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Multigames with social punishment and the evolution of cooperation
Accepted Manuscript Multigames with social punishment and the evolution of cooperation Zheng-Hong Deng, Yi-Jie Huang, Zhi-Yang Gu, Li-Gao
PII: DOI: Reference:
S0378-4371(18)30382-0 https://doi.org/10.1016/j.physa.2018.03.054 PHYSA 19388
To appear in:
Physica A
Received date : 31 October 2017 Revised date : 1 March 2018 Please cite this article as: Z.-H. Deng, Y. Huang, Z. Gu, Li-Gao Gu, Multigames with social punishment and the evolution of cooperation, Physica A (2018), https://doi.org/10.1016/j.physa.2018.03.054 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.
*Highlights (for review)
Highlights 1. We introduce multigames with social punishment and investigate the evolution of cooperation. 2. The population is randomly divided into two types. Players of type A choose to play the Prisoner’s Dilemma, while players of type B, select to play Snowdrift. Meanwhile, three strategies can be selected, including cooperation, defection, and punishment. 3. For the larger temptation to defect, the cooperation can be enhanced by the diversity of sucker's payoff.
*Manuscript Click here to view linked References
Multigames with social punishment and the evolution of cooperation Zheng-Hong Denga,*,Yi-Jie Huanga, Zhi-Yang Gub, Li-Gaoc a
School of Automation, Northwestern Polytechnical University, Xi’an 710072, China
b
Wenzhou Vocational&Technical College, Wenzhou 325035 ,China
c
School of Computer Science, Northwestern Polytechnical University, Xi’an 710072, China
ABSTRACT: Both the strategy social punishment defined as the special cooperation and playing multigames could lead to the enhancement of cooperation in social dilemmas. To take this into consideration, we introduce multigames with social punishment and investigate the evolution of cooperation. In our work, the population is randomly divided into two types. Players of type A, whose proportion is v , adopt a negative value of the sucker’s payoff to play the Prisoner’s Dilemma (PD), while players of type B, whose proportion is 1 v , adopt a positive value to play Snowdrift (SD). Meanwhile, three strategies can be selected, including cooperation, defection, and punishment. We show that for the larger temptation to defect, the cooperation can be enhanced by the diversity of sucker's payoff. In addition, when the contribution of sucker's payoff is larger or less players choose to play PD, the cooperators become more dominated, which can be explained by the special spatial distribution of the two types of players. Key words: social punishment; multigames; cooperation; two types; three strategies.
Corresponding to Email: [email protected]
1. Introduction The evolution of cooperation has been investigated in many domains such as biology systems, economic and social systems in the last few decades [1-4], which is usually not reconciled to the Darwin’s theory of natural selection [5]. And researching this issue has been a significant challenge for mankind for a long time. In order to study the mechanism of the emergence of cooperation effectively, the evolutionary game theory [6,7] as a powerful theory framework has been used in the analysis of diverse dilemmas. Based on this, more and more game models have been regarded as good metaphors of the real biological, human and economic behaviors, among which none has received as much attention as the Prisoner’s Dilemma (PD) [8] and the Snowdrift (SD) [9]. The Prisoner’s Dilemma game is usually regarded as the classical interpretation of conflict between the selfish individuals and the collective interests. Usually, the selfish players [10-16] going after short-term individual benefits, to a certain degree, might give rise to the tragedy of the commons [17]. In PD, two individuals have to simultaneously decide whether they want to cooperate (C) or defect (D). They both receive the reward R for mutual cooperation and the punishment P for mutual defection. And when confronting a defector, the cooperator will obtain the suck’s payoff S , while the defector will get the temptation T confronting a cooperator. The payoff ranking is set as T R P S with 2R T S so that defection is the optimal strategy to choose regardless of the opponent’s strategies in a finite well-mixed population, which always leads to the extinction of cooperation. In SD, the individuals interact in a similar way with the payoff ranking as T R S P , which results in the significant scenario where both the cooperation and defection will exist in the well-mixed population. In the last few years, there has been so many mechanisms to enhance the cooperation, which can be expressed as fellows: kin selection [18], “tit for tit” strategy [19], “win stay and lose shift” [8], direct and indirect reciprocity [20], group selection [21], as well as other special factors [22,23]. As is well known, the spatial reciprocity [24-26] is a mechanism by which cooperators can form the clusters that prevent the invasion of defectors to survive, which further enlightened such a few other mechanisms to improve the cooperation on the same structures. For example, high values of the clustering coefficient [27], different rewiring mechanisms [28,29] and social diversity [30-34]. However, it rarely contains the case of social punishment, where cooperators can spend some resources punishing defectors for their free-rider behavior to improve the cooperation [35-44]. So it will be of great significance to investigate the evolution of cooperation of PD and SD regarding the social punishment (Pu) as the independent strategy. In real life, it’s also worth mentioning that the individuals always simultaneously encounter more than one social dilemmas, which results in the birth of the mechanism of multigames [45,46]. And a few scholars have demonstrated that the cooperation can be enhanced by playing multigames [47-49]. In accordance with these, both social punishment and multigames could induce the improvement of cooperation, if we combine the two mechanisms, how will the cooperation evolve?
Now, in our paper, we will study multigames with social punishment and the evolution of cooperation on the square lattice [50]. In detail, the population is divided into two types, denoted by type A and type B. Players of type A, whose proportion is v , adopt a negative value of the sucker’s payoff to play the Prisoner’s Dilemma (PD), while players of type B, whose proportion is 1 v , adopt a positive value to play Snowdrift (SD). Meanwhile, three strategies can be selected, including cooperation (C), defection (D), and punishment (Pu). The rest of this paper is organized as follows: firstly, we proposed our model of multigames; subsequently, the main simulation results are shown and discussed in section 3; lastly we summarize our conclusions in section 4. 2. Methods In the model, the third strategy social punishment (Pu) defined as the special cooperation is appended to the PD and SD. When encountering the cooperator or the punisher itself, the punisher acts as the cooperator. When playing against the defector, the punisher pays a small cost and the defector gets the fine . And in the table I dose list the payoff matrix of PD and SD. For simplicity and not loss of the generality, In PD the payoffs can be rescaled as R 1 , P 0 , T b , S . In SD the payoffs can be rescaled as R 1 , P 0 , T b , S . As proposed in literature [35], even a smaller fine at a less cost could well enhance the cooperation, the values can be set as 0.3 and 0.1 . Furthermore, the population is randomly divided into two types denoted by type A and type B. While type A of players select to play the PD, type B of players select to play SD. The proportion of the two types are denoted by v (type A) and 1 v (type B). In our simulation, the individuals are located on the square lattice with periodic boundary conditions. Each player interacts with its four nearest neighbors. Initially, the three strategies of cooperation (C), defection (D) and punishment (Pu) are randomly distributed in the individuals with equal probability. Then, in each Monte Carlo step (MCS), the player at site x is randomly selected and plays games with all its neighbors to get the utility U x of player x . And player y is randomly selected from the one neighbor of player x and gets the utility U y in the same way as player
x . At last, player x adopts the strategy S y from player y with the transition probability determined by the Fermi function [50]:
W sx s y
1
1 exp U x U y K
, (1)
Where K 0.1 quantifies the uncertainty during the process of the strategy transition [51]. The selected value could ensure that the strategy of better-performing player will be adopted, although there is also the rare exception that the strategy of
worse-performing player will be adopted. Each full MCS makes every player change its strategy once on average. All simulation results are obtained on the square lattice with N 4 104 players. Allowing for social punisher as the special cooperator, taking c , c , d , p as the fraction of cooperators and publishers, cooperators, defectors, punishers respectively, and all of the values are determined in the stationary state with 3 105 MCS. To further improve accuracy, the final results are averaged over the last 3 10 4 independent runs. 3. Results and discussion Now let’s begin to discuss our simulation results by observing the influence of parameter on the evolution of cooperation. In Fig. 1, we display the total fraction of cooperators and punishers c in dependence on the temptation to defect b for
v 0.5 and different values of parameter . When 0 , it returns to the weak PD where no sucker’s payoff is involved, and the cooperation starts to decrease as about b 1.2 and become zero as about b 1.28 . However, it can be observed that the positive could improve the cooperation for the larger values of b . The larger the magnitude of sucker’s payoff, the more cooperators exist. Furthermore, even a relatively smaller parameter could ensure the cooperators survive for a larger values of b . In a word, the results show that the diversity of sucker’s payoff has a positive effective on the cooperation for the larger values of temptation to defect b . To further understand the influence of both the parameter and v on the evolution of cooperation. In Fig. 2, we present color map encoding fraction of cooperation c on the parameter v plane. At first glance, we can find that the cooperators will dominate the strategies for the about value of 0.5 0.95 and 0 v 0.4 . Then, after the closer inspection to Fig. 2, an interesting phenomenon appears. When the contribution of sucker’s payoff is larger, or less players playing the PD, the cooperators become more dominated. These results suggest that when SD acts as more important role or larger magnitude of sucker’s payoff is contained, we can observe more impressive facilitating effective. Knowing the fact that less players of type A and larger magnitude of the sucker’s payoff both favor the cooperation, it is necessary to further inspect the spatial distribution of these strategies. Therefore, Fig.3 depicts characteristic snapshots of strategy distributions in multigames, in which cooperators, defectors and punishers are colored red, green and blue respectively. From (a) to (d), what first attracts us is that very small is not able to avoid the extinction of cooperators, leading to the dominance of defectors. Then with the increment of , we can find that punishers gradually begin to prevail by forming more and more compact clusters to protect themselves from exploitation by defectors. Interestingly, when is large enough, the cooperators survive maybe resulting from the fact that clusters of punishers are
compact and large enough to leave less space for defectors. From (e) to (h), it’s evident that there are no cooperators, because the ability of punishers to protect themselves from exploitation by defectors is not strong enough. Also with the increment of v , the clusters of punishers are more and more less and sparser. That is to say, less players playing the PD could improve the cooperation. To further shed light on how the mechanism reacts on the series of strategy evolution, in Fig. 4, the time evolution of cooperation and average payoff in multigames is subsequently carried out. In Fig. 4 (a), at first glance, we can observe that the smaller is not able to prevent the cooperation from extinction at the stationary state, conversely, the higher could lead to cooperation survival, which is due to the distribution of the three strategies. Then in a more specific way, we can find that there is a downward trend at early stages for each value of , that is to say, defection is much better than cooperation for pursuing the higher individual payoff as illustrated in Fig. 4 (b), and the smaller , the faster decline of cooperation. Moreover, when the is large enough, the cooperation will recover from the valley bottom where the individual payoff is the lowest, because some of the cooperators turn into the punishers to form large and compact clusters that prevent the invasion of the defectors and even attract some defectors to change into the punishers for obtaining the higher individual payoff. Based on the fact that there are two types of individuals involved in the games, it’s necessary to explore how each type of players react on the cooperation respectively. In FIG. 5, it depicts each type’s fraction of cooperation (cooperators and punishers) ct in dependence on the temptation to defect b . Although the proportion of the two types is equal, from (a) to (c), the fractions of cooperation of type B are all above the type A. It can be concluded that individuals of type B play a more significant role than the type A to improve the total cooperation. In order to understand how the two types of individuals act on the evolution of cooperation, it is also necessary to further inspect the spatial distribution of the two types of strategies. In Fig.6, evolution snapshots of these strategies are expressed in multigames on the square lattice. Cooperators, defectors and punishers of type A are colored red, green and black respectively. Also cooperators, defectors and punishers of type B are colored blue, yellow and white respectively. From (a) to (f), it can be observed that the clusters formed by punishers of type A and B become more and more compact and larger, while the clusters of cooperators become more and more smaller and scattered, which directly demonstrates that part of the clusters of punishers are transformed from the clusters of cooperators. Except for these, we can also find that punishers of type B are located at the centers of clusters, while punishers of type A are distributed on the edges of the clusters. Therefore, we can draw a conclusion that the punishers of type B play a crucial role in the formation of large clusters, which can attract the punishers of type A surrounding them so as to be robust enough to prevent the invasion of defectors eventually resulting in the promotion of cooperation of the whole population. 4. Conclusion
We have investigated multigames with social punishment and the evolution of cooperation. First of all, the population is randomly divided into two types. Players of type A, whose proportion is v , adopt a negative value of the sucker’s payoff to play the Prisoner’s Dilemma (PD), while players of type B, whose proportion is 1 v , adopt a positive value to play Snowdrift (SD). Meanwhile, three strategies can be selected, including cooperation, defection, and punishment. Then through the numerical simulations, we can find that the diversity of sucker’s payoff has a positive effective on the cooperation for the larger values of temptation to defect. What’s more, when the contribution of sucker’s payoff is larger, or less players playing the PD, the cooperators become more dominated, which can be explained by punishers forming more and more compact clusters to improve the cooperation. In addition, we also find that individuals of type B play a more significant role than the type A to improve total cooperation resulting from the special spatial distribution of the two types of players. At last, we hope our work could motivate further research on multigames in the field of some human behavior [52-58]. Acknowledgment This work is partly supported by the National Natural Science Foundation of China Grant no. 61471299. References [1] J.E. Bone, B. Wallace, R. Bshary, N.J. Raihani, Power asymmetries and punishment in a prisoner’s dilemma with variable cooperative investment, PLoS One 11 (5) (2016) e0155773. [2] X.Y. Deng, Q. Zhang, Y. Deng, Z. Wang, A novel framework of classical and quantum prisoner's dilemma games on coupled networks, Sci. Rep. 6 (2016) 23024. [3] S.J. Gould, Darwinism and the expansion of evolutionary theory, Science 216 (4544) (1982) 380-387. [4] Z. Wang, C.T. Bauch, S. Bhattacharyya, et al., Statistical physics of vaccination, Phys. Rep. 664 (2016) 1-113. [5] N. Johnson, T. Lux, Financial systems: Ecology and economics, Nature 469 (7330) (2011) 302-303. [6] R.C. Lewontin R C. Evolution and the theory of games, J. Theoret. Biol. 1 (3) (1961) 382-403. [7] J. Hofbauer, K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. [8] M. Nowak, K. Sigmund, A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner's Dilemma game, Nature 364 (6432) (1993) 56-58. [9] C. Hauert, M. Doebeli, Spatial structure often inhibits the evolution of cooperation in the snowdrift game, Nature 428 (6983) (2004) 643-646. [10] G.Q. Zhang, T.P. Hu, Z. Yu. An improved fitness evaluation mechanism with noise in prisoner’s dilemma game, Appl. Math. Comput. 276 (2016) 31-36. [11] Z. Wang, A. Szolnoki, M. Perc, Evolution of public cooperation on interdependent networks: The impact of biased utility functions, Europhys. Lett. 97 (4) (2012) 48001.
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C
D
Pu
C
R
S
R
D
T
P
T
Pu
R
S
R
Table I. payoff matrix of the studied evolutionay game. The three strategies are coperation (C), defection (D) and punishment (Pu). Here, reprents the cost of punishment for punishers and
reprents the fine of defectors.
FIG. 1. (Color online). Total fraction of cooperators and punishers c in dependence on the temptation to defect b in multigames on the square lattice for v 0.5 and different values of parameter , as indicated in the legend. v 0.5 means that half of the individuals play the PD, the remaining half play SD.
FIG. 2. Color map encoding the faction of cooperation (cooperators and punishers)
c on the v parameter plane in multigames on the square lattice for value of the temptation to defect b 1.44 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
FIG. 3. Characteristic snapshots of strategy distributions in multigames on the square lattice. From (a) to (d) the value of v is 0.5, and the values of are 0.2, 0.3, 0.6,
0.8. From (e) to (h) the value of is 0.5, and the values of v are 0.2, 0.3, 0.6, 0.8. Cooperators, defectors and punishers are colored red, green and blue respectively. All the results are obtained for b 1.44 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
FIG. 4. (Color online). Time evolution of cooperation and average payoff in multigames on the square lattice. (a) depicts the fraction of cooperation (cooperators and punishers) c in dependence on the Monte Carlo step. (b) depicts the average payoff P in dependence on the Monte Carlo step. All the results are obtained for v 0.5 , b 1.44 and different values of , as indicated in the legend.
FIG. 5. (Color online). Evolution of cooperation for individuals playing PD (type A) and SD (type B) respectively in multigame on the square lattice. Each type’s fraction of cooperation (cooperators and punishers) ct in dependence on the temptation to defect b . ca represents the individuals (type A) playing PD, cb represents the individuals (type B) playing SD, as indicated in the legend. From (a) to (c) the value of v is 0.5 and the values of are 0.3, 0.6, 0.8 respectively.
FIG. 6. Evolution snapshots of cooperators of type A (red), defectors of type A (green), punishers of type A (black), cooperators of type B (blue), defectors of type B (yellow), punishers of type B (white) in multigames on the square lattice. From (a) to (f) the values of steps are 0, 10, 10000, 50000, 100000 and 300000. All the results are obtained for b 1.44 , v 0.5 and 0.8 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
PII: DOI: Reference:
S0378-4371(18)30382-0 https://doi.org/10.1016/j.physa.2018.03.054 PHYSA 19388
To appear in:
Physica A
Received date : 31 October 2017 Revised date : 1 March 2018 Please cite this article as: Z.-H. Deng, Y. Huang, Z. Gu, Li-Gao Gu, Multigames with social punishment and the evolution of cooperation, Physica A (2018), https://doi.org/10.1016/j.physa.2018.03.054 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.
*Highlights (for review)
Highlights 1. We introduce multigames with social punishment and investigate the evolution of cooperation. 2. The population is randomly divided into two types. Players of type A choose to play the Prisoner’s Dilemma, while players of type B, select to play Snowdrift. Meanwhile, three strategies can be selected, including cooperation, defection, and punishment. 3. For the larger temptation to defect, the cooperation can be enhanced by the diversity of sucker's payoff.
*Manuscript Click here to view linked References
Multigames with social punishment and the evolution of cooperation Zheng-Hong Denga,*,Yi-Jie Huanga, Zhi-Yang Gub, Li-Gaoc a
School of Automation, Northwestern Polytechnical University, Xi’an 710072, China
b
Wenzhou Vocational&Technical College, Wenzhou 325035 ,China
c
School of Computer Science, Northwestern Polytechnical University, Xi’an 710072, China
ABSTRACT: Both the strategy social punishment defined as the special cooperation and playing multigames could lead to the enhancement of cooperation in social dilemmas. To take this into consideration, we introduce multigames with social punishment and investigate the evolution of cooperation. In our work, the population is randomly divided into two types. Players of type A, whose proportion is v , adopt a negative value of the sucker’s payoff to play the Prisoner’s Dilemma (PD), while players of type B, whose proportion is 1 v , adopt a positive value to play Snowdrift (SD). Meanwhile, three strategies can be selected, including cooperation, defection, and punishment. We show that for the larger temptation to defect, the cooperation can be enhanced by the diversity of sucker's payoff. In addition, when the contribution of sucker's payoff is larger or less players choose to play PD, the cooperators become more dominated, which can be explained by the special spatial distribution of the two types of players. Key words: social punishment; multigames; cooperation; two types; three strategies.
Corresponding to Email: [email protected]
1. Introduction The evolution of cooperation has been investigated in many domains such as biology systems, economic and social systems in the last few decades [1-4], which is usually not reconciled to the Darwin’s theory of natural selection [5]. And researching this issue has been a significant challenge for mankind for a long time. In order to study the mechanism of the emergence of cooperation effectively, the evolutionary game theory [6,7] as a powerful theory framework has been used in the analysis of diverse dilemmas. Based on this, more and more game models have been regarded as good metaphors of the real biological, human and economic behaviors, among which none has received as much attention as the Prisoner’s Dilemma (PD) [8] and the Snowdrift (SD) [9]. The Prisoner’s Dilemma game is usually regarded as the classical interpretation of conflict between the selfish individuals and the collective interests. Usually, the selfish players [10-16] going after short-term individual benefits, to a certain degree, might give rise to the tragedy of the commons [17]. In PD, two individuals have to simultaneously decide whether they want to cooperate (C) or defect (D). They both receive the reward R for mutual cooperation and the punishment P for mutual defection. And when confronting a defector, the cooperator will obtain the suck’s payoff S , while the defector will get the temptation T confronting a cooperator. The payoff ranking is set as T R P S with 2R T S so that defection is the optimal strategy to choose regardless of the opponent’s strategies in a finite well-mixed population, which always leads to the extinction of cooperation. In SD, the individuals interact in a similar way with the payoff ranking as T R S P , which results in the significant scenario where both the cooperation and defection will exist in the well-mixed population. In the last few years, there has been so many mechanisms to enhance the cooperation, which can be expressed as fellows: kin selection [18], “tit for tit” strategy [19], “win stay and lose shift” [8], direct and indirect reciprocity [20], group selection [21], as well as other special factors [22,23]. As is well known, the spatial reciprocity [24-26] is a mechanism by which cooperators can form the clusters that prevent the invasion of defectors to survive, which further enlightened such a few other mechanisms to improve the cooperation on the same structures. For example, high values of the clustering coefficient [27], different rewiring mechanisms [28,29] and social diversity [30-34]. However, it rarely contains the case of social punishment, where cooperators can spend some resources punishing defectors for their free-rider behavior to improve the cooperation [35-44]. So it will be of great significance to investigate the evolution of cooperation of PD and SD regarding the social punishment (Pu) as the independent strategy. In real life, it’s also worth mentioning that the individuals always simultaneously encounter more than one social dilemmas, which results in the birth of the mechanism of multigames [45,46]. And a few scholars have demonstrated that the cooperation can be enhanced by playing multigames [47-49]. In accordance with these, both social punishment and multigames could induce the improvement of cooperation, if we combine the two mechanisms, how will the cooperation evolve?
Now, in our paper, we will study multigames with social punishment and the evolution of cooperation on the square lattice [50]. In detail, the population is divided into two types, denoted by type A and type B. Players of type A, whose proportion is v , adopt a negative value of the sucker’s payoff to play the Prisoner’s Dilemma (PD), while players of type B, whose proportion is 1 v , adopt a positive value to play Snowdrift (SD). Meanwhile, three strategies can be selected, including cooperation (C), defection (D), and punishment (Pu). The rest of this paper is organized as follows: firstly, we proposed our model of multigames; subsequently, the main simulation results are shown and discussed in section 3; lastly we summarize our conclusions in section 4. 2. Methods In the model, the third strategy social punishment (Pu) defined as the special cooperation is appended to the PD and SD. When encountering the cooperator or the punisher itself, the punisher acts as the cooperator. When playing against the defector, the punisher pays a small cost and the defector gets the fine . And in the table I dose list the payoff matrix of PD and SD. For simplicity and not loss of the generality, In PD the payoffs can be rescaled as R 1 , P 0 , T b , S . In SD the payoffs can be rescaled as R 1 , P 0 , T b , S . As proposed in literature [35], even a smaller fine at a less cost could well enhance the cooperation, the values can be set as 0.3 and 0.1 . Furthermore, the population is randomly divided into two types denoted by type A and type B. While type A of players select to play the PD, type B of players select to play SD. The proportion of the two types are denoted by v (type A) and 1 v (type B). In our simulation, the individuals are located on the square lattice with periodic boundary conditions. Each player interacts with its four nearest neighbors. Initially, the three strategies of cooperation (C), defection (D) and punishment (Pu) are randomly distributed in the individuals with equal probability. Then, in each Monte Carlo step (MCS), the player at site x is randomly selected and plays games with all its neighbors to get the utility U x of player x . And player y is randomly selected from the one neighbor of player x and gets the utility U y in the same way as player
x . At last, player x adopts the strategy S y from player y with the transition probability determined by the Fermi function [50]:
W sx s y
1
1 exp U x U y K
, (1)
Where K 0.1 quantifies the uncertainty during the process of the strategy transition [51]. The selected value could ensure that the strategy of better-performing player will be adopted, although there is also the rare exception that the strategy of
worse-performing player will be adopted. Each full MCS makes every player change its strategy once on average. All simulation results are obtained on the square lattice with N 4 104 players. Allowing for social punisher as the special cooperator, taking c , c , d , p as the fraction of cooperators and publishers, cooperators, defectors, punishers respectively, and all of the values are determined in the stationary state with 3 105 MCS. To further improve accuracy, the final results are averaged over the last 3 10 4 independent runs. 3. Results and discussion Now let’s begin to discuss our simulation results by observing the influence of parameter on the evolution of cooperation. In Fig. 1, we display the total fraction of cooperators and punishers c in dependence on the temptation to defect b for
v 0.5 and different values of parameter . When 0 , it returns to the weak PD where no sucker’s payoff is involved, and the cooperation starts to decrease as about b 1.2 and become zero as about b 1.28 . However, it can be observed that the positive could improve the cooperation for the larger values of b . The larger the magnitude of sucker’s payoff, the more cooperators exist. Furthermore, even a relatively smaller parameter could ensure the cooperators survive for a larger values of b . In a word, the results show that the diversity of sucker’s payoff has a positive effective on the cooperation for the larger values of temptation to defect b . To further understand the influence of both the parameter and v on the evolution of cooperation. In Fig. 2, we present color map encoding fraction of cooperation c on the parameter v plane. At first glance, we can find that the cooperators will dominate the strategies for the about value of 0.5 0.95 and 0 v 0.4 . Then, after the closer inspection to Fig. 2, an interesting phenomenon appears. When the contribution of sucker’s payoff is larger, or less players playing the PD, the cooperators become more dominated. These results suggest that when SD acts as more important role or larger magnitude of sucker’s payoff is contained, we can observe more impressive facilitating effective. Knowing the fact that less players of type A and larger magnitude of the sucker’s payoff both favor the cooperation, it is necessary to further inspect the spatial distribution of these strategies. Therefore, Fig.3 depicts characteristic snapshots of strategy distributions in multigames, in which cooperators, defectors and punishers are colored red, green and blue respectively. From (a) to (d), what first attracts us is that very small is not able to avoid the extinction of cooperators, leading to the dominance of defectors. Then with the increment of , we can find that punishers gradually begin to prevail by forming more and more compact clusters to protect themselves from exploitation by defectors. Interestingly, when is large enough, the cooperators survive maybe resulting from the fact that clusters of punishers are
compact and large enough to leave less space for defectors. From (e) to (h), it’s evident that there are no cooperators, because the ability of punishers to protect themselves from exploitation by defectors is not strong enough. Also with the increment of v , the clusters of punishers are more and more less and sparser. That is to say, less players playing the PD could improve the cooperation. To further shed light on how the mechanism reacts on the series of strategy evolution, in Fig. 4, the time evolution of cooperation and average payoff in multigames is subsequently carried out. In Fig. 4 (a), at first glance, we can observe that the smaller is not able to prevent the cooperation from extinction at the stationary state, conversely, the higher could lead to cooperation survival, which is due to the distribution of the three strategies. Then in a more specific way, we can find that there is a downward trend at early stages for each value of , that is to say, defection is much better than cooperation for pursuing the higher individual payoff as illustrated in Fig. 4 (b), and the smaller , the faster decline of cooperation. Moreover, when the is large enough, the cooperation will recover from the valley bottom where the individual payoff is the lowest, because some of the cooperators turn into the punishers to form large and compact clusters that prevent the invasion of the defectors and even attract some defectors to change into the punishers for obtaining the higher individual payoff. Based on the fact that there are two types of individuals involved in the games, it’s necessary to explore how each type of players react on the cooperation respectively. In FIG. 5, it depicts each type’s fraction of cooperation (cooperators and punishers) ct in dependence on the temptation to defect b . Although the proportion of the two types is equal, from (a) to (c), the fractions of cooperation of type B are all above the type A. It can be concluded that individuals of type B play a more significant role than the type A to improve the total cooperation. In order to understand how the two types of individuals act on the evolution of cooperation, it is also necessary to further inspect the spatial distribution of the two types of strategies. In Fig.6, evolution snapshots of these strategies are expressed in multigames on the square lattice. Cooperators, defectors and punishers of type A are colored red, green and black respectively. Also cooperators, defectors and punishers of type B are colored blue, yellow and white respectively. From (a) to (f), it can be observed that the clusters formed by punishers of type A and B become more and more compact and larger, while the clusters of cooperators become more and more smaller and scattered, which directly demonstrates that part of the clusters of punishers are transformed from the clusters of cooperators. Except for these, we can also find that punishers of type B are located at the centers of clusters, while punishers of type A are distributed on the edges of the clusters. Therefore, we can draw a conclusion that the punishers of type B play a crucial role in the formation of large clusters, which can attract the punishers of type A surrounding them so as to be robust enough to prevent the invasion of defectors eventually resulting in the promotion of cooperation of the whole population. 4. Conclusion
We have investigated multigames with social punishment and the evolution of cooperation. First of all, the population is randomly divided into two types. Players of type A, whose proportion is v , adopt a negative value of the sucker’s payoff to play the Prisoner’s Dilemma (PD), while players of type B, whose proportion is 1 v , adopt a positive value to play Snowdrift (SD). Meanwhile, three strategies can be selected, including cooperation, defection, and punishment. Then through the numerical simulations, we can find that the diversity of sucker’s payoff has a positive effective on the cooperation for the larger values of temptation to defect. What’s more, when the contribution of sucker’s payoff is larger, or less players playing the PD, the cooperators become more dominated, which can be explained by punishers forming more and more compact clusters to improve the cooperation. In addition, we also find that individuals of type B play a more significant role than the type A to improve total cooperation resulting from the special spatial distribution of the two types of players. At last, we hope our work could motivate further research on multigames in the field of some human behavior [52-58]. Acknowledgment This work is partly supported by the National Natural Science Foundation of China Grant no. 61471299. References [1] J.E. Bone, B. Wallace, R. Bshary, N.J. Raihani, Power asymmetries and punishment in a prisoner’s dilemma with variable cooperative investment, PLoS One 11 (5) (2016) e0155773. [2] X.Y. Deng, Q. Zhang, Y. Deng, Z. Wang, A novel framework of classical and quantum prisoner's dilemma games on coupled networks, Sci. Rep. 6 (2016) 23024. [3] S.J. Gould, Darwinism and the expansion of evolutionary theory, Science 216 (4544) (1982) 380-387. [4] Z. Wang, C.T. Bauch, S. Bhattacharyya, et al., Statistical physics of vaccination, Phys. Rep. 664 (2016) 1-113. [5] N. Johnson, T. Lux, Financial systems: Ecology and economics, Nature 469 (7330) (2011) 302-303. [6] R.C. Lewontin R C. Evolution and the theory of games, J. Theoret. Biol. 1 (3) (1961) 382-403. [7] J. Hofbauer, K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. [8] M. Nowak, K. Sigmund, A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner's Dilemma game, Nature 364 (6432) (1993) 56-58. [9] C. Hauert, M. Doebeli, Spatial structure often inhibits the evolution of cooperation in the snowdrift game, Nature 428 (6983) (2004) 643-646. [10] G.Q. Zhang, T.P. Hu, Z. Yu. An improved fitness evaluation mechanism with noise in prisoner’s dilemma game, Appl. Math. Comput. 276 (2016) 31-36. [11] Z. Wang, A. Szolnoki, M. Perc, Evolution of public cooperation on interdependent networks: The impact of biased utility functions, Europhys. Lett. 97 (4) (2012) 48001.
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C
D
Pu
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S
R
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T
P
T
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R
S
R
Table I. payoff matrix of the studied evolutionay game. The three strategies are coperation (C), defection (D) and punishment (Pu). Here, reprents the cost of punishment for punishers and
reprents the fine of defectors.
FIG. 1. (Color online). Total fraction of cooperators and punishers c in dependence on the temptation to defect b in multigames on the square lattice for v 0.5 and different values of parameter , as indicated in the legend. v 0.5 means that half of the individuals play the PD, the remaining half play SD.
FIG. 2. Color map encoding the faction of cooperation (cooperators and punishers)
c on the v parameter plane in multigames on the square lattice for value of the temptation to defect b 1.44 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
FIG. 3. Characteristic snapshots of strategy distributions in multigames on the square lattice. From (a) to (d) the value of v is 0.5, and the values of are 0.2, 0.3, 0.6,
0.8. From (e) to (h) the value of is 0.5, and the values of v are 0.2, 0.3, 0.6, 0.8. Cooperators, defectors and punishers are colored red, green and blue respectively. All the results are obtained for b 1.44 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
FIG. 4. (Color online). Time evolution of cooperation and average payoff in multigames on the square lattice. (a) depicts the fraction of cooperation (cooperators and punishers) c in dependence on the Monte Carlo step. (b) depicts the average payoff P in dependence on the Monte Carlo step. All the results are obtained for v 0.5 , b 1.44 and different values of , as indicated in the legend.
FIG. 5. (Color online). Evolution of cooperation for individuals playing PD (type A) and SD (type B) respectively in multigame on the square lattice. Each type’s fraction of cooperation (cooperators and punishers) ct in dependence on the temptation to defect b . ca represents the individuals (type A) playing PD, cb represents the individuals (type B) playing SD, as indicated in the legend. From (a) to (c) the value of v is 0.5 and the values of are 0.3, 0.6, 0.8 respectively.
FIG. 6. Evolution snapshots of cooperators of type A (red), defectors of type A (green), punishers of type A (black), cooperators of type B (blue), defectors of type B (yellow), punishers of type B (white) in multigames on the square lattice. From (a) to (f) the values of steps are 0, 10, 10000, 50000, 100000 and 300000. All the results are obtained for b 1.44 , v 0.5 and 0.8 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)