Evolutionary Cooperation in Networked Public Goods Game with Dependency Groups

Zhu, Xuzhen; Su, Xin; Ma, Jinming; Tian, Hui; Liu, RunRan

doi:https://doi.org/10.1155/2019/8670802

Complexity

On this page

Abstract Introduction Model Analysis Discussion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 8670802 | https://doi.org/10.1155/2019/8670802

Evolutionary Cooperation in Networked Public Goods Game with Dependency Groups

Xuzhen Zhu,¹Xin Su,²Jinming Ma,¹Hui Tian,¹and RunRan Liu³

Academic Editor: Guido Caldarelli

Received31 May 2019

Revised25 Aug 2019

Accepted13 Sept 2019

Published13 Oct 2019

Abstract

Either in microlevel organizations or macrolevel societies, the individuals acquire benefits or payoffs by forming interdependency groups linked by common interests. Conducting research on the effects of interdependency groups on the evolution of cooperation could have a better understanding of the social dilemma problem. In this paper, we studied a spatial public goods game with nonlocal interdependency groups where each of participants is located in a two-dimensional square lattice or Watts–Strogatz small-world network with payoffs obtaining from the interactions with nearest neighbors. In terms of the enhancement factor, the effects of group density on the evolutionary cooperation can be quite different. For a low enhancement factor, the cooperation level is a nonmonotonic function with the varying density of interdependency groups in the system, which means a proper density of interdependency groups can best promote the cooperative level. For a moderate enhancement factor, a higher density of interdependency groups can always correspond to a higher cooperative level. However, if the enhancement factor is too high, a high density of interdependency groups can impede the evolutionary cooperation. We give the explanations for the different roles of group density of interdependency by using the transition probabilities of C players into D players as well as the reverse. Our findings are very helpful for the understanding of emergence cooperation as well as the cooperation regulation in the selfish individuals.

1. Introduction

Cooperation is a ubiquitous phenomenon in biological societies, referring the process that individuals or groups working together for common or mutual benefits, such as marital relations and alliances. However, it is still a challenge to understand the emergence and maintenance of cooperation. Evolutionary game theory provides an abundant framework to address this issue and attract attention from many researchers, including physicists, biologists, and economists [1–3]. The results of this study can promote recognitions on the mechanism of cooperative emergence and imply that interdependency is an important mechanism to enhance cooperation. The prisoner’s dilemma game has been used to conduct extensive exploration on the characteristics of conflict among different groups with various interest preferences, and it can be used to explain cooperative behaviors among interactions between two-player groups [4, 5].

Taking a large number of interacting players into consideration, the same conflict is also able to be seized by the public goods game [6–11]. Based on the classical , N players are able to determine whether an amount c should be invested into a common pool (cooperate) or not (defect) concurrently and separately. Multiplied by an enhancement level r with , the obtained investment is able to be regarded as the groups production and management degree or the synergy impacts of collaboration. The obtained investment is then distributed evenly again among all participants, regardless of their factual strategy. Based on these processes, the whole group will obtain the most benefit when all players put all their property into the public pool. However, defectors do not have to take up any costs when they get the same benefits as the cooperators. Therefore, defection is frequently the first choice and seems to be the natural strategy to select for selfish players when .

Nowak and May have indicated that spatial structure is capable of promoting the progress of collaboration on the basis of the well-renowned model which represents network reciprocity [12]. Subsequently, based on the seminal work, evolutionary game has been extended by researchers to diverse topology structures, finding that it is possible to introduce agents’ heterogeneity in the form of heterogeneous interaction networks [13–21], which is also demonstrated to play a vital role in the smooth evolution of collaboration [15, 22–31]. Then, other core mechanisms responsible for stimulating cooperation, including migration [32], time-scale heterogeneity [33, 34], multilayered networks [35, 36] or groups [37, 38], aspirations [39–43], punishment [44], group intelligence [45], tolerance [46], and dynamic groupings [47], have been shown in a number of innovative studies. Besides, coevolutionary games, in which, the communications or several particular properties of players are also influenced by evolution, have helped us understand the occurrence of system properties facilitating the evolutionary collaboration more deeply [48].

What should be noted is that when it comes to Fermi function, Szabó and Tőke have introduced the noise in strategy adoption [49] representing players’ bounded rationality, finding that appropriate noise degree [50, 51] or players’ bounded rationality in strategy restriction [28] is able to promote collaboration. Besides, Perc has explored the effects imposed by stochastic payoff changes on the development of collaboration in the spatial , finding that Gaussian noise is able to promote collaboration resonantly, which is similar to classical coherence resonance [52]. Later, the framework has investigated the correlation between the “dynamical” coherence resonances induced by the payoff noise and the so-called “evolutionary” coherence resonance induced by the noise of strategy application [53]. Nevertheless, in terms of interactions by multiple players, it remains unclear in terms of the impacts imposed by the group-payoff variation on the evolutionary collaboration or the accurate mechanism.

A lot of attention has been paid for the evolutionary cooperation in terms of interdependency network reciprocity [35, 54–56], where players can acquire payoffs from the interactions in each layer of network or share information about strategic choices between players residing in different network layers. Besides, Zhang and Yang studied the effects of random partnerships on the evolution of cooperation in spatial game theory on a square lattice, in terms of [57]. Poncela et al. explored the effect of limiting the number of interactions that a node can establish per round of a prisoner’s dilemma game [58]. It has been shown that the diverse utilities of players maintain healthy public cooperativeness even in the face of adverse conditions [54], and for a proper interdependence level across network layers, the cooperation is promoted best [35]. Similarly, the information sharing across layers also reinforce the cooperation level significantly [55]. In real human or animal societies, individuals not only share information but also share payoffs or material well-being by the interdependent relationships (such as marriage or other connections with common interests) among them. In this case, the gains that individuals made in the social or economic activities are shared with the rest of the dependent group. Here, we explore the density of payoff-sharing groups on the evolution of cooperation by using the public goods game. We find that the density of payoff-sharing groups can play both positive and negative role for the evolutionary cooperation depending on the enhancement level of the game. We give explanations for the different roles of group density of interdependency by using the transition probabilities of C players into D players as well as the reverse.

Hereafter, we proceed to describe in detail the public goods game with payoff-sharing groups, followed by the presentation of the main results. We round off the discussion with concluding remarks.

2. Model

We applied the simplified version of the , in which, the core factors of the social plight are reserved, whose strength, however, is decided by the enhancement degree [9]. We take this simplified version of into consideration, in which, the player is in a two-dimensional lattice of four neighbors with periodic boundary conditions, with individual player and their nearest neighbor forming a group. At first, with equal likability of random choice, a collaborator or defector takes up each position. At every time step, the payoff of a focal player i is determined by its strategy and the quantity of partners in the neighborhood. Thus, the player i’s payoff is

is 1 if the player i is a collaborator. Otherwise, it is 0. In our study, in the convenience of simplicity, the investment c of a collaborator is set to 1, with the parameter r denoting the enhancement degree of the group. A defective strategy, compared with a collaborative strategy, will obtain more benefits when , characterizing the social dilemma. Nevertheless, for , it will be a better choice for all individuals to use a collaborative strategy.

Here, we introduce the payoff-sharing groups in the public goods game. We randomly choose a fraction p of players and match them one by one as paired interdependent groups, and therefore, the parameter p controls the density of payoff-sharing groups in the system. At each round of game, we reset the payoff of players in each interdependency group to the average of group payoff due to interdependencies, i.e., for two interdependent players i and j.

It is possible for every player to imitate the strategy of a freely selected neighbor in updating strategies, and a probability is determined by the payoff disparity synchronously, i.e., the probability for one player i to use the neighbor j’s strategy can be expressed aswhere κ features the bounded rationality of a person [49], indicating the unsureness or mistakes in the strategy application. (In the restrain, the stochastic impacts are able to be ignored and people are of excellent rationality, while for restrain, the stochasticity is the largest and people turn to be irrational comprehensively.)

It should be noted that the parameter p serves as a core parameter in this model, and we will systematically discuss its influence on evolutionary collaboration in the next section. We perform the simulations on both regular square lattice and Watts–Strogatz (WS) small-world network with rewiring probability 0.1. Simulations are conducted for a population of people. The core number of cooperator concentration is studied in the steady condition. Equilibrium frequencies of collaborators are acquired by more than 20,000 Monte Carlo time steps from 120,000 steps in total on average, with each data point outcome derived from more than 200 realizations on average.

3. Simulation and Analysis

Firstly, the evolution of cooperation affected by the density of payoff-sharing groups is studied. Clearly, it can be found in results indicated in the right panel of Figure 1 that the equilibrium frequencies as functions of the density of payoff-sharing groups p for diversified r when . No matter what value of p is, the cooperator concentration increases continuously from 0 to 1, whereby we can find that a larger value of p always corresponds to a higher cooperator concentration for some lower values of r, and a larger value of p can correspond to a lower cooperator concentration for some high values of r. This result implies that the density of payoff-sharing groups can play both positive and negative roles for the persistence of cooperation. In order to account for this phenomenon more clearly, we have plot the cooperator concentration as functions of p for different enhancement level of r in Figure 2. For some small values of r (i.e., ), the cooperator concentration versus p is a nonmonotonic function and there exists one optimal density of payoff-sharing groups that can best support the emergence of cooperation. For some moderate values of r, the cooperator concentration increases monotonously with the increase of p, and the cooperator concentration decreases monotonously with the increase of p for some high values of p. Similar results for small-world networks can be also found in Figure 2(b). All these results indicate that the effects of the density of payoff-sharing groups on the cooperation level depends on the enhancement level of the system for both regular square lattice and WS small-world networks.

(a)

(b)

(a)

(b)

We aim to explore the underlying mechanism of the nonmonotonic phenomenon of versus p for some small values of r. We studied the transition probability of C players into D players and the reverse probability as functions of p in Figure 3. For a lower value of p, it is benefit for the change of D players into C players and thus leads the emergence of cooperation; however, it also provides the possibilities for the C players changing into D players. Therefore, the system maintains a low level of cooperation when the number of D players that changed from C players and the number of C players that changed from D players reach equilibrium. With the increase of p, the transition probabilities and increase collectively, and the cooperation reaches its highest level when equals .

(a)

(b)

Figure 4 shows the evolution of cooperation frequency, the payoff advantage of cooperators (the payoff of cooperators minus the payoff of defectors), and the payoffs of players engaged in the interdependency groups is studied over time. In the early stages of evolution, cooperators and defectors are evenly distributed around each player, and defectors are highly profitable. Since the introduction of interdependency groups in the system, a defector and a cooperator can form an interdependent group and the cooperators can share some payoffs from their defective partners, which avoid the elimination of cooperators in the early stage of evaluation. After that, the cooperators that are connected together are able to earn a higher payoff than defectors. However the payoffs of defectors will be reduced gradually because of mutual betrayal, and the payoff advantage of cooperation will increase. A cooperator and a cooperator form an interdependent group with a probability , and a defector and a cooperator form an interdependency group with a probability . Therefore, most cooperators are exploited by defectors, and their returns are reduced when the cooperation level is increased. Cooperators who form dependent groups have stable returns. and are not stable in the evolution. Therefore, in the early stage of early evolution, the cooperation level is low and the interdependency group will lead to a relatively stable cooperative cluster formed by interdependent groups. Later on, interdependent group will reduce the stability of cooperative benefits and accelerate the collapse of cooperative clusters, which always has a negative impact on the formation of cooperative clusters. See Figures 4 and 5. This is the very reason why the cooperation frequency is not monotonous when r is small.

(a)

(b)

(c)

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

With the increase of r, such as closing to 5, the payoff gap between cooperators and defectors decreases and the cooperation becomes the dominant strategy in the system. In this case, interdependency groups have little influence on the stability of payoff collaborators. Thus, the early promotion of cooperation led the way. When the value of r is greater than 5, the best strategy in the system is cooperation, but the influence of the interdependency groups on the instability of the cooperator’s payoff still exists, but the impact is small.

To verify the results reported above for various densities of interdependency groups in the strategy application, we figure out depending on different p and r on regular square lattice and WS small-world network for a fixed in Figure 6. It is found that there exists an optimal value of p leading to the highest cooperation level for some small values of r. In addition, we can also find that the cooperation level can be increased continuously for medium values of r, indicating that the cooperation can be impacted by p in different ways depending on the specific value of r.

(a)

(b)

Furthermore, we investigate the influence of noise level κ on the cooperation level in Figure 7. In Figure 7, in spite of the minor differences in the specific curves, with the increase of the noise level κ, the cooperation level identically changes nonmonotonically in the convex pattern, with on both the regular square lattice (a) and the WS small-world network (c). Furthermore, the cooperation level versus the noise level κ decreases monotonically with on both the regular square lattice (b) and the WS small-world network (d). In addition, from Figure 7, we also can discover that the group density p of players influences the cooperation level prominently for and insignificantly for .

(a)

(b)

(c)

(d)

4. Discussion

In this paper, we have studied the effects of interdependency groups on the evolutionary collaboration in the public goods game. Based on the extensive simulations on regular square lattice and Watts–Strogatz (WS) small-world networks, we can find that the effects of interdependency groups on the evolutionary cooperation can be quite different for different values of enhancement levels. For a low enhancement factor, the cooperation level is a nonmonotonic function with the varying of density of interdependency groups in the system, which means a proper density of interdependency groups can best promote the cooperative level. For a moderate enhancement factor, a higher density of interdependency groups can always correspond to a higher cooperative level. However, if the enhancement factor is too high, a high density of interdependency groups can impede the evolutionary cooperation. We give explanations for the different roles of group density of interdependency by using the transition probabilities of C players into D players as well as the reverse. Our findings are very helpful for the understanding of emergence cooperation as well as the cooperation regulation in the selfish individuals.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Nos. 61602048 and 61773148).

References

S. J. Maynard, Evolution and the Theory of Games, Cambridge University Press, Cambridge, UK, 1982.
A. Macarthy, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.
H. Gintis, Game Theory Evolving, Princeton University Press, Princeton, NJ, USA, 2000.
R. Axelrod, The Evolution of Cooperation, Basic Books, New York, NY, USA, 1984.
G. Szabó and G. Fáth, “Evolutionary games on graphs,” Physics Reports, vol. 446, no. 4–6, pp. 97–216, 2007.
View at: Publisher Site | Google Scholar
G. Hardin, “The tragedy of the commons,” Science, vol. 162, pp. 1243–1248, 1968.
View at: Google Scholar
R. M. Isaac, K. F. McCue, and C. R. Plott, “Public goods provision in an experimental environment,” Journal of Public Economics, vol. 26, no. 1, pp. 51–74, 1985.
View at: Publisher Site | Google Scholar
C. Hauert, S. De Monte, J. Hofbauer, and K. Sigmund, “Volunteering as red queen mechanism for cooperation in public goods games,” Science, vol. 296, no. 5570, pp. 1129–1132, 2002.
View at: Publisher Site | Google Scholar
G. Szabó and C. Hauert, “Phase transitions and volunteering in spatial public goods games,” Physical Review Letters, vol. 89, Article ID 118101, 2002.
View at: Publisher Site | Google Scholar
A. Chaudhuri, “Sustaining cooperation in laboratory public goods experiments: a selective survey of the literature,” Experimental Economics, vol. 14, no. 1, pp. 47–83, 2011.
View at: Publisher Site | Google Scholar
M. Perc, J. Gómez-Gardeñes, A. Szolnoki, L. M. Floría, and Y. Moreno, “Evolutionary dynamics of group interactions on structured populations: a review,” Journal of the Royal Society Interface, vol. 10, no. 80, Article ID 20120997, 2013.
View at: Publisher Site | Google Scholar
M. A. Nowak and R. M. May, “Evolutionary games and spatial chaos,” Nature, vol. 359, no. 6398, pp. 826–829, 1992.
View at: Publisher Site | Google Scholar
F. C. Santos and J. M. Pacheco, “Scale-free networks provide a unifying framework for the emergence of cooperation,” Physical Review Letters, vol. 95, Article ID 098104, 2005.
View at: Publisher Site | Google Scholar
J. Gómez-Gardenes, M. Campillo, L. M. Floría, and Y. Moreno, “Dynamical organization of cooperation in complex topologies,” Physical Review Letters, vol. 98, Article ID 108103, 2007.
View at: Publisher Site | Google Scholar
F. C. Santos, M. D. Santos, and J. M. Pacheco, “Social diversity promotes the emergence of cooperation in public goods games,” Nature, vol. 454, no. 7201, pp. 213–216, 2008.
View at: Publisher Site | Google Scholar
Z.-G. Huang, Z.-X. Wu, A.-C. Wu, L. Yang, and Y.-H. Wang, “Role of collective influence in promoting cooperation,” Europhysics Letters, vol. 84, no. 5, p. 50008, 2008.
View at: Publisher Site | Google Scholar
H.-X. Yang, W.-X. Wang, Z.-X. Wu, Y.-C. Lai, and B.-H. Wang, “Diversity-optimized cooperation on complex networks,” Physical Review Letters, vol. 79, Article ID 056107, 2009.
View at: Publisher Site | Google Scholar
Z. Rong and Z.-X. Wu, “Effect of the degree correlation in public goods game on scale-free networks,” Europhysics Letters, vol. 87, no. 3, p. 30001, 2009.
View at: Publisher Site | Google Scholar
Z. Rong, H.-X. Yang, W.-X. Wang et al., “Feedback reciprocity mechanism promotes the cooperation of highly clustered scale-free networks,” Physical Review Letters, vol. 82, Article ID 047101, 2010.
View at: Publisher Site | Google Scholar
F. C. Santos, F. L. Pinheiro, T. Lenaerts, and J. M. Pacheco, “The role of diversity in the evolution of cooperation,” Journal of Theoretical Biology, vol. 299, pp. 88–96, 2012.
View at: Publisher Site | Google Scholar
L.-X. Zhong, D.-F. Zheng, B. Zheng, C. Xu, and P. M. Hui, “Networking effects on cooperation in evolutionary snowdrift game,” Europhysics Letters, vol. 76, no. 4, pp. 724–730, 2006.
View at: Publisher Site | Google Scholar
M. Perc and A. Szolnoki, “Social diversity and promotion of cooperation in the spatial prisoners dilemma game,” Physical Review Letters, vol. 77, no. 1, Article ID 011904, 2008.
View at: Publisher Site | Google Scholar
M. Perc and Z. Wang, “Heterogeneous aspirations promote cooperation in the prisoner’s dilemma game,” PLoS One, vol. 5, no. 12, Article ID e15117, 2010.
View at: Publisher Site | Google Scholar
H. Fort, “On evolutionary spatial heterogeneous games,” Physica A: Statistical Mechanics and Its Applications, vol. 387, no. 7, pp. 1613–1620, 2008.
View at: Publisher Site | Google Scholar
Y. Yao and S.-S. Chen, “Multiplicative noise enhances spatial reciprocity,” Physica A: Statistical Mechanics and Its Applications, vol. 413, pp. 432–437, 2014.
View at: Publisher Site | Google Scholar
A. Szolnoki, M. Perc, and G. Szabó, “Diversity of reproduction rate supports cooperation in the prisoner’s dilemma game on complex networks,” The European Physical Journal B, vol. 61, no. 4, pp. 505–509, 2008.
View at: Publisher Site | Google Scholar
P. Zhu and G. Wei, “Stochastic heterogeneous interaction promotes cooperation in spatial prisoner’s dilemma game,” PLoS One, vol. 9, no. 4, Article ID e95169, 2014.
View at: Publisher Site | Google Scholar
A. Szolnoki, J. Vukov, and G. Szabó, “Selection of noise level in strategy adoption for spatial social dilemmas,” Physical Review Letters, vol. 80, no. 5, Article ID 056112, 2009.
View at: Publisher Site | Google Scholar
W.-B. Du, X.-B. Cao, M.-B. Hu, and W.-X. Wang, “Asymmetric cost in snowdrift game on scale-free networks,” Europhysics Letters, vol. 87, no. 6, p. 60004, 2009.
View at: Publisher Site | Google Scholar
C.-Y. Xia, S. Meloni, M. Perc, and Y. Moreno, “Dynamic instability of cooperation due to diverse activity patterns in evolutionary social dilemmas,” Europhysics Letters, vol. 109, no. 5, p. 58002, 2015.
View at: Publisher Site | Google Scholar
Z.-X. Wu, Z. Rong, and M. Z. Q. Chen, “Diverse roles of the reduced learning ability of players in the evolution of cooperation,” Europhysics Letters, vol. 110, no. 3, p. 30002, 2015.
View at: Publisher Site | Google Scholar
H.-X. Yang, Z.-X. Wu, and B.-H. Wang, “Role of aspiration-induced migration in cooperation,” Physical Review Letters, vol. 81, no. 6, Article ID 065101, 2010.
View at: Publisher Site | Google Scholar
Z. Rong, Z.-X. Wu, and G. Chen, “Coevolution of strategy-selection time scale and cooperation in spatial prisoner’s dilemma game,” Europhysics Letters, vol. 102, no. 6, p. 68005, 2013.
View at: Publisher Site | Google Scholar
Z. Rong, Z.-X. Wu, D. Hao, M. Z. Q. Chen, and T. Zhou, “Diversity of timescale promotes the maintenance of extortioners in a spatial prisoner’s dilemma game,” New Journal of Physics, vol. 17, no. 3, Article ID 033032, 2015.
View at: Publisher Site | Google Scholar
Z. Wang, A. Szolnoki, and M. Perc, “Optimal interdependence between networks for the evolution of cooperation,” Scientific Reports, vol. 3, no. 1, p. 2470, 2013.
View at: Publisher Site | Google Scholar
Z. Wang, A. Szolnoki, and M. Perc, “Self-organization towards optimally interdependent networks by means of coevolution,” New Journal of Physics, vol. 16, no. 3, Article ID 033041, 2014.
View at: Publisher Site | Google Scholar
L.-L. Jiang and M. Perc, “Spreading of cooperative behaviour across interdependent groups,” Scientific Reports, vol. 3, no. 1, p. 2483, 2013.
View at: Publisher Site | Google Scholar
A. Szolnoki and X. Chen, “Cooperation driven by success-driven group formation,” Physical Review Letters, vol. 94, no. 4, Article ID 042311, 2016.
View at: Publisher Site | Google Scholar
X. Chen and L. Wang, “Promotion of cooperation induced by appropriate payoff aspirations in a small-world networked game,” Physical Review Letters, vol. 77, no. 1, Article ID 017103, 2008.
View at: Publisher Site | Google Scholar
J. Zhang, Y.-P. Fang, W.-B. Du, and X.-B. Cao, “Promotion of cooperation in aspiration-based spatial prisoner’s dilemma game,” Physica A: Statistical Mechanics and its Applications, vol. 390, no. 12, pp. 2258–2266, 2011.
View at: Publisher Site | Google Scholar
Z.-X. Wu and Z. Rong, “Boosting cooperation by involving extortion in spatial prisoner’s dilemma games,” Physical Review Letters, vol. 90, no. 6, Article ID 062102, 2014.
View at: Publisher Site | Google Scholar
M. Hetzer and D. Sornette, “An evolutionary model of cooperation, fairness and altruistic punishment in public good games,” PLoS One, vol. 8, no. 11, Article ID e77041, 2013.
View at: Publisher Site | Google Scholar
R.-R. Liu, C.-X. Jia, and Z. Rong, “Effects of enhancement level on evolutionary public goods game with payoff aspirations,” Applied Mathematics and Computation, vol. 350, pp. 242–248, 2019.
View at: Publisher Site | Google Scholar
X. Chen, A. Szolnoki, and M. Perc, “Competition and cooperation among different punishing strategies in the spatial public goods game,” Physical Review Letters, vol. 92, no. 1, Article ID 012819, 2015.
View at: Publisher Site | Google Scholar
A. Szolnoki, Z. Wang, and M. Perc, “Wisdom of groups promotes cooperation in evolutionary social dilemmas,” Scientific Reports, vol. 2, no. 1, p. 576, 2012.
View at: Publisher Site | Google Scholar
A. Szolnoki and M. Perc, “Competition of tolerant strategies in the spatial public goods game,” New Journal of Physics, vol. 18, no. 8, Article ID 083021, 2016.
View at: Publisher Site | Google Scholar
M. Ji, C. Xu, and P. Hui, “Effects of dynamical grouping on cooperation in n-person evolutionary snowdrift game,” Physical Review Letters, vol. 84, Article ID 036113, 2011.
View at: Publisher Site | Google Scholar
M. Perc and A. Szolnoki, “Coevolutionary games-a mini review,” BioSystems, vol. 99, no. 2, pp. 109–125, 2010.
View at: Publisher Site | Google Scholar
G. Szabó and C. Tőke, “Evolutionary prisoner’s dilemma game on a square lattice,” Physical Review E, vol. 58, no. 1, pp. 69–73, 1998.
View at: Publisher Site | Google Scholar
G. Szabó, J. Vukov, and A. Szolnoki, “Phase diagrams for an evolutionary prisoners dilemma game on two-dimensional lattices,” Physical Review Letters, vol. 72, no. 4, Article ID 047107, 2005.
View at: Publisher Site | Google Scholar
J.-Y. Guan, Z.-X. Wu, and Y.-H. Wang, “Effects of inhomogeneous activity of players and noise on cooperation in spatial public goods games,” Physical Review Letters, vol. 76, no. 5, Article ID 056101, 2007.
View at: Publisher Site | Google Scholar
M. Perc, “Coherence resonance in a spatial prisoner’s dilemma game,” New Journal of Physics, vol. 8, no. 2, p. 22, 2006.
View at: Publisher Site | Google Scholar
M. Perc and M. Marhl, “Evolutionary and dynamical coherence resonances in the pair approximated prisoner’s dilemma game,” New Journal of Physics, vol. 8, no. 8, p. 142, 2006.
View at: Publisher Site | Google Scholar
Z. Wang, A. Szolnoki, and M. Perc, “Interdependent network reciprocity in evolutionary games,” Scientific Reports, vol. 3, no. 1, p. 1183, 2013.
View at: Publisher Site | Google Scholar
A. Szolnoki and M. Perc, “Information sharing promotes prosocial behaviour,” New Journal of Physics, vol. 15, no. 5, Article ID 053010, 2013.
View at: Publisher Site | Google Scholar
C. Luo and X. Zhang, “Effect of self-organized interdependence between populations on the evolution of cooperation,” Communications in Nonlinear Science and Numerical Simulation, vol. 42, pp. 73–82, 2017.
View at: Publisher Site | Google Scholar
M. Zhang and J. Yang, “Random partnerships in spatial game theory,” Physical Review Letters, vol. 79, no. 1, Article ID 011121, 2009.
View at: Publisher Site | Google Scholar
J. Poncela, J. Gómez-Gardenes, and Y. Moreno, “Cooperation in scale-free networks with limited associative capacities,” Physical Review Letters, vol. 83, no. 5, Article ID 057101, 2011.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2019 Xuzhen Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

771

Downloads

1118

Citations