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Cooperation is enhanced by inhomogeneous inertia in spatial prisoner’s dilemma game
Accepted Manuscript Cooperation is enhanced by inhomogeneous inertia in spatial prisoner’s dilemma game Shuhua Chang, Zhipeng Zhang, Yu’e Wu, Yunya Xie
PII: DOI: Reference:
S0378-4371(17)30767-7 http://dx.doi.org/10.1016/j.physa.2017.08.034 PHYSA 18480
To appear in:
Physica A
Received date : 24 April 2017 Revised date : 30 July 2017 Please cite this article as: S. Chang, Z. Zhang, Y. Wu, Y. Xie, Cooperation is enhanced by inhomogeneous inertia in spatial prisoner’s dilemma game, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.08.034 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Cooperation is enhanced by inhomogeneous inertia in spatial prisoner’s dilemma game Shuhua Changa,∗, Zhipeng Zhanga , Yu’e Wua , Yunya Xiea a Coordinated
Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China
Abstract Inertia is an important factor that can not be ignored in the real world for some lazy individuals in the process of decision making. In this work, we introduce a simple classification mechanism of strategy changing in evolutionary prisoner’s dilemma games on different topologies. In this model, a part of players update their strategies according to not only the payoff difference, but also the inertia factor, which makes nodes heterogeneous and the system inhomogeneous. Moreover, we also study the impact of the number of neighbors on the evolution of cooperation. The results show that the evolution of cooperation will be promoted to a high level when the inertia factor and the inhomogeneous system are combined. In addition, we find that the more neighbors one player has, the higher density of cooperators is sustained in the optimal position. This work could be conducive to understanding the emergence and persistence of cooperative behavior caused by the inertia factor in reality. Keywords: evolutionary game, cooperation, inhomogeneous inertia, prisoner’s dilemma game
∗ Corresponding
author Email address: [email protected] (Shuhua Chang)
Preprint submitted to Journal of LATEX Templates
August 28, 2017
1. Introduction Cooperation has been one of the most pressing and interesting problems in evolutionary biology and public goods problems like global warming [1, 2, 3, 4]. To reveal the emergence and persistence of cooperative behavior among selfish 5
individuals, scientists are used to investigating it by means of evolutionary game theory [5, 6, 7, 8, 9, 10]. The prisoner’s dilemma game (PDG) [11, 12, 13, 14, 15, 16, 17, 18, 19], as a general paradigm to understand cooperative phenomena, is often used to study pairwise interaction in theory and experiment within this framework. In an original PDG, two players (agents) make a decision
10
simultaneously between cooperation (C) and defection (D). If both players cooperate, they will get the reward R equally, but only the punishment P when two defectors encounter. If two partners have different actions, the defector could get the highest temptation T and the cooperator only obtain the Sucker’s payoff S. For the PDG, the payoff must satisfy the necessary rankings T > R >
15
P > S and 2R > (T + S). A stable state or a Nash equilibrium of the game is mutual defection. In other words, although mutual cooperation yields the most collective payoff, rational players should always defect regardless of what the opponent chooses. However, it is inconsistent with the fact that there are a large number of cooperative phenomena in nature and society [20]. Therefore,
20
it is necessary to understand the mechanism among selfish individuals so as to solve our problems. Over the past decades, many schemes have been proposed to understand the emergence and maintenance of cooperation in many disciplines. Examples include kin selection [21], direct and indirect reciprocity [22, 23, 24], group
25
selection [25], spatiality reciprocity [26] and so on. Furthermore, some effective strategies are used, including the tit-for-tat [27, 28] or win-stay lose-shift [29, 30]. Particularly, spatial structure [26], introduced by Nowak and May, could protect cooperators from being invaded by defectors through forming clusters. In spatial evolutionary game, players occupy the vertices and only interact with their
30
neighbors on a regular lattice. They collect the payoffs which are from gaming
2
with their nearest neighbors. Interestingly, it has been attested that network structure has an important effect on improving cooperation. After that, the combination of networks and evolutionary PDG has received much attention. A lot of mechanisms have been introduced, such as social diversity [31, 32], 35
reward and punishment [33, 34, 35, 36, 37, 38, 39], reputation [40, 41], partner selection [42], the effect of aspiration [43, 44, 45], coevolution [46], inhomogeneity [47] and so on. Moreover, a lot of works have also been done based on complex networks that supply a framework to understand the cooperative behavior [48, 49, 50, 51, 52, 53].
40
The impact of inertia on the evolution of cooperation has attracted much interest in the last decades [54]. Inertia can measure the degree of changing current states of players. Liu et al. take advantage of inertia index to control players’ learning activity, and find that cooperation can be improved significantly. Besides, it is found that there is an optimal value to lead to a highest
45
cooperation level in scale-free networks [55]. Du et al. study how the inertia affects the cooperative behavior on square lattices and show that a moderate inertia can induce the highest cooperation level [56]. In addition, Szolnoki and Szab´o introduce the inhomogeneous activity of teaching in the PDG and emphasize the importance of inhomogeneity [8]. Following this line, in this
50
work we study the effect of inhomogeneous inertia and its distribution on the maintenance of cooperative behavior in some evolutionary PDGs on different topologies. In the present work, inertia is introduced as a factor to affect the capacity of strategy imitation. In addition, the effect of group size on the evolution of cooperation has been studied widely [57, 58, 59]. Wang et al. indicate
55
that the individuals with more neighbors have a trend to preserve their initial strategies [60]. Moreover, we also observe the influence of neighbor size on the evolution of cooperation. The aim of our research is to explore the consequence of inhomogeneous updating mechanism effects by inertia on the evolution of cooperation under different group size. The results show that the new mechanism
60
can enhance cooperation markedly. The rest of this paper is organized as follows. Section 2 shows the basic 3
model. In Section 3, the simulation results are presented and discussed. Section 4 concludes this paper.
2. Methods 65
For simplicity but without loss of generality, here we consider a weak evolutionary PDG [26], which is a special game with the temptation to defect T = b, mutual cooperation R = 1, the punishment for mutual defection P = 0, and the Suckers’ payoff S = 0. Consequently, it is not hard to see that the outcome of the game is only dependent on the parameter b. In addition, 1 < b < 2 ensures
70
an essential payoff ranking for this dilemma. Meanwhile, an L × L square lattice with periodic boundary conditions and Erd¨ os-R´enyi (ER) random graphs represent the network structures, and every node is occupied by a player. The edges among individuals denote the pairwise interaction. Initially, each player is designed as a cooperator or a defector with equal probability and all the players
75
are divided into two types (ni = A or B) randomly before simulation. All the players can not change their type during the evolutionary processes. Consequently, the proportions of players A and B, denoted by u and 1 − u, remain unchanged, and this division of agents is non-uniform. The game is simulated with the following elementary steps: first, player i gets
80
his total payoff Pi by gaming with his nearest neighbors. In the present paper, three different sizes of neighbors are compared to analyze the impact of inertia factor and the related proportion distribution on the evolution of cooperation. Specially, these include the von Neumann neighborhood (k = 4, and k is the degree), the Moore neighborhood [61] (k = 8) and the case of 12 neighbors.
85
Next, player i will choose a neighbor j randomly from its neighbors as the reference target, who acquires payoff Pj in the same way. The random initial strategy distribution is the start of Monte Carlo simulations (MCS). Finally, the process of evolution transfers into the period of changing own strategy. Player i will adopt j’s strategy according to the following probability:
4
1 , 1 + exp[α + (Pi − Pj )/(kK)] 1 , HB = 1 + exp[(Pi − Pj )/(kK)]
HA =
90
if ni ∈ A,
(1)
if ni ∈ B,
(2)
relying on the normalized payoff difference, where K represents the intensity of selection, α is the inertia parameter, and k is the degree of player i. If player i is affected by inertia when he makes decision, he would use the formula (1). Otherwise, player i has to adopt the formula (2). From individuals’ points of view, everybody deals with things in different styles. For example,
95
there are a part of people usually to change their states and the others stay at a position for a long time. According to this fact, we put some agents into type A to observe how this factor affects the evolution of cooperation. Evidently, this model can keep all agents homogeneous when u = 0 or 1. However, for 0 < u < 1, the Fermi updating rule and the modified Fermi updating rule are
100
mixed in the network, which forms an inhomogeneous environment. Moreover, the model returns to the traditional version when α = 0 or u = 0. Ultimately, a whole MCS is finished if the above-mentioned fundamental procedures are implemented. Every player has a chance to change his strategy in each round on average.
105
3. Results The results of MCS presented in the following discussions are comprised of 200 × 200 agents on a square lattice. Additionally, the fraction of cooperators is obtained by averaging the last 1000 full MCS over the total 61000 steps, and the final results are averaged by 10 independent runs to guarantee the accuracy.
110
The purpose of this paper is to study the evolution of cooperators under the new mechanism. Therefore, we start by showing the vital fraction of cooperation under different proportions of type A in Figure 1. Here, we set b = 1.1. Three curves represent three sizes of neighbors, respectively. One can find that all curves have the same changing tendency under the new mechanism. It can be 5
1 .0 k = 4 k = 8 k = 1 2
0 .8
c
0 .6 0 .4 0 .2 0 .0
0 .0
0 .2
0 .4
u
0 .6
0 .8
1 .0
Figure 1: Concentration of cooperators ρc vs the parameter u for different values of neighbors size k. Apparently, there is a maximum ρc in every curve. It is a traditional case when u = 0. The results are obtained for α = 5.0, b = 1.1, and K = 0.1.
115
observed that there exists an optimal value of proportion in all of the three curves under the new mechanism. The model degrades to the traditional PDG case and cooperators die out when u is equal to 0 because the temptation is high. On the contrary, all the agents are inertial if u is equal to 1, which loses the inhomogeneity, but still promotes cooperation. The results for the intermediate
120
value u demonstrate that although the temptation is large for the selfish agents, the new mechanism can also sustain a high level of cooperators for parts of u. Figure 1 shows clearly that there is an optimum composition of types A and B. In addition, the existence of inertial individuals results in firm faith for keeping cooperation strategy. In addition, Figure 1 testifies that the bigger group sizes
125
are, the more cooperations and the cooperations diffuse widely as u approaches to the optimal value. Under the new mechanism, the fraction of cooperators has an analogous trend compared with the previous study when α is large enough, and the concentration of cooperators stays in an interval. So, we do
6
1 .0 tra d = = = =
0 .8
1 3 5
c
0 .6
itio n 0 .5
0 .4 0 .2 0 .0 1 .0 0
1 .0 5
1 .1 0
1 .1 5
1 .2 0
1 .2 5
1 .3 0
b Figure 2: The frequency of cooperators ρc in dependence on the temptation b. The cooperation level is enhanced as the inertia ability increases obviously. The results are obtained for K = 0.1, u = 0.7 and k = 8.
not exhibit it. In short, we can observe obviously that cooperators can spread 130
their territory in varying degrees when the mechanism is introduced. Moreover, the other parameters are set as α = 5.0 and K = 0.1. It can be observed from Figure 1 that the results of eight neighbors perform much better than the other two cases. Consequently, we study the resistance ability of agents at different temptations under the situation of eight neighbors
135
in Figure 2. To observe the obvious effect, we fix u = 0.7 around the optimal position. The black curve represents the traditional version. With the help of spatial structure, the fraction of cooperators can be sustained at a level when b = 1. However, once the temptation increases, the cooperation will disappear quickly. This is consistent with the previous work.
140
However, when taking the new mechanism into account, the cooperation is enhanced clearly. For example, for small α = 0.5, the curve moves right slightly. However, as α getting larger, the amplitude of movement become more and more big. Specially, cooperation will become the dominant strategy for some b values. 7
1 .4 0 tra d = = =
1 .3 5
3 5
b
c
1 .3 0
itio n 1
1 .2 5 1 .2 0 1 .1 5 1 .1 0 0 .0
0 .2
0 .4
u
0 .6
0 .8
1 .0
Figure 3: The critical threshold value bc (cooperators die out) makes the system into the pure D phase, depending on the proportion of type A for different values of α. It is evident that cooperators can live in the shelter of higher u and α. The results are obtained for K = 0.1 and k = 8.
These results indicate that the introduction of the inertia factor promotes the 145
evolution of cooperation. Figure 2 shows the relation between the temptation b and the density of cooperators ρc for different values of inertia α. Based on the above preliminary results, it is interesting to pay attention to the critical threshold value bc , at which all agents choose defect strategy. For α = 0 or u = 0, that is to say
150
losing the inertia influence, it makes the cooperators extinct when b = 1.095. Consequently, we could see a straight line (black) at the bottom of Figure 3. However, once the inertia factor is considered even if it is small, for example, α = 1, bc will increase obviously when u = 1. In addition, bc monotonically increases with increase of u when α = 1. In other words, u = 1 is the most
155
powerful condition to promote cooperation if all agents are inertial in the whole system for small α. With the increase of α, another interesting phenomenon appears. There is an optimal combination of α and u (defined by the maximum
8
( a )
( b )
( c )
Figure 4: Time courses of the fraction of cooperation and characteristic snapshots of players on the regular lattice. The red points are defectors and the blue points are cooperators in panels (b) and (c). The snapshots correspond to the distributions of the tradition case (panel (b)) and k = 8 (panel (c)) at the 60000th round in panel (a), respectively. The results are obtained for K = 0.1, u = 0.5, α = 3.0, and b = 1.08.
of bc ), which is clear for α = 5. This illustrates that u = 1 may weaken the improvement from inertia if all players are lazy. 160
This enhancement of survival for cooperators is robust for different values of α. The result is consistent with Figure 1, and it shows that the situation in which all individuals are affected by inertia are not beneficial for cooperators’ resisting the intrusion from defectors for some α. This loses the advancement from inertia because everyone is the same around each player, which is the
165
reason for the decline of cooperators fraction. It remains of interest to elucidate how this new mechanism promotes cooperation. To provide answers, we show time courses of cooperation and some characteristic snapshots on the square lattice in Figure 4. Recalling the initial distribution of cooperators and defectors, as shown in panel (a), two types of
170
strategies have approximately the same fraction. However, it can be observed that all curves sharply decrease, as is expected, because the defectors are more successful than the cooperators initially. The cooperative fraction in traditional case (k = 4) decreases to 0 although the spatial reciprocity exists. Interestingly, the cooperation will rebound after the trough and keep fluctuation at a stable
175
value under the new mechanism. In particular, an appropriate combination of
9
1 .0 tra d = = = =
0 .8
1 3 5
c
0 .6
itio n 0 .5
0 .4 0 .2 0 .0
1 .0
1 .2
1 .4
1 .6
1 .8
b Figure 5: Frequency of cooperator ρc in dependence on the parameter b in ER network. The results are obtained for K = 0.1 and u = 0.7.
neighbors’ size and proportion of u can increase the velocity of rebounding of cooperators, for example, k = 8 and u = 0.5. This is not only due to the network reciprocity, but also due to the revision of players’ behaviors by the new mechanism. Spatially, we show the characteristic spatial configurations at the 180
stable state for the traditional case and the case of k = 8 in panels (b) and (c), respectively. This hints that the inhomogeneous inertia promotes cooperation. As shown in panel (b), cooperative clusters will vanish and the defectors become dominant in the whole network without the new mechanism. On the contrary, the inhomogeneous inertia can enhance the network reciprocity. Panel
185
(c) shows that cooperators can form bigger clusters to protect themselves under this mechanism. Lastly, we consider the mechanism in ER random graphs with the size of 20000 to test the robustness of the results. The fraction of cooperators is obtained by averaging the last 10000 steps of the full MCS over the total 100000
190
steps, and the final results are averaged by 10 independent runs as shown in Figure 5. Similar to Figure 2, compared with the traditional version, as the increasing of α, cooperation can be promoted effectively. These results indicate
10
that the updating rule of inhomogeneous inertia is robust for sustaining and promoting cooperation.
195
4. Conclusion In summary, we have studied the impact of the inhomogeneous inertia on the evolution of cooperation in the spatial prisoner’s dilemma game. In the model, the probability of changing strategy for some individuals depends on both the payoff difference and the inertia level α, and the impact makes the
200
players heterogeneous and the network inhomogeneous. Compared with the traditional version (namely α = 0 or u = 0), the introduction of the new simple mechanism can improve cooperation remarkably for different sizes of neighbors. In addition, to validate the generality of this mechanism, we also explore the evolution of cooperation in ER graphs and the results are consistent with the
205
results on the square lattice. In reality, individual behaviors may be guided in a certain direction by inertia just as that in the article. Our results indicate that the inertia factor has a positive influence on the evolution of cooperation, which may be helpful to understand the emergency and maintenance of cooperation.
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Highlights
Inhomogeneous inertia promotes cooperation in network population. (2) Inertia factor can facilitate the evolution of cooperation on different topologies. (3) Inhomogeneous inertia can enhance spatial reciprocity. (1)
PII: DOI: Reference:
S0378-4371(17)30767-7 http://dx.doi.org/10.1016/j.physa.2017.08.034 PHYSA 18480
To appear in:
Physica A
Received date : 24 April 2017 Revised date : 30 July 2017 Please cite this article as: S. Chang, Z. Zhang, Y. Wu, Y. Xie, Cooperation is enhanced by inhomogeneous inertia in spatial prisoner’s dilemma game, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.08.034 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Cooperation is enhanced by inhomogeneous inertia in spatial prisoner’s dilemma game Shuhua Changa,∗, Zhipeng Zhanga , Yu’e Wua , Yunya Xiea a Coordinated
Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China
Abstract Inertia is an important factor that can not be ignored in the real world for some lazy individuals in the process of decision making. In this work, we introduce a simple classification mechanism of strategy changing in evolutionary prisoner’s dilemma games on different topologies. In this model, a part of players update their strategies according to not only the payoff difference, but also the inertia factor, which makes nodes heterogeneous and the system inhomogeneous. Moreover, we also study the impact of the number of neighbors on the evolution of cooperation. The results show that the evolution of cooperation will be promoted to a high level when the inertia factor and the inhomogeneous system are combined. In addition, we find that the more neighbors one player has, the higher density of cooperators is sustained in the optimal position. This work could be conducive to understanding the emergence and persistence of cooperative behavior caused by the inertia factor in reality. Keywords: evolutionary game, cooperation, inhomogeneous inertia, prisoner’s dilemma game
∗ Corresponding
author Email address: [email protected] (Shuhua Chang)
Preprint submitted to Journal of LATEX Templates
August 28, 2017
1. Introduction Cooperation has been one of the most pressing and interesting problems in evolutionary biology and public goods problems like global warming [1, 2, 3, 4]. To reveal the emergence and persistence of cooperative behavior among selfish 5
individuals, scientists are used to investigating it by means of evolutionary game theory [5, 6, 7, 8, 9, 10]. The prisoner’s dilemma game (PDG) [11, 12, 13, 14, 15, 16, 17, 18, 19], as a general paradigm to understand cooperative phenomena, is often used to study pairwise interaction in theory and experiment within this framework. In an original PDG, two players (agents) make a decision
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simultaneously between cooperation (C) and defection (D). If both players cooperate, they will get the reward R equally, but only the punishment P when two defectors encounter. If two partners have different actions, the defector could get the highest temptation T and the cooperator only obtain the Sucker’s payoff S. For the PDG, the payoff must satisfy the necessary rankings T > R >
15
P > S and 2R > (T + S). A stable state or a Nash equilibrium of the game is mutual defection. In other words, although mutual cooperation yields the most collective payoff, rational players should always defect regardless of what the opponent chooses. However, it is inconsistent with the fact that there are a large number of cooperative phenomena in nature and society [20]. Therefore,
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it is necessary to understand the mechanism among selfish individuals so as to solve our problems. Over the past decades, many schemes have been proposed to understand the emergence and maintenance of cooperation in many disciplines. Examples include kin selection [21], direct and indirect reciprocity [22, 23, 24], group
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selection [25], spatiality reciprocity [26] and so on. Furthermore, some effective strategies are used, including the tit-for-tat [27, 28] or win-stay lose-shift [29, 30]. Particularly, spatial structure [26], introduced by Nowak and May, could protect cooperators from being invaded by defectors through forming clusters. In spatial evolutionary game, players occupy the vertices and only interact with their
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neighbors on a regular lattice. They collect the payoffs which are from gaming
2
with their nearest neighbors. Interestingly, it has been attested that network structure has an important effect on improving cooperation. After that, the combination of networks and evolutionary PDG has received much attention. A lot of mechanisms have been introduced, such as social diversity [31, 32], 35
reward and punishment [33, 34, 35, 36, 37, 38, 39], reputation [40, 41], partner selection [42], the effect of aspiration [43, 44, 45], coevolution [46], inhomogeneity [47] and so on. Moreover, a lot of works have also been done based on complex networks that supply a framework to understand the cooperative behavior [48, 49, 50, 51, 52, 53].
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The impact of inertia on the evolution of cooperation has attracted much interest in the last decades [54]. Inertia can measure the degree of changing current states of players. Liu et al. take advantage of inertia index to control players’ learning activity, and find that cooperation can be improved significantly. Besides, it is found that there is an optimal value to lead to a highest
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cooperation level in scale-free networks [55]. Du et al. study how the inertia affects the cooperative behavior on square lattices and show that a moderate inertia can induce the highest cooperation level [56]. In addition, Szolnoki and Szab´o introduce the inhomogeneous activity of teaching in the PDG and emphasize the importance of inhomogeneity [8]. Following this line, in this
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work we study the effect of inhomogeneous inertia and its distribution on the maintenance of cooperative behavior in some evolutionary PDGs on different topologies. In the present work, inertia is introduced as a factor to affect the capacity of strategy imitation. In addition, the effect of group size on the evolution of cooperation has been studied widely [57, 58, 59]. Wang et al. indicate
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that the individuals with more neighbors have a trend to preserve their initial strategies [60]. Moreover, we also observe the influence of neighbor size on the evolution of cooperation. The aim of our research is to explore the consequence of inhomogeneous updating mechanism effects by inertia on the evolution of cooperation under different group size. The results show that the new mechanism
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can enhance cooperation markedly. The rest of this paper is organized as follows. Section 2 shows the basic 3
model. In Section 3, the simulation results are presented and discussed. Section 4 concludes this paper.
2. Methods 65
For simplicity but without loss of generality, here we consider a weak evolutionary PDG [26], which is a special game with the temptation to defect T = b, mutual cooperation R = 1, the punishment for mutual defection P = 0, and the Suckers’ payoff S = 0. Consequently, it is not hard to see that the outcome of the game is only dependent on the parameter b. In addition, 1 < b < 2 ensures
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an essential payoff ranking for this dilemma. Meanwhile, an L × L square lattice with periodic boundary conditions and Erd¨ os-R´enyi (ER) random graphs represent the network structures, and every node is occupied by a player. The edges among individuals denote the pairwise interaction. Initially, each player is designed as a cooperator or a defector with equal probability and all the players
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are divided into two types (ni = A or B) randomly before simulation. All the players can not change their type during the evolutionary processes. Consequently, the proportions of players A and B, denoted by u and 1 − u, remain unchanged, and this division of agents is non-uniform. The game is simulated with the following elementary steps: first, player i gets
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his total payoff Pi by gaming with his nearest neighbors. In the present paper, three different sizes of neighbors are compared to analyze the impact of inertia factor and the related proportion distribution on the evolution of cooperation. Specially, these include the von Neumann neighborhood (k = 4, and k is the degree), the Moore neighborhood [61] (k = 8) and the case of 12 neighbors.
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Next, player i will choose a neighbor j randomly from its neighbors as the reference target, who acquires payoff Pj in the same way. The random initial strategy distribution is the start of Monte Carlo simulations (MCS). Finally, the process of evolution transfers into the period of changing own strategy. Player i will adopt j’s strategy according to the following probability:
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1 , 1 + exp[α + (Pi − Pj )/(kK)] 1 , HB = 1 + exp[(Pi − Pj )/(kK)]
HA =
90
if ni ∈ A,
(1)
if ni ∈ B,
(2)
relying on the normalized payoff difference, where K represents the intensity of selection, α is the inertia parameter, and k is the degree of player i. If player i is affected by inertia when he makes decision, he would use the formula (1). Otherwise, player i has to adopt the formula (2). From individuals’ points of view, everybody deals with things in different styles. For example,
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there are a part of people usually to change their states and the others stay at a position for a long time. According to this fact, we put some agents into type A to observe how this factor affects the evolution of cooperation. Evidently, this model can keep all agents homogeneous when u = 0 or 1. However, for 0 < u < 1, the Fermi updating rule and the modified Fermi updating rule are
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mixed in the network, which forms an inhomogeneous environment. Moreover, the model returns to the traditional version when α = 0 or u = 0. Ultimately, a whole MCS is finished if the above-mentioned fundamental procedures are implemented. Every player has a chance to change his strategy in each round on average.
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3. Results The results of MCS presented in the following discussions are comprised of 200 × 200 agents on a square lattice. Additionally, the fraction of cooperators is obtained by averaging the last 1000 full MCS over the total 61000 steps, and the final results are averaged by 10 independent runs to guarantee the accuracy.
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The purpose of this paper is to study the evolution of cooperators under the new mechanism. Therefore, we start by showing the vital fraction of cooperation under different proportions of type A in Figure 1. Here, we set b = 1.1. Three curves represent three sizes of neighbors, respectively. One can find that all curves have the same changing tendency under the new mechanism. It can be 5
1 .0 k = 4 k = 8 k = 1 2
0 .8
c
0 .6 0 .4 0 .2 0 .0
0 .0
0 .2
0 .4
u
0 .6
0 .8
1 .0
Figure 1: Concentration of cooperators ρc vs the parameter u for different values of neighbors size k. Apparently, there is a maximum ρc in every curve. It is a traditional case when u = 0. The results are obtained for α = 5.0, b = 1.1, and K = 0.1.
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observed that there exists an optimal value of proportion in all of the three curves under the new mechanism. The model degrades to the traditional PDG case and cooperators die out when u is equal to 0 because the temptation is high. On the contrary, all the agents are inertial if u is equal to 1, which loses the inhomogeneity, but still promotes cooperation. The results for the intermediate
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value u demonstrate that although the temptation is large for the selfish agents, the new mechanism can also sustain a high level of cooperators for parts of u. Figure 1 shows clearly that there is an optimum composition of types A and B. In addition, the existence of inertial individuals results in firm faith for keeping cooperation strategy. In addition, Figure 1 testifies that the bigger group sizes
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are, the more cooperations and the cooperations diffuse widely as u approaches to the optimal value. Under the new mechanism, the fraction of cooperators has an analogous trend compared with the previous study when α is large enough, and the concentration of cooperators stays in an interval. So, we do
6
1 .0 tra d = = = =
0 .8
1 3 5
c
0 .6
itio n 0 .5
0 .4 0 .2 0 .0 1 .0 0
1 .0 5
1 .1 0
1 .1 5
1 .2 0
1 .2 5
1 .3 0
b Figure 2: The frequency of cooperators ρc in dependence on the temptation b. The cooperation level is enhanced as the inertia ability increases obviously. The results are obtained for K = 0.1, u = 0.7 and k = 8.
not exhibit it. In short, we can observe obviously that cooperators can spread 130
their territory in varying degrees when the mechanism is introduced. Moreover, the other parameters are set as α = 5.0 and K = 0.1. It can be observed from Figure 1 that the results of eight neighbors perform much better than the other two cases. Consequently, we study the resistance ability of agents at different temptations under the situation of eight neighbors
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in Figure 2. To observe the obvious effect, we fix u = 0.7 around the optimal position. The black curve represents the traditional version. With the help of spatial structure, the fraction of cooperators can be sustained at a level when b = 1. However, once the temptation increases, the cooperation will disappear quickly. This is consistent with the previous work.
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However, when taking the new mechanism into account, the cooperation is enhanced clearly. For example, for small α = 0.5, the curve moves right slightly. However, as α getting larger, the amplitude of movement become more and more big. Specially, cooperation will become the dominant strategy for some b values. 7
1 .4 0 tra d = = =
1 .3 5
3 5
b
c
1 .3 0
itio n 1
1 .2 5 1 .2 0 1 .1 5 1 .1 0 0 .0
0 .2
0 .4
u
0 .6
0 .8
1 .0
Figure 3: The critical threshold value bc (cooperators die out) makes the system into the pure D phase, depending on the proportion of type A for different values of α. It is evident that cooperators can live in the shelter of higher u and α. The results are obtained for K = 0.1 and k = 8.
These results indicate that the introduction of the inertia factor promotes the 145
evolution of cooperation. Figure 2 shows the relation between the temptation b and the density of cooperators ρc for different values of inertia α. Based on the above preliminary results, it is interesting to pay attention to the critical threshold value bc , at which all agents choose defect strategy. For α = 0 or u = 0, that is to say
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losing the inertia influence, it makes the cooperators extinct when b = 1.095. Consequently, we could see a straight line (black) at the bottom of Figure 3. However, once the inertia factor is considered even if it is small, for example, α = 1, bc will increase obviously when u = 1. In addition, bc monotonically increases with increase of u when α = 1. In other words, u = 1 is the most
155
powerful condition to promote cooperation if all agents are inertial in the whole system for small α. With the increase of α, another interesting phenomenon appears. There is an optimal combination of α and u (defined by the maximum
8
( a )
( b )
( c )
Figure 4: Time courses of the fraction of cooperation and characteristic snapshots of players on the regular lattice. The red points are defectors and the blue points are cooperators in panels (b) and (c). The snapshots correspond to the distributions of the tradition case (panel (b)) and k = 8 (panel (c)) at the 60000th round in panel (a), respectively. The results are obtained for K = 0.1, u = 0.5, α = 3.0, and b = 1.08.
of bc ), which is clear for α = 5. This illustrates that u = 1 may weaken the improvement from inertia if all players are lazy. 160
This enhancement of survival for cooperators is robust for different values of α. The result is consistent with Figure 1, and it shows that the situation in which all individuals are affected by inertia are not beneficial for cooperators’ resisting the intrusion from defectors for some α. This loses the advancement from inertia because everyone is the same around each player, which is the
165
reason for the decline of cooperators fraction. It remains of interest to elucidate how this new mechanism promotes cooperation. To provide answers, we show time courses of cooperation and some characteristic snapshots on the square lattice in Figure 4. Recalling the initial distribution of cooperators and defectors, as shown in panel (a), two types of
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strategies have approximately the same fraction. However, it can be observed that all curves sharply decrease, as is expected, because the defectors are more successful than the cooperators initially. The cooperative fraction in traditional case (k = 4) decreases to 0 although the spatial reciprocity exists. Interestingly, the cooperation will rebound after the trough and keep fluctuation at a stable
175
value under the new mechanism. In particular, an appropriate combination of
9
1 .0 tra d = = = =
0 .8
1 3 5
c
0 .6
itio n 0 .5
0 .4 0 .2 0 .0
1 .0
1 .2
1 .4
1 .6
1 .8
b Figure 5: Frequency of cooperator ρc in dependence on the parameter b in ER network. The results are obtained for K = 0.1 and u = 0.7.
neighbors’ size and proportion of u can increase the velocity of rebounding of cooperators, for example, k = 8 and u = 0.5. This is not only due to the network reciprocity, but also due to the revision of players’ behaviors by the new mechanism. Spatially, we show the characteristic spatial configurations at the 180
stable state for the traditional case and the case of k = 8 in panels (b) and (c), respectively. This hints that the inhomogeneous inertia promotes cooperation. As shown in panel (b), cooperative clusters will vanish and the defectors become dominant in the whole network without the new mechanism. On the contrary, the inhomogeneous inertia can enhance the network reciprocity. Panel
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(c) shows that cooperators can form bigger clusters to protect themselves under this mechanism. Lastly, we consider the mechanism in ER random graphs with the size of 20000 to test the robustness of the results. The fraction of cooperators is obtained by averaging the last 10000 steps of the full MCS over the total 100000
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steps, and the final results are averaged by 10 independent runs as shown in Figure 5. Similar to Figure 2, compared with the traditional version, as the increasing of α, cooperation can be promoted effectively. These results indicate
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that the updating rule of inhomogeneous inertia is robust for sustaining and promoting cooperation.
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4. Conclusion In summary, we have studied the impact of the inhomogeneous inertia on the evolution of cooperation in the spatial prisoner’s dilemma game. In the model, the probability of changing strategy for some individuals depends on both the payoff difference and the inertia level α, and the impact makes the
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players heterogeneous and the network inhomogeneous. Compared with the traditional version (namely α = 0 or u = 0), the introduction of the new simple mechanism can improve cooperation remarkably for different sizes of neighbors. In addition, to validate the generality of this mechanism, we also explore the evolution of cooperation in ER graphs and the results are consistent with the
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results on the square lattice. In reality, individual behaviors may be guided in a certain direction by inertia just as that in the article. Our results indicate that the inertia factor has a positive influence on the evolution of cooperation, which may be helpful to understand the emergency and maintenance of cooperation.
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Highlights
Inhomogeneous inertia promotes cooperation in network population. (2) Inertia factor can facilitate the evolution of cooperation on different topologies. (3) Inhomogeneous inertia can enhance spatial reciprocity. (1)