory effects [22–24], migration [25–33], learning and teaching ability [34–38], punishment and reward [39–42], aspiration [11,43–46], and reputation [47–50], to name but a few.

Among most works mentioned above, individuals are assumed to select their neighbors to learn from in a completely random way. That is, for a focal individual, all his neighbors have an equal probability to be selected. However, this assumption apparently contradicts the real situation. In real social systems, different neighbors might have different attractiveness to a focal individual. Thus, the rational individuals may not choose a neighbor randomly to learn from. In [51], Wu et al. first introduced a dynamic preferential neighbor selection rule. Then Guan et al. introduced nonlinear attractive effects into a spatial PDG on square lattices [52] and regular small-world networks [53]. In [54], Yang et al. introduced a degree-related preferential selection mechanism to the spatial public-goods game on scale-free networks, while Du et al. [55] studied this mechanism in the PDG and the snowdrift game. In [56], Shi et al. introduced a payoff-related preferential selection mechanism in a spatial public-goods game where players are located on a square lattice. Wang and Perc [57,58] found that increasing the probability of adopting the strategy from the fittest opponent promotes the evolution of cooperation. In [59], Wang et al. introduced an age-related preferential selection mechanism into the PDG, which can promote cooperation. More recently, Ye et al. [60] introduced a rational selecting mechanism based on radical evaluation into evolutionary snowdrift game. They found that the selection based on radical evaluation significantly enhances the level of cooperation. And then they investigated the memory-based PDG with conditional selection on networks [61].

Inspired by these studies, we investigate the evolutionary PDGs on Erdös-Rényi (ER) networks [62] and Barabási-Albert scale-free (SF) networks [63] considering the preferential selection based on degree difference and focus on the effects of this preferential selection mechanism on cooperation. In particular, we introduce a preference parameter δ into the evolutionary games. Positive δ means that the neighbors who have small degree difference with the focal individual are more likely to be selected by the focal individual to learn from, whereas negative δ means the focal individual prefers to choose the neighbors with large degree difference. The preference strength could be adjusted through the variation of $|\delta|$ (see the model below). We find that, regardless of the network topology, positive δ always hurts cooperation, while there exists an optimal value of δ to promote cooperation to a maximal level for negative δ.

The paper is organized as follows. In the second section, the model of evolutionary games incorporating preferential selection mechanism is introduced in detail. Then, numerical results and discussion are presented in the third section. Lastly, we summarize the main results and conclusions in the fourth section.

We consider the evolution of cooperation in the PDGs on two classic types of complex networks with degree heterogeneity, the ER network and the SF network. In ER networks, individuals' degrees follow a Poisson distribution [64]. While in SF networks, individuals' degrees have a power-law distribution and exhibit a high heterogeneity level [63]. We set the population size as N = 1000. Initially, each individual is designated either as a cooperator (C) or defector (D) with equal probability. The game is iterated forward in accordance with the standard Monte Carlo simulation procedure comprising the following elementary steps.

First, a randomly selected individual i cumulates his payoff U_i by playing the games with all his neighbors. For simplicity but without loss of generality, the payoffs of PDG can be rescaled as R = 1, $P=S=0$, and $T=b\ (1 \leqslant b \leqslant 2)$. Next, the individual i selects one of his neighbors j according to the probability p_ij in terms of a preferential selection rule

$$\begin{eqnarray} &&p_{ij}=\frac{A_{ij}}{\sum_{k\in\Omega_i} A_{ik}}, \end{eqnarray} \tag{ 1 }$$

where $\Omega_i$ is the community composed of i's nearest neighbors. A_ij denotes the attractiveness of the neighbor j, which can be defined as

$$\begin{eqnarray} &&A_{ij}=\frac{1}{(1+\delta)^{\Delta k_{ij}}}, \end{eqnarray} \tag{ 2 }$$

where $\Delta k_{ij}=| k_i-k_j|$ denotes the absolute value of degree difference between the degrees of individual i and j. The adjustable parameter δ, ranging from −1 to 1, characterizes the preference of selection. Evidently, for $\delta=0$, the attractiveness of the neighbors is uniform so that the preferential selection is reduced to the completely random one. For $\delta<0$, neighbors with larger degree difference with the individual i have higher attractiveness, and they are more likely to be selected by i according to eq. (1). Contrarily, for $\delta>0$, neighbors with lower degree difference with the individual i are more likely to be selected. Meanwhile, the preference strength is denoted by the absolute value of δ, $|\delta|$. After the neighbor j is selected by the individual i, he acquires his payoff U_j in the same way as the individual i. Lastly then, the individual i adopts the strategy s_j from the selected neighbor j with the probability

$$\begin{eqnarray} &&W(s_{i}\leftarrow s_{j})=\frac{1}{1+\exp[(U_{i}-U_{j})/\kappa]}, \end{eqnarray} \tag{ 3 }$$

where κ characterizes the intensity of the noise related to the adoption of strategy. If $\kappa\rightarrow0$, the selection is strong and fitter individuals are readily adopted. While $\kappa\rightarrow+\infty$ denotes the completely random adoption. In one full Monte Carlo step (MCS), each individual has a chance to adopts a strategy from one of his neighbors once on average. During the evolution, we focus on the evolution of individuals' strategies and monitor the fraction of cooperators F_C. All simulations are run for 50000 MCSs to ensure that the system reaches a steady state, and F_C is obtained by averaging over the last 1000 MCSs. Each data point is obtained from 100 independent realizations.

We first present the fraction of cooperators F_C as a function of temptation to defect b for different preference parameter δ on ER and SF networks in figs. 1(a) and (b), respectively. From fig. 1, for each δ, F_C decreases as b increases no matter on ER or SF networks. Furthermore, it could be found that δ affects cooperation significantly. As shown in fig. 1(a), compared with the results for the situation without preferential selection $(\delta=0)$, F_C is obviously suppressed for $\delta>0$, where individuals are more likely to learn from the neighbors who have lower degree differences with them. With the increase of the preference strength, the suppression of F_C is further aggravated. In contrast, for $\delta<0$ when individuals prefer to choose the neighbors who have larger degree differences with them, F_C first increases then decreases with the increase of the preference strength $|\delta|$. It means that there exists an optimal $|\delta|$, at which the cooperation level could be promoted to a maximum. For too small or too large $|\delta|$, cooperation will be inhibited. The similar effects of δ on cooperation could be found on SF networks, as shown in fig. 1(b).

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In order to give a clearer picture of the effects of δ on cooperation, we present the cooperation level F_C against δ for different game parameter b in fig. 2. From fig. 2(a) for ER networks, F_C is significantly enhanced in most cases when $\delta<0$. In particular, for an intermediate b, there exists an optimal value of the preference strength, $|\delta|\approx0.8$, leading to a maximum of the cooperation level. When b is low, for example, $b=1.1$, all-C state could be maintained at a wide range when $|\delta|$ increasing from 0. At $|\delta|\approx 0.8$, F_C begins to decrease. When b is high, F_C could also be promoted by the preferential selection for $\delta<0$, whereas the optimization does not appear. Contrarily, for $\delta>0$, cooperation is inhibited and F_C decreases with the increase of δ. The results of F_C for SF networks shown in fig. 2(b) display similar behaviors. When $\delta<0$, for relatively small and intermediate values of b (e.g., $b=1.1,1.3$ and 1.5), F_C reaches 1 at about $|\delta|<0.1$ and it jumps down as $|\delta|>0.1$. For the larger b (e.g., $b=1.8$ or 1.9), there exists an optimal preference strength of $\delta\ (|\delta|\approx0.1)$ leading to a maximal cooperation level. When $\delta>0$, the results are qualitatively similar to those obtained on ER networks, except the faster drop of F_C with the increase of δ.

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To investigate the mechanism of the preferential selection during the evolution process, we show the time series of F_C for different preference parameter δ on ER and SF networks by figs. 3(a) and (b), respectively. One could find that, in the beginning stage of evolution (t < 10), F_C increases with the decrease of δ. It means that, for $\delta>0$, cooperators are more likely to be converted to defectors with the increase of the preference strength $|\delta|$, whereas for $\delta<0$ the things are the opposite. For most δ except strong preference strength (for example, $\delta=-0.8$ or −0.9 for ER networks), given the initial conditions of randomly distributed strategies in the population and cooperators have not yet organized into compact clusters, F_C decreases at the beginning. Then, with more and more C-clusters formed, F_C starts to increase due to mutual reciprocity among cooperators. Bearing in mind that network heterogeneity favors cooperation [4,21,65,66] and $\delta>0$ means that individuals choose the neighbors with small degree difference, which weakens the effects of network heterogeneity in the strategy transmission, cooperation is inhabited. In contrast, the effects of network heterogeneity could be enhanced by $\delta<0$, where individuals choose the neighbors with large degree differences. Thus, cooperation could be promoted. On the other hand, some clues could be found for the optimization when $\delta<0$ from the time series of F_C during the evolution. In the range of $\delta<0$, a certain cooperation level might be maintained even at the very beginning and then increase with the evolution. However, too strong preference strength (for example, $\delta=-0.9$ for ER networks and $\delta=-0.2,-0.5,-0.9$ for SF networks) will inhibit the spread of cooperation strategy at the ascent stage compared with other smaller preference strength at $\delta<0$. For the inhibition of cooperation spreading, we could provide a plausible explanation as follows. Under the situation of an extremely strong selection strength at $\delta<0$, individuals always choose the neighbors who have the largest degree differences with them. It leads to some fixed configurations, in which individuals happen to have the same strategy with their largest degree difference neighbors. For example, such a pair of defectors, who are the largest degree difference neighbors for each other, would not be converted to cooperators even when they are located close to the C-clusters. Thus, cooperation is deterred from further spreading and F_C decreases at relatively strong preference when $\delta<0$.

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To show the effects of the fixed configurations mentioned in the above explanation more clearly, we examine the robustness of the results against the perturbation when δ approaches −1. The perturbation is introduced by letting focal individuals choose their neighbors randomly instead of by preferential selection with the probability p_u. According to the speculation, when δ approaches −1, the extremely strong preferential selection results in some fixed configurations, which inhibit the spread of cooperation strategy. The perturbation will damage these fixed configurations and cooperation could further spread out. From fig. 4, one could find that larger p_u leads to higher cooperation level. The speculation is confirmed. As larger p_u causes more fixed configurations to break, cooperation could be further transmitted through pairwise imitations. On the other hand, fig. 4 shows that, the temptation to defect b has only small effects on the cooperation, which is consistent with the results for $\delta<0$ shown in fig. 2(b).

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As discussed in previous studies, in SF networks, individuals with high degrees are more likely to be cooperative and have significant positive effects on the evolution of cooperation [67]. It is an interesting question of whether the cooperative property of these high-degree individuals remains with the preferential selection. To address this question, we investigate the relationship between the distribution of strategies and the individuals' degree. The results are shown in fig. 5. Four different δ have been considered. When $\delta=-0.9$, the distribution of strategies is comparatively uniform. For $\delta=-0.1$ which leads to an extremely high cooperation level, almost all individuals are cooperators and there is no difference between individuals with different degrees. For $\delta=0$ when the preferential selection mechanism does not work, high-degree individuals show high cooperative level as expected. For $\delta=0.1$, the low- and high-degree individuals have relatively low cooperation levels, whereas the medium-degree ones have high cooperation levels. In comparison with the results obtained at $\delta=0$, the cooperative property of these high-degree individuals are changed by the preferential selection and different δ results in different strategy distribution related to individuals' degree.

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Furthermore, we present the contour plots for F_C in the parameter space $b\text{-}\delta$ in fig. 6 to provide a comprehensive description of the effects of b and δ on F_C. From fig. 6(a), one can observe that for ER networks, F_C displays different behaviors for $\delta>0$ and for $\delta<0$. At the medium b where cooperators and defectors coexist, for $\delta>0$, F_C slightly decreases with the increase of δ. However, for $\delta<0$, in a large range of $|\delta|$ when increasing from 0, F_C could be significantly promoted. When $|\delta|$ is extremely large, cooperation is greatly inhibited as $\delta\rightarrow-1$. There exist optimal values of the preference strength $|\delta|$ leading to the maximal F_C when b < 1.65. For very large $b~(b\ge1.65)$, the optimal phenomena disappear, and cooperators could be survived only at extremely strong preference strength. When $\delta\rightarrow-1$, those neighbors with the largest degree differences are definitely selected, which resembles the deterministic selection rule. It results in the fact that strategy transmission is limited among those certain pairs of individuals. Consequently, once some cooperators happen to have the most fascinating cooperative neighbor with each other, a certain level of cooperation could be maintained even under the high temptation to defect. Compared with fig. 6(a), the decline of F_C is much faster when δ increasing from 0 in fig. 6(b) for SF networks. For $\delta>0$, individuals are more likely to learn from the neighbors with low degree differences, which weakens the effects of the degree heterogeneity. Given that previous studies attribute the high cooperation level obtained in SF networks to degree heterogeneity, F_C is obviously inhibited in the range of $\delta>0$. On the other hand, similar to the results shown in fig. 6(a), the optimal behaviors of F_C against the preference strength $|\delta|$ still remain, except that the optimum of $|\delta|$ is much smaller. Taking into account of the own topology heterogeneity of SF networks, the further heterogeneity enhancement caused by $\delta<0$ are not needed as much as that in ER network any more.

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Then, we consider the effects of different intensities of noise, κ, on cooperation in the model. The cooperation level F_C as a function of the preference parameter δ on ER and SF networks for different noise levels are shown in fig. 7. We can see the similar phenomena induced by the preferential selection on ER or SF networks with different amplitudes of noise κ. Although different cooperation levels [10,68] are obtained under different values of κ, the results remain qualitatively unaffected no matter what the network topology is. Furthermore, we further investigate the effects of different average degree $\langle k\rangle$ and network size N on cooperation. From fig. 8, one can observe that, compared with the average degree $\langle k\rangle=4$ (see fig. 2), although different average degrees induce different levels of cooperation [13], the interesting phenomena are preserved for different average degrees. Moreover, one can see that the qualitative features of the results are unchanged under the different network sizes. Therefore, we can conclude that the results are robust against the different noise intensities, average degrees and network sizes.

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In summary, we have explored the cooperative behaviors in the context of evolutionary PDGs with preferential selection mechanism on ER and SF networks. The individual i selects one of his neighbors j to learn from via the probability proportional to the attractiveness of j on i, A_ij, which is based on the degree difference between i and j. The results show that, for ER networks, compared with the traditional version that individuals with homogeneous influence $(\delta=0)$, cooperation can be promoted significantly when individuals are more likely to choose the neighbors who have large degree differences with them $(\delta<0)$, and there exists an optimal preference strength $|\delta|$ resulting in the maximal cooperation level. However, if the low-degree-difference neighbors are preferentially selected $(\delta>0)$, the evolution of cooperation will be inhibited. When the preferential selection mechanism is introduced in SF networks where high degree heterogeneities exist, the optimal behavior of F_C with the variation of the preference strength can also be obviously found in the range of $\delta<0$, except that the optimal $|\delta|$ is much smaller when compared with that for ER networks. Similarly, the lower-degree-difference neighbors with the larger attractiveness $(\delta>0)$ will greatly inhibit the evolution of cooperation on SF network, and even a weak preference strength leads to a dramatic decrease of the cooperation level. In addition, we have studied the influence of noise κ, average degree and network size, and found that qualitative features of the results remain unchanged.

Our results suggest that, as a more realistic mechanism, the introduction of the preferential selection plays an important role in the evolution of cooperation. The proposed preferential selection mechanism in our work is based on the degree difference between two individuals. As a fundamental trait of an individual in a structured population, the degree is used to measure the attractiveness. We have to point out that, there are many other indicators of node's traits, such as betweenness centrality [69], coreness [70], and collective influence [38,71]. The consideration of these indicators may enrich the work and bring some interesting results, and it is worth further investigations. Overall, our research highlights the importance of the preferential selection in evolutionary games, and it should receive more attention.

This work was supported by the National Natural Science Foundation of China under Grants No. 11575036 and No. 11505016.